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We investigate the QCD resummation for the Higgs boson plus a high $P_T$ jetproduction with large rapidity separations in proton-proton collisions at theLHC. The relevant Balitsky-Fadin-Kuraev-Lipatov (BFKL) and Sudakov logs areidentified and resummed. In particular, we apply recent developments of thetransverse momentum dependent factorization formalism in the impact factors,which provides a systematic framework to incorporate both the BFKL and Sudakovresummations.

BFKL and Sudakov Resummation in Higgs Boson Plus Jet Production with Large Rapidity Separation