Two-loop Integrand Decomposition into Master Integrals and Surface Terms

TL;DR

The paper introduces a fully numerical unitarity framework for multi-loop amplitudes by organizing loop integrands into a basis of master integrals plus surface terms that integrate to zero. It constructs a complete set of off-shell surface terms using specialized integration-by-parts (IBP) vectors in adapted coordinates, linking them to on-shell maximal cuts through a Lie-algebra structure. By counting master integrands on-shell via holomorphic forms on the unitarity-cut spaces, the authors provide a consistency check for the off-shell construction and present an algorithm applicable to planar two-loop topologies (with extensions to non-planar cases). The work aims to bypass analytic integral reduction, enabling robust numerical multi-loop computations with direct relevance to precision QCD predictions at the LHC and beyond.

Abstract

Loop amplitudes are conveniently expressed in terms of master integrals whose coefficients carry the process dependent information. Similarly before integration, the loop integrands may be expressed as a linear combination of propagator products with universal numerator-tensors. Such a decomposition is an important input for the numerical unitarity approach, which constructs integrand coefficients from on-shell tree amplitudes. We present a new method to organise multi-loop integrands into a direct sum of terms that integrate to zero (surface terms) and remaining master integrands. This decomposition facilitates a general, numerical unitarity approach for multi-loop amplitudes circumventing analytic integral reduction. Vanishing integrals are well known as integration-by-parts identities. Our construction can be viewed as an explicit construction of a complete set of integration-by-parts identities excluding doubled propagators. Interestingly, a class of 'horizontal' identities is singled out which hold as well for altered propagator powers.

Figures (3)

• Figure 1: A generic two-loop integral topology is displayed with the naming conventions used in the main text. In order to reuse structures well known at one-loop level we interpret the two-loop topology as three rungs. The rungs carry loop momentum $\ell$, $\hat{\ell}$ and $\tilde{\ell}$, and have external momenta $p_i$, $\hat{p}_i$ and $\tilde{p}_i$ exiting. The rungs are joined in two four-point vertices on the top and bottom with external momenta $p_t$ and $p_b$ leaving, respectively. Most of our considerations will be focused on the planar case with no external momenta $\hat{p}_i$ attached to the central rung at all. Many considerations work analogously for the three rungs and we will often refer to the joint variables by dropping 'hat' and 'tilde' super scripts.
• Figure 2: Conventions for coordinate change for one-loop diagram. Propagator masses and external momenta enter as parameters. The loop momentum $\ell$ is parametrized by the inverse propagators and additional internal variables, in case further parameters are required.
• Figure 3: The junction of internal lines of a generic multi-loop diagram is displayed. The loop momenta $\ell_i$ and the external momentum $p_v$ join at the vertex. The momenta are constrained by momentum conservation. Each of the loop legs (here referred to as rungs) may have external momenta entering, which is indicated by small attached arrows. IBP vectors generate rotations in the respective transverse spaces of the individual rungs. Vertices impose interesting compatibility conditions between the rotations of the individual rungs. Via momentum conservation individual rotations of the rungs with loop momenta $\{\ell_1,...,\ell_{k-1}\}$ lead to a resulting transformation of $\ell_k$. The resulting transformation must be a rotation within this rung's transverse space to give a valid IBP vector.