Semi-numerical power expansion of Feynman integrals

I present an algorithm based on sector decomposition and Mellin-Barnes techniques to power expand Feynman integrals. The coefficients of this expansion are given in terms of finite integrals that can be calculated numerically. I show in an example the benefit of this method for getting the full analytic power expansion from differential equations by providing the correct ansatz for the solution. For method of regions the presented algorithm provides a numerical check, which is independent from any power counting argument.

[1]  V. Smirnov Asymptotic expansions in limits of large momenta and masses , 1990 .

[2]  T. Seidensticker Automatic application of successive asymptotic expansions of Feynman diagrams , 1999, hep-ph/9905298.

[3]  S. Laporta,et al.  difference equations , 2001 .

[4]  T. Binoth,et al.  Numerical evaluation of multi-loop integrals by sector decomposition , 2004 .

[5]  F. Tkachov,et al.  Integration by parts: The algorithm to calculate β-functions in 4 loops , 1981 .

[6]  A. V. Smirnov,et al.  Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA) , 2008, Comput. Phys. Commun..

[7]  M. Beneke,et al.  Asymptotic expansion of Feynman integrals near threshold , 1997, hep-ph/9711391.

[8]  Virtual O( alpha s) corrections to the inclusive decay b-->s gamma. , 1996, Physical review. D, Particles and fields.

[9]  F. Tkachov A theorem on analytical calculability of 4-loop renormalization group functions , 1981 .

[10]  T. Binoth,et al.  An automatized algorithm to compute infrared divergent multi-loop integrals , 2000 .

[11]  Differential equations for Feynman graph amplitudes , 1997, hep-th/9711188.

[12]  Christian Bogner,et al.  Operating system: Unix , 1983 .

[13]  Volker Pilipp Hard spectator interactions in B→ππ at order αs2 , 2007, 0709.3214.

[14]  G. Heinrich A numerical method for NNLO calculations , 2002, hep-ph/0211144.

[15]  A. Kotikov Differential equations method. New technique for massive Feynman diagram calculation , 1991 .

[16]  M. Czakon,et al.  NNLO fermionic corrections to the charm quark mass dependent matrix elements in B̄ → Xsγ , 2007, 0707.3090.