Elucidating the phase diagram of lattice gauge theories with fermionic matter in 2+1 dimensions has become a problem of considerable interest in recent years, motivated by physical problems ranging from chiral symmetry breaking in high-energy physics to fractionalized phases of strongly correlated materials in condensed matter physics. For a sufficiently large number $N_f$ of flavors of four-component Dirac fermions, recent sign-problem-free quantum Monte Carlo studies of lattice quantum electrodynamics (QED$_3$) on the square lattice have found evidence for a continuous quantum phase transition between a power-law correlated conformal QED$_3$ phase and a confining valence-bond-solid phase with spontaneously broken point-group symmetries. The critical continuum theory of this transition was shown to be the $O(2)$ QED$_3$-Gross-Neveu model, equivalent to the gauged Nambu-Jona-Lasinio model, and critical exponents were computed to first order in the large-$N_f$ expansion and the $epsilon$ expansion. We extend these studies by computing critical exponents to second order in the large-$N_f$ expansion and to four-loop order in the $epsilon$ expansion below four spacetime dimensions. In the latter context, we also explicitly demonstrate that the discrete $mathbb{Z}_4$ symmetry of the valence-bond-solid order parameter is dynamically enlarged to a continuous $O(2)$ symmetry at criticality for all values of $N_f$.
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