Regge field theory in zero transverse dimensions: loops versus “net” diagrams

Abstract

Toy models of interacting Pomerons with triple and quaternary Pomeron vertices in zero transverse dimension are investigated. Numerical solutions for eigenvalues and eigenfunctions of the corresponding Hamiltonians are obtained, providing the quantum solution for the scattering amplitude in both models. The equations of motion for the Lagrangians of the theories are also considered and the classical solutions of the equations are found. Full two-point Green functions ("effective" Pomeron propagator) and amplitude of diffractive dissociation process are calculated in the framework of RFT-0 approach. The importance of the loops contribution in the amplitude at different values of the model parameters is discussed as well as the difference between the models with and without quaternary Pomeron vertex. *The complete understanding of the high energy scattering processes is impossible without the calculation of the contributions of the different unitarization corrections to the amplitude. In the QCD Pomeron framework, [1,2,3], these corrections are arose due the self Pomeron interactions via the triple Pomeron interactions vertices [4,5]. These self Pomeron interactions lead to the very complicated picture of the amplitude's evolution with rapidity, see for example [6,7,8,9,10].There are few main approaches which claim that they properly describe the Pomeron self interactions. The first one is presented as a chain of the evolution equations with Balitsky-Kovchegov (BK) equation as a main field equation of the hierarchy, [11]. The BK equation may be formulated in the framework of the Color Glass Condensate Approach (CGK), [10], and may be properly analyzed in the terms of s-channel interacting color dipoles, [12]. This equation correctly describes the interaction of two non identical objects, for example such interactions as DIS on nuclei, see [13,14,15,16,17,18,19,20]. Another approach is the QCD Reggeon Field Theory (RFT-QCD), which is formulated in the momentum space and based on the standard diagrammatic calculus, [6,7,8,9,21,22,23]. In this approach the BK equation describes a resummation of the "fan" diagrams of the type depicted in Fig.1a with only Pomerons merging vertices considered.Usual BK equation, describing "fan" diagrams, does not include Pomeron's splitting vertex. Therefore, the next natural step toward the unitarization of the amplitude is a symmetrical consideration of the scattering process with both vertices included. In this case the splitting vertex may be accounted differently in two different approximations. In the first approach the semi-classical problem of the interest is considered and there the sum of the diagrams depicted in Fig.1b is calculated, neglecting Pomeron loops contribution into amplitude. The second approach is concentrated on the solution of the full quantum problem basing on some effective high-energy QCD inspired models, accounting diagrams of Fig.1c type, see for example [24]. So far several attempts of the calculations were done in both directions, on the basis of QCD-RFT t...

One-loop light-cone QCD, effective action for reggeized gluons and QCD RFT calculus

“…Calculating these vertices one after another we will reconstruct the expression similar to introduced in [8], see also [57][58][59][60][61][62][63][64][65][66][67][68], which can be considered as QCD Hamiltonian for reggeized gluon fields. There are the following well-known vertices of the interactions of reggeon fields.…”

On the one-dimensional reggeon model: eigenvalues of the Hamiltonian and the propagator

“…In [13] the eigenvalues of the Hamiltonian (4) were found for values of µ/λ = 1, 3, 5 and the propagator at µ/λ = 5 was calculated, although neither the calculational procedure nor the precision were reported. In this paper we concentrate on the theoretical problems related to the Hamiltonian in the complex plane and on the method of calculation of its eigenvalues and eigenfunctions.…”

On asymptotic solutions of Regge field theory in zero transverse dimensions

“…The duality of the systems Eq. (7) and Eq. (73) could be also understood from the point of conditions needed to apply on the systems in order to obtain the solution.…”

“…In general this task is very complicated, and, therefore, the simplified models which have dynamics similar to QCD ones could serve as a good polygon for the different ideas and methods check. This is the reason why RFT-0 (Reggeon Field Theory in zero transverse dimensions) model is attracted much interest during few last years, see [6,7]. The approach, which was formulated and studied a long time ago, even before the QCD era ( see [8]), has a dynamics which is very similar to the solutions obtained in framework of interacting QCD Pomerons, see [9,10].…”