Mandelstam Representation Research Articles

Scattering amplitudes from dispersive iterations of unitarity

We present and numerically implement a computational method to construct relativistic scattering amplitudes that obey analyticity, crossing, elastic and inelastic unitarity in three and four spacetime dimensions. The algorithm is based on the Mandelstam representation of the amplitude and iterations of unitarity. The input for the iterative procedure is given by the multi-particle double spectral density, the S-wave inelasticity, and the value of the amplitude at the crossing-symmetric point. The output, obtained at the fixed point of the iteration of unitarity, is a nonperturbative scattering amplitude. The amplitudes we obtain exhibit interesting features, such as non-zero particle production, intricate high-energy and near the two-particle threshold behavior. Scattering amplitudes obtained by initializing the iteration process with zero (or small) multi-particle input end up close to saturating the S-matrix bounds derived by other methods. There is a version of the iterative algorithm that is directly related to Feynman diagrams: it effectively re-sums infinitely many two-particle reducible planar Feynman graphs in the ϕ4 theory, which remarkably produces a unitary nonperturbative scattering amplitude function. Finally, we discuss how the algorithm can be further refined by including multi-particle unitarity.

Lower bound on inelasticity in pion-pion scattering

Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in $t$- plane with foci $t=0$, $u=0$ and right extremity $t=4 m_{\pi}^2 +\epsilon $, ($\epsilon > 0$), except for cuts prescribed by Mandelstam representation for $t\geq 4 m_{\pi}^2$, $u\geq 4 m_{\pi}^2$ , and bounded by $s^N$ on the boundary of this domain, we prove that for $s\rightarrow \infty$, \sigma_{inel} (s) > \frac{Const}{s^{5/2} }\exp {[-\frac{\sqrt{s}}{4} (N+5/2) \ln {s} ]}.

Proton–Proton On Shell Optical Potential at High Energies and the Hollowness Effect

We analyze the usefulness of the optical potential as suggested by the double spectral Mandelstam representation at very high energies, such as in the proton-proton scattering at ISR and the LHC. Its particular meaning regarding the interpretation of the scattering data up to the maximum available measured energies is discussed. Our analysis reconstructs 3D dynamics from the effective transverse 2D impact parameter representation and suggests that besides the onset of gray nucleons at the LHC there appears an inelasticity depletion (hollowness) which precludes convolution models at the attometer scale.

About the lower bound on the fixed-angle scattering amplitude at high energy

Assuming that the phase is small we prove that the fixed-angle scattering amplitude in potential scattering cannot decrease faster than exp (− Ckθlog k), where θ is the scattering angle and k the centre-of-mass momentum. This result agrees with the explicit calculation of Wu for the gaussian potential [ exp(−Cθk logk )] and with a previous bound derived from the Mandelstam representation with a finite number of subtractions. The applicability of this result outside the framework of potential scattering is discussed.

A partial wave dispersion relation analysis of pion-nucleon scattering amplitudes

Partial wave dispersion relations have been evaluated, using various new πN partial wave solutions and our results for the ππN\(\bar N\) amplitudes. The Karlsruhe-Helsinki 80 and CMU-LBL-79 solution forS, P, D, F waves are well compatible with this constraint, which follows from the Mandelstam representation. The VPI 80 solution leads to serious discrepancies in some cases, for instance for the P11 wave.—In both, the KH 80 and CMU-LBL 79, solutions the higher partial waves are not yet satisfactory. They can be improved by adding constraints based on the information on the nearby part of the left hand cuts.—Properties of nucleon resonances are discussed in connection with our results for the “generalized potentials”.

Comparison betweenNNpotential derived from dispersion relations and a model field theory

We compare the solution of a model field theory which relates off-shell amplitudes for elastic $\mathrm{NN}$ and $\ensuremath{\pi}N$ scattering to a similar relation between on-shell amplitudes derived from a dispersive approach. The first is a dynamical realization of $s$-channel unitarity, whereas the second starts with and exploits $t(u)$ channel unitarity. Provided that ${T}_{\mathrm{NN}}$ satisfies a Mandelstam representation, equivalent approximations ought to produce equivalent results and we perform the comparison for the actually calculated ${T}_{\mathrm{NN}}$ and the subsequently extracted potentials ${V}_{\mathrm{NN}}$. Interpreting ${V}_{\mathrm{NN}}$ we caution against the simultaneous retention of contributions with pions (crossed and uncrossed) linking intermediate nucleons as well as nucleons and physical isobars.NUCLEAR REACTIONS $\mathrm{NN}$ scattering, comparison between elastic amplitude and potential derived from model field theory and dispersive approach.

Unitary and analyticity bounds on average multiplicity and average transverse momentum

An upper bound on the partially integrated inclusive cross section is obtained at high energy from unitarity and analyticity assumptions which are to be expected for a two-body scattering amplitude that satisfies the Mandelstam representation. This result is analogous to the fixed-angle bound on two-body scattering amplitudes derived by Kinoshita, Loeffel and Martin. The bound leads to strong constraints on the average multiplicity, the average transverse momentum and the total cross section; in particular 〈n〉〈p T〉σ T 1 2 ⩽ const( ln s) 2 ( ln ln s) 1 2 . According to this, the energy dependence of average multiplicity and average transverse momentum favors logarithmic growth, at most.

Inverse scattering problem in nonrelativisticS-matrix theory

On the basis of unitarity and the Mandelstam representation, the inverse scattering problem is solved in a nonrelativistic S-matrix framework. Specifically, it is shown that the s-wave scattering amplitude, A/sub 0/(s), will uniquely determine the full scattering amplitude, A (s,t), provided no more than one subtraction in t is required and provided only the s wave has bound states. Generalizations of this result are also discussed.

Angular dependence of the differential cross section of hadronic exclusive processes at asymptotic energies

The angular dependence of the differential cross section is derived for meson-baryon and baryon-baryon elastic scattering and for some related processes, in the framework of the constituent interchange model, assuming the validity of dimensional scaling laws and using the Mandelstam representation of the amplitude.

Mandelstam Representation and Large Angle Scattering

Mandelstam Representation and Large Angle Scattering

Dispersion Calculation of the S-Matrix in Gauge Theories

We study the gauge independence of dispersion relations based on the special properties of the Feynman gauge and its non-linear variants in spontaneously broken gauge theory. It is shown that the discontinuity associated with the leading singularity of a Feynman diagram has a gauge independent meaning. In particular the double spectral function associated with each box diagram is gauge independent, and this property provides a gauge independent basis for writing the Mandelstam representation for four-point functions. We also study how to fix induced vertex terms such as the muon anomalous magnetic moment in the dispersion approach. It is indicated that we can treat the finite renormalization factors in analogy with the determination of the induced terms. We then attempt to under­ stand the renormalization program of the gauge theory as a self-consistent dispersion calcu­ lation of the S-matrix without recourse to the Ward identity. We briefly sketch how this program proceeds in the one-loop level calculation.

Off-shell Mandelstam representation for the second Born term of a Yukawa potential

The well-known result for the on-shell Mandelstam representation of the nonrelativistic second Born term of a Yukawa potential is extended to the general off-shell case for real-valued momentum and positive energies. The resulting representation is shown to differ from the on-shell case in two aspects. The first is a simple extension of the spectral function and corresponding Mandelstam boundary to the off-shell case which includes two additional parameters. These parameters simply reflect the extension to the case of arbitrary magnitudes for the initial- and final-state momentum off-shell. The second aspect is a change of sign of the principal-value (real) part of the representation off-shell which is not reflected by the imaginary part alone. This sign remains the same as that of the on-shell amplitude for all values of off-shell momentum only at sufficiently low energies. The critical scattering energy, above which the off-shell results differ in sign in a low-momentum region from those of the on-shell case, is determined by the mass of the exchanged particle. For the case of pion-exchange between nucleons, this energy corresponds to about 40 MeV in the laboratory. Potential implications for the calculation of two-pion-exchange contributions to nuclear potentials are discussed.

The subtracted Mandelstam representation in the Higgs model

The subtracted Mandelstam representation is presented for the Higgs model. Two subtractions are required for the double dispersion integrals, and one subtraction for the single dispersion integrals. A special and simple form of subtraction for the double dispersion integral is adopted, and there remains no arbitrary parameter except the coupling constant. High-energy behaviors are carefully investigated for both absorptive and dispersive parts and in general configurations of elastic scattering.

Sommerfeld-Watson representation for double-spectral functions

We discuss, for the case of pion-pion scattering, a closed system of equations which may be used for self-consistent calculation of partial-wave amplitudes. It is shown that, for a given sufficiently small input function, the equations have a locally unique solution in a particular Banach space of doubly Hoelder continuous partial wave amplitudes. At a fixed point, the scattering amplitude is shown to satisfy both a crossing symmetric unsubtracted Mandelstam representation and the elastic unitarity condition. In this initial study the partial-wave amplitudes are holomorphic in the right half complex angular- momentum plane. (orig.)

Convergent expansions of two-body scattering amplitudes as functions of two complex variables

The expansions of scattering amplitudes as functions of two complex variables are proposed. The expansions converge in the whole analyticity region prescribed by the Mandelstam representation. The method may be useful for the reconstruction of scattering amplitudes from the data. A possible connection to the Atkinson approach for the construction of scattering amplitudes satisfying first principles is briefly discussed.

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.

The two-pion exchange contribution to the nucleon-nucleon potential

The 2π exchange contribution to the nucleon-nucleon potential is calculated following the Mandelstam representation method of Cottingham and Vinh Mau. The \({\rm {N}} {\rm {\overline N}} \to {\rm {\pi \pi }}\) amplitude which is central to the calculation is obtained by analytic continuation of the low energy πN scattering amplitude which is constructed from nucleon and N* pole terms in a way consistent with the πN scattering data and soft pion theorems, ππ interactions are treated using the up-down solution for the S wave and ρ meson dominance for the P wave. The uncorrelated 2π contribution is obtained from the work of Partovi and Lomon.

Normal threshold approximation to analycity and Mandelstam representation

Normal threshold approximation to analyticity of two-body scattering amplitude formulated in the spaceC3 of independent complex variables\(\hat s,\hat t,\hat u\) and restricted by the assumption of polynomial boundedness in all three variables leads to a three-dimensional integral representation which is shown to reproduce Mandelstam representation on the complex plane\(\hat s + \hat t + \hat u = \sum\limits_{i = 1}^4 {m_i^2 } \).

Mandelstam representation for the Bethe-Salpeter amplitudes

Mandelstam representation is shown to be valid for the total Bethe-Salpeter scattering amplitude in the ladder approximation.

Application of Moffat's Model to η → 3π and K → 3π Decays

Moffat's model for meson–meson scattering, satisfying the Mandelstam representation, crossing symmetry, and Regge behavior, is applied to η → 3π and K → 3π decays. It is found that agreement with the data is excellent when the model is used with Lovelace's trajectory, but not as good with Moffat's trajectory. The model is also applied to the process [Formula: see text], and fair agreement is found in the low energy region; however, since the model is unitary only at low energies, no strong conclusions can be reached in this case.