1. A Calculational Formalism for One- Loop Integrals Introduction Goals Tensor Integrals as Building Blocks Numerical Evaluating of Tensor Integrals Outlook. W. Giele, Loops and Legs, 29/04/04 Collaborators: Formalism: E.W.N. Glover (hep-ph/0402152) Numerics: G. Zanderighi and E.W.N. Glover

2. Introduction We need NLO for: First estimate of normalization Better understanding of shape of distributions

3. 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

4. Tensor Integrals as Building Blocks Any one-loop amplitude can be decomposed in where the tensor integral is given by and

5. 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

6. 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 like for which we need determine the coefficient in terms of scalar integrals and evaluate

7. 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 rank where the tensor integrals are symmetric

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

9. 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 integrals we can numerically construct the tensor integral

10. 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 1965;W.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 )

11. 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

12. 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”: No dimensional dependence Only UV/IR divergent 2- and 3-point integrals

14. 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

15. We are currently implementing the recursion algorithm up to 6 external legs (including arbitrary mass configurations) -0.847448212761-7.72950724975j0.847448212761-7.72950724975j -0.773231936093-7.38189821256j0.773231936093-7.38189821256j -0.00436614882507-0.553570755011j0.00436614882487-0.55357075501j AnalyticRecursive G. Zanderighi, W.G & E.W.N. Glover

16. 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.: