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We study conformal field theories (CFTs) with Yukawa 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−ϵ expansion using the Gross–Neveu–Yukawa model, and the first few terms in the 2+ϵ expansion using the four-fermion interaction. We then apply Pade 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−ϵ 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.

Yukawa conformal field theories and emergent supersymmetry