Forward Dispersion Relations Research Articles

The dpn and tdn coupling constants from forward neutron-deuteron dispersion relations

Forward dispersion relations for nd scattering using unpolarized total cross sections are evaluated at low energies ( E lab < 20 MeV). Comparison with real parts deduced from near forward differential cross sections allows us to determine the coupling strengths of the nearlying p- and t-pole in an attractively direct way, free from theoretical information on nd amplitudes, in principle. The present best fit gives a very consistent picture. The dnp strength is right on the value deduced from triplet np scattering, indicating a quite accurate determination of the (less well known) tnd coupling constant.

S- and P-wave K +p scattering below 600 MeV/ c

Values for the s-wave K +p scattering length and curvature coefficient are obtained by extrapolating the low-energy amplitude to threshold using a parametric form derived from a forward dispersion relation. In addition, we present a calculation of the p-wave scattering lengths from forward dispersion relation sum rules. The calculations are performed in an iterative manner starting from the pure s-wave solution of Goldhaber et al.

Pion-Pion Scattering Below 850 MeV. A Unique Solution

Forward dispersion relations are used as the basis of an analysis of the information on pion-pion scattering available up to 850 MeV. These are shown to result in a model-independent relation between the scattering lengths ${\ensuremath{\alpha}}_{0}$ and ${\ensuremath{\alpha}}_{2}$, valid for all possible behaviors of ${{\ensuremath{\delta}}_{0}}^{0}$ in the $\ensuremath{\rho}$-region, and taking the value $\frac{{a}_{0}}{{a}_{2}}=\ensuremath{-}3.2\ifmmode\pm\else\textpm\fi{}1.0$ from recent low-energy work, the values ${a}_{0}=0.16\ifmmode\pm\else\textpm\fi{}0.04$, ${a}_{2}=\ensuremath{-}0.05\ifmmode\pm\else\textpm\fi{}0.01$ uniquely result. The ambiguity in ${{\ensuremath{\delta}}_{0}}^{0}$ in the $\ensuremath{\rho}$ region is also studied, by four different methods, and uniquely resolved, allowing a detailed picture of all the phases up to 850 MeV to be given.

Determination of the K ±p forward amplitudes and coupling constants using a finite-energy relation

The real parts of the amplitudes for K ±p forward elastic scattering are estimated from the available differential cross section data. A least-squares fit, based on a finite-energy dispersion relation, is performed to these values and the optimum real parts are calculated as a function of energy. From the fit we find that the KN coupling constants satisfy g Λ 2 + 0.84 g Σ 2 = 8.8 ± 3.0, a value smaller than that required by SU(3) invariance. Unlike the value of the couplings obtained using the conventional forward dispersion relations, the prediction is not particularly dependent on the model used to extrapolate the K −p amplitude into the unphysical region. The least-squares fit also provides a sensitive consistency test of the various parameterizations which have been used to analyse the low-energy K −p scattering data. The test favours the constant scattering length parameterization over the more recent K-matrix parameterizations. In particular, the consistency condition rules out the effective-range K-matrix parameterization due to Kim.

Hadron-nucleus forward dispersion relations

Forward elastic scattering of pions and nucleons from complex nuclei is investigated by dispersion relations (FDR). Explicit applications cover πD, π 4He , π 9Be , π 12C , π 4He and n 12C . Partly unpublished numerical evaluations are given. Results are expected to be typical for a wide range of nuclei. The unphysical region for π-scattering on I = 0 nuclei is unimportant. For the I = 1 2 nucleus 9Be, an effective π-nuclear coupling constant (ƒ 2) eff ≈ 0.06 is determined. With simple assumptions, general sum rules are shown to give universally (ƒ 2) eff = ƒ 2 nucleon as well as small unphysical contributions to the isospin-independent amplitude. Important exchange contributions from exchange diagrams (poles and cuts) are shown to appear in nucleon scattering at lower energies. This is shown to be consistent with single-particle potential scattering (optical model). The phenomenological potential scattering background term in slow neutron scattering is largely due to such exchange contributions. Antiparticle cross sections are shown to give small effects in low-energy nucleon scattering. A non-relativistic approximation to the relativistic FDR is compared to FDR for potential scattering and the Born term is empirically determined for n 4He. Good agreement is obtained by comparison with the empirical potential for differential n 4He scattering at low energy. A self-contained elementary discussion and review of FDR is given. The relation between asymptotic wave functions and pole residues is discussed in detail. Particular attention is drawn to the importance of an exponential dependence of residues on binding and radius, and it is illustrated by simple models.

Forward Compton Scattering

An analysis of the recently measured photon-proton total cross sections is performed. Smooth fits to the cross sections are obtained and used to calculate, by means of the forward dispersion relation, the real part of the spin-averaged forward amplitude. The resulting predictions for the real part are given. At high energies, the fits to the present total cross-section data, together with the calculated real part, suggest the presence in the high-energy behavoir of an extra real constant in addition to what one would have predicted from Regge theory and the high-energy behavior of the imaginary part. This extra real constant, which is consistent in sign and magnitude with the Thomson limit, $\ensuremath{-}\frac{\ensuremath{\alpha}}{{M}_{N}}$, could correspond to a fixed pole at $J=0$ in Regge-pole language. Possible ways to test the forward dispersion relation are discussed.

K-matrix analysis of the low-energy data for K −p and K 2op reactions

A nine-parameter zero-range K-matrix S-wave fit is made to the available experimental data for K −p and K 2 op reactions below 280 MeV/ c. A unique solution is obtained and compared with the effective-range solution obtained by Kim. It is found that both solutions give good agreement with the data for processes involving only the K N and Σπ channels, but that for processes involving the Λπ channel the present fit improves on the effective-range fit over the momentum range 0–280 MeV/ c. The scattering lengths corresponding to the K-matrix solution are given and compared with those obtained in previous analyses. The K-matrix solution is used to compute the unphysical region contribution to KN forward dispersion relations and the ΛKN and ΣKN coupling constants are calculated. In addition a multichannel analysis of the K −p and K 2 op data below 400 MeV/ c is performed with the inclusion of K N isospin one P-wave amplitudes. In the region up to 400 MeV/ c the P 1 2 and P 3 2 waves are found to be equally important. Argand plots of the various K N partial-wave amplitudes are given.

N, Δ and ϱ-exchange in pion-nucleon scattering

The N, Δ and ϱ-exchange contributions to the πN charge exchange amplitude as derived from fixed- t and fixed angle θ = 180° dispersion relations are compared with the corresponding terms in a chiral Lagrangian. A new evaluation of the backward dispersion relation combined with the forward dispersion relation gives information on the physical origin and the magnitude of the corrections to the simple ϱ-dominance and N, Δ, ϱ-exchange models. The result for the πNΔ-coupling constant g ∗2/4π = 0.26 differs from that of other authors and the determination of the ϱNN coupling constants leads to difficulties which were ignored in earlier work. The equality between the forward and backward amplitudes at threshold gives a relation between coupling constants similar to a result which was recently derived from chiral Lagrangians.

Derivative sum rules and inconsistencies in the low-energy [formula omitted]N scattering parameters

The KNΛ and KNΣ coupling constants predicted by kaon-nucleon forward dispersion relations are particularly sensitive to the parametrization used to extrapolate the K N scattering amplitude into the unphysical region. The dispersion relations which have hitherto been considered do not provide a sufficiently stringent and unambiguous consistency test to discriminate between alternative parametrizations which have been proposed. New sum rules are derived from the condition that the coupling constants defined by the dispersion relations should be energy-independent. Although these sum rules utilize the same input data as the conventional dispersion relations, they are entirely independent of the values of the coupling constants and provide a strong consistency requirement. Numerical tests of the sum rules show that all recent parametrizations of the low-energy K N scattering amplitude are inconsistent with the remainder of our knowledge of kaon-nucleon interactions. It is suggested that future parametrizations of the experimental data should incorporate more theoretical constraints.

Experimental Test of the Pion-Nucleon Forward Dispersion Relations at High Energies

The small-angle differential scattering cross sections of protons for pions have been measured to high precision at the Brookhaven AGS. The range of incident momenta was 8-20 GeV/c for ${\ensuremath{\pi}}^{+}$, and 8-26 GeV/c for ${\ensuremath{\pi}}^{\ensuremath{-}}$. The real part of the pion-nucleon forward scattering amplitude was determined by observing its interference with the known Coulomb amplitude. Combining these results with precision measurements of pion-proton total cross sections over this energy range provided a critical test of the predictions of the forward dispersion relations. The results demonstrate the validity of the dispersion relations up to at least 20 GeV/c laboratory momentum. The predictions of charge independence are also verified by comparing these experimental measurements with forward charge-exchange scattering cross sections. Furthermore, if microscopic causality is violated, this occurs at distances less than ${10}^{\ensuremath{-}15}$ cm.

Consistency tests of the low-energy K −p parametrizations

The value of the ΛKN coupling constant g Λ 2 predicted by forward KN dispersion relations is particularly sensitive to the model used to continue the if K N amplitude into the unphysical region. The various parametrizations which have been used to analyse the low-energy K −p scattering data are reviewed and compared, and a simple consistency test of these parametrizations is proposed. The consistency condition is satisfied by the existing constant scattering length parameters and by a new constant K-matrix solution obtained by Martin and Sakitt, but not by the effective-range K-matrix parametrization of Kim. Other recently proposed consistency tests are discussed, and it is shown in particular that an argument used by Chan and Meiere to rule out the determination of g Λ 2 using constant scattering lengths is not valid, since the inconsistency found by them in this case is due entirely to the incorrect behaviour of the extrapolated amplitude at the Σπ threshold. It is concluded that, of the existing low-energy K −p parametrizations, only the constant scattering length and constant K-matrix solutions are consistent. Using either solution, the forward dispersion relations predict g Λ 2 ≈ 4, a value considerably smaller than thant suggested by SU(3) invariance.

Dispersion relations and Glauber model for pion-deuteron scattering

Forward dispersion relations are used to calculate the real part of the elastic amplitude for forward pion-deuteron scattering. The same real part is also calculated from the Galuber multiple scattering model. The results of the two approaches are critically compared in the light of present experiments. A theoretically deduced value for the pion-deuteron scattering length is also given. A theoretical prediction for the pion-deuteron total cross section for energies below 450 MeV is found to agree rather well with experiments.

Computation of forward pp dispersion relations

We compute forward pp dispersion relations by expanding the unphysical part in a series of Chebyshev polynomials, and fitting the coefficients to the experimental low-energy values of Ref pp. The fit is very good and the results are consistent with experiment when extrapolated at high energy. As a by-product, values of real and imaginary parts of the $$\bar pp$$ amplitude at zero energy are obtained.

Low-energy pion-pion parameters from forward dispersion relations

Values of the ππ scattering lengths and low-energy phase shifts are tabulated corresponding to various assumptions on the mass and width of the sigma meson. The method uses forward and first derivative dispersion relations. The I = 0 S-wave scattering length a o comes out in the range −0.3 to 0.6 m π −1. The P-and D-wave scattering lengths are relatively independent of the sigma parameters. Solutions are compared with results on Ke 4 decay and with the predictions of current algebra. Super broad sigmas seem to be preferred. Such solutions are also more self consistent in a bootstrap sense.

A new analysis of the low-energy K +p scattering data

The elastic K +p scattering data are analysed with the help of K ±p derivative forward dispersion relations. A unique solution is obtained. From this solution the K +p s wave scattering length and effective range are found to be respectively a s = −0.32 ± 0.01 fm, r s = 0.31 ± 0.15 fm . A consequence of this analysis is that the K ±p dispersion relation prediction for the strange particle coupling constants is somewhat increased.

Adler's self-consistency condition for nuclei

It is shown that Adler's self-consistency condition for light nuclei ( 4He, 12C, 9Be, etc.) implies the smallness of the contribution from the unphysical region to charge symmetric π-nucleus forward dispersion relations.

A soluble model for the πΛ interaction

A simple model for the πΛ interaction, based on a forward dispersion relation for the inverse amplitude, is formulated. The assumption that the amplitude is dominated in the appropriate energy range by the Y 1 ∗ (1385) resonance leads to definite restrictions on the possible behaviour of the amplitude at lower and at higher energies. The asymptotic total cross section calculated from the model is consistent with the quark-model prediction. However, a negative S-wave scattering length is required, in disagreement with each of three alternative (mutually incompatible) predictions.

Recent Applications of KN Forward Dispersion Relations and Sum Rules

AbstractRecent applications of kaon‐nucleon forward dispersion relations, superconvergence relations, finite energy sum rules, derivative sum rules, and the Adler‐Weisberger sum rule are reviewed. These dispersion relations and sum rules have been used for the determination of the KNA and KN coupling constants, the phases of the kaon‐nucleon forward scattering amplitudes, and the Regge pole parameters, and have allowed tests of the PCAC hypothesis for strangeness‐changing currents. They also provide important constraints which help to resolve ambiguities in the phenomenological analysis of the scattering data. A critical comparison of various methods which have been proposed is given, the theoretical predictions are compared with experiment when possible, and our present knowledge of the kaon‐nucleon interaction parameters is summarized. Some new results are presented.

K+pScattering and a PossibleZ*Resonance

An energy-dependent phase-shift analysis in the kaon laboratory-momentum range below 2.0 BeV/c has been made of ${K}^{+}p$ scattering data, supplemented by information from ${K}^{+}p$ forward dispersion relations. Evidence for a possible resonance in the ${P}_{\frac{1}{2}}$ state is presented.

Nucleon-nucleus forward dispersion relations

Forward dispersion relations are illustrated by n − α elastic scattering. The evaluation at low energies shows the great importance of exchange effects. This result should be valid in general for light nuclei.