Bounds in 4D Conformal Field Theories with Global Symmetry

TL;DR

This work extends the conformal bootstrap program to 4D CFTs with continuous global symmetry by deriving vectorial crossing sum rules for the 4-point function $\langle \Phi \Phi^{\dagger} \Phi \Phi^{\dagger} \rangle$ and showing the number of independent constraints matches the number of symmetry channels. It provides explicit constructions for $SO(N)$, $U(1)$, and $SU(N)$, and develops a general framework for arbitrary groups and representations using Fierz identities and invariant tensors. A central result is a universal upper bound on the dimension of the lowest singlet scalar $d_S$ appearing in the $\Phi\times\Phi^{\dagger}$ OPE, with $d_S$ bounded by a function $f_S(d_\phi,R_G)$ that satisfies $d_S\le f_S(d_\phi,R_G)$ and approaches $2$ as $d_\phi\to 1$, with numerical exploration showing the bound weakening for larger groups. The findings have implications for Conformal Technicolor scenarios and point to the need for improved numerical methods to obtain stronger bounds in the globally symmetric case.

Abstract

We explore the constraining power of OPE associativity in 4D Conformal Field Theory with a continuous global symmetry group. We give a general analysis of crossing symmetry constraints in the 4-point function <Phi Phi Phi* Phi*>, where Phi is a primary scalar operator in a given representation R. These constraints take the form of 'vectorial sum rules' for conformal blocks of operators whose representations appear in R x R and R x Rbar. The coefficients in these sum rules are related to the Fierz transformation matrices for the R x R x Rbar x Rbar invariant tensors. We show that the number of equations is always equal to the number of symmetry channels to be constrained. We also analyze in detail two cases - the fundamental of SO(N) and the fundamental of SU(N). We derive the vectorial sum rules explicitly, and use them to study the dimension of the lowest singlet scalar in the Phi x Phi* OPE. We prove the existence of an upper bound on the dimension of this scalar. The bound depends on the conformal dimension of Phi and approaches 2 in the limit dim(Phi)-->1. For several small groups, we compute the behavior of the bound at dim(Phi)>1. We discuss implications of our bound for the Conformal Technicolor scenario of electroweak symmetry breaking.

Figures (3)

• Figure 1: For any 4-point configuration, there exists a conformal transformation which maps it onto a parallelogram.
• Figure 2: Geometric interpretation of the sum rule: (a) the sum rule has a solution $\Leftrightarrow$$\mathbf{y}$ belongs to the cone; (b) the assumed spectrum is such that the sum rule does not allow for a solution $\Leftrightarrow$$\mathbf{y}$ does not belong to the cone; (c) for $\Delta_{S}=\Delta_{S}^{\text{cr}}$, the $\mathbf{y}$ belongs to the cone boundary.
• Figure 3: This figure gives a geometric interpretation of the proof that at $d_{\phi}=1$ the sum rule has no solution unless the $\Delta=2$ singlet scalar is included in the spectrum. The solid-contour plane represents the zero set of the functional $\Lambda_{0}$. The vector $\mathbf{y}$ and all the twist $2$ vectors (black dots) lie in the $\Lambda_{0}=0$ plane. On the other hand, all twist $\neq2$ vectors, which for varying $\Delta$ trace separate curves labeled by spin and representation, lie on one side of this plane ($\Lambda_{0}>0)$. The $\Lambda_{0}=0$ plane can be slightly rotated so that the $\mathbf{y}$ vector and the twist 2 singlet scalar lie on one side of the rotated plane, while the rest of the twist $2$ vectors lie on the opposite side. The rotated plane (dashed contour) can be described by an equation $\Lambda_{0}+\varepsilon\Lambda _{1}=0$ for a small $\varepsilon$.