Crossing symmetric amplitude with Regge poles

We obtain a representation for a pion scattering amplitude that incorporates analyticity in the Mandelstam variables, and meromorphy in the right half of the l-plane. Full crossing symmetry and elastic unitarity in the three elastic regions are built into the equations; and we show how to derive the expected Regge asymptotic behavior as one Mandelstam variable tends to infinity, while another is held fixed at any real or complex value. Moreover, we demonstrate that the inelastic unitarity bound is respected by the partial wave amplitudes, even though the Regge trajectories are allowed to rise indefinitely.     

Bounds on the elastic scattering amplitude at finite energies

A bound on the ratio of differential cross sections of particle and antiparticle at finite energies has been derived using the relation between phase and modulus of the scattering amplitude which follows from analyticity, unitarity and crossing symmetry.Presented by Yu. S.Vernov at the Symposium on Hadron-Hadron Scattering at High Energies, Liblice, Czechoslovakia, June 16–21, 1975.     

Rigorous absolute bounds for pion-pion scattering (1). Solving a first extremum problem

This paper is the first of a series of three where we reconsider the derivation of rigorous absolute bounds for strong interactions. It is devoted to the solution of a preliminary extremum problem. From the knowledge, at a given physical energy s and unphysical angle cos θ₀ > 1, of the absorptive part A (s, cos θ₀) of the elastic scattering amplitude for two spinless particles, we find, by exploiting the unitarity condition, the least upper bound of the modulus |F(s, cos θ₁)| of the scattering amplitude for the same energy s and various angles θ₁. Upper (but not least upper) bounds given by previous authors are numerically compared with ours.     

Pion–nucleon amplitude near threshold: the sigma-term and scattering lengths beyond few loops

The pion–nucleon amplitude is considered in the vicinity of the elastic scattering threshold within a relativistic dynamical model dressing the πNN and πNΔ vertices self-consistently with an infinite number of meson loops. The dressing is formulated as solution of a system of coupled integral equations incorporating unitarity, crossing symmetry and analyticity constraints. The calculated scattering lengths and the sigma-term agree with recent data analyses. The dressing is important in this model both below and at threshold. The contribution of the Δ resonance is discussed, including effects of the consistent dressing of the πNΔ vertex. A comparison with the approaches of chiral perturbation theory and the Bethe–Salpeter equation is outlined.     

关于π-π共振态

本文从解析性和么正性出发,对具有共振行为的π-π散射分波振幅的形式作了普遍的讨论。理论中包含了一个标志与Breit-Wigner共振形式偏离的函数F_(l)^I)(v),当F_(l)^I)(v)=1时,振幅恰好具有Breit-Wigner共振形式。具体计算表明,这个函数与1有一定的偏离。利用ND^-1方法、交叉对称性、ρ与f⁰的Bootstrap近似,将所得到的振幅与-90的共振解。计算中考虑了非弹性过程和负半轴色散积分的贡献。计算结果得到,对于J=1,I=1态,v_R=6.4,Г₁=0.12;对于J=2,I=0态,v_(R2)=20,Г₂=0.016。这相当于m_ρ=762MeV,m_f⁰=1283MeV,ρ的半宽度约为45MeV。它们与目前的实验数据是很好符合的。     

Partial Wave Crossing Relations for Meson-Baryon Scattering

Three different types of relations for meson-baryon partial wave amplitudes following from the s ↔ u crossing symmetry of the invariant amplitudes are investigated: i) the partial wave crossing relations directly derived from the s ↔ u crossing symmetry and the partial wave projection, ii) the partial wave crossing relations following from a combined use of s ↔ u crossing symmetry and analyticity, and iii) the crossing sum rules, which are a direct consequence of the s ↔ u crossing symmetry. The kernels entering the crossing relations i) and ii) are given in an analytic form, valid for arbitrary angular momentum. Their properties are studied in detail. By means of the partial wave crossing relations ii) it is demonstrated that u-channel exchange of a resonance leads to a resonance-like behaviour of the partial wave amplitudes on the crossed physical cut. This phenomenon is described in terms of “pseudoresonances”. Finally the crossing sum rules iii) and their practical use are thoroughly discussed, and they are compared with the partial wave crossing relations i) and ii). Special attention is paid to the crossing symmetry constraints on the S-wave amplitudes. As a result of these investigations we present a crossing symmetric ansatz for these amplitudes which gives a more reliable parametrization than the usual effective range approximation.     

A soluble realistic model for low energy ππ scattering

Using analyticity, crossing and unitarity and making a few approximations, we predict the ππ scattering amplitudes in terms of the S-wave scattering lengths from threshold up to 800 MeV.     

A multiperipheral model with continued cross channel unitarity

We derive, by using a spectral representation in momentum transfer, t, an integral equation, similar in structure to a multipheral equation, with continued cross channel unitarity, for the absorptive part for a composite particle scattering amplitude from a Bethe-Salpeter equation describing composite particle scattering in the s channel. At high energy in the t channel, the equation becomes homogeneous and has a Reggeized solution. We indicate how this equation may be solved using determinental techniques. We also show how the composite particle amplitude resulting from the original equation may be used to construct production and three body amplitudes. We also infer the possibility of studying, using the amplitude from the cross channel problem, the effect of extra unitarity on Reggeon-Reggeon-particle vertices.     

General laws of hadron scattering at very high energies

We review the general properties of particle scattering at extremely high energy and fixed momentum transfer, concentrating on features which can be derived from analyticity, crossing symmetry, unitarity and other general principles. The progress since 1973 is reported; it is pointed out that the rigorous approach yields fairly tight high-energy correlations between the phase and the modulus of the crossing-even as well as the crossing-odd forward scattering amplitude, without assuming any particular model of the interaction mechanism. The approach becomes particularly predictive when the formalism is supplemented by the phenomenological assumption that the total cross section rises unboundedly at high energies. Relation of the results to existing hadron-hadron scattering data and to future experiments is discussed.     

Asymptotics and zeros of the imaginary part of the elastic scattering amplitude

The s-channel unitarity condition for the imaginary part of the hadronic elastic scattering amplitude outside the diffraction peak is studied within different assumptions about the behavior of its real part. The integral equation for the imaginary part is derived with the asymptotical expression for the real part inserted in the unitarity condition. The conclusions about the asymptotical approach to the black disk limit and possible zeros of the imaginary part of the amplitude are obtained. Their relation to the present day experiments is discussed.     

Unitarity in Dirichlet Higgs model

We show that a five-dimensional Universal Extra Dimension model, compactified on a line segment, is consistently formulated even when the gauge symmetry is broken solely by non-zero Dirichlet boundary conditions on a bulk Higgs field, without any quartic interaction. We find that the longitudinal W ⁺ W ⁻ elastic scattering amplitude, under the absence of the Higgs zero mode, is unitarized by exchange of infinite towers of KK Higgs bosons. Resultant amplitude scales linearly with the scattering energy μ ?s\propto\sqrt{s}, exhibiting five-dimensional nature. A tree-level partial-wave unitarity condition is satisfied up to 6.7 (5.7) TeV for the KK scale m _KK=430 (500) GeV, favored by the electroweak data within 90% CL.     

An Approximation Scheme for Constructing π°π° Amplitudes from ACU Requirements

The requirements of analyticity, crossing symmetry and unitarity (ACU) are used as input to construct amplitudes for the scattering of neutral π mesons. They are obtained explicitly as polynomials in certain conformally transformed variables which ensure the correct analyticity structure. Crossing symmetry is expressed as a set of linear relations between the expansion coefficients. The same is true for the threshold behaviour of the partial waves and the behaviour for large energies of either the partial waves and the amplitudes itself. To enforce the positivity of the partial waves (up to a certain l) a new set of variables is introduced. It is obtained by solving the system of linear inequalities expressing the positivity condition. In this way one gets explicit expressions for all amplitudes satisfying the ACU requirements to some approximation. The rigorous constraints obtained from these ACU requirements are used to test the accuracy of this construction. The properties of all amplitudes are investigated in detail. It is indicated how this approximation scheme may be improved to any degree of accuracy and how it may be generalized to the full isospin case.     

Thepp —p \bar p difference and the behaviour of phases at high energies

Predictions based on axiomatic field theory are obtained from the assumption that the difference between thepp and thep \(\bar p\) elastic scattering observed at the CERN ISR is caused by an “odderon-like” term in the scattering amplitude. A wide class of amplitudes violating the Pomeranchuk relation is considered. Contrary to common opinion, severe restrictions on the high-energy behaviour of the phase of the crossing-odd amplitude are shown to follow for the whole class. Less pronounced but still visible consequences follow for the phases of thepp andp \(\bar p\) amplitudes separately. We derive the corresponding high-energy bounds and correlations using the general frame of analyticity, crossing symmetry and unitarity. As a special case, the maximal odderon amplitude is discussed.     

Bounds on the π0π0 amplitude

A new approach to the problem of constraining the scattering amplitudes by using axiomatically proved properties of unitarity, analyticity and crossing is developed. It is applied to obtain absolute and semi-phenomenological bounds on the π⁰π⁰ amplitude at points lying on the Mandelstam triangle, in particular at the physical threshold (scattering length) and at the symmetric point (which is used to define the coupling constant). We have already reported a less general and powerful approach which improved the previous results in this field. The present version, which constitutes a noticeable generalization of the method, allows a new improvement of the bounds, both absolute and semi-phenomenological.     

Semi-phenomenological bounds for the pion-pion coupling constant

Upper and lower bounds on the Chew-Mandelstam ππ coupling constant λ are derived as functions of the D-wave scattering length a₂. The only information used is the axiomatic analyticity domain of the ππ scattering amplitude, and the crossing and unitarity properties. For a₂ = 7 × 10^?4, one obtains ?0.164<λ<0.162. This is compared with previous bounds.     

Asymptotics of the S-matrix and unitarisation

The manner in which the elastic scattering amplitude obeys unitarity, how it enters the circle of unitarity, and what its asymptotic limit is, remains a problem for models which include terms that rise fast with s. We have checked that the features of cross sections which come from unitarisation are present for most unitarisation schemes, e.g. those that saturate the profile function or those that describe multiple exchanges via an analytic formula. We have also obtained a scheme which interpolates between different classes of the unitarisation and found corresponding non-linear equations. Considering different forms of energy dependence of the scattering amplitude, and a variety of unitarisation schemes, we show that, in order to reproduce the data, the fits choose an amplitude that corresponds to an asymptotic value S = 0.     

Improvement of Multichannel Amplitudes for the Pion-Pion Scattering Using the Dispersion Relations

The multichannel S- and P-wave amplitudes for the ππ scattering, constructed requiring analyticity and unitarity of the S-matrix and using the uniformization procedure, are elaborated using the dispersion relations with imposed crossing symmetry condition. The amplitudes are modified in the low-energy region to improve their consistency with experimental data and the dispersion relations. Agreement with data is achieved for both amplitudes from the threshold up to 1.8 GeV and with dispersion relations up to 1.1 GeV. Consequences of the applied modifications, e.g. changes of the S-wave lowest-pole positions, are presented.     

Mesons of the f0 family in processes ππ↦ππ, K¯K up to 2 GeV and the chiral-symmetry breaking

In a combined analysis of the experimental data on the coupled processes ππ↦ππ, KˉK in the channel with I ^G J ^PC = 0⁺0^{+ +}, the various scenarios of these reactions (with different numbers of resonances) are considered. In a model-independent approach, based only on analyticity and unitarity, a resonance is represented by a pole cluster (poles on the Riemann surface) of the definite type that is defined by the state nature. The best scenario contains the resonances f ₀(665) (with properties of the σ-meson), f ₀(980) (with a dominant sˉs component), f ₀(1500) (with a dominant flavour-singlet, e.g., glueball component) and the f ₀(1710) (with a considerable sˉs component). If the f ₀(1370) exists, it has a dominant sˉs component. The coupling constants of the observed states with the considered channels and the ππ and KˉK scattering lengths are obtained. The conclusion on the linear realization of chiral symmetry is drawn. Received: 25 April 2002 / Accepted: 4 June 2002 / Published online: 26 November 2002 RID="a" ID="a"e-mail: surovcev@thsun1.jinr.ru RID="b" ID="b"e-mail: krupa@savba.sk RID="c" ID="c"e-mail: fyzinami@nic.savba.sk Communicated by V.V. Anisovich     

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.     

Consequences of Crossed Channel Structure for High Energy Elastic Scattering

This review describes a field in the S-matrix theory of elastic stattering. Analyticity and elastic unitarity in the crossed t-channel are shown in a rigorous manner to yield constraints on s-channel quantities. Strong restrictions on s-channel partial wave with angular momenta l ⩾√s/2(b₀ + λ In s) follow from realistic assumptions on the amplitude in the unphysical domain (Mandelstam analyticity and Regge behaviour). The possibility of phenomenological application of these results is discussed.