The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.     

K −-Proton atomic transitions

The relation between theK ^}-P scattering length and the X-ray spectrum for the 2p → 1s electromagnetic transition inK ^?-P atoms is examined. A coupled-channel potential model is used to explicitly calculate the energy of theS-matrix pole in the 1s channel, which is then compared with the energy obtained from the scattering lengths via the standard equation. The X-ray spectrum is calculated and compared with the Lorentzian shape associated with the complex energy of theS-matrix pole. In addition, theK ^?p branching ratios are compared at threshold and at the complexS-matrix pole energy.     

Weak LSZ limits and reduction formula for the non-relativistic coulombic interaction

The asymptotic condition is stated in the case of a non-relativistic Coulombic potential (long-range interaction), and the corresponding LSZ reduction formula is established for the S-matrix elements of the theory. Since the Coulombic Green function is known in closed (non-perturbative) form, we are able to rigorously prove that the resulting S-matrix is free of infinite Coulombic phases and coincides with Dollard's S-matrix. A brief discussion of the forward-scattering amplitude is also given.     

A new approach to potential scattering

An approximate integral-representation of theS-matrix in partial-wave expansion is derived for a scalar Schrödinger particle in a central field. The method consists of linearizingCalogero's Riccati equation for the interpolatingS-matrix in such a way that the solution of the linearized equation deviates as little as possible from the exact one. TheS-matrix thus obtained exhibits exact crossing-symmetry and uniform convergence independent of the coupling constant of the scattering potential. In the weak coupling limit it is especially shown thatour method is more accurate than the second Born approximation. In the second part of the paper we specialize ourS-matrix to low and large energies. At low energies, a general integral for the scattering length is obtained and at large energies the summation over all angular momenta is carried out yielding an expression for the scattering amplitude.     

Anl-independent representation of a Majorana potential

The local andl-independent potential which isS-matrix equivalent to the parity dependent potential of Michel and Reidemeister forα scattering from²⁰Ne has been determined by inversion. The two potentials which fit elastic scattering produce radically different wavefunctions where the nuclei overlap. Phenomenological consequences are discussed.     

Complete S-matrix of the massive thirring model

On the basis of Zamolodchikov's S-matrix for two sine-Gordon solitons we derive the S-matrix for the scattering of an arbitrary number of particles including bound states.     

Relativistic effects in the 3N continuum and the A y puzzle

The three-nucleon (3N) Faddeev equation is solved in a Poincaré-invariant model of the three-nucleon system. Two-body interactions are generated so that when they are added to the two-nucleon invariant mass operator (rest energy) the two-nucleon S-matrix is identical to the non-relativistic S-matrix with a CD Bonn interaction. Cluster properties of the three-nucleon S-matrix determine how these two-nucleon interactions are embedded in the three-nucleon mass operator. Differences in the predictions of the relativistic and corresponding non-relativistic models for elastic and breakup processes are investigated. Of special interest are the lowering of the A _y maximum in elastic nucleon-deuteron (Nd) scattering below ≈25?MeV caused by the Wigner spin rotations and the significant changes of the breakup cross sections in certain regions of the phase space.     

Quantum scattering from a class of anisotropic potentials

The nonrelativistic quantum scattering problem for a class of 3-dimensional anisotropic, or explicitly angle-dependent potentials, is evaluated. The S-matrix is expressed in terms of the time-dependent propagator and evaluated using earlier results of a path integral solution. The scattering amplitude is then obtained from the S-matrix.     

On the regular holonomic character of theS-matrix and microlocal analysis of unitarity-type integrals

The previously proved results that every analytically renormalized Feynman integral is a regular holonomic function suggests that theS-matrix should be locally expressible as an infinite sum of regular holonomic functions. A regularity propertyR is formulated that expresses the condition that theS-matrix be locally expressible near each physical pointp as a convergent sum of regular holonomic functions, with each term enjoying some of the regularity properties of a corresponding Feynman integral. This propertyR holds at every physical pointp that has yet been analyzed by the methods of axiomatic field theory orS-matrix theory. Some analyticity properties of unitarity-type integrals are then examined under the assumption that theS-matrix satisfies propertyR and a weak integrability condition. These results rest heavily on some recently proved properties of regular holonomic functions.     

Large-angle scattering of heavy ions: (I). General structure of elastic cross section

Closed-form expressions are derived for the differential cross section of elastic heavy-ion scattering at large angles. The derivation is based on the general form of the elastic partial-wave S-matrix in real l-space. By a generalization of analytic techniques developed in earlier work, it is shown that the large-angle scattering cross section has a universal structure involving combinations of Bessel functions and the Fourier transforms of the rapidly varying parts of the S-matrix, irrespective of their dynamical origin. Anomalous large-angle scattering is attributed to deviations of the S-matrix from its “normal strong-absorption profile”, and general conditions for backward-angle enhancement are given. Our model-independent formulation provides the framework for an “inductive” method of analyzing experimental angular distributions and excitation functions aimed at identifying, as uniquely as possible, the dynamical mechanisms that operate in large-angle heavy-ion scattering. Extensions of the formalism to inelastic scattering and transfer reactions, and applications of the analytic method, will be described in subsequent papers.     

Integrability of the S-matrix versus integrability of the Hamiltonian

This report reviews the relations between the integrability properties of the S-matrix and of the Hamiltonian. Particular emphasis is put on the situation where the Hamiltonian has a conserved quantity which is not compatible with the asymptotics and where correspondingly the integrability does not transfer to the S-matrix. As questions of integrability are more readily handled in classical dynamics, all developments are first performed classically. Several examples are discussed to illustrate the main points. The quantum mechanical discussion reveals that the eigenphase statistics of the S-matrix depends principally on the chaoticity of the scattering map while basis dependent quantities such as the distribution of matrix elements tend to have random matrix behaviour only in the presence of topological chaos. The relevance of these considerations to the evaluation of scattering data is discussed.     

Time-dependent mean field S-matrix theory

By using the path integral approach to many-body systems, we formulate a time-dependent mean field S-matrix theory of nuclear reactions. Many-body channel eigenstates are constructed by using projection techniques. In this way the S-matrix between the channel eigenstates is expressed as a superposition of S-matrix elements between wave-packet-like states localized in space and time. A field operator representation of the interaction picture S-matrix is derived which enables one to apply the path integral approach. Applying the stationary phase approximation to the path integral representation of the interaction picture S-matrix between the localized states an asymptotically constant time-dependent mean field approximation to this S-matrix is obtained. Finally, the S-matrix between the projected channel eigenstates is obtained by evaluating the integral, arising from the projections, over the space-time positions of the localized states in the stationary phase approximation. The stationary phase conditions select those localized states from the projected channel states for which the mean field values of energy and momentum coincide with their corresponding channel eigenvalues.     

Three-nucleon calculations without the explicit use of two-body potentials

Using as two-nucleon interaction input the ³S₁-³D₁ and ¹S₀ partial waves, the Faddeev equations are solved for the three-nucleon bound state. The ³S₁³D₁T-matrix is calculated from the Reid potential. Avoiding the usual potential fit, the ¹S₀T-matrix is directly continued off-shell and is constructed consistent with the ¹S₀ phase shift of elastic two-nucleon scattering. The off-shell part of the ¹S₀T-matrix is parametrized and with this parametrization the dependence of the three-nucleon bound-state properties is studied. As a result it is found that the binding energy varies only between 6.2 MeV and 6.8 MeV, while the minimum in the charge form factor for electron scattering from ³He lies between 12.9 fm^?2 and 18.7 fm^?2. The larger (smaller) ³He binding energy is accompanied by a ³He charge form factor whose minimum is at larger (smaller) momentum transfers.     

Analytical properties of theS-matrix in many-channel-scattering

This work is an extension of previous work byNewton ^1–3 andWeidenmüller ^4,5 to a more realistic case of many-channel scattering: We consider arbitrary orbital angular momenta and a potential matrix of the Yukawa type. The analytical properties of the modifiedS-matrix are investigated. The modifiedS-matrix is meromorphic in certain strips around the real axes of the Riemann surface. This surface is determined solely by the kinematical branchpoints. The region of analyticity can be extended further for the diagonal and the squared nondiagonal elements of theS-matrix to include the entire Riemann surface except cuts on the imaginary axes. These cuts can possibly include part of the real axes. The one-pole approximation of theS-matrix is of a Breit-Wigner form. An exact expression for the partial widths and a sum rule for the partial widths are derived. A generalized Levinson theorem is proved.     

Eigenfunction expansions for stationary scattering theory in spaces with negative norm

The space with “negative” norm $\text{L}$_- with which the Hilbert space $\text{L}$ of wave-pockets is equipped is used as a natural carrier of non-square integrable functions which appear in scattering theory, such as the plane waves and distorted plane waves in potential scattering. The concept of a complete set of observables is formulated within this framework. The S-matrix formulae leading to the expression for the scattering cross-section are derived on a general level, and then are applied to the case of potential scattering.     

Multiphoton detachment from a zero-range potential revisited

Direct detachment of an electron bound by a regularized zero-range potential is reinvestigated. For the definition of the S-matrix amplitude for detachment, the wave function of the electron after the laser pulse has passed through is projected on an exact scattering state of the zero-range potential rather than a plane-wave state, which is normally used. The exact scattering state generates an additional contribution to the amplitude, which leads to an isotropic term in the angle-resolved energy spectrum.     

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^? (where ? is arbitrarily small). In particular, the experimentally observed linear rise of the neutrino proton cross section cannot continue indefinitely.     

On the Regge-pole description of nucleus-nucleus scattering

The properties of the Regge poles of theS-matrix for scattering of strongly-absorbed nuclear particles are considered. Simple formulae are obtained for describing the Regge trajectories in terms of the nuclear radius, the quasi stationary levels in the combined nuclear-Coulomb-potential and the widths of these levels. The predictions of these formulae are compared with the Regge trajectories obtained previously, for a Woods-Saxon potential, and with those required to fit¹⁶O-¹²C backward scattering.