Recurrence Relations and Dispersive Techniques for Precision Multi-Loop Calculations

TL;DR

The paper tackles the challenge of computing precision two-loop electroweak corrections for multi-scale processes by marrying Tarasov dimension-recurrence with a dispersive, two-point basis for tensor reductions. The authors develop a tensor-decomposition formalism that maps N-point integrals to shifted-dimension master two-point functions, and they implement recurrence relations to reduce complex integrals to a minimal set of building blocks, complemented by a dispersion framework that handles sub-loops via spectral densities. They provide explicit procedures for one-loop cases up to four points and demonstrate numerical consistency with Collier for C- and D-type functions, while outlining a scalable roadmap to full two-loop amplitudes in planar topologies. This approach offers a semi-analytic, stable path toward automated ab initio multi-loop calculations applicable to current and future precision experiments like MOLLER, P2, Belle II, and EIC. The work significantly contributes to reducing computational cost and increasing numerical precision, bridging analytic reductions and numerical integration in a way that supports the next generation of high-precision collider phenomenology.

Abstract

Ab initio predictions of two-loop electroweak contributions to observables are increasingly essential for precision collider experiments, yet their evaluation remains very challenging. We connect recurrence techniques and dispersive method in order to evaluate complex multi-loop Feynman diagrams. By expressing multi-point Passarino-Veltman functions in a two-point basis and using shifted space-time dimensions with recurrence relations, we minimize the number of required dispersive integrals. This approach reduces computation time and enables a precise and efficient analysis of one- and two-loop diagrams.

Figures (6)

• Figure 1: The one-loop $N$-point diagram in the notation corresponding to tensors $T^{(N)}_{\mu_1\ldots\mu_M}$
• Figure 2: The one-loop $N$-point diagram in the notation corresponding to tensors $J^{(N)}_{\mu_1\ldots\mu_M}$
• Figure 3: Numerical results for the three-point functions $C_{0}$, $C_{1}$, $C_{2}$, $C_{00}$, $C_{11}$ and $C_{12}$ ($k_{2}^{2}=-1.5~{\rm GeV}^{2}$, $(k_{1}+k_{2})^{2}=m_{3}^{2}$, $m_{1}=0.5~{\rm GeV}$, $m_{2}=1.0~{\rm GeV}$, $m_{3}=1.5~{\rm GeV}$). The functions $C_{2l,n_{1},n_{2}}$ are defined in Eq. (\ref{['C_to_J_mapping_2']}). Crossed dots are results based on this work and solid lines are produced from Collier library.
• Figure 4: Numerical results for the four-point functions $D_{2l,n_1,n_2,n_3}$ ($k_{2}^{2}=-1.5~{\rm GeV}^{2}$, $k_{3}^{2}=-2.5~{\rm GeV}^{2}$, $k_{4}^{2}=m_{4}^{2}$, $(k_{1}\cdot k_{3})=4.0~{\rm GeV}^{2}$, $(k_{2}\cdot k_{3})=-1.0~{\rm GeV}^{2}$$m_{1}=1.5~{\rm GeV}$, $m_{2}=0.5~{\rm GeV}$, $m_{3}=2.0~{\rm GeV}$, $m_{4}=2.5~{\rm GeV}$). The functions $D_{2l,n_{1},n_{2},n_{3}}$ are defined in Eq. (\ref{['D_to_J_mapping_2']}). Crossed dots are results based on this work and solid line is produced from Collier library.
• Figure 5: The two-loop planar $N$-point diagram ($k_{1:j}\equiv k_{1}+\ldots+k_{j}$).