Thresholds in Reggeon field theory

It is explicitly shown that thresholds in rapidity do not modify the critical behaviour of Reggeon Field Theory (RFT). A rule to determine the threshold for any diagram is given.     

Recent developments in Reggeon field theory

In the last few years Reggeon Field Theory (RFT) has been developed into a very powerful tool for analyzing the complex angular momentum structure of high energy scattering amplitudes. In a previous review of this subject (Abarbanel et al. [8]) RFT was motivated, formulated, and its early results were discussed. These issues will be mentioned only briefly in the introduction to the present review in order to make it self-contained. It will then be followed by a discussion of the recent developments and progress in the field. A major part of this review is devoted to the discussion of various efforts made to elucidate the s-channel content of Reggeon Field Theory. The formulation of RFT on a lattice and the analysis of production amplitudes are emphasized as the main approaches towards understanding the implication of RFT on s-channel unitarity issues. The path integral and the Hamiltonian formulations of RFT on a lattice, along with their results, are treated in detail. The work done on production amplitudes, in particular cut Reggeon field theory, its derivation and applications, are presented. The problem of the approach to scaling, the evaluation of the transition energy and scaling functions are discussed as they emerge from the representations of Reggeon Green's functions. Practical problems such as the confrontation of the RFT results with the experimental data and the relevance of different approximations are also analyzed.     

An asymptotically free Reggeon field theory

Renormalization group techniques are used to examine Reggeon field theories with derivative triple Pomeranchuk interactions in accord with the decoupling theorems. The simplest asymptotically free “weak coupling” solution has total cross sections which behave as σ_tot^AB(s) ∽ g_Ag_B?c/log s +…, and the triple Pomeranchuk vertex has a nonsense wrong signature zero asymptotically ～(t?t₁?t₂). Another “strong coupling” solution reproduces the results of Migdal, Polyakov, Ter-Martirosyan, Abarbanel and Bronzan to lowest order in the ?-expansion.     

Reggeon field theory: Formulation and use

We formulate and discuss Reggeon field theory, which enables one to systematically analyze the exchange of Regge poles and associated branch points in high energy hardron scattering. The field theory is first motivated by a consideration of hybrid Feynman graphs, and then a more general derivation from crossed-channel multiparticle unitarity relations is given. Rules for Reggeon interaction and propagation are formulated. We treat in some detail the problem of the Pomeron or vacuum pole which has α(0) = 1 and is responsible for diffractive processes. In particular the renormalization group analysis of Reggeon field theory is presented and the structure of Pomeron partial wave amplitudes is elucidated. Also the question of Pomeron or absorptive corrections to secondary trajectories (both fermion and boson) is considered. We make some comments on important problems yet remaining in Reggeon field theory; in particular, we stress the study of its s-channel content.     

Spontaneous symmetry breaking in Reggeon field theory

We consider a Reggeon field theory when the bare or input Regge intercept α_O is greater than one. This corresponds to a negative mass squared term in conventional field theory and allows for a spontaneous symmetry break-down. A theory with Regge intercept at one emerges, restoring the Froissart bound by t-channel considerations alone. In our elementary example the resulting bare trajectory is nearly of the square root variety familiar from s-channel eikonalization of models which violate the Froissart bound.     

Scaling and the approach to scaling in Reggeon field theory

We address ourselves to the problem of the approach to asymptotic scaling in Reggeon field theory. We present a method of integration of the renormalization group equations and obtain the asymptotic scaling functions, as well as the corrections which occur at non-asymptotic energies.     

Reggeon field theory on a lattice: A formulation

The principle of universality is taken from statistical mechanics and applied to the reggeon field theory. Following a pedagogical review of the implications of universality for the reggeon calculus, a lattice formulation of the reggeon field theory is presented and discussed. Scaling laws for the thermodynamic quantities of such systems are described and relations among critical indices are derived. These relations enable high-energy scattering behaviour to be calculated using lattice methods.     

Higher order ϵ-terms in the renormalization group approach to Reggeon field theory

We calculate the O(?²) terms in a Wilson ?-expansion of the scaling exponents in Reggeon field theory. We find that these corrections are comparable to the O(?) terms.     

Elliptical flows in Reggeon theory

A new mechanism for the generation of an azimuthal anisotropy of particles produced in heavy-ion collisions is analyzed. An analysis is performed within Reggeon theory and is general. The resulting effect can be interpreted in terms of the partonic structure of a fast nucleus. This mechanismcan contribute significantly to the observed value of the elliptic flow υ ₂. The dynamics in question differs significantly from a standard mechanism of the collective motion of a hadron medium after a collision.

     

Reggeon field with non-vanishing vacuum expectation value

It is argued that if the vacuum expectation value of the reggeon field does not vanish, the solution of the reggeon calculus corresponds to the leading singularity of intercept one. The physical interpretation of the reggeon field theory with unstable vacuum is discussed.     

Self-consistent Pomeranchuk singularities in Reggeon field theories

A self-consistency or bootstrap principle is suggested to determine the structure of the Pomeranchuk singularity in the neighborhood of J = 1 and t = 0. The ingredients in this are a Reggeon field theory which we require to be renormalizable and infrared free in the sense of the renormalization group. This guarantees that the input singularity reproduces itself near J=1, t=0 with small computable corrections. Several examples of physical significance are discussed.     

Reggeon field theory for α(0) > 1

We obtain the asymptotic behaviour of the scattering amplitude when the pomeron has intercept α(0) larger than one. The reggeon field theory is studied by introducing a lattice in impact parameter space. Use is made of a previous result showing asymptotically the dynamics is controlled at each lattice site (α′ = 0 case) by a two-level structure. This leads to a non-Hermitean Hamiltonian expressed in terms of spin operators in which the intersite interaction terms is proportional to the pomeron slope α′. The spectrum of such a system shows a degenerate ground state for α(0) > α_c >~ 1 and a continuum with vanishing excitation gap at α(0) = α_c. The vacuum does not change structure at the critical value. The critically is shown by an order parameter which is given by the matrix element of a field operator between the vacuum and its degenerate companion. The nature of this critical phenomenon is better understood by continuously transforming the Hamiltonian into that of an Ising model with a transverse field which shows a well-known second-order phase transition. By defining the S-matrix so as to preserve the formal perturbation expansion, we find that for α(0) > α_c, the zero gap state contributes a non-trivial asymptotic constant. The final asymptotic picture is that of a gray disc expanding like log s, so that the total cross section behaves as (log s)². For α(0) < α_C, the vacuum is non-degenerate and correspondingly the total cross section drops to zero as an inverse power of s.     

Quantum field theory in the infinite temperature limit

The T = ∞ limit for renormalizable 4-dimensional Euclidean QFT is considered. A general argument is presented in three examples: φ³, QED, QCD. Using an expansion of the Green's functions generating functional, it is shown at T = ∞ quantum dynamics generally becomes 3 dimensional. All superficially divergent diagrams survive at T = ∞ and ensure renormalization of effective dynamics. The correction to naive dimensional reduction is studied; appearance of “electric” masses in QED and QCD is shown to be the result of such a correction. A curious symmetry of the generating functional in QED and QCD, its implications and breaking by the thermal corrections of heavy modes are discussed. Presence of the symmetry implies survival of some fermion modes at T = ∞.     

Reggeon scattering in an external field: A solitonic model

Solitonic solutions of the nonlinear Schrödinger equation are considered to represent reggeons. The effect of a constant electric field on reggeons is formulated and the modified energy levels due to the interaction are calculated using the usual semiclassical approach. In summing overall stability angles a simple subtraction procedure is adopted for regularization.     

Finite temperature effects in quantum field theory

We consider quantum electrodynamics at finite temperatures. By making use of the real time formalism we compute, on the one-loop level, the finite-temperature correction to the mass of the electron and to the anomalous magnetic moment a_e^th. The gauge-invariant correction to the electron mass is found to be a ten percent effect at a temperature of the order of 2×10¹⁰ K. Some astrophysical implications of this result are briefly discussed. The leading temperature correction to the anomalous magnetic moment of the electron is, at a temperature of 300 K, found to be one order of magnitude smaller than the τ-lepton contribution to a_e^th.     

Finite temperature field theory with boundaries: The photon field

We calculate the deviations from Planckian form of the photon field finite temperature stress tensor in a manifold with boundary, due to scattering from the boundary. Familiar non-integrable divergences are found in the photon stress tensor as the boundary is approached and these are shown to be an inescapable consequence of initial calculational assumptions. Modifications of these assumptions are discussed which serve to remove the divergences and to illustrate the importance of the role played by surface gravitational actions.     

Studying the perturbative Reggeon

We consider the flavour non-singlet Reggeon within the context of perturbative QCD. This consists of ladders built out of “reggeized” quarks. We propose a method for the numerical solution of the integro-differential equation for the amplitude describing the exchange of such a Reggeon. The solution is known to have a sharp rise at low values of Bjorken-x when applied to non-singlet quantities in deep-inelastic scattering. We show that when the running of the coupling is taken into account this sharp rise is further enhanced, although the dependence is suppressed by the introduction of the running coupling. We also investigate the effects of simulating non-perturbative physics by introducing a constituent mass for the soft quarks and an effective mass for the soft gluons exchanged in the t-channel. Received: 30 June 1999 / Published online: 16 November 1999