Effective Range Formula Research Articles

Kπ SCATTERING

The s-wave Kπ phase shifts for the two isospin states T = 1/2 and T = 3/2 are calculated on the basis of a variational method due to Schiff by assuming a quartic boson–boson interaction. The trial forms for the state functions are optimized by making the expectation value of the field Hamiltonian stationary. The trial functions used are of separable form and it is hoped that with the development of more elaborate functions, this variational method may lead to a description of the Kπ resonances. In particular, the vacuum, single-particle, and two-particle (scattering and bound) states are investigated. Finally, the results are compared with the effective-range formula for Kπ scattering developed by Lee (1960).

Extrapolation of Electron-Rare Gas Atom Cross Sections to Zero Energy

Modified effective-range formulas recently developed for electron-atom scattering are employed first to analyze the Ramsauer-Kollath (RK) electron-rare gas atom cross sections, and then to compare them with determinations of the same quantities made by different and newer methods. It is found that these cross sections can be extrapolated unambiguously down to zero energy. The extrapolated RK scattering lengths are found to be approximately 1.19, 0.24, -1.70, -3.7, and $\ensuremath{-}6.5{a}_{0}$ for He, Ne, Ar, Kr, and Xe, respectively. Good agreement is found, in general, both in sign and magnitude with the pressure shift estimates of the scattering lengths. Momentum transfer cross sections are also extended down to zero energy and are compared with some drift velocity and microwave results. The best agreement is found with the Gilardini and Brown curve for Ne, and with those of Pack, Phelps, Frost, and Voshall for the other rare gases, although there are some differences with the latter. In particular, a "pressure effect" is discussed in connection with these results.

S-wave interaction in a three-pion system and the decay of Kπ3

A model of s-wave ( T = 0,2) π-π interaction is proposed from certain formal analogies with field theory, on lines similar to a recent paper (16) by one of us (ANM). The model leads in an exact fashion to a relativistic effective range formula of the two phase shifts δ 0 0 and δ 0 2, from which a simple relation is obtained between the s-wave scattering lengths and the strength parameters of the corresponding interactions. The model is then used to solve a three-body Schrödinger equation in momentum space for the isovector 3π wave function in a purely outgoing state so as to make the results applicable to K π3-decay. The possible symmetries of the 3π system (with two pions identical) are fully taken into account. This wave function is then employed for evaluation of the energy distribution of the unlike pion in τ-decay, on the usual assumption of zero range for the weak interaction. The results are qualitatively similar to those known from earlier dispersion theoretic treatments. However attractive π-π interactions seem to have little effect on the spectrum, suggesting that the structure of the weak interaction itself plays the dominant role.

K+−pScattering; the Phenomenology of a Repulsive Interaction

Recent experimental data on K/sup +/-proton elastic scattering in the energy range 20 to 450 Mev were analyzed to give the energy dependence of the s- wave phase shift delta , and the energy dependence was fitted to the effective range formula k cot delta = a/sup -1/ + 1/2 r/sub 0/k/sup 2/. It is pointed out that the k cot dd expansion is not appropriate for dominantiy repulsive irteractions, and that when it is used the parameters a and r/sub 0/ do not have their usual connotation. The expansion delta approx equal -kR + 1/3 k/sup 3/ (R - R')/sup 3/ is presented for obtaining a more detailed, but shape- independent, picture of the scattering. The behavior of delta at very low energies is described by the constant length R, and R' is a lengih dependent on k/ sup 2/. The sign of the k/sup 3/ term gives the sign of the potential in the outer fringe, and its magnitude measures the softness'' of the potential there. The new data on K/sup +/-p scattering predict that the k/sup 3/ term is negative, and thus that the tail is attractive. (H.D.R.)

On the Y*-resonances

On the assumption that the recently observed A-π and Σ-π resonances are in theS-wave,J=1/2 state, effective range formulae are derived for the Λ-π and Σ-π scattering amplitudes in this state using a static approximation.

Final State Interaction in Kaon Production

>The final state approach is applied to the reaction pi - + P → K + Y + pi . The pi -Y and K-Y interactions are neglected and only the K- pi interaction is considered in the final state. The matrix elements for the primary interaction are determined by the use of Fermi's statistical model. The secondary interaction is handled by the use of effective range formulae, where the parameters are fitted to the experimental full width at half maximum of 23 Mev and an effective mass of 878 Mev for an I = 1/2 resonance. Both an S-wave and P-wave resonance are considered. The results indicate a κ 0 π+ Σ-/ κ + π0 Σ-- ratio of 1.7 with a small amount of K0 pi - Σ + reaction for both the S- and P-wave cases. This result is in agreement with experimental results. The calculations also indicate a κ + π0 Σ-/ κ 0 π0 Σ0 ratio of 107 for an S-wave resonance and a ratio of 0.7 for a P-wave resonance. Unfortunately, the reaction κ 0π0 Σ0is difficult to determine experimentally and no experimental results have been obtained on them as yet.

Nature of the resonance in the static pion-nucleon scattering in one-meson approximation

The solution of the Chew-Low integral equation for the static p-wave meson-nucleon scattering is studied as a function of the coupling constant to deduce the nature of the 33-resonance. It is shown that a dynamical resonance can occur for a suitable choice of the cut off function. The possibility of a kinematical resonance is also discussed and the effective range formula is derived directly from the dispersion relation.

Pion-Pion Interaction

The modified effective range formulas for S- and P-wave pion-pion scattering which contain the effect of nucleon closed loops are derived using the chain approximation. These are extensions along the lines of Chew and Mandelstam, and Okubo and Miyamoto. The resonance at approximates 700 Mev is predicted with a broad width of T/2 approximates 400 Mev for the I = l = l state and the resonance at approximates 300 Mev with a small width of T/2 approximates 20 Mev for the I = l = 0 state, consistent with experiment corresponding to p and omega 0 mesons. respectively. (auth)

Applications of Conformal Mapping to the Phenomenological Representation of Scattering Amplitudes

A series representation of scattering amplitudes analytic in cut planes is derived by means of a conformal mapping. The new series has the advantage that in general it converges more rapidly than the conventional power series. An example is considered in which an angular distribution which requires ${(cos\ensuremath{\theta})}^{4}$ terms for a good fit requires only second-order terms in the new series. The resulting advantages in using this series in extrapolations to poles are discussed. As a second example, a modified effective-range formula is derived.

Energy Dependence of the Low-EnergyK−-Proton andK+-Proton Cross Sections

The ${K}^{\ensuremath{-}}$-proton and ${K}^{+}$-proton $S$-wave scattering is analyzed by using a relativistic effective-range formula derived by studying the analytic properties of partial-wave scattering amplitudes. The influence of the pion-pion interaction on the elastic scattering and reaction cross sections is discussed.

On the Semi-Classical Treatments of the Meson-Nucleon Scattering

The quantum mechanical treatment of the static meson-nucleon scattering which has close connection with the semi-classical one is presented. On this footing the semi-classical treatments of the p-wave pion-nucleon scattering, recently developed independently by Jackson and by Weisskopf, are examined. As a result it is shown that their treatments are not complete and that, if treated correctly, their effective range formula should be modified largely. This would result in the deviation from the experimental behavior. The applications to the charged and neutral scalar theories are also given and compared with the Castillejo-Dalitz-Dyson theory.

Low-Energy Neutral Pion Photoproduction inH2and He

The results of an experiment on the photoproduction of neutral pions from ${\mathrm{H}}_{2}$ and He in the very low-energy region are compared with the dispersion relation calculations of Chew, Goldberger, Low, and Nambu (C.G.L.N.). For ${\mathrm{H}}_{2}$, agreement was found for incident $\ensuremath{\gamma}$-ray energies above 155 Mev for ${N}^{+}\ensuremath{\simeq}+0.065$ and disagreement below for all constant values of ${N}^{+}$. The He results give support to the C.G.L.N. effectiverange formulas for the small $p$-wave phase shifts.

Effective range for K +-nucleon interaction from dispersion relations

An exact formula for the effective range for K +-P interaction, in the S-wave approximation, has been deduced using dispersion theoretic techniques. It is shown that if the assumption of zero effective range which is generally used for a phenomenological fit of K-N data, is valid then our analysis clearly indicates that the K-meson-hyperon relative parity must be odd (the Σ- Λ relative parity being even). The contribution of the dispersion integral over the unphysical region has been fully taken into account in the present analysis. A general discussion of the nature of the coupling is given for a non-vanishing but small effective range. The advantage of the use of the effective range formula to determine the sign of the relative parities is also pointed out.

Partial-Wave Dispersion Relations for Pion-Nucleon Scattering

Partial-wave dispersion relations for pion-nucleon scattering are derived from the Mandelstam representation. The symmetry of the representation is used to obtain expressions for the discontinuities across the unphysical branch cuts. These expressions involve scattering amplitudes for pion-nucleon scattering and for the process $\ensuremath{\pi}+\ensuremath{\pi}\ensuremath{\rightarrow}N+\overline{N}$. In the approximation of neglecting all but the nearest singularities it is shown that the Chew-Low effective-range formula is a solution to the equations.

Vergleich der Photoerzeugung von ?0 am Proton mit der Voraussage nach der Dispersionstheorie

The real parts of the photoamplitudesE 1S1/2,M 1P1/2,M 1P3/2 have been calculated from the angular distribution of the reactionγ+p→p+π0 recently measured byGoldansky et al. at 160 to 240 MeV. One of the solutions fits pretty well to the theoretical prediction for theM 1P3/2-amplitude according to the dispersion method ofChew, Goldberger, Low andNambu. There is a discrepancy forM 1P1/2 ifα11 is taken from the effective range formula, but the positive values ofα11, necessary to give agreement, are not excluded by the results of the phase shift analysis, especially sincePontecorvo et al. have recently found positive values at higher energies. The prediction for the real part of theE 1 S1/2-amplitude agrees with the experimental data, if pretty large recoil corrections are added which had been neglected byChew et al.

Modification of the Effective-Range Formula for Nucleon-Nucleon Scattering

The effective-range formula is welt known to be a usefal description of low-energy scattering phase shifts. However, except at the limit of zero kinetic energy, the effective-range formula is, in general, not exact. A studv is reported of the deviation from the effectiverange formula in light of the analytic structure of pantial-wave amplitudes as suggested by Mandelstam (Phys. Rev. 112, 1344(1958)). Consideration is given to the s-wave N-N scattering in both the singlet and the triplet states. (W.D.M.)

�ber die Vertr�glichkeit einer Dispersionsrelation f�r die Pion-Nukleon-Streuung mit den experimentellen Daten

The dispersion relation for the 3-3-amplitude given byChew, Goldberger, Low andNambu cannot be tested presently because of the experimental uncertainties of the scattering phase, but it discriminates between several interpolations of the experimental data. The only conclusion which can be drawn from the discrepancy found byTsuchida andKakazawa is that the Breit-Wigner formula is not suitable. We have shown that the experimental values are consistent with the dispersion relation if an effective range formula is used up to 120 MeV. The result for the coupling constantf2=0.097 is somewhat higher than the value used in the effective range formula, but the difference is probably not larger than the terms which are neglected in the derivation of the dispersion relation.

Dispersion Relations forp−nScattering

The application of dispersion relations to low-energy $p\ensuremath{-}n$ scattering is examined. It is shown that Khuri's dispersion relation can be extended to include tensor forces, but serious difficulties appear on attempting to include exchange forces. The application of the relativistic field theory dispersion relations to low-energy scattering is made by using the effective range formula. The spurious poles of the $S$-matrix are related to the two- and three-pion terms in the unphysical region contribution for forward scattering.

Indication forK,Λ,ΣRelative Parities from Effective-Range Analysis ofK+pScattering

From dispersion relations an effective range formula is derived for ${K}^{+}p$ scattering. In the expression for the effective range, the integrals over the cross sections are certainly convergent and weighted against the contributions from the unphysical region. This expression is then analyzed under the experimentally suggested hypothesis of rather constant ${K}^{+}p$ cross sections up to \ensuremath{\sim}110 Mev; it is observed that the effective range is rather energy independent and the integrals contributing to it are estimated to be all of the same sign. The expression for the effective range is then quantitatively evaluated, and it is shown that the comparison with the low-energy dependence of ${\ensuremath{\sigma}}^{+}$ indicates equal $\ensuremath{\Lambda}$ and $\ensuremath{\Sigma}$ parities with opposite $K$ parity ($K$ pseudoscalar). The possibility of evaluating the coupling constants from the low-energy behavior of the ${K}^{+}p$ cross section is then briefly discussed.

On the Interpretation of Resonance Levels and Their Widths in Terms of the Scattering Length and the Effective Range

The so-called effective range formula for nuclear scattering is deduced from the nuclear dispersion formalism for a fairly general class of potentials. It is shown that only in the case of zero angular momentum scattering is the angular momentum uniquely determined from the low energy variation of the cross section. The relation between the scattering length and effective range on the one hand and the parameters of nuclear dispersion theory (resonance levels and reduced widths) is briefly discussed.