Cutting massless four-loop propagators
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Springer
Received: October 18, 2019 Accepted: November 4, 2019 Published: December 4, 2019
Vitaly Mageryaa and Andrey Pikelnerb a
II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Luruper Chaussee 149, Hamburg 22761, Germany b Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna 141980, Russia
E-mail: [email protected], [email protected] 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. Keywords: Perturbative QCD, Scattering Amplitudes ArXiv ePrint: 1910.07522
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2019)026
JHEP12(2019)026
Cutting massless four-loop propagators
Contents 1 Introduction
2
2 4-loop virtual integrals (VVVV)
5 5 5 7 10 13
4 2-loop 3-particle cut integrals (VVRR, VRRV) 4.1 1-loop 1→3 amplitudes 4.2 2-loop 1→3 amplitudes 4.3 Cross-checks
13 16 17 23
5 3-loop 2-particle cut integrals (VVVR, VVRV)
23
6 5-particle phase-space integrals (RRRR)
24
7 Relations from Cutkosky rules
24
8 Dimensional recurrence relations for 3-particle cut integrals 8.1 DRA method by example: VRRV 8.2 Solving DRR for VVRR integrals
29 29 33
9 Conclusions
35
A Results A.1 VRRR A.2 VVRR A.3 VRRV
36 36 45 49
B Table of loop integrals
51
C Multiple zeta values basis up to weight 12
52
D Supplementary material
53
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3 1-loop 4-particle cut integrals (VRRR) 3.1 Direct integration over the phase space 3.2 An overview of dimensional recurrence relations 3.3 Solving DRR for VRRR integrals 3.4 Cross-checks
1
Introduction
What are cut integrals? To calculate a total cross section of an (off-shell or massive) particle decay, one needs to integrate the probability density of the final state over its phase
–2–
JHEP12(2019)026
Inclusive physical observables like total scattering cross sections and related quantities are naturally defined within the perturbation theory in terms of cut Feynman integrals. Particularly, particle decay cross sections at the level of N3 LO in massless QCD require the knowledge of cuts of
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