Critical coupling and $\Delta ⟨{\varphi }^2⟩$ in three-dimensional ${\varphi }^4$ theory

Abstract

We consider the ${\varphi }^4$ theory with $\varphi (x)\in \mathbb{R}$ in three Euclidean dimensions. For a variety of self-couplings $\hat{\lambda }$, the critical (bare) parameter ${\hat{\mu }}_{0\mathrm{c}}^2(\hat{\lambda })$ is determined where the lattice-regulated system changes from the symmetric phase to the broken phase. Next, we send the box volume to infinity and switch to a universal scheme; this yields ${\hat{\mu }}_{\mathrm{c}}^2(\hat{\lambda })$ for each simulated $\hat{\lambda }$. Finally, a continuum extrapolation gives $f_{\mathrm{c}}={\lim }_{\hat{\lambda }\to 0}{\hat{\lambda }}^2/{\hat{\mu }}_{\mathrm{c}}^2(\hat{\lambda })$, and we find $f_{\mathrm{c}}=18.808(64{)}_{\mathrm{stat}}(45{)}_{\mathrm{sys}}=18.808(78{)}_{\mathrm{tot}}$. In addition, $\Delta ⟨{\varphi }^2{⟩}_{\mathrm{c}}/\lambda =-0.00303(12{)}_{\mathrm{stat}}(12{)}_{\mathrm{sys}}=-0.00303(17{)}_{\mathrm{tot}}$ is determined in the combined continuum and infinite-volume limit.

22 pages, 11 figures, 14 tables