Mandelstam Representation Research Articles

On the violation of cp-invariance in three-particle decays of K mesons

The derivation and the solution of integral equations for 3 π decay of K° are performed. Equations are obtained using a one-dimensional Mandelstam representation for the transition amplitude of two particles with one of the particle masses analytically continued up to the decaying values and in the case of two-body unitarity condition written taking into account a possible violation of CP invariance. The solutions of the equations obtained for the neutral case are analysed taking into account the contribution of anomalous singularities.

Uniqueness of Amplitudes Satisfying the Mandelstam Representation

It is shown that within the framework of the Mandelstam representation, the scattering amplitudes, if they exist, are uniquely determined by the absorptive parts over the elastic region of one channel. It is illustrated in detail with pion-pion and pion-nucleon scattering.

Infinitely rising trajectories and the existence of the Mandelstam representation

A simple Regge dominated model based on Mandelstam analyticity is constructed which imposes an asymptotic constraint on an infinetly rising trajectory function, namely Im α(s) ⩾ 1 π In ( s s 0 ) Re α (s) . Using this constraint as input, we solve an integral equation for the trajectory which is then compared with data. Finally, we show that this constraint is sufficient to guarantee the existence of the Mandelstam representation without substractions.

Invariant Amplitudes for Photon Processes

A method for constructing independent invariant amplitudes for photon processes which satisfy gauge invariance and are free from both kinematic singularities and zeros (constraints) is proposed. The method is based on the use of a simple projection operator which automatically ensures gauge invariance at every stage. The invariant amplitudes for pion photoproduction, pion Compton scattering, and nucleon Compton scattering are found. Previously known results in the first two cases are reproduced and their interpretation is clarified. For nucleon Compton scattering the result is new. Comparisons are made between these invariant amplitudes, the Hearn and Leader amplitudes, and the regularized helicity amplitudes. The Mandelstam representation may be applied to these kinematic-singularity-free and zero-free amplitudes without any ad hoc substractions. All known low-energy theorems on photon processes can be derived most directly from these invariant amplitudes.

The two-pion exchange contribution to meson-nucleon scattering in the low and intermediate energy region

The pion-nucleon and kaon-nucleon partial wave amplitudes with large angular momenta are estimated using the Mandelstam representation.

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.

Application of a Crossing-Symmetric Model Satisfying the Mandelstam Representation toπ−πandK−πScattering

A model for meson-meson scattering satisfying the Mandelstam representation, crossing symmetry, and Regge behavior and including a Pomeranchukon amplitude is applied to $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ and $K\ensuremath{-}\ensuremath{\pi}$ scattering. Solutions are found for the $K\ensuremath{-}\ensuremath{\pi}$ partial waves that satisfy unitarity approximately at low energies, give a satisfactory fit to the on-mass-shell data, and predict scattering lengths consistent with current algebra. Apart from a change in the coupling constant, effectively the same parameters are then used to predict the low-energy $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering, and the solutions are found to satisfy unitarity approximately up to 900 MeV. The predicted on-mass-shell results agree well with the available data. The general conditions below threshold for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering that follow from crossing symmetry and positivity are well satisfied. The extrapolated $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ and $K\ensuremath{-}\ensuremath{\pi}$ amplitudes off the mass shell are found to agree satisfactorily with the data for $\ensuremath{\pi}N\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}N$ and $\mathrm{KN}\ensuremath{\rightarrow}K\ensuremath{\pi}N$ when a phenomenological form factor is used in the extrapolation. The total and differential cross sections at high energy are found to have characteristic Regge behavior. The Pomeranchukon amplitude produces total $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ and $K\ensuremath{-}\ensuremath{\pi}$ cross sections consistent with factorization in the asymptotic region.

Dual, Crossing-Symmetric Amplitude with Mandelstam Analyticity

We present a class of scattering amplitudes which are dual and crossing symmetric, have Regge asymptotic behavior and second-sheet poles for resonances in all channels, and satisfy a Mandelstam representation with correct double-spectral-function boundaries. The Regge trajectories and residues are essentially arbitrary.

ππ Scattering Amplitudes that Satisfy Inelastic Unitarity Constraints

AbstractWe construct amplitudes which are represented by a Mandelstam representation with a finite number of subtractions and that satisfy ππ crossing symmetry and the unitarity constraints Im flI(s) ≧|flI (s)|2, l=0, 1, 2,…, for all energies above threshold s > 4, in the three isospin channels I=0, 1, 2.The following types of solutions are derived. The amplitudes have a positive double spectral function ϱ(s, t) ≧ 0. The total cross section decreases like σT(s) ∼ (log s)‐δ for arbitrary δ ≧ 1, including the limiting case δ=1. The amplitudes are dominated by Regge poles, the total cross section can reach a constant asymptotic value, σT(s) → const. The amplitudes are dominated by Regge cuts, the total cross section can increase logarithmically σT(s) → log s.

A proof of the existence of functions that satisfy exactly both crossing and unitarity IV. nearly constant asymptotic cross sections

A proof of the existence of functions that satisfy a Mandelstam representation with the crossing symmetry appropriate to pion-pion scattering, elastic unitarity below the inelastic threshold, and the inelastic inequalities above it, is generalized to a restricted class of amplitudes that need an arbitrary, finite number of subtractions. Although the leading angular momentum singularity is constrained not to rise above unity in the elastic region, it is possible for the total cross section to display an almost constant asymptotic behaviour.

Mandelstam Representation in Potential Scattering

A simpler and more general proof of the Mandelstam representation based on Hartog's theorem is presented within the framework of potential scattering; the potential is chosen to be a superposition of the Yukawa type with no further restrictions.

The Mandelstam representation from the partial-wave amplitudes

We write down the necessary and sufficient conditions for the partial-wave scattering amplitudesa(l, s) in order that the complete amplitude\(A\left( {s, t, u} \right) = \sum\limits_l {\left( {2l + 1} \right)a\left( {l,s} \right)P_l \left( z \right)} \) have the correct analytic properties ins, t andu and the Mandelstam representation be valid. This is done for the case of equal-mass scattering.

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 construction of functions that satisfy crossing symmetry and unitarity

We construct π0π0 amplitudes with arbitrary polynomial increase in the double spectral region that satisfy crossing symmetry and the inelastic-unitarity bounds Imf l(s)−|fl(s)|2⩾0,l=0, 1, 2, ..., for alls>4. Exact elastic unitarity is then obtained by an iteration procedure which was proposed by Mandelstam and discussed in detail by Atkinson. The resulting amplitude which fulfils crossing symmetry, elastic unitarity and the unitarity bounds in the inelastic region is the sum of Regge terms with bounded trajectories and a once-subtracted Mandelstam representation which guarantees the unitarity of the whole amplitude.

Relativistic Partial-Wave Analysis in Two Variables and the Crossing Transformation

The properties with respect to the crossing transformation of previously suggested relativistic two-variable expansions are considered. Signature amplitudes with definite symmetries are introduced, and it is shown that if the total amplitude is an analytic function satisfying a subtractionless Mandelstam representation, and if the signature amplitudes are square-integrable functions (with a suitable measure) in the physical region of each channel, then two-variable expansions of one type can be continued into expansions of another type, from one channel into the other. The expansion coefficients, called Lorentz amplitudes, in both physical regions, carrying all the dynamics, are shown to be two separate pieces of a general analytic function of the complex angular momentum $l$. It is suggested that these analyticity properties should be made use of to relate low-energy and high-energy scattering data.

The connection between Toller and Froissart-Gribov signatured amplitudes

Formulae are derived relating the Toller signatured O(2, 1) amplitude to the conventional Froissart-Gribov amplitude and vice versa. This suggests that the amplitudes have equivalent dynamical content. We consider only equal mass spinless particles but need only rigorous analyticity domains and not necessarily the Mandelstam representation. A vital ingredient of our work is the hermitian analyticity property.

Theory of Complex Angular Momentum: Past and Future

Three fundamental subjects on Regge poles are discussed in this paper. They are: (i) the relationship between the Mandelstam representation and the theory of complex angular momentum; (ii) the validity of the Regge asymptotic form for a general, complex s, (which includes the physical region of the t channel as a special case); (iii) a major obstacle in the future of Regge poles as a complete scheme of dynamical calculations.

Mandelstam representation in potential scattering

A new simplified proof of the Mandelstam representation in potential scattering is given. It is stressed that only on the mass shell amplitudes are needed throughout the proof.

A proof of the existence of functions that satisfy exactly both crossing and unitarity (III). Subtractions

A proof is given of the existence of functions that satisfy the Mandelstam representation with an arbitrary, finite number of subtractions, the crossing symmetry appropriate to pion-pion scattering, and the elastic unitarity condition below the four-pion threshold.

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.