Remarks on Defining the DLCQ of Quantum Field Theory as a Light-Like Limit

TL;DR

This paper investigates the challenges of defining discrete light-cone quantization (DLCQ) as a light-like limit in quantum field theory, showing it is finite at one loop in QED but may diverge in other theories or higher loops.

Contribution

It provides a detailed analysis of the light-like limit in DLCQ, demonstrating finiteness in QED at one loop and discussing potential divergences in other cases.

Findings

One-loop QED has a well-defined light-like limit.

In general, the limit involves replacing an alpha integral with a discrete sum.

Divergences may occur in other theories or at higher loops.

Abstract

The issue of defining discrete light-cone quantization (DLCQ) in field theory as a light-like limit is investigated. This amounts to studying quantum field theory compactified on a space-like circle of vanishing radius in an appropriate kinematical setting. While this limit is unproblematic at the tree-level, it is non-trivial for loop amplitudes. In one-loop amplitudes, when the propagators are written using standard Feynman $\alpha$-parameters we show that, generically, in the limit of vanishing radius, one of the $\alpha$-integrals is replaced by a discrete sum and the (UV renormalized) one-loop amplitude has a finite light-like limit. This is analogous to what happens in string theory. There are however exceptions and the limit may diverge in certain theories or at higher loop order. We give a rather detailed analysis of the problems one might encounter. We show that quantum electrodynamics at one loop has a well-defined light-like limit.