Regularization of Feynman 4-Loop Integrals with Numerical Integration and Extrapolation

Abstract

In this paper we continue our recent work on evaluating numerical approximations for a set of 4-loop self-energy integrals required in the computation of higher orders in perturbation theory. The results are given by a Laurent expansion in the dimensional regularization parameter, $\varepsilon ,$ where $\varepsilon$ is related to the space-time dimension as $\nu = 4-2\varepsilon .$ Although the leading-order coefficients for the diagrams with massless internal lines are given in analytic form in the literature, we obtain them using a numerical approach and with modern computational techniques. In a similar manner, we derive results for diagrams with massive lines. The SIMD (Single Instruction, Multiple Data) nature of the computation of loop integrals based on composite lattice rules lends itself to an efficient GPU implementation. We further apply double exponential numerical integration layered over the message passing interface (MPI) parallel platform. Limits of integral sequences (as the regularization parameter tends to zero) are implemented numerically using linear and nonlinear extrapolation procedures. Numerical results are given to illustrate the versatility of the methods and show some robustness with regard to the selection of sequence parameters.

Appendices

Appendix

A C and D Functions

In the code of the C and D functions below for the diagrams of Fig. 1, we denote $x_{i_1\ldots i_k} = \sum _{j = i_1}^{i_k} x_j.$

1.1 A.1 $M_{43}$ Massive Diagram

1.2 A.2 $M_{44}$

For $M_{44}$ (and $M_{45}$) we use the expression for W from [8], and further

$$\begin{aligned}&U = C \\&V = M^2 - \frac{W}{U} \\&D = UV = U M^2 - W \\&M^2 = \sum _{r=1}^N m_r^2\, x_r \end{aligned}$$

Thus $M^2 = 1$ for massive diagrams with all masses $m_r = 1,$ and $D = U - W = C - W.$

For massless diagrams, $M^2 = 0,$ so $D = -W.$

1.3 A.3 $M_{45}$