Boson-dominated quantum critical metals at Lorentz symmetric point

We study the strongly coupled theory of a Fermi surface in 2+1 dimensions coupled to a massless scalar in a general limit in which fermionic loops are suppressed such as a matrix large $N$ limit or a vector small $N_f$ limit. The case where the Fermi velocity equals the boson velocity gives extra symmetry to the low energy limit of all diagrams which allows us to make general statements about the non-perturbative fermion two-point function. We find that these theories generally have a non-monotonic dispersion across the Fermi surface resulting in splitting of the Fermi surface.