Application of the negative-dimension approach to massless scalar box integrals

TL;DR

The paper develops and applies the negative-dimension NDIM framework to massless one-loop box integrals, enabling region-specific hypergeometric representations that hold for arbitrary propagator powers and dimensions $D$. It demonstrates that the integrals can be expressed as finite sums of generalized hypergeometric functions (notably Appell $F_2$ for two scales, with additional multi-variable functions for more complex kinematics), organized by convergence regions and related through analytic continuation. The authors check limiting cases (e.g., $ u_i o0$, $M^2 o0$, on-shell) and show how these results extend to two-loop graphs with one-loop insertions by modifying propagator powers, offering explicit reductions to polylogarithms in certain limits. Overall, NDIM provides an efficient, integration-free analytic toolkit for one-loop box integrals and relevant multi-loop insertions, with potential broader impact on higher-loop computations.

Abstract

We study massless one-loop box integrals by treating the number of space-time dimensions D as a negative integer. We consider integrals with up to three kinematic scales (s, t and either zero or one off-shell legs) and with arbitrary powers of propagators. For box integrals with q kinematic scales (where q=2 or 3) we immediately obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with q-1 variables, valid for general D. Because the power each propagator is raised to is treated as a parameter, these general expressions are useful in evaluating certain types of two-loop box integrals which are one-loop insertions to one-loop box graphs. We present general expressions for this particular class of two-loop graphs with one off-shell leg, and give explicit representations in terms of polylogarithms in the on-shell case.

Figures (2)

• Figure 1: The one-loop box diagram.
• Figure 2: The kinematic regions for the one-loop box with one off-shell leg. The solid line shows the phase-space boundary $|s|+|t|=M^2$, together with the reflections $|s|=|t|+M^2$ and $|t|=|s|+M^2$. The reflections are relevant for the convergence properties of the hypergeometric functions which only involve the absolute values of ratios of the scales. The dashed lines show the boundaries $|s|=M^2$ and $|t|=M^2$.