multi-Regge Theory Research Articles

Triangle anomaly in triple-Regge limits

Reggeized gluon interactions due to a single quark loop are studied in the full triple-regge limit and in closely related helicity-flip helicity-pole limits. Triangle diagram reggeon interactions are generated that include local axial-vector effective vertices. It is shown that the massless quark triangle anomaly is present as a chirality-violating infra-red divergence in the interactions generated by maximally non-planar Feynman diagrams. An asymptotic dispersion relation formalism is developed which provides a systematic counting of anomaly contributions. The asymptotic amplitude is written as a sum over dispersion integrals of triple discontinuities, one set of which is unphysical and can produce chirality transitions. The physical-region anomaly appears in the generalized real parts, determined by multi-regge theory, of the unphysical discontinuities. The amplitudes satisfy a signature conservation rule that implies color parity is not conserved by vertices containing the anomaly. In the scattering of elementary quarks or gluons the signature and color parity of the exchanged reggeon states are such that the anomaly cancels. At lowest-order, it cancels in individual diagrams after the transverse momentum integrations are performed.

ANALYTIC MULTI-REGGE THEORY AND THE POMERON IN QCD II: GAUGE-THEORY ANALYSIS

The high-energy Regge behavior of gauge theories is studied via the formalism of analytic multi-Regge theory. Perturbative results for spontaneously broken theories are first organized into Reggeon diagrams. Unbroken gauge theories are studied via a Reggeon-diagram infrared analysis of symmetry restoration. Massless fermions play a crucial role and the case of QCD involves the supercritical Pomeron as an essential intermediate stage. An introductory review of the buildup of transverse-momentum diagrams and Reggeon diagrams from leading-log calculations in gauge theories is presented first. It is then shown that the results closely reproduce the general structure for multi-Regge amplitudes derived in Part I of the article, allowing the construction of general Reggeon diagrams for spontaneously broken theories. Next it is argued that, with a transverse-momentum cutoff, unbroken gauge theories can be reached through an infrared limiting process which successively decouples fundamental-representation Higgs fields. The first infrared limit studied is the restoration of SU(2) gauge symmetry. The analysis is dominated by the exponentiation of divergences imposed by Reggeon unitarity and the contribution of massless quarks to Reggeon interactions. Massless quarks also produce “triangle anomaly” transverse-momentum divergences which do not exponentiate but instead are absorbed into a Reggeon condensate — which can be viewed as a “generalized winding-number condensate.” The result is a Reggeon spectrum consistent with confinement and chiral-symmetry breaking, but there is no Pomeron. The analysis is valid when the gauge coupling does not grow in the infrared region, i.e. when a sufficient number of massless quarks is present. An analogy is drawn between the confinement produced by the Reggeon condensate and that produced by regularization of the fermion sea, in the presence of the anomaly, in the two-dimensional Schwinger model. When the analysis is extended to the case of QCD with the gauge symmetry restored to SU(2), the Reggeon condensate can be identified with the Pomeron condensate of supercritical Pomeron theory. In this case, the condensate converts an SU(2) singlet Reggeized gluon to a Pomeron Regge pole — which becomes an SU(3) singlet when the full gauge symmetry is restored, The condensate disappears as SU(3) symmetry is recovered, and in general this limit gives the critical Pomeron at a particular value of the transverse cutoff. If the maximal number of fermions consistent with asymptotic freedom is present, no transverse-momentum cutoff is required. For SU (N) gauge theory it is argued that, when the theory contains many fermions, there are N–2 Pomeron Regge poles of alternating signature. This spectrum of Pomeron trajectories is in direct correspondence with the topological properties of transverse flux tubes characterized by the center ZN of the gauge group. The corresponding Reggeon-field-theory solution of s-channel unitarity should include a representation of ZN in the cutting rules. Finally, the implications of the results for the phenomenological study of the Pomeron as well as for the construction of QCD with a small number of flavors are discussed. Also discussed is the attractive possibility that a flavor doublet of color-sextet quarks could both produce the critical Pomeron in QCD and be responsible for electroweak dynamical-symmetry breaking.

ANALYTIC MULTI-REGGE THEORY AND THE POMERON IN QCD I: THE SUPER-CRITICAL POMERON

The formalism of analytic multi-Regge theory is developed as a basis for the study of abstract critical and super-critical pomeron high-energy behavior and for related studies of the Regge behavior of spontaneously broken gauge theories and the pomeron in QCD. Asymptotic domains of analyticity for multiparticle amplitudes are shown to follow from properties of field theory and S-matrix theory. General asymptotic dispersion relations are then derived for such amplitudes in which the spectral components are described by the graphical formalism of hexagraphs. Further consequences are distinct Sommerfeld-Watson representations for each hexagraph spectral component, together with a complete set of angular momentum plane unitarity equations which control the form of all multi-Regge amplitudes. Because of this constraint of “reggeon unitarity” the critical pomeron solution of the reggeon field theory gives the only known “non-trivial” unitary high-energy S-matrix. By exploiting the full structure of multi-Regge amplitudes as the pomeron becomes super-critical, one can study the simultaneous modification of hadrons and the pomeron. The result is a completely consistent description of the super-critical pomeron appearing in hadron scattering. Reggeon unitarity is satisfied in the super-critical phase by the appearance of a massive “gluon” (Reggeized vector particle) coupling pair-wise to the pomeron.

S-channel discontinuities and the two-Reggeon cut

We make a detailed calculation of the $s$-channel discontinuities of the two-Reggeon cut in order to see how the singularities of the Reggeon-particle vertices affect the weight with which an intermediate state contributes to the cutlike part of the amplitude. We confirm the counting of Halliday and Sachrajda for the Mandelstam graph and show the extent to which this result can be applied to more general cut diagrams so that the results of Abramovskii, Gribov, and Kancheli hold. We also consider amplitudes with vertices constructed from the precepts of multi-Regge theory (for instance, from a dual model) in which the counting of terms is different: In particular, the simultaneous discontinuity through both Reggeons does not contribute to the cut. The weight of a given $n$-particle intermediate state in forming the cut depends crucially on the analytic properties of the vertex one chooses.

Signature, factorization and unitarity in multi-Regge theory: The five-point function

We develop the multi-Regge theory of the five-point function from the Sommerfeld-Watson approach and emphasize the use that can be made of t-channel unitarity. We include signature systematically so that signature phase factors come out naturally and discontinuities can be taken in a straightforward way. We give a general method for defining Froissart-Gribov continuations. Using the unitarity relations they satisfy we prove factorization of the individual helicity amplitudes contributing to Regge pole residues. This is sufficient to prove factorization of the full amplitude and its discontinuities in multi-Regge limits as well as the factorization of discontinuities in helicity-pole limits. We define reggeon scattering amplitudes which are continuations of t-channel helicity amplitudes to complex helicity. We show that the t-channel unitarity relation for these amplitudes contains an extra term, besides the usual phase-space integral, which vanishes when the trajectory function goes through integer points. We give a generalized crossing relation which relates s- and t-channel reggeon amplitudes.

Fixed-angle lower bounds for production amplitudes and inclusive experiments

The application of the Cerulus-Martin theorem to production amplitudes is considered and a lower bound of the form e −Cs γ logs 1 2 ⩽ γ ⩽ 1 is obtained in a certain subset of the fixed-angle limits of such amplitudes defined in a previous paper. The value of γ obtained depends on the assumptions made which are related to those of multi-Regge theory. It is also shown that the experimentally observed exponential decrease of inclusive differential cross sections in the fixed-angle, fixed scaled longitudinal momentum limit implies a similar decrease for production amplitudes in fixed- angle limits. As a result the lower bound derived for production amplitudes will also apply to inclusive differential cross sections in such a limit.

Landau singularities in multi-Regge theory and fixed-angel behaviour

A study of the implications of the Landau singularity structure in multi-Regge theory is begun. The incorporation of a physical-region pole in the three-Reggeon description of the six-particle amplitude is considered. Triple-Regge limits are found to involve fixed-angle limits of the four-particle amplitudes appearing in the residue of the pole. It is shown that under certain conditions the presence of the physical-region pole necessitates an « undesirable » set of angular-nmmeutum singularities of the forml1+l2+l3 = const. Exponential decrease of thc four-particle amplitude at fixed angle is then considered and it is shown that some examples of such behaviour including those most favoured by experimental results, do not necessitate these singularities.

The fixed-angle limit in inclusive experiments

The fixed-angle, fixed scaled longitudinal momentum, limit of the inclusive single particle distribution is related to a fixed-angle limit of the six-particle amplitude. A straightforward derivation of the lower bound exp [- c slog s] for the inclusive distribution, is such a limit, can then be given, based on the assumptions of multi-Regge theory. Possible generalisations of the Orear and Krisch fits to the four-particle amplitude are also discussed.

Application of multi-Regge theory to production processes

This paper describes and reviews both kinematic (phase space) and dynamical aspects of production processes within the framework of multi-Regge theory. Those techniques that rely on additional theoretical assumptions are not treated. As an illustrative example, the case of three-body final states is developed in considerable detail. A simplified multi-Regge amplitude is assumed and formulae for the resulting differential cross sections are given. The emphasis of this work is the analysis of experimental data containing potential multi-Regge events.