Partial-wave Dispersion Relations Research Articles

Consequences of analyticity and unitarity for partial-wave amplitudes

Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with l ≧ 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission factor and the real part of the phase shift. (d) If the transmission factor goes to zero (maximal inelasticity), the rate at which it approaches this limit is sometimes bounded by a known function. In these cases, the unphysical singularities and the transmission factor together determine the leading asymptotic term of the phase shift. (e) In some situations, a sign inconsistency in the asymptotic constraints implies the existence of poles in any N D solution with residues whose signs prevent their interpretation as bound states. These poles are then “ghosts.” (f) In an exact partial-wave dispersion relation, at least when l ≧ 2, there is no simple way by which to determine, from the sign of the unphysical function at threshold or at infinity, whether the corresponding “force” is attractive or repulsive. This sign is fixed at these energies by general arguments. As a result of these constraints, one can conclude that the usual approximations of the unphysical cut in terms of the singularities of pole diagrams have a chance of satisfying consistency requirements only for the s- and p-waves with spin zero and one particle exchanges. For p-waves, the exchange of a spin one particle is required. Certain rapidly convergent integral identities are developed which can be used to test for the presence of ghosts or of bound states in N D solutions or to test the accuracy of solutions. Finally, the relativistic Chew-Low model is used to demonstrate numerically the importance of some of the restrictions obtained.

Threshold Behaviour of Partial-Wave Dispersion Relation

Threshold Behaviour of Partial-Wave Dispersion Relation

Connection Between the Asymptotic Behavior and the Sign of the Discontinuity in One-Dimensional Dispersion Relations

Starting from one-dimensional dispersion relations (either fixed-transfer or partial-wave), information on the sign of absorptive part which follows from unitarity, and in some cases analyticity in the Mandelstam ellipse in the $t$ plane and polynomial boundedness, we derive various consequences of physical interest, e.g., a high-energy lower bound on the forward scattering amplitude, the minimum fluctuation of the sign of the discontinuity across the left-hand cut in the partial-wave dispersion relations, etc. In the derivations, the positiveness of the absorptive part plays an essential role by allowing us to construct a Herglotz function which has a well-known asymptotic behavior.

Application of the method of moments to partial-wave dispersion relations: I. A. P. Balachandran, The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois

Application of the method of moments to partial-wave dispersion relations: I. A. P. Balachandran, The Enrico Fermi Institute for Nuclear Studies, The University of Chicago, Chicago, Illinois

The threshold behaviour of partial wave scattering amplitudes and theN/D-method

It is shown that in partial wave dispersion relations the weight function on the unphysical cut must have a certain number of zeros in order to permit the correct threshold behaviour of the amplitude. Assuming a solution — not necessarily with correct threshold behaviour — of the once-subtractedN/D-equations to exist, the role of the subtraction parameters in repeatedly subtractedN/D equations is studied with particular reference to the threshold behaviour.

S-wave meson-nucleon interactions and unitary symmetry

Low-energyS-wave meson-micleon scattering is studied by making use of unitary symmetry. The parameters characterizing the reactions are explained in terms of the vector-meson exchange mechanism. The validity of the model is investigated by theS-wave partial-wave dispersion relation.

Bound States and Regge Trajectories in a Vector Meson Exchange Model. I. Application to theKK¯System

Partial-wave dispersion relations, extended to noninteger angular momenta are utilized, together with assumptions on the dominance of one-meson exchange to compute the properties of bound states. The exchanged mesons are represented by Regge poles, which lead to a set of equations of generalized Fredholm type when the $\frac{N}{D}$ technique is applied. Bound-state energies in a one-channel system of two spinless particles are computed, as well as the slope of the Regge trajectory which passes through each bound state; the latter is accomplished by an extension of the $\frac{N}{D}$ formalism to angular momenta in the neighborhood of the positive integers. Threshold questions are treated by an approximation for more complex diagrams. The integral equations are solved without further approximations by electronic computer methods. The model is applied to the $\ensuremath{\phi}$ meson as a nearly bound state of the $K\overline{K}$ system in the present work and yields information on $S$-wave $K\overline{K}$ interactions. Application to future "bootstrap" calculations, the reason for computing the Regge slopes, is discussed, as well as the relationship to the strip approximation.

Existence of solutions of partial-wave dispersion relations

High-energy boundary conditions and the convergence properties and existence of solutions of low-energy partial-wave calculations are studied. The physical significance of the inverse-amplitude solution for low-energy pion-pion scattering is discussed.

Semiphenomenological Solutions of Pion-Nucleon Partial-Wave Dispersion Relations

A variational method of solving $\ensuremath{\pi}\ensuremath{-}N$ partial-wave dispersion relations is developed. Analytic trial functions are used, and their parameters are varied to obtain a unitary solution. A good solution for the ($\frac{3}{2},\frac{3}{2}$) resonance is obtained by using the Layson function. The shape of the resonance, as well as its position, is obtained. The only problem about the solution is the validity of the short-range interaction term which is used. Crossing of the real part of the amplitude verifies that it is accurate, but it is not obvious why the variation method should give such a good result. The explanation appears to be that in many cases the low-energy behavior of a partial wave is dominated by the long-range interactions, and a comparatively simple analytic function will give a good solution. An application of the variational method to confirm an earlier analysis of $s$-wave $\ensuremath{\pi}\ensuremath{-}N$ scattering is also given.

Composite Particle Model for the Nucleon and the (3,3) Resonance

Partial-wave dispersion relations are used with interaction terms arising from the exchange of a nucleon, the ${N}^{*}$ and the $\ensuremath{\rho}$ meson to produce integral equations for the partial-wave amplitudes for pion-nucleon scattering. The solutions of these equations depend on a single arbitrary parameter, the energy at which the dispersion integrals are "cutoff." It is shown that when the value of the cutoff is adjusted to produce the ${N}^{*}$ at the correct energy, a bound state, the nucleon, appears in the $p$ wave, $I=J=\frac{1}{2}$ amplitude. Thus, the nucleon mass, the pion-nucleon coupling constant, the width of the ${N}^{*}$, and mass of the ${N}^{*}$ are all determined by the cutoff parameter. It is shown that when the ${N}^{*}$ mass has the experimental value the other quantities are in reasonable agreement with experiment. The effect of variations in the coupling constants controlling the interaction terms is also studied. In each case all of the $J\ensuremath{\le}\frac{3}{2}$ partial-wave amplitudes are calculated and compared with experiment.

Complex Singularities in Production Amplitudes

It is shown for a simple model of production that Watson's scheme for final-state interaction may be good even though the amplitude has complex singularities. The method used consists in continuing analytically partial-wave dispersion relations. Stress is put upon the necessity of correctly choosing the cuts starting at the complex singularities. The safest choice in this approach is to take the cut along the line on which the series of partial waves diverges. A simple geometrical recipe is given to see the evolution of this line, the appearance of an anomalous threshold and a complex singularity.

Low-Energy Pion-Pion Scattering

This calculation starts with the partial-wave dispersion relations for pion-pion scattering as deduced from the Mandelstam representation for the scattering amplitude. The contribution of the "left-hand cut" was calculated from $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ perturbation theory (to second order in $\ensuremath{\lambda}$). The $s$-wave and $p$-wave amplitudes were obtained numerically and found to agree closely with those obtained by Chew, Mandelstam, and Noyes who proceeded from a different viewpoint.

Theory of theJ=32,I=32πNResonance

We calculate the position ${W}_{R}$ and width ${\ensuremath{\gamma}}_{33}$ of the $J=\frac{3}{2}$, $I=\frac{3}{2}$ $P$-wave $\ensuremath{\pi}N$ resonance, using partial-wave dispersion relations. In the present calculation we treat as given the nucleon and $\ensuremath{\rho}$-meson masses and coupling constants, which determine the long-range part of the forces. The parameters, which characterize the distant part of the left-hand cut, are fixed by using the expressions for the ($\frac{3}{2}, \frac{3}{2}$) $P$-wave $\ensuremath{\pi}N$ state given by fixed energy dispersion relations, in a region where they are valid without subractions, in a way used by Bal\'azs for the $\ensuremath{\pi}\ensuremath{\pi}$ problem. We then impose the self-consistency demand that the position and width of the ($\frac{3}{2}, \frac{3}{2}$) resonance used as input values in the crossed channel in the fixed-energy dispersion relation be the same as the calculated values of the position and width. The preliminary results of the calculation are ${W}_{R}\ensuremath{\approx}m+2.35$ and ${\ensuremath{\gamma}}_{33}\ensuremath{\approx}0.14$. The experimental values are ${W}_{R}=m+2.17$ and ${\ensuremath{\gamma}}_{33}\ensuremath{\approx}0.12$, (where $m$ is the nucleon mass and we use units in which $\ensuremath{\hbar}=c={m}_{\ensuremath{\pi}}=1$). These results constitute the first part of the intended selfconsistent calculation of the nucleon mass and ($\frac{3}{2}, \frac{3}{2}$) resonance position, exploiting the "reciprocal bootstrap" mechanism discussed by Chew.

Analysis of Partial-Wave Dispersion Relations

Questions of existence and uniqueness of solutions of partial-wave dispersion relations are studied, with particular attention to the N/D method. The interaction, assumed to be given, is represented by (i) the strengths and locations of unphysical singularities and (ii) the inelastic partial-wave cross section. A generalization of the N/D method to include part (ii) of the interaction leads to a nonsingular integral equation for ImD. This equation is amenable to the Fredholm theory only if there is a correlation between items (i) and (ii) of the interaction, and only if the increase of inelastic processes at high energies is not too rapid. Certain Cauchy integrals associated with (i) and (ii) must be nonzero at threshold if there is to exist a solution with the normal threshold momentum dependence. Thus, there is no solution for any model in which (i) is constructed from a few partial waves in the two crossed channels. For certain interactions the real part of the phase shift approaches a multiple of pi at large energy, just as in potential scattering. The Castillejo-DalitzDyson (CDD) ambiguity is analyzed in some detail. A uniqueness theorem is proved which asserts that if a solution of a particular type exists, itmore » is the only solution of the problem within the class usually considered. Thus the CDD ambiguity is partially bypassed. In certain cases tue unique solution is found by the ordinary N/D method without subtractions. Some useful results on principal value integrals are obtained. The discussion is carried out for the example of pion- nucleon scattering in the complex plane of w, the certer-of-mass energy. The behavior of the amplitude near w = plus or minus (M-m) is derived from crossing symmetry. (auth)« less

On Mandelstam's program in potential scattering

Mandelstam's program for constructing the scattering amplitude from its analytic properties and unitarity is analysed in the case of nonrelativistic scattering by a cutoff potential or by a hard sphere. The asymptotic behavior of the scattering amplitude in the momentum transfer plane is obtained, leading to a double dispersion representation for the amplitude. The usefulness of this representation is limited by an essential singularity at infinity in the momentum transfer plane. An infinite system of dispersion relations, connecting each partial wave with all succeeding ones, is derived from the dispersion relation for fixed momentum transfer. The partial-wave amplitudes must be constructed from this system together with the unitarity condition. Possible ambiguities in the solution of this problem are investigated. It is shown that ambiguities in the exact solution affecting only a finite number of partial waves (Castillejo, Dalitz and Dyson ambiguities) do not exist. They would arise, however, in approximate solutions and it would be very hard, in practice, to eliminate them from the exact solution. The ambiguities can be formulated in terms of the positions of the poles of the S-matrix. A series of sum rules which must be fulfilled by the poles is derived. The solution of the system is investigated in the particular case of scattering by a hard sphere. In this case, if one assumes that the exact solution is known for angular momenta larger than some (arbitrarily given) valae, each partial-wave dispersion relation for smaller values of the angular momentum can be exactly solved, and it follows from the sum rules that the solution is unique.

Use of the Generalized Jost Function in Quantum Field Theory

The structure of the S-matrix is investigated in terms of the so-called generalized Jost function. Starting from the partial wave dispersion relation, the analytic properties of this function can be derived in the single-channel scattering, and it is shown that there exists a close connection between the analyticity of this function and the generalized Levinson relation. In the case where the one-particle singularity in scattering amplitudes is caused by the elementary particle, the corresponding Jost function has a pole, while in the case where this singularity is caused by the composite bound) state, the Jost function does not have the pole. Singularities corresponding to the Castillejo- Dalitz-Dyson ambiguity are completely separated. Thus, by making the conjecture that the analyticity in the low-momentum region still remains in the multichannel case, the Levinson theorem can be extended to the multichannel case and to the S- matrix theory where the Hamiltonian is not used explicitly. A new representation of the S-matrix, and how the structure of the Smatrix changes according to each case mentioned above, is clarified.

Final state interaction andCP invariance in radiative Kπ2 decay

The effect of an I=1, J=1 pion-pion resonance on the matrix element of the decay K→ 2π + γ is studied. It is found that while the matrix element strongly favours low photon frequencies as a result of the final state interaction, the observable effects are much reduced by the phase-space factor. These effects are probably observable only in K20 decay. The usual final-state interaction theorem is found to be invalid in general because of the existence of a « radiative rescattering » channel. The use of partial wave dispersion relations is briefly discussed. Some selection rules for the photon states in K20 decay are derived on the basis ofCP invariance. The branching ratio R(K20→2π + 2e)/R(K20→ 2π + γ) is calculated for various values of the π-π resonance parameters. The correlation of the electron and the pion planes is calculated.

Pion-Nucleon Scattering and Pion-Pion Interactions

Low energy $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ scattering is analyzed by partial wave dispersion relations. From the experimental $\ensuremath{\pi}\ensuremath{-}N$ phase shifts we derive the in the physical energy region and on the crossed cut. We are able to separate the discrepancies into the short-range (\ensuremath{\lesssim}0.2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm) $\ensuremath{\pi}\ensuremath{-}N$ interactions and the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions to $\ensuremath{\pi}\ensuremath{-}N$ scattering. The $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions found in this way satisfy several stringent tests which show the validity of our method for deriving and separating the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ contributions.The (+) charge combination of $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ amplitudes yields considerable information about the $T=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction at low energies. The $T=0$, $J=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering is dominant, and we determine possible sets of the corresponding phase shift ${{\ensuremath{\delta}}_{0}}^{0}$. Several of our solutions for ${{\ensuremath{\delta}}_{0}}^{0}$ agree with recent solutions of the Chew-Mandelstam equations for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering. Comparison with the latter suggests that the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ coupling parameter is $\ensuremath{\lambda}=\ensuremath{-}0.18\ifmmode\pm\else\textpm\fi{}0.05$, and the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering length is ${a}_{0}=1.3\ifmmode\pm\else\textpm\fi{}0.4$ (in units where $\ensuremath{\hbar}=\ensuremath{\mu}=c=1$). Other information, from the $p+d$ and $\ensuremath{\pi}+N\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}+N$ experiments and from $\ensuremath{\tau}$ decay, is consistent with our proposed values of ${{\ensuremath{\delta}}_{0}}^{0}$.The (-) charge combination of $s$- and $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ amplitudes gives information about the $T=1$, $J=1$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction which is consistent with the observed $\ensuremath{\rho}$ resonance. However in the $T=1$, case, complete prediction of our $\ensuremath{\pi}\ensuremath{-}N$ results via the helicity amplitudes for $\ensuremath{\pi}+\ensuremath{\pi}\ensuremath{\rightarrow}N+\overline{N}$ is not yet satisfactory. Possible reasons for this are given.The $p$-wave $\ensuremath{\pi}\ensuremath{-}N$ interaction is separated into its constituent parts, and for example, it is seen that any attempt to determine the position of the ($\frac{3}{2}, \frac{3}{2}$) resonance must include the $T=0$, $J=0$ $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ interaction. The extent to which our analysis depends on assuming charge independence is examined. We also discuss how our results can be regarded as a fairly good physical proof of the Mandelstam representation.

Partial wave dispersion relations for photon-nucleon scattering and nucleon-antinucleon production by photons

Partial wave dispersion relations for the processes γ Open image in new window →→γ+ Open image in new window and γ+γ→ Open image in new window + Open image in new window to the ordere2 are derived with the help of the Mandelstam representation. The angular momentum decomposition for both processes is carried out and the analytic properties of the partial wave amplitudes are determined. The absorptive parts of these amplitudes along part of the physical branch cuts are calculated by means of the unitarity condition. For γ+ Open image in new window →γ+ Open image in new window the resulting expressions involve amplitudes for single pion photoproduction; for γ+γ→ Open image in new window + Open image in new window they involve theT=0s-wave pion-pion phase shift and the total cross-section for the process γ+π→π+π. The discontinuities along the unphysical branch cuts are calculated by means of crossing symmetry and of analytic continuation through Legendre polynomials. The limits of convergence of the Legendre expansions are discussed and reasons for the importance of the two-pion exchange contribution in photon-nucleon scattering are presented.

Partial wave dispersion relations for lambda-nucleon scattering

Partial wave dispersion relations are derived for the process Λ+N→ Λ+N and the associated processes Λ+ % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXafv3ySLgzGmvETj2BSbWexLMB% bXgBcf2CPn2qVrwzqf2zLnharCWqVvNCPvMCG4uz3bqehqvATv2CG4% uz3bIuV1wyUbqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0x% bbL8F4rqqrFfpiZxe9LqFHe9Lqpepeea0xd9q8as0-LqLs-Jirpepe% ea0-as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaGaaiaadaWaamaaceGa% aqaafaGbaaGcbaacbaGaf8xdX7Kbaebaaaa!4CD9! $$\bar {\mathcal{N}}$$ →Λ+ % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXafv3ySLgzGmvETj2BSbWexLMB% bXgBcf2CPn2qVrwzqf2zLnharCWqVvNCPvMCG4uz3bqehqvATv2CG4% uz3bIuV1wyUbqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0x% bbL8F4rqqrFfpiZxe9LqFHe9Lqpepeea0xd9q8as0-LqLs-Jirpepe% ea0-as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaGaaiaadaWaamaaceGa% aqaafaGbaaGcbaacbaGaf8xdX7Kbaebaaaa!4CD9! $$\bar {\mathcal{N}}$$ and Λ+ % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXafv3ySLgzGmvETj2BSbWexLMB% bXgBcf2CPn2qVrwzqf2zLnharCWqVvNCPvMCG4uz3bqehqvATv2CG4% uz3bIuV1wyUbqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0x% bbL8F4rqqrFfpiZxe9LqFHe9Lqpepeea0xd9q8as0-LqLs-Jirpepe% ea0-as0Fb9pgea0lrP0xe9Fve9Fve9qapdbiGaaiGadiWacmaaceGa% aqaafaGbaaGcbaacdaGaf83MdWKbaebaaaa!4C81! $$\bar \Lambda $$ →N+ % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXafv3ySLgzGmvETj2BSbWexLMB% bXgBcf2CPn2qVrwzqf2zLnharCWqVvNCPvMCG4uz3bqehqvATv2CG4% uz3bIuV1wyUbqeeuuDJXwAKbsr4rNCHbGeaGqiVu0Je9sqqrpepC0x% bbL8F4rqqrFfpiZxe9LqFHe9Lqpepeea0xd9q8as0-LqLs-Jirpepe% ea0-as0Fb9pgea0lrP0xe9Fve9Fve9qapdbaGaaiaadaWaamaaceGa% aqaafaGbaaGcbaacbaGaf8xdX7Kbaebaaaa!4CD9! $$\bar {\mathcal{N}}$$ .