  We develop two approaches to the problem of soft fragmentation of hadrons in
a gauge theory for high energy processes. The first approach directly adapts
the standard resummation of the parton distribution function's anomalous
dimension (that of twist-two local operators) in the forward scattering regime,
using $k_T$-factorization and BFKL theory, to the case of the fragmentation
function by exploiting the mapping between the dynamics of eikonal lines on
transverse-plane to the celestial-sphere. Critically, to correctly resum the
anomalous dimension of the fragmentation function under this mapping, one must
pay careful attention to the role of regularization, despite the manifest
collinear or infra-red finiteness of the BFKL equation. The anomalous
dependence on energy in the celestial case, arising due to the mismatch of
dimensionality between positions and angles, drives the differences between the
space-like and time-like anomalous dimension of parton densities, even in a
conformal theory. The second approach adapts an angular-ordered evolution
equation, but working in 4-2$\epsilon$ dimensions at all angles. The two
approaches are united by demanding that the anomalous dimension in
4-2$\epsilon$ dimensions for the parton distribution function determines the
kernel for the angular-ordered evolution to all orders.
