Computing one-loop corrections to effective vertices with two scales in the EFT for Multi-Regge processes in QCD

Abstract

The computation of one-loop corrections to Reggeon-Particle-Particle effective vertices with two scales of virtuality is considered in the framework of gauge-invariant effective field theory for Multi-Regge processes in QCD. Rapidity divergences arising in loop integrals are regulated by "tilted Wilson lines" prescription. General analysis of rapidity divergences at one loop is given and necessary scalar integrals with one and two scales of virtuality are computed. Two examples of effective vertices at one loop are considered: the effective vertex of interaction of (space-like) virtual photon with one Reggeized and one Yang-Mills quark and the effective vertex of Reggeized gluon to Yang-Mills gluon transition with an insertion of the operator tr [G µν G µν ] carrying the (space-like) off-shell momentum. All terms ∼ r ± in the rapidity-regulator variable r cancel between diagrams and only log r-divergence is left. It is checked on several examples, that obtained results indeed allow one to reproduce Regge limit of one-loop QCD scattering amplitudes.(NLLA) [28,29,30]. It is natural to ask for the systematic tool, which makes this property of QCD scattering amplitudes manifest. The gauge-invariant Effective Field Theory (EFT) for Multi-Regge processes in QCD [31,32] is such a tool. In the original papers [31,32], the Reggeization of gluon and quark was shown to hold in a framework of EFT in the LLA. Computing loop corrections in this EFT one finds a new type of divergences of loop integrals, the so-called rapidity divergences, which will be discussed in the forthcoming sections of the present paper. Similar divergences also arise in the context of Soft-Collinear Effective Theory (SCET) 1 [35,36] and Transverse-Momentum-Dependent (TMD) factorization [37,38] and several prescriptions to regularize them, such as analytic regularization [35], δ-regularization [39] and "tilted Wilson lines" regularization [37,38] where introduced in the literature. Among mentioned prescriptions, only the last one does not modify the standard definition of the Wilson lines, and therefore it can be relatively straightforwardly applied to the High-Energy EFT [31,32]. First such applications where performed in the Refs. [40,41,42] where one-loop corrections to the propagator of Reggeized gluon and effective vertices of interaction of on-shell quark and gluon with one Reggeized gluon where computed in the framework of EFT, and found to be consistent with the results obtained earlier from QCD. Later, the two-loop Regge trajectory of a gluon was extracted from a rapidity-divergent part of the two-loop correction to a Reggeized gluon propagator in the EFT [43] and it coincides with known QCD result. Also, the calculation of the one-loop correction to the propagator of Reggeized quark (Q) and vertex of interaction of on-shell photon with one Reggeized and one QCD quark (Qγq-vertex) has been done in the Ref. [44] and it was shown, that this results allow one to reproduce the Multi-Regge asymptotics of the positive-signature part of the one-loop QC...

Towards stability of NLO corrections in high-energy factorization via modified multi-Regge kinematics approximation

“…The paper is organized as follows: in section 2 we formulate the basic formalism of PRA for the particular process we have chosen to study and derive the evolution equation for UPDF in MRK approximation, then in section 3 we formulate our MMRK approximation, analyze it's performance in comparison to an exact QCD amplitude with one additional emission and write-down UPDF-evolution equation in MMRK approximation and corresponding NLO double-counting subtraction terms; in section 4 we describe our phase-space slicing strategy, compute corresponding analytic integrals and double-counting subtraction integral in the soft limit; in section 5 we recall the virtual part of NLO-correction under consideration, computed in ref. [44], and derive corresponding virtual subtraction terms; finally in section 6 we present and discuss some numerical results and formulate our conclusions. In the appendix A we derive an approximate doubly-logarithmic solution for our UPDF evolution equation, which we use for illustrative numerical calculations throughout this paper and in the appendix B we take a few iterations of evolution kernel (2.30) to demonstrate it's properties.…”

“…which has been used in ref. [44]. The logarithm log(1/r) can be identified with the rapidity difference between two adjacent real emissions (see the discussion in the end of section 1 of ref.…”

“…The logarithm log(1/r) can be identified with the rapidity difference between two adjacent real emissions (see the discussion in the end of section 1 of ref. [44]), and therefore, the virtual part of the evolution equation will be proportional to the coefficient in front of this logarithm -the one-loop Regge-trajectory of a gluon:…”

“…refs. [4,7,44,59] for the detailed presentation) and is rather long and non-instructive expression, so we refrain from presenting it here. Some relevant limits of it are given below and in section 4.…”

“…We have computed the one-loop correction to an amplitude of the process (2.12) in ref. [44]. Apart from IR and UV divergences, which we regularize dimensionally, it contains rapidity divergence, physically corresponding to rapidity of a gluon in a loop going far negative.…”