# Vacuum loops in light-front field theory

## TL;DR

This paper shows that vacuum diagrams in light-front field theory are non-zero and explores their behavior, revealing subtle roles of zero modes and the connection to traditional equal-time field theory results.

## Contribution

It demonstrates that vacuum diagrams in genuine light-front field theory are non-zero and analyzes their behavior using Hamiltonian perturbation theory and DLCQ, clarifying the role of zero modes.

## Key findings

- Vacuum diagrams are non-zero despite simple kinematical arguments.
- Vacuum amplitudes behave as $C\,\lambda^2\mu^{-2}$ in D=2 and diverge in D=4.
- DLCQ analysis confirms vacuum amplitudes converge to continuum values.

## Abstract

We demonstrate that vacuum diagrams in the genuine light front (LF) field theory are non-zero, in spite of simple kinematical counter-arguments (positivity and conservation of the LF momentum $p^+$, absence of Fourier zero mode). Using the light-front Hamiltonian (time-ordered) perturbation theory, the vacuum amplitudes in self-interacting scalar $\lambda\phi^3$ and $\lambda\phi^4$ models are obtained as $p=0$ limit of the associated self-energy diagrams, where $p$ is the external momentum. They behave as $C\lambda^2\mu^{-2}$ in D=2, with $\mu$ being the scalar-field mass, or diverge in D=4, in agreement with the usual "equal-time" form of field theory, and with the same value of the constant $C$. The simplest case of the vacuum bubble with two internal lines is analyzed in detail, displaying the subtle role of the small $k^+$ region and its connection to the $p=0$ limit. However, the vacuum bubbles in the genuine light-front field theory are non-vanishing not due to the Fourier mode carrying LF momentum $k^+=0$ (as is the case in the LF evaluation of the covariant Feynman diagrams), in full accord with the observation that the LF perturbation theory formula breaks down in the exact zero-mode case. This is made explicit using the DLCQ method - the discretized (finite-volume) version of the theory, where the light-front zero modes are manifestly absent, but the vacuum amplitudes still converge to their continuum-theory values with the increasing "harmonic resolution" K.