Cutting massless four-loop propagators

Andrey Pikelner,Vitaly Magerya

doi:10.1007/jhep12(2019)026

Andrey Pikelner, Vitaly Magerya

Open Access

https://doi.org/10.1007/jhep12(2019)026

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Abstract

Among the unitarity cuts of 4-loop massless propagators two kinds are currently fully known: the 2-particle cuts with 3 loops corresponding to form-factors, and the 5-particle phase-space integrals. In this paper we calculate master integrals for the remaining ones: 3-particle cuts with 2 loops, and 4-particle cuts with 1 loop. The 4-particle cuts are calculated by solving dimensional recurrence relations. The 3-particle cuts are integrated directly using 1→3 amplitudes with 2 loops, which we also re-derive here up to transcendentality weight 7. The results are verified both numerically, and by showing consistency with previously known integrals using Cutkosky rules. We provide the analytic results in the space-time dimension 4 − 2ε as series in ε with coefficients being multiple zeta values up to weight 12. In the supplementary material we also provide dimensional recurrence matrices and SummerTime files suitable for numerical evaluation of the series in arbitrary dimensions with any precision.

Highlights

Inclusive physical observables like total scattering cross sections and related quantities are naturally defined within the perturbation theory in terms of cut Feynman integrals
We have presented the calculation of the previously unknown master integrals for 3- and 4-particle cuts of massless four-loop propagators
Both direct integration over the phase space and the solution of dimensional recurrence relations were used in the calculation, with the latter resulting in expressions that allow the numerical evaluation of the integrals as ε-series to arbitrary order with arbitrary precision via the SummerTime package

Summary

Introduction

Inclusive physical observables like total scattering cross sections and related quantities are naturally defined within the perturbation theory in terms of cut Feynman integrals. For an integral in d dimensions with n cut lines, mL loops to the left of the cut, mR loops to the right, and p propagators we shall factor out the full n-particle phase space PSn, mL + mR one-loop bubbles, and the full q2 dependence as follows: Note that both normalization factors—B and PSn — are dimensionless, so the power of q2 directly corresponds to the dimensionality of the integral. The master integrals for 4-loop massless propagators (VVVV) have been first calculated in [23] up to transcendentality weight seven, and updated to weight twelve in [24] The latter provides the results in terms of quickly convergent nested infinite sums, suitable for numerical evaluation to arbitrary precision in arbitrary space-time dimension d with the SummerTime package [25]. Integrals 19, 34, and 35 have the one-loop amplitude factorized, and can be expressed via the four-particle phase-space integrals from [29] (recomputed to weight 12 in [16])

Direct integration over the phase space

An overview of dimensional recurrence relations

Solving DRR for VRRR integrals

Cross-checks

Relations from Cutkosky rules

Dimensional recurrence relations for 3-particle cut integrals

DRA method by example

Solving DRR for VVRR integrals

Conclusions

A Results

B Table of loop integrals

C Multiple zeta values basis up to weight 12