  In metals, low-energy effective theories are characterized by a set of
coupling functions. Among them, the angle-dependent Fermi momentum specifies
the size and shape of Fermi surface. Since the Fermi momentum grows incessantly
under the renormalization group flow, a metallic fixed point is defined only
modulo a rescaling of Fermi momentum. In this paper, we discuss the physical
consequences of this projective nature of fixed points for non-Fermi liquids
with hot Fermi surfaces. The first is the absence of a unique dynamical
critical exponent that dictates the relative scaling between energy and
momentum. The second is mismatches between the scaling dimensions of couplings
and their relevancy. Nonetheless, each projective fixed point is characterized
by a few marginal and relevant coupling functions, and the notion of
universality survives. We illustrate our findings by charting the space of
projective fixed points and extracting their universal properties for the
Ising-nematic quantum critical metal beyond the patch theory. To control the
theory, we use the dimensional regularization scheme that tunes the
co-dimension of Fermi surface. Near the upper critical dimension, two exactly
marginal coupling functions span the space of stable projective fixed points:
functions that specify the shape of the Fermi surface and the angle-dependent
Fermi velocity. All other coupling functions, including the Landau functions
and the universal pairing interaction, are fixed by those two marginal
functions. With decreasing dimensions, the forward scattering remains
irrelevant while the pairing interaction becomes relevant near two dimensions.
In two dimensions, it is expected that the universal superconducting
fluctuations lower the symmetry of the non-Fermi liquid realized above the
superconducting transition temperatures from the loop U(1) group to a proper
subgroup.
