One-loop inelastic amplitudes from tree-level elasticity in 2d

TL;DR

This paper explores one-loop inelastic amplitudes in 1+1 dimensional quantum field theories, showing how they relate to tree-level processes and demonstrating one-loop integrability in affine Toda models through specific counterterms.

Contribution

It derives a formula for one-loop inelastic amplitudes from tree-level elasticity and proves one-loop integrability for affine Toda theories via a simple potential scaling.

Findings

One-loop inelastic amplitudes equal their tree-level counterparts with corrected masses.

Counterterms are required to restore integrability at one loop.

Affine Toda theories are shown to be one-loop integrable through potential scaling.

Abstract

We investigate the perturbative integrability of different quantum field theories in 1+1 dimensions at one loop. Starting from massive bosonic Lagrangians with polynomial-like potentials and absence of inelastic processes at the tree level, we derive a formula reproducing one-loop inelastic amplitudes for arbitrary numbers of external legs. We show that any one-loop inelastic amplitude is equal to its tree-level version, in which the masses of particles and propagators are corrected by one-loop bubble diagrams. These amplitudes are nonzero in general and counterterms need to be added to the Lagrangian to restore the integrability at one loop. For the class of simply-laced affine Toda theories, we show that the necessary counterterms are obtained by scaling the potential with an overall multiplicative factor, proving in this way the one-loop integrability of these models. Even though we focus on bosonic theories with polynomial-like interactions, we expect that the on-shell techniques used in this paper to compute amplitudes can be applied to several other models.