Towards stability of NLO corrections in High-Energy Factorization via Modified Multi-Regge Kinematics approximation

The perturbatively-stable scheme of Next-to-Leading order (NLO) calculations of cross-sections for multi-scale hard-processes in DIS-like kinematics is developed in the framework of High-Energy Factorization. The evolution equation for unintegrated PDF, which resums $\log 1/z$-corrections to the coefficient function in the Leading Logarithmic approximation together with a certain subset of Next-to-Leading Logarithmic and Next-to-Leading Power corrections, necessary for the perturbative stability of the formalism, is formulated and solved in the Doubly-Logarithmic approximation. An example of DIS-like process, induced by the operator ${\rm tr}\left[G_{\mu\nu}G^{\mu\nu}\right]$, which is sensitive to gluon PDF already in the LO, is studied. Moderate ($O(20\%)$) NLO corrections to the inclusive structure function are found at small $x_B<10^{-4}$, while for the $p_T$-spectrum of a leading jet in the considered process, NLO corrections are small ($<O(20\%)$) and LO of $k_T$-factorization is a good approximation. The approach can be straightforwardly extended to the case of multi-scale hard processes in $pp$-collisions at high energies.