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

Abstract

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 $\hat{I}\mu$-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 $N\ge 4$, but that nevertheless the stable fixed point inevitably emerges at a large number of fermion flavors $N_f$. For $N_f<N_{f,c1}\approx N/2$, 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 $N_f>N_{f,c2}\approx N$, the fixed point satisfies the stability conditions as well. In the present work we compute the $O(\hat{I}\mu )$ (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.

12 page, 3 figures