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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 $N_f=2$ 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 $N_f$, which disappears for a broad range of flavor numbers around the physical case, $N_f=2$, 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 $4-\epsilon$ spacetime dimensions. We determine the $\epsilon$ correction to the upper and lower critical flavor numbers for the fixed-point annihilation and find that they both go towards the physical case $N_f=2$. However, this only happens very slowly, such that an extrapolation to $\epsilon=1$ 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., Comment: 13 pages, 6 figures, 2 tables

We investigate a family of four-dimensional quantum field theories with weakly interacting ultraviolet fixed points up to four loop order in perturbation theory. Key new ingredients are the three loop gauge contributions to quartic scalar beta functions, which we compute in the $\overline{\text{MS}}$ scheme for a template $SU(N_c)$ gauge theory coupled to $N_f$ fundamental fermions and elementary scalars. We then determine fixed point couplings, field and mass anomalous dimensions, and universal scaling exponents up to the first three non-trivial orders in a small Veneziano parameter. The phase diagram and UV-IR connecting trajectories are found and contrasted with asymptotic freedom. Further, the size of the conformal window, unitarity, and mechanisms leading to the loss of conformality are investigated. Our results provide blueprints for concrete 4d non-supersymmetric conformal field theories with standard model-like field content, and invite further model building., Comment: 17 pages, 2 external figures; v2: finite-N RGEs provided in appendix, minor adjustments, v3: new figure; match published version

We consider theories with fermionic degrees of freedom that have a fixed point of Wilson-Fisher type in non-integer dimension $d = 4-2\epsilon$. Due to the presence of evanescent operators, i.e., operators that vanish in integer dimensions, these theories contain families of infinitely many operators that can mix with each other under renormalization. We clarify the dependence of the corresponding anomalous-dimension matrix on the choice of renormalization scheme beyond leading order in $\epsilon$-expansion. In standard choices of scheme, we find that eigenvalues at the fixed point cannot be extracted from a finite-dimensional block. We illustrate in examples a truncation approach to compute the eigenvalues. These are observable scaling dimensions, and, indeed, we find that the dependence on the choice of scheme cancels. As an application, we obtain the IR scaling dimension of four-fermion operators in QED in $d=4-2\epsilon$ at order $\mathcal{O}(\epsilon^2)$., Comment: 30 pages, 4 figures

We study the fixed point that controls the IR dynamics of QED in $d = 4 - 2\epsilon$. We derive the scaling dimensions of four-fermion and bilinear operators beyond leading order in $\epsilon$-expansion. For the four-fermion operators, this requires the computation of a two-loop mixing that was not known before. We then extrapolate these scaling dimensions to $d = 3$ to estimate their value at the IR fixed point of QED$_3$ as function of the number of fermions $N_f$. The next-to-leading order result for the four-fermion operators corrects significantly the leading one. Our best estimate at this order indicates that they do not cross marginality for any value of $N_f$, which would imply that they cannot trigger a departure from the conformal phase. For the scaling dimensions of bilinear operators, we observe better convergence as we increase the order. In particular, $\epsilon$-expansion provides a convincing estimate for the dimension of the flavor-singlet scalar in the full range of $N_f$., Comment: 39 pages, 5 figures

We study Quantum Electrodynamics in d=3 (QED_3) coupled to N_f flavors of fermions. The theory flows to an IR fixed point for N_f larger than some critical number N_f^c. For N_f<= N_f^c, chiral-symmetry breaking is believed to take place. In analogy with the Wilson-Fisher description of the critical O(N) models in d=3, we make use of the existence of a perturbative fixed point in d=4-2epsilon to study the three-dimensional conformal theory. We compute in perturbation theory the IR dimensions of fermion bilinear and quadrilinear operators. For small N_f, a quadrilinear operator can become relevant in the IR and destabilize the fixed point. Therefore, the epsilon-expansion can be used to estimate N_f^c. An interesting novelty compared to the O(N) models is that the theory in d=3 has an enhanced symmetry due to the structure of 3d spinors. We identify the operators in d=4-2epsilon that correspond to the additional conserved currents at d=3 and compute their infrared dimensions., Comment: 6 pages, 3 figures. v2: references added, minor corrections