Exact integral positivity and crossing constraints on S and P wave pion-pion scattering amplitudes

S- and P-wave pion-pion scattering amplitudes are constrained by the fact that they have to be compatible with crossing symmetry and the positivity of the absorptive parts of all partial waves. The construction of the necessary and sufficient conditions ensuring the compatibility with positivity and an arbitrary finite set of Balachandran-Nuyts crossing conditions is described. Two sets of constraints are derived explicitly. The first set contains inequalities involving integrals of the S-waves over the unphysical interval 0?s?4m_π². The inequalities of the second set relate integrals of S-waves over the unphysical interval to integrals of S- and P-wave absorptive parts extended to the physical region.     

Rigorous sum rules and bounds on pion-pion scattering amplitudes with positivity

We derive the complete set of crossing symmetric forward pion-pion amplitudes with non-negative absorptive parts obeying twice-subtracted dispersion relations. We give a method to eliminate subtraction constants for these amplitudes by making a simultaneous use of subtracted dispersion relations for the amplitude and its inverse, and unitarity. We thus deduce from axiomatic field theory (i) lower bounds on forward real parts in terms of physical forward amplitudes, (ii) lower bounds on S- and P-wave scattering lengths in terms of forward amplitudes in any part of the physical region, (iii) sum rules on scattering lengths in terms of physical forward amplitudes which provide a complete test of forward dispersion relations, (iv) upper bounds on scattering lengths in terms of total cross sections and (v) absolute sum rule inequalities on total and forward differential cross sections in terms of the pion mass alone which provide new tests of dispersion relations not involving scattering lengths.     

Quasilinear models of low-energy pion-pion scattering and the rigorous constraints below threshold

Models of S- and P-wave pion-pion scattering are investigated. Their amplitudes satisfy elastic unitarity and are quasilinear in the sense that they are nearly linear functions of s in the unphysical interval 0 < s < 4m_π². The models are submitted to rigorous constraints holding in this interval and resulting from crossing symmetry and positivity of the absorptive parts. If one imposes only the linear parts of the amplitudes as input, these constraints have nearly no restrictive effect. However, if the ?-meson is required and if the shape of the I = 2 S-wave is fixed, the I = 0 S-wave is strongly restricted in a small but finite domain above threshold (280 MeV < m_ππ ? 450 MeV). In the ?-region, the limitations resulting from the constraints are no more significant. One concludes that the rigorous constraints below threshold have a physical relevance which is restricted to a small energy interval above threshold.     

Exact representation of meson-meson scattering amplitudes consistent with current-algebra constraints

All four-point Green functions connected to the vector and axial-vector currents by Ward identities are constructed, consistent with the constraints of an arbitrary current algebra. The structure of the four-point functions is separated into one-particle reducible and irreducible parts, without approximations. That is, no single-particle approximation is made, nor is it assumed that the strangeness-changing vector currents are conserved. The result is a representation of the meson-meson scattering amplitude that explicitly satisfies all the constraints of the current algebra. In particular, it satisfies: (i) threshold theorems, (ii) crossing symmetry and (iii) reduction to the usual tree-approximation in the single-particle approximation. Application is made to kaon-pion scattering, obtaining a representation of the scattering amplitude that is suitable for the investigation of the constraints of elastic unitarity.     

Dispersion relations for forward pion-pion scattering and their consistency with Wolf pion-pion phase shifts

It is shown that the Wolf pion-pion phase shifts are in reasonable agreement with dispersion relations for forward scattering. The extent to which the inaccuracies of the experimental data might modify this agreement is examined.At attempt is made to explain some discrepancies which arise probably from too broad a -meson peak.The author is indebted to Dr. M. Petrá for valuable discussions, to P. Bóna for help with some calculations and to Dr. J. Mikloko from the Computing Centre of the Slovak Academy of Sciences for programming and computation. He is also indebted to Dr. G. Wolf and to Dr. F. Meiere for sending a copy of their papers prior to publication.     

Analysis of the constraints and correlations the Roy equations impose on S- and P-wave pion-pion scattering

The solutions of the S- and P-wave Roy equations which are infinitesimally close to the physical solutions are analysed. This gives an insight into the deviations from the physical phase shifts which are allowed by analyticity and crossing symmetry.     

Inequalities on left-hand discontinuities of pion-pion partial wave amplitudes

The partial wave crossing relations derived by Balachandran and Nuyts are used to derive inequalities among integrals over left-hand discontinuities of elastic pionpion scattering, weighted by Jacobi functions of the second kind. The only other input for these inequalities are — besides isospin — positivity and boundedness of the right-hand imaginary parts. It is pointed out that similar inequalities follow directly from recent inequalities of Martin.     

Phenomenology of S and P pion-pion partial wave amplitudes

A scheme of parametrization ofππ S andP partial wave amplitudes is proposed. Analytic, crossing and unitarity properties are rigorously satisfied. The analyticity property is made explicit by the use of a conformally mapped variable. Our expressions fit excellently with both phase shift and inelastic curves for centre of mass energy up to 1·4 GeV.     

A possibility of a pion-pion scattering experiment

A possibility of a pion-pion collision experiment is discussed. This experiment permits one to perform very precise measurements of the s-wave scattering lengths at isospin state of I = 0 and 2 and a pion energy of 200 MeV-1 GeV. Quark-antiquark condensate, effective Lagrangian theory, and some features of the QCD lattice approach can be tested with very high precision. The text was submitted by the authors in English.     

The manifold of solutions of Roy's S- and P-wave equations for pion-pion scattering: The neighbourhood of the physical amplitudes

We study the solutions of the coupled S- and P-wave Roy equations for low-energy pion-pion scattering which are in the neighbourhood of a given solution. We give a general method for constructing these solutions for fixed inelasticities and driving terms and variable S-wave scattering lengths. In the case of the neighbourhood of the physical amplitudes, our procedure leads to a seven-dimensional manifold of solutions. If we omit variations affecting only the I = 0 S-wave around the $\text{K}\text{K}$ threshold, the number of dimensions is reduced to five, in accordance with the results of the phenomenological analysis of low-energy pion-pion scattering. Our solutions allow a study of the correlations between S- and P-waves implied by the Roy equations. This work completes a previous one dealing with the single-channel problem of the I = 1 P-wave.     

Quark scattering amplitudes with quasi-elastic unitarity

We consider quark-quark scattering at high energies and fixed momentum transfer. In a model where in the s and u channel intermediate states only gluons with sufficiently small transverse momenta are emitted, the scattering amplitudes are expressed in terms of the S-matrix elements for exactly soluble two-dimensional field theories.     

Consistency constraints on from QCD dispersion relations and chiral perturbation theory in decays

We use both old and new theoretical developments in QCD dispersion relation constraints on the scalar form factor in the decay to obtain constraints on the strange quark mass. The perturbative QCD side of the calculation incorporates up to four-loop corrections, while the hadronic side uses a recently developed parameterization constructed explicitly to satisfy the dispersive constraints. Using chiral perturbation theory ( PT) as a model for soon-to-be measured data, we find a series of lower bounds on increasing with the accuracy to which one believes PT to represent the full QCD result.     

Unitarity structure of scattering amplitudes

As a first step towards constructing scattering amplitudes satisfying unitarity, analyticity and crossing symmetry, we derive a linear non-singular integral equation for the total scattering amplitude which is equivalent to the unitarity condition. For this purpose we use the partial-waveN/D representation (with inelasticity) and the convolution theorem for Legendre transforms. We also discuss briefly the choice of two functionsN(s, cos Θ),C(s, cos Θ) which determine the unitary scattering amplitude through the integral equation. These functions may hopefully be chosen so that the analyticity and crossing symmetry requirements are satisfied.

     

Determination of Regge trajectories in a bootstrap model of pion-pion scattering

A simple expansion is used for calculating the zero-energy intercepts of the leading Regge trajectories of the pion-pion scattering amplitude. The results agree well with those of other determinations.     

Consistency tests of Ampcalculator and chiral amplitudes in SU(3) Chiral Perturbation Theory: A tutorial-based approach

The positive-parity yrast bands of ^{79, 81, 83, 85, 87, 89}Y isotopes have been studied using the projected shell model (PSM). Nuclear-structure properties like yrast spectra, transition energies, band diagrams, kinetic moment of inertia, rotational frequencies and reduced transition probabilities (B(M1) and B(E2) are calculated. The results obtained from the PSM calculations are also compared with the available experimental as well as theoretical data and, in general, a reasonable agreement is obtained between them. Calculations in the present work also predict that these isotopes have multi-quasiparticle structure.     

On the low-energy πN scattering amplitudes

The pion-nucleon scattering amplitudes with current algebra constraints involving “remainder terms” are calculated by saturating them with the N¹(1236) contribution having off-mass shell effects and compared with experiments.     

On the superconvergence of meson-nucleon invariant scattering amplitudes

Using two models, i.e. perturbation theory with the usual PS-PS Hamiltonian and a “relativistic” version of the Lee model with the same type of interaction, it has been tried to clarify the question if there are superconvergent amplitudes in the (π, K) meson-nucleon forward scattering. Does the impressive validity of the well-known sum rule for theB ⁽⁺⁾ amplitude makes a serious support of the assumed superconvergence (becauseB describes the spin-flip scattering)? Indications, obtained from perturbation theory, including all Feynman diagrams up to the sixth order in the coupling constant and from the Lee model, where all allowed diagrams can be summed up, do not support the superconvergence assumption. Therefore the impressive prediction of theB ⁽⁺⁾ amplitude sum rule seems to be accidental.     

Unitary amplitudes with cut-plane analyticity from given scattering data

We continue the study of constructing the scattering amplitude for scalar particles from elastic scattering data at a given energy. Here we require the scattering amplitude, f (z), to be analytic in a suitably cut z-plane; (z is the cosine of the scattering angle). We find that, in the elastic domain, the unitarity constraint is satisfied by a continuum of amplitudes, all corresponding to the same elastic scattering data. This continuum ambiguity in determination of f (z), which was noted earlier in formulations based on a smaller domain on analyticity, is associated with lack of knowledge of the contribution to unitarity from inelastic channels. We discuss the problem of incorporating smooth energy dependence in the determination of the scattering amplitude over some range of energy, and show that, under certain restrictions on the scattering data, there is a continuum of amplitudes having smooth energy dependence.     

Compact multigluonic scattering amplitudes with heavy scalars and fermions

Combining the Berends-Giele and on-shell recursion relations we obtain an extremely compact expression for the scattering amplitude of a complex massive scalar-antiscalar pair and an arbitrary number of positive helicity gluons. This is one of the basic building blocks for constructing other helicity configurations from recursion relations. We also show explicitly that the scattering amplitude of massive fermions to gluons, all with positive helicity, is proportional to the scalar one, confirming in this way the recently advocated SUSY-like Ward identities relating both amplitudes.