The Vacuum Structure of Light-Front $\phi^4_{1+1}$-Theory

TL;DR

This paper investigates the vacuum structure of 1+1 dimensional light-front $^4$-theory, revealing non-perturbative spontaneous symmetry breaking through a mean-field approach and proper renormalization, despite a trivial vacuum in perturbation theory.

Contribution

It introduces a non-perturbative analysis of the zero mode constraint in light-front $^4$-theory, demonstrating spontaneous symmetry breaking with a non-zero vacuum expectation value.

Findings

No symmetry breaking in perturbation theory.

Non-zero vacuum expectation value appears non-perturbatively.

Correct renormalization is crucial for consistent results.

Abstract

We discuss the vacuum structure of $\phi^4$-theory in 1+1 dimensions quantised on the light-front $x^+ =0$. To this end, one has to solve a non-linear, operator-valued constraint equation. It expresses that mode of the field operator having longitudinal light-front momentum equal to zero, as a function of all the other modes in the theory. We analyse whether this zero mode can lead to a non-vanishing vacuum expectation value of the field $\phi$ and thus to spontaneous symmetry breaking. In perturbation theory, we get no symmetry breaking. If we solve the constraint, however, non-perturbatively, within a mean-field type Fock ansatz, the situation changes: while the vacuum state itself remains trivial, we find a non-vanishing vacuum expectation value above a critical coupling. Exactly the same result is obtained within a light-front Tamm-Dancoff approximation, if the renormalisation is done in the correct way.