A field-theoretic study of the Regge-eikonal model. II

A previous study of the Regge-eikonal model in φ³ is extended to QED. First we define a reggeon amplitude that is built up by tower diagrams, and then study multiregge exchange by use of Gribov's reggeon calculus. The situation is essentially the same as in φ³: The eikonal approximation is the true high-energy and weak coupling limit (s → ∞, α ln s = fixed), but it breaks down outside of the weak-coupling limit. This confirms that the eikonal approximation as a model for Regge-cuts is not justifiable by field-theoretic arguments because of the neglect of inelastic intermediate states.     

A non-perturbative study of regularized bosonic string amplitudes

The Wick-rotated, light-cone gauge U(n) Veneziano model of open and closed strings is examined in the limit as n → ∞, with g₀²n fixed. The amplitudes in both the “pomeron sector” (whose graphs are cylinders with holes punched out) and the “reggeon sector” (whose graphs are rectangles with holes punched out) are regularized using the lattice method of Giles and Thorn, as well as an alternative method. It is found that this string model is trivial. When the cut-off is removed, the pomeron and reggeon sector spectra are those of the free closed and open strings, respectively. This result is independent of the choice of g₀²n. A possible extension to fermionic strings, and the implications for large-n gauge theories are briefly discussed.     

Phase transitions and the high temperature expansion for the reggeon calculus

We study a recently proposed lattice version of the reggeon field theory. First a simple one-dimensional system possessing many of the features of the reggeon calculus (Ising model in an imaginary magnetic field) is solved. Surprisingly, the system is found to undergo a phase transition at a non-zero critical temperature, which, although of first order obeys the universality and scaling laws previously postulated for the reggeon calculus. Returning to the full lattice theory, the machinery for performing a high temperature expansion is set up, and initial calculations carried out to order T^?3. In this order, best estimates for the critical indices η and v in the asymptotic elastic amplitude A(s,t) ～ is (ln s)^ηf(t(lns)^v) yield $\text{η = 0 (}\text{1}\text{2}\text{～ 1), v = 0(1}\text{1}\text{2}\text{～ 2)}$. Check on the method, including comparisons with known Ising model results, are also discussed.     

Higher order reggeon field theory results for dσ/dt

The critical reggeon field theory (RFT) asymptotic scaling law for dσ/dt can be evaluated using the ε expansion for renormalization group (RG) quaatities at t ≠ 0 along with the two variable RG equations found by Abarbanel, Bartels, Bronzan and Sidhu (ABBS). The amplitude thus obtained has an ε-exponentiated form. Interestingly, it reduces in a certain limit to the Bessel function pomeron model advocated by Kane. We perform a systematic numerical study here.Our results are as follows. Although the asymptotic RFT scaling is violated by the data, introduction of a mild external particle vertex with one parameter improvesthe situation considerably at current energies. Moreover, the shape of the ABBS scaling curve predicted by the O(ε²) results of Dash and Grandou is in at least qualitative agreement with the data, including the absence of a second dip. Two additional parameters (the t = 0 normalization and the scale in t) have to be fixed by hand.The overall conclusion is that in spite of many a priori objections the asymptotic RFT does a surprisingly good job of describing both the s and t dependence of dσ/dt at current energies.     

General features of the reggeon calculus with α > 1

We discuss some general properties of the reggeon calculus with input intercept greater than one, in an attempt to formulate a model yielding indefinitely rising cross sections which is consistent with t-channel unitarity. In general the leading behaviour of the amplitude always factorises. If interactions between reggeons are ignored the scattering is equivalent to that from a grey absorbing disk of radius O (ln s). Diagrams involving such interactions are subject to large-scale cancellations. These enable the model to give sensible results for inclusive cross sections. In quantum number exchange reactions, factorisation of the Regge pole residues is weakly broken in the model.     

The content of unitarity constraints on a pomeron pole of intercept one

We present a consistent picture of a pomeron pole with intercept one, together with its cuts, which evades the decoupling arguments. We use the reggeon cut discontinuity formulae to introduce Gribov's reggeon calculus as an exact solution of multiparticle t-channel unitarity. We show how, within the calculus, two-pomeron iterations of a singular kernel can be responsible for the zero in the triple-pomeron vertex. Using the concept of a bare pomeron pole as a multiperipheral production process which is subsequently renormalised by other effects, we apply the reggeon calculus analysis to inclusive cross sections. We find that the inclusive sum rule decoupling arguments are avoided because of the addition of enhanced absorptive corrections to the conventional Regge pole contributions. However, we show that in this picture the combined pole and two-pomeron cut contribution to the total cross section factories to order (ln s)^?2.We also show that, when the correct helicity structure of the pomeron is taken into account, the s-channel unitarity condition for pomeron scattering amplitudes does not lead to any serious decouplings.     

Cut contributions in the triple-pomeron limit

We investigate pomeron cut contributions in the triple-pomeron limit of one-particle inclusive cross sections, near t = 0 where the triple-pole coupling vanishes. We find that at t = 0 the cuts themselves are suppressed, contributing factors (lnM²)^?2, rather than the single logarithms characteristic of cut contributions in two-body processes. We construct a simple reggeon calculus model in which all important cuts near t = 0 can be calculated, and suggest a simple way of parametrizing the data in that region.     

Three gluon integral equation and odd C singlet Regge singularities in QCD

The integral equation for a three gluon system in the odd C colour singlet channel is derived utilising reggeon calculus based on QCD. This equation is shown to be free from infrared divergences. It generates a fixed branch point near j = 1 as the singularity.     

Reggeon phases and the dual bootstrap program

Analyticity requirements lead to complex Regge vertices in multiparticle production amplitudes. Their effect is discussed within a dual, unitary and exchange-degenerate multi-Regge model. A resulting complex Regge vertex is incorporated in fixed-t model calculations of the reggeon bootstrap and the topological pomeron. The bootstrap condition for particle-particle scattering is unaffected but new j-plane singularities appear in Regge-Regge scattering amplitudes. The pomeron and reggeon intercepts determine the two parameters of our Regge vertex and admit two possible solutions: one is close to the naive phase rule (real vertex) and the other is complex and leads to a high lying non-leading pole in the pomeron channel.     

Exploiting the equivalence of reggeon field theory in zero transverse dimensions and reaction–diffusion processes

The reggeon field theory in zero transverse dimensions is investigated. Two versions of the theory are considered: one that allows for at most triple pomeron interactions and the other that embodies an additional 2→2 quartic reggeon coupling. The behavior of the scattering amplitude at asymptotic rapidities is obtained in both cases. In an s-channel picture of the high energy scattering both models can be viewed as reaction–diffusion processes. We derive known results in reggeon field theory rather easily using the reaction–diffusion formalism. We find that some results which are surprising from the reggeon field theory point of view turn out to have a simple interpretation from the reaction–diffusion point of view.     

On the analytic properties of the reggeon-reggeon amplitude

The analytic properties of the reggeon-reggeon amplitude in the missing-mass like variable are discussed. (This amplitude is defined through the suitable high-energy limit of the eight-point amplitude). It is shown on the basis of simple Feynman diagrams that it is possible to define reduced amplitudes which are free from kinematical singularities provided that the eight-point amplitude is expressed in a form explicitly consistent with Steinmann relations. In a simple model which is considered in this paper, the reduced amplitudes have the singularity structure of ordinary two-body scattering amplitudes. The finite energy sum rules and four reggeon vertices are also briefly discussed.     

Diffractive scattering in the dual model

A method is suggested for calculating the elastic amplitude as the shadow of multiparticle processes. This does not require explicit formulae for the production amplitudes, but depends only on the following assumptions: (a) Regge behaviour, (b) semi-local duality, (c) exchange degeneracy. The system is multiperipheral in general structure but because of the assumption (a)-(c), it incorporates such features as the clustering of final particles and provides a proper treatment of quantum numbers, phases, and resonance spins. As a result, the unitarity sum separates automatically into two components corresponding respectively to pomeron and ordinary reggeon exchange, each of which is seen to have the correct qualitative dependence on the quantum numbers, the energy s, and the momentum transfer t. Quantitatively, the elastic amplitude is given in terms of a triple-Regge vertex, for which we suggest a simple parametization based on a detailed study of existing experiment and the Veneziano model. The method is then applied to calculate the elastic amplitudes for π^±p, K^±p and pp within the range 6 < s < 50 and 0 > t > ?0.5 GeV². Semi-quantitative agreement is obtained with experiment with essentially no free parameter. Although the investigation is restricted at present to elastic scattering as the shadow of only the non-diffractive component satisfying (a)-(c), the method is believed to apply also to other diffractive processes, and may be regarded as the first step in an iterative solution of the full unitarity equation.     

A topological expansion for high-energy hadronic collisions: (I). General properties and connection with the reggeon calculus

In this and in a companion paper a “topological expansion” for high-energy hadronic processes is proposed and discussed. In this first paper the general properties of the expansion and its connection with Gribov's reggeon calculus are presented. The topological expansion is first defined mathematically for a large class of theories and is shown to be equivalent to a “large N expansion” in some theories which include planar dual models and non-Abelian gauge theories. Next, the definition of the bare parameters is given in terms of graphs on a sphere. The bare pomeron pole and its couplings are thus introduced. The (inclusive) form of s-channel unitarity and its consequences fo the above couplings are recalled. It is then shown how the expansion in the number of “handles” of the graph can be related to Gribov's reggeon calculus and how, with the aid of discontinuity equations in the J-plane, scaling solutions can be obtained and critical indices can be computed to yield known results. Our main new results, including the study of s-channel discontinuities and of positivity constraints as well as the definition of a new fireball expansion, and the discussion of the relevance of this theory at finite (present) energies are presented in the second paper.     

High energy pp data and the reggeon theory of diffraction scattering

Recent proton-proton scattering experiments from the CERN ISR show several striking features. There is strong evidence that the total cross section is rising. The elastic differential cross section shows a break in the forward slope, as well as a pronounced dip-bump structure, with the dip located at t ? ? 1.3 (GeV/c)². We demonstrate that all these features and their energy dependence can be simultaneously and quantitatively very well described within the framework of Gribov reggeon calculus.     

Multi-pomeron couplings to external particles in reggeon field theory

We show that, contrary to previous assertions, multi-pomeron couplings to the external particles are equally as important as single couplings in the asymptotic behaviour of the amplitude in the reggeon calculus. Our arguments require a careful discussion of the relation of the reggeon calculus to a renormalizable field theory and of the renormalization of composite operators in reggeon field theory.     

Perturbation theory at large order in reggeon field theory

We estimate the behaviour of the perturbation expansion and the ? expansion at large orders in reggeon field theory. The perturbation expansion is divergent but Borel summable for α₀ < 1. At fixed total rapidity, the Borel summability extends into α₀ > 1.     

The nature of confined states

We show that in spite of charge confinement in the Schwinger model¹ and its nonconfinement in (QED)₄, the charged states in the two theories have many features in common. A convenient infrared regularization procedure is introduced to facilitate the study of large-distance behaviors in the Schwinger model, particularly those properties that are relevant to the question of when a charged state is physical. One difference that emerges between the two theories is that when a charged state in the Schwinger model is made physical while its energy is kept bounded, the charge goes off to infinity. The end-product could be considered neutral if the charge is defined as the limit of local measurements. On the other hand, if one attempts to change a local charged state in the Schwinger model into a physical state by transporting the localization region to asymptotic distances, the state may end up in either a θ-sector or the corresponding (θ+π)-sector, depending on the direction of transport. A possible generalization of this θ-mixing property to quark-like states in QCD is commented upon.     

Flavouring the Gribov process

The exchange of flavour carrying trajectories is studied in the non-covariant parton interpretation of reggeon field theory. While pomeron exchange is described by wee partondensities, i.e. diagonal elements of the density matrix of a fast hadron, meson exchange is described by density matrix elements which are diagonal in parton number but off-diagonal in flavour. The reggeon field theory “hamiltonian” describes a markoffian evolution of this block-diagonal density matrix during a boost. This interpretation is possible both if there are two distinctf and ? trajectories and in case of ? identity. The meson trajectories are superpositions of odd and even signature trajectories.     

Sister trajectories in the dual resonance model

The existence of a Regge tranjectory γ = α/3 ? 1 in the eight-point amplitude of the Dual Resonance Model set of demonstrated. It is suggested that the three presently known trajectories α, β and γ are part of an infinite set of “sister” trajectories, the expected properties of which are discussed.