Regge Terms Research Articles

Parametrization of quark and gluon generalized parton distributions in a dynamical framework

We present a parametrization of the chiral even generalized parton distributions, $H$, $E$, $\stackrel{\texttildelow{}}{H}$, $\stackrel{\texttildelow{}}{E}$, for the quark, antiquark, and gluon, in the perturbative QCD (pQCD)-parton framework. Parametric analytic forms are given as a function of two equivalent sets of variables $x$, $\ensuremath{\xi}$, $t$ (symmetric frame) and $X$, $\ensuremath{\zeta}$, $t$ (asymmetric frame), at an initial scale, ${Q}_{o}^{2}$. In the $X>\ensuremath{\zeta}$ region, a convenient and flexible form is obtained as the product of a Regge term $\ensuremath{\propto}{X}^{\ensuremath{-}\ensuremath{\alpha}+{\ensuremath{\alpha}}^{\ensuremath{'}}t}$, describing the low $X$ behavior, times a spectator model-based functional form depending on various mass parameters; the behavior at $X<\ensuremath{\zeta}$ is determined using the generalized parton distributions symmetry and polynomiality properties. The parameters are constrained using data on the flavor separated nucleon electromagnetic elastic form factors, the axial and pseudoscalar nucleon form factors, and the parton distribution functions from both the deep inelastic unpolarized and polarized nucleon structure functions. For the gluon distributions we use, in particular, constraints provided by recent lattice QCD moments calculations. The parametrization's kinematical range of validity is $0.0001\ensuremath{\le}X\ensuremath{\le}\phantom{\rule{0ex}{0ex}}0.85$, $0.01\ensuremath{\le}\ensuremath{\zeta}\ensuremath{\le}0.85$, $0\ensuremath{\le}\ensuremath{-}t\ensuremath{\le}1\text{ }\text{ }{\mathrm{GeV}}^{2}$, $2\ensuremath{\le}{Q}^{2}\ensuremath{\le}100\text{ }\text{ }{\mathrm{GeV}}^{2}$. With the simultaneous description of the quark, antiquark, and gluon sectors, this parametrization represents a first tool enabling a global QCD analysis of deeply virtual exclusive experiments.

Regge analysis of the ππ scattering amplitude

The theoretical predictions for the subtraction constants lead to a very accurate dispersive representation of the ππ scattering amplitude below 0.8 GeV. The extension of this representation up to the maximum energy of validity of the Roy equations (1.15 GeV) requires a more precise input at high energies. In this paper we determine the trajectories and residues of the leading Regge contributions to the ππ amplitude (Pomeron, f and ρ), using factorization, phenomenological parametrizations of the πN and NN total cross sections at high energy, and a set of sum rules which connect the high and low energy properties of ππ scattering. We find that nonleading Regge terms are necessary in order to achieve a smooth transition from the partial waves to the Regge representation at or below 2 GeV. We obtain thus a Regge representation consistent both with the experimental information at high energies and the Roy equations for the partial waves with ℓ≤4. The uncertainties in our result for the Regge parameters are sizable, but in the solutions of the Roy equations these only manifest themselves above the $K\hspace {0.03cm}\overline {\rule [0.7em]{0.6em}{0em}}\hspace {-0.8em}K$ threshold.

Classical string solutions inAdS5×S5

We discuss some new simple closed bosonic string solutions in ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ that may be of interest in the context of AdS/CFT duality. In the first part of this work we consider solutions with two spins $({S}_{1},{S}_{2})$ in ${\mathrm{AdS}}_{5}$. Starting from the flat-space solutions and using perturbation theory in the curvature of ${\mathrm{AdS}}_{5}$ space, we construct leading terms in the small two-spin solution. We find corrections to the leading Regge term in the classical string energy and uncover a discontinuity in the spectrum for a certain type of solution. We then analyze the connection between small-spin and large-spin limits of string solutions in ${\mathrm{AdS}}_{5}$. We show that the ${S}_{1}={S}_{2}$ solution in ${\mathrm{AdS}}_{5}$ found in earlier papers admits both limits only in the simplest cases of the folded and rigid circular strings. In the second part of the paper, we construct a new class of chiral solutions in ${R}_{t}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ for which embedding coordinates of ${S}^{5}$ satisfy the linear Laplace equations. They generalize the previously studied rigid string solutions. We study in detail a simple nontrivial example.

Status of the high-energy elastic scattering: Total cross-section, real part and dσ/dt

A complete database of experimental results on high-energy elastic scattering including the most recent reports from UA4/2, CDF and E710 is fitted by both the standard and non-standard analytic parametrizations for the elastic pp and $$\bar pp$$ amplitudes. The standard theoretical models of the Pomeron and Regge terms can reproduce the data well, while the new CDF total cross-section can support the existence of some non-standard physics.

A quantitative investigation of the pomeron

A comparative investigation of various Pomeron models is carried out through the study of their predicted values of $ \sigma_{tot}$, B, and $\frac{\sigma_{el}}{\sigma_{tot}}$ in high energy pp and p$\bar{p}$ scattering. Our results strongly support a picture of the Pomeron in which we have both moderate blackening and expansion of the p($\bar{p}$) - p amplitude in impact parameter space as a function of energy in the ISR-SSC domain. In particular, we obtain an excellent reproduction of the data with a hybrid eikonal model which combines the hard Lipatov-like QCD Pomeron with the old fashioned soft Pomeron and Regge terms. Our analysis shows that the additive quark model, at least in the naive form, is not compatible with the data.

Two-component Pomeron and hadron total cross sections and real parts

The predictions from a five-parameter formula obtained from a two-component Pomeron model and fit to hadron-nucleon total cross sections from 2 to 200 GeV/c are in remarkable agreement with new CERN ISR data on total proton-proton cross sections and real parts up to an equivalent ${P}_{\mathrm{lab}}$ of 2000 GeV/c and with total-cross-section data from cosmic rays up to 40000 GeV/c. This oversimplified formula with a Regge term varying as ${s}^{\ensuremath{-}\frac{1}{2}}$ two Pomeron-like terms with slightly increasing and slightly decreasing power behavior, and dependence upon quantum numbers given by simple quark-counting rules is adequate to fit all available data and can be useful for analysis of future data. Predictions for the ratios of real to imaginary parts of ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$, ${K}^{\ifmmode\pm\else\textpm\fi{}}p$, and ${p}^{\ifmmode\pm\else\textpm\fi{}}p$ forward amplitudes are given.

Sum rules and non-regge terms in photon-photon scattering

We discuss whether resonances are dual to the Regge terms in both real and virtual photon-photon scatterings. Various sum rule considerations lead us to suggest that this is not the case.

Tracing back resonances to sets of exponents which differ by one unit. New finite-energy sum rules

Supposing that an amplitude $A$ can be expressed for large enough energies as a sum of contributions of Regge terms, we find new finite-energy sum rules which involve partial sums of these Regge terms. A correspondence is given between the resonances of the amplitude $A$ and the terms which appear in these partial sums.

Crossing relations for infinitely rising trajectories: An asymptotic expression of local duality

Asymptotic crossing relations for the Regge amplitudes are proposed as a model-independent expression of local duality. They allow local duality to be formulated solely in terms of Regge trajectories without any violation of unitarity or neglect of resonance widths. The crossing relations connect the direct- and crossed-channel Regge-pole amplitudes for the elastic scattering of two equal-mass particles as $s\ensuremath{\rightarrow}\ensuremath{\infty}$ in a given domain of the real $\mathrm{st}$ plane. The main purpose of this paper is to determine the type of domain in which such crossing relations will possess nontrivial solutions that are both self-consistent and in keeping with the known properties of the Regge trajectories and residues. An extensive and systematic study of all domains involving large, positive $s$ and values of $t$ in some interval [${t}_{1}, {t}_{2}$] along the positive $t$ axis is presented. We consider both intervals of fixed length and intervals whose length is increasing with $s$. Only one type of interval is found that is satisfactory, namely, intervals consisting of values of $t$ that are increasing in proportion to $s$, i.e., for any given $s$, $t\ensuremath{\in}[\frac{s}{{r}_{1}}, s]$ for some ${r}_{1}>1$. The reasons are the following. It is the only type of interval (1) in which we can provide an existence proof for the solutions to the crossing relations, the proof being valid for asymptotically parallel trajectories; (2) for which the generalization to unequal-mass scattering encounters no obvious inconsistencies; and (3) for which the residues calculated from the crossing relations have an asymptotic form of the type found in the Veneziano model. Furthermore, we can prove the validity of the proposed crossing relations over this type of interval if the modified $s$-channel background integral ${B}_{s}^{\ensuremath{'}}$ satisfies the bound $\mathrm{ln}|{B}_{s}^{\ensuremath{'}}|<\frac{\ensuremath{\surd}2a{s}^{p}}{p}$, where $\mathrm{Re}\ensuremath{\alpha}(s)\ensuremath{\rightarrow}a{s}^{p}$ as $s\ensuremath{\rightarrow}\ensuremath{\infty}$ for $p>0.802$. The analogous proof for any of the other types of domains considered requires the bound $\mathrm{ln}|{B}_{s}^{\ensuremath{'}}|<N\mathrm{ln}s$ for some $N$, which is a much stronger bound. The proof to which we refer assumes only that the contributions of Regge branch cuts to the scattering amplitude are negligible compared to the contributions of Regge poles, ${\ensuremath{\alpha}}_{n}(s)$, for $s$ positive and sufficiently large. In the indicated domain, the crossing relations imply certain homogeneous integral equations that the residues must satisfy. Although a complete solution is not given, we show that as $s\ensuremath{\rightarrow}\ensuremath{\infty}$ the residues behave as $\mathrm{exp}(\ensuremath{-}\ensuremath{\gamma}a{s}^{p})$, where $\ensuremath{\gamma}$ is a logarithmic function of ${r}_{1}$ and $p$. From the expression for $\ensuremath{\gamma}$ we obtain the upper bound $\ensuremath{\gamma}<\mathrm{ln}(3+2\ensuremath{\surd}2)\ensuremath{-}\frac{\ensuremath{\surd}2}{p}$ (for $p=1$, $\ensuremath{\gamma}<0.348$). The constant analogous to $\ensuremath{\gamma}$ in the Veneziano model is 0.38. We add that if the crossing relations proposed here are not valid, then either the background integrals must generate the crossed-channel Regge terms or the background integrals in concert with the direct-channel Regge terms must do so.

Regge extrapolation to the resonance region

The structure of dual amplitudes (beta functions) suggests that in non-exotic channels one should add to the usual Regge terms a second term which drops off as exp (− π Im α( s)) at large s but which is important for s≲5 GeV 2. This “dual correction” multiplies the usual Regge imaginary part by an oscillating complex factor. With inclusion of such extra terms, it is possible to match πp amplitudes from phase-shift analyses significantly better than if the terms are omitted. The partial waves of a modified fit using the leading N, Δ, ϱ, f trajectories with a fixed-pole pomeron show much more of the correct resonant structure than do those of an unmodified fit.

Deep-Inelastic Scattering and Static Properties of the Baryons in the Quark-Gluon Model

The longitudinal structure function [${W}_{L}({q}^{2},\ensuremath{\nu})$] of inelastic neutrino and electron scattering is studied in the deep-inelastic limit in the canonical quark-gluon model. Although ${W}_{L}({q}^{2},\ensuremath{\nu})$ vanishes asymptotically in this model, $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ should scale. Sum rules are derived which relate integrals over the scaling limit of $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ and the well-known structure function ${F}_{2}(\frac{\ensuremath{-}{q}^{2}}{2\ensuremath{\nu}})$ to octet baryon masses, the Gell-Mann---Oakes---Renner parameter (c), and the pion-nucleon sigma term (${\ensuremath{\sigma}}_{\ensuremath{\pi}}$). The sum rules are convergent, since leading Regge terms are to be subtracted off according to a well-known prescription and contain no arbitrary constants if the residues of $\ensuremath{\alpha}=0$ singularities in forward current-hadron scattering are polynomials in ${q}^{2}$. The sum rules are derived using light-cone techniques. It is shown that the parton model and Bjorken-Johnson-Low commutators yield identical results. Similar sum rules are presented for other interactions and scalar "quarks." Estimates of $c$ and ${\ensuremath{\sigma}}_{\ensuremath{\pi}}$ allow numerical evaluation of the sum rules indicating that the integrals over $\ensuremath{\nu}{W}_{L}({q}^{2},\ensuremath{\nu})$ are small. The pattern of chiral-symmetry breaking in the vector-gluon model is discussed. It is shown that the dictum that scaling laws may be abstracted from free-field theory leads to difficulties (in that it generates too trivial a theory) if applied to the chiral-symmetry-breaking structure functions of neutrino scattering. Abstraction from gluon models, however, remains adequate.

Range, light-cone dominance and scaling at low Q2 in electroproduction

The analysis of light cone singularities is used to connect the y · P, or range, dependence of the current correlation function with the Q 2 dependence of the inelastic electroproduction structure functions. We study for what regions of the Q 2, ν plane and for what y · P dependence the leading light-cone singularity dominates contributions from less singular terms with the same y · P dependence. When the leading singularity can be shown to dominate for a particular region of Q 2 and ν, we study whether this implies scaling for νW 2 in that kinematic region. It is shown that a division of the current correlation function into short and long range contributions is fundamentally ambiguous and not related to scaling at low Q 2. Short range terms which are shown to be light-cone dominated for all Q 2 so long as ν → ∞, are found but are shown not to scale at low Q 2 and to be indistinguishable from corrections to long range terms which produce the leading Regge behavior. We show that leading Regge terms may receive contributions far away from the light cone for small virtual photon mass, but that light-cone dominance and scaling are recovered when the photon mass is taken very large.

Polarization and Dip-Bump Structure inπ−pCharge-Exchange Scattering

We show that the new ${\ensuremath{\pi}}^{\ensuremath{-}}p$ charge-exchange polarization and differential cross-section data at large angles can be explained in terms of a simple model which includes a $\ensuremath{\rho}$ Regge pole plus a background term which one may interpret as either a secondary pole or a cut. We show that the rise after the dip is primarily due to the background term and not the spin-flip Regge term recovering from $\ensuremath{\alpha}$ passing through zero near $t=\ensuremath{-}0.6$ ${\mathrm{GeV}}^{2}$. We discuss the effect of this insight on dip systematics.

Pomeron structure from low-energy K +p scattering

Using phase-shift analysis and high-energy Regge terms as input, we evaluate the pomeron amplitude in K +p elastic scattering below 2.5 GeV/ c taking into account pomeron-Regge cut contributions. Our results show linear s-dependence for this amplitude and exponential t-dependence with step-wise increasing slope at each strongly coupled inelastic threshold. Spin conservation is favoured though s-channel helicity conservation cannot be excluded for some phase-shift solutions.

A construction of functions that satisfy crossing symmetry and unitarity

We construct π0π0 amplitudes with arbitrary polynomial increase in the double spectral region that satisfy crossing symmetry and the inelastic-unitarity bounds Imf l(s)−|fl(s)|2⩾0,l=0, 1, 2, ..., for alls>4. Exact elastic unitarity is then obtained by an iteration procedure which was proposed by Mandelstam and discussed in detail by Atkinson. The resulting amplitude which fulfils crossing symmetry, elastic unitarity and the unitarity bounds in the inelastic region is the sum of Regge terms with bounded trajectories and a once-subtracted Mandelstam representation which guarantees the unitarity of the whole amplitude.

A dispersion approach to interference model and local duality principle

The dispersion relations are used to obtain an interpolation formula for the amplitude which would have low-energy resonant behavior and high-energy Regge behavior. The method may be applied to both single- and multi-Regge exchange cases. A sum rule relating the Regge terms and the resonant terms is obtained, and it holds well throughout the low-energy and the intermediate-energy regions.

Sum rules for π scattering in the backward direction

We write finite-energy sum rules (FESR) at fixedu forthe π scattering amplitudesAiu,DIu, ,Iu=1/2, 3/2 (Iu isospin exchanged in theu-channel) and we verify the consistency between the Regge terms, as obtained by high-energy data, and the integrals over the low-energy region. We use then our FESR to obtain a better determination of the Regge parameters for Δδ-trajectory exchange. We find, in particular, a negative sign for the residue nearu=0. Finally our results are employed to discuss the validity of the «mean-value principle», developed by Dolen, Horn and Schmid for the Regge boson exchange, in the case of fermion exchange (u≃0); we find that the generalization of this principle does not seem possible for the amplitudes we studied, at least if one tries to represent the amplitude of a single channel with the fermion Regge term alone.

Regge poles and large-angle scattering

We study the limiting form of a Regge amplitude for large values of both s and | t|. If such a term dominates the amplitude, a limit on the behavior of the trajectory as t → −∞ is obtained. Results obtained by fitting simply parametrized Regge terms to the pp scattering data near 90° are presented.

Analytic Continuation of Divergent Integrals with Respect to a Parameter—The Case of Perturbations to Strong Interactions

A principle is proposed to overcome divergent integrals in various formulas by continuing them analytically with respect to an appropriately chosen parameter. The analyticity is examined on the complex plane of the parameter in the case of the formula for the electromagnetic mass difference, as an example. The contributions from the Regge terms including the Regge cuts are discussed in detail, specifically in connection with the tadpole model.

Generation of Parallel Daughters from Superconvergence

A dynamical model based on supercnvergence equations and Regge asymptotics generates parallel daughter trajectories for positive and negative $t$. The results are found by studying different reactions both by means of sum rules and the partial-wave analysis of Regge terms proposed by Schmid.