Critical coupling in $\phi_2^4$ theory

We consider $\phi^4$ theory with $\phi(x)\in\mathbb{R}$ in two Euclidean dimensions. We determine for a variety of self-couplings $\hat{\lambda}$ the (negative) critical bare mass $\hat{\mu}_{0\mathrm{c}}^2(\hat{\lambda})$ where the lattice-regularized system changes from the symmetric to the broken phase. Based on these data, the transition to infinite volume and a universal scheme with the renormalized parameter $\hat{\mu}_\mathrm{c}^2(\hat{\lambda})$ is made. Finally, $f_\mathrm{c}=\lim_{\hat{\lambda}\to0}\hat{\lambda}/\hat{\mu}_\mathrm{c}^2(\hat{\lambda})$ is determined, with a judicious choice of the parameterizations considered. Our final result reads $f_\mathrm{c}=11.1097(20)_\mathrm{stat}(09)_\mathrm{sys}=11.1097(22)_\mathrm{tot}$.