  We study the antiferromagnetic quantum critical metal in $3-\epsilon$ space
dimensions by extending the earlier one-loop analysis [Sur and Lee, Phys. Rev.
B 91, 125136 (2015)] to higher-loop orders. We show that the
$\epsilon$-expansion is not organized by the standard loop expansion, and a
two-loop graph becomes as important as one-loop graphs due to an infrared
singularity caused by an emergent quasilocality. This qualitatively changes the
nature of the infrared (IR) fixed point, and the $\epsilon$-expansion is
controlled only after the two-loop effect is taken into account. Furthermore,
we show that a ratio between velocities emerges as a small parameter, which
suppresses a large class of diagrams. We show that the critical exponents do
not receive corrections beyond the linear order in $\epsilon$ in the limit that
the ratio of velocities vanishes. The $\epsilon$-expansion gives critical
exponents which are consistent with the exact solution obtained in $0 <
\epsilon \leq 1$.
