Beyond one-loop calculation: Higher-order effects on Gross-Neveu-Yukawa tensorial criticality

We study the Gross-Neveu-Yukawa field theory for the $\mathrm{SO}(N$) symmetric traceless rank-2 tensor order parameter coupled to Majorana fermions using the $\ensuremath{\epsilon}$-expansion around upper critical dimensions of $3+1$ to two loops. Previously we established in the one-loop calculation that the theory does not exhibit a critical fixed point for $N\ensuremath{\ge}4$, but that nevertheless the stable fixed point inevitably emerges at a large number ${N}_{f}$ of fermion flavors. For ${N}_{f}<{N}_{f,c1}\ensuremath{\approx}N/2$, no critical fixed point exists; for ${N}_{f,c1}<{N}_{f}<{N}_{f,c2}$, a real critical fixed point emerges from the complex plane but fails to satisfy the additional stability conditions necessary for a continuous phase transition; and finally only for ${N}_{f}>{N}_{f,c2}\ensuremath{\approx}N$ does the fixed point satisfy the stability conditions as well. In the present work we compute the $O(\ensuremath{\epsilon})$ (two-loop) corrections to the critical flavor numbers ${N}_{f,c1}$ and ${N}_{f,c2}$. Most importantly, we observe a sharp decrease in ${N}_{f,c2}$ from its one-loop value, which brings it closer to the point ${N}_{f}=1$ relevant to the standard Gross-Neveu model. Some three-loop results are also presented and discussed.