Scattering Length Approximation Research Articles

Threshold cusp in the s-wave neutron strength function

The energy dependence of the l = 0, S-matrix neutron strength function is calculated in the scattering length approximation, both above and below neutron threshold and is found to exhibit a cusp exactly at threshold. Although only its upper half is accessible to neutron-resonance experiments, it should be observable in the spectroscopic factors extracted from (d, p) reactions leading to states of the final nucleus in the neighbourhood of neutron thresholds. It should be especially prominent for nuclei with A ≈ 55 and A ≈ 155, where the 3s and 4s single-particle states are virtual.

Dispersion theoretic impulse approximation: KD elastic scattering

Dispersion theory techniques are applied to construct an impulse approximation model for K +-meson-deuteron elastic scattering. The discontinuity of the annihilation amplitude on the N N cut is examined for small deuteron mass. Analytic continuation in the deuteron mass yields an anomalous cut; it is this contribution which is isolated and interpreted as the impulse approximation. For low energies an s-wave scattering length approximation is used for the KN amplitude. The spins of the particles are taken seriously throughout. The angular distribution for a lab momentum of 230 MeV/ c is calculated and compared with existing data.

Multimeson Resonances and Nucleon-Nucleon Interaction

Relativistic partial-wave dispersion relations are formulated for elastic nucleon-nucleon scattering. These dispersion relations are integral equations with an inhomogeneous term taken from single-particle exchange contributions. The particles under consideration are $\ensuremath{\pi}(I=1, \mathrm{pseudoscalar})$, $\ensuremath{\eta}(I=0, \mathrm{pseudoscalar})$, $\ensuremath{\rho}(I=1,\mathrm{vector})$, $\ensuremath{\omega}(I=0, \mathrm{vector})$, $\ensuremath{\phi}(I=0, \mathrm{vector})$, and $\ensuremath{\sigma}(I=0, \mathrm{scalar})$. The existence of a $\ensuremath{\sigma}$ meson is not well established. Two possibilities are considered: (i) The $\ensuremath{\sigma}$ meson exists, in which case the mass and coupling constants are taken to be two parameters of the present problem. (ii) The $\ensuremath{\sigma}$ meson does not exist but the $I=0$, $J=0$ two-pion continuum is taken into account. This two-pion continuum can be treated as a superposition of scalar particles with a mass spectrum determined by pion-nucleon and pion-pion interactions. Information on the $\ensuremath{\pi}N$ interaction is obtained from $\ensuremath{\pi}N$ scattering data, while the $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ interaction is represented with a relativistic scattering-length approximation. In addition to the $\ensuremath{\pi}\ensuremath{\pi}$ scattering length, a cutoff on the two-pion spectrum is introduced. Thus two parameters are introduced in either (i) or (ii). Aside from the masses and coupling constants of the particles mentioned, a cutoff parameter is needed for each of the vector mesons $\ensuremath{\rho}$, $\ensuremath{\omega}$, and $\ensuremath{\phi}$. These are taken to be coefficients in an exponentially decreasing factor suggested by the Regge-pole behavior of composite particles. A total of twelve adjustable parameters is used and a search program is formulated to fit 560 $\mathrm{pp}$ and $\mathrm{np}$ data collected by the Livermore group ranging from 9.68 to 388 MeV. In both cases (i) and (ii), a fit is obtained with a "goodness to fit" value of approximately 8%, meaning that the ${\ensuremath{\chi}}^{2}$ is \ensuremath{\sim}548 if the uncertainty inherent in the theory is assumed to be 8%.

Study ofKe4Decays

It is assumed that the $K\ensuremath{\rightarrow}\ensuremath{\pi}+\ensuremath{\pi}+e+\ensuremath{\nu}$ process has a Born term dominated by ${K}^{*}$ exchange. By assuming a "nearly conserved" axial current we obtain a Goldberger-Treiman-type relation by which the ${K}_{e4}$ amplitude is related to the ${K}^{*}$ width and $K\ensuremath{\rightarrow}\ensuremath{\mu}+\ensuremath{\nu}$ amplitude. This is used to determine the left-hand cut for a set of partial-wave dispersion relations. With the assumption of elastic unitarity for pion-pion scattering, integral equations of the Muskhelishvili-Omn\`es type are obtained, which are then solved using various assumed forms for the pion-pion $T=0$ interaction. The effect of the $\ensuremath{\rho}$ resonance is included in the $T=1$ $P$-wave amplitude. We obtain agreement with the experimental rate for ${{K}_{e4}}^{+}$ decay when the $S$-wave pion-pion interaction is described by a scattering-length approximation with a scattering length of ${\ensuremath{\alpha}}_{0}=(1\ifmmode\pm\else\textpm\fi{}0.3)$ pion Compton wavelengths. With this value of ${\ensuremath{\alpha}}_{0}$, the two-pion invariant mass distribution is in good agreement with experiment, and the total $P$-wave contribution to the total rate is predicted to be 18%. If a $\ensuremath{\sigma}$ meson ($T=0$, $S$-wave resonance at 400 MeV) is assumed to dominate the $S$-wave pion-pion interaction, the calculated rate becomes larger than the experimental one by two orders of magnitude. The possibility of $T$ violation is discussed.

The pion-pion interaction in τ decay

The momentum dependence of the τ-decay rate deviates considerably from that predicted by the relativistic phase space factor and Coulomb corrections. The difference is attributed here to the final state pion pion interaction. Three different phenomenological analyses are made to determine the T = 0 and T = 2 s-state pion-pion force required for consistency with τ and τ′ data: a scattering length approximation, an independent pair approximation for an exponential potential, and a Born approximation for a Yukawa potential. The results of all three approximations agree where they are applicable and indicate a weak or repulsive T = 0 force and an attractive T = 2 force.

The excited hyperon and pion-hyperon resonances

Recently Good, in a study of momentum spectra for pi /sup +/ and pi / sup -/ in K/sup -/ + p yields LAMBDA + pi /sup +/ + pi /sup -/ at K/sup -/ lab momentum Bev/c, reported the existence of an excited hyperon Y* in the I = 1 state, strongly produced, and of mass 1370 plus or minus 25 Mev. The Good resonance is discussed qualitatively in terms of zero effective range theory or the so-called scattering length approximation. (T.R.H.)