Double Spectral Functions Research Articles

Regge amplitudes through solution of S-matrix equations

This work is a first step in a program for construction of meson–meson scattering amplitudes with analyticity, crossing symmetry, and unitarity. The construction is to be carried out by solving a nonlinear integral equation for the partial wave amplitude a (l,s) at complex l and physical s. The program is intended to overcome the difficulties encountered in the traditional approach based on the Mandelstam iteration of double-spectral functions. An important initial step is to analyze nonrelativistic potential scattering from this autonomous S-matrix viewpoint, in which the Schrödinger equation is replaced by a nonlinear equation for the partial wave amplitude. In the present paper, it is demonstrated that the partial-wave equation has a locally unique solution, provided the potential is of a suitably restricted Yukawa type. This result indicates the feasibility of a pure S-matrix approach to dynamics. In the present report, the potential is so restricted in strength as to preclude bound states or resonances. The extension of the method to the case of strong potentials will be pursued in a later publication.

Topics in the S-matrix theory of massless particles

We discuss how massless particle reactions may be incorporated into standard S-matrix theory. The crucial element for doing so is a low-energy zero. Examples of reactions where such zeros occur are weak interaction processes involving neutrinos, chirally symmetric massless pion scattering, and two-photon exchange between neutral systems. These zeros make two-body unitarity a good approximation for sufficiently low energy despite the coalescence of multiparticle thresholds. Through two-body unitarity, these zeros produce lines of zeros in the absorptive parts and double spectral functions. These lines of zeros are the S-matrix analog of the requirement of an infrared finite field theory. Not only do they produce finite total cross sections at finite energies, but they also allow both upper and lower bounds to be derived for these cross sections at high energies. This upper bound is our main result. If a plausible smoothness assumption is made, we find σ tot < s ϵ (where ϵ is arbitrarily small). In particular, the experimentally observed linear rise of the neutrino proton cross section cannot continue indefinitely.

Absorptive parts and double spectral functions associated with dual amplitudes

A phenomenologically successful dual amplitude with logarithmic trajectories is studied. The delta function imaginary parts of the amplitude are approximated by an “averaged” imaginary part. As expected, the resulting average s-channel discontinuity vanishes when α( t) = 0, −1, −2, … and also gives rise to a double spectral function with a correct hyperbolic boundary. It is argued that the dual amplitude has properties which are very similar to those expected from a composite particle model.

Extension of Roy’s exact partial-wave equations to any energy

A method to continue analytically Roy’s equations to any energy is investigated. The supplementary conditions to perform such a continuation are Mandelstam representation with a finite number of subtractions and some other rather weak conditions on the double-spectral functions. The fact that the continuation has to be performed after quite tedious calculations and the difficulties with the estimation of errors are the main disadvantages of the method. On the other hand, it provides a complete test of analyticity, crossing and unitarity for a partial-wave set without any limitations in energy.

Asymptotic Behavior of Dual Amplitudes with Mandelstam Analyticity

The asymptotic ($|s|\ensuremath{\rightarrow}\ensuremath{\infty}$) behavior of a number of dual, crossing-symmetric amplitudes with nonzero Mandelstam double-spectral functions is studied. The difficulties in obtaining Regge behavior in the right-hand half of the $s$ plane are analyzed in detail.

Self-Consistent Dispersion-Theoretic Electromagnetic Scattering Amplitudes

A procedure for calculating self-consistent electromagnetic scattering amplitudes when there are bound states, applied in a previous paper to the electron-positron system, is extended to include spin-0-spin-0 and spin-0-spin-$\frac{1}{2}$ elastic scattering. The second-order amplitudes which result have the correct double-spectral functions and are cutoff-independent, analytic, and crossing-symmetric. The Regge trajectory functions which are required by self-consistency imply the usual Coulomb bound-state spectrum with reduced mass and recoil corrections, and the appropriate Regge asymptotic behavior. The correspondence between the amplitudes obtained by this procedure and by other means is discussed briefly.

Selection rules for Regge cuts

We combine the expectation that diagrams which generate Regge cuts must contain third double-spectral functions with the requirements of exchange degeneracy to arrive at a selection rule for Regge cuts. This rule prohibits Regge-Regge cuts from contributing to KN scattering, and we speculate that this prohibition might explain the observation that the K+p total cross-section is such a remarkably constant function of energy, and that the cross-sections for KN andK-N charge-exchange scattering rapidly become equal as the energy increases.

Self-Consistent Dispersion-Theoretic Electron-Positron Scattering Amplitude

A method is outlined for the construction of dispersion-theoretic electromagnetic scattering amplitudes for processes in which bound states may appear; in particular, the electron-positron system is examined. A Lorentz-covariant spinor amplitude is exhibited which is analytic, cutoff independent, and crossing symmetric, which has the correct double-spectral functions through second order in the fine-structure constant $\ensuremath{\alpha}$, and which reduces to the usual Born approximation in lowest order. Moreover, the amplitude displays Regge asymptotic behavior and the positronium Regge poles. Self-consistency requires that the positronium poles appear at the correct position and with the proper residue, and that the amplitude possesses a well-defined Jacob-Wick expansion. It is found that this second-order amplitude provides a suitable basis for iterative calculation of the higher-order terms using the same procedure.

Model forπ−πScattering Satisfying Analyticity and Crossing Symmetry

A crossing-symmetric model for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering is presented which satisfies the Mandelstam representation, has a finite number of resonances associated with an exchange-degenerate trajectory that turns over at high energies, and which has Regge asymptotic behavior in all channels. The Pomeranchukon amplitude is nonresonating and has background cuts. The total amplitude satisfies the Adler condition. Satellites are included that eliminate all the odd-daughter (ghost) resonances. The double-spectral functions are calculated and shown to have, except for the lack of curvature, the correct boundaries determined by elastic unitarity. The structure of the second-sheet singularities is briefly discussed. The $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths are calculated and found to be consistent with those obtained from current algebra, when terms of order ${{m}_{\ensuremath{\pi}}}^{2}$ and unitarity corrections are neglected.

Compositeness, Feynman Diagrams, and the Reggeized Absorption Model

In this paper we derive the Reggeized absorption model from field-theoretic diagrams. This model has been used to describe a large number of quasi-two-body reactions. It involves a Regge-cut correction to Regge-pole amplitude which is generated by the exchange of the Regge pole and a Pomeranchuk pole. The cut features the product of the Reggeon and Pomeranchon (without complex conjugation of either) and a large magnitude for the cut (coherent inelastic effects add to the original cut term). The fundamental physical assumption of our derivation is that physical particles are composite objects of constituent pieces of matter. In a scattering process, some of the constituent matter takes part in the scattering while the rest stands by as a spectator. These ideas lead us to describe double-scattering processes by a class of diagrams involving exchange of two Reggeons in the crossed channel and propagation of composite physical particles in the direct channel. When the direct-channel particles are Reggeized, we obtain an expression for the Regge box diagram. We begin our analysis of diagrams by discussing the Amati-Fubini-Stanghellini diagram and similar diagrams to demonstrate how the absence of third double-spectral functions leads to the absence of a cut. For simple diagrams, we find that we are forced to invoke properties of form factors to show absence of the cut, but that for sufficiently composite diagrams the absence of the cut rests solely on the absence of the third double-spectral functions. Next we discuss the Mandelstam diagram and similar diagrams to demonstrate how the presence of third double-spectral functions leads to cuts. For each diagram we bring the expression for the amplitude to the form of the absorption model. Finally, we study the general class of diagrams referred to above. These diagrams feature compositeness in the direct channel (physical particles are composite), third double-spectral functions (physical particles have definite signature; no exchange degeneracy), and two-Reggeon exchange (double scattering and the Glauber spectator approximation). By assuming saturation of direct-channel amplitudes by physical states, we are led to an absorption formula (no complex conjugations) that includes the coherent inelastic factor $\ensuremath{\lambda}$ (diffraction production of direct-channel resonances).

Amplitudes with Mandelstam Analyticity and Dual Structure

A general representation for scattering amplitudes is proposed which is crossing symmetric, satisfies Regge behavior and duality, and possesses double spectral functions. As the width parameter goes to zero, the resonance poles on the second sheet move onto the real axis and the amplitude reduces to the Veneziano model. Partial waves generated from this amplitude exhibit correct threshold behavior in both the real and imaginary parts. A generalization to more-point functions is also proposed.

Estimates of the unitarity integral

The elastic unitarity integral is studied for amplitudes which satisfy a Mandelstam representation without subtraction. The double spectral functions are taken to belong to function spaces which allow local, even non-integrable, singularities. The existence of fixed point solutions is derived and the additional restrictions due to inelastic unitarity are discussed.

A class of functions that satisfy crossing symmetry and unitarity

Atkinson's proof of the existence of functions that satisfy crossing symmetry and unitarity is extended to amplitudes given by a Mandelstam representation with one subtraction. The inelastic unitarity bounds Im ƒ l(s) ⩾ | ƒ l(s) | 2 are obtained only for a finite interval of the energy, for higher values of s we only get 1 ⩾ Im ƒ l(s) ⩾ 0 . The proof extends to positive and to oscillating double spectral functions.

Scattering amplitudes that satisfy a mandelstam representation with one subtraction and unitarity

It is proved that there exist functions which satisfy a Mandelstam representation with one subtraction elastic unitarity and the inelastic unitarity bounds Im ƒ l(s) ⩾ |ƒ l(s)| 2 for all energies. We give solutions with positive and with oscillating double spectral functions.

Bootstrap of Pion-Pion Scattering in the Unitarized Strip Approximation

We give the result of a bootstrap of the $\ensuremath{\rho}$ and Pomeranchuk trajectories in pion-pion scattering, using the unitarized strip approximation. In this scheme, we construct an analytic, crossing-symmetric, unitary amplitude with Regge asymptotic behavior. This is achieved by starting with strips in which the double spectral function is parametrized by Regge poles, in the way devised by Chew and Jones, and adding the elastic double spectral functions in each channel calculated by the Mandelstam iteration method. Unitarity is imposed using the $\frac{N}{D}$ method with inelasticity. We find self-consistent trajectories which bear a fair resemblance to those found experimentally, at least for small $|t|$, and obtain a $\ensuremath{\rho}$ of width 155 MeV, which is very much better than earlier calculations. This and related improvements are due to the proper inclusion of Pomeranchon exchange. The requirement of self-consistency does not fix the trajectories uniquely, but the range of solutions is comparatively narrow, with $0.32\ensuremath{\le}{\ensuremath{\alpha}}_{\ensuremath{\rho}}(0)\ensuremath{\le}0.69$. There are still some unsatisfactory features in comparison with the phenomenological trajectories: We always find that ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(\ensuremath{\infty})&gt;\ensuremath{-}1$, there is a too rapid increase of $\mathrm{Im}\ensuremath{\alpha}$ just above threshold, and the trajectories do not rise much above $\mathrm{Re}\ensuremath{\alpha}=1$. We draw some conclusions about the prospects for this sort of $S$-matrix dynamics.

Positive double spectral functions are incompatible with the existence of the f(1260) resonance

We show that the existence and the known properties of the 2 +, I = 0 f(1260) resonance make it very unlikely that the double spectral function ϱ( s, t) of the Mandelstam representation for the isospin-zero ππ scattering amplitude be positive. In particular, Atkinson's parametrization ϱ = ϱ elastic + ϱ inelastic with ϱ inelastic ⩾ 0 is completely ruled out.

‘Impossibility’ of positive double spectral functions

Strictly positive double spectral functions are shown to be inconsistent with the observed diffraction peaks. But examples of oscillating spectral functions consistent with unitarity and the diffraction peaks are easily produced.

Implications of the Optical Model for theS-Matrix Approach to High-EnergyppScattering

The optical model is shown to indicate that the high-energy large-momentum-transfer $\mathrm{pp}$ elastic scattering amplitude obeys the Mandelstam representation and that the corresponding absorptive parts and double spectral functions are highly oscillatory.

Threshold Behavior of Partial-Wave Dispersion Relation and Short-Range Force

In a paper on the existence of a ghost-free solution of the unsubtracted partial-wave despersion relation, Frye and Warnock conjectured that the positive-definite value at threshold of the unitarity integral on the physical region must be canceled by the short-range force (i.e., multiparticle-exchange contribution) arising from the first and second double spectral functions in order to guarantee the required $p$ wave and higher threshold behavior. We examined this conjecture explicitly in an unsubtracted dispersion relation for the $I=l=1$ pion-pion scattering amplitude. With certain choices of the $\ensuremath{\rho}$ Regge parameters, we show in our approximation scheme that this conjecture is nearly satisfied, and we can conclude that the so-called subtraction constant or threshold factor which is usually arbitrarily introduced to correct the threshold behavior could be expressed in terms of known physical quantities. However, with some other choice of $\ensuremath{\rho}$ Regge parameters, our numerical result shows that this conjecture is not satisfied by our one-$\ensuremath{\rho}$-Regge-pole approximation and indicates a need for additional Regge poles or Castillejo-Dalitz-Dyson poles.

Theory of Pion-Pion Scattering

Strong-interaction dynamics is formulated on two main assumptions. The first is similar to the one used in the Regge-pole-resonance interference model, and the other is that the potentials arising from the exchange of a few dominant particles and resonances in crossed channels can be accepted physically in the low-energy regions and that the convergent high-energy behavior of the potentials should be guaranteed by the Regge-pole theory. A long-range potential which satisfies both the low-energy and the high-energy boundary conditions stated above is introduced. The dynamical model with only Regge-pole exchange potentials (as in the new form of the strip approximation) does not produce low-energy resonance behavior as a dynamical output. We show that a potential which does produce the main low-energy resonance behavior as output can be constructed by an appropriate superposition of single-particle exchange potentials and Regge-pole exchange potentials according to our second assumption. Also, two other potentials arising from the large-$t$ parts of the Mandelstam double spectral functions are defined under the first assumption.