Crossing symmetric amplitude with Regge poles

We obtain a representation for a pion scattering amplitude that incorporates analyticity in the Mandelstam variables, and meromorphy in the right half of the l-plane. Full crossing symmetry and elastic unitarity in the three elastic regions are built into the equations; and we show how to derive the expected Regge asymptotic behavior as one Mandelstam variable tends to infinity, while another is held fixed at any real or complex value. Moreover, we demonstrate that the inelastic unitarity bound is respected by the partial wave amplitudes, even though the Regge trajectories are allowed to rise indefinitely.     

Theory of Nucleon-Nucleon Scattering

We present a new study of kinematical problems of relativistic nucleon-nucleon interactions, more complete than the previous studies. Two-component spinor formalism is used throughout. Several choices of spinor bases, their relations, crossing relations in all channels, helicity amplitudes, partial-wave amplitudes, the absence of kinematical singularities, and unitarity are discussed. The contributions of arbitrary intermediate states J in the unitarity are summed and the Regge forms for all five scalar amplitudes are obtained.     

Constraints on reggeon amplitudes from analyticity and planar unitarity

Using planar dual amplitudes as a guide, we discuss some features of reggeon amplitudes which are relevant in the context of the topological expansion. We look into the analytic properties and, in particular, discuss the validity of finite-mass sum rules for reggeon-reggeon scattering. We investigate the form taken by planar unitarity when a multiperipheral assumption is added. The integral equations obtained are not of the standard Chew-Goldberger-Low type. We find that pure pole-type solutions (i.e. without Regge cuts) to planar unitarity are possible in a way consistent with the symmetry and factorization properties of reggeon-reggeon amplitudes. The appearance of “good” FMSR in the unitarity integrals follows from a careful treatment of phase space — all possible configurations are counted uniquely — and is crucial in achieving the cut cancellation. Throughout the paper we emphasize various subtle points that have been overlooked in the literature.     

Light cone,Regge contributions and large-angle scattering

The asymptotic behaviour of various three-particle vertices involving reggeons far off-mass-shell, and that of various exclusive reactions is found by using the light-cone algebra of the local and bilocal operators and supposing that the Regge contributions survive in the amplitude in the limit of large momentum transfer. The scaling behaviour (29), (34) is obtained in this way for the exclusive amplitudes. Besides, the asymptotic behaviour of hadronic form factors is obtained. The asymptotic behaviour of Regge-cut contributions and the region of applicability of the scaling behaviour (29) are found for the 2 → 2 amplitude. The relations between the dispersion contributions, Regge contributions, and those of quark diagrams, as well as the role of the signature are discussed. Predictions are given for the energy and the angular distributions for a number of two-particle reactions.     

Off-mass shell dual amplitudes (II)

A new method for continuing dual model amplitudes off the mass shell was recently proposed. This paper explores some of the properties of the resulting amplitudes. It is demonstrated that one-current amplitudes contain fixed poles in the J-plane at positions that are correlated with the asymptotic power behavior of form factors. The two-point function is explicitly calculated and shown to fall asymptotically as a power provided that a certain condition involving the dimension of space-time and a parameter (that is believed to be correlated with the leading Regge intercept) is satisfied. Certain formulas required for future investigation of more complicated amplitudes are also derived.     

The vertex bootstrap conditions in the dual unitarisation scheme

The bootstrap conditions for the triple- and double-Regge vertices are derived utilising the planar unitarity condition for six- and five-point amplitudes. The problem of symmetry breaking is considered within the one-dimensional approximation. It is shown that the vertex and propagator equations together predict the equal spacing rule for Regge trajectory intercepts and a well defined pattern of symmetry breaking for vertices.     

Check of the gluon-reggeization condition in the next-to-leading order: Quark part

The bootstrap condition for gluon production in multi-Regge kinematics is considered in the next-to-leading order. Bootstrap conditions follow fromthe requirement that the Regge formof amplitudes in QCD be consistent with s-channel unitarity and are nonlinear relations between the Reggeized-gluon trajectory and vertex functions. Their fulfillment ensures the Reggeization of gluons—that is, the Regge form of both elastic and inelastic amplitudes. This condition is the only one that has not yet been verified. The demonstration of its fulfillment is the ultimate step in proving the Reggeization of gluons in the next-to-leading logarithmic approximation. In the present article, this is done for the quark part of the bootstrap condition.     

Asymptotics of 4d spin foam models

We study the asymptotic properties of four-simplex amplitudes for various four-dimensional spin foam models. We investigate the semi-classical limit of the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the boundary data. For some classes of geometrical boundary data, the asymptotic formulae are given, in all three cases, by simple functions of the Regge action for the four-simplex geometry.     

Double logarithmic asymptotics and regge singularities of quark amplitudes with flavour exchange

We derive simple equations in terms of definite signature partial waves for the quark scattering and annihilation amplitudes in the double logarithmic approximation of QCD. We apply the method of isolating the softest virtual particle in the graphs which is based on gauge invariance and unitarity. Besides the particle Regge trajectories and corresponding cuts, there are further singularities in the angular momentum plane generated by perturbative double logarithmic contributions and which may be of phenomenological relevance. We also consider amplitudes with external currents and discuss how to include ultraviolet single logarithms and non-perturbative contributions.     

Regge poles in multiparticle scattering amplitudes

A study is made of the high-energy limits of multi-particle scattering amplitudes in a φ³ theory. A limited set of chekerboard diagrams dominate the asymptotic behavior .The diagrams sum to produce Regge poles. If N is the number of particles bound together, the trajectory functions of the leading Regge poles are given by α(N)=? (2N ? 3) +g²8π²m². Comparison is made with the generalized multiperipheral model of Auerbach, Aviv, Sugar and Blankenbecler in which the corresponding trajectories have the form α(N)=a?bN+cN².     

Application of K-matrix CMSR'S to single charged pion photoproduction

The background terms, required in addition to Regge poles with some definite properties, in order to describe the high energy behavior of the t-channel single pion photoproduction amplitudes F_t^(?)(v, t), i = 1, …, 4, are investigated in the framework of a reggeized K-matrix model. To this end continuous moment sum rules (CMSR's) are applied, not to the full amplitudes, but just to the two-particle K-matrix amplitudes assumed to show pure Regge pole asymptotic behavior. The K-matrix amplitudes are defined by taking into account in the Heitler equation the elastic (πN) intermediate state alone. Such a definition corresponds to a weak-cut model with only final-state interactions included. It turns out that strong background contributions are still present on the high energy side of the K-matrix CMSR's. The Regge cut contributions generated within our formalism are found to be too weak to account for the background terms in the full amplitudes. This is presumably to a great extent due to the neglect of the (?N) diffraction-dissociation intermediate state in the definition of the K-matrix amplitudes.     

Leading logarithmic approximation in QCD

The leading logarithmic approximation (LLA) for the scattering amplitudes in QCD is reviewed. The double-logarithmic asymptotics of scattering amplitudes is obtained as a solution to nonlinear evolution equations in the infrared cutoff. The DGLAP equation describes an evolution of parton distributions with increasing parton virtuality. The evolution of the amplitudes with respect to the scale in the longitudinal subspace is given by the BFKL equation. The gluon and quarks in QCD lie on the Regge trajectories calculable in perturbation theory. Mesons and baryons are composite states of Reggeized quarks. Similarly the Pomeron and Odderon are colorless ground states of Reggeized gluons. In the case of multicolor QCD, the Reggeon field theory in LLA is completely integrable. The Reggeon interactions in QCD are derived from a gauge-invariant effective action. In particular, next-to-leading corrections to the BFKL equation in QCD and in supersymmetric gauge models are obtained in this way.     

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.     

Confining potentials and Regge poles

A study is made of scattering models which are described by a set of two coupled Schrödinger equations, containing a confining and a nonconfining potential. The partial wave amplitudes are compared with those of conventional potential scattering. Notably the behavior of Regge poles and Regge trajectories is compared.     

Relativistic generalization of dynamic equation for Jost function

A system of dynamic equations for the simultaneous determination of the scattering amplitude and the relativistic Jost function is derived from the Mandelstam representation and the unitarity condition. The relativistic generalization of the de Alfaro, Regge and Rossetti dynamic equation for the Jost function is given.     

Analytic model of Regge trajectories

A model for a Regge trajectory compatible with the threshold behavior required by unitarity and asymptotics in agreement with analyticity constraints is given in explicit form. The model is confronted in the time-like region with widths and masses of the mesonic resonances and, in the space-like region, the ρ trajectory is compared with predictions derived from the π-N charge-exchange reaction. Breaking of the exchange degeneracy is studied in the model and its effect on both the masses and widths is determined. Received: 23 January 2001 / Accepted: 9 February 2001     

Unified treatment: analyticity,Regge trajectories,Veneziano amplitude,fundamental regions and Moebius transformations

In this paper we present a unified treatment that combines the analyticity properties of the scattering amplitudes, the threshold and asymptotic behaviors, the invariance group of Moebius transformations, the automorphic functions defined over this invariance group, the fundamental region in (Poincaré) geometry, and the generators of the invariance group as they relate to the fundamental region. Using these concepts and techniques, we provide a theoretical basis for Veneziano type amplitudes with the ghost elimination condition built in, related the Regge trajectory functions to the generators of the invariance group, constrained the values of the Regge trajectories to take only inverse integer values at the threshold, used the threshold behavior in the forward direction to deduce the Pomeranchuk trajectory as well as other relations. The enabling tool for this unified treatment came from the multi-sheet conformal mapping techniques that map the physical sheet to a fundamental region which in turn defines a Riemann surface on which a global uniformization variable for the scattering amplitude is calculated via an automorphic function, which in turn can be constructed as a quotient of two automorphic forms of the same dimension. Received: 22 October 2002 / Revised version: 19 February 2003 Published online: 23 May 2003 RID="a" ID="a" e-mail: arc@lucent.com     

Non perturbative calculation of the imaginary part of vacuum polarization

We derive a formula for the unitarity relation of the vacuum polarization which includes arbitrary number of soft photons in the vertex functions intermediate states. This quasi-elastic approximation has a spectral form, the soft photon contribution appearing as a broadened delta function. An interpolating formula for the vertices is given which incorporates the asymptotic behavior found by many authors.     

ππ Scattering Amplitudes that Satisfy Inelastic Unitarity Constraints

We construct amplitudes which are represented by a Mandelstam representation with a finite number of subtractions and that satisfy ππ crossing symmetry and the unitarity constraints Im f_l^I(s) ≧|f_l^I (s)|², l=0, 1, 2,…, for all energies above threshold s > 4, in the three isospin channels I=0, 1, 2. The following types of solutions are derived.

• 1 The amplitudes have a positive double spectral function ϱ(s, t) ≧ 0. The total cross section decreases like σ_T(s) ∼ (log s)^-δ for arbitrary δ ≧ 1, including the limiting case δ=1.
• 2 The amplitudes are dominated by Regge poles, the total cross section can reach a constant asymptotic value, σ_T(s) → const.
• 3 The amplitudes are dominated by Regge cuts, the total cross section can increase logarithmically σ_T(s) → log s.