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.     

What can we learn about the Regge plane from pp cross sections?

We have analysed pp total and different cross sections from s = 10 (GeV)² upwards in terms of singularities in the Regge plane. The data can be fitted by poles alone but a considerably more continuous discontinuity structure is possible. An interpretation of the pole residues in terms of expected Regge poles runs into difficulties at some values of t.     

The structure of t-channel exchanges in KN scattering

Recent phase-shift solutions are used to evaluate KN FESR integrals in order to examine zeros and phases of the t-channel exchange amplitudes in the most model-independent way. The results suggest that the line reversal breaking in KN CEX observed for pL ? 5.5 GeV/c is due to a large EXD breaking component in the A₂ exchange flip amplitude. The ? exchange flip amplitude is well described by a Regge pole amplitude with NWSZ at ?t ≈ 0.5 (GeV/c)². The imaginary parts of ? and ω exchange non-flip amplitudes are both peripheral.     

Kaon-nucleon finite energy sum rules revisited

Kaon-nucleon finite energy and continuous moment sum rule integrals are evaluated using recent phase-shift solutions to extract the t-channel exchange amplitudes and the results are compared with Regge pole models. The exchange degeneracy breaking previously observed in the ? and A₂ amplitudes is confirmed, and it is suggested that this may be due to an exotic P₀₁ KN resonance. Regge trajectories are extracted from the integrals and are in good agreement with those found from direct fits to high-energy data.     

Complex Regge poles and high energy scattering data

For the high energy data on σ_tot^pp the proton-proton total cross section, and on ?, the ratio of real to imaginary part of the forward elastic amplitude, including the recent ISR results, an excellent fit is obtained with P, P', ω Regge poles plus a pair of complex conjugate Regge poles. The complex poles describe the possible oscillatory behavior of the quantities σ_tot^pp and ?. The physical meaning of these complex poles and extrapolation to future measurements of the above quantities are discussed.     

Regge theory and QCD in large-angle scattering

We argue that exclusive processes whose cross-sections are determined by Regge pole trajectory exchanges, α(t), at small momentum transfers (t), are controlled by these same exchanges at very large |t| too. The analysis is based on a Reggeized version of the constituent interchange model (CIM), which predicts that α(t)→?1 ast→?∞, and that the asymptotic form of the Regge residues is fixed by the dimensional counting rules and CIM constraints. Between the two regimes of soft and hard scattering there is an intermediate region where Regge cuts are important, but these higher-twist effects die away as |t| is increased. We show that this hypothesis is consistent with high energy nucleon-nucleon and pion-nucleon scattering at all angles.     

Saddles and trajectories,and poles and surface waves in heavy-ion elastic scattering

The terms I_m^±(θ) of the Poisson summation formula are used to describe and classify the oscillatory structures of the heavy-ion elastic cross sections coming from a strong-absorption model. The method of stationary phase, which relates these terms to branches of a deflection function, is shown to be inadequate and is replaced by the saddle-point method in the plane of complex angular momenta. The positions of the saddle points are found to be fairly independent of the parametrisation and define certain “active” regions which contribute to the scattering amplitude. The complex saddle points may be thought of as corresponding to complex trajectories in the nuclear potential. If the strong-absorption parametrisation possesses poles many saddle-point contributions can be simply expressed as the residues of the poles nearest the real axis. One of these leading parametric poles has an energy dependence similar to that of the dominant Regge pole generated by a complex optical potential. Poles lying above the real axis give rise to terms which may be thought of as surface waves, i.e. they are damped as they progress around the nuclear surface. These terms possess classically equivalent paths in the sense that they correspond to deflections less than π. Terms coming from poles below the real axis have no classical equivalent, increase as they progress and can, then, only be thought of as being diffractive. Some useful formulae for the cross section are given and the limiting case of a sharp cut-off is discussed.     

Finite energy sum rules for π-N backward scattering

Finite energy sum rules for the A and S,π-N scattering amplitudes are saturated in the region ?0.4 GeV²< u <0 with a number of s andt channel resonances. It turns out that one needs a rather large π—π s wave contribution, to get agreement with the high energy Regge pole analysis, which explains the dip structure in π⁺p scattering atu ₀=?0.15 GeV² by a zero in theN _α contribution. The coupling of theN _α trajectory seems to be much stronger than the coupling of theN _γ andΔ _δ trajectories.     

Analysis of the πN invariant amplitudes in terms of algebraic functions and a new determination of the πN low-energy parameters

We have analysed the energy dependence of the πN invariant amplitudes in terms of algebraic functions which automatically fulfill fixed-t dispersion relations. The amplitudes were taken from several phase shift solutions. The analysis was repeated for nine t-values between 0 ? t ? ?0.3 GeV². By evaluating the resulting analytic amplitudes at threshold and at ν = 0 we recalculate the S, P, D scattering lengths and the current algebra σ term. We also discuss the trajectories of the zeros and poles in the complex energy plane.     

Real parts of scattering amplitudes and Regge phenomenology

High energy scattering amplitudes are computed using the Harari model for the imaginary parts and fixed t dispersion relations for the real parts. This procedure induces a corrective term to the real part of the Regge amplitudes whose effects are investigated in π-nucleon and K-nucleon charge exchange reactions.     

Impact parameter analysis of KN and πN scattering in the intermediate energy region

We have evaluated the s-channel-helicity partial-wave amplitudes for $\text{K}\text{N and πN}$ scattering as functions of the impact parameter using partial-wave data in the energy range P_L≈1.0–2.0 GeV/c. We find that the $\text{K}\text{N}$ background and resonance amplitudes exhibit features consistent with the dual absorptive picture for pomeron and f + ω and A₂ + ? exchanges. Comparison of the πN low-energy amplitude with the partial-wave decomposition of a quantitative Regge model gives evidence for local duality between the s-channel resonance and t- and u-channel Regge exchanges.     

High energy pion photoproduction amplitudes from fixed-t dispersion relations and duality

Contributions to the high energy imaginary parts of the charged and neutral pion photoproduction amplitudes from degenerate ? and A₂ exchanges and ω and B exchanges are parametrized similarly to the dual absorptive model. These contributions together with the imaginary parts of the amplitudes found at low energies from partial-wave analyses are then used to evaluate the high energy real parts of the photoproduction amplitudes from fixed-t dispersion relations. A fit is made to data on pion photoproduction at 3.4 and 16 GeV incident photon energies, for momentum transfers up to ?1(GeV/c)². It is shown that much of the data is reproduced both qualitatively and quantitatively by our model. The high energy imaginary parts of the photoproduction amplitudes given by the fit are shown to be in accord with their average low energy behaviour.     

The s-channel dual absorptive model as a consequence also of t-channel absorption

The dual absorptive model applied to t-channel predicts that Regge residues above threshold are I_n-functions which for t ? 0, for the leading trajectory, are simply J_n(R_o√−t). Thus s-channel absorptive models can be expressed entirely in terms of t-channel quantities. Implications to complex Regge poles is discussed.     

On the behavior of the πN partial wave amplitudes ats=0

The behavior of theπN partial wave amplitudes in the limitss→+0 ands→? 0 is related to backward scattering in thes- andt-channel, respectively. Assuming Mandelstam analyticity we prove with the aid of the Phragmen-Lindelöf theorem that the amplitudes for high energy backward scattering inπN→πN andππ→N¯N are equal and therefore dominated by the same exchange mechanism, namely Reggeized Fermion exchange. The dominating Regge trajectory is the Δ_δ-trajectory, and it is shown that theπN partial wave amplitudes diverge fors→±0 as \(s^{ - \alpha \Delta _\delta (0) - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} \) . — Reduced amplitudes are defined which are regular ats=0. — Using recent results ofπN backward scattering real and imaginary part of thes-wave amplitudef ₀₊ ^(?) are calculated in the interval 0≦s≦7.     

Phenomenological characteristics of absorption in charge- and hypercharge-exchange reactions

We try to explain in a unique framework charge- and hypercharge-exchange reactions above 3 GeV/c. Our approach is based on exchange degenerate and SU(3) symmetric Regge poles to which absorption corrections are applied.In the absence of a firm theoretical basis for these corrections we adopt the approach of fixing the Regge poles through theoretical and phenomenological considerations; we then determine what kind of absorption we should apply. At intermediate energy (around 5 GeV/c our results indicate complete absorption of the s-wave amplitude. Of the two terms 1 and e^?iπα of the Reggesignature factor, the latter should be more strongly absorbed. This effect may explain especially the breaking of line-reversal symmetry for hypercharge-exchange reactions, π^?p→π⁰n polarization data, and the difference of peripherality of ??ω and A₂?f₀ contributions.In addition, K-exchange in $\text{p}\text{p}\text{→}\text{Λ}\text{Λ}$ has been studied.     

Model independent analysis of the structure in pp scattering

An empirical study of the structure at $\text{t≈?1.1}\text{in}\text{d}\text{σ}\text{d}\text{t}$ (pp → pp), in terms of two exponential amplitudes plus interference, gives the following results. The steeper exponential shows standard Regge shrinkage, corresponding to a trajectory α ≈1+0.3 t. The other exponential slope stays constant; the overall energy dependence of this amplitude is approximately s^?1 for 10–30 GeV, leveling off possibly to a constant at ISR. The relative phase is about 2.1 rad at the lower momenta, becoming 2.5 at ISR. These results suggest a simple physical picture, in which a relatively unstructured shrinking Regge amplitude interferes with an even-signature non-Regge background. This interpretation predicts that the interference minimum in $\text{d}\text{σ}\text{d}\text{t}$ is deepest at an intermediate momentum near 175 GeV where measurements will soon be made at NAL.     

Elementary scalar mesons and Regge poles

We show, for the annihilation amplitude of π⁺π^? → two vector mesons, that parity conservation and rotational symmetry demand the existence of a triplet of Regge singularities crossing at t = 0. In the usual scalar-vector field theory, calculations of the leading terms of this amplitude up to the eight order show that these singularities are moving Regge poles. Contrary to common belief, the elementary scalar meson lies on one of them.     

Factorization and the couplings of the vacuum trajectory

The decoupling theorems associated with an isolated factorizable pomeron pole of unit intercept are re-examined. It is found that the coupling of three such poles, Γ(t, t, 0), need not vanish, precisely at the point t = 0. This is demonstrated by summing only over states in the appropriate unitarity sum, and sum rule, which are consistent with the M², s/M² → ∞ limit. The triple-Regge region then makes a constant contribution to σ_total, insteadsb of the ln lns result obtained if the isolated pole is assumed to couple also to states such that s/M² = constant. The physical implications regarding factorization and the pole-cut relationship are discussed. The relationship between higher order optical theorems (Mueller discontinuities) and particular terms in the unitarity sum for the two → two absorptive part A₂₂ is exploited. Consistent contributions to the triple-Regge region contribute constant vertex corrections to pure pole behaviour in A₂₂. There is no cut contribution and the magnitude of the vertex corrections reflects the relative amount of diffractive production. The analysis is extended to multiple fireball production where pure multipole structures emerge. The series naturally terminates if the diffractive component is sufficiently small. The implications for the behaviour of the total cross section at machine energies are discussed.