Left-hand Discontinuity Research Articles

Exchange forces in dispersion relations investigated using circuit relations.

We propose a novel method to compute in an exact manner the left-hand cut discontinuity of the electron-atom partial wave scattering amplitude in the complex energy plane within the static exchange approximation. Zero energy dispersion relations for electron-hydrogen scattering are computed numerically for illustration.

Partial-wave dispersion relations: Exact left-handE-plane discontinuity computed from the Born series

We show that for a superposition of Yukawa potentials, the exact left-hand cut discontinuity in the complex-energy plane of the (S-wave) scattering amplitude is given, in an interval depending on n, by the discontinuity of the Born series stopped at order n. This establishes an inverse and unexpected correspondence of the Born series at positive high energies and negative low energies. With the discontinuity on the left-hand axis elucidated, we can construct a viable dispersion relation (DR) for the partial (S-) wave amplitude. The DR is numerically verified for the exponential potential at zero scattering energy. Generalization to higher partial waves, and extension of these ideas to field theory are discussed.

Unitarity corrections to kaon-pion scattering from current algebra

A method is presented to unitarize the hard-meson current algebra kaon-pion amplitude derived by Ford. Elastic unitarity is satisfied in the lowest-order correction to current algebra. The amplitude contains̀ three real parameters and does not have any arbitrary function for simulating left-hand discontinuities. Once the isospin-half P-wave is given in the low-energy region, this method leads to a sharp determination of the S-wave phase shifts. Our solution favors the “down” type for the I = 1 2 S-wave phase shifts.

Real Part of theK+pAmplitude Based on Duality

The concept of duality is applied to the real part of ${K}^{+}p$ scattering. We study this part of the amplitude via fixed-$t$ dispersion relations. We are interested in seeing what effect replacing the left-hand discontinuities (${K}^{\ensuremath{-}}p$ scattering) by a low-energy extrapolation of Regge-pole amplitudes will have on the real part or the right-hand side (${K}^{+}p$ scattering).

Unitarity corrections to pion-pion scattering from current algebra

A method is presented to unitarize the hard-pion current-algebra pion-pion amplitude derived by Schnitzer. A three-parameter crossing symmetric amplitude is obtained that does not have any arbitrary function for simulating the left-hand discontinuities. Elastic unitarity is satisfied in the lowest-order correction to current algebra. Once the P-wave is given in the low-energy region, this method leads to a sharp determination of the S-wave phase shifts. Our solution favors the down type for the I = 0 S-wave phase shifts.

An improvedN/D method for inelastic partial-wave amplitudes

The problem of constructing the partial-wave amplitudes exhibiting a given left-hand cut and a given inelasticity is reformulated in terms of an improvedN/D method. This method presents the following advantages. The form of the basic equations is independent of the asymptotic behaviour of the left-hand discontinuity. The effects of inelasticity are evaluated explicitly and disconnected from the effects of the left-hand cut. The inherent ambiguities of anN/D decomposition do not affect the final expression of the scattering amplitude. The CDD pole ambiguity is reduced to the arbitrariness in the choice of a meromorphic function in this final expression. The description of elastic or inelastic resonances serves as illustration of the method.

Unitarized pion-pion Veneziano partial-wave amplitudes

A unitarization method is investigated which is based on dispersion relations for the inverse amplitude; the zeros of the partial-wave amplitudes are taken from the Veneziano model and the discontinuity across the right-hand cut is given by elastic unitarity, whereas crossing symmetry yields the left-hand cut discontinuity (for large negatives-values the latter is assumed to be constant). Two free subtraction constants are fixed by minimizing the violation of crossing symmetry-expressed in terms of a number of exact conditions on the partial-wave amplitudes. It was not possible to satisfy all of the crossing constraints simultaneously, but we arrived at a unique solution of the dispersion relations (with a very broadI = 0S-wave resonance) which is unitary and approximately crossing symmetric.

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.

N/D equations: Self-damping mechanism, threshold behaviour and subtracted equations, Regge-pole hypothesis

Forl≠0 we study, for «regular» left-hand cut discontinuities, theN/D equations of the Fredholm type. The tool used is the inversion formalism from which we getN/D equations, if the exist, as limiting equations. On the one hand analysing the problem of correct threshold behaviour for theS l matrix reconstructed fromN/D, we study the corresponding avalable subtracted equations. We find that for thel-th wave there exist (l+1) equations. For thep-th subtracted equation (p=0,…,l) we investigate according to different discontinuities: i) the corresponding potentials, ii) the asymptoticD behaviour and the Levinson theorem, iii) the intrinsic conditions satisfied by the discontinuity, iv) the problem of ghosts and bound states. From the considered discontinuities we get two different kinds of interactions: i) leading only to Regge poles in complexl, ii) where fixed branch points occur also in complexl, andD does not go to constant asymptotically, (subtracted equation). On the other hand we discuss the problem of a self-damping mechanism for «regular» discontinuities approximated by poles (or more general approximations) and we distinguish the interactions according as the corresponding states satisfy or not usual spectrum requirement. In the residues of the pole space the self-damping domain of physically available interactions for thel-th wave is a connected domain and the boundary belongs to the surface such that the solutions of thel-th subboundary equations are also solutions of the (l-1)-th subtracted one.

The DoubleN/DMethod

A method of obtaining the partial-wave amplitude by two steps is proposed, if the amplitude is written as T = t+r where t satisfies the unitarity relation by itself: first t is solved by the N/D method, and then r is solved with the solution of t and the left-hand discontinuity of r as inputs by using the N/D method again. It is proved that the particle pole of T appears in r, not in t. As an interesting application, three examples are discussed: these are the addition of the modified Born term, the channel coupling effects, the effects of the small change of the left-hand discontinuity. For the last case the shifts of the mass and the coupling constant are practically the same as those derived by Dashen and Frautschi, although their derivation is very questionable.

Existence of solutions of partial-wave dispersion relations and singularN/D equations

It is shown that, for left-hand discontinuities which asymptotically exceed certain limits imposed by unitarity, the partial-wave dispersion relation admits no ghost-free solutions. However, for discontinuities which on the left tend to a limit λ, 0<λ<1, the singular integral equation that arises by application of the usualN/D approach is shown to lead to a one-parameter manifold of ghost-free solutions; in particular, a specific solution which is analytic at λ=0 belongs to this manifold. The variation of the zeros of theD function (for varying λ) is shown explicitly in a simple example.

Bootstrap solutions in a vector-meson model

In a model with a left-hand discontinuity defined by the exchange of a vector meson with a sharp cut-off elastic-unitary bootstrap solutions are derived for a number of values of the vector meson mass, by solving the Fredholm integral equation obtained from the usualN/D decomposition. The corresponding self-consistent pseudoscalar-vector couplings are found to be larger than the experimental ones. In all cases the iteration expansion (in powers of the coupling) is shown to converge. It is then proved that the denominator functions are Herglotz; and that all the self-consistent solutions obtained correspond to ghost-free amplitudes. A sum rule derived on grounds of the asymptotic and threshold properties is found to be fulfilled. Finally, Levinson's theorem is shown to be satisfied.

Zeros of the Partial-Wave Scattering Amplitude

Under the general assumptions of analyticity, unitarity, temperedness, and normal threshold behavior, the relation between the number of zeros and the high-energy behavior of the partial-wave scattering amplitude is studied. If the left-hand-cut discontinuity makes a finite number of sign changes, the maximum and minimum numbers of zeros can be determined by the high-energy upper and lower bound of the partial-wave respectively. In particular, for $C{|s|}^{\ensuremath{-}1+\ensuremath{\epsilon}}&lt;|{f}_{l}(s)|&lt;C$, the number of zeros is determined. After having determined the number of zeros, the number of subtractions as well as the sign of the scattering length is obtained for given number and character of zeros of the left-hand discontinuity $\ensuremath{\Delta}{f}_{l}(s)$.

Возможность нестабильности парциальных волновых амплитуд относительно нарушения непрерывности слева

Nonrelativistic, purely elasticS- andP-wave two-pole models are discussed. The location of the poles of the partial-wave amplitudes is studied as a function of the input parameters. Some values of these parameters lead to unphysical results. The amplitudes are instable with respect to the input parameters for critical values of these parameters.

Sur la formulation de modèles de l'amplitude de diffusion élastique à partir de la méthode ND−1

The ND −1 equations for the elastic scattering of two identical particles without spin are written under the hypothesis that the partial wave amplitude satisfies a dispersion relation with one subtraction. The N and D functions satisfy a set of two coupled singular integral equations and depend on the left hand discontinuity and on the inelasticity factor. These quantities are the basic parameters for model theories, and if they satisfy suitable asymptotic conditions, the solution of the ND −1 equations exists and is unique. The asymptotic behaviour of the real part of the phase shift is studied and gives a possible generalization of Levinson's theorem. Some simple models are examined, which illustrate various points of the method.

Analytic and asymptotic properties of solubleN/D models

A class of relativistic models, defined in terms of theN/D equations for asymptotically varying left-hand discontinuities and constant inelasticity, is considered. This is shown to present an alternative, analogous in certain respects to, but simpler than, the class of models based on the ladder approximation of the Bethe-Salpeter equation. In terms of anN/D amplitude with an asymptotically vanishing left-hand discontinuity an approximate strong-coupling calculation of the Pomeranchuk trajectory in agreement with the Regge pole analysis of the diffraction scattering experimental data is given; the corresponding residue function is also determined. AnN/D amplitude with an asymptotically constant left-hand discontinuity is shown to be associated with a nonshrinking diffraction peak. Certain analogies to the solutions for nonrelativistic scattering by regular or singular energy-independent potentials are also discussed.

Bound on the Coupling Constant from Unitarity and Analyticity

Unitarity of the $S$ matrix is used to bound the sum of the pole term and the left-hand contribution appearing in a dispersion relation for a partial-wave scattering amplitude. An upper bound for the coupling constant is then constructed for the case of an arbitrary distribution of any finite number of sign changes in the left-hand cut discontinuity.

Bounds for the correction to the Born term and applications to p- p scattering for a generalized dispersion model

The unitarizing corrections to the Born term, as given by the Chew-Arndt, MacGregor-Arndt ( j 0 even), and Scotti-Wong models, are shown to be bounded above and below. The bounds follow from the mathematical forms of the models, to all of which forms a theorem (proved in this paper) applies. The correction bounds are given for several p- p amplitudes for the range 0 to 300 MeV. The results are shown to explain why the apparently different models mentioned above give similar Born terms after subtraction of the correction from the experimentally determined ( p- p) amplitude. A generalization of these models for the unitarizing correction is proposed, which has properties leading to partial-wave dispersion relations, and which has a specified relationship between the asymptotic behavior and the fluctuation of the sign of the left-hand discontinuity. Upper and lower bounds are found for the correction term prescribed by this generalized model (which is not restricted, in its application, to nucleon-nucleon scattering).

TheN/D method in potential scattering

The Chew-Mandelstam procedure of constructing the partial-wave scattering amplitude from analytic properties and unitarity is investigated in nonrelativistic potential scattering. The left-hand discontinuity is kept to lowest order in the strength of the interaction. It is shown that the existence of a bound-state pole in the scattering amplitude implies the existence of a ghost state for the corresponding potential with the opposite sign of the strength. The role of the subtraction point when the scattering amplitude is constructed by an iterative method is discussed. Numerical results for the Yukawa potential are given and compared with the corresponding quantities obtained from the Schrodinger equation.

Self-Consistent Calculation of theρ-Meson Regge Pole

The left-hand discontinuities in the partial-wave amplitudes for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering are assumed to be dominated by the exchange of the $\ensuremath{\rho}$ meson in a form suggested by the Regge representation for a resonance. This Regge behavior provides the necessary high-energy cutoff and allows the $\frac{N}{D}$ equations to be solved. The partial-wave $I=1$ amplitudes are calculated for noninteger angular momenta $l&lt;1$ as well as $l=1$. The trajectory ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(s)$ as well as the residue ${\ensuremath{\beta}}_{\ensuremath{\rho}}(s)$ of the $\ensuremath{\rho}$-meson Regge pole are evaluated. An attempt is made to obtain a self-consistent solution for the relevant parameters, namely the position and width of the $\ensuremath{\rho}$ resonance and ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(0)$. The results of this calculation give ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(0)\ensuremath{\gtrsim}0.9$. The $I=0$ vacuum trajectory is also discussed.