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.
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