Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach

TL;DR

The paper investigates four-dimensional CFTs with an SU($N$) global symmetry by applying the numerical conformal bootstrap to the four-point function of a spin-0 adjoint operator $\phi_i^{\Bar{k}}$. By deriving and solving the SU($N$) crossing relations, it shows that for $N=12$ the absence of a spin-0 SU($12$)-breaking relevant operator in the $[11,11,1,1]$ representation implies a lower bound $Δ_{φ_i^{\Bar{k}}} \ge 1.71$, which translates to an upper bound on the mass anomalous dimension at the fixed point: $\gamma_m^* = 3 - Δ_{φ_i^{\Bar{k}}} \le 1.29$. The results rely on SDPB-based semidefinite programming and reveal a kink-like feature in the $[N-2,2]$ channel, suggesting a potential related CFT. When combined with lattice QCD insights at finite flavors, the work provides a rigorous, model-independent constraint on $\gamma_m^*$ and points to directions for strengthening bounds with improved numerics. The approach is extendable to other $N$ and representations, as demonstrated by Appendix results for $N=8$ and $N=16$.

Abstract

We study four-dimensional conformal field theories with an $SU(N)$ global symmetry by employing the numerical conformal bootstrap. We consider the crossing relation associated with a four-point function of a spin~$0$ operator~$φ_i^{\Bar{k}}$ which belongs to the adjoint representation of $SU(N)$. For~$N=12$ for example, we found that the theory contains a spin~$0$ $SU(12)$-breaking relevant operator when the scaling dimension of~$φ_i^{\Bar{k}}$, $Δ_{φ_i^{\Bar{k}}}$, is smaller than~$1.71$. Considering the lattice simulation of many-flavor quantum chromodynamics with $12$~flavors on the basis of the staggered fermion, the above $SU(12)$-breaking relevant operator, if it exists, would be induced by the flavor-breaking effect of the staggered fermion and prevent an approach to an infrared fixed point. Actual lattice simulations do not show such signs. Thus, assuming the absence of the above $SU(12)$-breaking relevant operator, we have an upper bound on the mass anomalous dimension at the fixed point~$γ_m^*\leq1.29$ from the relation~$γ_m^*=3-Δ_{φ_i^{\Bar{k}}}$. Our upper bound is not so strong practically but it is strict within the numerical accuracy. We also find a kink-like behavior in the boundary curve for the scaling dimension of another $SU(12)$-breaking operator.

Figures (4)

• Figure 1: Restriction of the smallest scaling dimension of a spin $0$ operator in the $[N-1,N-1,1,1]$ representation of $SU(N)$ with $N=12$. The horizontal axis is the scaling dimension of the spin $0$ adjoint operator $\phi_i^{\Bar{k}}$, $d=\Delta_{\phi_i^{\Bar{k}}}$, and the vertical axis is the scaling dimension of the operator in the $[N-1,N-1,1,1]$ representation. Boundary curves are obtained by setting, from left to right, $(\texttt{derivativeOrder}=N_{\text{max}},\texttt{keptPoleOrder},\texttt{Lmax})=(10,14,24)$, $(12,14,24)$, $(14,16,24)$, and $(16,20,24)$, respectively. We see that the operator becomes relevant, i.e., the scaling dimension becomes smaller than $4$, when $d=\Delta_{\phi_i^{\Bar{k}}}<1.71$.
• Figure 2: Restriction on the smallest scaling dimension of a spin $0$ operator in the $[N-2,2]$ representation of $SU(N)$ with $N=12$. The horizontal axis is the scaling dimension of the spin $0$ adjoint operator $\phi_i^{\Bar{k}}$, $d=\Delta_{\phi_i^{\Bar{k}}}$, and the vertical axis is the scaling dimension of the operator in the $[N-2,2]$ representation. Boundary curves are obtained by setting, from left to right, $(\texttt{derivativeOrder}=N_{\text{max}},\texttt{keptPoleOrder},\texttt{Lmax})=(10,14,24)$, $(12,14,24)$, $(14,16,24)$, and $(16,20,26)$, respectively. We see that the operator becomes relevant when $d=\Delta_{\phi_i^{\Bar{k}}}<1.41$.
• Figure A1: Restriction on the smallest scaling dimension of a spin $0$ operator in the $[N-1,N-1,1,1]$ representation of $SU(N)$ with $N=8$, $N=12$, and $N=16$ (from left to right). The horizontal axis is the scaling dimension of the spin $0$ adjoint operator $\phi_i^{\Bar{k}}$, $d=\Delta_{\phi_i^{\Bar{k}}}$, and the vertical axis is the scaling dimension of the operator in the $[N-1,N-1,1,1]$ representation. We see that the operator becomes relevant when $d<1.67$ for $N=8$, and when $d<1.71$ for $N=16$.
• Figure A2: Restriction on the smallest scaling dimension of a spin $0$ operator in the $[N-2,2]$ representation of $SU(N)$ with $N=8$, $N=12$, and $N=16$ (from left to right). The horizontal axis is the scaling dimension of the spin $0$ adjoint operator $\phi_i^{\Bar{k}}$, $d=\Delta_{\phi_i^{\Bar{k}}}$, and the vertical axis is the scaling dimension of the operator in the $[N-2,2]$ representation. We see that the operator becomes relevant when $d<1.34$ for $N=8$, and when $d<1.42$ for $N=16$.