Regularization of IR divergent loop integrals

We report results of a new numerical regularization technique for infrared (IR) divergent loop integrals using dimensional regularization, where a positive regularization parameter ?, satisfying that the dimension d = 4 + 2?, is introduced in the integrand to keep the integral from diverging as long as ? > 0. A sequence of integrals is computed for decreasing values of ?, in order to carry out a linear extrapolation as ? ? 0. Each integral in the sequence is calculated according to the Direct Computation Method (DCM) to handle (threshold) integrand singularities in the interior of the domain. The technique of this paper is applied to one-loop N-point functions. In order to simplify the computation of the integrals for small ?, particularly in the case of a threshold singularity, a reduction of the N-point function is performed numerically to a set of 3-point and 4-point integrals, and DCM is applied to the resulting vertex and box integrals.

[1]  J. N. Lyness,et al.  Van der Monde systems and numerical differentiation , 1966 .

[2]  J. Fujimoto,et al.  Numerical Evaluation of Feynman Integrals by a Direct Computation Method , 2008 .

[3]  Elise de Doncker,et al.  Transformation, Reduction and Extrapolation Techniques for Feynman Loop Integrals , 2010, ICCSA.

[4]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[5]  B. Burrows,et al.  Lower bounds for quartic anharmonic and double‐well potentials , 1993 .

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

[7]  Elise de Doncker,et al.  Dimensional recursion for multivariate adaptive integration , 2010, ICCS.

[8]  D. Shanks.  Non‐linear Transformations of Divergent and Slowly Convergent Sequences , 1955 .

[9]  Elise de Doncker,et al.  Recursive box and vertex integrations for the one-loop hexagon reduction in the physical region , 2010 .

[10]  J.A.M. Vermaseren,et al.  New features of FORM , 2000 .

[11]  Y. Kurihara.  Dimensionally regularized one-loop tensor integralswith massless internal particles , 2005 .

[12]  R. Bulirsch,et al.  Bemerkungen zur Romberg-Integration , 1964 .

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

[14]  Robert Piessens,et al.  Quadpack: A Subroutine Package for Automatic Integration , 2011 .

[15]  Shujun Li,et al.  On Iterated Numerical Integration , 2005, International Conference on Computational Science.

[16]  Marko Ljucovic,et al.  Extrapolation Algorithms for Infrared Divergent Integrals , 2011, 1110.3587.

[17]  Elise de Doncker,et al.  Computation of loop integrals using extrapolation , 2004 .

[18]  Junpei Fujimoto,et al.  New implementation of the sector decomposition on FORM , 2009, 0902.2656.

[19]  T. Binoth,et al.  A numerical evaluation of the scalar hexagon integral in the physical region , 2003 .