Critical exponent $eta$ at $O(1/N^3)$ in the chiral XY model using the large $N$ conformal bootstrap

We compute the $O(1/N^3)$ correction to the critical exponent $eta$ in the chiral XY or chiral Gross-Neveu model in $d$-dimensions. As the leading order vertex anomalous dimension vanishes, the direct application of the large $N$ conformal bootstrap formalism is not immediately possible. To circumvent this we consider the more general Nambu-Jona-Lasinio model for a general non-abelian Lie group. Taking the abelian limit of the exponents of this model produces those of the chiral XY model. Subsequently we provide improved estimates for $eta$ in the three dimensional chiral XY model for various values of $N$.