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We analyze general properties of the perturbation expansion fortwo-dimensional quantum critical metals with singular forward scattering, suchas metals at an Ising nematic quantum critical point and metals coupled to aU(1) gauge field. We derive asymptotic properties of fermion loops appearing assubdiagrams of the contributing Feynman diagrams -- for large and smallmomenta. Substantial cancellations are found in important scaling limits, whichreduce the degree of divergence of Feynman diagrams with boson legs.Implementing these cancellations we obtain improved power-counting estimatesthat yield the true degree of divergence. In particular, we find thatperturbative contributions to the boson self-energy are generally ultravioletconvergent for a dynamical critical exponent $z<3$, and divergent beyondthree-loop order for $z \geq 3$.

Fermion loops and improved power-counting in two-dimensional critical metals with singular forward scattering