Renormalization of Quantum Fields on the Lightcone Worldsheet I: Scalar Fields

TL;DR

This paper demonstrates that the lightcone worldsheet formalism for scalar  field theory remains renormalizable in space-time dimensions up to 6, with all counter-terms expressible as local worldsheet actions, despite Lorentz symmetry breaking.

Contribution

It shows the renormalization of the lightcone worldsheet formalism for scalar  theory, including the necessary non-covariant counter-terms, extending its validity up to six dimensions.

Findings

Renormalization is consistent in dimensions   6.

All counter-terms can be represented as local worldsheet terms.

Two non-covariant counter-terms are required due to Lorentz symmetry breaking.

Abstract

We show that the lightcone worldsheet formalism, constructed to represent the sum of the bare planar diagrams of scalar \phi^3 field theory, survives the renormalization procedure in space-time dimensions D not greater than 6. Specifically this means that all the counter-terms, necessary to produce a successful renormalized perturbation expansion to all orders, can be represented as local terms in the lightcone worldsheet action. Because the worldsheet regulator breaks Lorentz invariance, we find the need for two non-covariant counter-terms, in addition to the usual mass, coupling and wave function renormalization. One of these can be simply interpreted as a rescaling of transverse coordinates with respect to longitudinal coordinates. The second one introduces couplings between the matter and ghost worldsheet fields on the boundaries.