Time-Dependent Variational Principle for $\phi^4$ Field Theory: RPA Approximation and Renormalization (II)

TL;DR

This paper applies the time-dependent variational principle to analyze the  field theory, exploring static solutions, renormalization, and meson modes, revealing two viable non-trivial phases and their physical properties.

Contribution

It introduces a Gaussian-time-dependent variational approach to  theory, identifying two stable phases and analyzing their oscillation modes and renormalization conditions.

Findings

Identified two non-trivial stable solutions in  theory.

Derived conditions for renormalization in the continuum limit.

Found a closed-form solution for small oscillations, revealing zero modes and scattering amplitude behavior.

Abstract

The Gaussian-time-dependent variational equations are used to explored the physics of $(\phi^4)_{3+1}$ field theory. We have investigated the static solutions and discussed the conditions of renormalization. Using these results and stability analysis we show that there are two viable non-trivial versions of $(\phi^4)_{3+1}$. In the continuum limit the bare coupling constant can assume $b\to 0^{+}$ and $b\to 0^{-}$, which yield well defined asymmetric and symmetric solutions respectively. We have also considered small oscillations in the broken phase and shown that they give one and two meson modes of the theory. The resulting equation has a closed solution leading to a ``zero mode'' and vanished scattering amplitude in the limit of infinite cutoff.