  We study conformal field theories 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 $4-\epsilon$ expansion using the
Gross-Neveu-Yukawa model, and the first few terms in the $2+\epsilon$ expansion
using the four-fermion interaction. We then apply Pade approximants to produce
estimates in $d=3$. For $N=1$, which corresponds to one 2-component Majorana
fermion, it has been suggested that the Yukawa theory flows to a ${\cal N}=1$
supersymmetric CFT. We provide new evidence that the $4-\epsilon$ 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
2-component Dirac fermion, this theory has an emergent supersymmetry with 4
supercharges, and we provide new evidence for this.
