  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~$\phi_i^{\Bar{k}}$ 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~$\phi_i^{\Bar{k}}$, $\Delta_{\phi_i^{\Bar{k}}}$, 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~$\gamma_m^*\leq1.29$ from the
relation~$\gamma_m^*=3-\Delta_{\phi_i^{\Bar{k}}}$. 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.
