Yukawa CFTs and Emergent Supersymmetry

Abstract

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 $4-ε$ 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 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 $4-ε$ 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 2-component Dirac fermion, this theory has an emergent supersymmetry with 4 supercharges, and we provide new evidence for this.

Figures (7)

• Figure 1: RG flow and fixed point structure for the GNY model with $N=1$ (one Majorana fermion in $d=3$) and the NJLY model with $N=2$ (one Dirac fermion in $d=3$), obtained from the one-loop $\beta$-functions (\ref{['betasGNY']}) and (\ref{['NJLYbeta']}) in $d=4-\epsilon$. The attractive IR fixed points have "emergent" supersymmetry with 2 and 4 supercharges respectively. The red triangles denote unstable fixed points with negative quartic potential which can be seen in the one-loop analysis in $d=4-\epsilon$; their fate in $d=3$ is unclear.
• Figure 2: Padé estimates in $d=3$ of $\Delta_{\sigma}$ versus $\Delta_{\psi}$ for $N=1,2,3,4,5,6,8,20$, compared to the large $N$ results (\ref{['lN-3d']}). The $N=1$ value corresponds to the SUSY fixed point discussed in Section \ref{['emergentSUSY']}. The black dotted line is the SUSY relation $\Delta_{\sigma}=\Delta_{\psi}-1/2$.
• Figure 3: Padé estimates of $\tilde{F}-N\tilde{F}_f$ in $2<d<4$ compared to the large $N$ result (\ref{['tFLN']}).
• Figure 4: Padé estimates in $d=3$ of $\Delta_{\sigma}$ versus $\Delta_{\psi}$ for $N=4,6,8,10,12,20,100$, compared to the large $N$ results (\ref{['lN-3d-NJL']}).
• Figure 5: Padé estimates of $\tilde{F}-N\tilde{F}_f$ in $2<d<4$ compared to the large $N$ result (\ref{['tFLNNJL']}).