Integrands of loop amplitudes within loop-tree duality

Zoltán Szőr,Robert Runkel,Juan Pablo Vesga,Stefan Weinzierl

doi:10.1103/physrevd.101.116014

Zoltán Szőr, Robert Runkel + Show 2 more

Open Access

https://doi.org/10.1103/physrevd.101.116014

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Journal: Physical Review D	Publication Date: Jun 22, 2020
Citations: 34	License type: CC BY 4.0

Affiliation: Johannes Gutenberg University Mainz

Abstract

Using loop-tree duality, we relate a renormalised $n$-point $l$-loop amplitude in a quantum field theory to a phase-space integral of a regularised $l$-fold forward limit of a UV-subtracted $(n+2l)$-point tree-amplitude-like object. We show that up to three loops the latter object is easily computable from recurrence relations. This defines an integrand of the loop amplitude with a global definition of the loop momenta. Field and mass renormalisation are performed in the on-shell scheme.

Highlights

Precision physics at the LHC requires next-to-next-toleading order (NNLO) calculations for various processes
We show that the integrand for the renormalized n-point l-loop amplitude within the loop-tree duality approach is related to the regularized l-fold forward limit of a UVsubtracted ðn þ 2lÞ-point tree-amplitude-like object, if field renormalization and mass renormalization are performed in the on-shell scheme
We present the equivalence between the renormalized npoint l-loop amplitude and the phase space integral of a regularized l-fold forward limit of a UV-subtracted ðn þ 2lÞ-point tree-amplitude-like object as a general property of quantum field theory

Summary

INTRODUCTION

Precision physics at the LHC requires next-to-next-toleading order (NNLO) calculations for various processes. We show that the integrand for the renormalized n-point l-loop amplitude within the loop-tree duality approach is related to the regularized l-fold forward limit of a UVsubtracted ðn þ 2lÞ-point tree-amplitude-like object, if field renormalization and mass renormalization are performed in the on-shell scheme. The main result of this paper is given in Eq (99), which relates the renormalized n-point l-loop amplitude to a phase space integral of a regularized l-fold forward limit of a UVsubtracted ðn þ 2lÞ-point tree-amplitude-like object.

BASIC NOTATION

GRAPHS

CUTTING AND SEWING

LOOP-TREE DUALITY

THE REGULARIZED FORWARD LIMIT

VIII. DIAGRAMS WITH HIGHER POWERS OF THE PROPAGATORS

THE INTEGRAND OF THE RENORMALIZED LOOP AMPLITUDE

RECURRENCE RELATIONS

FIELDS WITH NONVANISHING VACUUM EXPECTATION VALUES

CHECKS

Findings

XIII. CONCLUSIONS AND OUTLOOK