From LO to NLO in the parton Reggeization approach

We present recent developments of the parton Reggeization approach (PRA), which is based on high-energy factorization of hard processes in the multi-Regge kinematics and Lipatov’s effective theory of Reggeized gluons and Reggeized quarks. The scheme of calculations in the leading order (LO) of the PRA is discussed. We present important examples of LO PRA applications for cross section calculations of multi-scale hard processes, such as pair production of BB̄−mesons, pair production of photons, and pair production of jets. Also the problem of matching of NLO calculations in PRA with NLO results in the Collinear Parton Model is discussed.

similar structure in the non − emission probability in ISR PS |APRA| 2 is obtained from Lipatov's gauge-invariant effective theory for MRK processes in QCD [Lipatov 1995;Lipatov, Vyazovsky, 2001]. Some Feynman rules for Reggeized gluons: Feynman rules for Reggeized quarks:

Implementation in FeynArts.
The Feynman rules of Lipatov's EFT, up to the order O(g 3 s , eg 2 s , e 2 gs, e 3 ) are implemented in our model-file ReggeQCD for the package FeynArts.
The approach to derive gauge-invariant scattering amplitudes with off-shell initial-state partons, using the spinor-helicity techniques and Britto-Cachazo-Feng-Witten-like recursion relations for such amplitudes, was introduced in Dijet production at the LHC ( √ S = 7 TeV). Open points -only 2 jets with pT > 30 GeV at |y| < 0.8, Closed points -inclusive data.

Complicated kinematical situation!
The BB-pair is searched in the events with a hard jet. Our solution -"merging" of two contributions: 1 The hard jet = b-jet: The hard jet = b-jet: Instead, the double-counting of region kT → 0 or ∆y → ∞ should be subtracted from NLO contribution to the hard-scattering coefficient.

Comparison with CMS data
Vertices of Lipatov's theory are nonlocal: contain eikonal denominators -1/k ± ⇒ Rapidity divergences in real and virtual corrections! Rapidity divergences are related with BFKL resummation of contributions ∼ log 1/x. So some elements of BFKL resummation necessarily enter at NLO.
Motivation for PRA is the multi-scale correlational observables. ⇒ NLO CPM accuracy for single-scale observables is enough.
We DO NOT want to do our own fit of unPDFs ⇒ using (M S) PDFs of CPM as collinear input. But NLO PDFs are scheme-dependent! For the single-scale observables (e.g. F 2/L (x, Q 2 )) collinear factorization is a theorem (up to corrections ∼ (Λ 2 QCD /Q 2 ) # ). ⇒ Physical normalization condition at NLO: +O(α 2 s (Q 2 )) + Higher twist, Basic idea: Collect all terms in PRA, contributing O(αs) to the normalization condition and match them to the NLO CPM result.
Other corrections: ∆F contain only fq(x1, Q 2 ), and will be important for the γ + Q → q + g subprocess and one-loop correction.
Factorization + cancellation of artificial divergences ⇒ Renormalization group. The latter allows to resum large logarithms of scale ratios.

Rapidity divergences and regularization.
Due to the presence of the 1/q ± -factors in the induced vertices, the loop integrals in EFT contain the light-cone (Rapidity) divergences: The regularization by explicit cutoff in rapidity was proposed by Lipatov [Lipatov, 1995] (q ± = q 2 + q 2 T e ±y ): The dependence on the regulators yi have to cancel between the contributions of the neighbouring regions order-by-order in αs, building up the 1 n! (log(s)ω(t)) n -terms.
The regularization of the light-cone divergences is achieved by the shifting the n ± vectors from the light-cone: and for the lowest-order(Rgg, Qqg) induced vertices the PV prescription is at work: For the higher-order induced vertices, there is the nontrivial interplay between color and kinematics in the pole prescription.
Recently, this prescriptions has been tested at one loop for the case of where k 2 = 0, p 2 = −t1, (k + q) 2 = 0. The e ρ -divergences cancel in the sum of diagrams. The terms proportional e − ρ → 0 for > 0 ⇒ only logarithmic singularities∼ ρ are left.
From LO to NLO in the parton Reggeization approach.
The subtracted result: The subtraction term: From LO to NLO in the parton Reggeization approach.
The cross-check.