Expansion by regions with pySecDec

TL;DR

The paper presents a geometric formulation of expansion by regions and its implementation in pySecDec, enabling automated asymptotic expansions and efficient evaluation of multi-loop amplitudes. It harnesses the Newton polytope geometry and Cheng-Wu invariance to identify regions and to cover the integration domain, including handling of thresholds and regulators. Key contributions include automated region-based expansions, amplitude computation via weighted sums, and adaptive contour deformation, all demonstrated across a range of one- and two-loop examples. The approach significantly improves performance in scale-hierarchical regimes and provides a practical toolkit for phenomenological applications in high-energy physics.

Abstract

We discuss the technique of expansion by regions from a geometric perspective, and its implementation within pySecDec, a toolbox for the evaluation of dimensionally regulated parameter integrals. The program offers an automated way to perform asymptotic expansions and provides a new mechanism for efficiently evaluating amplitudes, as well as individual integrals. The usage of the new features available within pySecDec is illustrated with several examples.

Figures (12)

• Figure 1: One loop bubble with a squared massive propagator. Dashed lines denote massless propagators. The "dot" means that the corresponding propagator occurs with power two in the integral.
• Figure 2: Newton polytope for $P(x,t) = t+x+x^2$, together with the directions of the region vectors (drawn as normal vectors to the facets in positive $t$-direction).
• Figure 3: Newton polytope for $P(x,t) = t+x+x^2$, along with an example vector $\mathbf{u}'$.
• Figure 4: Example of subdivisions generated by expansion by region of integrals as defined in \ref{['eq:int']} with $P_1 = 1 + t x_1 + x_1 x_2 + t x_2$ (left) and $P_2 = t + x_1 + t x_1 x_2 + x_2$ (right) in the limit $t\rightarrow0$.
• Figure 5: Scan over different orders of magnitude of $r = m^2/s$ for the first of the three two-loop triangles given in Table \ref{['tab:timings']}. Integration times (in minutes) are plotted against $r$. The relative accuracy goal is $10^{-3}$; the wall clock limit has been set to 5 hours. The dashed lines denote the evaluation with standard pySecDec, the solid lines denote the evaluation with the expansion by regions option of pySecDec.