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

TL;DR

This work develops a subgraph-based diagrammatic framework to compute subleading $1/N$ corrections to conformal 3-point correlators in momentum space. By decomposing multi-scale loop integrals into sums of one-scale subgraphs, the approach leverages conformal symmetry constraints to fix subleading corrections with only the first few terms, greatly simplifying the analysis. The authors apply the method to $raket{JJJ}$ in the critical $O(N)$ vector model and in the Gross-Neveu-Yukawa model, and to $raket{JJullet}$ correlators with $ullet= ext{σ}_T, ext{σ}$, obtaining explicit $1/N$ corrections and matching conformal-symmetry expectations and certain 3D exact results. They also discuss implications for finite-temperature conductivity near quantum critical points and outline extensions to other CFTs and higher-point functions, highlighting potential bootstrap applications in momentum space.

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 $\langle JJσ_{T}\rangle$ and $\langle JJσ\rangle$ 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.

Figures (11)

• Figure 1: Diagrammatic representation of $\mathcal{I}(\nu_1,\nu_2,\nu_3)$ and the two subgraphs in the expansion: $\Gamma$ and $\gamma$ marked in red color.
• Figure 2: Vertices of the conserved currents $J^a_\mu(p)$ and $T_{\mu\nu}(p)$.
• Figure 3: Feynman diagram for the current 3-point correlator $\langle J^a_\mu J^b_\nu J^c_\rho\rangle$ and its expansion with two subgraphs: $\Gamma$ and $\gamma$ marked in red color. The Feynman diagrams for the correlators $\langle TJJ \rangle$ and $\langle TTT \rangle$ are similar but with different vertices for the stress tensor.
• Figure 4: Feynman diagram for the conformal current 3-point correlator $\langle J^a_\mu J^b_\nu J^c_\rho\rangle$ up to the order $O(1/N)$.
• Figure 5: Subgraphs $\gamma_1$ (1234) and $\gamma_2$ (16) of the Feynman diagram $D_4$ in Figure \ref{['fig:JJJ1N']}. The diagram $D_4$ itself constructs a trivial subgraph $\Gamma$. In addition, there are extra subgraphs of $D_4$ which relate to scaleless integrals.