Evaluation of Feynman integrals with arbitrary complex masses via series expansions

TL;DR

The paper tackles the challenge of evaluating multiloop Feynman integrals with arbitrary complex masses by developing a series-expansion approach to the differential equations obeyed by Master Integrals, implemented in the SeaSyde Mathematica package. It provides an algorithm for analytic continuation in the complex plane of kinematic variables, including careful treatment of branch cuts and path choices, and solves the system one variable at a time or in triangular form when possible. The authors apply the method to two-loop mixed EW-QCD corrections for neutral-current Drell-Yan, computing 36 Master Integrals (32–36 requiring a semi-analytical, complex-mass treatment) and validating results against multiple public tools with very high precision, including a demonstration of CMS consistency as $\Gamma_V\to0$. The work enables reliable, high-precision predictions in scenarios with unstable intermediate states and complex masses, facilitating improved phenomenology for LHC and future colliders by providing a robust framework for complex-mass Feynman integral evaluation.

Abstract

We present an algorithm to evaluate multiloop Feynman integrals with an arbitrary number of internal massive lines, with the masses being in general complex-valued, and its implementation in the \textsc{Mathematica} package \textsc{SeaSyde}. The implementation solves by series expansions the system of differential equations satisfied by the Master Integrals. At variance with respect to other existing codes, the analytical continuation of the solution is performed in the complex plane associated to each kinematical invariant. We present the results of the evaluation of the Master Integrals relevant for the NNLO QCD-EW corrections to the neutral-current Drell-Yan processes.

Figures (6)

• Figure 1: Example of analytic continuation.
• Figure 2: Example of the effect of branch-cuts on the convergence of the expanded solution: reduced convergence area (left) and different path for the analytic continuation (right).
• Figure 3: The two possible approaches in the definition of the path: the one requiring a logarithmic expansion (left) and the one only relying on the Taylor expansion (right).
• Figure 4: Examples of two possible paths for linking two points laying on the same branch-cut: no singularities between them are present (left) or at least one is (right).
• Figure 5: Comparison between real- and complex-valued masses for MI10 and MIs from 32 to 36. The plots show the value of the $\varepsilon^0$ order for $130 \; \text{GeV} \le \sqrt{s} \le 190\;\text{GeV}$, $\cos\theta=0$ and $M^2=m_W^2$ or $M^2=\mu_W^2$. In the MI10 plot, the red crosses represent the analytical value obtained with complex masses with GiNaC.