  We analyze the scaling theory of two-dimensional metallic electron systems in
the presence of critical bosonic fluctuations with small wave vectors, which
are either due to a U(1) gauge field, or generated by an Ising nematic quantum
critical point. The one-loop dynamical exponent z=3 of these critical systems
was shown previously to be robust up to three-loop order. We show that the
cancellations preventing anomalous contributions to z at three-loop order have
special reasons, such that anomalous dynamical scaling emerges at four-loop
order.
