Bounds on OPE Coefficients in 4D Conformal Field Theories

TL;DR

This work applies the conformal bootstrap with semidefinite programming to derive bounds on OPE coefficients and current two-point functions in 4D CFTs. It extends the analysis to tensor operators and to product global symmetries $SO(N)\times SO(M)$ (and related structures), incorporating a gap in the scalar singlet channel ($\Delta_S\ge4$) motivated by composite Higgs models and the absence of relevant deformations. The results show that imposing the scalar-gap strengthens lower bounds on the vector central charge $\kappa$ and that, across group structures, the bounds approach free theory values as external dimensions approach the unitarity limit, while larger $d$ yields tighter constraints. These findings offer phenomenological guidance for strongly coupled sectors in beyond-the-Standard-Model scenarios and demonstrate the viability of product-group bootstrap analyses for constraining CFT data.

Abstract

We numerically study the crossing symmetry constraints in 4D CFTs, using previously introduced algorithms based on semidefinite programming. We study bounds on OPE coefficients of tensor operators as a function of their scaling dimension and extend previous studies of bounds on OPE coefficients of conserved vector currents to the product groups SO(N)xSO(M). We also analyze the bounds on the OPE coefficients of the conserved vector currents associated with the groups SO(N), SU(N) and SO(N)xSO(M) under the assumption that in the singlet channel no scalar operator has dimension less than four, namely that the CFT has no relevant deformations. This is motivated by applications in the context of composite Higgs models, where the strongly coupled sector is assumed to be a spontaneously broken CFT with a global symmetry.

Figures (8)

• Figure 1: Upper bounds on the three-point function coefficient $\lambda_0$ between two scalar operators of dimension $d=1.6$ and a scalar operator ${\cal O}$ of dimension $\Delta$ calculated at $k=11$ with no assumptions on the spectrum (blue line) and assuming that no scalar operator in the OPE is present below $\Delta_0 = 2$ (red line). For illustrative purposes, we show the free-theory value for $d=1$ (in which case $\Delta = 2$), $\lambda^{\rm free}_0 = \sqrt{2}$, as a black dashed line.
• Figure 2: Upper bounds on the three-point function coefficient $\lambda_2$ between two scalar operators of dimension $d$ and a tensor operator ${\cal O}$ with spin $l=2$ and dimension $\Delta$ calculated at $k=11$. (a) Starting from below, the lines correspond to the values $d=1.01, 1.1,1.2,1.3,1.4,1.5,1.6$. No assumption on the spectrum is made. (b) For $d=1.62$ with no assumption on the spectrum (blue line) and assuming that no scalar operator in the OPE is present below $\Delta_0 = 2$ (red line), $\Delta_0 = 3$ (brown line) and $\Delta_0 = 4$ (green line). For illustrative purposes, we show the free-theory value for $d=1$ (in which case $\Delta = 4$), $\lambda^{\rm free}_2 = 1/\sqrt{3}$, as a black dashed line in both panels.
• Figure 3: Upper bounds on the three-point function coefficient $\lambda_4$ between two scalar operators of dimension $d$ and a tensor operator ${\cal O}$ with spin $l=4$ and dimension $\Delta$ calculated at $k=11$. (a) Starting from below, the lines correspond to the values $d=1.01, 1.1,1.2,1.3,1.4,1.5,1.6$. No assumption on the spectrum is made. (b) For $d=1.62$ with no assumption on the spectrum (blue line) and assuming that no scalar operator in the OPE is present below $\Delta_0 = 2$ (red line), $\Delta_0 = 3$ (brown line) and $\Delta_0 = 4$ (green line). For illustrative purposes, we show the free-theory value for $d=1$ (in which case $\Delta = 6$), $\lambda^{\rm free}_4 = 1/\sqrt{35}$, as a black dashed line in both panels.
• Figure 4: Lower bounds on the two-point function coefficient $\kappa$ between two conserved SO$(N)$ or SU$(N/2)$ adjoint currents as obtained from a four-point function of scalar operators in the fundamental representation with dimension $d$ calculated at $k=10$. From below, the lines which start at $d=1$ correspond to $N=2$ (blue), $N=6$ (red), $N=10$ (brown), $N=14$ (green), $N=18$ (black), with no assumption on the spectrum. In the same order and using the same color code, the lines which start at $d\simeq 1.58$, $d\simeq 1.46$, $d\simeq 1.37$, $d\simeq 1.31$ and $d\simeq 1.29$ show the bound which is obtained under the assumption that no scalar operator in the singlet channel has dimension $\Delta_S<4$. For illustrative purposes, we show the free-theory value $\kappa_{\rm free} = 1/6$ as a black dashed line.
• Figure 5: Lower bounds on the two-point function coefficient $\kappa$ between two conserved $SO(10)$ or $SU(5)$ adjoint currents as obtained from a four-point function of scalar operators in the fundamental representation with dimension $d$ calculated at $k=10$. From below, the lines correspond to the case with no assumption on the spectrum (blue) and assuming that no scalar operator in the singlet channel has dimension $\Delta_S<2$ (red), $\Delta_S<3$ (brown), $\Delta_S<4$ (green). For illustrative purposes, we show the free-theory value $\kappa_{\rm free} = 1/6$ as a black dashed line.