Baryons and dual unitarization

Processes involving baryons are discussed in the scheme of dual unitarization. In particular, the topological expansion is generalized to any hadronic S-matrix elements involving baryons and/or mesons. Our expansion is based on a model for the baryon propagator, which is a set of three planar Feynman diagrams joined at a junction line. The resulting expansion is a double expansion in 1/N (N= the number of quark flavours) and in the number of baryon loops. Based on this, several new observations are made in phenomenological problems, and a unifying point of view is stressed. The scheme is evidently crossing invariant, and unitarity constraints are imposed order by order in 1/N and in the baryon loop number.     

Evaluation of ensemble averages for simple Hamiltonians perturbed by a GOE interaction

Using an expansion in powers of N^?1, where N is the dimension of the Hamiltonian matrix, we evaluate ensemble averages of the resolvent, of products involving several resolvents, and of the moments of the Hamiltonian H₀ + λV. Here, H₀ is arbitrary but fixed, and V is a GOE ensemble. The nature of the N^?1 expansion is also 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.     

Replica variables,loop expansion,and spectral rigidity of random-matrix ensembles

The replica trick of statistical mechanics is used to derive integral representations of n-point Green's functions both for the GOE and the EGOE. These integral representations are particularly suited for perturbative evaluation (loop expansion). Using the one-loop correction to the GOE one-point function, it is found that the density of states at the edge of the semicircle scales is $\text{～N}{}^{\mathrm{<mtext>?1</mtext><mtext>3</mtext>}}\text{?}\text{(N}{}^{\mathrm{<mtext>2</mtext><mtext>3</mtext>}}\text{δ}\text{)}$ where N is the dimension of the matrix ensemble. For the n-point functions with n ≥ 2, the existence of the microscopic limit to all orders in N^?1 is proved by decomposing the integration variables into massive (i.e., macroscopic) and massless (microscopic) components. Evaluation of the EGOE two-point function to leading order in the inverse local distance variable yields the first analytic evidence that the long-range correlations of EGOE spectra are similar to the GOE but not-stationary.     

Factorizable Z(N) models

The factorizable S-matrix with Z(N) symmetry is constructed. It is speculated that the field theory belonging to this S-matrix matrix is related to the scaling limit of Z(N) generalizations of the Ising model.     

Modification of Hauser-Feshbach calculations by direct-reaction channel coupling

If open channels are strongly coupled by direct reactions, the traditional Hauser-Feshbach method of calculating fluctuation cross sections is invalid, because of non-statistical correlations which the direct channel-coupling induces between resonance partial widths in different channels. The fluctuation cross sections can still be computed from the optical S-matrix elements, however, and the formulas necessary for doing so are obtained here with the aid of an “optical background” representation of the full S-matrix. The resulting compound-elastic cross section is increased over the Hauser-Feshbach expression by a factor of 2(Γ ? D) or 3(Γ ? D) in the large-N limit, and compound-reaction cross sections are increased by roughly a factor of $\text{(N + 1)}\text{N}$, where N is the number of directly-coupled open channels.     

Applications of the Dotsenko-Fateev integral in random-matrix models

The characteristic multi-dimensional integrals that represent physical quantities in random-matrix models, when calculated within the supersymmetry method, can be related to a class of integrals introduced in the context of two-dimensional conformal field theories by Dotsenko and Fateev. Known results on these Dotsenko-Fateev integrals provide a means by which to perform explicit calculations (otherwise difficult) in random-matrix theory. We illustrate this by (i) an evaluation of the mean squared S-matrix elements for the Gaussian orthogonal ensemble coupled with M external channels, and (ii) a direct derivation of the asymptotic behaviour of the dynamical density-density cotrelator in the limit of large spatial and temporal separation for the Calogero-Sutherland model which, at certain couplings, is known to map onto the parameter-dependent random-matrix ensembles.     

A dynamical theory for the off-shell continuation of the πNt-matrix

A dynamical theory, based on analyticity and dispersion theory, for the half-off-shell continuation of the on-shell πNt-matrix is proposed and developed. The resulting half-shell t-matrix is covariant, unitary, crossing symmetric, and based on a field-theoretic foundation. The dynamical information required to continue half off shell is obtained from field theory and consists of the off-shell amplitudes corresponding to the exchanges of the nucleon in the s- and u-channels and the ? and σ mesons in the t-channel. A coupled system of integral equations is derived for the partial wave half-shell t-matrix, which is truncated at the S- and P-waves and solved numerically. The results are compared with those obtained from various separable models of the πNt-matrix. The half-shell t-matrix is examined for separability and is found to be approximately separable in the P₃₃ and P₃₁ states. The dynamical content of the half-shell t-matrix is further illustrated by modeling the dynamical equation.     

On the role of unitarity in statistical theories of nuclear reactions

It is argued that the approximate statistical S-matrix with energy independent background S-matrix, residues, poles, and with fixed dimensions need not satisfy analytic unitarity but should in general obey the average unitarity condition. The freedom obtained by relaxing analytic unitarity allows a representation where level-level correlations are not present. Different approaches to statistical theories of nuclear reactions employing the pole decomposition of the S-matrix are compared. It is seen that any such approach is characterized by the assumed form of two (matrix) parameters. A model is developed which gives the expected results for the compound cross sections in the limits of strong absorption and weak absorption with statistically equivalent channels, and interpolates between the two extremes. The model depends, however, on the parameter $\text{πΓ}\text{D}$. The possibility of extracting the value of this parameter from experimental data for the variance of cross sections is also investigated.     

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.     

Analyticity and ergodicity in the maximum-entropy approach to statistical nuclear reactions

In the maximum-entropy approach to statistical nuclear reactions one imposes naturally the constraints of unitarity and symmetry of theS-matrix, and of a fixed expectation value ofS. We show that the analytical structure of theS-matrix and the requirement that the problem be ergodic (so that energy averages can be replaced by ensemble averages) impose certain restrictions on the distribution of statisticalS-matrices. Some of these additional constraints are then imposed numerically in a two-channel calculation, and are shown to improve the results for the fluctuation cross sections, the elastic enhancement factor, etc.     

Analytic Structure and Power-Series Expansion of the Jost Matrix

For the Jost-matrix that describes the multi-channel scattering, the momentum dependencies at all the branching points on the Riemann surface are factorized analytically. The remaining single-valued matrix functions of the energy are expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain an analytic expression for the Jost-matrix (and therefore for the S-matrix) near an arbitrary point on the Riemann surface (within the domain of its analyticity) and thus to locate the resonant states as the S-matrix poles. This approach generalizes the standard effective-range expansion that now can be done not only near the threshold, but practically near an arbitrary point on the Riemann surface of the energy. Alternatively, The semi-analytic (power-series) expression of the Jost matrix can be used for extracting the resonance parameters from experimental data. In doing this, the expansion coefficients can be treated as fitting parameters to reproduce experimental data on the real axis (near a chosen center of expansion E ₀) and then the resulting semi-analytic matrix S(E) can be used at the nearby complex energies for locating the resonances. Similarly to the expansion procedure in the three-dimensional space, we obtain the expansion for the Jost function describing a quantum system in the space of two dimensions (motion on a plane), where the logarithmic branching point is present.     

High temperature collective spin-photon coupling in a microwave cavity

An ensemble of N noninteracting spins being in thermal equilibrium and coupled to the resonant mode of a lossless microwave cavity is studied as the function of the spin temperature τ. Near τ = 0 the system is known to be in a coupled spin-photon state that manifests itself by the splitting of the cavity mode (vacuum Rabi splitting). The cavity emission spectrum is simulated for arbitrary τ. A critical temperature τ _C = ω _S √N/2, where ω _S is the spin excitation energy, is related to the destruction of the strong coupling regime as a consequence of thermal excitations arising within the spin ensemble.     

Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models

The general properties of the factorized S-matrix in two-dimensional space-time are considered. The relation between the factorization property of the scattering theory and the infinite number of conservation laws of the underlying field theory is discussed. The factorization of the total S-matrix is shown to impose hard restrictions on two-particle matrix elements: they should satisfy special identities, the so-called factorization equations. The general solution of the unitarity, crossing and factorization equations is found for the S-matrices having isotopic O(N)-symmetry. The solution turns out to have different properties for the cases N = 2 and N 3. For N = 2 the general solution depends on one parameter (of coupling constant type), whereas the solution for N 3 has no parameters but depends analytically on N. The solution for N = 2 is shown to be an exact soliton S-matrix of the sine-Gordon model (equivalently the massive Thirring model). The total S-matrix of the model is constructed. In the case of N 3 there are two “minimum” solutions, i.e., those having a minimum set of singularities. One of them is shown to be an exact S matrix of the quantum O(N)-symmetric nonlinear σ-model, the other is argued to describe the scattering of elementary particles of the Gross-Neveu model.     

Low-energy three-body scattering in a hyperspherical adiabatic basis

We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particles in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exactS-matrix analytically, in an expansion in powers of the wave number. We were able to associate the definite powers ofq in the expansion of theS-matrix to the corresponding inverse powers of in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the usual approximations found in the literature (extreme and uncoupled adiabatic approximations), when calculating the diagonal and off-diagonalS-matrix elements. Finally, we show that the coupled adiabatic equations uncouple as the energy goes to zero.     

Statistical reactions with memory and thermalized-nonequilibrated nuclear states

Starting from the Feshbach S-matrix pole expansion we modify the standard statistical model for compound reactions by introducing correlations between fluctuating S-matrix elements with different J (total spin) and π (parity) values. The S-matrix (J, π)-correlations are obtained at the expense of introducing infinitesimally small entrance-exit channel off-diagonal (J, π)-correlations between the random variables of the statistical model. Although later on these correlations are switched off by means of a properly applied limiting procedure, the S-matrix (J, π)-correlations do not vanish and can be strong. The physical origin of the S-matrix (J, π)-correlations resembles the effect of spontaneous symmetry breaking while S-matrix (J, π)-decoherence is due to quantum chaos. Novel reaction mechanism results in the excitation of peculiar nuclear states: The intermediate system is thermalized so that the shape of the spectrum is angle-independent and Maxwellian with angle-independent slope, yet the intermediate nucleus is not equilibrated since the angular distribution is forward-peaked, i.e., memory of the direction of the initial beam is not lost. The existence of thermalized-nonequilibrated nuclear states is supported by data on the 50–100% forward peaking of neutrons in the typically evaporation (1–3.5 MeV) part of the spectrum observed in the ⁹³ N b(n, n′) scattering with E _n = 7 MeV.     

More about the S-matrix of the chiral SU(N) Thirring model

We fix the bound state poles of the S-matrix of the chiral SU(N) Thirring model by general arguments. Avoiding an infrared problem by using a modified $\text{1}\text{N}$ expansion, the result is confirmed in leading order.     

N=2 extended superconformal algebra with “central” charges

We show how central charges may be incorporated in a superconformal (D = 4) algebra for N = 2. The charges are no longer truly central and so are at variance with the well-known theorems on (super-) symmetries of the S-matrix. We discuss the possible relevance of the algebra and justify our interest in it.