A Calculational Formalism for OneLoop Integrals

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A Calculational Formalism for One-Loop Integrals
Collaborators Formalism E.W.N. Glover
(hep-ph/0402152) Numerics G.
Zanderighi and E.W.N. Glover

• Introduction
• Goals
• Tensor Integrals as Building Blocks
• Numerical Evaluating of Tensor Integrals
• Outlook.

W. Giele, Loops and Legs, 29/04/04
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Introduction

• We need NLO for
• First estimate of normalization
• Better understanding of shape of distributions

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Goals

• An automated evaluation of one loop virtual
  contributions
• The number of external legs should be limited by
  computer resources
• No numerical integration in loop space but
  numerical reduction to basis set of integrals

For pure numerical approaches see G.J. van
Oldenborgh J.A.M. Vermaseren 1990 D.Soper
2000 D. Soper Z. Nagy 2003
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Tensor Integrals as Building Blocks

• Any one-loop amplitude can be decomposed
  inwhere the tensor integral is given byand

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• The calculation of the tensor integral causes
  algebraic problems
• Next we would express the coefficients into
  scalar integrals
• As we increase the number of external lines this
  quickly becomes unmanageable

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• For example the 6 photon amplitude is a sum over
  the different permutations of
• This goes up to rank 6 six-point tensor integrals
• The rank 6 on itself would create 49 terms
  likefor which we need determine the coefficient
  in terms of scalar integrals and evaluate

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• Clearly for an automated evaluation of loop
  graphs we need to numerically evaluate the tensor
  integrals
• This leaves us only with the calculation of the
  coefficient for each rankwhere the tensor
  integrals are symmetric

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Numerical Evaluating of Tensor Integrals
(See also A. Ferroglia, M. Passera, G. Passarino
S. Uccirati, 2003
F. del Aguila R.Pittau, 2004)

• To accomplish this we need to
• Separate the divergent part of the tensor
  integral
• Calculate the finite part numerical
• Calculate the divergent part algebraic (if not
  known)E.g. the leading color divergent part of a
  color ordered amplitude is simply(this
  requires a careful definition of finite)

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• The decomposition to generalized scalars does
  fulfill all the requirements (Davydychev 1991)
• It translates a tensor integral into higher
  dimensional scalar integrals. E.g.
• If we can numerically evaluate the scalar
  integralswe can numerically construct the
  tensor integral

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• A recursive scheme to calculate
  is well established in the literature
• Recursion relations between scalar integrals have
  been known for a long time in 4 dimensions
  (Melrose 1965W.L. van Neerven J.A.M.
  Vermaseren 1984)
• The extension to arbitrary dimensions was first
  formalized by Z. Bern, L.J. Dixon D.A. Kosower
  1993
• And further developed by many groups into the
  formulation we will use (O.V. Tarasov 1996 J.M.
  Campbell, E.W.N. Glover and D.J. Miller 1997 J.
  Fleischer, F. Jegerlehner O.V. Tarasov 2000
  T. Binoth, J.P. Guillet G. Heinrich 2000 G.
  Duplancic B. Nizic 2003)

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• The basic identity behind the recursion relations
  is the integration by part identity (K.G.
  Chtyrkin, A.L. Kataev F.V. Tkachov 1980)
• This leads to the base equation (for
  )

Evaluated numerical
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• Sequential application of recursive relations
  eventually lead to a basis set of known
  integrals
• The 4-dimensional finite part is calculated by
  the numerical recursion algorithm
• The divergent part is either known or calculated
  by the singular decomposition

Only UV/IR divergent2- and 3-pointintegrals
No dimensional dependence
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• We need to be able to calculate the divergent
  contribution analytical
• For this we extended a method (S. Dittmaier,
  2003) to work with tensor integrals.
• The singular decomposition decomposes a tensor
  integral instantaneously into a sum over
  divergent triangles

Finite parts of triangles in numerical part
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G. Zanderighi, W.G E.W.N. Glover

• We are currently implementing the recursion
  algorithm up to 6 external legs (including
  arbitrary mass configurations)

Analytic
Recursive
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Outlook

• We have constructed an explicit algorithmic
  method to calculate NLO corrections to processes
  with large number of legs
• A numerical evaluation of the finite part of rank
  m N-point tensor integrals
• An analytic singular decomposition to 3-point
  functions for rank m N-point tensor integrals
• We are in the middle of implementing the
  algorithm for arbitrary masses configurations
• This will lead shortly to first applications,
  e.g.

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