Yukawa conformal field theories and emergent supersymmetry

We study conformal field theories (CFTs) withYukawa interactions in dimensions between 2 and 4; they provide UV completions of the Nambu-Jona-Lasinio and Gross-Neveu models which have four-fermion interactions.We compute the sphere free energy and certain operator scaling dimensions using dimensional continuation. In the Gross-Neveu CFT with N fermion degrees of freedom we obtain the first few terms in the 4-∈ expansion using the Gross-Neveu-Yukawa model, and the first few terms in the 2 + ∈ expansion using the four-fermion interaction. We then apply Padé approximants to produce estimates in d = 3. For N = 1, which corresponds to one two-component Majorana fermion, it has been suggested that the Yukawa theory flows to an N = 1 supersymmetric CFT. We provide new evidence that the 4 - ∈ expansion of the N = 1 Gross-Neveu-Yukawa model respects the supersymmetry. Our extrapolations to d = 3 appear to be in good agreement with the available results obtained using the numerical conformal bootstrap. Continuation of this CFT to d = 2 provides evidence that the Yukawa theory flows to the tri-critical Ising model. We apply a similar approach to calculate the sphere free energy and operator scaling dimensions in the Nambu-Jona-Lasinio-Yukawa model, which has an additional U(1) global symmetry. For N = 2, which corresponds to one two-component Dirac fermion, this theory has an emergent supersymmetry with four supercharges, and we provide new evidence for this.