  Conserved surface roughening represents a special case of interface dynamics
where the total height of the interface is conserved. Recently, it was
suggested [F. Caballero et al., Phys. Rev. Lett. 121, 020601 (2018)] that the
original continuum model known as `Conserved Kardar-Parisi-Zhang'(CKPZ)
equation is incomplete, as additional non-linearity is not forbidden by any
symmetry in $d > 1$. In this work, we perform detailed field-theoretic
renormalization group (RG) analysis of a general stochastic model describing
conserved surface roughening. Systematic power counting reveals additional
marginal interaction at the upper critical dimension, which appears also in the
context of molecular beam epitaxy. Depending on the origin of the surface
particle's mobility, the resulting model shows two different scaling regimes;
If the particles move mainly due to the gravity, the leading dispersion law is
$\omega \sim k^2$, and the mean-field approximation describing a flat interface
is exact in any spatial dimension. On the other hand, if the particles move
mainly due to the surface curvature, the interface becomes rough with the
mean-field dispersion law $\omega \sim k^4$ , and the corrections to scaling
exponents must be taken into account. We show, that the latter model consist of
two sub-class of models that are decoupled in all orders of perturbation
theory. Moreover, our RG analysis of the general model reveals that the
universal scaling is described by a rougher interface than CKPZ universality
class. The universal exponents are derived within the one-loop approximation in
both fixed $d$ and $\varepsilon$-expansion schemes, and their relation is
discussed. We point out all important details behind these two schemes which
are often overlooked in the literature, and their misinterpretation might lead
to inconsistent results.
