Physical Sheet Research Articles

Correlated Wishart matrices and critical horizons

We discuss a practical method to determine the eigenvalue spectrum of the empirical correlation matrix. The method is based on the analysis of the behavior of a conformal map at a critical horizon which is defined as a border line of the physical Riemann sheet of this map. The map is a convenient representation of the Mar\v{c}enko-Pastur equation.

Collective modes of an anisotropic quark-gluon plasma: II

We continue our exploration of the collective modes of an anisotropic quark gluon plasma by extending our previous analysis to arbitrary Riemann sheets. We demonstrate that in the presence of momentum-space anisotropies in the parton distribution functions there are new relevant singularities on the neighboring unphysical sheets. We then show that for sufficiently strong anisotropies that these singularities move into the region of spacelike momentum and their effect can extend down to the physical sheet. In order to demonstrate this explicitly we consider the polarization tensor for gluons propagating parallel to the anisotropy direction. We derive analytic expressions for the gluon structure functions in this case and then analytically continue them to unphysical Riemann sheets. Using the resulting analytic continuations we numerically determine the position of the unphysical singularities. We then show that in the limit of infinite contraction of the distribution function along the anisotropy direction that the unphysical singularities move onto the physical sheet and result in real spacelike modes at large momenta for all "out-of-plane" angles of propagation.

Integral representation of the Coulomb Green function derived from the Sturmian expansion

We recast the Sturmian expansion of the one-particle Coulomb Green function as an integral which, through an appropriate choice of contour, is applicable anywhere on the Riemann energy surface (excluding the neighborhood of the negative real axis on the physical energy sheet, a region to which the series representation is ideally suited). As a numerical test we have used this integral representation, in conjunction with a two-particle convolution of one-particle Coulomb Green functions, to calculate the cross sections for both partial and complete breakup of the negative hydrogen ion by one-photon absorption.

Unified treatment: analyticity, Regge trajectories, Veneziano amplitude, fundamental regions and Moebius transformations

In this paper we present a unified treatment that combines the analyticity properties of the scattering amplitudes, the threshold and asymptotic behaviors, the invariance group of Moebius transformations, the automorphic functions defined over this invariance group, the fundamental region in (Poincare) geometry, and the generators of the invariance group as they relate to the fundamental region. Using these concepts and techniques, we provide a theoretical basis for Veneziano type amplitudes with the ghost elimination condition built in, related the Regge trajectory functions to the generators of the invariance group, constrained the values of the Regge trajectories to take only inverse integer values at the threshold, used the threshold behavior in the forward direction to deduce the Pomeranchuk trajectory as well as other relations. The enabling tool for this unified treatment came from the multi-sheet conformal mapping techniques that map the physical sheet to a fundamental region which in turn defines a Riemann surface on which a global uniformization variable for the scattering amplitude is calculated via an automorphic function, which in turn can be constructed as a quotient of two automorphic forms of the same dimension.

Polarization-Free Quantum Fields and Interaction

A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Complex Scaling of the Faddeev Equations

In this work we compare two different approaches to calculation of the three-body resonances on the basis of Faddeev differential equations. The first one is the complex scaling approach. The second method is based on an immediate calculation of resonances as zeros of the three-body scattering matrix continued to the physical sheet.

Riemann Surfaces of Some Static Dispersion Models and Projective Spaces

We show that the analytic continuation of the S-matrix elements, which are meromorphic functions of the energy ω in the complex plane with the cuts (-∞,-1] and [+1,+∞), from the physical sheet to nonphysical ones results in a system of nonlinear difference equations. A global analysis of this system is performed in the projective spaces P_N and PN+1. We discuss the connection between the spaces PN and PN+1 and obtain some particular solutions of the initial system.

Behavior of solutions to the pion dispersion equation in the complex frequency plane

Solutions to the pion dispersion equation in the complex plane of the pion frequency ω are obtained for symmetric nuclear matter. Three well-known solution branches—a sound, a pion, and an isobar one—exist on the physical sheet at the medium density below its critical value (ρ ρc, the fourth branch ωc appears on the physical sheet. The condition ω c 2 ≤0 is valid for this branch (in general, Re ω c 2 ≤0). This suggests ground-state instability, possibly associated with pion condensation. The behavior of these solutions is analyzed for various medium densities. The appearance of each solution on the physical sheet of the complex frequency plane and its disappearance are also studied.

Complex scaling of the Faddeev operator

The work is devoted to comparison of two different approaches to calculation of three-body resonances on the basis of the Faddeev differential equations. The first one is the well known complex scaling approach. The second method is based on an immediate calculation of the zeros of the scattering matrix continued to the physical sheet.

Analysis of the low-energy ηNN-dynamics within a three-body formalism

The interaction of an $\eta$-meson with two nucleons is studied within a three-body approach. The major features of the $\eta NN$-system in the low-energy region are accounted for by using a s-wave separable ansatz for the two-body $\eta N$- and $NN$-amplitudes. The calculation is confined to the $(J^\pi;T)=(0^-;1)$ and $(1^-;0)$ configurations which are assumed to be the most promising candidates for virtual or resonant $\eta NN$-states. The eigenvalue three-body equation is continued analytically into the nonphysical sheets by contour deformation. The position of the poles of the three-body scattering matrix as a function of the $\eta N$-interaction strength is investigated. The corresponding trajectory, starting on the physical sheet, moves around the $\eta NN$ three-body threshold and continues away from the physical area giving rise to virtual $\eta NN$-states. The search for poles on the nonphysical sheets adjacent directly to the upper rim of the real energy axis gives a negative result. Thus no low-lying s-wave $\eta NN$-resonances were found. The possible influence of virtual poles on the low-energy $\eta NN$-scattering is discussed.

Operator interpretation of the resonances generated by 2×2 matrix Hamiltonians

We consider the analytic continuation of the transfer function for a 2x2 matrix Hamiltonian into the unphysical sheets of the energy Riemann surface. We construct a family of non-selfadjoint operators which reproduce certain parts of the transfer-function spectrum including resonances situated on the unphysical sheets neighboring the physical sheet. On this basis, completeness and basis properties for the root vectors of the transfer function (including those for the resonances) are proved.

Blood Calcium and Phosphorus Relationships in the Parturient Paresis Syndrome in Mature Holstein Cows

An observational study was conducted in 79 mature Holstein cows from 28 dairy farms selected from Ontario Veterinary College, Ruminant Field Service clients that participated in a monthly herd health program. The objectives were to measure and describe the serum levels of calcium and phosphorus, to evaluate the relationships between these two values in the parturient paresis syndrome of dairy cows and to find specific cut-off values to better predict clinical milk fever cows. All eligible third plus lactation cows entered the study if the calving occurred between February 15 and May 15, 1992. All cows were examined within the first 24 hours after parturition by a farm service clinician. A standard questionnaire and physical examination sheet was completed for every cow. At the end of the evaluation, a serum sample was taken from the caudal vein with a vacutainer vial.

Lax-Phillips theory for sound waves scattering by thin infinite elastic planes

The simplest acoustic systems with sound and vibration—fluid-loaded infinite thin elastic planes—are investigated in frames of operator theory. The operators for such systems are derived, which is of interest by itself, since the initial equations for such systems involve frequency-dependent boundary conditions. The spectral analysis of the operators obtained is presented in explicit form. We study sound waves scattering by infinite elastic planes using Lax-Phillips scattering theory. We show that the singularities of the reflection coefficient form the spectrum of the contracting semigroup in the Lax-Phillips scheme. Their part on the physical sheet describes the well-known head (side) waves, when the velocity of the energy transfer through the elastic boundary is supersonic.

Dynamic Form of Static Dispersion Relations and Projective Spaces

The S-matrix in the static limit of a dispersion relation has a finite order N and is a matrix of meromorfic functions of energy ω in the plane with cuts (-∞, -1],[+1, = ∞). In the elastic case it reduces to N functions Si(ω) conncted by the crossing symmetry matrix A. The problem of analytical continuation of Si(ω) from the physical sheet to unphysical ones can be treated as a nonlinear system of difference equations. It is shown that a global analysis of this system can be carried out effectively in projective spaces PN and PN+1. The connection between spasec PN and PN+1 is discussed.

Representations for the Three‐Body T‐Matrix, Scattering Matrices and Resolvent in Unphysical Energy Sheets

AbstractExplicit representations for the Faddeev components of the three‐body T‐matrix continued analytically into unphysical sheets of the energy Riemann surface are formulated. According to the representations, the T‐matrix in unphysical sheets is explicitly expressed in terms of its components taken in the physical sheet only. The representations for the T‐matrix are then used to construct similar representations for the analytic continuation of the three‐body scattering matrices and the resolvent. Domains on unphysical sheets are described where the representations obtained can be applied. A method for finding three‐body resonances based on the Faddeev differential equations is proposed.

Representations for the three-bodyT-matrix on nonphysical sheets. I

Explicit representations are given for the components of the three-body Faddeev T-matrix analytically continued to nonphysical sheets of the Riemann energy surface. This allows us to express the T-matrix on nonphysical sheets in terms of its physical sheet components only. The representations for the T-matrix are used to construct similar representations for an analytic continuation of the scattering matrix and its resolvent. Domains on nonphysical sheets are described where these representations are valid.

Potential Scattering on R3 ⊗ S1

In this paper we consider non-relativistic quantum mechanics on a space with an additional internal compact dimension, i.e., R3 ⊗ S1 instead of R3. More specifically we study potential scattering for this case and the analyticity properties of the forward scattering amplitude, Tnn(K), where K2 is the total energy and the integer n denotes the internal excitation of the incoming particle. The surprising result is that the analyticity properties which are true in R3 do not hold in R3 ⊗ S1. For example, Tnn(K) is not analytic in K for Im K>0, for n such that (|n|/R)>μ, where R is the radius of S1, and μ−1 is the exponential range of the potential, V(r, φ) = O(e−μr) for large r. We show by explicit counterexample that Tnn(K) for n ≠ 0, can have singularities on the physical energy sheet. We also discuss the motivation for our work and briefly the lesson it teaches us.

Summation of the eigenvalue perturbation series by multi-valued Pade approximants: application to resonance problems and double wells

Quadratic Pade approximants are used to obtain energy levels both for the anharmonic oscillator x2/2- lambda x4 and for the double well -x2/2+ lambda x4. In the first case, the complex-valued energy of the resonances is reproduced by summation of the real terms of the perturbation series. The second case is treated formally as an anharmonic oscillator with a purely imaginary frequency. We use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function. Then, we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants. In this way we can reconstruct the real energy by summation of the complex series. Such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory (around the minima of the double well potential) incorrect.

Σ-hypernuclear response by 4He(K stopped−, π∓) reaction

Σ-hypernuclear production spectra by 4He(K stopped −, π ∓) reaction are theoretically demonstrated so as to be compared with experimental data. The result suggests that there exists a Σ 4He bound state, of which pole lies in the physical sheet.

The analytic structure of the anomalous dimension of the four-gluon operator in deep inelastic scattering

In the double logarithmic approximation of perturbative QCD we show that the anomalous dimension of the four-gluon operator in DIS has a rich analytic structure. In addition to a pole on the physical sheet we find singularities on second and other nonphysical sheets. We attempt to interpret these singularities as bound states and resonances.