Conformal 3-point correlators in momentum space, method of subgraphs and the $1/N$ expansion

Abstract

Conformal 3-point correlators of conserved currents play important roles in numerous directions. These correlators are fixed by conformal symmetry up to a few parameters, which are known only at leading order in perturbative expansions. The major challenges come from the multi-loop Feynman integrals with three external momenta. In this work, we employ the method of subgraphs to compute the subleading order corrections to the conformal current 3-point correlators in the large $N$ expansion. We show that the method of subgraphs generates diagrammatic expansions for the conformal 3-point correlators, and that it is closely related to the operator product expansions in momentum space. We verify the subgraph expansions of conserved current 3-point correlators using exact results in 3D. We demonstrate that multi-loop 3-point Feynman integrals can be significantly simplified by taking the subgraph expansions. Due to constraints from conformal symmetry, it suffices to keep only the first few terms in the subgraph expansions to completely fix the subleading order corrections. We apply this method to compute the $1/N$ corrections to current correlators $\langle JJJ\rangle$ in the critical $O(N)$ vector model and the Gross-Neveu-Yukawa model. We also compute the $1/N$ corrections to the coefficients in the current-current-scalar correlators $\langle JJ\sigma_{T}\rangle$ and $\langle JJ\sigma\rangle$ in the critical $O(N)$ vector model. We compare the perturbative results with the bootstrap data and discuss their application to conductivity near the quantum critical point.