Roy equation analysis of ππ scattering

With the aim of generating new constraints on the OZI suppressed couplings of chiral perturbation theory a set of six equations of the Roy and Steiner type for the S- and P-waves of the $\pi K$ scattering amplitudes is derived. The range of validity and the multiplicity of the solutions are discussed. Precise numerical solutions are obtained in the range $E\lesssim 1$ GeV which make use as input, for the first time, of the most accurate experimental data available at $E\gtrsim 1$ GeV for both $\pi K\to\pi K$ and $\pi\pi\to K\overline{K}$ amplitudes. Our main result is the determination of a narrow allowed region for the two S-wave scattering lengths. Present experimental data below 1 GeV are found to be in generally poor agreement with our results. A set of threshold expansion parameters, as well as sub-threshold parameters are computed. For the latter, a matching with the SU(3) chiral expansion at NLO is performed.

A new analysis of pi K scattering from Roy and Steiner type equations

Roy-Steiner-equation analysis of pion-nucleon scattering

We review the structure of Roy-Steiner equations for pion-nucleon scattering, the solution for the partial waves of the t-channel process $\pi\pi\to \bar N N$, as well as the high-accuracy extraction of the pion-nucleon S-wave scattering lengths from data on pionic hydrogen and deuterium. We then proceed to construct solutions for the lowest partial waves of the s-channel process $\pi N\to \pi N$ and demonstrate that accurate solutions can be found if the scattering lengths are imposed as constraints. Detailed error estimates of all input quantities in the solution procedure are performed and explicit parameterizations for the resulting low-energy phase shifts as well as results for subthreshold parameters and higher threshold parameters are presented. Furthermore, we discuss the extraction of the pion-nucleon $\sigma$-term via the Cheng-Dashen low-energy theorem, including the role of isospin-breaking corrections, to obtain a precision determination consistent with all constraints from analyticity, unitarity, crossing symmetry, and pionic-atom data. We perform the matching to chiral perturbation theory in the subthreshold region and detail the consequences for the chiral convergence of the threshold parameters and the nucleon mass.

Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As a first step towards the solution of the full system we cast the equations for the $\pi \pi \to \overline N N$
 partial waves into the form of a Muskhelishvili-Omnes problem with finite matching point, which we solve numerically in the single-channel approximation. We investigate in detail the role of individual contributions to our solutions and discuss some consequences for the spectral functions of the nucleon electromagnetic form factors.

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Roy-Steiner equations for pion-nucleon scattering

We study the implications of crossing and analyticity properties on partial-wave amplitudes in the physical region. We first show the physical interest of the ππ equations recently written by Roy; comparison with the Chew-Mandelstam equations is particularly instructive, and we show how these equations can be used:a) to construct low-energy amplitudes in practice, andb) to check the consistency of experimental data (adirect method for solving the «up-down» ambiguity and calculatings-wave scattering lengths is obtained). We then reconsider the Martin inequalities and the sum rules from which they are derived, and show that they can be used in a more powerful way. Our considerations and results give strong support to the model of Le Guillou, Morel and Navelet for ππs-waves. We finally give the general method for deriving physical-region crossing equations for arbitrary processes, in particular πN→πN.

/pdf/crossing-and-physical-partial-wave-amplitudes-27rh9ytky0.pdf

Crossing and physical partial-wave amplitudes

We present a calculation ofS-wave ππ amplitudes based upon known rigorous results due to analyticity, unitarity and crossing symmetry; inequalities and crossing sum rules. The experimentalP-wave phase shifts between 500 and 1100 MeV are used as an input, and it is shown that the method leads to sharpS-wave phase-shift predictions for a given very-low-energyP-wave amplitude. Uncertainties in the extrapolation of the experimentalP-wave phase shifts downwards are mainly responsible for a reasonable dispersion ofS-wave phase shifts. Much care is taken in order to introduce correctly long-range forces into the game. The results, in very good agreement with experiment, are physically interpreted, and compared with those of other models, in particular current algebra and field theory calculations.S-wave scattering lengths and zeros are found to be in agreement with current algebra models. A well-defined σ-meson is predicted under the form of a second-sheet pole inI=0S-wave amplitude. Its mass,m
σ=420 MeV, is remarkably independent of any variation of all parameters, in particular of theP-wave scattering length, and it exists even if the phase shift does not pass through 90°. This phase shift is found to be of the «down» type above the ρ mass in the absence of inelasticity. Our conclusion is that, if there are no strongI=2D-wave forces, the relationship betweenS andP waves is well determined by analyticity, crossing symmetry and unitarity.

/pdf/unambiguous-pps-waves-from-general-principles-and-the-4skht5y3a3.pdf

J. C. Le Guillou

Papers

Crossing and physical partial-wave amplitudes

Unambiguous ππS-Waves from general principles and the existence of the ρ-meson