One-Loop Tensor Integrals in Dimensional Regularisation

TL;DR

This paper introduces a Gram-determinant–robust framework for evaluating one-loop tensor integrals in dimensional regularisation by forming finite combinations of scalar integrals. These combinations arise naturally from differentiating scalar integrals with respect to external parameters or by expressing them in higher dimensions (D+2, etc.), effectively removing spurious singularities and reducing numerical instability. The authors provide explicit constructions for 3-, 4-, and 5-point integrals (triangles, boxes, and pentagons) with massless internal lines, introducing finite building blocks such as Lc, Ld, and Le, and demonstrate how to treat adjacent and opposite box configurations differently to ensure stability. The approach yields compact, systematically computable results that maintain correct infrared and ultraviolet pole structure and can be extended to general masses and kinematics.

Abstract

We show how to evaluate tensor one-loop integrals in momentum space avoiding the usual plague of Gram determinants. We do this by constructing combinations of $n$- and $(n-1)$-point scalar integrals that are finite in the limit of vanishing Gram determinant. These non-trivial combinations of dilogarithms, logarithms and constants are systematically obtained by either differentiating with respect to the external parameters - essentially yielding scalar integrals with Feynman parameters in the numerator - or by developing the scalar integral in $D=6-2\e$ or higher dimensions. As an explicit example, we develop the tensor integrals and associated scalar integral combinations for processes where the internal particles are massless and where up to five (four massless and one massive) external particles are involved. For more general processes, we present the equations needed for deriving the relevant combinations of scalar integrals.

Figures (8)

• Figure 1: The triangle graph and each of the three pinchings obtained by omitting the internal line associated with $\alpha_m$ for $m=1,2$ and 3. In each case, the internal line is shrunk to a point and the momenta at either end are combined. The relation between the external momenta and the $\alpha_i$ can be seen by cutting the loop; $\alpha_i\alpha_j = -1/p^2$ where $p$ is the momentum on one side of the cut and $\alpha_i$, $\alpha_j$ are associated with the cut lines.
• Figure 2: The finite functions for the triply massive triangle graph with $s_{12} = 1,~~p_1^2 = 0.2$ as a function of $\Delta_3/\Delta_3^{{\rm max}}$ where $\Delta_3^{{\rm max}} = -(s_{12}-p_1^2)^2$. The functions have been evaluated using double precision Fortran. The dashed lines show the approximate form for the function in the limit $\Delta_3 \to 0$, retaining only the first term of the Taylor expansion.
• Figure 3: The finite functions for the triangle graph with two external masses with $s_{12} = 1$ evaluated in double precision Fortran as a function of $(s_{12}-p_1^2)/s_{12}$. The dashed lines show the approximate form for the function in the limit $p_1^2 \to s_{12}$, retaining only the first term of the Taylor expansion.
• Figure 4: The box graph and each of the four pinchings obtained by omitting the internal line associated with $\alpha_m$ for $m=1,2,3$ and 4.
• Figure 5: The finite functions for the one mass box graph as a function of $\Delta_4/\Delta_4^{{\rm max}}$ where $\Delta_4^{{\rm max}} = 2 s_{12}s_{23}(s_{123}-s_{12}-s_{23})$. The phase space point is $s_{123} = 1$, $s_{12} = 0.4$, $s_{23} = 0.08$ and $p_1^2$ altered so the limit is approached and the functions have been evaluated in double precision Fortran. The dashed lines show the approximate form for the function in the limit $\Delta_4 \to 0$, retaining only the first term of the Taylor expansion as given in Appendix C.