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

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~${\varphi }_i^{\text{Bar}k}$ 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~${\varphi }_i^{\text{Bar}k}$, ${\mathrm{\Delta }}_{{\varphi }_i^{\text{Bar}k}}$, 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~${\gamma }_m^{\ast }\le 1.29$ from the relation~${\gamma }_m^{\ast }=3-{\mathrm{\Delta }}_{{\varphi }_i^{\text{Bar}k}}$. 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.