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An extensive study of the compact $U(1)$ lattice gauge theory with a higherderivative gauge-fixing term and a suitable counter-term has been undertaken todetermine the nature of the possible continuum limits for a wide range of theparameters, especially at strong gauge couplings ($g>1$), adding to ourprevious study at a single gauge coupling $g=1.3$ \cite{DeSarkar2016}. Ourmajor conclusion is that a continuum limit of free massless photons (with theredundant pure gauge degrees of freedom decoupled) is achieved at any gaugecoupling, not necessarily small, provided the coefficient $\tilde{\kappa}$ ofthe gauge-fixing term is sufficiently large. In fact, the region of continuousphase transition leading to the above physics in the strong gauge couplingregion is found to be analytically connected to the point $g=0$ and$\tilde{\kappa} \rightarrow \infty$ where the classical action has a globalunique minimum, around which weak coupling perturbation theory in bareparameters is defined, controlling the physics of the whole region. A secondmajor conclusion is that, local algorithms like Multihit Metropolis fail toproduce faithful field configurations with large values of the coefficient$\tilde{\kappa}$ of the higher derivative gauge-fixing term and at largelattice volumes. A global algorithm like Hybrid Monte Carlo, although at timesslow to move out of metastabilities, generally is able to produce faithfulconfigurations and has been used extensively in the current study.

Non-perturbatively gauge-fixed compact $U(1)$ lattice gauge theory