###### Abstract

A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for tensor integrals as well as for scalar ones appearing in loop calculations of the standard model. Examples of 3- and 4-point diagrams in 1-loop integrals and 2- and 3-point diagrams in 2-loop integrals with arbitrary masses are shown.

Moreover it is shown that numerical evaluations of the Hypergeometric function, which often appears in the loop integrals, can be performed using the numerical contour-integration method.

KEK-CP-159

Numerical Contour Integration for Loop Integrals

Y. Kurihara and T. Kaneko

High Energy Accelerator Research Organization,

Oho 1-1, Tsukuba, Ibaraki 305-0801, Japan

## 1 Introduction

In future collider experiments such as at the LHC and ILC, the standard model will be checked with very high precision and a signal of new physics will be searched for through a tiny difference between experimental measurements and theoretical predictions. The theoretical uncertainty must be, at least, one order of magnitude smaller than the experimental one. The theoretical prediction is obtained based on the perturbative calculation of the quantum field theories. The theoretical calculation of a higher order correction with one- or two-loops must be performed in order to meet with experimental requirements. A loop integration is one of the critical issues of a computation of these higher order corrections. Basically there are two methods to perform the loop integration, an analytical method and a numerical one. Though the analytical method gives, in principle, fast and stable results, for many cases it is hard to give a compact expression for the one-loop integral and it is very difficult to perform two-loop computations for the standard model with multiple energy scales.

Numerical methods have also been investigated to perform loop integrals. The symmetrical sampling method was proposed in [1] by Oyanagi et al. in 1988 and has been investigated through a series of papers[2, 3, 4]. Another numerical method, the hybrid method[5], has been proposed. It is to perform a part of the multi-dimensional integrations analytically up to the integrand being a logarithmic form. A remaining integration was performed numerically by Monte Carlo method. Both methods are tuned to calculate up to two-loop/three-point functions with arbitrary masses[6].

Recently another numerical methods using the Sato-Bernstein-Tkachov (SBT) relation 111This relation was introduced at first by Sato[7] relating to prehomogeneous vector spaces and was generally proved by Bernstein[8]. An application of this relation to the loop integral was proposed by Tkachov[9]. is proposed and utilized intensively[10, 11, 12]. A recent review of the method using the SBT relation can be found in [13].

Yet another numerical method has been proposed by de Doncker[14], an ‘-algorithm’. This method takes the infinitesimal imaginary parameter appearing in a denominator of the loop integral at finite values, and takes its extrapolation to zero. It is demonstrated that this method can give precise results for a non-scalar one-loop/three-point digram[14].

We propose a new method of loop integrations such as a ‘numerical contour-integration (NCI) method’ in this report. The loop integral can be interpreted as a contour integral in a complex plane for integrands with multi-poles in the plane. We show that the stable and efficient numerical integrations along an appropriate contour for tensor integrals as well as scalar ones appear in loop calculations of the standard model. This method is applicable, so far, up to two-loop/three-point functions with arbitrary masses and arbitrary polynomials of Feynman parameters in the numerator of the integrand. The basic idea of this method and some numerical results are shown in this report.

## 2 Feynman parameter representation of tensor integrals

The tensor integral of a massless one-loop N-point graph with rank in a space-time dimension of can be written as,

 T(N)μ⋯νM = ∫dnk(2π)nikμ⋯kνA1A2⋯AN,

where

 Ai = (k+si)2−m2i+i0, si = i∑j=1pj,  s0=0,

and is a four momentum of an ’th external particle (incoming), a loop momentum, and the internal masses. An infinitesimal imaginary part () is included to obtain analyticity of the integral . The momentum integration can be done using Feynman’s parameterization which combines the propagators. After the momentum integration, an ultra-violet pole is subtracted under some renormalization scheme. Finally the tensor integral can be expressed using integrals of the type

 I(N)i⋯kM = ∫[dx]xi⋯xk(DN−i0)N−2, (1)

where

 ∫[dx] = ∫10dx1∫1−x1dx2⋯∫1−∑N−1i=1xidxN−1.

An explicit form of the numerator, and a relation between and , can be obtained straightforwardly depending on the diagram[15]. The remaining task is to perform the parametric integration of in Eq.(1), which is called a tensor integral in this report. (When the numerator of the integrand is unity, it is called a scalar integral.)

## 3 one-loop/three-point

A general formula of the tensor integral for the one-loop/three-point function with arbitrary masses is[1]

 I(3) = ∫10dx∫1−x0dyf(x,y)D3−i0, D3 = (M1x−M2y)2−rxy+m231x+m232y+m22,

where

 r = s−(M1+M2)2, m231 = −M21+m21−m22, m232 = −M22+m23−m22, M21 = p21, M22 = p22, s = (p1+p2)2.
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