multipoint feynman diagrams at the one loop level

Multipoint Feynman diagramsat the one loop level

PhD Thesis

Krzysztof Kajda

Department of Field Theory and Particle Physics

University of Silesia - Institute of Physics

Katowice Poland

3rd september 2009

Supervisor

Dr hab Janusz GluzaDepartment of Field Theory and Particle PhysicsUniversity of Silesia Katowice Poland

AcknowledgementsMy special thanks go to my supervisordr hab Janusz Gluza for his helpfulsuggestions and important advice duringthe course of this work

Contents

1 Introduction 7

2 Evaluating Feynman integrals 9

21 Feynman parametrisation 922 Mellin-Barnes representation theoretical background 1323 Numerical calculations in the Euclidean region 19

3 Construction of Mellin-Barnes representations by AMBRE package 23

31 Loop by Loop algorithm 2332 Remarks on derivation of Mellin-Barnes representations 27

321 Order of integration in the loop by loop method 27322 Importance of F polynomial in derivation of representations 32323 Using Barnes lemmas 34

33 One-loop integrals with tensor numerator 3934 Cross-checks in Euclidean region with CSectors 4235 Applications and perspectives of AMBRE 45

4 Tensor reduction of five and six point one loop integrals 47

41 Passarino-Veltman reduction 4742 Gram determinants and algebra of signed minors 4943 Tensor integrals and shifted space-time dimensions 5144 Reduction of pentagons up to rank two 5145 Reduction of six point functions 5746 Numerical results and discussion 59

5 Applications five point integrals in QED 65

51 Infrared divergences connected with e+eminus rarr micro+microminusγ 6752 Results and numerical cross-checks for virtual contributions 70

6 Summary and conclusions 81

A Gamma function 83

B Software 85

B1 AMBRE and generation of Mellin-Barnes representations 85B2 Reduction of six and five point integrals in hexagon 89B3 Sector decomposition with CSectors 92

5

Contents

Bibliography 95

6

1 Introduction

Research is an organized method

for keeping you reasonably

dissatisfied with what you have

(Charles F Kettering)

High energy physics a branch of science that studies elementary particles of matterand the interactions among them developed over the span of XXth century Many gen-erations of researchers examined basic properties of subatomic particles [1] Notablyduring the 70rsquos the Standard Model of particle physics [2 3] evolved as a modern classi-fication of elementary particles and description of the three of four basic interactions Inorder to increase our knowledge about elementary particles many international acceler-ator collaborations exist like CERN DESY(Hamburg) Fermilab KEK SLAC Highenergy physics experiments may confirm Standard Model just like the recent discoveryof Ωminus

b baryon in DZero experiment at the Fermilab [4] and future search for Higgs par-ticle in LHC or can hopefully give a clue about rdquonew physicsrdquo beyond the StandardModel All these experiments due to accuracy they achieve often require rather painfuland difficult theoretical calculations at certain leading order of corrections The currentneed of precision goes at large extent beyond the so called tree order meaning that loopeffects must be taken into account

Nowadays there are numerous methods of involving loop integrals [5] Some of thembecame a standard let us mention here the FF [6] or LoopTools [7] packages Latelyseveral new tools appeared like CutTools [8] which uses Ossola Papadopoulos andPittau (OPP) [9] method or BlackHat [10] using unitary cut method and on-shell recur-sion [11] to construct one-loop amplitudes Many of them are still under developmentAll available loop methods are quite advanced and in this thesis we focus only on a fewwhich allow us to examine loop integrals

The title of this thesis refers to one-loop integrals and although we focus mostly onsuch cases multi-loop calculus is also possible within part of the methods and developedtools we discuss Firstly we focus on Mellin-Barnes method [5] which is described inchapter two Because it is possible to obtain analytic results using it M-B can be usedto analyse infrared (IR) divergences that may appear in loop integrals Quite recentlywe have developed a special tool AMBRE [12] which produces M-B representations ofFeynman diagrams in an automatic way It is the first public tool already used in aresearch work by several groups The detailed discussion of AMBRE is in the chapterthree Let us note that it is not always easy to solve derived M-B integrals into the final

7

1 Introduction

analytical form For such cases numerical integration can be made for purely numericaltests However due to an oscillatory behaviour of M-B integrals [13] numerical resultsneed non-physical phase space points to be stable numerically There is no way tocalculate physical processes numerically using directly Mellin-Barnes representations ofFeynman diagrams

During practical calculations of virtual corrections one needs a method to computetensor loop integrals Often some reduction scheme is applied eg the well knownPassarino-Veltman [14] for one loop diagrams Problem with numerical instability re-lated to the so called Gram determinants might appear In chapter four we present acure for the problem of Grams a reduction scheme worked out by us in [15 16] forsix and five point integrals based on algebra of rdquosigned minorsrdquo The advance over theprevious schemes based on this algebra is that these determinants were cancelled Thescheme itself is a continuous work based on [17] where Grams were still present Herewe also show a special software that uses this scheme hexagon [16] Although its usagewithin Monte Carlo event generator would not be a good idea (the program was writtenin MATHEMATICA [18] more suitable for analytic manipulations) it is still a valuable toolfor performing many cross-checks eg IR tests described in the last chapter We haveused it to perform checks and tests to the well known LoopTools program which is basedon the early FF package Quite recently the tensor structure for a five point diagramshas been also added to LoopTools [19] using work of [20 21]

The last chapter covers results which are not public yet Somehow it merges togetherall the previous chapters Here we have calculated loop corrections (four and five pointdiagrams) of the QED e+eminus rarr micro+microminusγ process As a primary tool for numerical calcula-tions we used LoopTools The same calculations can be repeated using rdquosigned minorsrdquoreduction scheme Here instead of hexagon we would use our reduction scheme encodedinto Fortran This program is still under development and so far we used it to test andcheck single integrals only

Apart from purely numerical calculations we have also used tools and techniques tocalculate infrared divergences and do proper checks and verifications Here we usedAMBRE to obtain IR analytic results and CSectors [22] which uses sector decompositionalgorithm [23 24 25] to perform proper numerical cross-checks

This thesis ends with two appendices In the Appendix A a short description of thegamma function is given while in the Appendix B the software (AMBRE hexagon andCSectors packages) are presented

8

2 Evaluating Feynman integrals

Modern experiments in particle physics need precise theoretical predictions whichare related to the higher order Quantum Field Theory corrections going beyond the socalled tree level Loop corrections can be treated in different ways First we describeFeynman parametrisation which is important for the next loop method described andexplored in detail ie Mellin-Barnes representations Finally we focus on numericalevaluation techniques in the Euclidean region in particular we give a description of thesector decomposition method

21 Feynman parametrisation

As an introductory example we begin with an integration of the following simple twodimensional parametric integral

int 1

0

dx1

int 1

0

dx2δ(1 minus x1 minus x2)

P1x1 + P2x2

=

int 1

0

dx11

P1x1 + P2(1 minus x1)

=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)

where Dirac delta properties and integration by substitution were used In this unso-phisticated example we see that product in the denominator can be replaced by its sumtogether with integration over additional parameters This is the so called Feynmanparametrisation in its simplest two dimensional form1 Eq211 can be generalized in astraightforward way

1

P1 Pn

= Γ(n)

int 1

0

int 1

0

dx1 dxnδ(1 minus x1 minus minus xn)

(P1x1 + + Pnxn)n (212)

Here factorial (n minus 1) was replaced by the gamma function Γ(n) (see EqA03) Itis also easy to check above equation for n gt 2 The only precaution which must be

1 Feynman parametrisation is closely related to the so called alpha parametrisation and the so calledSchwinger trick [5]

1

P1 Pn

=

int

infin

0

int

infin

0

dα1 dαneminusP1α1 eminusPnαn

9


overtaken is setting an appropriate upper limit for the integral after rdquocancellingrdquo Diracdelta For example

int 1

0

dx1

int 1

0

dx2

int 1

0

dx3δ(1 minus x1 minus x2 minus x3)f(x1 x2 x3)

=

int 1

0

dx1

int 1minusx1

0

dx2f(x1 x2 1 minus x1 minus x2) (213)

where f is some general function Note that in case of Eq211 it was not significantIt is also worth observing how Feynman parametrisation looks like in case of general

powers of terms (propagators) in a denominator

1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j

δ(1 minus x1 minus minus xn)

(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)

We would like to use Feynman parametrisation in case of loop integrals which in turn canbe defined in d dimensions in the sense of dimensional regularisation in the followingway

G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2

1 + i0)ν1 (q2n minus m2

n + i0)νn (216)

where T (k) is a tensor numerator of some rank In particular in case of a scalar integralT (k) = 1 The momentum q2 is a composition of internal k and external p momentaIn this notation single propagator is of the form

Pi = q2i minus m2

i + i0 (217)

For simplicity we start with a scalar case It will be expanded to include the case ofgeneral m-rank numerator afterwards Let us insert above definition into Eq214 Onemay see that

nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)

where M is a LtimesL matrix containing Feynman parameters Q is an L-dimensional vectorcomposed of external momenta and Feynman parameters and J contains kinematicinvariants and Feynman parameters At this point integral G(1) is of the form

G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL

[kT Mk minus 2kT Q + J ]Nν (219)

10


After eliminating linear terms in Eq218 by performing shift

kl = kl +Lsum

i=1

Mminus1li Qi (2110)

it is relatively easy to perform momentum integration resulting in the following parametri-sation

G(1) = (minus1)NνΓ(Nν minus

d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)

F = minusdet(M)J + QMQ (2112)

Matrix M is defined as M = det(M)Mminus1 According to the graph theory the polyno-mials U and F can also be constructed from the topology of a given Feynman diagramas follows [26 27 28]

bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)

The sum runs over 1-trees of the given graph ie maximal connected sub-graphswithout loops

bull F - polynomial

F =sum

TisinT2

prod

l isinT

xl(minussT )2 (2114)

Here the sum is over 2-trees ie sub-graphs that do not involve loops and consistsof two connectivity components sT is the sum of the external momenta that flowinto the connectivity components of the 2-tree T

In the case of four point massless on-shell Feynman diagram the construction of U andF functions using above rules was demonstrated on Fig21 and Fig22 resulting U andF to be as

U = x1 + x2 + x3 + x4

F = minussx1x3 minus tx2x4 (2115)

It is interesting to observe that all this mathematical apparatus together with interpret-ing these results from the graph theoretical point of view is the same as that used in the

11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1

Figure 21 1-trees contribiuting to the U polynomial

3

4

2

1 3

4

2

1

Figure 22 2-trees contributing to the F polynomial

problem of the solution of Kirchhoffrsquos laws for electrical circuits Feynman parametersxi play the role of ohmic resistance and the U polynomial is a Kirchhoff result [29]

The Eq2111 can be generalized to L-loop m-rank integral resulting in (see eg [25])

G(T (k)) =(minus1)Nν

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2L minus r

2

)

(minus2)r2

UNνminusd2(L+1)minusm

FNνminusd2Lminus r

2

ArPmminusr[micro1microm]

(2116)

where a strange looking ArPmminusr

[micro1microm]object represents the tensor structure of the

numerator illustrated by the following example

bull m=2

sum

rle2

ArP2minusr[micro1micro2]

=

A0P2 + A1P

1 + A2P0[micro1micro2]

= P micro1P micro2 + gmicro1micro2 (2117)

12

22 Mellin-Barnes representation theoretical background

bull m=3

sum

rle3

ArP3minusr[micro1micro2micro3]

=

A0P3 + A1P

2 + A2P1 + A3P

0[micro1micro2micro3]

= P micro1P micro2P micro3 + gmicro1micro2P micro3 + gmicro2micro3P micro1 + gmicro3micro1P micro2 (2118)

where A0 P0 is one Ar is zero for r odd and Ar = g[micro1micro2 middot middot middot g microrminus1micror] for r even P micro and

gmicroν are defined as

P microi rarrsum

l

[MalQl]microi

gmicroimicroj rarr (Mminus1)abgmicroimicroj (2119)

The indices ab correspond to different loop momenta for example

G(kmicro1

1 kmicro2

2 ) rarr P micro1P micro2 + gmicro1micro2 rarrsum

l

[M1lQl]micro1[M2lQl]micro2

+ (Mminus1)12gmicro1micro2 (2120)

Of course in this dissertation we focus on the one loop integrals (L = 1) which obviouslyleads to

M = det(M)Mminus1 = 1 (2121)

if we bear in mind that M is L times L matrix


Mellin integrals [30] are integrals over contours in a complex plane along the imaginaryaxis of a product and a ratio of gamma functions defined in EqA01 The main featureof the method presented in this chapter is the Mellin-Barnes formula used to representa sum of terms raised to some power by a product of these terms This operation allowsto create the M-B representation for a given Feynman integral

The backbone of the procedure to build up Mellin-Barnes representations is the fol-lowing relation

1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin

dzΓ(λ + z)Γ(minusz)AzBminusλminusz (221)

where

bull the integration contour separates the poles of Γ(minusz) from those of Γ(λ + z)

bull A and B are complex numbers such that |arg(A) minus arg(B)| lt π

13


p p

k

k + p

Figure 23 Two-point (self-energy) diagram

The Mellin-Barnes relation Eq221 can be iterated and easily extended into a sum ofseveral terms

1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i

times Aminusλminusz2minusminuszn

1 Γ(λ + z2 + + zn)nprod

i=2

Γ(minuszi) (222)

As the example of usage of formula Eq222 the following self-energy example Fig23was chosen

G(1)SE1l2m =

int

ddk

(k2 minus m2 + i0)ν1((k + p)2 minus m2 + i0)ν2 (223)

where all the typical constant factors were omitted index SE1l2m stands for self-energy(SE) one-loop (1l) with two massive (2m) internal lines We will use analogous nomen-clature later on After calculating U and F polynomials (see the previous section) wehave

U = x1 + x2 F = m2(x1 + x2)2 minus sx1x2 minus i0 (224)

Feynman parametrisation for this diagram reads

G(1)SE1l2m =Γ(ν1 + ν2 minus

d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd

(m2(x1 + x2)2 minus sx1x2 minus i0)ν1+ν2minusd2 (225)

We see that the Dirac δ function causes the U polynomial which is simply a sum ofFeynman parameters to be one In general every one-loop n-point diagram has the Upolynomial of the form x1 + + xn so U = 1 for all one-loop cases

At this point Eq221 can be used to start constructing a Mellin-Barnes representationWe use it to replace a sum of Feynman parameters in the F polynomial into its product

14


with additional integration over the complex space

1

F λ=

1

(m2(x1 + x2)2 minus sx1x2 minus i0)λ

=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin

dz1Γ(λ + z1)Γ(minusz1)(m2 minus i0)z1(minuss minus i0)minusλminusz1

times (x1x2)minusλminusz1 [x1 + x2]

2z1 (226)

where λ = ν1 + ν2 minus d2 The term [x1 + x2]2z1 again can be changed according to

Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin

dz2Γ(λ + z1)Γ(minusz1)Γ(minus2z1 + z2)

times Γ(minusz2)(m2 minus i0)z1(minuss minus i0)minusλminusz1xminusλminusz1+z2

1 xminusλ+z1minusz2

2 (227)

Next step is to insert Eq227 back into Eq225 and collect powers of Feynman param-eters which in our case are

xa1minus11 = x

(minusλminusz1+z2+ν1)minus11

xa2minus12 = x

(minusλ+z1minusz2+ν2)minus12 (228)

Finally the integration over Feynman parameters is performed using the following for-mula

int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)

which in practice is restricted to relevant collecting powers of Feynman parameters Thefinal Mellin-Barnes representation for the self-energy diagram Eq223 is2

G(1)SE1l2m =(minus1)ν1+ν2

Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin

dz2(m2 minus i0)z1(minuss minus i0)d2minusν1minusν2minusz1

times Γ(minusd2 + ν1 + ν2 + z1)Γ(minusz1)Γ(minus2z1 + z2)Γ(minusz2)

timesΓ(d2 minus ν1 + z1 minus z2)Γ(d2 minus ν2 minus z1 + z2)

Γ(minus2z1)Γ(d minus ν1 minus ν2) (2210)

For more complicated cases F polynomial often contains more than two terms In suchcases it is straightforward to use general Eq222 formula

2 Usually in Mellin-Barnes representations infinitesimal complex part i0 is omitted eg in [5] andlater in this thesis It does not mean that i0 is irrelevant Final analytical results for Feynman

integrals in general contain kinematic terms like log( ts) log(minusm2

s) see eg [31] Even if i0

is omitted at the beginning it must be recreated later to make the analytic continuation to thephysical domain possible eg log(minusxminus i0) = log(|x|) minus Θ(x)iπ

15


The structure of F polynomial affects form of the final M-B representation This isobviously due to the fact that Eq222 changes sum of n terms raised to some powerinto nminus1 dimensional integral over the complex space This observation is helpful whenone wants to estimate the dimension of the final M-B representation only by looking atF polynomial It will be discussed in the next chapter in more detail

During derivation of a Mellin-Barnes representation one is usually interested in sim-plifying the final representation (dimensionality of integrals) as much as possible It isvery important if one wants to obtain an analytical result from it One of the possibilitiesis to apply one of the following lemmas

bull First Barnes lemma

int c+iinfin

cminusiinfin

dz Γ(a + z)Γ(b + z)Γ(c minus z)Γ(d minus z) =

Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)

bull Second Barnes lemma

int c+iinfin

cminusiinfin

dzΓ(a + z)Γ(b + z)Γ(c + z)Γ(d minus z)Γ(e minus z)

Γ(a + b + c + d + e + z)=

Γ(a + d)Γ(a + e)Γ(b + d)Γ(b + e)Γ(c + d)Γ(c + e)

Γ(a + b + d + e)Γ(a + c + d + e)Γ(b + c + d + e) (2212)

We can now apply first Barnes-Lemma to Eq2210 and simultaneously substitutingpowers of propagators equal one ν1 = ν2 = 1 and d = 4 minus 2ǫ we get

G(1)SE1l2m =

int c+iinfin

cminusiinfin

dz1(m2)z1(minuss)minusǫminusz1

Γ(1 minus ǫ minus z1)2Γ(minusz1)Γ(ǫ + z1)

Γ(2 minus 2ǫ minus 2z1)(2213)

At this point ie after defining the Mellin-Barnes representation for the Feynman in-tegral we are interested in further processing Eq2213 result so that calculation ofthe Laurent expanded ǫ terms would be possible We can use the Cauchy theorem toget representation in terms of a sum of contour integrals valid at ǫ = 0 Importantis that the Mellin-Barnes representation is only well defined if the integration contourseparates the left poles Γ( + z) from the right poles Γ( minus z) and in general for thecombination of Γ functions with ǫ = 0 that will not be the case (if the contour is chosento be a straight line parallel to the imaginary axis) In practice two solutions to thisproblem have originated

bull rdquoTausk methodrdquo - which fixes the contours parallel to the imaginary axis andaccounts for the poles crossing in the analytic continuation [32] The idea of it is

16


GΕ + z1D

G-z1D

G1- Ε - z1Dnot separated

-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z

Figure 24 Left and right poles of Eq2213 Left plot shows situation when ǫ rarr 0which leads to non-separated left and right poles The right figure presentsa proper shift which separates the poles (ǫ rarr 1 Re(z1) rarr minus12)

presented on Fig24 This method was implemented in MB program3 [13] writtenin the MATHEMATICA [18]

bull rdquoSmirnov methodrdquo - is based on deforming the contour [33] and then shifting thempast the poles of the Γ functions which results in residue integrals This algorithmhas recently been implemented in the MBresolve program [34]

Because of applicability of the rdquoTausk methodrdquo in MB we focus on this method and showhow it is implemented in the MB program by executing two simple commands Eq2214and Eq2215 below Using this scheme the final result (ǫ = 0) will be obtained fromthe case where the poles are separated (ǫ 6= 0) Depending whether the poles crossedthe contours from left or right (when ǫ rarr 0) one should add or subtract the residue ofthe integrand on that pole

Let us focus on the example of Eq2213 In this case we have the following gammafunctions contributing to left and right poles

bull Left poles Γ(ǫ + z1)

bull Right poles Γ(minusz1) Γ(1 minus ǫ minus z1)

Graphically we can see what happens if we put ǫ rarr 0 the left plot of Fig24 We notethat left and right poles are not separated from each other To counteract that thearbitrary choice of ǫ rarr 1 according to the rdquoTausk methodrdquo is chosen the right plotof Fig24 Separation of left and right poles can be done using the MB software thenintegration contour is found4

rules = MBoptimizedRules[final eps-gt0 eps] (2214)

3 It allows to analytically continue any MB integral in a given parameter and to resolve the singularitystructure in this parameter The package can also perform numerical integrations at specifiedkinematic points

4 MBoptimizedRules procedure is based on a powerful MATHEMATICA function FindInstance whichliterally finds instance of variables that makes the expression true

17


As an output the following choice is calculated ǫ rarr 1 Re(z1) rarr minus12 We can see thatwith such a choice arguments of gamma functions are positive (integral is well defined)The very next step in the calculation of Eq2213 is the analytic continuation ie wehave to finally go down to the case where ǫ rarr 0 Again in MB it is done automaticallyby executing the following command

integrals = MBcontinue[final eps-gt0 rules] (2215)

and as an output we will get the sum of two terms

1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2ǫ minus 2z1) (2217)

Let us note that Eq2216 is just a residue of Eq2217 As it has been already writtenit appears due to the shift of poles The MBcontinue function implemented in the MB

program cares about extracted residues in an automatic way Now Eq2217 is welldefined in the limit ǫ rarr 0

If expanded as a Laurent series in ǫ they give contributions to 1ǫ and the constantpart (the last zero argument in the curly brackets below cuts Laurent series at ǫ0)respectively

MBexpand[integrals Exp[epsEulerGamma] eps00] (2218)

where the result was multiplied by an exponent of the Euler gamma γe to get rid of itin the final result

11

ǫminus ln(m2) (2219)

2

int minus 12+iinfin

minus 12minusiinfin

dz1(m2)z1(minuss)minusz1

Γ(1 minus z1)2Γ(minusz1)Γ(z1)

Γ(2 minus 2z1) (2220)

Obviously the integral which gives contribution to the constant part must be furthercalculated analytically or integrated numerically The first is done by summing theresidues the latter can be done within the MB package Let us now focus on the analyticalevaluation of the Eq2220 This integral can be calculated by closing the contour tothe left ie it can be written as an infinite sum over the residues by making use of theresidue theorem

∮

dzf(z) = 2πisum

Res[f(z)] (2221)

Taking for simplicity m = 1 the sum of Eq2220 and Eq2219 is equal

G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18

23 Numerical calculations in the Euclidean region

This analytic result can be cross-checked in Euclidean region by numerical integrationEq2220 it can be done within MB package

We would like to note that ln(x) presented in Eq2222 can be expressed in the lan-guage of harmonic polylogarithms [35] here we would have ln(x) = H0(x) Analogouslywe can treat higher orders of expanded terms in ǫ An alternative way to obtain thesame result is differential equation method [36] In fact there are problems where M-Band differential equation methods support each other nicely [37]

We went through some simple one dimensional case where series can be even summeddirectly by the newest versions of MATHEMATICA (starting with version 60) But ingeneral multidimensional Mellin-Barnes integrals will be present after using Cauchyrsquostheorem as nested sums [38] The state of art research in loop integrals shows thatup to four (five) dimensional massive (massless) M-B integrals can be summed up intoanalytical form [33 5 32 39 40] If there is a problem with summation of complicatednested sums an expansion in a ratio of some kinematic variables can be useful and theonly remedy [31]


This section is intended to present the sector decomposition method [41 42 23 25]which isolates divergences from parameter integrals It is still an active field of researchfor the recent idea see [43]

In case of loop integrals we are interested in the Laurent expansion in ǫ of the Feyn-man parameter integral Eq2116 so that the coefficients of the series can be computednumerically later on The major difficulty is separation of overlapping singularities Letus look at the following example

int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)

This integral contains the singular region when both x1 and x2 vanish simultaneouslythey are overlapping for x1 rarr 0 and x2 rarr 0 The first one decomposes the integrationrange into N sectors in each l-sector xl is the largest

int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)

where θ-function is defined as

θ(x ge y) =

1 if x ge y is true0 otherwise

19


y

x

minusrarr + minusrarr(2)

(1)

+

y

x

t

t

Figure 25 The basic idea of sector decomposition method The figure was taken from[25]

Also the variables are transformed in each l-sector often called rdquoprimary sectorrdquo accord-ing to the following scheme

xj =

xltj for j lt lxl for j = lxltjminus1 for j gt l

(233)

Graphically the situation can be illustrated by Fig25 Because of Eq23 xl factorizesalso in case of Eq2116 from F and U ie U(x) rarr U(x)xL

l F (x) rarr F (x)xL+1l This

allows to useint 1

0

dxlxl δ(1 minus xl(1 +Nminus1sum

k=1

tk)) = 1 (234)

which eliminates xl in other words the singular behaviour leading to poles in ǫ comesfrom the region of small ti In general after eliminating xl we end up with

int 1

0

Nminus1prod

j=1

tj tνjminus1j

UNνminus(L+1)d2l (t )

FNνminusLd2l (t )

l = 1 N (235)

Of course in general the separation of the singularities is not obtained after first step andall the procedure has to be repeated iteratively (starting from Eq235) until a completeseparation of overlapping regions is achieved Finally one obtains a form where allsingularities are factorised in terms of parameters tj and all poles can be extractedleading to integrals which are finite and can be integrated numerically

Recently public computer implementation of sector decomposition algorithm has ap-peared [44] The sector decomposition set of libraries allows to calculate numericallygiven Feynman integral Because the input of U and F polynomial must be providedthe idea of special interface written in MATHEMATICA emerged Such a program shouldbe capable of preparing all the necesary input for the sector decomposition also ex-ecution and output had to be done for the routines in a fully automatic way SoonCSectors [22 45] was developed allowing easily and quickly obtaining numerical resultsfor a given Feynman integral Computation is not bounded to scalar L-loop integralsonly but allows to calculate tensor m-rank diagrams as well Without such an interface

20


as CSectors user would spend a lot of time preparing U and F polynomials and anappropriate tensor structure The main purpose of CSectors in this thesis is numericalcross-check for the results obtained using Mellin-Barnes method Let us stress againthat numeric works properly for them in Euclidean region

This section concludes chapter devoted to the evaluation of loop integrals usingMellin-Barnes and sector decomposition methods precedent by introduction to Feyn-man parametrisation which is the backbone of both methods In the following chapterwe will focus on derivation of Mellin-Barnes representations using AMBRE package andits important issues

21


22

3 Construction of Mellin-Barnes representations by

AMBRE package

A brief sketch of Mellin-Barnes techniques presented in the last chapter made it clearthat process of derivation of M-B representations should be easily introduced in fullyautomatic way in a form of some set of program routines Such an idea came up in 2006and was practically realized by developing the first public program for constructingMellin-Barnes representations AMBRE [12] The name itself is an acronym for AutomaticMellin-Barnes Representation The procedure is designed to calculate1

bull scalar multi-loop multi-leg integrals

bull tensor m-rank one-loop integrals

AMBRE was written in Computer Algebra System language provided by MATHEMATICA andits very favourable point is that the final result of M-B representation can be used as aninput in the MB [13] program at once

More detailed technical description of AMBRE is provided within this dissertation inthe Appendix B1

31 Loop by Loop algorithm

First section of this chapter will introduce rdquoloop by looprdquo algorithm which we useto construct Mellin-Barnes representations Also we will show relevant examples as anoutput of AMBRE and the simplicity of the program usage

The Feynman parametrisation Eq2116 contains the so called U function which is apolynomial depending on Feynman parameters In the previous chapter it was pointedout that in case of one-loop integrals U can be set as equal one and only F polynomialmatters The scheme we want to use the so called rdquoloop by looprdquo or rdquore-insertionrdquoalgorithm means that one treats every sub-loop part of multi-loop diagram as one-loop integral for which the very same Mellin-Barnes methods as already presented areapplied The algorithm will be presented using two-loop four point master integral2

diagram as an example

1 Apart from the public program (ver 12) there is a version of AMBRE which is able to deriveMellin-Barnes representations for two-loop m-rank integrals It has already been used in practicalcalculations [46]

2 It is one of master integrals appearing in two-loop Bhabha e+eminus rarr e+eminus process [36]

23

3 Construction of Mellin-Barnes representations by AMBRE package

p1

p2

p3

p4k2 minus p3 minus p4

k2 minus p3k2 minus p1

k2 + k1

k1

rarr

p1

p2

p3



k2

Figure 31 The left picture shows two-loop four point master integral (B5l2m2) Afterapplying loop by loop algorithm on the first sub-loop (the one over k1) thediagram presented on the right side emerges with the modified power of thepropagator with momentum k2 (double internal line)

GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2

P5 = (k2 minus p3 minus p4)2 minus m2 (311)

From now on AMBRE notation will be used in forthcoming examples The detailed de-scription of this convention is presented in Appendix B1 where appropriate completeexample was given In order to input Eq311 into AMBRE one has to execute the follow-ing

In[]= Fullintegral[1PR[k10n1]PR[k1+k2mn2]PR[k2-p10n3]

PR[k2-p30n4]PR[k2-p3-p4mn5]k1k2]

Figure 32 Eq311 as an input in the AMBRE package Symbol In[]= stands for theMATHEMATICA input From now on we will keep this notation

where propagators have the following general notation PR[kmn1]equiv (k2 minus m2)minusn1After defining our initial diagram one has to decide about the order in which one-

loop sub-loops will be worked out using Mellin-Barnes Eq222 and Feynman parametersintegration Eq229 formulas In our example we have two sub-loops in which internalmomenta flow over k1 and k2 Let us pick up the one over k1 as a first sub-loop tocompute In AMBRE demanded order is indicated by the sequence of the list in Fig32in this case k1k2 The importance of the right choice of ordering of sub-loops willbe discussed later

24


Using IntPart function AMBRE is able to choose right propagators for a given sub-loopwithout user help Fig33

In[]= IntPart[1]

numerator=1

integral=PR[k10n1]PR[k1+k2mn2]

momentum=k1

Figure 33 IntPart separates first sub-loop of Eq311 The lines below In[] is anauxiliary output produced by AMBRE

At this point we are ready to calculate F polynomial for the propagators of Fig33It results in the polynomial of such a form

F = minusk22x1x2 + m2x1x2 + m2x2

2

= minus[k22 minus m2]x1x2 + m2x2

2 (312)

We see that calculation of the sub-loop over k1 causes k2 momentum to appear in theF function At this stage it is treated as a normal momentum which appears becauseof its presence in one of the propagator in Fig33 After simple rearrangement of F k2

can be used to form a new propagator see right diagram of Fig31 for comparison Thiscan be always done in this way because of momentum conservation of the nested loops

Now Mellin-Barnes formula Eq222 followed by the integration over Feynman pa-rameters Eq229 are both applied Propagator k2

2 minus m2 which appeared in F willbe raised into the power containing integration variables of complex integrals emerg-ing from M-B formula From the Fig34 we see how Mellin-Barnes representation isautomatically obtained for the first sub-loop

To remove any doubts concerning AMBRE notation the first sub-loop result is repeatedin less technical notation below

GB5l2m2(1) =

int

ddk2

[(k2 minus p1)2]n3 [(k2 minus p3)2]n4 [(k2 minus p3 minus p4)2 minus m2]n5 [k22 minus m2]minusz1

times

int c+iinfin

cminusiinfin

dz1(minus1)n1+n2+z1(m2)2minusǫminusn1minusn2minusz1Γ(minusz1)Γ(n1 + z1)

timesΓ(4 minus 2ǫ minus 2n1 minus n2 minus z1)Γ(minus2 + ǫ + n1 + n2 + z1)

Γ(n1)Γ(n2)Γ(4 minus 2ǫ minus n1 minus n2)(313)

The remaining step is to repeat the same procedure for the second sub-loop ie the oneover k2 which is now just a one-loop box diagram In AMBRE again it is done in a fullyautomatic way Fig35

Thus all the planar multi-loop scalar integrals can be treated in this way in orderto get Mellin-Barnes representation3 Preceding discussion on the example of master

3 The case of multi-loop tensor integral is more problematic although technically possible It isdiscussed later in this thesis

25


In[]= SubLoop[integral]

Iteration nr1 gtgtIntegrating over k1ltlt

F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2

Representation after integrating over k1

Out[]=SubLoop1[((-1)^(n1+n2+z1)(m^2)^(2-eps-n1-n2-z1)

Gamma[4-2eps-2n1-n2-z1]Gamma[-z1]Gamma[n1+z1]

Gamma[-2+eps+n1+n2+z1])

(Gamma[n1]Gamma[4-2eps-n1-n2]Gamma[n2]) PR[k2 m z1]]

Figure 34 An intermediate result of M-B representation after the first sub-loop wasworked out (see Fig31) Note the new propagator appearing in F polyno-mial Again Out[]= should be understood as an output from the pro-gram The rest between In[]= and Out[]= has to be treated as anauxiliary AMBRE comment output

In[]= IntPart[2]

numerator=1

integral=

PR[k2m-z1]PR[k2-p10n3]PR[k2-p30n4]PR[k2-p3-p4mn5]

momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]

Final representation

Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^(2-eps-n1-n2-z1+z2)

(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3

Gamma[4-2eps-2n1-n2-z1]Gamma[n1+z1]Gamma[-2+eps+n1+n2+z1]

Gamma[-z2]Gamma[-z3]Gamma[n3+z3]Gamma[n4+z3]

Gamma[-2+eps+n3+n4+n5-z1+z2+z3]Gamma[2-eps-n3-n4+z1+z2-z3-z4]

Gamma[-z4]Gamma[-2z2+z4]Gamma[2-eps-n3-n4-n5-z2-z3+z4])

(Gamma[n1]Gamma[4-2eps-n1-n2]Gamma[n2]Gamma[n3]Gamma[n4]

Gamma[n5]Gamma[4-2eps-n3-n4-n5+z1]Gamma[-2z2])

Figure 35 A part of AMBRE calculation for the second sub-loop of B5l2m2 leading to thefull result The object FX[] appearing in F function is used to indicateappearance of (x1 + x4)

2 and that it is required to perform another M-Btransform on objects appearing in the squared bracket

26

32 Remarks on derivation of Mellin-Barnes representations

integral B5l2m2 can be summarized in the following steps the so called rdquoloop by looprdquoor rdquore-insertionrdquo technique

1 Define kinematic invariants which depend on the external momenta4

2 Make a decision about the order in which L one-loop sub-loops (L ge 1) will beworked out sequentially

a) Construct a Feynman integral for the chosen sub-loop and perform manipu-lations on the corresponding F -polynomial to make it optimal for later useof the M-B representations

b) Apply M-B formula Eq222 on F -polynomial

c) Integrate over Feynman parameters using equation Eq229 For integralswith more than one-loop repeat steps a-c for the remaining sub-loops

3 Collect the results for all the sub-loops obtain the final M-B representation result

In the next section we are going to present some interesting issues related to the con-struction of Mellin-Barnes representations


As it has been already signalized in the previous section there are interesting issuesconcerning derivation of Mellin-Barnes representations Some of them appeared to be aproblem during development of AMBRE which had to be solved in order to get a workingprogram which could perform loop by loop techniques in an automatic way

321 Order of integration in the loop by loop method

During construction of Mellin-Barnes final result it is possible to choose different orderof sub-loops over which representation will be worked out In this section it will be shownthat a free choice of such order although possible is not always proper from practicalpoint of view Wrong sequence leads to significant increase in number of dimensionsin the final M-B representation and makes the final result impossible for use in latercalculations

Before we begin with our discussion on the order of iterations let us consider howthe selection of internal momenta flow inside the diagrams impacts derivation of M-Brepresentations As an example we have massless two-loop vertex diagram presented onFig36 The first of two diagrams marked by a) has internal momenta chosen in such away that k1 and k2 flow around sub-loops only None of them is present all around theouter lines as in case of diagram b) To see the difference between these two diagrams

4 On the web page httppracusedupl~gluzaambre there is a file called KinematicsGenwhich generates kinematics for 3- 4- 5- 6- outgoing particles automatically

27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)

Figure 36 Massless vertex diagrams with two different choices of internal momentaflow a) k1 and k2 flow around each of the two sub-loops separately b) herek1 flows partly on the same sub-loop as k2

let us take the first of these two graphs and see the process of derivation of Mellin-Barnesrepresentation in AMBRE for different order of iteration As it was mentioned the abovevertices are treated as massless diagrams ie

p21 = p2

2 = 0 (p1 + p2)2 = s (321)

In order to prevent reader from being lost when analysing the examples in this sectionwe introduce the following notation

bull diagram a)

ndash example a1 k1 rarr k2


bull diagram b)

ndash example b1 k1 rarr k2


First we begin with the example a1 where we choose order k1 rarr k2 The process ofcalculating the first sub-loop is presented on Fig37

We see that the F polynomial will be of the following form which can be collected interms of propagators k2 dependent

F = minusk22x1x2 minus (k2

2 + 2k2 middot p2 + p22)x2x3 minus (p2

1 + 2p1 middot p2 + p22)x1x4

minus(k22 + 2k2 middot p1 + p2

1 + 2k2 middot p2 + 2p1 middot p2 + p22)x2x4 minus p2

1x3x4 minus p22x1x3

= minusk22x1x2 minus [k2 + p2]

2x2x3 minus sx1x4 minus [k2 + p1 + p2]2x2x4 (322)

Four terms in this polynomial stand for three dimensional complex integral related tok1 sub-loop The obtained propagators in the F polynomial will be connected with the

28


In[]= IntPart[1]

numerator=1

integral=

PR[k10n1]PR[-k1+k20n4]PR[k1+p20n2]PR[k1+p1+p20n3]

momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]


Out[]=SubLoop1[((-1)^(2-eps-z3)(-s)^z3

PR[k20z1]PR[k2+p20z2]

PR[k2+p1+p202-eps-n1-n2-n3-n4-z1-z2-z3]]

Figure 37 The first sub-loop of example a1 Fig36 worked out in AMBRE package Apart of the output (dots) was skipped

In[]= IntPart[2]

numerator=1

integral=PR[k20n6-z1]PR[k2+p20-z2]

PR[k2+p1+p20-2+eps+n1+n2+n3+n4+n5+z1+z2+z3]

momentum=k2



F polynomial

-sX[1]X[3]


Out[]=((-1)^(n1+n2+n3+n4+n5+n6)(-s)^(4-2eps-n1-n2-n3-n4-n5-n6)

Gamma[-z1]Gamma[2-eps-n1-n2-n4-z1-z2]Gamma[-z2]

Gamma[n2+z2]Gamma[2-eps-n6+z1+z2]Gamma[2-eps-n1-n2-n3-z3]

Gamma[4-2eps-n1-n2-n3-n4-n5-z1-z3]Gamma[-z3]

Gamma[-4+2eps+n1+n2+n3+n4+n5+n6+z3]Gamma[n1+z1+z3]

Gamma[-2+eps+n1+n2+n3+n4+z1+z2+z3])(Gamma[n1]Gamma[n2]

Gamma[n3]Gamma[4-2eps-n1-n2-n3-n4]Gamma[n4]Gamma[n6-z1]

Gamma[6-3eps-n1-n2-n3-n4-n5-n6-z3]

Gamma[-2+eps+n1+n2+n3+n4+n5+z1+z2+z3])

Figure 38 The second iteration (ie over k2) of example a1 Fig36 The final result isa three dimensional Mellin-Barnes representation

29


k2 sub-loop part during second phase of calculations Fig38 Unlike as in the previousBhabha two-loop example Fig31 now we see more than one propagators in F duringfirst iteration

If the order of iteration will now be reversed ie we move towards example a2 wherefirst we start with k2 and then k1 we notice that obtained Mellin-Barnes representationis one dimension smaller than the one presented in the previous example The completescheme of AMBRE for this calculation is presented in Fig39

The crucial point which makes example a1 and example a2 representations differentis the structure of F polynomial during first iteration In the latter case F has threeterms which lead to two dimensional M-B result The structure of F polynomial in thelast iteration of both examples does not increase dimensionality of the final M-B result

To convince ourself of the equivalence of these two Mellin-Barnes representations thenumerical cross-check can be made using MB program [13] It will be presented later inthis chapter At this moment let us also notice that equivalence between obtained Mellin-Barnes representations (respectively two and three dimensional) imply (after Laurentseries expansion in ǫ) non-trivial relations among individual integrals To our knowledgethis kind of rdquoexperimental mathematicsrdquo has not yet been explored in the literature

Another very interesting example is presented on Fig36 diagram b) Let us see whathappens when the order k1 rarr k2 is chosen (example b1) We limit the following exampleto show only the F polynomials which are

F1 = minus[k2 + p1]2x1x3 minus k2

2x2x3 minus sx2x4 minus [k2 + p1 + p2]2x3x4

minus[k2 minus p2]2x1x5 minus [minusk2 + p1 + p2]

2x2x5 minus sx3x5 minus k22x4x5 (323)

for the first sub-loop

F2 = minussx2x3 minus sx1x4 minus 2sx2x4 minus sx1x5 minus 2sx3x5 minus 4sx4x5 (324)

for the second sub-loop One may easily see that the number of terms in both polyno-mials will cause large (in terms of number of dimensions) Mellin-Barnes representationsThis is very unfavourable situation which makes getting analytic ǫ expand result practi-cally impossible and numerical evaluation must be done on more dimensional complexintegrals which significantly impacts the precision of numerical calculation In case ofexample b2 of Fig36 the reverse situation ie k2 rarr k1 is much more optimal Suchan order generates final 2-dim representation only As a short summary the follow-ing examples are presented Fig310 shows a two loop box diagram where the orderof iteration is not important in both cases one derives optimal (with small number ofdimensions) Mellin-Barnes representation In contradiction rdquocourtrdquo diagram Fig311must be worked out according to the following order k2 rarr k3 rarr k1 or k3 rarr k2 rarr k1It is straightforward to notice that starting with k1 would lead to the same problemsas in the example on Fig36 The following shows F polynomials that appear during

30


In[]= IntPart[1]

numerator=1

integral=PR[k20n6]PR[-k1+k20n4]PR[k2+p1+p20n5]

momentum=k2



F polynomial

-PR[k10]X[1]X[2]-sX[1]X[3]-PR[k1+p1+p20]X[2]X[3]


In[]= IntPart[2]

numerator=1

integral= PR[k10n1-z1]PR[k1+p20n2]

PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]



Gamma[2-eps-n4-n6-z1]Gamma[-z1]Gamma[2-eps-n1-n2+z1]

Gamma[2-eps-n5-n6-z2]Gamma[4-2eps-n2-n3-n4-n5-n6-z1-z2]

Gamma[-z2]Gamma[-4+2eps+n1+n2+n3+n4+n5+n6+z2]

Gamma[n6+z1+z2]Gamma[-2+eps+n4+n5+n6+z1+z2])(Gamma[n4]

Gamma[n5]Gamma[4-2eps-n4-n5-n6]Gamma[n6]Gamma[n1-z1]


Gamma[-2+eps+n3+n4+n5+n6+z1+z2])

Figure 39 Process of derivation of M-B representation for example a2 Fig36 in AMBREWe see that reversing the order of iteration gives two dimensional represen-tation It is one dimension less comparing to the case of example a1

31


k2k1

Figure 310 Double box diagram the so called rdquoladderrdquo diagram If we choose internalmomenta flow as shown the order of iteration produces optimal represen-tation in both cases

k3

k1

k2

Figure 311 Three loop diagram rdquocourtrdquo diagram In this case k2 rarr k3 rarr k1 andk3 rarr k2 rarr k1 orders of iterations are optimal

derivation of the massless rdquocourtrdquo diagram

F1 = minus[k3]2x1x2 minus [k1 + k3 + p1 + p2]

2x2x4 minus [k1 + k3 + p2]2x2x3

minus[k1 + p1 + p2]2x1x4 minus [k1 + p2]

2x1x3

F2 = minus[k1 + p2]2x1x2 minus [k1 + p1 + p2]

2x1x3 minus [k1 minus p3]2x1x4 minus tx2x4

F3 = minussx1x3 minus tx2x4 (325)

where index numerates an appropriate iteration (worked out sub-loop) The final Mellin-Barnes representation has eight dimensions

Note that propagators involving only k1 internal momenta (ie propagators supposedto be used during last iteration) appear already at the first sub-loop Of course this isnormal if one realizes that k1 momentum flows all the way around outer lines

322 Importance of F polynomial in derivation of representations

It is not a big surprise that structure of F polynomial affects the number of dimensionof the final Mellin-Barnes representation The formula Eq222 causes to produce nminus 1dimension complex integral from n term F polynomial Let us introduce the followingfive point example with two different masses Fig312 The Feynman integral introduced

32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5

Figure 312 QED-like five point diagram

together with this diagram is as follows

G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)

with the independent kinematic variables chosen in the cyclic way ie s12 s23 s34s45 s15 where sij = (p1 + pj)

2 Eq326 can be easily connected with QED Bhabhae+eminus rarr e+eminusγ process if one replaces m by electron mass and massless propagator byphoton The polynomial in this case has the following form

F = m2x22 + 2m2x2x4 + m2x2

4 + 2m2x2x5 + 2m2x4x5 + m2x25 minus s12x1x3

minus s15x2x4 + m2x3x4 minus s34x3x4 + m2x1x5 minus s45x1x5 minus s23x2x5 (327)

Numerous number of terms leads to Mellin-Barnes representation which is completelyuseless due to a huge number of integration variables Simplification of F polynomialcan be described in the following steps

bull simplification of all the mass terms into m2(xi + + xj)2 which in AMBRE is

indicated by m^2 FX[xi++xj]^2

bull constructing propagators which will be used in the next iteration This step doesnot affects above one loop five point integral

bull collecting all the terms in respect to products of Feynman parameters xixj

33


By default AMBRE automatically performs first two steps In any moment it can bechanged by switching to manual mode for calculating F polynomial for a given sub-loopusing Fauto[0] switch

Applying first and last step we obtain simplified form of F polynomial

F = m2(x2 + x4 + x5)2

minus s12x1x3 minus s15x2x4 + s34x3x4 + s45x1x5 minus s23x2x5 (328)

where

s34 = (m2 minus s34)

s45 = (m2 minus s45) (329)

From the above polynomial one sees that final M-B result is 7-dim integral Using firstBarnes-Lemma Eq2211 it can be reduced to 5-dim representation

G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz5(minus((m2)z1(minuss12)z2(minuss15)

z3(minuss23)minus3minusǫminusz1minusz2minusz3minusz4minusz5

times (s34)z4(s45q)

z5Γ(minusz1)Γ(minusz2)Γ(minusz3)Γ(minus2 minus ǫ minus z1 minus z2 minus z3 minus z4)

times Γ(minusz4)Γ(1 + z2 + z4)Γ(1 + z3 + z4)Γ(minus3 minus 2ǫ minus 2z2 minus z4 minus z5)

times Γ(minus2 minus ǫ minus z1 minus z2 minus z4 minus z5)Γ(minusz5)Γ(1 + z2 + z5)

times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))

times (Γ(minus1 minus 2ǫ)Γ(minus3 minus 2ǫ minus 2z1 minus 2z2 minus z4 minus z5)) (3210)

where typical constant factors are omitted and powers of propagators are equal oneWe can see that choice of independent kinematic variables also takes effect on numberof terms in F polynomial The substitutions of Eq329 could be made on kinematicinvariants although it is relatively easier to do the simplification directly on F polyno-mial

Calculations made using AMBRE are presented on Fig313 and Fig314 The lattershows application of first Barnes-Lemma The usage of Barnes-Lemmas as well as someissues related to them will be discussed in the next section

323 Using Barnes lemmas

In the introductory chapter about derivation of Mellin-Barnes representations it wasmentioned that such complex integrals can be simplified using the so called Barneslemmas Eq2211 and Eq2212 ie relations which change integration over complexvariable into a product of gamma functions only Such a simplification means reductionin number of dimensions in a final result The example of scalar five point functionwas already discussed in the previous section There dimension of final Mellin-Barnesrepresentation was reduced by two by applying first Barnes-Lemma twice Fig314

In AMBRE the Barnes-Lemma application can be made using two functions which tryto apply Eq2211 Eq2212 on M-B result These functions are

34


In[]= IntPart[1]

numerator=1

integral=PR[k0n1]PR[k+p1mn2]PR[k+p1+p20n3]

PR[k-p5mn5]PR[k-p4-p5mn4]

momentum=k

In[]= Fauto[0]

F polynomial

fupc=m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =

Out[]=m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]

In[]= repr=SubLoop[integral]

Iteration nr1 gtgtIntegrating over kltlt

U amp F polynomial was computed by user gtgtFauto[0]ltlt


Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5

Gamma[-z1]Gamma[-z2]Gamma[-z3]Gamma[-z4]

Gamma[n3+z2+z4]Gamma[-z5]Gamma[n1+z2+z5]

Gamma[-2+eps+n1+n2+n3+n4+n5+z1+z2+z3+z4+z5]

Gamma[-z6]Gamma[2-eps-n1-n3-n4-n5-z1-z2-z4-z5+z6]

Gamma[2-eps-n1-n2-n3-n5+z1-z2-z3-z4-z6-z7]Gamma[-z7]

Gamma[n5+z3+z4+z7]Gamma[-2z1+z6+z7])

(Gamma[n1]Gamma[n2]Gamma[n3]Gamma[n4]

Gamma[4-2eps-n1-n2-n3-n4-n5]Gamma[n5]Gamma[-2z1])

Figure 313 Derivation of Mellin-Barnes representation for the scalar five point integralHere Fauto[0] function was used to allow for the manual modification ofF polynomial (fupc)

35


In[]= BarnesLemma[repr1] n1-gt1n2-gt1n3-gt1n4-gt1n5-gt1

gtgt Barnes 1st Lemma will be checked for z7z6 ltlt

Starting with dim=7 representation

1 Checking z10Barnes Lemma was applied


gtgt Representation after 1st Barnes Lemma ltlt

1st Barnes Lemma was applied for z6z7

Obtained representation has dim=5

Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(-3-eps-z1-z2-z3-z4-z5)(s34p)^z4(s45p)^z5

Gamma[-z1]Gamma[-z2]Gamma[-z3]Gamma[-2-eps-z1-z2-z3-z4]

Gamma[-z4]Gamma[1+z2+z4]Gamma[1+z3+z4]

Gamma[-3-2eps-2z2-z4-z5]Gamma[-2-eps-z1-z2-z4-z5]

Gamma[-z5]Gamma[1+z2+z5]Gamma[3+eps+z1+z2+z3+z4+z5])

(Gamma[-1-2eps]Gamma[-3-2eps-2z1-2z2-z4-z5]))

Figure 314 Simplification of Mellin-Barnes QED-like pentagon representation usingAMBRE Application of Eq2211 was possible on integration variables z7

and z6 After eliminating two integration variables final result is 5-dimrepresentation In the end all the powers of propagators were set to one

36


bull First Barnes-Lemma BarnesLemma[repr_1]

bull Second Barnes-Lemma BarnesLemma[repr_2]

s

m2

m1

m3

p22 = M2

2

p21 = M2

1

Figure 315 One loop vertex diagram with different masses

In this section we will focus on a situation when normal application of lemmas is notpossible due to integration variables zi appearing in the exponents Let us follow theexample of M-B representation for one-loop vertex diagram Fig315

G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)

minus1minusǫminusz1minusz2minusz3minusz4minusz5minusz7

times MM z2

1 MM z4

2 (minuss)z7

times Γ(minusz1)Γ(minusz2)Γ(minusz3)Γ(minusz4)Γ(1 + 2z1 + z2 + z4)Γ(minusz5)Γ(minusz6)Γ(minusz7)

times Γ(1 + z2 + 2z3 + z5 + z6 + z7)Γ(1 + ǫ + z1 + z2 + z3 + z4 + z5 + z6 + z7)

times Γ(minus1 minus 2ǫ minus 2z1 minus 2z2 minus 2z3 minus z4 minus z5 minus z6 minus z7))Γ(1 minus 2ǫ)) (3211)

As it can be seen after carefully analysing Eq3211 above Barnes first lemma can beapplied to z6 only ie it appears in gammas twice with negative sign and twice withpositive sign and it is absent in the powers and in the gammas of the denominatorAll the remaining integration variables zi appear in powers which does not allow tointroduce first lemma One can counteract this situation by applying a shift to variablesz3 z5 ie z3 rarr z3 minus z5 This cancels all the z5 appearance in the powers Note that itwas possible only because z3 + z5 was present in one of the powers and minusz3 minus z5 in thesecond one

Technically the algorithm implemented in the AMBRE function responsible for Barnes-Lemmas searches for the pairs of two integration variables zi + zj and zi minus zj whichafter application of the appropriate shift cancels them from the powers This behaviouris implemented in the program as an option which can be turned on or off using booleanparameters All the procedure of doing first Barnes-Lemma on M-B representationpresented Eq3211 is presented in Fig316

Quite recently the special MATHEMATICA package barnesroutines specially dedicatedto Barnes-Lemma has been released and is publicly available [47]

37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7


Gamma[1+2z1+z2+z4]Gamma[-z5]Gamma[-z6]

Gamma[-1-2eps-2z1-2z2-2z3-z4-z5-z6-z7]

Gamma[-z7]Gamma[1+z2+2z3+z5+z6+z7]

Gamma[1+eps+z1+z2+z3+z4+z5+z6+z7])Gamma[1-2eps])

In[]= BarnesLemma[repr 1 Shifts-gtTrue]

gtgt Shifting z3-gtz3-z5








Out[]=-(((m1^2)^z1(m2^2)^z3(m3^2)^(-1-eps-z1-z2-z3-z4-z7)

MM1^z2MM2^z4(-s)^z7Gamma[-z1]Gamma[-z2]

Gamma[-eps-z1-z2-z3]Gamma[-z3]Gamma[-2eps-2z1-z2-z4]

Gamma[-z4]Gamma[1+2z1+z2+z4]Gamma[-z7]

Gamma[1+z2+z3+z7]Gamma[1+eps+z1+z2+z3+z4+z7])

(Gamma[1-2eps]Gamma[1-eps-z1+z7]))

Figure 316 AMBRE calculations of applying first Barnes-Lemma on Eq3211 The ap-propriate shift allows to do lemma substitution not only on z6 but on z5 aswell

38

33 One-loop integrals with tensor numerator


Calculation of a given process containing loop integrals involve tensor Feynman inte-grals Very nice feature of AMBRE is that it allows to derive Mellin-Barnes representationsnot only for scalar loop diagrams but also for m-rank one loop integrals The result canbe obtained not only for the tensor numerator contracted with some external momentumandor metric tensor ie k middotp k middotk etc but also for tensor structure involving internalmomentum with Lorentz index only ie without contracting it with a projector

In case of one-loop integrals Eq2116 can be re-written into the following form


Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)

where T (k) = kmicro1 kmicrom One can also note the Eq2121 property for one-loop caseswhich simplifies generation of a tensor structure Let us see the following rank two tensorexample (compare with Eq2120)

G(kmicro1kmicro2) rarr P micro1P micro2 + gmicro1micro2 rarr Qmicro1Qmicro2 + gmicro1micro2 (332)

In AMBRE tensor structure generation is implemented in the general way so it works forany rank Practically it was tested up to rank nine on some diagrams5

The practical application of AMBRE in this thesis is calculation of infrared parts comingfrom five and four point diagrams for the QED e+eminus rarr micro+microminusγ process (the last chapter)As it is shown later during generation of amplitudes for this process we deal withFeynman integrals up to rank three The same maximum rank we have for the Bhabhae+eminus rarr e+eminusγ reaction Simultaneously in the previous section the one-loop scalarfive point QED diagram was presented by Eq326 Now this example is going to beextended on the case of rank two tensor integral

G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)

5 The mentioned highest rank m = 9 was tested on self-energy diagram although such a rank is notphysical it gives confidence that algorithm was implemented correctly More complicated diagramswere tested as well Firstly against IBP relations generated by IdSolver program [48] and verifiedby CSectors

39


From the technical point of view construction of Mellin-Barnes representation for one-loop tensor integrals is very similar to the scalar ones In both cases we will have thesame F polynomial The only difference is the numerator represented by the objectArP

mminusr[micro1microm]

which is present in Eq331 It inserts additional Feynman param-eters via Qmicro vectors (which is a sum of products of external momentum and Feynmanparameter) These new parameters must be included during integration over Feynmanparameters Eq229 Also because of the sum over r in Eq331 the AMBRE result isoutputed as a list divided according to this sum This allows to separate terms contain-ing metric tensor and combinations of external momenta (the so called chords) AMBRE

result for this tensor pentagon integral Eq333 is presented on Fig317At the end of this section a brief discussion about implementing ability of deriva-

tion multi-loop tensor integrals into AMBRE is discussed The main difficulty of suchimplementation compared to one-loop cases is the rdquoloop by looprdquo algorithm itself

We will show this considering the following two-loop self-energy diagram where allthe fields were set to be massless and numerator is of the rank three (note differentinternal momenta in the numerator)

int

(k1 middot p)(k1 middot p)(k2 middot p)

[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)

The calculation starts by working out sub-loop over k1 which lead to the following Fpolynomial

F = minus[k2]2x1x2 minus sx1x3 minus [(k2 + p)2]x2x3 (335)

Before we proceed to next sub-loop let us see the structure related to the numerator(tensor structure)

P micro1P micro2 + gmicro1micro2 rarr Qmicro1Qmicro2 + gmicro1micro2

rarr (kmicro1

2 x2 minus pmicro1x3)(kmicro2

2 x2 minus pmicro2x3) + gmicro1micro2

rarr kmicro1

2 kmicro2

2 x22minuskmicro2

2 pmicro1x2x3 kmicro1

2 pmicro2x2x3 pmicro1pmicro2x2

3 gmicro1micro2 (336)

We clearly see that from that point we will have to perform our second sub-loop calcula-tions separately for all above parts because of different tensor ranks in the next iteration(the one over k2) Situation after first sub-loop can be presented in the following way

int

p1micro1p1micro2

(k2 middot p)

[k22]

minusz1 [(k2 + p)2]minus3+ǫ+n1+n2+n3+z1+z2

times kmicro1

2 kmicro2

2 MB1minuskmicro2

2 pmicro1MB2 kmicro1

2 pmicro2MB3 pmicro1pmicro2MB4 g

micro1micro2MB5ddk2 (337)

where MBi states for a Mellin-Barnes part of the integral for a given part of expressionAll these parts are slightly distinct due to different Feynman parameters in Eq336Here as an explicit example we show MB1

MB1 = ((minus1)2minusǫminusz2(minuss)z2Γ(2 minus ǫ minus n1 minus n2 minus z1)Γ(minusz1)Γ(4 minus ǫ minus n1 minus n3 minus z2)

times Γ(minusz2)Γ(n1 + z1 + z2)Γ(minus2 + ǫ + n1 + n2 + n3 + z1 + z2))

times (Γ(n1)Γ(n2)Γ(6 minus 2ǫ minus n1 minus n2 minus n3)Γ(n3)) (338)

40


In[]= Fullintegral[k[mu1]k[mu2]PR[k0n1]PR[k+p1mn2]

PR[k+p1+p20n3]PR[k+p4+p5mn4]PR[k+p5mn5]k]

In[]= IntPart[1]

numerator=k[mu1]k[mu2]


PR[k+p5mn5]PR[k+p4+p5mn4]

momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]

In[]= repr = SubLoop[integral]




ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(3-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)(s34p)^z4

(s45p)^z5g[mu1mu2]Gamma[-z1]Gamma[-z2]Gamma[-z3]

Gamma[-z4]Gamma[n3+z2+z4]Gamma[-z5]Gamma[n1+z2+z5]

Gamma[-3+eps+n1+n2+n3+n4+n5+z1+z2+z3+z4+z5]Gamma[-z6]

Gamma[3-eps-n1-n3-n4-n5-z1-z2-z4-z5+z6]



(2Gamma[n1]Gamma[n2]Gamma[n3]Gamma[n4]


Figure 317 Fragment of derivation of M-B representation for QED Bhabha pentagonwith rank two numerator in AMBRE Note that calculation was made withnot contracted numerator The final result is by default shortened in thespecial function ARint Index of it corresponds to the sum over r in Eq331ie in this example rdquo1rdquo is for Qmicro1Qmicro2 term and rdquo2rdquo for gmicro1micro2 The latterwas extracted using in-build ARint[2repr] function

41


It is clear that during more than two loops additional third iteration can cause to furtherrdquofragmentationrdquo of the expression The above situation was visualised on Fig318 whereit is clear that during calculation we get separate Mellin-Barnes terms which must beprocessed separately

int (k1middotp)(k1middotp)(k2middotp)

[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2

int p1micro1p1micro2

(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2


2 pmicro2MB3 pmicro1pmicro2MB4 gmicro1micro2MB5

Figure 318 Process of calculation of rank three two-loop self-energy diagram Eq334Second step corresponds to Eq336 We see that expression fragments toseparate M-B terms which must be treated separately

The last two sections of this chapter summarises work on Mellin-Barnes representationand AMBRE package In the following numerical cross-checks with CSectors in Euclideanregion and discussion about applications and perspectives is going to be presented

34 Cross-checks in Euclidean region with CSectors

Up to now all presented Mellin-Barnes examples were illustrated without independentfinal result comparison Current section covers this gap and provides numerical resultsfor M-B representations derived using AMBRE and numerically calculated in MB and thevery same diagrams computed using sector decomposition method in CSectors As itwas already pointed out MB package is able to perform numerical integration of complexintegrals which appear after analytic continuation and expansion in ǫ One dimensionalintegrals are calculated using MATHEMATICA in-build functions and higher dimensional

42


ones are integrated via Fortran6 The numerical integration in MB is executed using thefollowing function

MBintegrate[integrals kinematics options] (341)

where integrals is a set of integrals to evaluate and kinematics is a set of rulesfor kinematic variables providing numerical input to the computation routines egs-gt-2 t-gt-3 The last one is a set of options which in detail are described in [13] Incase of CSectors numerical computation was done using sector decomposition [44]set of routines7

We begin numerical cross-checks by going back to the two-loop vertex example a)in Fig36 (V6l0m case) Here we will show that order of iteration in rdquoloop by looprdquoalgorithm leads to the same numerical results Also additional verification was madewith the help of CSectors Below we show numerical output up to constant part ie ǫ0

V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mCSectors = (79974 plusmn 39) +(24515 plusmn 06)

ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)

where point s = minus 111

was used as an input parameter The output of the first two resultsgiven by MB package may seem incorrectly typed due to a number of digits left and lack oferrors coming from numerical integration Here we indicated common parts of numericalresults by vertical lines In fact terms ǫminus4 and ǫminus3 are exactly the same because theircontribution comes from non-integrals (some analytic expression) Here MATHEMATICA

can provide huge number of digits of precision for such cases For ǫminus2 ǫminus1 and constantparts the contribution is from one and two dimensional complex integrals which arecalculated in MATHEMATICA (within MB) and Fortran respectively In contrast to the

6 The code is linked with CUBA libraries [49] which is a library offering a choice of four independentroutines for multidimensional numerical integration Vegas Suave Divonne and Cuhre CERNlib

[50] implements complex gamma (Γ) functions and its derivative (ψ) By default deterministicCuhre algorithm is used for integration when dimension lt 5 Above this dimension thresholdMonte Carlo Vegas is used This behaviour can be changed by an appropriate option

7 It is an implementation in C++ using GiNaC libraries for the symbolic part and the GNU ScientificLibrary for the numerical part (Monte Carlo integration)

43


second one the first one does not provide errors which should come within numericalintegration In other words error information is useless In case of CSectors this is not aproblem at all All the uncertainties are calculated via sector decomposition libraries(see Appendix B)

The rest of numerical cross-checks is presented in the following tables During numer-ical calculations all the masses and powers of propagators were set to one

AMBRE and MB CSectors

ǫ0 minus03933253674084889 minus0393325 plusmn 56 middot 10minus7

ǫminus1 1 1 plusmn 14 middot 10minus6

T [s] 43 76s = minus3

Table 31 Numerical results for the self-energy diagram Fig23 All the computationshave been made on Intel Core Duo 266GHz with 1GB memory


ǫ0 minus20314 plusmn 0001 minus20315 plusmn 0001ǫminus1 minus53229(62554815002) minus53231 plusmn 00002

T [s] 38 154s = minus11 t = minus12

Table 32 Results for two-loop Master Integral (B5l2m2) Fig31 appearing in two loope+eminus rarr e+eminus Bhabha process


ǫ0 022858 plusmn 000005 0228637 plusmn 000004ǫminus1 0136722 0136726 plusmn 99 middot 10minus6

T [s] 223 163s12 = minus3 s23 = minus7 s34 = minus12 s45 = minus111 s15 = minus6

Table 33 Numerics for the one loop QED pentagon diagram Fig312

If we analyse these results and time of calculation we can define the following findingNumerical calculations using Mellin-Barnes method often takes longer for diagrams in-volving masses than in case of the very same massless cases Such situation is due tohigher number of terms appearing in F polynomial which in turn translates to moredimensional M-B representation and more difficult integration in the end In case ofsector decomposition (CSectors) situation is reversed Here massless cases often causesserious troubles The number of generated sectors easily can cause to fill completelyRAM and SWAP memory For example calculation of the three-loop massless ladder

44

35 Applications and perspectives of AMBRE


ǫ0 minus0103415 plusmn 6 middot 10minus6 minus0103354 plusmn 00004ǫminus1 minus0109074 minus0109271 plusmn 000009ǫminus2 minus000965743 minus000966186 plusmn 000002ǫminus3 00831878 0083189 plusmn 4 middot 10minus6

ǫminus4 minus00228571 minus0022858 plusmn 1 middot 10minus6


Table 34 Numerical result for the massless double box Fig310

box (the extension of the diagram in Fig310) is relatively easy from numerical pointof view using Mellin-Barnes method But it is a serious challenge when using sectordecomposition method


Modern high energy physics requires performing sometimes difficult calculations whichoften are impossible without computer Many aspects of numerical and analytical com-putations can be dressed into appropriate algorithms and in the end automatised usingsome programming language The automatisation of derivation of Mellin-Barnes repre-sentations was the most important reason in developing AMBRE package The existence ofsuch software shortens time spend on M-B calculations and eliminates possible humanmistakes during intermediate steps The AMBRE was entirely written in MATHEMATICA

Computer Algebra System language which in natural way connects its output with MB

package also written in the very same CASAMBRE has already been used in a few important calculations in QED and QCD It

helped to solve analytically many new Master Integrals eg for three loop massless QCDform factors [39 40 51] in massive QED Bhabha [52] and Maximally SupersymmetricYang-Mills theory [53]

Presently we are using it for numerical checks of new algorithm which solves MasterIntegrals using relations among two-loop integrals which follows from Gram determi-nants [46] It is eg used in calculations of the so called rdquodouble pentagonrdquo Fig319Here we meet sixteen dimensional integrals For that we have introduced an AMBRE ver-sion capable of computing two-loop tensor integrals Relations among these integralsinclude dozens of tensor double pentagons We use powerful computers to solve themnumerically The obvious perspective of AMBRE is generalisation from tensor m-rank one-and two- loop integrals to L-loop cases Another interesting extension could be tried incase of non-planar diagrams

This chapter ends discussion about AMBRE application of Mellin-Barnes In the lastchapter we will present IR divergences calculated with the help of this program for five

45


Figure 319 The topology of the so called rdquodouble pentagonrdquo as an example of one ofthe various applications of AMBRE

and four point one-loop diagrams presented in the e+eminus rarr micro+microminusγ QED process Forthat the following chapter describes reduction of six and five point one-loop integrals

46

4 Tensor reduction of five and six point one loop

integrals

Reduction of n point one-loop Feynman diagrams to a sum of its reduced diagrams hasa long history going back in time to the early 60rsquos of the last century [54] It was thenindicated that the amplitude for n-point one-loop diagram with n gt 6 can be writtenas a sum of the reduced diagrams containing only five external legs [55] Later thiswork was generalized and five point one-loop diagrams were introduced as a sum of boxdiagrams [56] In fact authors of that time also proved that triangle one-loop diagramcould be reduced to the sum of three self-energy diagrams if the space-time would havebeen two dimensional [57] Later work was focused on reduction schemes for five and sixpoint integrals so that unwanted Gram determinant could be avoided [17 20 21 16]

In this chapter we focus on evaluation of tensor one-loop integrals especially the oneswith five and six external legs Firstly the well known Passarino-Veltman reductionscheme is shortly introduced in order to brought in Gram determinants which often arethe source of numerical instability Later the scheme of reduction based on algebra ofrdquosigned minorsrdquo [54] in which unwanted determinants were cancelled is presented [1516] The chapter is summarized by the numerical cross-checks made with the speciallydeveloped new software along this scheme hexagon1 and non-public Fortran program

This chapter can be treated as a continuation of [17] We not only focus on cancellationof Gram determinants for high ranks of tensor five point integrals but we also finalisedour work by developing reduction software This is an important step towards workingout good numerically stable software for this kind of numerical calculations Becausediscussion about presented reduction method could fill all the thesis we focus only onsome basic ideas on which this reduction scheme is based on More results can be foundin our main paper [16]

41 Passarino-Veltman reduction

The Passarino-Veltman tensor reduction [14] was historically first systematic proce-dure which could be implied for one-loop integrals This chapter will briefly introducethis method which is to be the base for the later discussion As an example let us first

1 It is available here httppracusedupl~gluzahexagon

47

4 Tensor reduction of five and six point one loop integrals

write the basic one-loop Feynman integrals up to three legs and up to rank two

A(m1) =

int

ddk

iπd2

1

P1

B0micromicroν(p1m1m2) =

int

ddk

iπd2

1 kmicro kmicroν

P1P2

C0micromicroν(p1 p2m1m2m3) =

int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21

P2 = (k minus p1)2 minus m2

2

P3 = (k minus p1 minus p2)2 minus m2

3 (412)

Obviously more external legs and higher rank tensor generalisation is possible herePassarino-Veltman reduction uses the fact that due to the Lorentz symmetry the resultcan only depend on tensor structures built from the external momenta pmicro

i and the metrictensor gmicroν Therefore tensor integrals are written in terms of coefficients multiplied byexternal momenta andor metric tensor ie

Bmicro = pmicro1B1

Bmicroν = pmicro1p

micro1B11 + gmicroνB00

Cmicro = pmicro1C1 + pmicro

2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν

1pmicro2)C12 + qmicro

2 qν2C22 + gmicroνC00 (413)

By contracting both sides of above expressions with external momenta and the metrictensor it is possible to find a solution for the coefficients Bi Bij After contractionof left hand sides scalar products appear which can be easily expressed in terms ofpropagators eg

k middot p1 =1

2

(

[k2 minus m21] minus [(k minus p1)

2 minus m22] + m2

1 minus m22 + p2

1

)

(414)

In the case of the two point vector integral we would end up with the following formulaallowing to find B1 coefficient

p21B1 =

1

2

(

(p21 + m2

1 minus m22)B1 + A(m2) minus A(m1)

)

(415)

Note that A integrals appeared after one of the propagators had been cancelled in thetwo point B function For the other case ie vector three point integral where explicitcalculation of pimicroC

micro was omitted for simplicity we have(

p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48

42 Gram determinants and algebra of signed minors

We see that when trying to solve this system of equations the inverse matrix togetherwith determinant appear in the denominator

2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)

The most important drawback of this algorithm is related to these determinants Insome phase space regions Gram determinants can tend to zero resulting in coefficientstaking large values with possible cancellations among them This is the main difficultyin development of numerically stable program for automated evaluation of tensor loopintegrals


In the previous section we have seen how Gram determinants appear in the Passarino-Veltman tensor reduction scheme the three point diagram example was given Eq417Such determinants are obviously generalized to the case of n-point one-loop diagrams

Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣

p21 p1 middot p2 p1 middot pnminus1

p1 middot p2 p22 p2 middot pnminus1

p1 middot pnminus1 p2 middot pnminus1 p2nminus1

∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)

At this point we are going to introduce the basic of algebra used in our reduction schemeFor a diagram with internal lines 1 n the so called rdquomodified Cayley determinantrdquocan be introduced

()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)

with Yij = minus(qi minus qj)2 + m2

i + m2j where qi are chords of a given diagram Both

determinants are related by()n = minusGnminus1 (423)

and from now on we will name ()n the Gram determinant of the Feynman integralBy cutting from ()n rows j1 j2 and columns k1 k2 we get the so-called rdquosignedminorsrdquo The sign of this object is determined by the sum of indices of excluded rowsand columns and by taking into account the appropriate signatures of the permutationstaken separately from excluded rows and columns ie

(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣

rows j1j2 deletedrows k1k2 deleted

∣

∣

∣

∣

(424)

49


Excluded rows and columns of Cayley determinant are numbered from zero in this waywe have

(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)

Using rdquosigned minorsrdquo one can write tensor reduction formulas in a compact and elegantway but before we proceed further let us see some interesting properties related to themThey are crucial if we want to write reduction formulas without Gram determinantsFirst of all one evidently sees that due to the symmetric Cayley determinant the rdquosignedminorrdquo objects are symmetric in replacing excluded rows by columns

(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)

The other two very important properties which will be needed are

nsum

i=1

(

0i

)

n

=

( )

n

(427)

together with its extensionnsum

i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)

and similar to Eq427 but with j 6= 0 inserted

nsum

i=1

(

ji

)

n

= 0 (429)

The last two that will be needed are

(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)

All of these properties have been already discussed in [54] where also the detailed andcomplete introduction to the algebra of rdquosigned minorsrdquo was given

50

43 Tensor integrals and shifted space-time dimensions


At first we give the reduction of tensor integrals to a set of scalar integrals for arbitraryn-point functions Following [17 58] and setting all the powers of propagators to beequal to one one has

Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)

where [d+] is an operator shifting the space-time dimension by two units and

I[d+]lstumiddotmiddotmiddotp i j kmiddotmiddotmiddot =

int [d+]l nprod

r=1

1

P1+δri+δrj+δrk+middotmiddotmiddotminusδrsminusδrtminusδruminusmiddotmiddotmiddotr

int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)

with propagator denominators

Pj = (k minus qj)2 minus m2

j (434)

By combining integration by parts identities with relations connecting integrals in dif-ferent space-time dimensions [28] one obtains the following basic recurrence relations[17]

()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)

44 Reduction of pentagons up to rank two

We begin the following reduction scheme based on the algebra of rdquosigned minorsrdquowith the example of scalar pentagon diagrams For the scalar five point integral weuse recursion relation Eq436 The operator kminus decreases the k-th propagator power

51


by one Because in our example we have all powers of propagators equal to one andbecause we are interested in the case with n = 5 Eq436 changes into

(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)

With I[d+]5 finite in the limit d rarr 4 we get as a final reduction formula for the scalar five

point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)

ie five point integral is expressed in terms of scalar four point functions Is4 which are

obtained by scratching s-th line in the pentagon diagram In the case of vector five pointintegral (rank m = 1) we have

Imicro5 =

4sum

i=1

qmicroi I5i (443)

with

I5i equiv minusI[d+]5i = (d minus 4)

(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)

where again in the limit d rarr 4 the I[d+]5 disappears and finally

I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)

Is4 as in the case of scalar integrals are constructed by cancelling s-th line in five point

diagram Scalar and vector cases are simple and lead to a direct reduction to scalar fourpoint integrals without the Gram determinant ()5 In the following we reduce tensorintegrals of rank two and show like in [20 59] that also in these cases the unwantedGram determinant can be cancelled

Let us now progress with more complicated case which is rank two five point integralThis tensor integral can be written Eq432

Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52


By replacing the metric tensor gmicroν by2

gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)

and again using recurrence relation Eq435 we obtain

Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)

Now I[d+]5i can be replaced by rank two coefficient ie Eq444 and I

[d+]s4i is constructed

using recurrence relation Eq435 Explicit formulas for recurrence relations of specificI5 and I4 cases can be found in [16] Finally we end up with the following result

I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)

where four point Is4 and three point Ist

3 integrals were factorised Now we would like touse algebra of signed minors to somehow cancel unwanted Gram determinant ()5 Beforewe do this let us change Eq449 by adding and subtracting into coefficient of Is

4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)

2 The Eq447 is based on the assumption that the chord qmicro5 = 0 (otherwise the summation variables

i and j would run from 1 to 5) With this choice one can prove Eq447 by contracting both sidesof the equation with eg qlmicro That is sufficient because any (four dimensional) vector must be alinear combination of qlmicro To work out the rhs one needs to write 2qi middot ql = Yil minus Yi5 minus Y5l + Y55

[54] and then use some identities for sums likesum5

i=1

(

ij

)

5

Yil

53


and for the expression in the bracket of Ist3 we add the following zero term

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)

which also uses the fact that

sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)

After simple modifications we get such an intermediate expression

I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)

At this stage we would like to show that Gram determinant ()5 really factorises fromthe bracket terms so that it cancels with the one in the denominator So we postulate

54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)

We start from proving Eq4414 To begin we show that rhs of is symmetric in theindices i and j for fixed s Obviously the third term is symmetric due to the symmetricCayley determinant Eq426 The symmetry of the first two terms means

(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)

By using relation Eq4210 and basic properties of signed minors we do the followingsubstitution

(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)

where we used Eq426 Eq4210 Inserting above into Eq4416 results in

(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)

which is easily proved using Eq4211 all the terms will immediately cancel

Now let us see the simplest case of Eq4414 ie As0ss In this case we see that As0

ss = 0which in turn implicates

Xs0ss = 0 (4419)

If we move further we see that by applying Eq427 and Eq429 to Eq4414 we get

5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55


and due to the symmetry in i and j we also have

5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)

Both results give us a hint of how Xs0ij might look like namely due to Eq4420 it should

contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)

A further contribution must vanish after summing over i Due to Eq4419 it mustcontain a factor3

(

0 js i

)

5

(4423)

The second factor of this contribution can only depend on s and has been determinedby explicit calculation to be

(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)

We limit to prove only Xs0ij all the necessary explanation for the Eq4415 can be found

in [16] The final result of Xstij is as follows

Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)

Now we see that Gram determinant was cancelled partially in I5ij

I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56

45 Reduction of six point functions

It is still presented in the remaining two terms but this is not a problem at all Againusing expression for metric tensor Eq447 we finally end with a compact reductionformula for rank two five point integral without ()5

Imicro ν5 =

4sum

ij=1

qmicroi qν

j Eij + gmicroνE00 (4428)

Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)

The higher rank tensor integrals were derived using the same recurrence relations andrdquosigned minorsrdquo algebra In case of five point integrals the reduction formulas werederived up to rank three in [16]


In case of hexagons there is a nice property which states that tensors of rank m can bereduced to a sum of six five point tensor integrals of rank mminus 1 This property has alsobeen derived in [20] an earlier demonstration of this property has been given already in[17] This simplification is due to the fact that their Gram determinant vanishes ()6 = 0[17] Apart from that the above results for the five point tensor integrals can be directlyused thus reducing the six point tensors to scalar four three and two point integralsAccording to recurrence relations we write

I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)

Here we see already the general scheme of reducing six point functions to five pointfunctions In general in any rdquosigned minorrdquo a further column

(

rr

)

6

(453)

is added That is because we now use Cayley determinant of dimension six and additionalrminuspart is scratched As in Eq443 and Eq444 we obtain

Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)

While in Eq444 the first part vanishes in the limit d rarr 4 here its disappearance isdue to Eq427 and ()6 = 0

5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)

Eq456 will play a crucial role for the higher tensor reduction The resulting form inEq455 is already the generic form for the higher tensors too Therefore it is useful tointroduce a vector

vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)

With this definition we can write vector integral in a compact way

Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58

46 Numerical results and discussion

We see that vector six point integral depends on scalar five point function In facttensor rank two and three hexagon integrals will be reduced according to the schemewhere dependence on rank one and two respectively will be observed All the reductionformulas for the hexagons up to rank four were given in [16]


In order to check the correctness of the presented reduction scheme the hexagon

package was written As it allows to obtain also fully analytic result it appeared to bevery helpful tool during process of derivation of reduction formulas During the stage ofcross-checking we have made the following verifications

bull internal checks were used for tensor integrals and consisted mainly in writing ascalar product of internal and external momenta in terms of lower rank tensorintegrals which had been checked before

bull external checks were made with use of LoopTools [7] (five point integrals) AMBREwith MB (five point integrals) Csectors (five and six point integrals)

We present them in Table 41 For some cross-checks the coefficient of tensor integralswas only calculated Our notation is presented in the following

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1

g[microνqλ]i E00i (461)

59


Figure 41 Momenta flow used in the numerical examples for five and six point integrals

where F stands for six point integrals and E for five point functions The momentaflow convention in six and five point diagrams is presented on Fig41 In all thesechecks we used LoopTools to calculate the finite parts of the scalar four three andtwo point functions which appear after the reduction by hexagon We note that ingeneral the functions defined directly in LoopTools may not be sufficient to cover thewhole kinematic phase space obtained from reduction of six point functions But as analternative it is possible to use other available libraries eg QCDLoop [60]

Numerical results in Table 41 are divided in to parts described below

1) Here we used AMBRE and MB to check the decomposition in the Euclidean kinematicregion including tensor structures Because LoopTools does not provide resultcontaining ǫ terms so we have limited our result which is contracted tensor of rankthree to constant part only At the level typical for Monte-Carlo calculations it isin agreement with the result from Mellin-Barnes integration which was 0218885

2) The second example comes from [61] (Table 2 region I) Here we use CSectors

and show for the case of the scalar six point function agreement with the calculationusing the sector decomposition method (typically five digits accuracy)

3) In this example we give cross-check for scalar five point function between hexagonLoopTools and [61] (Table 1 region III) The other tensor calculated tensor coef-ficients agree directly with LoopTools

Because two independent implementations of reduction scheme were developed hexagonand the one in Fortran the cross-checks were also made among these two packages Inthe Table-42 we give phase space points corresponding to the reaction gg rarr ttqq withexternal momenta generated by Madgraph [62 63] In the other tables we give numericalresults (which is presented as tensor components) for this phase space point All theresults are the same for two programs

60


1) Comparison with AMBRE amp MBm pmicro1p

ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1

s12 = minus3 s23 = minus6 s34 = minus5 s45 = minus7 s15 = minus2In RedE3[ p2

1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741

2) Comparison with sector decomposition and [61] F0

Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1

s12 = s23 = s34 = s45 = s56 = s16 = s123 = s234 = minus1 s345 = minus52In RedF0[ p2

1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526

3) Comparison with LoopTools E0 [61] E1 E2 E3 E4 E34 E123 E002

Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2

5 = 811600s12 = 4 s23 = minus15 s34 = 15 s45 = 310 s15 = minus12In RedE0[ p2

1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I

In RedEget[rank1 p21 p

25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I

In RedEcoef[ee34 p21 p

25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I

Table 41 Numerical cross-checks

61


p1 021774554 E+03 00 00 021774554 E+03p2 021774554 E+03 00 00 ndash 021774554 E+03p3 ndash 020369415 E+03 ndash 047579512 E+02 042126823 E+02 084097181 E+02p4 ndash 020907237 E+03 055215961 E+02 ndash 046692034 E+02 ndash 090010087 E+02p5 ndash 068463308 E+01 053063195 E+01 029698267 E+01 ndash 031456871 E+01p6 ndash 015878244 E+02 ndash 012942769 E+02 015953850 E+01 090585932 E+01

m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600

Table 42 The components of external four-momenta for the six point numerics whereall internal particles are massive

F0

ndash 0223393 Endash18 ndash i 0396728 Endash19micro F micro

0 0192487 Endash17 + i 0972635 Endash171 ndash 0363320 Endash17 ndash i 011940 Endash172 0365514 Endash17 + i 0106928 Endash173 0239793 Endash16 + i 0341928 Endash17micro ν F microν

0 0 0599459 Endash14 ndash i 0114601 Endash140 1 0323869 Endash15 + i 0423754 Endash150 2 ndash 0294252 Endash15 ndash i 0375481 Endash150 3 ndash 0255450 Endash14 ndash i 0195640 Endash141 1 ndash 0164562 Endash14 ndash i 0993796 Endash161 2 0920944 Endash16 + i 0706487 Endash171 3 0347694 Endash15 ndash i 0127190 Endash162 2 ndash 0163339 Endash14 ndash i 0994148 Endash162 3 ndash 0341773 Endash15 + i 0818678 Endash173 3 ndash 0413909 Endash14 + i 0670676 Endash15

Table 43 Tensor components for scalar vector and rank m = 2 six-point functionskinematics defined in Table 42 and Fig41

62


micro ν λ F microνλ

0 0 0 ndash 0227754 Endash11 ndash i 0267244 Endash120 0 1 0140271 Endash13 ndash i 0119448 Endash120 0 2 ndash 0201270 Endash13 + i 0101968 Endash120 0 3 0102976 Endash12 + i 0624467 Endash120 1 1 0183904 Endash12 + i 0142429 Endash120 1 2 ndash 0131028 Endash13 ndash i 0610343 Endash140 1 3 ndash 0543316 Endash13 ndash i 0158809 Endash130 2 2 0181352 Endash12 + i 0141686 Endash120 2 3 0506408 Endash13 + i 0163568 Endash130 3 3 0600542 Endash12 + i 0130733 Endash121 1 1 ndash 0563539 Endash13 + i 0178403 Endash131 1 2 0210641 Endash13 ndash i 0584990 Endash141 1 3 0120482 Endash12 ndash i 0574688 Endash131 2 2 ndash 0201182 Endash13 + i 0620591 Endash141 2 3 ndash 0686164 Endash14 + i 0205457 Endash141 3 3 ndash 0447329 Endash13 + i 0193180 Endash132 2 2 0582201 Endash13 ndash i 0163889 Endash132 2 3 0119659 Endash12 ndash i 0570084 Endash132 3 3 0457464 Endash13 ndash i 0181141 Endash133 3 3 0557081 Endash12 ndash i 0374359 Endash12

Table 44 Tensor components for a massive rank R = 3 six-point function kinematicsdefined in Table 42 and Fig41

63


micro ν λ ρ F microνλρ

0 0 0 0 0666615 Endash09 + i 0247562 Endash090 0 0 1 ndash 0200049 Endash10 + i 0294036 Endash100 0 0 2 0200975 Endash10 ndash i 0237333 Endash100 0 0 3 0645477 Endash10 ndash i 0162236 Endash090 0 1 1 ndash 0116956 Endash10 ndash i 0516760 Endash100 0 1 2 0160357 Endash11 + i 0222284 Endash110 0 1 3 0792692 Endash11 + i 0729502 Endash110 0 2 2 ndash 0111838 Endash10 ndash i 0513133 Endash100 0 2 3 ndash 0681086 Endash11 ndash i 0708933 Endash110 0 3 3 ndash 0804454 Endash10 ndash i 0801909 Endash100 1 1 1 0100498 Endash10 ndash i 0151735 Endash130 1 1 2 ndash 0348984 Endash11 ndash i 0195436 Endash120 1 1 3 ndash 0211111 Endash10 + i 0295212 Endash110 1 2 2 0357455 Endash11 + i 0662809 Endash140 1 2 3 0121595 Endash11 ndash i 0807388 Endash130 1 3 3 0825803 Endash11 ndash i 0142086 Endash110 2 2 2 ndash 0958961 Endash11 ndash i 0585948 Endash120 2 2 3 ndash 0209232 Endash10 + i 0289031 Endash110 2 3 3 ndash 0802359 Endash11 + i 0994701 Endash120 3 3 3 ndash 0102576 Endash09 + i 0378476 Endash101 1 1 1 ndash 0246426 Endash10 + i 0276326 Endash101 1 1 2 0915670 Endash12 ndash i 0660629 Endash121 1 1 3 0303529 Endash11 ndash i 0287480 Endash111 1 2 2 ndash 0822697 Endash11 + i 0919635 Endash111 1 2 3 ndash 0116294 Endash11 + i 0100024 Endash111 1 3 3 ndash 0146918 Endash10 + i 0183799 Endash101 2 2 2 0908296 Endash12 ndash i 0654735 Endash121 2 2 3 0109510 Endash11 ndash i 0100875 Endash111 2 3 3 0717342 Endash12 ndash i 0557293 Endash121 3 3 3 0450661 Endash11 ndash i 0485065 Endash112 2 2 2 ndash 0245154 Endash10 + i 0274313 Endash102 2 2 3 ndash 0318500 Endash11 + i 0279750 Endash112 2 3 3 ndash 0146317 Endash10 + i 0182912 Endash102 3 3 3 ndash 0477335 Endash11 + i 0477368 Endash113 3 3 3 ndash 0730168 Endash10 + i 0112865 Endash09


64

5 Applications five point integrals in QED

We have reached to the final chapter which is devoted to the practical applicationof the techniques presented in the previous chapters Here we focus on the NLO QEDprocess e+eminus rarr micro+microminusγ which involves a set of five point integrals This process is a morechallenging calculation when compared to the e+eminus rarr e+eminusγ case due to additionalscale (mmicro) However at least number of diagrams and channels is smaller What isconsidered here can be directly applied to the e+eminus rarr e+eminusγ reaction We have decidedto focus on two important low energy experiments KLOE [64] which is a detector inaccelerator DAΦNE located in Frascati Italy and BaBar which is located at the SLACNational Accelerator Laboratory (California USA) The latter ceased operation in 2008but data analysis is still ongoing

Low energy physics is presently an important field of activity Let us mention mea-surement of (g minus 2)micro and its theoretical calculation [65] but also low energy hadronphysics eg determination of form-factors In this context an important quantity isthe pion form factor [66] To describe it properly experimental data are needed and forthat the process e+eminus rarr micro+microminusγ serves as normalization reaction [67]

Rexp =σ(e+eminus rarr π+πminusγ)

σ(e+eminus rarr micro+microminusγ) (501)

To our knowledge so far five point functions have not been included in Monte Carlogenerators for the considered process As an additional motivation these integrals canbe important for measurement of charge asymmetry As a matter of fact KLOE mea-sures it Results are not published so far There are some discrepancy between dataand theoretical calculations [68] that is why especially at such small energies five pointfunctions are worth considering For this process interference contributions (ie loop

e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Figure 51 FSR and ISR diagrams at the tree level

65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

Figure 52 FSR four point diagrams

e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

Figure 53 ISR four point diagrams

times tree) were evaluated for both four and five point integrals (final and initial stateradiation) see Fig51 Fig52 Fig53 and Fig54 Before any amplitude could be cal-culated the appropriate diagrams had to be generated For this step DIANA [69] programwas used Then multiplication of loop and tree amplitudes and further processing usingtrace method was done using symbolic manipulation program FORM [70] The last twosteps were made using MATHEMATICA and consisted of expressing loop integrals in termsof LoopTools tensor coefficients Here due to number of fermion lines the maximumrank of tensor integrals was three Final result was transferred then to the Fortran 77

code We have decided to use such a code format because all the numerical calculationswere performed using Phokhara-60 [71] (a Monte Carlo event generator) which is alsowritten in Fortran 77 Our interference amplitudes had been implemented in a sep-arate subroutine which then was linked with the main program All these steps wereautomatised using Bash scripting As an independent check we used FeynArts [72] andFormCalc [7] to perform the very same calculations independently The first softwaregenerates necessary diagrams with amplitudes and second one is able to do loop timestree computation (here we were also using trace method) Both use MATHEMATICA tooperate although in case of FormCalc the FORM program is used during computationsto take advantage of its speed and perform fermion trace calculations Because of therelation

MtreeMdaggerloop + MloopM

daggertree = 2Re(MloopM

daggertree) (502)

66

51 Infrared divergences connected with e+eminus rarr micro+microminusγ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

Figure 54 FSR and ISR five point diagrams

only MloopMdaggertree had to be calculated From the cross-check reasons relation Eq502 was

also checked numerically on computed amplitudes Further aspects concerning numericalcomputations will be discussed in the last section Meanwhile in the next section we aregoing to focus on infrared divergences which appear in diagrams of Fig52 Fig53 andFig54


In this section we discuss infrared divergences which appear in diagrams that wepresented in the introduction of this chapter All four and five point diagrams weconsider here ie Fig52 Fig53 and Fig54 are UV finite1 The only divergences wehad to calculate were IR singularities They appear when there is an appearance ofmassless line eg photon propagator between two on-shell massive lines If we look atloop diagrams which we want to calculate it is clear how many IR terms we might haveIn case of five point integrals two virtual photons generate infrared divergences and incase of four point only one will be involved in IR terms (because the other virtual photonis connected with off shell line)

The main task is to calculate 1ǫ divergences which has only infrared nature Theprocedure of evaluation IRrsquos was the following

bull first Mellin-Barnes representation was derived for a given integral using the AMBRE

package

bull then MB tool was used to regularise M-B representation and perform analytic con-tinuation process

bull finally integrals containing gamma functions (one dimensional integrals) were eval-uated using Cauchy theory

1 That can be also checked using FormCalc which has in-build function calledUVDivergentPart[expr] which returns UV divergent terms (see eg Appendix of [20] orFormCalc manual where UV terms were written)

67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2

Figure 55 Four and five point schematic diagrams here pa + pb = pc + pd + pe

Before we show the final IR result we would like to show explicitly Mellin-Barnes rep-resentations as well as form of F polynomials We limit ourself to scalar four and fivepoint diagram cases here All the notation refers to the Fig55

We begin with showing F polynomial for the four point diagram which is of the form

F = m2microx

21 + saex1x3 + m2

ex23 + scdx1x4 + sabx2x4 (511)

where

sab = minus(pa + pb)2 scd = minus(pc + pd)

2 + m2micro sae = minus(pa minus pe)

2 + m2e + m2

micro (512)

are the kinematic variables which should be compared with four point diagram presentedin Fig55 Application of Mellin-Barnes apparatus on the above polynomial leads to thefollowing four dimensional M-B representation

MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz4(m2micro)z1(sae)

z2(m2e)

z3(scd)z4(sab)

minus2minusǫminusz1minusz2minusz3minusz4

times Γ(minusz1)Γ(minusz2)Γ(minusz3)Γ(minusz4)Γ(minus1 minus ǫ minus z1 minus z2 minus z3)Γ(1 + z2 + 2z3)

times Γ(minus1 minus ǫ minus z1 minus z2 minus z3 minus z4)Γ(1 + 2z1 + z2 + z4)

times Γ(2 + ǫ + z1 + z2 + z3 + z4)Γ(minus2ǫ) (513)

In the very same way we proceed with five point diagrams firstly one has to calculateF function

F = m2micro(x1 + x5)

2 + sbdx1x3 + m2ex

23 + scex1x4 + sabx2x4 + scdx2x5 + saex3x5 (514)

where again some choice of kinematic variables which optimises F polynomial wereintroduced

sij = minus(pi + pj)2 sij = minus(pi minus pj)

2 + m2e + m2

micro (515)

68


The Mellin-Barnes representation for the scalar five point diagram Fig55 is the follow-ing

MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6

times minus((m2micro)z1(sbd)

z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)


times Γ(minusz1)Γ(minusz2)Γ(minusz3)Γ(minusz4)Γ(minusz5)Γ(minusz6)Γ(1 + z2 + z4)

times Γ(minus1 minus ǫ + z1 minus z3 minus z5)Γ(minus2 minus ǫ minus z1 minus z2 minus z3 minus z4 minus z5)

times Γ(1 + z4 + z5)Γ(minus2 minus ǫ minus z1 + z3 minus z4 minus z5 minus z6)Γ(1 + z5 + z6)

times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))

times (Γ(minus1 minus 2ǫ)Γ(minus1 minus ǫ minus z1 minus z3 minus z5)) (516)

Because we deal with loop integrals up to rank three (in case of five point functions) andrank two (for four point integrals) 1ǫ divergences were calculated for integrals with anappropriate tensor numerator as well Here the ability of AMBRE for constructing M-Brepresentations for one-loop integrals with numerator not contracted was used FinallyMB package was used to determine divergent parts This procedure was repeated forall integrals although the latter is not necessary if one analyses the structure of IRand applies appropriate momenta mapping for Eq517 and Eq519 In case of tensorintegrals with rank two and three there is no IR contribution for terms containing metrictensor

IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)

wheresab = minus(pa + pb)

2 sae = minus(pa minus pe)2 + m2

e + m2micro (518)

The index m indicates a rank of an integral eg m = 0 stands for a scalar integral inthat case we do not have chords qmicro which is obvious In the case of these four pointdiagrams only one photon line gives rise to IR singularities On Fig55 it is the linebetween momenta pa and pe In case of vector rank two and three integrals the chordqmicro of this line appears in the final IR result

For the five point integrals infrared divergences come from two photon propagatorsThis is the reason why IR result is a sum of two terms

IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69


Evaluated infrared contributions Eq517 and Eq519 were checked in the followingways

bull analytic result (Eq517 and Eq519) was cross-checked among numerical integra-tion using MB package (here MBintegrate function discussed in one of the previouschapters was used)

bull above results were also verified using CSectors package by sector decompositioncalculations

bull finally analytic result was transformed to λ dependence2 ie Aǫ = A ln λ+C [73]and in this form IR part was subtracted from four and five point loop integralsleading to invariance on shifting λ parameter during numerical calculation

Technically the subtraction of IR part was only done for the purpose of cross-checks andwas not necessary during KLOE and BaBar numerical simulation because λ parametercould be set to one in LoopTools directly

In the context of IR divergences we have made also the following test Five pointdiagrams were taken as an input to hexagon program Here the big advantage of thissoftware is that it can do the reduction of five point diagrams in a fully analytical wayAfter the reduction had been done we calculated IR parts both for five (lhs) and fourpoint diagrams (rhs) and both sides were compared according to

I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)

where ai is some coefficient which contains kinematic parameters (see Fig56) It comesfrom rdquosigned minorsrdquo (see eg Eq442 for a scalar case) This method of checks wasalso presented in [74] for e+eminus rarr e+eminusγ

52 Results and numerical cross-checks for virtual contributions

In this section we focus on numerical aspects of performed calculations The infraredparts which we discussed in the previous section are subtracted from the final resultThe results presented here should be treated as a basis for the future work in which hardpart will be added [75] ie our virtual IR will be necessary here The λ dependent termswill be reintroduced and only after phase space integration final λ independence will berestored Here we only touch contributions from virtuality of discussed loop integralssimilar analysis in the context of two-loop Bhabha studies have been made eg in [52]

2 We are working in the dimensional regularisation but LoopTools uses photon mass λ as a regulatorIt was checked that in this case C = 0 The argumentation is the following We have IR divergentfive point integrals from LoopTools we get E0 which is λ dependent (E0 = LT(λ)) We alsoevaluated with CSectors and AMBREMB that E0 = Aǫ which we rewrite as E0 = A ln(λ) =MB(λ) Now if MB(λ) = LT(λ) for any λ we can say that both constant and divergent part arethe same so C = 0

70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5

Figure 56 Figure refers to the Eq5111 Here five point diagram was reduced usinghexagon to the four point functions presented below Each number i) shouldbe understood as an i-th internal line which has been cut in the pentagondiagram

The Phokhara program comes with a special input file which is used to change be-haviour of the program eg photon angle range CMS energy and other necessary cutsand settings We have chosen cuts so that our numerical calculations simulate behaviourof KLOE and BaBar experiments (see Table 51) But before any proper numerical eval-uations have been made several purely technical tests had to be performed We wereinterested in examining the numerical stability using different compilers and differentreal number declarations In the tests g77 [76] and ifort [77] compilers were used Inthe end the second one gave us much better results so eventually it was chosen later formain calculations An additional advantage of this compiler is that it has also implemen-tation of floating point numbers in quadrupole precision At this point we summarisethe main difference between two floating point formats and show the number of precisiondigits one can expect

bull double precision floating point number is represented by 64 bits (1 sign 11 expo-

71


KLOE BaBarECMS 102 GeV 1056 GeV

Q2 00447 minus 50 GeV2 00447 minus 106 GeV2

Eminγ 002 GeV 3 GeVθmicro 50 minus 130 40 minus 140

θγ 0 minus 15 and 165 minus 180 20 minus 138

Table 51 Parameters used as an input in Phokhara event generator Both sets werechosen to match two experiments KLOE and BaBar ECMS is a CMS collisionenergy Q2 a squared invariant mass of the muonic system and Eminγ is aminimal energy of tagged photon The angles are measured between directionof a particle and the z axis

nent and 52 fraction) leading to 15-16 digits of precision

bull quad precision floating point number is represented by 128 bits (1 sign 15 expo-nent and 112 fraction) It means we have approximately 34 digits of precision

Quad precision was only used within our subroutine and LoopTools3 libraries the restof Phokhara program remained in the standard double precision We did not wantto change the floating numbers declaration everywhere in the well tested software iePhokhara and accidentally generate some unwanted bugs leading to software errors

Since version 22 LoopTools allows to choose in which decomposition five point in-tegrals should be calculated Implementation of both Passarino-Veltman [14] and Den-nerDitmaier [20 21] schemes exist Numerical calculations can be switched betweenthese two schemes using so called rdquoversion keyrdquo [19] After we had performed numericaltests we decided to use the latter scheme as it gave us the best accuracy for five pointintegrals

Apart from the numerical tests several physical checks have been made to ensure thatcalculated amplitudes are correct One of it was verification of gauge invariance It isa well known property which states that some diagrams must cancel among each otherwhen amplitudes are contracted with photon momenta instead of polarisation vectorsIn our case such invariance is presented among tree diagrams and also four and fivepoint loop integrals Fig57 The first case is very simple and was checked analyticallyand numerically the second one was checked only numerically Because we have usedtrace method (where one uses algebra of gamma traces) the numerical checks of gaugeinvariance required sum of loop diagrams in Fig57 to be contracted with one of thetree diagrams Fig51

During the process of preparation of the interference amplitudes (loops times tree) wedevoted attention to simplify results as much as possible Finally we managed to obtain

3 During compilation of these libraries it is possible to compile them with quadrupole precisionsetting using an appropriate Makefile file

72


+

+ +

a b c

Figure 57 FSR gauge invariance between tree diagrams (upper picture) and gauge in-variance among four and five point one-loop integrals (below) Here diagramswere limited to FSR cases the same property is present for ISR amplitudesIt was checked numerically by contracting with appropriate tree diagrams

KLOE BaBardouble precision 10minus2 10minus5

quadrupole precision 10minus12 10minus10

Table 52 Gauge invariance for loop diagrams Fig57 for KLOE and BaBar settingand different real number declarations The numbers give relative accuracy

defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))

Indices a b c refer to a b c diagrams

in Fig57

about 200 lines of Fortran 77 code for a single four point interference amplitude andabout 700 lines for a single pentagon (it is slightly less for gauge invariant amplitudes)Obviously such lengthy expressions might cause numerical instability That was thereason why we wanted to test calculations in quadrupole precision as well As it isshown in Table 52 numerical results for gauge invariance look much better in quadrupoleprecision than in double one This is clear especially in case of KLOE (see Table 51)where photon angle is chosen to be around 0 and 180 We have observed that settingθγ out of these values improved results significantly for the numerical gauge invarianceresults (see BaBar results for comparison) In the figures Fig58 Fig59 and Fig510we show explicitly which diagrams were used to form interference amplitudes MloopM

daggertree

Note that here we calculate ISR and FSR parts only ie we have total number of verticesin tree and loop diagrams odd for every fermion line The remaining parts would be

73


INT (an interference among ISR and FSR) amplitudes where the number of vertices iseven These contributions are important for the charge asymmetry

74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

Figure 58 Interference amplitudes (loop times tree) involving four point and tree FSRdiagrams

75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Figure 59 Four point one loop and tree diagrams all ISR All of them form interferenceamplitudes

76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Figure 510 Interference amplitudes formed with five point integrals and tree diagrams

As it was pointed out at the beginning of this chapter the interference amplitudeswere prepared using different softwares Thereafter both results were cross-checkedthat was an additional check for gauge invariance and rdquonormalrdquo numerical calculationsThe end of this chapter covers discussion about numerical results for KLOE and BaBarexperiments Some of the numerical results were prepared for tree diagrams only inthis case we used ISR FSR and INT diagrams which are encoded in Phokhara During

77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]

Figure 511 Two upper plots show results simulated for BaBar in which the left onepresents θγ dependence for the complete tree level and the right one isa result for loop contributions based on Fig58 Fig59 and Fig510 TheKLOE distribution over photon angle for the complete tree level is presentedin the lower plot

calculations 25middot106 points were generated from which about 3middot105 points were acceptedwithin KLOE cuts For BaBar it was 6 middot 106 and 1 middot 106 points respectively

All the plot points were histogramed using one of subroutines which is encoded inPhokhara program and provides calculation of Monte-Carlo errors All values ie θmicroθγ and Q2 were distributed into two hundred bins We begin our numerical analysis byshowing dependence on the photon angle θγ Fig511 Plots obtained for BaBar cuts fortree and loop amplitudes are presented there We see that the θγ dependence obtainedfor the tree is not symmetric due to non symmetric cut which was chosen (see Table 51)By comparing the values of tree and virtual corrections we see that the latter are verysmall compared to the LO ones The same situation we have observed for the KLOEcase but because of symmetric cut we see symmetric dependence Here we have limitedourselves to the tree contribution plot only (NLO result is again much smaller)

Next we present dependence on θmicro in Fig512 We see that muon angle dependence

78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+

Figure 512 Results for BaBar θmicro angle dependence (NLO contributions based onFig58-Fig510) Plots were generated both for microminus and micro+

is antisymmetric between microminus and micro+ This fact is expected but can be interpreted asan additional verification of made calculations

Finally we present plots showing dependence on the squared invariant mass of thesystem formed by the muons and the tagged photon Figures were generated for θmicro gt 90

and θmicro lt 90 also figure with full range of θmicro (Table 51) were presented for KLOEFig513 and BaBar Fig514 Careful analysis of these plots reveal that for BaBarplotted dependence goes rapidly to zero for about 47 GeV2 (that is why the plot is cutat 50 GeV2) and in case of KLOE the zero value is present up to about 05 GeV2That behaviour is dictated by kinematics and implemented cuts If we had two particlesin the final state only the particles would fly away from the collision point in exactlyopposite direction and each of them would carry half of the total energy Here we havethree body problem so the energetic spectrum for every particle is continuous If bothmuons fly away in the same direction the photon would go in opposite direction Nowif we add to that different photon cuts for KLOE and BaBar from Table 51 the effectseen on plots Fig513 and Fig514 will appear

Numerical results for the virtual corrections presented here should be treated as aprelude to the forward backward asymmetry analysis

79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

Figure 513 KLOE NLO results for θmicro gt 90 and θmicro lt 90 (left plot) Q2 dependencefor the full θmicro range allowed by cuts in Table 51 (right plot) we see thatpoints subtract to zero We have chosen θmicro connected with micro+

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

Figure 514 NLO results for BaBar for θmicro gt 90 and θmicro lt 90 If we plot Q2 dependencefor the full θmicro range we see that points subtract to zero (right plot) Wehave chosen θmicro for the micro+

80

6 Summary and conclusions

In spite of the amount of contents showed within this thesis we have been able tocover only a part of modern and constantly developing loop techniques One of themost important projects that we have done was a research within the Mellin-Barnesmethod which started during calculations of the massive two-loop Bhabha scattering[78] and ended by developing a very valuable tool AMBRE From the beginning sectordecomposition algorithm was used as a method for doing cross-checks with Mellin-Barnescalculations Here a helpful program was written CSectors which soon is planned tobe publicly available Another tool hexagon was developed during research work onreduction scheme of six and five point integrals

We have used all these techniques to work out NLO virtual corrections involving fiveand four point one-loop diagrams in the process e+eminus rarr micro+microminusγ Infrared divergenceshave been computed using AMBRE and MB As an independent check CSectors programwas used Reduction using hexagon was very helpful and allowed us to additionallycheck IRs As far as numerical calculations are concerned we have shown λ independentvirtual contributions to the e+eminus rarr micro+microminusγ process Here we have chosen to use theQED process involving muons in the final state But there are no circumstances to useexactly the same procedure to the Bhabha e+eminus rarr e+eminusγ reaction The same procedurecan be also used for calculations within the QCD theory

It would be interesting to repeat all the numerical calculations with the usage of otherthan LoopTools tools eg QCDloop or even more modern methods ie involving OPPor unitary cuts Another future perspective point is the usage of helicity method forpreparation of amplitudes

Our results should be treated as the first benchmark for the further developmentVirtual corrections considered here are not enough to judge on experimental observablesFig511-Fig514 Adding real corrections may change results substantially

81


82

A Gamma function

The basic function of Mellin-Barnes representation is a special function called Gammafunction FigA1 One of possible definitions [79] is the following one (Re(x) gt 0)

Γ(x) =

int infin

0

eminusttxminus1dt (A01)

The Gamma function is an extension of the factorial function to real and complex num-bers It also fulfils the function equation

Γ(x + 1) = xΓ(x) (A02)

EqA02 can be used with Γ(n+1) where n isin N and n gt 0 Then the following is true

Γ(n + 1) = nΓ(n) = n(n minus 1)Γ(n minus 1) = = nΓ(1) = n (A03)

clearly showing connection to the factorial function It is very important to note thatGamma function Γ(x) has poles located on the negative real axis at x = 0minus1minus2minus3 as seen on FigA1 In other words for Γ(z) z isin C it is analytic everywhere except points

-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren

Figure A1 Gamma function Γ(x) plot for x-real (left plot) and Γ(z) for z-complex (rightplot)

z = 0minus1minus2minus3 and the residue at z = minusk is

Res(Γminusk) =(minus1)k

k (A04)

83

A Gamma function

84

B Software

B1 AMBRE and generation of Mellin-Barnes representations

Recently Mellin-Barnes (M-B) representations of Feynman integrals have been usedextensively in various phenomenological and theoretical studies of quantum field theoryIn order to automatize process of creating such representations special (semi-) automaticMATHEMATICA program was developed [12] The AMBRE 1 stands for Automatic Mellin-Barnes Representation It can be used to construct planar Mellin-Barnes representationsfor

bull scalar multi-loop multi-legs integrals


Fullintegral[numerator propagators internal momenta]

Figure B1 The input function of AMBRE package

The arguments are as follows

bull numerator numerator which must be given in the scalar form see also FigB2

bull propagators product of propagators of the form PR[qmn1]equiv (q2 minus m2)minusn1

bull internal momenta list of internal momenta The order of internal momenta inthe list must be taken with utmost attention By changing order in this list oneforces AMBRE to calculate sub-loops in user defined custom way

k1p2k1p2k1k1

Figure B2 Proper way of introducing (k1 middot p2)2k1 middot k1 numerator in the AMBRE

Another two functions provided by this package which are necessary to be able to con-struct Mellin-Barnes representations are IntPart and SubLoop presented on FigB3

First of the above functions prepares a sub-loop of the full integral by collecting allpropagators which carry a given loop momentum ki It will display a piece of the fullintegral with

1 It can be downloaded from httpprojectshepforgeorgmbtools or directly fromhttppracusedupl~gluzaambre which also contains example files

85

B Software

IntPart[iteration options]

SubLoop[integral options]

Figure B3 Proper way of using IntPart and SubLoop functions

bull the numerator associated with the given sub-loop

bull sub-loop for a given internal momentum

bull internal momentum for which AMBRE will integrate the sub-loop

It must be stressed that execution of IntPart[iteration] function proceeds in theorder IntPart[1] IntPart[2] then IntPart[3] and so on If there is a need tochange the ordering of integrations one has to change the order in the starting listof internal momenta Once again compare FigB1 and it description Inserting egIntPart[2] before IntPart[1] would not be a proper way to do this In the output ofIntPart[iteration] the so called MATHEMATICA tag message will be displayed

Fautomode U and F polynomials will be calculated

in AUTO mode In order to use MANUAL mode execute Fauto[0]

Figure B4 The tag message giving information that F -polynomial modification is pos-sible

By running Fauto[0] AMBRE will calculate the F -polynomial (with name fupc) fora given sub-loop At this stage a user might wish to modify fupc manually eg byapplying some changes in kinematics

The second function presented on B3 is SubLoop It is a basic function used forderiving Mellin-Barnes representation for a given sub-loop This function takes outputgenerated by IntPart and performs the following calculations

1) calculates the F -polynomial for the sub-loop (only if Fauto[0] was not set)

2) determines the M-B representation for the F -polynomial

3) integrates over Feynman parameters

It must be executed always with argument integral As a result the Mellin-Barnesrepresentation for a given sub-loop integral will be displayed In multi-loop calculationsone will notice additional propagators (marked in red in the output of AMBRE) whichappear from the intermediate F -polynomial (and are added into propagators of nextsub-loop)The last optional argument in both functions allows to insert additional option thepossibilities are

86


bull Text -gt TrueFalse displays or not information text

bull Result -gt TrueFalse

bull Xintegration -gt TrueFalse performs or not integration over Feynman pa-rameters

It is possible to apply Barnes lemmas Eq2211 and Eq2212 on obtained Mellin-Barnes representations using BarnesLemma function FigB5

BarnesLemma[M-B representation number options]

Figure B5 Function used to apply Barnes lemmas

The meaning of arguments is following

bull M-B representation a Mellin-Barnes representation obtained after execution ofSubLoop

bull number accepts two possible values 1 for first Barnes lemma and 2 for secondone

bull options the following additional options

ndash Text -gt TrueFalse displays or not information text

ndash Shifts -gt TrueFalse searches for the pairs of two integration variableszi + zj and zi minus zj which after application of the appropriate shift one of itis cancelled

The following pedagogical example FigB6 shows usage of AMBRE software

87

B Software

In[1]= ltltAMBREm

by KKajda ver 12

last modified 9 Apr 2008

last executed on 09042009 at 1944

In[2]= invariants = p^2 -gt s

Fullintegral[1 PR[k m n1] PR[k + p m n2] k]

In[3]= IntPart[1]

numerator=1

integral=PR[kmn1] PR[k+pmn2]

momentum=k

Fautomode U and F polynomials will be calculated in

AUTO mode In order to use MANUAL mode execute Fauto[0]

In[4]= MBrepr = SubLoop[integral]


Computing U amp F polynomial in AUTO mode gtgtFauto[1]ltlt

U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]


((-1)^(n1+n2) (m^2)^z1 (-s)^(2-eps-n1-n2-z1) Gamma[-z1]

Gamma[-2+eps+n1+n2+z1] Gamma[2-eps-n1+z1-z2] Gamma[-z2]

Gamma[-2 z1+z2] Gamma[2-eps-n2-z1+z2])(Gamma[n1]

Gamma[4-2 eps-n1-n2] Gamma[n2] Gamma[-2 z1])

In[5]= BarnesLemma[MBrepr n1 -gt 1 n2 -gt 1 1]

gtgt Barnes 1st Lemma will be checked for z2 ltlt




1st Barnes Lemma was applied for z2


((m^2)^z1 (-s)^(-eps - z1) Gamma[1 - eps - z1]^2

Gamma[-z1] Gamma[eps + z1])Gamma[2 - 2 eps - 2 z1]

Figure B6 Scalar one loop self-energy example

88

B2 Reduction of six and five point integrals in hexagon


The hexagon2 is MATHEMATICA package for reducing six point Feynman integrals Ad-ditionally package allows to obtain results for pentagons as they are connected withhexagons Results that can be obtained are listed below

bull six point functions up to rank four

bull five point functions up to rank three

These tensor ranks are sufficient to get results within QED model Output can benumerical or analytical depending on user needs and is presented in a basis formed frommetric tensor gmicroν and the chords qmicro

i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1

g[microνqλ]i E00i (B21)

Kinematics that was used inside this package is presented in the following diagramsFig B7

This package is able to output full result of tensor integrals list of all coefficients ora specific coefficient for a given rank Result can be numerical or analytical dependingon user inputTo obtain full result for scalar six point diagram the following function must be executedinside MATHEMATICA environment

RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software

Figure B7 Momenta flow used in the package in six and five point diagrams

where sij = (pi + middot middot middot + pj)2 Results for higher ranks (onetwothree and four) can be

obtained by RedF1 RedF2 RedF3 RedF4 respectively with the same arguments as forRedF0 Results for five point function will be generated using

RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)

and RedE1 RedE2 RedE3 List of all coefficients for a given rank in case of six pointfunction will be printed after executing

RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)

Here argument rank must be replaced by one of these rank0 rank4 AdditionallyrankALL displays all coefficients In similar way list of coefficients of five point functionscan be displayed using RedEget Package can also produce specific tensor coefficient fora given rank

RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)

Needed coefficient must be inserted instead of coef ex ff00 For five point functionone has to call function RedEcoef

List of hexagon functions

bull RedF0[arguments] RedF1[arguments] RedF2[arguments] RedF3[arguments]RedF4[arguments] - displays full result for six point functions scalar vectortensor 2nd order tensor 3rd order and tensor 4th order respectively

bull RedE0[arguments] RedE1[arguments] RedE2[arguments] RedE3[arguments] -displays full result for five point functions scalar vector tensor 2nd order tensor3rd order

90


bull RedFget[rankarguments] - prints list of rules of all coefficients for a given rankof six point function Possible options for rank are the following rank0 rank1rank2 rank3 rank4 rankALL

bull RedEget[rankarguments] - prints list of rules of all coefficients for a given rankof five point function Possible options for rank are the following rank0 rank1rank2 rank3 rankALL

bull RedFcoef[coefarguments] - gives specific tensor coefficient of six point function

bull RedEcoef[coefarguments] - gives specific tensor coefficient of five point function

91

B Software

B3 Sector decomposition with CSectors

The CSectors[22 45] is a MATHEMATICA package which uses sector decomposition

libraries [44] for numerical calculation of Lminusloop mminusrank tensor integrals in d dimen-sions using sector decomposition method [41 42 23 24] It significantly simplifies gen-eration of numerical result for given loop integral even with tensor numerator Thebasic function of this package DoSectors All arguments of this function are presentedon FigB8

DoSectors[numerator propagators internal momenta options][min max]

Figure B8 The basic function of CSectors package

The arguments of DoSectors are the following


bull propagators product of propagators of the form PR[q m n1]equiv (q2 minus m2)n1

bull internal momenta list of internal momenta

bull options this package allows to modify its behaviour by adding additional optionsThe use of this options is identical to the usage of options in the MATHEMATICA

environment List of additional functions is as follows

ndash SetStrategy sets one of strategies available in sector decomposition pro-gram [44]

ndash SourceName prefix for source binary and log files It just makes possible tochoose any name for the files connected with calculation of a given integral

ndash TempFileDelete by default it is set to TempFileDelete-gtTrue When setto False it does not delete c++ source and binary files as well as log file

ndash LogFile forces package to create or not log file in which numerical resultsfor given integral and epsilon term are stored

ndash ShowErrors controls whether errors of numerical calculation will be dis-played or not in the end of calculation Errors are calculated using sector

decomposition function resget_error() [44]

ndash IterationsLow IterationsHigh CallsLow CallsHigh are Monte-Carloparameters more detailed description can be found in [44]

ndash compiler allows to choose other compiler than the default which is g++

ndash cppflags libs contain paths to header and library files required by sector

decomposition program

Default values of options used by CSectors can also be displayed by executingOptions[DoSectors]

92


bull min max indicate minimum and maximum Laurent series expansion

k1p2k1p2k1k2

Figure B9 Proper way of introducing (k1 middot p2)2k1 middot k2 numerator in the CSectors

The CSectors computation might be described by the following steps

1 Main function of this package DoSectors is executed in MATHEMATICA environ-ment It causes to prepare all the necessary input for later evaluation U and Fpolynomials as well as numerator tensor structure if present

2 Then routine generates c++ source codes to be linked with sector decomposition

libraries When this is done all the generated source code files are automaticallycompiled

3 All the binaries are executed in sequence and the intermediate results for thesingle ǫ terms (including the errors) are stored in log files

4 Finally the results are transferred back to MATHEMATICA where are outputted Theoverall error for the result is

∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)

A simple scalar one loop self-energy example presented on FigB10 shows usage ofCSectors package Obtained output is presented in the list form where first elementis a numerical result followed by sub list of errors calculated for the appropriate epsilonterm

93

B Software

In[1]= ltltCSectorsm

by KKajda ver09

last modified 12 feb 2009

In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3

invariants = p^2 -gt s

DoSectors[1 PR[k m n1] PR[k + p m n2] k][-4 0]

Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2

Generating c++ sourceInt 1done

Compiling source codeInt 1done

Running binary fileInt 1done

Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography

[1] Particle Data Group Collaboration C Amsler et al ldquoReview of particlephysicsrdquo Phys Lett B667 (2008) 1

[2] S Weinberg ldquoA Model of Leptonsrdquo Phys Rev Lett 19 (1967) 1264ndash1266

[3] S Salam ed N Svartholm (Almqvist and Wiksells Stockholm) (1968) p367 inElementary Particle Physics

[4] CDF Collaboration T Aaltonen et al ldquoObservation of the Ωminusb and Measurement

of the Properties of the ξminusb and Ωminusb rdquo arXiv09053123 [hep-ex]

[5] V A Smirnov ldquoEvaluating Feynman integralsrdquo Springer Tracts Mod Phys 211(2004) 1ndash244

[6] G J van Oldenborgh and J A M Vermaseren ldquoNew Algorithms for One LoopIntegralsrdquo Z Phys C46 (1990) 425ndash438

[7] T Hahn and M Perez-Victoria ldquoAutomatized one-loop calculations in four and Ddimensionsrdquo Comput Phys Commun 118 (1999) 153ndash165arXivhep-ph9807565

[8] G Ossola C G Papadopoulos and R Pittau ldquoCutTools a programimplementing the OPP reduction method to compute one-loop amplitudesrdquoJHEP 03 (2008) 042 arXiv07113596 [hep-ph]

[9] G Ossola C G Papadopoulos and R Pittau ldquoReducing full one-loopamplitudes to scalar integrals at the integrand levelrdquoNucl Phys B763 (2007) 147ndash169 arXivhep-ph0609007

[10] C F Berger et al ldquoAn Automated Implementation of On-Shell Methods for One-Loop Amplitudesrdquo Phys Rev D78 (2008) 036003 arXiv08034180 [hep-ph]

[11] Z Bern L J Dixon D C Dunbar and D A Kosower ldquoOne-Loop n-PointGauge Theory Amplitudes Unitarity and Collinear LimitsrdquoNucl Phys B425 (1994) 217ndash260 arXivhep-ph9403226

[12] J Gluza K Kajda and T Riemann ldquoAMBRE - a Mathematica package for theconstruction of Mellin-Barnes representations for Feynman integralsrdquoComput Phys Commun 177 (2007) 879ndash893 arXiv07042423 [hep-ph]

95

Bibliography

[13] M Czakon ldquoAutomatized analytic continuation of Mellin-Barnes integralsrdquoComput Phys Commun 175 (2006) 559ndash571 arXivhep-ph0511200

[14] G Passarino and M J G Veltman ldquoOne Loop Corrections for e+eminus AnnihilationInto micro+microminus in the Weinberg Modelrdquo Nucl Phys B160 (1979) 151

[15] T Diakonidis J Fleischer J Gluza K Kajda T Riemann and J Tausk ldquoOnthe tensor reduction of one-loop pentagons and hexagonsrdquoNucl Phys Proc Suppl 183 (2008) 109ndash115 arXiv08072984 [hep-ph]

[16] T Diakonidis J Fleischer J Gluza K Kajda T Riemann and J Tausk ldquoAcomplete reduction of one-loop tensor 5- and 6-point integralsrdquoPhys Rev D80 (2009) 036003 arXiv08122134 [hep-ph]

[17] J Fleischer F Jegerlehner and O V Tarasov ldquoAlgebraic reduction of one-loopFeynman graph amplitudesrdquo Nucl Phys B566 (2000) 423ndash440arXivhep-ph9907327

[18] ldquoWolfram MATHEMATICArdquo httpwwwwolframcom

[19] T Hahn ldquoNew Features in FormCalc and LoopToolsrdquo talk at EuroGDR 2005avaliable online httpwwwfeynartsdelooptoolseurogdr05pdf

[20] A Denner and S Dittmaier ldquoReduction of one-loop tensor 5-point integralsrdquoNucl Phys B658 (2003) 175ndash202 arXivhep-ph0212259

[21] A Denner and S Dittmaier ldquoReduction schemes for one-loop tensor integralsrdquoNucl Phys B734 (2006) 62ndash115 arXivhep-ph0509141

[22] J Gluza K Kajda T Riemann and V Yundin ldquoNew results for loop integralsAMBRE CSectors hexagonrdquo arXiv09024830 [hep-ph]

[23] T Binoth and G Heinrich ldquoAn automatized algorithm to compute infrareddivergent multi-loop integralsrdquo Nucl Phys B585 (2000) 741ndash759arXivhep-ph0004013

[24] T Binoth and G Heinrich ldquoNumerical evaluation of multi-loop integrals by sectordecompositionrdquo Nucl Phys B680 (2004) 375ndash388 arXivhep-ph0305234

[25] G Heinrich ldquoSector Decompositionrdquo Int J Mod Phys A23 (2008) 1457ndash1486arXiv08034177 [hep-ph]

[26] N Nakanishi ldquoGraph Theory and Feynman Integralsrdquo Gordon and Breach NewYork (1971)

[27] O Zavialov ldquoRenormalized Quantum Field Theoryrdquo Kluwer (1990)

96

Bibliography

[28] O V Tarasov ldquoConnection between Feynman integrals having different values ofthe space-time dimensionrdquo Phys Rev D54 (1996) 6479ndash6490arXivhep-th9606018

[29] J D Bjorken and S D Drell ldquoRelativistic quantum mechanicsrdquo New York McGraw-Hill 070054932 (1964)

[30] R B Paris and D Kaminski ldquoAsymptotics and Mellin-Barnes IntegralsrdquoEncyclopedia of Mathematics and its Applications 85

[31] M Czakon J Gluza and T Riemann ldquoThe planar four-point master integralsfor massive two- loop Bhabha scatteringrdquo Nucl Phys B751 (2006) 1ndash17arXivhep-ph0604101

[32] J B Tausk ldquoNon-planar massless two-loop Feynman diagrams with four on-shelllegsrdquo Phys Lett B469 (1999) 225ndash234 arXivhep-ph9909506

[33] V A Smirnov ldquoAnalytical result for dimensionally regularized massless on-shelldouble boxrdquo Phys Lett B460 (1999) 397ndash404 arXivhep-ph9905323

[34] A V Smirnov and V A Smirnov ldquoOn the Resolution of Singularities of MultipleMellin- Barnes Integralsrdquo Eur Phys J C62 (2009) 445ndash449arXiv09010386 [hep-ph]

[35] E Remiddi and J A M Vermaseren ldquoHarmonic polylogarithmsrdquoInt J Mod Phys A15 (2000) 725ndash754 arXivhep-ph9905237

[36] M Czakon J Gluza and T Riemann ldquoMaster integrals for massive two-loopBhabha scattering in QEDrdquo Phys Rev D71 (2005) 073009arXivhep-ph0412164

[37] M Czakon J Gluza and T Riemann ldquoOn the massive two-loop corrections toBhabha scatteringrdquo Acta Phys Polon B36 (2005) 3319ndash3326arXivhep-ph0511187

[38] S Moch P Uwer and S Weinzierl ldquoNested sums expansion of transcendentalfunctions and multi-scale multi-loop integralsrdquoJ Math Phys 43 (2002) 3363ndash3386 arXivhep-ph0110083

[39] G Heinrich T Huber and D Maitre ldquoMaster Integrals for FermionicContributions to Massless Three-Loop Form FactorsrdquoPhys Lett B662 (2008) 344ndash352 arXiv07113590 [hep-ph]

[40] G Heinrich T Huber D A Kosower and V A Smirnov ldquoNine-PropagatorMaster Integrals for Massless Three-Loop Form FactorsrdquoPhys Lett B678 (2009) 359ndash366 arXiv09023512 [hep-ph]

97

Bibliography

[41] K Hepp ldquoProof of the Bogolyubov-Parasiuk theorem on renormalizationrdquoCommun Math Phys 2 (1966) 301ndash326

[42] M Roth and A Denner ldquoHigh-energy approximation of one-loop Feynmanintegralsrdquo Nucl Phys B479 (1996) 495ndash514 arXivhep-ph9605420

[43] T Kaneko and T Ueda ldquoA geometric method of sector decompositionrdquoarXiv09082897 [hep-ph]

[44] C Bogner and S Weinzierl ldquoResolution of singularities for multi-loop integralsrdquoComput Phys Commun 178 (2008) 596ndash610 arXiv07094092 [hep-ph]

[45] J Gluza K Kajda T Riemann and V Yundin ldquoNumerical Evaluation OfFeynman Tensor Integrals In Euclidean Kinematical Regionrdquo in preparation

[46] J Gluza K Kajda and D A Kosower ldquoTowards a Basis of Planar Two-LoopIntegralsrdquo in preparation

[47] D A Kosower ldquoA tool for automatic application of the first and second Barneslemmasrdquo avaliable online httpprojectshepforgeorgmbtools

[48] M Czakon ldquoIdSolver programrdquo not public

[49] T Hahn ldquoCUBA A library for multidimensional numerical integrationrdquoComput Phys Commun 168 (2005) 78ndash95 arXivhep-ph0404043

[50] ldquoCERN Program Libraryrdquo avaliable online httpcernlibwebcernchcernlib

[51] J Gluza A Mitov S Moch and T Riemann ldquoThe QCD form factor of heavyquarks at NNLOrdquo JHEP 07 (2009) 001 arXiv09051137 [hep-ph]

[52] S Actis M Czakon J Gluza and T Riemann ldquoVirtual Hadronic andHeavy-Fermion O(α2) Corrections to Bhabha ScatteringrdquoPhys Rev D78 (2008) 085019 arXiv08074691 [hep-ph]

[53] Z Bern et al ldquoThe Two-Loop Six-Gluon MHV Amplitude in MaximallySupersymmetric Yang-Mills Theoryrdquo Phys Rev D78 (2008) 045007arXiv08031465 [hep-th]

[54] D B Melrose ldquoReduction of Feynman diagramsrdquo Nuovo Cim 40 (1965) 181ndash213

[55] L M Brown Nuovo Cimento 22 (1961) 178

[56] F R Halpern Phys Rev Lett 10 (1963) 310

[57] G Kallen and A Wightman Dan Mat Fys Skv 1 (1958) no 6

98

Bibliography

[58] A I Davydychev ldquoA Simple formula for reducing Feynman diagrams to scalarintegralsrdquo Phys Lett B263 (1991) 107ndash111

[59] T Binoth J P Guillet G Heinrich E Pilon and C Schubert ldquoAn algebraic numerical formalism for one-loop multi-leg amplitudesrdquo JHEP 10 (2005) 015arXivhep-ph0504267

[60] R K Ellis and G Zanderighi ldquoScalar one-loop integrals for QCDrdquoJHEP 02 (2008) 002 arXiv07121851 [hep-ph]

[61] T Binoth G Heinrich and N Kauer ldquoA numerical evaluation of the scalarhexagon integral in the physical regionrdquo Nucl Phys B654 (2003) 277ndash300arXivhep-ph0210023

[62] T Stelzer and W F Long ldquoAutomatic generation of tree level helicityamplitudesrdquo Comput Phys Commun 81 (1994) 357ndash371arXivhep-ph9401258

[63] F Maltoni and T Stelzer ldquoMadEvent Automatic event generation withMadGraphrdquo JHEP 02 (2003) 027 arXivhep-ph0208156

[64] KLOE Collaboration F Ambrosino et al ldquoMeasurement of the DAΦNEluminosity with the KLOE detector using large angle Bhabha scatteringrdquoEur Phys J C47 (2006) 589ndash596 arXivhep-ex0604048

[65] F Jegerlehner ldquoThe Anomalous Magnetic Moment of the Muonrdquo Springer Tractsin Modern Physics (2007)

[66] A Hoefer J Gluza and F Jegerlehner ldquoPion pair production with higher orderradiative corrections in low energy e+eminus collisionsrdquoEur Phys J C24 (2002) 51ndash69 arXivhep-ph0107154

[67] M Davier ldquoPrecision Measurement of the e+eminus rarr micro+microminusγ cross section with theISR Methodrdquo 10th International Workshop On Tau Lepton Physics (Tau08) httptau08inpnsksu

[68] H Czyz Private discussion

[69] M Tentyukov and J Fleischer ldquoA Feynman diagram analyser DIANArdquoComput Phys Commun 132 (2000) 124ndash141 arXivhep-ph9904258

[70] J A M Vermaseren ldquoFORM Symbolic Manipulation Systemrdquo avaliable online httpwwwnikhefnl~form

[71] H Czyz A Grzelinska and J H Kuhn ldquoSpin asymmetries and correlations inLambda-pair production through the radiative return methodrdquoPhys Rev D75 (2007) 074026 arXivhep-ph0702122

99

Bibliography

[72] T Hahn ldquoGenerating Feynman diagrams and amplitudes with FeynArts 3rdquoComput Phys Commun 140 (2001) 418ndash431 arXivhep-ph0012260

[73] W J Marciano and A Sirlin ldquoDimensional Regularization of InfraredDivergencesrdquo Nucl Phys B88 (1975) 86

[74] K Kajda ldquoThe pentagon diagrams of Bhabha scatteringrdquo Matter To The Deepest(Ustron 2007)

[75] H Czyz M Gunia J Gluza K Kajda T Riemann and V Yundin Work inprogress

[76] ldquog77 - Fortran 77 compilerrdquo avaliable online httpwwwgnuorgsoftwarefortranfortranhtml

[77] ldquoIntel Fortran Compilerrdquo avaliable online httpsoftwareintelcomen-usintel-compilers

[78] M Czakon J Gluza K Kajda and T Riemann ldquoDifferential equations andmassive two-loop Bhabha scattering The B5l2m3 caserdquoNucl Phys Proc Suppl 157 (2006) 16ndash20 arXivhep-ph0602102

[79] E T Whittaker and G N Watson ldquoA Course of Modern Analysisrdquo SeriesCambridge Mathematical Library

100

Introduction

Evaluating Feynman integrals

Feynman parametrisation

Mellin-Barnes representation theoretical background

Numerical calculations in the Euclidean region

Construction of Mellin-Barnes representations by AMBRE package

Loop by Loop algorithm

Remarks on derivation of Mellin-Barnes representations

Order of integration in the loop by loop method

Importance of F polynomial in derivation of representations

Using Barnes lemmas

One-loop integrals with tensor numerator

Cross-checks in Euclidean region with CSectors

Applications and perspectives of AMBRE

Tensor reduction of five and six point one loop integrals

Passarino-Veltman reduction

Gram determinants and algebra of signed minors

Tensor integrals and shifted space-time dimensions

Reduction of pentagons up to rank two

Reduction of six point functions

Numerical results and discussion

Applications five point integrals in QED

Infrared divergences connected with e+ e- + -

Results and numerical cross-checks for virtual contributions

Summary and conclusions

Gamma function

Software

AMBRE and generation of Mellin-Barnes representations

Reduction of six and five point integrals in hexagon

Sector decomposition with CSectors

Bibliography

Supervisor

Dr hab Janusz GluzaDepartment of Field Theory and Particle PhysicsUniversity of Silesia Katowice Poland


Contents

1 Introduction 7












A Gamma function 83

B Software 85


5

Contents

Bibliography 95

6

1 Introduction









7

1 Introduction






8





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography


Contents

1 Introduction 7












A Gamma function 83

B Software 85


5

Contents

Bibliography 95

6

1 Introduction









7

1 Introduction






8





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography

Contents

Bibliography 95

6

1 Introduction









7

1 Introduction






8





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography

1 Introduction









7

1 Introduction






8





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography

1 Introduction






8





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography





int 1

0

dx1

int 1

0


P1x1 + P2x2

=

int 1

0

dx11


=1

P1 minus P2

int P2

P1

dt1

t2

=1

P1P2

(211)


1

P1 Pn

= Γ(n)

int 1

0

int 1

0


(P1x1 + + Pnxn)n (212)



1

P1 Pn

=

int

infin

0

int

infin

0


9



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography



int 1

0

dx1

int 1

0

dx2

int 1

0


=

int 1

0

dx1

int 1minusx1

0




1prodn

i=1 P νi

i

=Γ(Nν)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j


(P1x1 + + Pnxn)Nν (214)

with

Nν =nsum

i=1

νi (215)


G(T (k)) =1

(iπd2)L

int

ddk1 ddkLT (k)

(q21 minus m2


n + i0)νn (216)


Pi = q2i minus m2

i + i0 (217)


nsum

i=1

Pixi =nsum

i=1

(q2i minus m2

i + i0)xi =Lsum

ij=1

kTi Mijkj minus 2

Lsum

j=1

kTj Qj + J (218)


G(1) =Γ(Nν)

Γ(ν1) Γ(νn)

times

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

int

ddk1 ddkL


10



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography



kl = kl +Lsum

i=1

Mminus1li Qi (2110)



d2L)

Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minus

nsum

i=1

xi

)

UNνminusd(L+1)2

FNνminusdL2 (2111)

where

U = det(M)



bull U - polynomial

U =sum

TisinT1

prod

l isinT

xl (2113)


bull F - polynomial

F =sum

TisinT2

prod

l isinT




U = x1 + x2 + x3 + x4



11


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography


3

4

2

1 3

4

2

1

3

4

2

1 3

4

2

1


3

4

2

1 3

4

2

1





Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(


2

)

(minus2)r2


FNνminusd2Lminus r

2


(2116)




bull m=2

sum

rle2


=

A0P2 + A1P



12


bull m=3

sum

rle3


=

A0P3 + A1P

2 + A2P1 + A3P





P microi rarrsum

l

[MalQl]microi



G(kmicro1

1 kmicro2


l




M = det(M)Mminus1 = 1 (2121)





1

(A + B)λ=

1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin


where



13


p p

k

k + p



1

(A1 + + An)λ=

1

Γ(λ)

1

(2πi)nminus1

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz2 dzn

nprod

i=2

Azi

i



i=2

Γ(minuszi) (222)


G(1)SE1l2m =

int

ddk






d2)

Γ(ν1)Γ(ν2)

int 1

0

2prod

j=1

dxjxνjminus1j δ

(

1 minus

2sum

i=1

xi

)

times(x1 + x2)

ν1+ν2minusd




14



1

F λ=

1


=1

Γ(λ)

1

2πi

int c+iinfin

cminusiinfin



2z1 (226)


Eq221 resulting in

1

F λ=

1

Γ(λ)

1

(2πi)2

int c+iinfin

cminusiinfin

dz11

Γ(minus2z1)

int c+iinfin

cminusiinfin




2 (227)


xa1minus11 = x


xa2minus12 = x



int 1

0

nprod

i=1

dxjxajminus1j δ

(

1 minus

nsum

i=1

xi

)

=Γ(a1) Γ(an)

Γ(a1 + + an) (229)



Γ(ν1)Γ(ν2)

int c+iinfin

cminusiinfin

dz1

int c+iinfin

cminusiinfin










15





int c+iinfin

cminusiinfin


Γ(a + c)Γ(a + d)Γ(b + c)Γ(b + d)

Γ(a + b + c + d) (2211)


int c+iinfin

cminusiinfin


Γ(a + b + c + d + e + z)=




G(1)SE1l2m =

int c+iinfin

cminusiinfin






16


GΕ + z1D

G-z1D


-4 -2 2 4Re z

G1- Ε - z1D

G-z1D

GΕ + z1D

-4 -2 2 4Re z












17





1 (m2)minusǫΓ(ǫ) (2216)

2

int minus 12+iinfin

minus 12minusiinfin









11


2

int minus 12+iinfin

minus 12minusiinfin



Γ(2 minus 2z1) (2220)


∮

dzf(z) = 2πisum

Res[f(z)] (2221)


G(1)SE1l2m = 2 +1 + x

1 minus xln(x) +

1

ǫ where x = minus

1 minusradic

1 minus 4s

1 +radic

1 minus 4s

(2222)

18








int 1

0

dx1

int 1

0

dx2δ

(

1 minus2sum

i=1

xi

)

xminusǫ1 xminusǫ

2

x1(x1 + x2) (231)


int 1

0

dNx =Nsum

l=1

int 1

0

dNx

Nprod

j=1

j 6=l

θ(xl ge xj) (232)


θ(x ge y) =


19


y

x


(1)

+

y

x

t

t



xj =


(233)



allows to useint 1

0


k=1

tk)) = 1 (234)


int 1

0

Nminus1prod

j=1

tj tνjminus1j


FNνminusLd2l (t )

l = 1 N (235)



20




21


22


AMBRE package












23


p1

p2

p3



k2 + k1

k1

rarr

p1

p2

p3



k2


GB5l2m2(1) =

int

ddk1ddk2

1

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

=

int

ddk21

P n3

3 P n4

4 P n5

5

int

ddk11

P n1

1 P n2

2

P1 = (k1)2

P2 = (k1 + k2)2 minus m2

P3 = (k2 minus p1)2

P4 = (k2 minus p3)2








24



In[]= IntPart[1]

numerator=1


momentum=k1




2


2 (312)






GB5l2m2(1) =

int

ddk2


times

int c+iinfin

cminusiinfin







25




F polynomial

-PR[k2m]X[1]X[2]+m^2X[2]^2







In[]= IntPart[2]

numerator=1

integral=


momentum=k2



F polynomial

m^2FX[X[1] + X[4]]^2 - tX[2]X[3] - sX[1]X[4]



(-s)^(2-eps-n3-n4-n5+z1-z2-z3)(-t)^z3









26
















27


p2

p1

k1 + p2 k2 minus k1

k1

k2

k2 + p1 + p2

p1 + p2

k1 + p1 + p2

a)p2

p1

k1 k2

k1 minus p2

k1 + k2 minus p2

k1 + k2 + p1

p1 + p2

k1 + p1

b)



p21 = p2

2 = 0 (p1 + p2)2 = s (321)


bull diagram a)



bull diagram b)







1 + 2p1 middot p2 + p22)x1x4



1x3x4 minus p22x1x3




28


In[]= IntPart[1]

numerator=1

integral=


momentum=k1



F polynomial

-PR[k20]X[1]X[2]-PR[k2+p20]X[2]X[3]-sX[1]X[4]

-PR[k2+p1+p20]X[2]X[4]






In[]= IntPart[2]

numerator=1



momentum=k2



F polynomial

-sX[1]X[3]












29














30


In[]= IntPart[1]

numerator=1


momentum=k2



F polynomial



In[]= IntPart[2]

numerator=1


PR[k1+p1+p20-2+eps+n3+n4+n5+n6+z1+z2]

momentum=k1



F polynomial

-sX[1]X[3]











31


k2k1


k3

k1

k2




2x2x4 minus [k1 + k3 + p2]2x2x3


2x1x3








32


p1

p2

p3

p5

p4

k1

k1 + p1

k1 + p1 + p2

k1 + p4 + p5

k1 + p5



G(1) =

int

ddk

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (326)



F = m2x22 + 2m2x2x4 + m2x2








33




F = m2(x2 + x4 + x5)2


where


s45 = (m2 minus s45) (329)


G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin



times (s34)z4(s45q)




times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5))







34


In[]= IntPart[1]

numerator=1



momentum=k

In[]= Fauto[0]

F polynomial


m^2X[3]X[4]-s34X[3]X[4]+m^2X[1]X[5]-s45X[1]X[5]-

s23X[2]X[5]

In[]= fupc =


s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





Out[]=((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3

(-s23)^(2-eps-n1-n2-n3-n4-n5-z1-z2-z3-z4-z5)

(s34p)^z4(s45p)^z5










35










Out[]=-(((m^2)^z1(-s12)^z2(-s15)^z3









36




s

m2

m1

m3

p22 = M2

2

p21 = M2

1



G(1) =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz7(minus((m21)

z1(m22)

z3+z5(m23)


times MM z2

1 MM z4

2 (minuss)z7







37


In[]= repr=-(((m1^2)^z1(m2^2)^(z3+z5)

(m3^2)^(-1-eps-z1-z2-z3-z4-z5-z7)

MM1^z2MM2^z4(-s)^z7






















38






Γ(ν1) Γ(νn)

int 1

0

nprod

j=1

dxjxνjminus1j δ

(

1 minusnsum

i=1

xi

)

timessum

rlem

Γ(

Nν minusd2minus r

2

)

(minus2)r2

1

FNνminusd2minus r

2


(331)





G(kmicro1kmicro2) =

int

ddkkmicro1kmicro2

P n1

1 P n2

2 P n3

3 P n4

4 P n5

5

P1 = (k1)2

P2 = (k1 + p1)2 minus m2

P3 = (k1 + p1 + p2)2

P4 = (k1 + p4 + p5)2 minus m2

P5 = (k1 + p5)2 minus m2 (333)


39








int


[k21]

n1 [(k2 minus k1)2]n2 [(k1 + p)2]n3 [k22]

n4 [(k2 + p)2]n5ddk1d

dk2 (334)





rarr (kmicro1



rarr kmicro1

2 kmicro2

2 x22minuskmicro2





int

p1micro1p1micro2

(k2 middot p)

[k22]


times kmicro1

2 kmicro2

2 MB1minuskmicro2








40




In[]= IntPart[1]




momentum=k

In[]= Fauto[0]

In[]= fupc =

m^2FX[X[2]+X[4]+X[5]]^2-s12X[1]X[3]-s15X[2]X[4]+

s34pX[3]X[4]+s45pX[1]X[5]-s23X[2]X[5]





ARint[1]ARint[2]

In[]= ARint[2repr]

Out[]=-((-1)^(n1+n2+n3+n4+n5)(m^2)^z1(-s12)^z2(-s15)^z3











41




[k21 ]ν1 [(k2minusk1)2]ν2 [(k1+p)2]ν3 [k2

2 ]ν4 [(k2+p)2]ν5ddk1d

dk2


(k2middotp)

[k22 ]ν4 [(k2+p)2]ν5 kmicro1

2 kmicro2

2 MB1 minuskmicro2







42






V6l0mexample a1= 800428|6140449509 +

2452280156|9081353

ǫ

+596665332838|975

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4

V6l0mexample a2= 800428|5575775925 +

2452280156|8656705

ǫ+

+596665332838|1408

ǫ2+

14507266400430143

ǫ3+

3025

ǫ4


ǫ+

(596648 plusmn 005)

ǫ2+

+(145066 plusmn 0007)

ǫ3+

(30248 plusmn 0002)

ǫ4 (342)







43







T [s] 43 76s = minus3











44















45




46


integrals







47



A(m1) =

int

ddk

iπd2

1

P1


int

ddk

iπd2

1 kmicro kmicroν

P1P2


int

ddk

iπd2

1 kmicro kmicroν

P1P2P3

(411)

where

P1 = k2 minus m21


2


3 (412)



Bmicro = pmicro1B1

Bmicroν = pmicro1p



2C2

Cmicroν = pmicro1p

ν1C11 + (pmicro

1pν2 + pν




k middot p1 =1

2

(


2 minus m22] + m2

1 minus m22 + p2

1

)

(414)


p21B1 =

1

2

(

(p21 + m2


)

(415)



p1microCmicro

p2microCmicro

)

=

(

p21 p1 middot p2

p1 middot p2 p22

)(

C1

C2

)

(416)

48



2

∣

∣

∣

∣

p21 p1 middot p2

p1 middot p2 p22

∣

∣

∣

∣

(417)




Gnminus1 = 2

∣

∣

∣

∣

∣

∣

∣

∣

∣




∣

∣

∣

∣

∣

∣

∣

∣

∣

(421)


()n =

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

0 1 1 11 Y11 Y12 Y1n

1 Y12 Y22 Y2n

1 Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

(422)





(

j1 j2 k1 k2

)

n

equiv (minus1)P

i(ji+ki)sgnjsgnk

∣

∣

∣

∣


∣

∣

∣

∣

(424)

49



(

0

0

)

n

=

∣

∣

∣

∣

∣

∣

∣

∣

∣

Y11 Y12 Y1n

Y12 Y22 Y2n

Y1n Y2n Ynn

∣

∣

∣

∣

∣

∣

∣

∣

∣

(425)


(

i1 irj1 jr

)

n

=

(

j1 jr

i1 ir

)

n

(426)


nsum

i=1

(

0i

)

n

=

( )

n

(427)


i=1

(

j 0k i

)

n

=

(

jk

)

n

(428)


nsum

i=1

(

ji

)

n

= 0 (429)


(

α0

)

n

(

α βk l

)

n

+

(

αk

)

n

(

α βl 0

)

n

+

(

αl

)

n

(

α β0 k

)

n

= 0 (4210)

and(

i lj k

)

n

( )

n

=

(

ij

)

n

(

lk

)

n

minus

(

ik

)

n

(

lj

)

n

(4211)


50




Imicron =

int d

kmicro

nprod

r=1

Pminus1r = minus

nminus1sum

i=1

qmicroi I

[d+]ni (431)

Imicro νn =

int d

kmicro kν

nprod

r=1

Pminus1r =

nminus1sum

ij=1

qmicroi qν

j νijI[d+]2

nij minus1

2gmicroνI [d+]

n (432)



int [d+]l nprod

r=1

1


int d

equiv

int

ddk

iπd2

[d+]l = 4 + 2l minus 2ǫ (433)



j (434)


()n νjj+I [d+]

n =

[

minus

(

j

0

)

n

+nsum

k=1

(

j

k

)

n

kminus

]

In (435)

(d minusnsum

i=1

νi + 1) ()n I [d+]n =

[

(

0

0

)

n

minusnsum

k=1

(

0

k

)

n

kminus

]

In (436)

(

0

0

)

n

νjj+In =

nsum

k=1

(

0j

0k

)

n

[

d minusnsum

i=1

νi(kminusi+ + 1)

]

In (437)



51



(d minus 4)

( )

5

I[d+]5 =

(

00

)

5

I5 minus

5sum

s=1

(

0s

)

5

Is4 (441)


point function

I5 =1

(

00

)

5

5sum

s=1

(

0s

)

5

Is4 (442)



Imicro5 =

4sum

i=1

qmicroi I5i (443)

with


(

0i

)

5(

00

)

5

I[d+]5 minus

1(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (444)


I5i = minus1

(

00

)

5

5sum

s=1

(

0 i0 s

)

5

Is4 (445)




Imicro ν5 =

4sum

ij=1

qmicroi qν

j νijI[d+]2

5ij minus1

2gmicroνI

[d+]5 (446)

52



gmicro ν = 24sum

ij=1

(

ij

)

5( )

5

qmicroi qν

j (447)


Imicro ν5 =

4sum

ij=1

qmicroi qν

j I5ij

I5ij = νijI[d+]2

5ij = minus

(

0j

)

5( )

5

I[d+]5i +

5sum

s=1s 6=i

(

sj

)

5( )

5

I[d+]s4i (448)




I5ij = minus1

(

00

)

5

( )

5

5sum

s=1s 6=i

(

0j

)

5

(

0 i0 s

)

5

+

(

sj

)

5

(

0 si s

)

5

(

00

)

5(

ss

)

5

Is4

+1

( )

5

5sum

st=1st 6=i

(

sj

)

5

(

t si s

)

5(

ss

)

5

Ist3 (449)



4 thefollowing term

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5(

ss

)

5

(4410)




i=1

(

ij

)

5

Yil

53



(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

minus

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5(

00

)

5

(

ss

)

5

+

(

0j

)

5

(

t s0 i

)

5(

00

)

5

= 0 (4411)


sum

st

(

t s0 i

)

5

=sum

s

(

sum

t

(

0 st i

)

5

)

= 0 (4412)


I5ij =1

(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

Is4

minus1

(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

t s0 s

)

5

(

ij

)

5

Ist3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4413)


54


the following

As0ij = minus

(

0j

)

5

(

0 s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

0 si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xs0ij (4414)

Astij = minus

(

0j

)

5

(

t s0 i

)

5

(

ss

)

5

minus

(

sj

)

5

(

t si s

)

5

(

00

)

5

+

(

s0

)

5

(

0 s0 s

)

5

(

ij

)

5

=

( )

5

Xstij (4415)


(

ss

)

5

[(

0i

)

5

(

0 j0 s

)

5

minus

(

0j

)

5

(

0 i0 s

)

5

]

+

(

00

)

5

[(

si

)

5

(

0 sj s

)

5

minus

(

sj

)

5

(

0 si s

)

5

]

= 0 (4416)


(

0j

)

5

(

0 i0 s

)

5

=

(

0j

)

5

(

0 s0 i

)

5

= minus

(

0i

)

5

(

0 sj 0

)

5

minus

(

00

)

5

(

0 si j

)

5

=

(

0i

)

5

(

0 s0 j

)

5

minus

(

00

)

5

(

0 si j

)

5

(4417)


(

si

)

5

(

0 sj s

)

5

+

(

sj

)

5

(

0 ss i

)

5

+

(

ss

)

5

(

0 si j

)

5

= 0 (4418)




Xs0ss = 0 (4419)


5sum

j=1

As0ij = minus

( )

5

(

0 s0 i

)

5

(

ss

)

5

(4420)

55



5sum

i=1

As0ij = minus

( )

5

(

0 s0 j

)

5

(

ss

)

5

(4421)


contain a term

minus

(

0 s0 i

)

5

(

0 sj s

)

5

(4422)


(

0 js i

)

5

(4423)


(

0 s0 s

)

5

(4424)

Thus we conclude

Xs0ji = Xs0

ij = minus

(

0 s0 i

)

5

(

0 sj s

)

5

+

(

0 js i

)

5

(

0 s0 s

)

5

(4425)



Xstij = minus

(

0 s0 j

)

5

(

t si s

)

5

+

(

0 is j

)

5

(

t s0 s

)

5

(4426)


I5ij =1

(

00

)

5

5sum

s=1s 6=i

1(

ss

)

5

Xs0ij Is

4 minus1

(

00

)

5

5sum

st=1s 6=it

1(

ss

)

5

Xstij I

st3

minus

(

ij

)

5(

00

)

5

( )

5

5sum

s=1s 6=i

1(

ss

)

5

(

s0

)

5

(

0 s0 s

)

5

Is4

+

(

ij

)

5(

00

)

5

( )

5

5sum

st=1s 6=it

1(

ss

)

5

(

s0

)

5

(

t s0 s

)

5

Ist3 (4427)

3 Observe thatsum5

j=1

(

0 js i

)

5

= 0 butsum5

i=1

(

0 js i

)

5

= minussum5

i=1

(

0 ji s

)

5

= minus

(

js

)

5

56



Imicro ν5 =

4sum

ij=1

qmicroi qν


Eij =5sum

s=1

S4sij Is

4 +5sum

st=1

S3stij Ist

3 (4429)

where

S4sij =

1(

00

)

5

5sum

s=1

1(

ss

)

5

Xs0ij (4430)

S3stij = minus

1(

00

)

5

5sum

st=1

1(

ss

)

5

Xstij (4431)

E00 = minus1

2

1(

00

)

5

5sum

s=1

(

s0

)

5(

ss

)

5

[

(

0 s0 s

)

5

Is4 minus

5sum

t=1

(

t s0 s

)

5

Ist3

]

(4432)




I6 =6sum

r=1

(

0r

)

6(

00

)

6

Ir5 =

6sum

r=1

(

rk

)

6(

0k

)

6

Ir5 k = 1 6 (451)

57


and Eq442 now reads

Ir5 =

1(

0 r0 r

)

6

6sum

s=1s 6=r

(

0 rs r

)

6

Irs4 (452)


(

rr

)

6

(453)


Imicro6 =

5sum

i=1

qmicroi I6i (454)

I6i = minusI[d+]6i

= (d minus 5)

(

0i

)

6(

00

)

6

I[d+]6 minus

1(

00

)

6

6sum

r=1

(

0 i0 r

)

6

Ir5 (455)


5sum

i=1

qmicroi

(

0i

)

6

= 0 (456)


vmicror = minus

1(

00

)

6

5sum

i=1

(

0 i0 r

)

6

qmicroi

= minus1

(

0k

)

6

5sum

i=1

(

0 rk i

)

6

qmicroi k = 0 6 (457)


Imicro6 =

6sum

r=1

vmicror Ir

5 (458)

58









F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1


59










60



ν1p

λ1Emicroνλ

Phase space pointp2

1 = p22 = p2

3 = p25 = 1 p2

4 = 0 m21 = m2

3 = 0 m22 = m2

4 = m25 = 1


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0218741


Phase space pointp2

1 = p22 = p2

3 = p24 = p2

5 = p26 = minus1 m2

1 = m22 = m2

3 = m24 = m2

5 = m26 = 1


1 p26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26 ]

Out 0013526


Phase space pointp2

1 = p22 = 0 p2

3 = p25 = 49256 p2

4 = 9100m2

1 = m22 = m2

3 = 49256 m24 = m2


1 p25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 413403 - 459721I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out ee1 =-238605 + 527599I ee2 =-580875 + 0597891I

ee3 =-144931 + 208149I ee4 =-113362 + 181593I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 71964 + 310115I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out-0149527 - 031059I


25 s12 s23 s34 s45 s15m

21 m

25 ]

Out 0154517 - 0387727I


61



m1 = 1100 m2 = 1200 m3 = 1300 m4 = 1400 m5 = 1500 m6 = 1600


F0





62





63





64







e+

e-

Γ

ΓΜ-

Μ+ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ


65


e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ


e+

e-

Μ-

Μ+Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ






daggertree) (502)

66


e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ








package




67


pa

pb

pd

pe

pc

pc

pd

pe pa

pb

k + q

k + q1

k + q2




F = m2microx

21 + saex1x3 + m2


where



2 + m2e + m2

micro (512)


MB4pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin


z2(m2e)

z3(scd)z4(sab)







2 + sbdx1x3 + m2ex




2 + m2e + m2

micro (515)

68



MB5pt =

int c+iinfin

cminusiinfin

int c+iinfin

cminusiinfin

dz1 dz6


z2(m2e)

z3(sce)z4(sab)

z5(scd)z6(sae)





times Γ(3 + ǫ + z1 + z2 + z3 + z4 + z5 + z6))



IR4pt =1

2ǫ

ln[

1minusx1+x

]

sabsaex

mle2prod

i=0

qmicroi x =

radic

1 minus4m2

em2micro

s2ae

(517)



e + m2micro (518)



IR5pt =1

2ǫ

ln[

1minusx1+x

]

sbdscdx

mle3prod

i=0

q1microi+

ln[

1minusy1+y

]

saescey

mle3prod

i=0

q2microi

1

sab

(519)

where

x =

radic

1 minus4m2

em2micro

s2bd

y =

radic

1 minus4m2

em2micro

s2ae


2 + m2e + m2

micro (5110)

69








I IR5 = a1I

IR41 + a2I

IR42 + a3I

IR43 + a4I

IR44 (5111)





70


p1 + p2

p3

p3

p4

p5

p1

p2

2

5

1

3

4

1) 2)

3) 4)

p2

p1

p3 + p5

p4

p2 minus p3 p5

p1 p4

p5

p4

p2

p1

p3

p4 + p5




71














72


+

+ +

a b c





defined as maxP

i=abc Re(M iloop

Mdaggertree)

min(Re(M iloop

Mdaggertree))


in Fig57


daggertree


73



74


e-

e+

Μ+

Μ-

Γ

ΓΓ e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γe+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-


75


e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e+

e-

Μ-

Μ+Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γe-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ


76


e-

e+

Μ+

Μ-

Γ

Γ

Γ

e+

e-

Γ

ΓΜ-

Μ+

e-

e+

Μ+

Μ-

Γ

Γ

Γ e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

ΓΓ

e+

e-

Γ

ΓΜ-

Μ+e-

e+

Μ+

Μ-

Γ

ΓΓ

e-

e+

ΓΓ

Μ+

Μ-

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

ΓΓ

e-

e+

Μ+

Μ-

Γ

Γ

Γ

e-

e+

Μ+

Μ-

Γ

Γ



77


-00002

-00001

0

00001

00002

00003

00004

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

-4e-07

-3e-07

-2e-07

-1e-07

0

1e-07

2e-07

3e-07

4e-07

20 40 60 80 100 120

dσd

θ γ [n

b]

θγ [deg]

0

02

04

06

08

1

12

14

0 20 40 60 80 100 120 140 160 180

dσd

θ γ [n

b]

θγ [deg]





78


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

40 60 80 100 120 140

dσd

θ micro [n

b]

θmicro [deg]

θmicro-

θmicro+






79


-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-008

-006

-004

-002

0

002

004

006

008

03 04 05 06 07 08 09 1

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]

θmicro+lt90deg

θmicro+gt90deg

-2e-06

-15e-06

-1e-06

-5e-07

0

5e-07

1e-06

15e-06

2e-06

0 10 20 30 40 50

dσd

Q2 [n

bG

eV2 ]

Q2 [GeV2]


80






81


82

A Gamma function


Γ(x) =

int infin

0



Γ(x + 1) = xΓ(x) (A02)




-4 -2 2 4x

-5

5

GHxL

-4

-20

24

ReHzL-2

0

2

ImHzL

0

2

4

6

GHzLcurren




k (A04)

83

A Gamma function

84

B Software











k1p2k1p2k1k1





85

B Software

















86















87

B Software

In[1]= ltltAMBREm

by KKajda ver 12





In[3]= IntPart[1]

numerator=1


momentum=k






U polynomial

X[1]+X[2]

F polynomial

m^2 FX[X[1]+X[2]]^2-s X[1] X[2]
















88







i

F micro =5sum

i=1

qmicroi Fi

F microν =5sum

ij=1

qmicroi qν

i Fij

F microνλ =5sum

ijk=1

qmicroi qν

i qλkFijk +

5sum

i=1

gmicroνqλi F00i

F microνλρ =5sum

ijkl=1

qmicroi qν

i qλkqρ

l Fijkl +5sum

ij=1

qmicroi q

[νj gλρ]F00ij

Emicro =4sum

i=1

qmicroi Ei

Emicroν =4sum

ij=1

qmicroi qν

i Eij + gmicroνE00

Emicroνλ =4sum

ijk=1

qmicroi qν

i qλkEijk +

4sum

i=1




RedF0[p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B22)


89

B Software




RedE0[p21 p

25 s12 s23 s34 s45 s15m

21 m

25] (B23)


RedFget[rank p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B24)


RedFcoef[coef p21 p

26 s12 s23 s34 s45 s56 s16 s123 s234 s345m

21 m

26] (B25)





90






91

B Software























92



k1p2k1p2k1k2








∆I =

radic

radic

radic

radic

Nsum

i=1

(∆Ii)2 (B31)


93

B Software


by KKajda ver09


In[2]= n1 = 1 n2 = 1 n3 = 1

m = 1 s = -3



Using strategy C

U amp F polynomials

U = x1+x2

F = x1^2+5 x1 x2+x2^2




Out[3]= -0393325 + eps^(-1) 565622^-7 134887^-6eps


94

Bibliography














95

Bibliography
















96

Bibliography














97

Bibliography


















98

Bibliography















99

Bibliography









100

Introduction










Using Barnes lemmas















Gamma function

Software




Bibliography

multipoint feynman diagrams at the one loop level

Documents