Partial-wave Dispersion Relations Research Articles

The energy dependence of Zweig-rule-violating couplings. A dynamical calculation of ϕ → ρπ

It has been argued that the violation of the Zweig rule is strongly dependent on the kinematical region, especially that it should decrease for large timeliket (asymptotic planarity). We have calculated thet-dependence of the vertex ϕρπ with two different methods, the first one using partial-wave dispersion relations and unitarity and the second one based on FESR and duality. The decrease in the timelike region is confirmed by both calculations. In the spacelike region the energy dependence of the Zweig-rule-violating coupling depends on the method of continuation to off-shell values. We only find an energy dependence if the full amplitude πρ → K\(\bar K\) is taken into account.

Partial Wave Dispersion Relation with Inelasticf Cut and Pion-Pion Scattering in the 1 GeV Region

Partial Wave Dispersion Relation with Inelasticf Cut and Pion-Pion Scattering in the 1 GeV Region

Inelasticity effects inππ production inγγ collisions

The inelasticity effects in the production of pion pairs in the processγγ→ππ for real photons are investigated using the partial wave dispersion relations. The total cross sections for different photon helicities are calculated. It is observed that this process is dominated by theππ final state interaction. A prediction for S* (997) →γγ decay width is also made.

Proton-deuteron scattering near threshold: An on-shell two-body approach

A simple on-shell partial wave dispersion relation model is applied to p−d and n−d scattering near threshold. The results are compared with those obtained using a more sophisticated, but more complex, off-shell model, and it is shown that the Coulomb corrections to the effective range parameters are primarily determined by the properties of the long range nucleon exchange mechanism.

Dynamical calculation of the decay ψ′ → ψ + 2 π

We calculate the ψ′ → ψ + 2 π decay width on the basis of partial wave dispersion relations in the ππ channel. This is basically a coupled channel problem involving ππ, D D , etc. intermediate states. The calculation acquires anomalous thresholds due to the fact that the vertex ψ′ ψ (2π) is internally unstable. It is found that anomalous singularity contributions provide by far the most dominant contribution, which explains the rather large ψ′ → ψ + 2 π decay mode. Very good agreement with the experimental width and mass distribution is achieved.

Influence of anomalous thresholds in electron-positron annihilation into hadrons: The e +e − → ϱππ cross section

We calculate the cross section for e + e − → ϱππ on the basis of partial-wave dispersion relations in the ππ channel taking proper account of anomalous singularity contributions. The appearance of anomalous thresholds is due to the fact that the vertex γ V ϱ( ππ) becomes internally unstable as the virtual photon mass is increased. Reasonable agreement with existing data is found. The anomalous singularity contributions provide by far the dominant part of the cross section which is a warning to using naive vector dominance extrapolations in estimating the electron-positron cross section.

Regge Models and Dispersion Relations for Partial Waves

AbstractThe asymptotic behaviour of partial wave amplitudes is calculated supposing various Regge models for the total scattering amplitude A(s, t, u). The high energy partial wave behaviour obtained is combined with the validity of partial wave dispersion relations. It is shown that consistency of these assumptions can only be achieved by demanding.1) a definite asymptotic behaviour of the discontinuity of the left hand cut of partial wave amplitudes.2) the validity of partial wave sum rules of similar kind as the well‐known finite energy sum rules for the total amplitude.All steps of the derivation shall first be demonstrated for elastic scattering of identical scalar particles. Then within the helicity formalism the results are generalized for particles with arbitrary spin and different masses. Finally the question is studied whether the sum rules can be employed to determine unknown CDD‐pole parameters in an N/D approach for the I = J = 1/2 state in πN scattering. It is shown that the sum rules of highest order are able to do that.

Partial-wave analysis of K +p elastic scattering

We have analyzed K +p cross-section and polarization data in the momentum range 0.78–2.53 GeV/ c. Single energy fits were made using an ACE parametrization which had no cutoff in J, was analytic and had Regge-asymptotic behavior in the cut cos θ plane, and contained explicit contributions from P, ϱ, A 2, Λ, Σ, and low-mass di-pion exchange. Direct channel analyticity constraints were imposed by a partial-wave dispersion relation interpolation procedure, fitting expansions in regulated-norm optimal bases to each partial wave which entered as a search parameter in the single energy fits. Output from these two programs was then used for a simultaneous fit by a linearized least squares program which provided input for another iterative fitting cycle. We find no strong evidence for the existence of exotic resonances. The method of analytic smoothing, in comparison with “shortest path” methods, is found to allow more significant threshold structure to appear in the amplitudes.

Partial-wave dispersion relations for the nonrelativistic five-nucleon system

Partial-wave dispersion relations for the nonrelativistic five-nucleon system

Dispersion calculation of the transition form factorFπωγ(t)with cut contributions

In this paper we study the $t$ dependence of the $\ensuremath{\gamma}\ensuremath{\pi}\ensuremath{\omega}$ vertex functions on the basis of partial-wave dispersion relations and unitarity. The right-hand cut is approximated by the $2\ensuremath{\pi}$ contribution and the left-hand cut by the nearest $s$- and $u$-channel poles. The electromagnetic $\ensuremath{\pi}\ensuremath{\omega}$ transition form factor is calculated as a function of $t$ in the timelike and spacelike region and is compared with predictions of the $\ensuremath{\rho}$-dominance model and with recent experimental data for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\omega}{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ near threshold. The influence of the left-hand cut and the finite width of the $\ensuremath{\rho}$ resonance is explicitly shown.

New dispersion relations and their application to partial-wave amplitudes

After reviewing the commonly used dispersion relations, a systematic investigation of more generalized dispersion relations on parametrized curves in the Mandelstam plane fors-u crossing-symmetric amplitudes is made with the aim of obtaining dispersion relations which receive contributions from all three channels, however, in such a way that knowledge of the absorptive parts is only required in regions well inside the various Lehmann ellipses. In addition we require that the dispersion relations receive no contributions from kinematic singularities arising from the parametrization and that they allow partial-wave projections to be made in a relatively simple manner. It is found that dispersion relations on hyperbolic curves in the Mandelstam plane are a natural solution of the problem. The dispersion relations are written in a remarkably simple form similar to the usual fixed-t dispersion relation but with an additionalt-channel contribution. As an interesting application, we derive generalized partial-wave dispersion relations for elastic pion-nucleon scattering, where the left-hand cut contribution is explicitly given by convergent partial-wave series in the crossed channels.

Phenomenology of low-energy ππ scattering

Exact representations of the low-energy ππ amplitudes by (for the S-waves) generalized effective range formulas and by (for the P-wave) a resonance plus background expression are derived from inverse-amplitude partial-wave dispersion relations, and applied to the phenomenological analysis of a typical set of phase shifts deduced from peripheral pion production. With the aid of rigorous crossing conditions it is found that the simplest satisfactory account of the S-waves is provided by a current algebra type solution with (Adler) zeros and small scattering lengths. Solutions without zeros ( S-wave dominant) are completely ruled out. Inelasticity is important in describing the rise of the isoscalar phase shift above the σ resonance, but can be neglected below 1 GeV in other channels. The P-wave is ϱ-dominated, but its width is sensitive to background terms. The phenomenological importance of rigorous results is stressed.

Dispersion-relation analysis of K +p scattering below 2 GeV/ c

We have obtained K +p scattering amplitudes that fit a large body of data below 2 GeV/ c, satisfy phenomenological partial-wave backward dispersion relations, and are consistent with forward dispersion relations for the spin-non-flip amplitude, and forward dispersion-relation sum rules for the K +p P-wave scattering lengths. This has been achieved by firstly making an energy-dependent phase-shift analysis of 1 660 pieces of K +p data below 2 GeV/ c, using parameterised partial-wave dispersion relations, with the additional constraints of forward dispersion relations and P-wave scattering lengths obtained from forward dispersion-relation sum rules, and then checking the resulting solutions for consistency with backward K +p dispersion relations.

Theory of low-energy ππ scattering

A four-parameter family of solutions to the ππ partial-wave dispersion relations is obtained using the inverse amplitude method. Choosing the parameters consistent with phenomenological values and so as to maximize agreement with rigorous sum rules and inequalities leads to a firm prediction of the phase shifts for S and P wave scattering and for the amplitudes below threshold. Neglecting inelasticity, the I = 0 S-wave resonates near s = 540 MeV , the I = 2 S-wave falls to about −12° near 1 GeV, and the P-wave is ϱ-dominated. The S-waves have subthreshold zeroes consistent with the Adler condition. Inelastic effects are estimated and found to be small below the rho meson mass.

The Schur-Pick-Nevanlinna interpolation theory - an alternative to the N/ D equations

It is pointed out that the Schur-Pick-Nevanlinna interpolation theory is a natural framework for the study of partial-wave dispersion relations. It is an alternative to the usual treatment by N/ D equations. In this paper some simple cases are studied in detail. More general problems are briefly touched upon.

Remarks on the Regge-Like Behaviour of the One-Boson-Exchange Amplitude in Low-Energy Nucleon-Nucleon Scattering

For the nucleon-nucleon scattering amplitudes at low energies, we study Regge-like behaviour using the K-matrix unitarization method. Here the Regge-like behaviour is phenomenologically represented by the energy dependence of the coupling constant and mass of the exchanged boson in the one-boson-exchange (OBE) amplitudes. This behaviour is shown to be likely for the vector-meson-exchange part. It is consistent with the results of an earlier analysis in terms of the partial wave dispersion relation. For the scalar meson, however, no strong evidence is found for such a behaviour and its behaviour is not inconsistent with cut or fixed pole.

Consistency between the low-energy pion-pion interaction and pion-nucleon scattering

The ππ- N N helicity amplitudes f + 0 and f ± 1 are calculated assuming certain ππ phases δ 0 0 and δ 1 1 and taking careful account of errors. These values are then compared with πN data via partial wave and backward dispersion relations. Excellent agreement is obtained with a conventional ϱ-form for δ 1 1 but in the case of δ 0 0 we require the low-energy behaviour of the Morgan-Shaw favoured phase BD1 to be somewhat reduced to give a scattering length in the range −0.15 ⪅ a 0 0 ⪅ 0.10.

Partial Wave Amplitudes and Resonance Poles

This book is concerned with some of the mathematical problems associated with the partial wave dispersion relations of two body elastic scattering amplitudes. There are two separate themes: one concerns the uniqueness of the solutions when the left hand cut is specified.

Analyticity as a test for the ambiguities of ππ partial waves

In this note the consistency with analyticity properties of a recent Chew-Low extrapolation of single-pion production cross sections on hydrogen is explored with the help of phase representations and once subtracted partial wave dispersion relations. An inelastic, “down” solution for the I = 0 S-wave is favoured by this analysis at least in the range of c.m. energies from 700 to 1000 MeV.

K +p dispersion relations II. Epsilon exchange

The role of epsilon exchange in KN scattering is discussed in relation to a recent K +p phase shift analysis based on partial-wave dispersion relations. The effective ϵ coupling constant is calculated and found to be positive, and to lie in the range 2 ⪅ ƒ ϵ K K ƒ ϵ N N /4π ⪅ 5 .