  We study radiative $p_{\perp}$-broadening of high energy quarks passing
through hot and cold QCD matter. With $L$ the length of the matter and $l_{0}$
the size of constituents of the matter we find $\avg{p_{\perp}^{2}}$ has both
double logarithmic terms, $\ln^{2}(L/l_{0})$, and single logarithmic terms,
$\ln(L/l_{0})$, coming from gluon radiation induced by the matter. We use a
(slight) extension of a formalism developed by B. Zakharov for studying energy
loss, a formalism which, for much of our calculation, reduces to a simple
dipole scattering analysis. We estimate the radiative contribution to be a
sizable correction to the nonradiative value of $\avg{p_{\perp}^{2}}$. We also
carry out a resummation of the double logarithmic terms that we find, and we
briefly discuss running coupling effects which appear here in a rather unusual
way.
