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

We study the Gross-Neveu-Yukawa field theory for the SO(N) symmetric traceless rank-two tensor order parameter coupled to Majorana fermions using the ε-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≥4, but that nevertheless the stable fixed point inevitably emerges at a large number of fermion flavors N_f. For N_f<N_f,c1≈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≈N, the fixed point satisfies the stability conditions as well. In the present work we compute the O(ε) (two-loop) corrections to the critical flavour 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.