Interior Dispersion Research Articles

Tests of current algebra and partially conserved axial-vector current in the subthreshold region of the pion-nucleon system

Current algebra and partially controlled axial-vector current ~PCAC! make powerful predictions in the threshold and subthreshold regions of the pN system @1#. Some of the most important examples are the Adler consistency condition @2# ~which predicts a zero in the pole-subtracted isospin-even amplitude for one soft pion!, the Weinberg-Tomozawa prediction of pN scattering lengths @3#, the Adler-Weisberger sum rule @4# ~which constrains the isospin-odd amplitude at n5t50 for massless pions!, and the pN s term ~which measures the chiral-symmetry breaking in the pN system!. The values of these quantities are determined from amplitudes evaluated within the small subthreshold ‘‘crescent’’ region in the Mandelstam diagram shown in Fig. 1 @5#. Because the crescent lies below the physical threshold of all of the crossed reactions (pN→pN , pN→pN , and NN→pp!, the invariant amplitudes are real in this region. In this article we report on an application of a recent Virginia Polytechnic Institute phase-shift analysis SP98 @6# and interior dispersion relations ~IDR’s! @7# to map the relevant amplitudes within the entire crescent @8#. Hence, this analysis provides current tests of several predictions of chiral symmetry. The present work is an extension and update of work done at the beginning of the era of the meson factories @9#; since then, new high-precision pN data have been obtained, from which improved low-energy phase shifts have been extracted. The general structure of the pion-nucleon invariant amplitudes has long been known @10#, of course, but the IDR method is especially well suited for studies of the subthreshold region, especially when coupled with the high-precision VPI phase-shift analyses.

ππscattering amplitudes in the subthreshold region

$\ensuremath{\pi}\ensuremath{\pi}$ scattering amplitudes inside the subthreshold triangle $(0<~s<~\ensuremath{\Sigma}$, $0<~t<~\ensuremath{\Sigma}$, $0<~u<~\ensuremath{\Sigma}=4{\ensuremath{\mu}}^{2})$ are studied using interior dispersion relations. The dispersion integrals are evaluated above a center-of-mass energy ${W}_{\ensuremath{\pi}\ensuremath{\pi}}=0.6$ GeV by using a standard phase shift analysis and from this value down to threshold by using unitary models consistent with existing low-energy experimental data. It was found that by restricting the $s$-wave scattering lengths to lie on a ``universal curve'' all crossing properties and appropriated threshold sum rules were reasonably well satisfied. The invariant amplitudes are found to be in good qualitative agreement with the predictions of Weinberg's chiral model and its corrections derived from chiral perturbation theory.

ππ scattering amplitudes within the sub-threshold triangle

The on-shell ππ scattering amplitudes within and on the boundary of the subthreshold triangle (0 ≤ s ≤ Σ, 0 ≤ t ≤ Σ, 0 ≤ u ≤ Σ ≡ 4 μ 2) are investigated. Interior Dispersion Relations, when combined with experimental ππ phase shifts, inelasticities, and scattering lengths, provide an almost perfect tool for this study. The general results obtained are in striking agreement with the chiral model of Weinberg.

On difference schemes with artificial dispersion

The interior dissipative and dispersion properties of difference schemes for non-linear hyperbolic equations are studied by analyzing the differential approximation. The examples of the quasilinear transport equation and the system of equations of isothermal gas dynamics are considered. A class of schemes with artificial dispersion is studied, whereby the distorting effect of interior dispersion and dissipation on the difference solution is weakened. The theoretical results are confirmed by direct computations.

The isosinglet and isovector ?? scattering lengths

The ππ scattering lengthsa00,a20 anda11 are determined from πN elastic scattering data using interior dispersion relations. The importance of the Born-Term contribution, via unitarity, to the imaginary part of all amplitudes is discussed. Proper consideration of these contributions and the analytic properties of the amplitudes near threshold allows us to obtain from the recent πN partial wave analysis of Pietarinen the following scattering lengths $$\begin{gathered} \mu a_0^0 = 0.27 \pm 0.03,\mu ^3 a_1^1 = 0.032 \pm 0.005, \hfill \\ \mu ^5 a_2^0 = 0.002 \pm 0.001. \hfill \\ \end{gathered} $$

Applications of interior dispersion relations to pion photoproduction near threshold

Interior dispersion relations are applied in the boundary limit to the determination of certain threshold and subthreshold quantities for pion photoproduction, viz., the threshold s-wave multipole amplitudes E/sup i//sub 0//sub +/, and the amplitudes involved in the current-algebra theorems of Fubini and Furlan, and of Adler and others. Where comparisons are possible, our results are consistent with determinations by other methods. We also find further substantiation for conclusions reached elsewhere concerning the continuation of soft-pion theorems to their on-shell versions.

The I = 0, s-wave ππ scattering length

Interior dispersion relations are applied to πN scattering amplitudes to extract the I = 0, s-wave ππ scattering length, a 0 0. Two methods are used, the second of which also incorporates known ππ scattering phase shifts. We have used the πN amplitude analysis of Pietarinen and obtained values of a 0 0: (0.28 ± 0.09) μ − and (0.25 ± 0.08) μ − for the two methods respectively, consistent with other recent determinations from K e4 and πN → ππN data. The 1971 πN phase-shift analysis of Ahmehed and Lovelace was also used successfully in the first method to obtain the value a 0 0: (0.15 ± 0.14) μ −, consistent with the above. The second method was found to be ineffective in the face of relatively unsmooth discrepancy functions obtained from the Ahmehed-Lovelace phase shifts.

A study of [formula omitted] partial-wave amplitudes using interior dispersion relations

We investigate ππN N partial-wave amplitudes using a spin separation based on interior dispersion relations. The couplings of the f 0(1271) and g(1688) mesons are determined. Evidence is seen for J = 4 and J = 5 contributions from regions near and above the N N threshold. No evidence for p-wave contributions other than ϱ exchange is seen. The connection of our f 0 couplings with s-channel helicity conservation and tensor meson dominance is discussed.

Test of the partial conservation of axial-vector current limit in pion-nucleon scattering

A comparison is made of the on-shell amplitudes ${\overline{F}}^{(\ifmmode\pm\else\textpm\fi{})}(0, t)$, obtained using interior dispersion relations, to their off-shell current-algebra predictions in order to test the pion PCAC (partial conservation of axial-vector current) limit in pion-nucleon scattering. It is concluded that, when $\ensuremath{\nu}$ and $t$ are used as variables, the on-shell corrections to PCAC are on the order of 10% per pion.

Higher spin exchange in [formula omitted] scattering

An improved evaluation of the interior dispersion relation for πN isospin odd amplitudes C − B − leads to fairly well-defined g-meson exchange contributions. The J = 1 partial waves are practically saturated by ϱ-exchange, i.e. there is no significant ϱ′-exchange contribution in contrast to the result of an earlier simplified evaluation. Evidence for a spin 5 exchange contribution is seen in the C − amplitude.

Determinations of the πN σ term and s-wave scattering lengths

Determinations of the σ term and s-wave scattering lengths for pion-nucleon scattering are made using interior dispersion relations. The result for the σ term is obtained by extrapolating to the Cheng-Dashen point in only one variable. It is completely independent of the pion-nucleon coupling constant and insensitive to d and f waves in the resonance region. The use of interior dispersion relations allows a series of s-wave scattering length determinations to be made. It is observed that data at intermediate scattering angles gives more consistent results than data near the forward or backward directions, lending further support to the σ term calculation which utilizes only interior angle information.

Determination of the pion-nucleon coupling constant

A method for determining the pion-nucleon coupling constant ${f}^{2}$ by the application of an interior dispersion relation to the ${B}^{(+)}\ensuremath{\pi}N$ scattering amplitude is presented and its advantages over other dispersion relation techniques are discussed. The method is used to calculate ${f}^{2}$ for two sets of phase shifts: CERN 71 of Almehed and Lovelace and a modified version of the Carter, Bugg, and Carter solution, to obtain ${f}^{2}=(81.5\ifmmode\pm\else\textpm\fi{}1.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ and ${f}^{2}=(79.5\ifmmode\pm\else\textpm\fi{}1.0)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, respectively.

Spin content of crossed-channel contributions and interior dispersion relations

A method of using interior dispersion relations is presented to measure the relative importance and spin content of crossed-channel contributions to two-body scattering amplitudes in the physical region. As an illustration of the method, the ${B}^{(\ensuremath{-})}$ amplitude of $\ensuremath{\pi}N$ scattering is considered. It was found that in the physical region this amplitude receives contributions essentially only from the two lowest crossed-channel spin states. The higher of the two states ($J=3$) was found to be adequately described by the $g(1680)$ meson, while the lower ($J=1$) could not be described by the $\ensuremath{\rho}(760)$ meson alone, but could be described by a sum of the $\ensuremath{\rho}(760)$ and the ${\ensuremath{\rho}}^{\ensuremath{'}}(1600)$ meson.

An interior dispersion relation determination of the πN σ-term

We determine the value of the πN σ-term by interior dispersion relations to be σ π N = 68 MeV.