  I look at the renormalization of the medium structure function and a medium
induced jet function in a factorized cross section for jet substructure
observables in Heavy Ion collisions. This is based on the formalism developed
in \cite{Vaidya:2020lih}, which uses an Open quantum system approach combined
with the Effective Field Theory(EFT) for forward scattering to derive a
factorization formula for jet observables which work as hard probes of a long
lived dilute Quark Gluon Plasma(QGP) medium. I show that the universal medium
structure function that captures the observable independent physics of the QGP
has both UV and rapidity anomalous dimensions that appear due to medium induced
Bremsstrahlung. The resulting Renormalization Group(RG) equations correspond to
the BFKL equation and the running of the QCD coupling respectively. I present
the first results for the numerical impact of resummation using these RG
equations on the mean free path of the jet in the medium. I also briefly
discuss the prospects of extending this formalism for a short lived dense
medium.
