Absence of SO(4) quantum criticality in Dirac semimetals at two-loop order

Evidence for relativistic quantum criticality of antiferromagnetism and superconductivity in two-dimensional Dirac fermion systems has been found in large-scale quantum Monte Carlo simulations. However, the corresponding (2 + 1)-dimensional Gross-Neveu-Yukawa field theory with Nf = 2 four-component Dirac fermions coupled to two triplets of order parameters does not exhibit a renormalization group fixed point at one-loop order. Instead, the theory only features a critical point for a large or very small fractional number of fermion flavors Nf, which disappears for a broad range of flavor numbers around the physical case, Nf = 2, due to fixed-point annihilation. This raises the question on how to explain the observed scaling collapse in the quantum Monte Carlo data. Here, we extend previous renormalization-group analyses by studying a generalized model at two-loop order in 4 - e spacetime dimensions. We determine the e correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case Nf = 2. However, this only happens very slowly, such that an extrapolation to e = 1 still suggests the absence of criticality in 2 + 1 dimensions. Thereby, we consolidate the finding that the continuum field theory does not feature a stable renormalization-group fixed point and no true quantum criticality would be expected for the considered system. We briefly discuss a possible reconciliation in terms of a complex conformal field theory. Further, we also explore the fixed-point structure in an enlarged theory space and identify a candidate stable fixed-point solution.