Partial-wave Dispersion Relations Research Articles

Regge-Like Behaviour of Necleon-Nucleon Scattering Amplitudes at Low Energy

The possibility that the nucleon-nucleon scattering amplitudes have Regge-like behaviour at low energy is investigated phenomenologically by assuming energy dependent coupling constants and exchanged meson masses, which give the damping and shrinkage effects on scattering amplitudes, respectively. These effects may be compared with those of Reggeized amplitudes. Analysis is made in terms of the partial wave dispersion relation utilizing the Kantor amplitude. The results show that both damping and shrinkage effects act to improve the fit to the experimental results. A discussion is made about a simple model having infinite one-particle-exchange structure whether or not such effects are appreciable at low energy.

K +p dispersion relations. I. Scattering below 1.3 GeV/c

An investigation of K +p scattering has been made using the techniques of dispersion theory. A phase-shift analysis of data below 1.3 GeV/ c was performed using parameterized partial-wave dispersion relations. A unique physical solution was found, but this could be realised by a number of different sets of exchange forces. By examining the self consistency of the solutions with backward-angle K +p dispersion relations, preferred solutions were obtained. Some brief remarks on the dynamics of low-energy K +p scattering implied by these solutions are made.

Generalized Partial-Wave Dispersion Relation and theNDMethod

We show that a meaningful dispersion relation and associated $\frac{N}{D}$ integral equations can be formulated for any partial-wave amplitude, even though the ordinary partial-wave dispersion relation with a finite number of subtractions is violated by models like that of Veneziano. It is also shown that the arbitrary cutoff of the $\frac{N}{D}$ method, which is often introduced to represent the high-energy behavior of the input "potential," can be eliminated in some models (including that of Veneziano) in terms of Regge parameters.

Theoretical investigation of the reaction γγ → ππ and the ϵ → γγ and f → γγ decay widths

The γγ → ππ cross section and the ϵ → γγ decays width are calculated on the basis of partial-wave dispersion relations. The left hand cut integral is approximated by vector meson exchange diagrams. On the right hand cut elastic unitarity is taken into account which establishes a close correspondence between the cross section and the ππ phase shifts. The I = 0, J = 2 phase shift is parametrized in terms of the f(1260) and for the corresponding J = 0 phase shift a “down” and an “up” solution (ϵ Breit-Wigner) is considered. The cross section is found to depend sensitively on the form of the ππ final state interaction. The obtained ϵ → γγ and f → γγ couplings are compared with other estimates.

Pion-nucleon phase shifts below 270 MeV

Pion-nucleon phase shifts below 270 MeV are derived from fixed t-dispersion relations using as input the π +p total cross sections measured by Bugg et al. below 290 MeV and CERN phase shifts above that energy. No particular high-energy model is needed. It is pointed out that the predicted d-wave phase shifts — in contradiction to those belonging to earlier phase-shift solutions — are in good agreement with partial-wave dispersion relations and recent information on the helicity amplitudes describing the process ππ → N N . Furthermore our prediction for the P 13 wave has very recently got strong support from the work by Bugg et al. on pion-nucleon charge-exchange scattering.

Threshold Zeros and theNDMethod in Yukawa-Potential Scattering

The various methods which have been used to enforce the existence of the threshold zeros in the solutions of the partial-wave dispersion relations are compared. It is found that these methods are very closely related and are in fact equivalent under the proper conditions. The above are then studied in detail for the case of the potential scattering of equal-mass particles under the influence of a Yukawa interaction.

Unitarity Effects in the Veneziano Model

It is assumed that the Veneziano model in some ways represents a good approximation to the scattering amplitude and as such already contains the effects of numerous coupled, inelastic channels. This hypothesis is used to calculate the absorption parameters for pion-pion scattering below 2 GeV using a partial-wave dispersion relation, where some simple forms for the phase shifts are assumed. The results are compared with other analyses and extended to the case of pion-kaon scattering. Some implications of these results are discussed, and the problem of the presence or absence of the Pomeranchukon is outlined.

Information on low-energy ππ interaction from elastic pion-nucleon phase shifts

The well-known methods of partial-wave dispersion relations and backward amplitude dispersion relations for analysing πN phase shifts with a view to obtain the low-energy ππ interaction have been modified by means of recent theories for performing analytic continuation of experimental data in order to obtain a unified model independent treatment. Good agreement between the two methods can be obtained only after a most important correction term has been taken into account, a fact which was overlooked in all previous applications of backward dispersion relations. First results of application of the FESR method of ref. [15] for calculating a partial-wave amplitude in an extended region on the left-hand cut are presented. Results for the T = J = 1 ππ → N N interaction are in accordance with results of Höhler et al. [12]. Results for the T = J = 0 case are against a phase shift that resonates clearly below 900 MeV and in favour of the up-down or down-down solution of ref. [13]. Some information of ππ d-waves is given.

Dispersion relation approach to the nuclear structure calculation

Abstract Partial wave dispersion relations are used as dynamical equations describing nucleon-nucleon scattering. For each state, from the empirical values of the phase shifts for different relative energies of the two nucleons, the first term of the Born series is obtained. In this connection, approximate methods of solving the inverse of a dispersion relation are discussed, and the results are compared with those obtained from the Schrodinger equation. Off-diagonal elements of the Born term for energy non-conserving processes are found by extrapolating the diagonal elements. The matrix obtained in this way enables one to calculate the reaction matrix and also the harmonic oscillator matrix elements directly from the phase shifts. Numerical results for both types of calculation are presented and compared with recent works of Elliott and collaborators.

Simple Model for Low-Energy Pion-Pion Scattering

A simple model for low-energy pion-pion scattering is given, which expresses the $S$-wave phase shifts in terms of the $\ensuremath{\rho}$-meson parameters. A vital assumption is that the isospin amplitudes are as free as possible from singularities---that is, that the threshold branch points are weak, and can be treated by successive approximation. The present version of the model obeys elastic unitarity above threshold. If a parameter is chosen in a range such that the amplitudes are crossing-symmetric below threshold to a high degree of accuracy, and such that the ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ $S$-wave amplitude obeys a set of constraints proven from axiomatic field theory, then the numerical predictions agree reasonably with experiment. A method of improving the model using partial-wave dispersion relations is indicated. Comparison with other, more complicated, models suggests that these often contain many superfluous features which obscure the essential origins of their predictions. It is concluded, for example, that there is no real difference between duality and interference models, as far as prediction of $S$-wave $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths are concerned.

Solutions of Partial-Wave Dispersion Relations. II

The study of the partial-wave dispersion relation (PWDR) is continued, being extended to the relativistic problem in the presence of inelasticity. Less restrictive sufficient conditions on the left-hand-cut discontinuity $\ensuremath{\Delta}T$ are obtained to guarantee a solution of the PWDR problem when $\ensuremath{\Delta}T$ has both positive and negative parts. The effect of Castillejo-Dalitz-Dyson poles on solutions is also discussed.

Detailed dynamics of narrow resonance models for ππ scattering

We discuss the dynamical behavior of narrow resonance models for ππ scattering in the framework of what we call atonous duality, by which we mean that the pole locations and residues in a particular channel completely specify the narrow resonance amplitude as a Mittag-Leffler expansion containing no entire functions. We operate under the following assumptions: (i) Crossing symmetry, isospin conservation, and Bose statistics hold. (ii) The only singularities of the amplitude are simple poles at locations defined by a + bξ(≡ α( ξ)) = 0, 1, 2,… where ξ is a Mandelstam variable. (iii) There are no I = 2 poles. (iv) The pole at s = ξ has a residue which is a polynomial in t of order ⩽α(ξ). (v) Atonous duality holds. (vi) All partial widths are positive. Under these assumptions we find that the I = 1 amplitude A 1 t ( ν, t) has Regge behavior for ν → ∞ at fixed t, while A 2 t ( ν, t) has pure fixed pole behavior. There are no Regge cuts. Furthermore, we find it impossible to formulate partial wave dispersion relations for these models. As an example of such a model we discuss the one term Veneziano ansatz for ππ scattering.

Construction of theSMatrix from Its Left-Hand-Cut Discontinuity, When the Latter is Asymptotically Unbounded

A new procedure is proposed for constructing the partial-wave $S$ matrix, given its left-hand-cut discontinuity $\ensuremath{\lambda}(m)$ ($m=\ensuremath{-}2\mathrm{ik}$, where $k$ is the c.m. momentum). The advantage over the $\frac{N}{D}$ method is that this procedure works even when $\ensuremath{\lambda}(m)$ is unbounded as $m\ensuremath{\rightarrow}\ensuremath{\infty}$ (provided that it oscillates in a suitable way); in fact, the new technique works even when $\ensuremath{\lambda}(m)$ is not bounded by any finite power of $m$, so that not only does the usual $\frac{N}{D}$ decomposition break down, but there exists no partial-wave dispersion relation with a finite number of subtractions. An equivalent potential is introduced that provides considerable insight into the nature of elementary-particle, bound-state, ghost, and Castillejo-Dalitz-Dyson poles. The method is generalized to include inelastic effects.

Solution of Nonrelativistic Partial-Wave Dispersion Relations

The partial-wave dispersion-relation (PWDR) problem is studied in the nonrelativistic elastic case. In particular, a set of conditions on the left-hand-cut discontinuity is obtained which is sufficient to guarantee that the PWDR problem has a solution. The nature of the solutions so obtained and the possible extensions are discussed.

The πN partial waves at low energies and in the unphysical region

The real parts of theπN partial wave amplitudesfl±(−) have been calculated from fixed-t dispersion relations, using phase shifts up to 2 GeV and the reggeizedp-exchange model at higher energies. The comparison with Refl±(−) as calculated directly from phase shifts shows to what extent the input is consistent with the fixed-t relation. The result is also of interest for the test of simple models, in particular for the comparison with predictions from current algebra. — A new evaluation of the partial wave dispersion relation forf0+(−) is presented. Combined with the first method it shows that the determination of thep-coupling parameter is based on an assumption which is difficult to justify.

Ladder approximation in partial-wave dispersion relations

The left-hand cut of the 3/2, 3/2\(\pi - \mathcal{N}\) partial-wave amplitude is calculated by iterating the Bethe-Salpeter equation with nucleon exchange including a smooth cut-off. The partial-wave dispersion relation is approximately satisfied, if the cut-off is chosen properly. The magnetic-dipole excitation of the isobar as well as the transition form factor are determined through a dispersion sum rule retaining only a class of ladder diagrams.

Threshold Behavior of Partial-Wave Dispersion Relation and Short-Range Force

In a paper on the existence of a ghost-free solution of the unsubtracted partial-wave despersion relation, Frye and Warnock conjectured that the positive-definite value at threshold of the unitarity integral on the physical region must be canceled by the short-range force (i.e., multiparticle-exchange contribution) arising from the first and second double spectral functions in order to guarantee the required $p$ wave and higher threshold behavior. We examined this conjecture explicitly in an unsubtracted dispersion relation for the $I=l=1$ pion-pion scattering amplitude. With certain choices of the $\ensuremath{\rho}$ Regge parameters, we show in our approximation scheme that this conjecture is nearly satisfied, and we can conclude that the so-called subtraction constant or threshold factor which is usually arbitrarily introduced to correct the threshold behavior could be expressed in terms of known physical quantities. However, with some other choice of $\ensuremath{\rho}$ Regge parameters, our numerical result shows that this conjecture is not satisfied by our one-$\ensuremath{\rho}$-Regge-pole approximation and indicates a need for additional Regge poles or Castillejo-Dalitz-Dyson poles.

On semi-phenomenological calculations with partial-wave dispersion relations

The «variational» method of solving partial-wave dispersion relations suggested by Donnachie and Hamilton is discussed and shown to be equivalent to fitting the left-hand cut of a unitary trial function to that obtained from crossing over the low-energy physical region. A repetition of their numerical calculation for theN33* gives different results which are in close accord with those obtained by theN/D method; a two-parameter trial function is completely determined, but gives a solution with no resonance. A three-parameter function gives a solution with a one-parameter ambiguity; when the resonance position is given, the width and shape, which correspond to the effective position and strength of the nearby left-hand cut, are determined in agreement with experiment. The solutions are discussed and related to theN/D method.

Current algebra and π-π scattering lengths. A dispersion relation approach

The usual simple linear expansion of the scattering amplitude ins, t andu is used to continue on to the mass shell but not to physical values ofs. The continuation to physicals. values is made using a linear combination of the partial-wave dispersion relations on the first and second Riemann sheets, into which any desired effects of the unitarity cut may easily be inserted. A complicated spectrum ofT=0 andT=2 scattering lengths is obtained. Many of the qualitative features of the solution are in agreement with other calculations in which some account has been taken of the unitarity cut.

Evidence from πp phase shift analysis for nine more possible nucleon resonances

From πp phase shift analyses at 59 energies, linked with partial wave dispersion relation fits, we suggest the probable existence of nine new nucleon resonances in the mass range 1680–2190 MeV, all with very small branching ratios to the elastic channel.