Recursive Approaches to QCD Matrix

Elements

including work with

Z. Bern, S Bidder, E Bjerrum-Bohr, L. Dixon, H Ita, D

Kosower W Perkins K. Risager

RADCOR RADCOR 2005

David Dunbar,

Swansea University, Wales

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QCD Matrix Elements

-QCD matrix elements are an important part of calculating the QCD background for processes at LHC

-NLO calculations (at least!) are needed for precision

-One-Loop n-point on-shell amplitudes unknown for n>5

-In this talk we discuss how recent (> Dec 03) ideas of a “weak-weak” duality might help,

- apply to 2g 4g

2 4 talks of Binoth,Dittmaier

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Duality with String Theory

In Dec 03 Witten proposed a Weak-Weak duality between

A) Yang-Mills theory ( N=4 )

B) Topological String Theory

S-matrix of two theories should be identical

-True for tree level gluon scattering

Rioban, Spradlin,Volovich

proposal constructive for N=4 but… N<4..??

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Duality Details

Amplitude a function

Spinor helicity replaces polarisation by fermionic variables

Use fermionic coordinates and replace bosonic momenta by `twistors`

Amplitude now function entirely of fermionic variables

Fourier (Penrose transform) one of spinors,

Amplitude in twistor space is closest to string theory

Xu, Zhang,Chang 87

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Is the duality useful?

Theory A :Theory A :

hard, hard, interestinginteresting

Theory B: Theory B:

easyeasy

Perturbative QCD,Perturbative QCD,hard, interestinghard, interesting

TopologicalTopologicalString TheoryString Theory::

harder, uninteresting harder, uninteresting

-duality may be useful indirectly-duality may be useful indirectly

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Inspired by duality – the CSW/MHV-vertex construction

Promotes MHV amplitude to fundamental object by

-Off-shell continuation

-makes YM perturbative expansion match that of B-instantons

Cachazo Svercek Witten 04, (Nair)

Parke-Taylor, Berends-Giele(colour ordered amplitudes)

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+

_

___

_

+

+

+ +

+

+

_

_

_

-three point vertices allowed-number of vertices = (number of -)

-1

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MHV-vertex constructionWorks for gluon scattering tree amplitudes

Works for (massless) quarks

Works for Higgs and W’s

Works for photons

Works for gravity

-Works for one-loops in SUPERSYMMETRIC theories

-In trouble for non-SUSY theories

Bjerrum-Bohr,DCD,Ita,Perkins, Risager

Ozeren+Stirling

Badger, Dixon, Glover, Forde, Khoze, Kosower Mastrolia

Wu,Zhu; Su,Wu; Georgiou Khoze

Bedford,Brandhuber, Spence and Travaglini;

Qigley,Rozali

A(+++…++)

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Inspired by duality –BCFW construction for tree amplitudes

Return of the analytic S-matrix!

Shift amplitude so it is a complex function of z

Amplitude becomes an analytic function of z, A(z)

Full amplitude can be reconstructed from analytic properties

Britto,Cachazo,Feng (and Witten)

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Provided,

Residues occur when amplitude factorises on multiparticle pole

(including two-particles)

then

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-results in recursive on-shell relation

(three-point amplitudes must be included)

1 2

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CSW vs BCFDifference

CSW asymmetric between helicity sign

BCF chooses two special legs

Similarities-

both rely upon analytic structure

CSW can be derived from a type of analytic shift

Risager; Bjerrum-Bohr,Dunbar,Ita,Perkins and Risager, 05

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One-Loop Amplitudes

One Loop Gluon Scattering Amplitudes in QCD

-Four Point : Ellis+Sexton

-Five Point : Bern, Dixon,Kosower

-Six-Point and beyond--- present problem

n-point MHV amplitudes supersymmetric theories

Bern,Dixon,Dunbar and Kosower 94/95

Six-point N=4 amplitudes

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Supersymmetric Decomposition

Supersymmetric gluon scattering amplitudes are the linear combination of

QCD ones+scalar loop

-this can be inverted

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N=4 One-Loop Amplitudes –solved!

Amplitude is a a sum of scalar box functions with rational coefficients (BDDK,1994)

Coefficients are ``cut-constructable’’ (BDDK,1994)

….

….

Quadruple cuts turns calculus into algebra (Britto,Cachazo,Feng,2005)

Box Coefficients are actually coefficients of terms like

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Box-Coefficients

-works for massless corners (complex momenta)

S

Britto,Cachazo,Feng

or signature (--++)-works for non-supersymmetric

Bjerrum-Bohr,Bidder,DCD,Perkins

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N=1 One-Loop Amplitudes -????

Important to choose a good basis of functions

A) choose chiral multiplet

B) use D=6 boxes

Amplitude also cut constructible

-six gluon amplitudes now obtained using unitarity

Bidder,Bjerrum-Bohr,Dixon, Dunbar, PerkinsBritto, Buchbinder Cachazo, Feng, 04/05

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The Final Pieces : scalar contributions

-last component of QCD amplitudes- R is rational and not cut constructible (to

O())

cut construcible

recursive?recursive?

-can we avoid direct integration?

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Recursion for Rational terms

-can we shift R and obtain it from its factorisation?

1) Function must be rational

2) Function must have simple poles

3) We must understand these poles

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-understanding poles

-multiparticle factorisation

theoremsBern,Chalmers

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Complication,

Either R or the coefficients of integral functions may contain Spurious Singularities which are not present in the full amplitude

It is important and non-trivial to find shift(s) which avoid these spurious singularities whilst still affecting the full R/coefficient

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Example of Spurious singularities

Collinear Singularity Multi-particle

poleCo-planar singularity

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Splitting Amplitude into C and R is not unique

The integral functions can be defined to include rational pieces, e.g

rather than

avoids a spurious singularity as r1 (r=s/s’)

Spurious singularities spoil understanding of residues – can we avoid them?

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Results:

It has been demonstrated, using

A) (-++….+++), (+++++….++) and

B) (--++++) and (--++….+++)

that shifts can be found which allow calculation of rational parts recursivelyBern, Dixon Kosower

1)

2) Shifts can be found which allow the integral coefficients to be computed

recursivelyA(---..--+++…++)

Bern, Bjerrum-Bohr, Dunbar, Ita

Forde, Kosower

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State of Play Six Gluon Scattering

X

X X

XX

X

X

X

X

X X

X X

X

X

XX

2g- 4g

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Conclusions-Reasons for optimism in computing one-loop QCD matrix

elements

-Recent progress uses UNITARITY and FACTORISATION as key features of on-shell amplitudes

-Inspired by Weak-Weak duality but not dependant upon it

-after much progress in highly super-symmetric theories the (harder) problem of QCD beginning to yield results

-first complete result for a partial 2g ng amplitude!

-also fermions, masses, multiloop……..???.......