series, -expansion of the hypergeometric functions and
TRANSCRIPT
Series, ε-expansion of thehypergeometric functions and Feynman
diagrams
M.Yu.Kalmykov.
DESY,Zeuthen & JINR, Dubna
Based on the resultsA.I. Davydychev & M.Yu.K.
hep-th/0303162 (ver.3)F. Jegerlehner & M.Yu.K.
Nucl.Phys.B676 (2004)
Introduction
One of the most powerful tools used in loop calculations isdimensional regularization.
G. ’tHooft and M. Veltman, Nucl. Phys. B44 (1972) 189;
C.G. Bollini and J.J. Giambiagi, Nuovo Cimento 12B (1972) 20;
J.F. Ashmore, Lett. Nuovo Cim. 4 (1972) 289;
G.M. Cicuta and E. Montaldi, Lett. Nuovo Cim. 4 (1972) 329.
In some cases, one can derive results valid for an arbitrary space-
time dimension n=4−2ε usually in terms of various hypergeometric
functions. Dimensional regularization allows one to simultaneously
regulate the ultraviolet (UV) and infrared (IR) singularities.
For practical purposes the coefficients of the expansion in ε are
important.
In particular, in multiloop calculations higher order terms of the ε
expansion of one– and two–loop functions are needed, since one can
get contributions where these functions are multiplied by poles in ε.
Such poles may appear not only due to factorizable loop, but also as
IR-singularities or a result of application of the integration by part or
other techniques.
F.V. Tkachov, Phys. Lett. B100 (1981) 65;
K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159;
O.V. Tarasov, Phys. Rev. D54 (1996) 6479
Mikhail Kalmykov 2
What have been done before
D. Broadhurst (unpublished) ∼ 1991;
A. Grozin, 2000 (unpublished); Int.J.Mod.Phys. A (2004) 473;
S. Moch, P. Uwer & S. Weinzierl, J.Math.Phys. 43 (2002) 3363;
S. Weinzierl, hep-ph/0402131;
Expansion of hypergeometric functions of several variables around
integer values of parameters.
The result of expansion are expressible in terms of nested sums
or another new functions, like
harmonic polylogarithms
E. Remiddi & J.A.M. Vermaseren, Int. J. Mod. Phys. A15 (2000) 725.
2-d harmonic polylogarithms
T. Gehrmann and E. Remiddi, Nucl. Phys. B601 (2001) 248
multivariable polylogarithms
A.B. Goncharov, Math. Res. Lett. 5 (1998) 497.
For numerical evaluation of the new functions the proper subroutines
was elaborated
T. Gehrmann and E. Remiddi,
Comput. Phys. Commun. 141, 296 (2001)
Comput. Phys. Commun. 144, 200 (2002)
S. Weinzierl,
Comput. Phys. Commun. 145, 357 (2002)
Problem
The elaboration of the algorithm for analytical calculation of the higher
order terms of the ε-expansion of any hypergeometric functions of
several variables with arbitrary set of parameters.
Mikhail Kalmykov 3
Hypergeometric function
The generalized hypergeometric function can be written as series
PFQ
„
{A1 + a1ε}, {A2 + a2ε}, · · · {AP + aPε}{B1 + b1ε}, {B2 + b2ε}, · · · {BQ + bqε}
z
«
=∞X
j=0
zj
j!
ΠPs=1(As + asε)j
ΠQr=1(Br + brε)j
,
where (α)j ≡ Γ(α+ j)/Γ(α) is the Pochhammer symbol.
We want to construct the ε-expansion of this series.
PFQ =
8
<
:
P ≤ Q converges for all finite z
P = Q+ 1 converges for all |z| < 1
P > Q+ 1 diverges for all z 6= 0
Mikhail Kalmykov 4
Case of integer and half-integer valuesWe will concentrate on the case P+1FP .
Let us consider firstly the case of integer or half-integer values of
parameters: {Ai, Bj} ∈ {mi,mj + 12}.
To perform the ε-expansion we use the well-known representation
(1 + aε)j = j! exp
"
−∞X
k=1
(−aε)k
kSk(j)
#
,
which, for integer positive values, Ai ≡ mi > 1, yields
(m+aε)j=(m)j exp
(
−∞X
k=1
(−aε)k
k[Sk(m+j−1)−Sk(m−1)]
)
,
where Sk(j) =Pj
l=1 l−k is the harmonic sum satisfying the relation
Sk(j) = Sk(j − 1) +1
jk.
For half-integer positive values, Ai ≡ mi + 1/2 > 0, we use the
duplication formula
„
m+1
2+ aε
«
j
=(2m+ 1 + 2aε)j
4j (m+ 1 + aε)j
To work only with positive values we can apply several times the
Kummer relation:
PFQ
„
a1, · · · , aPb1, · · · , bQ
z
«
= 1 + za1 . . . aP
bi . . . bQP+1FQ+1
„
1, 1+a1, · · · , 1+aP2, 1+b1, · · · , 1+bQ
z
«
.
Mikhail Kalmykov 5
SumsAfter applying this procedure the original hypergeometric function can
be written as
P+1FP
„
{mi + aiε}J , {mj + 12 + djε}P+1−J
{ni + biε}K, {nj + 12 + cjε}P−K z
«
=
∞X
j=1
zj
j!
1
4j(K−J+1)
ΠJi=1(mi)j
ΠKl=1(nl)j
ΠP+1−Jr=1
(2mr + 1)j
(mr + 1)jΠP−Ks=1
(ns + 1)j
(2ns + 1)j∆
where
∆ = exp
2
4
∞X
k=1
(−ε)k
k
„ KX
l=1
bkl Sk(nl + j − 1)
−JX
i=1
akiSk(mi + j − 1)
+P−KX
s=1
cks
»
Sk(2ns + j) − Sk(ns + j)
–
−P+1−JX
r=1
dkr
»
Sk(2mr + j) − Sk(mr + j)
–«
3
5
with Sk(j) =Pj
l=11
lk
In this way, the analytical calculation of the ε-expansion is reduce to
analytical calculation of the multiple sums
∞X
j=1
zj
j!
1
4j(K−J+1)Πi,r,l,s
(mi − 1 + j)!(2mr + j)!
(nl − 1 + j)!(2ns + j)!×
[Sa1(m1+j−1)]i1 . . . [Sap(mp+j−1)]
ip ×[Sb1(2mr+2j)]j1 . . . [Sbq(2ns+2j)]jq,
where {mj, nk} - positive integer numbers.
Mikhail Kalmykov 6
Calculation methods
How to calculate this sums analytically?
• Integral representation for ψ-function
ψ(z) =d
dzln Γ(z) , ψ(m)(z) =
dm
dzmψ(z) ,
S1(j − 1) = ψ(j)+γE ,
Sk(j−1) = ζk−(−1)k
(k − 1)!ψ
(k−1)(j) ,
ψ(1 + z) = ψ(z) +1
z,
ψ(m)(1 + z) = ψ(m)(z) +(−1)mm!
zm+1,
ψ(z) + γ =
Z 1
0
1 − tz−1
1 − tdt ,
ψ(m)
(z) = −Z 1
0
tz−1
1 − tlnmt dt ,
where γE is Euler’s constant.
Multiple sums reduced to the multiple integral of the harmonic sums
=⇒ further simplifications in terms of Goncharov polylogartohms
or some more simple functions.
D. Borwein, J.M. Borwein, R. Girgensohn, (1995)
J.M. Borwein, R. Girgensohn, (1996)
P. Flajolet, B. Salvy, (1998)
O.M. Ogreid, P. Osland, (1998), (2001), (2002)
J. Fleischer, A.V. Kotikov, O.L. Veretin, (1999).
• Generating function approach
H.S. Wilf,Generatingfunctionology, Academic Press, London, 1994.
Mikhail Kalmykov 7
Generating functions approach
Let us rewrite the multiple sums
∞X
j=1
„
z
4(K−J+1)
«j 1
j!Πi,r,l,s
(mi − 1 + j)!(2mr + j)!
(nl − 1 + j)!(2ns + j)!×
[Sa1(m1+j−1)]i1 . . . [Sap(mp+j−1)]ip ×[Sb1(2mr+2j)]
j1 . . . [Sbq(2ns+2j)]jq,
in the following form
ΣA;B(u) =∞X
j=1
ujηA;B(j) u =
z
4(K−J+1),
where A ≡“
i1,...,ipa1,...,ap
”
and B ≡“
j1,...,jqb1,...,bq
”
denote the collective sets
of indices, whereas ηA;B(j) is the coefficient of uj.
ηA;B(j) =1
j!Πi,r,l,s
(mi − 1 + j)!(2mr + j)!
(nl − 1 + j)!(2ns + j)!×
[Sa1(m1+j−1)]i1 . . . [Sap(mp+j−1)]ip ×[Sb1(2mr+2j)]
j1 . . . [Sbq(2ns+2j)]jq,
The idea is to find a recurrence relation with respect to j, for the
coefficients ηA;B(j), and then transform it into a differential equation
for the generating function ΣA;B(u). In this way, the problem of
summing the series would be reduced to solving a differential equation.
Mikhail Kalmykov 8
Special case
Let us consider the following hypergeometric function
P+1FP
„
{32 + biε}J−1, {1 + aiε}K, {2 + diε}L
{32 + fiε}J , {1 + eiε}R, {2 + ciε}K+L−R−2 z
«
=1
2z
ΠK+L−R−2s=1 (1 + csε)Π
Jk=1(1 + 2fkε)
ΠLi=1(1 + diε)Π
J−1r=1 (1 + 2brε)
∞X
j=1
1“
2jj
”
(4z)j
jK−R−1∆ ,
where
∆ = exp
" ∞X
k=1
(−ε)k
k
“
TkSk(j − 1) + 2kUkSk(2j − 1) +Wkj−k”
#
where
Ak ≡X
aki , Bk ≡
X
bki , Ck ≡
X
cki , Dk ≡
X
dki ,
Ek ≡X
eki , Fk ≡
X
fki , Uk ≡ Fk −Bk, Wk ≡ Ck −Dk ,
Tk ≡ Bk + Ck + Ek −Ak −Dk − Fk,
The ε-expansion of hypergeometric function is reduced to studing of
the multiple inverse binomial sums∞X
j=1
1“
2jj
”
uj
jc[Sa1(j−1)]
i1 . . . [Sap(j−1)]ip [Sb1(2j−1)]
j1 . . . [Sbq(2j−1)]jq,
with u = 4z and
ηA;B;c(j)=1
jc[Sa1]
i1 . . . [Sap]ip [S̄b1]
j1 . . . [S̄bq]jq,
where Sa = Sa(j − 1) and S̄a = Sa(2j − 1).
Mikhail Kalmykov 9
Recurrence relation
The recurrence relation for this coefficient
ηA;B;c(j)=1
jc[Sa1]
i1 . . . [Sap]ip [S̄b1]
j1 . . . [S̄bq]jq,
can be written in the following form:
2(2j + 1)(j + 1)c−1ηA;B;c(j + 1) − j
cηA;B;c(j) = rA;B(j) ,
where the explicit form of the “remainder” rA;B(j) is given by
„
2j
j
«
rA;B(j) =
pY
k=1
h
Sak(j−1)+j−akiik
qY
l=1
h
Sbl(2j−1)+(2j)−bl+(2j + 1)−blijl
−pY
k=1
ˆ
Sak(j − 1)˜ik
qY
l=1
ˆ
Sbl(2j − 1)˜jl .
In other words, it contains all contributions generated by j−ak, (2j)−bl
and (2j + 1)−bl which appear because of the shift of the index j.
Multiplying both sides of this equation by uj, summing from 1 to
infinity, and using the fact that any extra power of j corresponds to the
derivative u(d/du), we arrive at the following differential equation for
the generating function ΣA;B;c(u):
„
4
u− 1
«„
ud
du
«c
ΣA;B;c(u) − 2
u
„
ud
du
«c−1
ΣA;B;c(u) =
2ηA;B;c(1) +RA;B(u) ,
where ηA;B;c(1) = 12δp0 and RA;B(u) ≡P∞
j=1 ujrA;B(j).
Mikhail Kalmykov 10
From definition it is easy to deduce
∞X
j=1
1“
2jj
”
uj
jcSa1 ⇒ Ra1;−(u) =
∞X
j=1
uj“
2jj
”
1
ja1,
∞X
j=1
1“
2jj
”
uj
jcS̄b1 ⇒ R−;b1(u) =
∞X
j=1
uj“
2jj
”
2
4
1
(2j)b1+
1
(2j + 1)b1
3
5 ,
· · ·
The differential equation for generating function ΣA;B;c(u) take the
most simple form in terms of the geometrical variable
u ≡ uθ = 4 sin2 θ2 ,
4
u− 1 = cot2 θ
2 , ud
du= tan θ
2
d
dθ,
1
2 sin2 θ2
„
2 sin θ2 cos θ2
d
dθ− 1
«„
tan θ2
d
dθ
«c−1
ΣA;B;c(uθ) =
δp0 +RA,B(uθ) .
Furthermore, it can be represented as
„
tan θ2
d
dθ
«c−1
ΣA;B;c(uθ) = tan θ2 σA;B(θ) ,
whered
dθσA;B(θ) = δp0 +RA,B(uθ)|
uφ=4 sin2 φ2
.
Mikhail Kalmykov 11
Introducing lθ ≡ ln`
2 sin θ2
´
, the differential equation for em
generating function can be rewritten as
„
1
2
d
dlθ
«c−kΣA;B;c(uθ) = ΣA;B;k(uθ) ,
or in integral form
∞X
j=1
uj
jcf(j) =
1
(c− 2)!
θZ
0
dφcos φ2
sin φ2
»
lnu− 2 ln“
2 sin φ2
”
–c−2 ∞X
j=1
“
4 sin2 φ2
”j
jf(j) ,
where f(j) stands for an arbitrary combination of the sums.
The iterative solution of this equation is
ΣA;B;c(uθ) = −kX
i=1
(−2)i
i!liθ ΣA;B;c−i(uθ)
+(−2)k
k!
θZ
0
dφ lkφdΣA;B;c−k(uφ)
dφ,
Statement. If for some k the derivative ΣA;B;c−k(uθ) is expressible
only in terms of the powers of θ and lθ, then the sum ΣA;B;c(uθ) can
be presented in terms of the generalized log-sine functions. Moreover,
according to a statement proven by [Davydychev & Kalmykov (2001)],
the analytic continuation of any generalized log-sine function Ls(k)j (θ)
can be expressed in terms of Nielsen polylogarithms.
Mikhail Kalmykov 12
Some definitions[L. Lewin, Polylogarithms and associated functions North-Holland,
Amsterdam, 1981.]
Ls(k)j (θ) = −
θZ
0
dφ φk lnj−k−1
˛
˛
˛
˛
2 sinφ
2
˛
˛
˛
˛
, Lsj(θ) = Ls(0)j (θ)
is the generalized log-sine function
TiN (z) = Im [LiN (iz)] =1
2i
»
LiN (iz) − LiN (−iz)
–
,
is the inverse tangent integral.
Lsci,j(θ) = −θZ
0
dφ lni−1
˛
˛
˛
˛
2 sin φ2
˛
˛
˛
˛
lnj−1
˛
˛
˛
˛
2 cos φ2
˛
˛
˛
˛
.
is the generalized log-sine-cosine function introduced by [Davydychev,
Kalmykov (2001)]. These functions satisfied to the following properties:
Lsci,1(θ) = Lsi(θ) , Lsc1,j (θ) = −Lsj(π − θ) + Lsj(π) .
Lsci,j(θ) = −Lscj,i(π − θ) + Lscj,i(π) ,k−1X
i=0
1
i!(k − 1 − i)!Lsci+1,k−i(θ) =
1
2(k − 1)!Lsk(2θ) ,
2Lsc2,2(θ) = 12Ls3(2θ) − Ls3(θ) + Ls3(π − θ) − Ls3(π) ,
6Lsc3,3(θ) + 4 [Lsc2,4(θ) − Lsc2,4(π − θ)]
= 12Ls5(2θ) − Ls5(θ) + Ls5(π − θ) + 15
8 πζ4 ,
Lsc2,3(θ) = 112Ls4(2θ) − 1
3Ls4(θ) + 2Ti4`
tan θ2
´
−2 ln`
tan θ2
´
Ti3`
tan θ2
´
+ ln2 `tan θ2
´
Ti2`
tan θ2
´
− 16θ ln3 `tan θ
2
´
.
Up to the level k = 5 only one new function is appeared Lsc2,4(θ).
Mikhail Kalmykov 13
Results
∞X
j=1
1“
2jj
”
uj
jc= −
c−2X
i=0
(−2)i
i!(c− 2 − i)!(lnu)c−2−i
Ls(1)i+2(θ) , c ≥ 2.
∞X
j=1
uj
jc1“
2jj
”S2 = −16
c−2X
i=0
(−2)i
i!(c− 2 − i)!(lnu)
c−2−iLs
(3)i+4(θ) ,
∞X
j=1
uj
jc1“
2jj
”
“
S22−S4
”
=−160
c−2X
i=0
(−2)i
i!(c−2−i)! (lnu)c−2−i
Ls(5)i+6(θ) ,
0 ≤ u ≤ 4, u = 4 sin2 θ2.
∞X
j=1
1“
2jj
”
uj
jS
21 = 4 tan θ
2
»
Ls3(π−θ) − Ls3(π)
−2Ls2(π−θ)Lθ + θL2θ + 1
24θ3
–
,
∞X
j=1
1“
2jj
”
uj
jS1S̄1 = tan θ
2
5 [Ls3(π−θ) − Ls3(π)] − Ls3(θ)
+12Ls3(2θ)−2Ls2(θ)Lθ+2Ls2(π−θ) lθ−8Ls2(π−θ)Lθ−2θlθLθ
+4θL2θ+
112θ
3
ff
,
∞X
j=1
1“
2jj
”
uj
jS3 = tan θ
2
6Cl4 (θ) − θ2Cl2 (θ) − 4θCl3 (θ) − 2θζ3
ff
,
∞X
j=1
1“
2jj
”
uj
j2S̄2 = 4θTi3
`
tan θ4
´
− 8ˆ
Ti2`
tan θ4
´˜2+ 1
96θ4 ,
· · ·
Lθ ≡ ln`
2 cos θ2´
, lθ ≡ ln`
2 sin θ2
´
.
Mikhail Kalmykov 14
Starting from series, depending on the angle θ2 we express the result
in terms of polylogarithmic functions of imaginary argument, which
depend on the
2θ, θ, π − θ,θ
2, π − θ
2,
θ
4
At the level 4 One needs to introduce a new function
Φ(θ) ≡θZ
0
dφ Ls2(φ) ln“
2 cos φ2
”
,
which obeys the following symmetry property:
Φ(θ) + Φ(π − θ) = Φ(π) + Ls2(π − θ) Ls2(θ) ,
where
Φ(π) = 16 ln
42 − ζ2 ln
22 + 7
2ζ3 ln 2 − 5316ζ4 + 4Li4
`
12
´
.
Φ(θ) can be related to the real part of a certain harmonic polylogarithm
of complex argument,
Φ(θ) = 196θ
2(2π − θ)
2 − LθCl3 (θ) + ζ3 ln 2 − H−1,0,0,1(1)
+12
»
H−1,0,0,1(eiθ) + H−1,0,0,1(e
−iθ)
–
,
where
H−1,0,0,1(y) =
yZ
0
dxLi3(x)
1 + x,
H−1,0,0,1(1) = − 112 ln4 2 + 1
2ζ2 ln2 2 − 34ζ3 ln 2 + 3
2ζ4 − 2Li4`
12
´
.
Mikhail Kalmykov 15
∞X
j=1
1“
2jj
”
uj
jcf
f c = 1 c = 2 c = 3 c = 4
1 ? ? ? ?
S1 + + +
S̄1 + + +
S2 ? ? ? ?
S21 + +
S1S̄1 + +
S̄2 + +
S̄21 + +
S3 + +
S1S2 + +
S31 +
S2S̄1 + +
S21S̄1 +
S1
`
S̄2 + S̄21
´
+
S̄31 + 3S̄1S̄2 + 2S̄3 +
S22 − S4 ? ? ? ?
Table 1: Equation index for the inverse binomial sums
An star ? means that the corresponding equation holds for general c.
Mikhail Kalmykov 16
Sums with shifted indices
∞X
j=1
1“
2jj
”
uj
2j + 1=
θ
sin θ− 1 =
2
sin θTi1`
tan θ2
´
− 1 ,
∞X
j=1
1“
2jj
”
uj
(2j + 1)2=
2
sin θ2
Ti2`
tan θ4
´
− 1 ,
∞X
j=1
1“
2jj
”
uj
(2j + 1)S1 = θ cot θ2 +
2
sin θ[Ls2(π − θ) − θLθ] − 2 ,
∞X
j=1
1“
2jj
”
uj
(2j + 1)S2 =
θ3
6 sin θ− 2θ cot θ2 − 1
2θ2 + 4 ,
∞X
j=1
1“
2jj
”
uj
(2j + 1)S̄1 =
1
sin θ[2Ls2(π−θ)−2θLθ+Ls2(θ)+θlθ]
+12θ cot θ2 − 2
sin θ2
Ti2`
tan θ4
´
− 1 ,
∞X
j=1
1“
2jj
”
uj
(2j + 1)S
21 = −1
2θ2+ 4 cot θ2 [Ls2(π − θ) − θLθ] − 4
+2θ cot θ2+θ3
6 sin θ+
4
sin θ
h
Ls3(π−θ)−Ls3(π)+θL2θ−2Ls2(π − θ)Lθ
i
,
∞X
j=1
1“
2jj
”
uj
(2j + 1)S1S2 = 8−θ2
lθ+2θCl2 (π−θ)−2θCl2 (θ)
−4Cl3 (π−θ)−2Cl3 (θ)−ζ3 + cot θ2
h
16θ
3−4θ+4θLθ−4Cl2 (π − θ)i
+1
sin θ
θ2 [Cl2 (π−θ)−Cl2 (θ)]−4θ [Cl3 (π−θ) + Cl3 (θ)]
−8Cl4 (π−θ)+6Cl4 (θ)+θζ3− 13θ
3Lθ
ff
,
Mikhail Kalmykov 17
Analytical continuationTo obtain results valid in other regions of variable u (for u < 0 and
u > 4), one needs to construct the proper analytical continuation
of the expressions presented in the previous section. For generalized
log-sine integrals it is described by [Davydychev, Kalmykov 2001].
Let us introduce a new variable
y ≡ eiσθ, ln(−y − iσ0) = ln y − iσπ,
where the choice of the sign σ = ±1 is related to the causal “+i0”
prescription for the propagators. The inverse relations are
u = −(1 − y)2
y, y =
1 −q
uu−4
1 +q
uu−4
, ud
du= −1 − y
1 + yy
d
dy,
In terms of this variable y, the analytic continuation of all generalized
log-sine integrals can be expressed in terms of Nielsen polylogarithms,
iσ [Lsj(π)−Lsj(θ)] =1
2jjlnj(−y)
h
1−(−1)ji
+(−1)j(j−1)!
j−2X
p=0
lnp(−y)2pp!
j−1−pX
k=1
(−2)−k
× [Sk,j−k−p(y)−(−1)pSk,j−k−p(1/y)] ,
whereas for the function Φ(θ) we get
Φ(θ) = ζ3 ln 2 + 12ζ4 − H−1,0,0,1(1)
+12
h
H−1,0,0,1(y) + H−1,0,0,1(y−1)i
−14
h
Li4(y) + Li4
“
y−1”i
−12
ˆ
ln(1 + y) − 12 ln y
˜
h
Li3(y) + Li3
“
y−1”i
,
Mikhail Kalmykov 18
For the cases involving the inverse tangent integrals the analytic
continuation are
TiN`
tan θ2
´
= −σ2i
[LiN (ω) − LiN (−ω)] ,
ω =1 − y
1 + y= −iσ tan θ
2 ,
TiN`
tan θ4
´
= −σ2i
[LiN (ωs) − LiN (−ωs)] ,
ωs =1 − √
y
1 +√y
= −iσ tan θ4 .
The results have simple form in term of “new” variables.
∞X
j=1
1“
2jj
”
uj
jS2(j − 1) = −1
6
1 − y
1 + yln3 y ,
∞X
j=1
1“
2jj
”
uj
jS2(2j − 1) =
1 − y
1 + y
»
2Li3(−ω) − 2Li3(ω) − 1
24ln3 y
–
,
where
y =1 −
q
uu−4
1 +q
uu−4
, ω =1 − √
y
1 +√y.
Mikhail Kalmykov 19
Binomial & harmonic sumsUsing the procedure described early, one can also construct the ε-
expansion of hypergeometric functions of the following types:
P+1FP
„
{32 + biε}J , {1 + aiε}K, {2 + diε}L
{32 + fiε}J−1, {1 + eiε}R, {2 + ciε}K+L−R u
«
=2
u
ΠK+L−Rs=1 (1 + csε)Π
J−1k=1(1 + 2fkε)
ΠLi=1(1 + diε)ΠJ
r=1(1 + 2brε)
∞X
j=1
„
2j
j
«
zj
jK−R−1∆ ,
P+1FP
„
{32 + biε}J , {1 + aiε}K, {2 + diε}L
{32 + fiε}J , {1 + eiε}R, {2 + ciε}K+L−R−1 z
«
=1
z
ΠK+L−R−1s=1 (1 + csε)Π
Jk=1(1 + 2fkε)
ΠLi=1(1 + diε)ΠJ
r=1(1 + 2brε)
∞X
j=1
zj
jK−R−1∆ ,
The ε-expansion of these hypergeometric function are reduced to
studing of the multiple binomial sums and multiple harmonic sums∞X
j=1
„
2j
j
«
uj
jc[Sa1(j−1)]
i1 . . . [Sap(j−1)]ip [Sb1(2j−1)]
j1 . . . [Sbq(2j−1)]jq,
∞X
j=1
uj
jc[Sa1(j−1)]i1 . . . [Sap(j−1)]ip [Sb1(2j−1)]j1 . . . [Sbq(2j−1)]jq,
Using the generating function approach all these sums can be calculated.
The result have simple form in terms of new variables:
• for multiple binomial sums
χ =1 −
√1 − u
1 +√
1 − u, u =
4χ
(1 + χ)2,
• for multiple harmonic sums
ξ =1 − √
u
1 +√u, u =
(1 − ξ)2
(1 + ξ)2,
Mikhail Kalmykov 20
Results
The first several terms (up to Li4) of ε-expansion for the hypergeometric
functions P+1FP are calculated analytically.
The hypergeometric functions of the special type have been considered:
P+1FP
„
{32+biε}
J−1, {1+aiε}K, {2+diε}L{3
2+fiε}J, {1+eiε}R, {2+ciε}K+L−R−2 z
«
{I}
P = K + L+ J − 2
P+1FP
„
{32+biε}
J, {1+aiε}K, {2+diε}L{3
2+fiε}J, {1+eiε}R, {2+ciε}K+L−R−1 z
«
{II}
P = K + L+ J − 1
P+1FP
„
{32+biε}
J+1, {1+aiε}K, {2+diε}L{3
2+fiε}J, {1+eiε}R, {2+ciε}K+L−R z
«
{III}
P = K + L+ J
For higher order terms of ε-expansion (above Li4) the one-fold integral
representation are available.
The algorithm can be extended on the function P+1FP with the arbitrary
set of parameters.
Up to weight 3 the result are expressible in terms of the ordinary
polylogarithms only (however, new variable should be introduced) At
the level weight 4, the harmonic polylogarithm H−1,0,0,1 it is necessary
to add.
For the particular values of variable z (z=1/4, z=1, z=4) the terms of
order (up to Li5) are available.
Mikhail Kalmykov 21
Examples
2F1 :⇒ K + L = 2, J = 1; 3; expansion up to O(ε4)
2F1
„
1 + a1ε, 1 + a2ε32 + fε,
z
«
, 2F1
„
1 + a1ε, 2 + d1ε32 + fε,
z
«
2F1
„
2 + d1ε, 2 + d2ε32 + fε,
z
«
3F2 :⇒ K + L+ J = 4, J = 1, 2; 3+8; expansion up to O(ε3)
3F2
„
1+a1ε, 1+a2ε, 1+a3ε32+f1ε, 1+e1ε
z
«
3F2
„
1+a1ε, 1+a2ε, 1+a3ε32+f1ε, 2+c1ε
z
«
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
3F2
„
2+d1ε, 2+d2ε, 2+d3ε32+fiε, 1+e1ε
z
«
3F2
„
2+d1ε, 2+d2ε, 2+d3ε32+fiε, 2+c1ε
z
«
3F2
„
32+b1ε, 1+a1ε, 1+a2ε32+f1ε,
32+f2ε
z
«
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
3F2
„
32+b1ε, 2+d1ε, 2+d2ε32+f1ε,
32+f2ε
z
«
4F3 :⇒ K + L+ J = 5, J = 1, 2, 3; 3+8+15;
expansion up to O(ε2)
Mikhail Kalmykov 22
Examples
A.Davydychev & M.K., hep-th/0303162
P126
p
1p
2
J 3
1
2
p
p
p3
K
p2
p3
1p
p3
= 0
= 0
m
m
= 0
= 0m
pp pp
m
M
m m
D4
2 = m2
m
m
(
2σ =
1ν
ν2
σ σ1
3
2
ν , ν , σ , σ , σ )21 1 2 3
m2
2
2
2
2
F10101
σ
ν2
ν1
1 2(σ,ν ,ν )J011
F.Jegerlehner, M.Kalmykov & O.Veretin, Nucl.Phys.B658 (2003) 49.
M
m m
M
m m
p^2=m^2 p^2=m^2
F00112F20110
Mikhail Kalmykov 23
Examples
Particular case z = 1
A.Davydychev & M.K., Nucl.Phys. B605 (2001) 266.
Master-integrals for packages
ONSHELL2
J.Fleischer & M.K. Comp.Phys.Comm. 128 (2000) 531.
MATAD
L.V. Avdeev, Comput. Phys. Commun. 98 (1996) 15
M. Steinhauser, Comput. Phys. Commun. 134 (2001) 335.
3E D3
D 32
ρ
1
ν
1
2σ2
σ
ν2
ν1
J011 1 2
σ1
σ2
α
β
V1001 (α,σ ,σ ,β)1 2F10101
1
(σ,ν ,ν )
ν
ν
νν
5
6
3 4
ν2
ν1
ν
σ
3
ν1
ν2
ρ
4 (ν ,ν ,ν ,ν ,ν ,ν ) (0,σ,0,ν ,ν ,ν )
σ
1 212(σ ,σ ,ρ ,ρ ,ν)1 2 3 4 65
Mikhail Kalmykov 24
Two-loop vertexA.Davydychev & M.K., hep-th/0303162
P126
p
1p
2
p3
= 0
= 02
2
(p2)2P126 =
∞X
j=1
1“
2jj
”
uj
j2
8
<
:
1
ε2+
1
ε
»
−S1 − log(−u)
–
+ 12 log
2(−u)
−S1 log(−u) − 32S2 − 15
2 S21 + 4S1S̄1 + 2
S1
j+ O(ε)
9
=
;
.
J. Fleischer, A. Kotikov, O.Veretin (1999)
(p2)2P126=− 1
2ε2ln2 y
+1
ε
»
2Li2(−y) ln y−4Li3(−y)+ln2 y ln(1 − y)− 13 ln3 y−ζ2 ln y−3ζ3
–
+8H−1,0,0,1(−y) + 2S2,2
“
y2”
− 8S2,2(−y) − 8S2,2(y) − 4Li4(y)
−12Li4(−y) − 8Li2(y) Li2(−y) − 14 [Li2(−y)]2 + 28S1,2(−y) ln y
−10Li3(−y) ln y + 7Li2(−y) ln2 y − 4Li2(−y) ln y ln(1 − y)
− ln2y ln
2(1 − y) − 1
6 ln4y + 2
3 ln3y ln(1 − y) + 2ζ2 ln y ln(1 − y)
−52ζ2 ln2 y + 6ζ3 ln(1 − y) − 11ζ3 ln y − 6Li2(−y) ζ2 − 27
4 ζ4 + O(ε) .
G. Passarino and S. Uccirati (2004)
R. Bonciani, P. Mastrolia and E. Remiddi (2004)
Mikhail Kalmykov 25
Two-loop vertexA.Davydychev & M.K., hep-th/0303162
K
p2
p3
1p
2 = m2
m
m
(
2σ =
1ν
ν2
σ σ1
3
2
ν , ν , σ , σ , σ )21 1 2 3
m2
K(1, 1, 1, 1, 0; p2,m) = (m
2)−2ε 1
2ε2
×Γ2(1 − ε)Γ(1 + 2ε)Γ(1 − 4ε)
Γ(1 − 2ε)Γ(1 − 3ε)Γ(1 + ε)
1
(1 − 2ε)(1 − 3ε)
×
8
<
:
1 − (1 + y)1−4ε
2(1 − y)(1 − y4ε) − 2ε
(1 + y)1−4ε
y−2ε(1 − y)
∞X
j=0
(−4ε)j
×j−1X
p=0
lnp y
2pp!
j−pX
k=1
(−2)−kh
Sk,j+1−k−p(−y)−(−1)pSk,j+1−k−p“
−y−1”i
9
=
;
,
Mikhail Kalmykov 26
Three-loop vacuum diagram D4A.Davydychev & M.K., hep-th/0303162
m
M
m m
D4
(m2)3ε
(1 − ε)(1 − 2ε)D4(1, 1, 1, 1, 1, 1;u)
=2ζ3
ε− 9ζ4 +
∞X
j=1
uj“
2jj
”
8
<
:
− 1
2j2ln
2u+
3
j3lnu− 5
j4− 1
j2ζ2
+4
j3S1−
4
j3S̄1+
2
j2S
21−
4
j2S1S̄1−
1
j2S2+
2
j2
“
S̄21 + S̄2
”
+O(ε)
9
=
;
=2ζ3
ε+2Ls
(1)4 (θ)+8lθ [Cl3 (θ)−ζ3]−2θLs3(θ)−6 [Ls2(θ)]
2
+112θ
4− 12ζ2θ
2−9ζ4 + O(ε), u =M2
m2
(m2)3ε(1 − ε)(1 − 2ε)D4(1, 1, 1, 1, 1, 1;u)
=2ζ3
ε+ 1
4 ln2u ln
2y
+ lnu
»
6Li3(y) − 6 ln yLi2(y) + 12 ln3 y − 3 ln2 y ln(1 − y) − 6ζ3
–
+4Li4(y) − 4S2,2(y) + 6 [Li2(y)]2 − 4 ln(1 − y)Li3(y)
+12 ln y ln(1 − y)Li2(y) − 3 ln2yLi2(y) + 5 ln
2y ln
2(1 − y)
Mikhail Kalmykov 27
−73 ln3 y ln(1 − y) + 1
4 ln4 y − 12ζ2Li2(y) − 8ζ2 ln y ln(1 − y)
+32ζ2 ln
2y + 4ζ3 ln(1 − y) + 3ζ4 + O(ε).
As a non-trivial check on these results we consider two particular values,
M2 = m2 and M2 = 0. In the first case (θ = π3), we reproduce the
known result for the master integral D4 ≡ D4(1, 1, 1, 1, 1, 1;u)|u→1,
(m2)3ε(1−ε)(1−2ε)D4|u→1 =2ζ3
ε−77
12ζ4−6ˆ
Ls2`
π3
´˜2+O(ε) .
In the second case, M2 = 0 (θ = 0, y → 1), the result for the master
integral BM is reproduced
(m2)3ε(1−ε)(1−2ε) D4|u→0 =2ζ3
ε− 9ζ4 + O(ε) .
In a similar manner, analytical results can be deduced for other diagrams
with two different mass scales, like D3 and E3 .
Mikhail Kalmykov 28
Two-loop sunsetA.Davydychev & M.K., hep-th/0303162
pp
m
m
0
;
;σ
ν2
ν1;
1 2(σ,ν ,ν )J011 For the integrals J011 with there are two master integrals of this
type, J011(1, 1, 1) and J011(1, 1, 2). However, two other independent
combinations of the integrals of this type happen to be more suitable
for constructing the ε-expansion, J011(1, 2, 2) and [J011(1, 2, 2) +
2J011(2, 1, 2)].
J011(1, 2, 2) =(m2)−1−2ε
(1 − ε)(1 + 2ε)3F2
1, 1 + ε, 1 + 2ε32 + ε, 2 − ε
p2
4m2
!
= 2(m2)−2ε
p2(1 − y)2ε y2ε
−12 ln2 y + ε
h
12 ln3 y + ζ2 ln y
−6 ln yLi2(−y) − 4 ln yLi2(y) + 3ζ3 + 12Li3(−y) + 6Li3(y)i
+ε2h
12H−1,0,0,1(−y) − 12 ln y
»
S1,2(−y) + S1,2
“
y2”
–
+ 134 ζ4
+8 ln yS1,2(y) − 12S2,2(−y) + 3S2,2
“
y2”
+ 6ζ2Li2(−y)−12 ln(1 − y)Li3(−y) − 7
24 ln4 y − 12ζ2 ln2 y + 2ζ3 ln y
+6 ln yLi3(−y) + 8 ln yLi3(y) + 4ζ2Li2(y) − 94Li4
“
y2”
+18 [Li2(−y)]2 + 12Li2(y) Li2(−y) + 4 [Li2(y)]2i
+ O(ε3)
ff
,
u ≡ p2
m2 = −(1−y)2y , y =
1−q
uu−4
1+q
uu−4
,
Mikhail Kalmykov 29
Two-loop sunsetF.Jegerlehner & M.K., hep-ph/0308216
J012
M
m p^2=m^2
;
;
;
(σ,β,α)
0
β
α
σ
For this integral J012 there are three master integrals of this type,
J012(1, 1, 1), J012(1, 2, 1) and J012(1, 1, 2). However, the other
independent combinations of the integrals of this type happen to
be more suitable for constructing the ε-expansion, J012(1, 2, 2),
J012(2, 2, 1), and [J012(1, 2, 2) + J012(2, 1, 2) + J012(2, 2, 1)].
J012(1, 2, 2) = −(M2)−1−2ε
ε(1 − ε)×
8
<
:
Γ(1 − ε)Γ(1 + 2ε)
Γ(1 + ε)3F2
1, 32, 1 + 2ε
2, 2 − ε
4m2
M2
!
−
m2
M2
!−ε
3F2
1, 1 + ε, 32 − ε
2 − ε, 2 − 2ε
4m2
M2
!
9
=
;
,
J012(2, 2, 1) =(M2)−1−2ε
ε2(1 − ε)×
8
<
:
(1 + ε)Γ(1 − ε)Γ(1 + 2ε)
Γ(1 + ε)4F3
1, 32, 1 + 2ε, 2 + ε
2, 2 − ε, 1 + ε
4m2
M2
!
−
m2
M2
!−ε
3F2
2, 1 + ε, 32 − ε
2 − ε, 2 − 2ε
4m2
M2
!
9
=
;
,
Mikhail Kalmykov 30
J012(1, 2, 2) + J012(2, 1, 2) + J012(2, 2, 1) =
(M2)−1−2ε 1
ε2
8
<
:
Γ(1 − ε)Γ(2 + 2ε)
Γ(1 + ε)
2
41 + 2m2
M2
1 + 2ε
1 − ε×
3F2
1, 32, 2 + 2ε
2, 2 − ε
4m2
M2
!
3
5
−
m2
M2
!−ε2
41 + 2m2
M2
1 + ε
1 − ε3F2
1, 2 + ε, 32 − ε
2 − ε, 2 − 2ε
4m2
M2
!
3
5
9
=
;
= (M2)−1−2ε1 + y
1 − y
8
<
:
1
εln y
−»
6 ln y ln(1 − y) +1
2ln
2y − 2ζ2 + 8Li2(−y) + 6Li2(y)
–
+ε
»
24S1,2
“
y2”
− 8S1,2(−y) − 12S1,2(y) − 12ζ2 ln(1 − y)
+ ln(1 − y)
„
48Li2(−y) + 36Li2(y)
«
− 44Li3(−y)
+ ln y
„
20Li2(−y) + 24Li2(y)
«
− 42Li3(y)
+18 ln y ln2(1 − y) + 3 ln
2y ln(1 − y) +
1
6ln
3y − 2ζ3
–
+O(ε2)
9
=
;
,
witch new variable
y =1 −
q
1 − 4m2
M2
1 +q
1 − 4m2
M2
,M2
m2=
(1 + y)2
y.
Mikhail Kalmykov 31