We present a prescription to calculate manifestly gauge invariant tree-level helicity amplitudes for arbitrary scattering processes with off-shell initial-state gluons within the kinematics of high-energy scattering. We show that it is equivalent to Lipatov's effective action approach, and show its computational potential through numerical calculations for scattering processes with several particles in the final state.

We study forward dijet production in dilute-dense hadronic collisions. By considering the appropriate limits, we show that both the transverse-momentum-dependent (TMD) and the high-energy factorization formulas can be derived from the Color Glass Condensate framework. Respectively, this happens when the transverse momentum imbalance of the dijet system, k t , is of the order of either the saturation scale, or the hard jet momenta, the former being always much smaller than the latter. We propose a new formula for forward dijets that encompasses both situations and is therefore applicable regardless of the magnitude of k t . That involves generalizing the TMD factorization formula for dijet production to the case where the incoming small-x gluon is off-shell. The derivation is performed in two independent ways, using either Feynman diagram techniques, or color-ordered amplitudes.

We study matrix elements of Fourier-transformed straight infinite Wilson lines as a way to calculate gauge invariant tree-level amplitudes with off-shell gluons. The off-shell gluons are assigned "polarization vectors" which (in the Feynman gauge) are transverse to their off-shell momenta and define the direction of the corresponding Wilson line operators. The infinite Wilson lines are first regularized to prove the correctness of the method. We have implemented the method in a computer FORM program that can calculate gluonic matrix elements of Wilson line operators automatically. In addition we formulate the Feynman rules that are convenient in certain applications, e.g. proving the Ward identities. Using both the program and the Feynman rules we calculate a few examples, in particular the matrix elements corresponding to gauge invariant $g^{*}g^{*}g^{*}g$ and $g^{*}g^{*}g^{*}g^{*}g$ processes. An immediate application of the approach is in the high energy scattering, as in a special kinematic setup our results reduce to the form directly related to Lipatov's vertices. Thus the results we present can be directly transformed into Lipatov's vertices, in particular into $RRRP$ and $RRRRP$ vertices with arbitrary "orientation" of reggeized gluons. Since the formulation itself is not restricted to high-energy scattering, we also apply the method to a decomposition of an ordinary on-shell amplitude into a set of gauge invariant objects.Comment: 30 pages, quite a few figures. Minor corrections, references updated. Submitted to JHE

We provide a description, within the High Energy Factorization formalism, of central-forward dijet correlation data measured by the CMS experiment and the predictions for nuclear modification ratio R p A in p + Pb collisions. In our study, we use the unintegrated gluon density derived from the BFKL and BK equations supplemented with subleading corrections and a hard scale dependence. The latter is introduced at the final step of the calculation by re-weighting the Monte Carlo generated events using suitable Sudakov form factors, without changing the total cross section. We achieve a good description of data in the whole region of the azimuthal angle.

Abstract:We study the production of forward di-jets in proton-lead and proton-proton collisions at the Large Hadron Collider. Such configurations, with both jets produced in the forward direction, impose a dilute-dense asymmetry which allows to probe the gluon density of the lead or proton target at small longitudinal momentum fractions. Even though the jet momenta are always much bigger than the saturation scale of the target, Q s , the transverse momentum imbalance of the di-jet system may be either also much larger than Q s , or of the order Q s , implying that the small-x QCD dynamics involved is either linear or non-linear, respectively. The small-x improved TMD factorization framework deals with both situations in the same formalism. In the latter case, which corresponds to nearly back-to-back jets, we find that saturation effects induce a significant suppression of the forward di-jet azimuthal correlations in proton-lead versus proton-proton collisions.

A framework for an improved TMD (iTMD) factorization scheme at small x, involving off-shell perturbative subamplitudes, was recently developed as an interpolation between the TMD k t ≪ Q regime and the BFKL k t ∼ Q regime. In this article, we study the relation between CGC and iTMD amplitudes. We first show how the dipolesize expansion of CGC amplitudes resembles the twist expansion of a TMD amplitude. Then, by isolating kinematic twists, we prove that iTMD amplitudes are obtained with infinite kinematic twist accuracy by simply getting rid of all genuine twist contributions in a CGC amplitude. Finally we compare the amplitudes obtained via a proper kinematic twist expansion to those obtained via a more standard dilute expansion to show the relation between the iTMD framework and the dilute low x framework. Discussions 32A Effective Feynman rules in the external CGC shockwave field 34 B CGC amplitudes for all channels 34 C The integral 35

Basing on the Slavnov-Taylor identities, we derive a new prescription to obtain gauge invariant tree-level scattering amplitudes for the process g * g → N g within high energy factorization. Using the helicity method, we check the formalism up to several final state gluons, and we present analytical formulas for the the helicity amplitudes for N = 2. We also compare the method with Lipatov's effective action approach.

We study saturation effects in the production of forward dijets in proton-lead collisions at the Large Hadron Collider, using the framework of High Energy Factorization. Such configurations, with both jets produced in the forward direction, probe the gluon density of the lead nucleus at small longitudinal momentum fraction, and also limit the phase space for emissions of additional jets. We find significant suppression of the forward dijet azimuthal correlations in proton-lead versus proton-proton collisions, which we attribute to stronger saturation of the gluon density in the nucleus than in the proton. In order to minimize model dependence of our predictions, we use two different extensions of the Balitsky-Kovchegov equation for evolution of the gluon density with sub-leading corrections.