  The transport coefficient $\hat{q}$ plays a pivotal role in describing the
phenomenon of jet quenching in the quark-gluon plasma (QGP) produced in
ultra-relativistic nucleus-nucleus collisions. It is challenging to compute
this coefficient from first principles due to its non-perturbative nature. In
this article, we present an $ab$-$initio$ formulation of $\hat{q}$ based on the
standard techniques of perturbative quantum chromodynamics (pQCD) and lattice
gauge theory. We construct $\hat{q}$ by considering a leading order (LO)
process where a hard parton produced from the hard scattering undergoes
transverse broadening due to scatterings with the thermal medium. We do an
analytic continuation to the Euclidean region and use the dispersion relation
to express $\hat{q}$ in terms of series of local Field-Strength-Field-Strength
(FF) operators. Each term in the series is suppressed by the hard scale
$q^{-}$. Finally, we compute the local operators on the quenched SU(3) lattice
and present our estimates for $\hat{q}$.
