Implications of some less-known scattering approximations in a deuteron-model bound state

We compare results obtained in potential-model testing for the³S bound state of the deuteron by some scattering approximations which are not familiar in literature. The test concerns: the plane-wave approximation used in the continuous-energy spectrum within the framework of the Gelfand-Levitan formalism, the Brysk approximation modified byMichalík (Czech. J. Phys.B 17 (1967), 757;B 18 (1968), 517) and the Jost method modified bySasakawa (Prog. Theoret. Phys. Suppl. No. 27 (1963), 1). It is pointed out which of these approximations can be preferred in similar model calculations, where the wave-function approach is demanded. In particular cases we also give contributions for the first-order energy shift due to a weak Yukawa perturbation on the deuteron, if its bound state is described by the mentioned approximative wave functions.     

Low-lying spectrum of the Y-string three-quark potential using hyper-spherical coordinates

We calculate the energies of three-quark states with definite permutation symmetry (i.e. of SU(6) multiplets) in the N=0, 1, 2 shells, confined by the Y-string three-quark potential. The exact Y-string potential consists of one term, the so-called three-string term, and three angle-dependent two-string terms. Due to this technical complication we treat the problem at three increasingly accurate levels of approximation: (1) the (approximate) three-string potential expanded to first order in trigonometric functions of hyper-spherical angles; (2) the (approximate) three-string potential to all orders in the power expansion in hyper-spherical harmonics, but without taking into account the transition(s) to two-string potentials; (3) the exact minimal-length string potential to all orders in a power expansion in the hyper-spherical harmonics, and taking into account the transition(s) to two-string potentials. We show the general trend of improvement of these approximations: the exact non-perturbative corrections to the total energy are of the order of one per cent, as compared with approximation (2), yet the exact energy differences between the [20,1⁺],[70,2⁺],[56,2⁺],[70,0⁺]-plets are shifted to 2:2:0.9, from the Bowler and Tynemouth separation rule 2:2:1, which is obeyed by approximation (2) at the one per cent level. The precise value of the energy separation of the first radial excitation (“Roper”) [56^′,0⁺]-plet from the [70,1⁻]-plet depends on the approximation, but does not become negative, i.e. the “Roper” remains heavier than the odd-parity [70,1⁻]-plet in all of our approximations.     

Three-Body Phase Shift in One-Dimensional 2 + 1 Scattering

For a model of three particles on a line, subject to attractive delta-function interactions, we consider the phase shift. We do this from the point of view of the calculation of the S-matrix in a hyperspherical adiabatic basis (an adiabatic S-matrix), and for energies ranging from the (negative) energy of the two-body bound state to a total energy of zero. We derive analytical expansions and present numerical work, for different approximations, and compare with the exact results that we obtain from the work of McGuire, whose model we have borrowed. We show that the simplest adiabatic approximation gives results that are qualitatively wrong, but that better approximations yield, for most of our range, excellent agreement with the exact result. Understanding the threshold behaviour, however, requires a zero-energy three-body bound state, or resonance, previously unsuspected for this model. The methods developed for the case of the simplest adiabatic approximation also yield threshold and low-energy results applicable to the two-body problem in two dimensions. Received December 23, 1996; revised May 13, 1997; accepted for publication June 19, 1997     

Nonlinear dynamics of soft boson collective excitations in hot QCD plasma III: Bremsstrahlung and energy losses

Within the framework of semiclassical approximation a general formalism for deriving an effective current generating bremsstrahlung of arbitrary number of soft gluons (longitudinal or transverse ones) in scattering of higher-energy parton off thermal parton in hot quark-gluon plasma with subsequent extension to two and more scatterers is obtained. For the case of static color centers, an expression for energy loss induced by usual bremsstrahlung of lowest-order with allowance for an effective temperature-induced gluon mass and finite mass of the projectile (heavy quark) is derived. The detailed analysis of contribution to radiation energy loss associated with existence of effective three-gluon vertex induced by hot QCD medium is performed. It is shown that in general, the bremsstrahlung associated with this vertex has no sharp direction (as in the case of usual bremsstrahlung) and therefore here, we can expect an absence of suppression effect due to multiple scattering. For the case of two-color static scattering centers it was shown that the problem of calculation of bremsstrahlung induced by four-gluon hard thermal loop (HTL) vertex correction can be reduced to the problem of the calculation of bremsstrahlung induced by three-gluon HTL correction. It was shown that for limiting value of soft gluon occupation number N_k 1/α_s all higher processes of bremsstrahlung of arbitrary number of soft gluons become of the same order in coupling, and the problem of resummation of all relevant contributions to radiation energy loss of fast parton, arises. An explicit expression for matrix element of two soft gluon bremsstrahlung in small angles approximation is obtained.     

Large constant terms from soft gluons

The vertex and soft gluon integrals in Drell—Yan and deep-inelastic processes are calculated without approximations, keeping the initial and final quarks on-shell. Asymptotic q²-expansion reveals log²-, log- and constant terms. The latter are small in deep-inelastic scattering but unexpectedly large in Drell—Yan production. Lowest order gluon diagrams provide a reliable approximation for the AF-corrections in deep-inelastic scattering but fail to do so in Drell—Yan processes.     

Effect of anomalous tbW vertex on decay-lepton distributions in e + e − → tt and CP-violating asymmetries

We obtain analytic expressions for the energy and polar-angle double differential distributions of a secondary lepton l ⁺ (l ⁻) arising from the decay of t(t) in e ⁺ e ⁻ → tt with an anomalous tbW decay vertex. We also obtain analytic expressions for the various differential cross-sections with the lepton energy integrated over. In this case, we find that the angular distributions of the secondary lepton do not depend on the anomalous coupling in the decay, regardless of possible anomalous couplings occurring in the production amplitude for e ⁺ e ⁻ → tt. Our study includes the effect of longitudinal e ⁻ and e ⁺ beam polarization. We also study the lepton energy and beam polarization dependence of certain CP-violating lepton angular asymmetries arising from an anomalous tbW decay vertex and compare them with the asymmetries arising due to CP-violation in the production process due to the top electric or weak dipole moment.     

Momentum scale expansion of sharp cutoff flow equations

We show how the exact renormalization group for the effective action with a sharp momentum cutoff, may be organized by expanding one-particle irreducible parts in terms of homogeneous functions of momenta of integer degree (Taylor expansions not being possible). A systematic series of approximations - the O(p^M) approximations - result from discarding from these parts, all terms of higher than the M^th degree. These approximations preserve a field reparametrization invariance, ensuring that the field's anomalous dimension is unambiguously determined. The lowest order approximation coincides with the local potential approximation to the Wegner-Houghton equations. We discuss the practical difficulties with extending the approximation beyond O(p⁰).     

Renormalization group approach to interacting fermion systems in the two-particle-irreducible formalism

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is invariant under the RG transformation, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex Φ⁽²⁾. Susceptibilities are calculated from the Bethe-Salpeter equation that relates them to Φ⁽²⁾. While the one-loop approximation breaks down at low energy in one-dimensional systems (for reasons that we discuss), it reproduces the exact results both in the normal and ordered phases in single-channel (i.e. mean-field) theories, as shown on the example of BCS theory. The possibility to continue the RG flow into broken-symmetry phases is an essential feature of the 2PI RG scheme and is due to the fact that the 2PI two-particle vertex, contrary to its 1PI counterpart, is not singular at a phase transition. Moreover, the normal phase RG equations can be directly used to derive the Ginzburg-Landau expansion of the thermodynamic potential near a phase transition. We discuss the implementation of the 2PI RG scheme to interacting fermion systems beyond the examples (one-dimensional systems and BCS superconductors) considered in this paper.     

Valence-bond studies on BH3

Valence-bond calculations using a minimal Slater basis set are reported for the ground state of BH₃. The energy obtained with various configurations is compared. A small build-up study involving configurations of natural atomic orbitals is described. The most important configuration is 1s²2sp _x p _y h ₁ h ₂ h ₃. Calculations using hybrid orbitals are described. The perfect pairing, resonance valence-state and molecular orbital approximations are compared with the valence-bond results. The perfect-pairing approximation involving neutral B gives a lower energy than the molecular orbital approximation.     

Variational methods for the ground state of liquid helium4

Assuming a Bijl-Jastrow-type wave function for the ground state of liquid He⁴, one can express the energy as a functional of the pair distribution functiong(r) when use is made of one of several “cluster approximations” known from the theory of classical fluids. The applicability of these approximations, and especially an integrodifferential equation forg(r) derived byAbe andHiroike, are discussed. It is shown that both the HNC and the PY approximations, when used consistently, yield the phonon behaviour of the liquid-structure factorS(k) for smallk. In the HNC approximation the energy as a function of density is calculated by a variational procedure. The velocity of sound following from \(\mathop {\lim }\limits_{k \to 0} \) S(k) is in good agreement with experiments and, at the equilibrium density, also with that calculated from the energy-versus-density curve. In the PY approximation a minimum of the energy expectation value does not exist without further restrictions on the trial wave function.     

Angular Velocity Distribution of the Electric Microfield in Plasma

In theory of the spectral line shapes, the conventional scheme use two approximations for the local electric field (microfield) due to all charged particles of the plasma. The quasi‐static approximations for the ions and the impact approximation for the electrons. The first approximation consists to say that the electric field is constant during the characteristic time. In this work we shall transpose the idea of the first approximation, to the angular velocity of the microfield whereas its strength is kept constant and equal to its mean value. We shall use the Holtsmark approach and the independent particles model (due to Margeneau and Lewis) to compute the static distribution function of the angular velocity of the microfield. In the first approach (Holtsmark), the distribution shows a Lorentzian behavior, whereas the second approach (Margenau and Lewis) shows a gaussian behavior. Subsequently, we have applied the obtained static distribution to show the effect on the broadening of Lyman‐alpha line for a plasma composed of He⁺ ions. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the phenomenological three-graviton vertex

In General Relativity, the graviton interacts in three-graviton vertex with a tensor that is not the energy-momentum tensor of the gravitational field. We consider the possibility that the graviton interacts with the definite gravitational energy-momentum tensor that we previously found in the G ² approximation. This tensor in a gauge, where nonphysical degrees of freedom do not contribute, is remarkable, because it gives positive gravitational energy density for the Newtonian center in the same manner as the electromagnetic energy-momentum tensor does for the Coulomb center. We show that the assumed three-graviton vertex does not lead to contradiction with the precession of Mercury’s perihelion. In the S-matrix approach used here, the external gravitational field has only a subsidiary role, similar to the external field in quantum electrodynamics. This approach with the assumed vertex leads to the gravitational field that cannot be obtained from a consistent gravity equation.     

The importance of gravitational self-field effects in binary systems with compact objects. I. The “static two-body problem” and the attraction law in a post-Newtonian approximation of general relativity

As a first step toward a proper treatment of compact objects in binary systems the attraction force of two massive bodies connected by a rod is calculated in a post-Newtonian expansion. Contrary to a calculation by Weyl und Bach we start without specializing the internal structure of the bodies. We consider general anisotropic pressures and do not require axial symmetry for the bodies. We calculate the attraction force first in a post-Newtonian approximation and then (in paper II) we shall be concerned with the post-post-Newtonian approximation. In both approximations we obtain Newton's attraction forceM _S1 M _S2/R ² plus terms of order 1/R ³ and higher, whereM _S1,M _S2 are the Schwarzschild masses of the bodies.     

Towards efficient methods for the study of pattern formation in ferrofluid films

Hexagonal and labyrinthine patterns appear in thin ferrofluid films after application of a magnetic field perpendicular to the film. The pattern size and the stability of the hexagonal and labyrinthine structures can be predicted by free energy approaches. Several approximations are used in the literature to accelerate the calculation of the magnetic energy. They are usually based on the use of a uniform, average or constant magnetization. In the uniform approximation the magnetization at all points in the pattern is assumed to be equal to its value at the center of the stripes or cylinders in the labyrinthine or hexagonal patterns. Recent papers indicate that this approximation gives qualitatively wrong results. This is corroborated here by a comparison with accurate results. When a volume-averaged magnetization is used during the calculation of the demagnetization field, from which the magnetic energy is evaluated, the theoretical results are only slightly modified with respect to the accurate results. Thus, we can propose a new method which gives results in good agreement with the accurate values and accelerates the calculations by a factor of 1000. The influence of the approximations is explained by a study of the evolution of the demagnetization field in the patterns. This study indicates that the volume-averaged approximation might only be reliable for patterns with a homogeneously distributed magnetic fluid. Another approximation of a constant magnetization, which is widely used in the literature, assumes that the magnetization does not change during the pattern formation in contrast to the uniform and average approximations. A different way of computing the constant magnetization than that usually employed markedly improves the agreement with the accurate results. This is explained by the derivation of a direct relationship between the approximations of a constant and an average magnetization.Received: 28 October 2003, Published online: 2 March 2004PACS: 47.54. + r Pattern selection; pattern formation - 47.65. + a Magnetohydrodynamics and electrohydrodynamics - 77.84.Nh Liquids, emulsions, and suspensions; liquid crystals     

{\hbar}-adic Quantum Vertex Algebras and Their Modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of (^h/_2p){\hbar}-adic nonlocal vertex algebra and (^h/_2p){\hbar}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free \mathbb C[[(^h/_2p)]]{{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}-module W, we study (^h/_2p){\hbar}-adically compatible subsets and (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subsets of (End W)[[x, x ⁻¹]]. We prove that any (^h/_2p){\hbar}-adically compatible subset generates an (^h/_2p){\hbar}-adic nonlocal vertex algebra with W as a module and that any (^h/_2p){\hbar}-adically S{\mathcal{S}}-local subset generates an (^h/_2p){\hbar}-adic weak quantum vertex algebra with W as a module. A general construction theorem of (^h/_2p){\hbar}-adic nonlocal vertex algebras and (^h/_2p){\hbar}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of \mathfrak s\mathfrak l₂{{\mathfrak s}{\mathfrak l}_{2}} to (^h/_2p){\hbar}-adic quantum vertex algebras.     

Radiative tau lepton pair production as a probe of anomalous electromagnetic couplings of the tau

We calculate the squared matrix element for the process e⁺e⁻ → τ⁺τ⁻γ allowing for anomalous magnetic and electric dipole moments at the ττγ vertex. No interferences are neglected and no approximations of light fermion masses are made. We show that anomalous moments affect not only the cross section, but also the shape of the photon energy and angular distributions. We also demonstrate that in the case of the anomalous magnetic dipole moment, the contribution from interference involving Standard Model and anomalous amplitudes is significant compared to the contribution from anomalous amplitudes alone. A program to perform the calculation is available and it may be employed as a Monte Carlo generator.     

Semiempirical valence bond potential energy surfaces for simple chemical reactions: 1 A′ states of H2O and H2F+

A semiemprical valence bond method is employed to study the potential energy surfaces (PESs) of the lowest ¹ A′ states of H₂O and isoelectronic H₂F⁺. The calculation is based on the search for the electronic configurations which play the most important rôle in the formation of the stable electronic states of the molecules and the relevant diatomic fragments. Several approximations are used to reduce the number of permutations in the calculation of the energy matrix. The Moffit atoms-in-molecule approximation is used to correct for the atomic errors arising from the poor basis set. The computed data for the lowest electronic states of OH, HF, HF⁺, H₂O and H₂F⁺, as well as the general behaviour of the PESs for different molecular geometries, are in good agreement with the available ab initio and experimental data.     

Anomalous <Emphasis Type="Italic">VVH</Emphasis> interactions at a linear collider

We examine, in a model independent way, the sensitivity of a linear collider to the couplings of a light Higgs boson to a pair of gauge bosons, including the possibility of CP violation. We construct several observables that probe the various possible anomalous couplings. For an intermediate mass Higgs, a collider operating at a center of mass energy of 500 GeV and with an integrated luminosity of 500 fb⁻¹ is shown to be able to constrain the ZZH vertex at the few per cent level, with even higher sensitivity for some of the couplings. However, lack of sufficient number of observables as well as contamination from the ZZH vertex limits the precision to which anomalous part of the WWH coupling can be probed.