  We analyze the interplay of longitudinal and transverse fluctuations in a
$U(1)$ symmetric two-dimensional $\phi^4$-theory. To this end, we derive
coupled renormalization group equations for both types of fluctuations obtained
from a linear (cartesian) decomposition of the order parameter field.
Discarding the longitudinal fluctuations, the expected
Berezinskii-Kosterlitz-Thouless (BKT) phase characterized by a finite stiffness
and an algebraic decay of order parameter correlations is recovered.
Renormalized by transverse fluctuations, the longitudinal mass scales to zero,
so that longitudinal fluctuations become increasingly important for small
momenta. Within our expansion of the effective action, they generate a
logarithmic decrease of the stiffness, in agreement with previous functional
renormalization group calculations. The logarithmic terms imply a deviation
from the vanishing beta-function for the stiffness in the non-linear sigma
model describing the phase fluctuations at three-loop order. To gain further
insight, we also compute the flow of the parameters characterizing longitudinal
and transverse fluctuations from a density-phase representation of the order
parameter field, with a cutoff on phase fluctuations. The power-law flow of the
longitudinal mass and other quantities is thereby confirmed, but the stiffness
remains finite in this approach. We conclude that the marginal flow of the
stiffness obtained in the cartesian representation is an artifact of the
truncated expansion of momentum dependences.
