The non-perturbative renormalization group in the ordered phase

We study some analytical properties of the solutions of the non-perturbative renormalization group flow equations for a scalar field theory with Z₂ symmetry in the ordered phase, i.e. at temperatures below the critical temperature. The study is made in the framework of the local potential approximation. We show that the required physical discontinuity of the magnetic susceptibility χ(M) at M=±M₀ (M₀ spontaneous magnetization) is reproduced only if the cut-off function which separates high and low energy modes satisfies to some restrictive explicit mathematical conditions; we stress that these conditions are not satisfied by a sharp cut-off in dimensions of space d<4.By generalizing a method proposed earlier by Bonanno and Lacagnina [Nucl. Phys. B 693 (2004) 36] to any kind of cut-off we propose to solve numerically the renormalization group flow equations for the threshold functions rather than for the local potential. It yields an algorithm sufficiently robust and precise to extract universal as well as non-universal quantities from numerical experiments at any temperature, in particular at sub-critical temperatures in the ordered phase. Numerical results obtained for the φ⁴ potential with three different cut-off functions are reported and compared. The data confirm our theoretical predictions concerning the analytical behavior of χ(M) at M=±M₀.Fixed point solutions of the adimensioned renormalization group flow equations are also obtained in the same vein, that is by solving the fixed points equations and the associated eigenvalue problem for the threshold functions rather than for the potential. We report high precision data for the odd and even spectra of critical exponents for different cut-offs obtained in this way.     

Spherically symmetric collapse in quantum gravity

The problem of classical singularities is revised on the basis of the quantum-gravity effective equations. We find a simple rule for establishing the Birkhoff theorem in spherically symmetric problems. All exact solutions of the lagrangian with C²_αβγσ are obtained. Spherically symmetric collapse of the thin null shell of mass M is considered in the framework of a local theory describing vacuum polarization effects. The boundary-value problem is set and the asymptotic solution is obtained. It is found that the shell collapses to r = 0 without the rise of a singularity, and begins expanding. The global behaviour of the solution is obtained for small M. For large M it is conjectured that the event horizon does not form, and the apparent horizon is closed. An object forms, possessing the observable properties of a black hole, but living a finite time.     

A Class of Super Dense Stars Models Using Charged Analogues of Hajj-Boutros Type Relativistic Fluid Solutions

We present a spherically symmetric solution of the general relativistic field equations in isotropic coordinates for perfect charged fluid, compatible with a super dense star modeling. The solution is well behaved for all the values of Schwarzschild parameter u lying in the range 0 < u < 0.1727 for the maximum value of charge parameter K = 0.08163. The maximum mass of the fluid distribution is calculated by using stellar surface density as ρ _b = 4.6888×10¹⁴g cm^?3. Corresponding to K = 0.08 and u _max = 0.1732, the resulting well behaved solution has a maximum mass M = 0.9324M _⊙ and radius R = 8.00 and by assuming ρ _b = 2×10¹⁴g cm^?3 the solution results a stellar configuration with maximum mass M = 1.43M _⊙ and radius R _b = 12.25 km. The maximum mass is found increasing with increasing K up to 0.08. The well behaved class of relativistic stellar models obtained in this work might has astrophysical significance in the study of internal structure of compact star such as neutron star or self-bound strange quark star like Her X-1.     

Bound state of solution of Dirac-Coulomb problem with spatially dependent mass

The bound state solution of Coulomb Potential in the Dirac equation is calculated for a position dependent mass function M(r) within the framework of the asymptotic iteration method (AIM). The eigenfunctions are derived in terms of hypergeometric function and function generator equations of AIM.     

On long-wave sound scattering by a Rankine vortex: Non-resonant and resonant cases

The well-known two-dimensional problem of sound scattering by a Rankine vortex at small Mach number M is considered. Despite its long history, the solutions obtained by many authors still are not free from serious objections. The common approach to the problem consists in the transformation of governing equations to the d’Alembert equation with right-hand part. It was recently shown [I.V. Belyaev, V.F. Kopiev, On the problem formulation of sound scattering by cylindrical vortex, Acoustical Physics 54(5) (2008) 603-614] that due to the slow decay of the mean velocity field at infinity the convective equation with nonuniform coefficients instead of the d’Alembert equation should be considered, and the incident wave should be excited by a point source placed at a large but finite distance from the vortex instead of specifying an incident plane wave (which is not a solution of the governing equations).Here we use the new formulation of Belyaev and Kopiev to obtain the correct solution for the problem of non-resonant sound scattering, to second order in Mach number M. The partial harmonic expansion approach and the method of matched asymptotic expansions are employed. The scattered field in the region far outside the vortex is determined as the solution of the convective wave equation, and van Dyke's matching principle is used to match the fields inside and outside the vortical region. Finally, resonant scattering is also considered; an O(M²) result is found that unifies earlier solutions in the literature. These problems are considered for the first time.     

Nonlinear development of instabilities in supersonic vortex sheets: I. The basic kink modes

Classical linearized stability analysis predicts (neutral) stability of supersonic vortex sheets for compressible flow with normalized Mach numbers, M > √2, while recent detailed numerical simulations by Woodward indicate the nonlinear development of instabilities for M > √2 through the development and interaction of propagating kink modes in the slip-stream. These kink modes are discontinuities in the slip-stream bracked by shock waves and rarefaction waves which grow self-similarly in time. In this paper, the apparent paradox is resolved by developing appropriate small amplitude high frequency nonlinear time-dependent asymptotic perturbed solutions which yield the response to a very small amplitude nonlinear planar sound wave incident on the vortex sheet. The analysis leads to three specific angles of incidence depending on M > √2 where nonlinear resonance occurs. For these three special resonant angles of incidence the perturbation expansions automatically yield simplified equations. These equations involve an appropriate Hamilton-Jacobi equation for the perturbed vortex sheet location; the derivative of the solution of this Hamilton-Jacobi equation provides boundary data for two nonlinear Burgers transport equations for the sound wave emanating from the two sides of the vortex sheet. These equations are readily solved exactly and lead to the quantitative time-dependent nonlinear development of three different types of kink modes with a structure similar to that observed by Woodward.     

On solutions of truncated Kramers-Moyal expansions; continuum approximations to the Poisson process

We discuss the solutions of a Kramers-Moyal-expansion-type master equation for a discrete Poisson process, truncated at an arbitrary orderM. As was shown some time ago solutions withM=3, 7, 11 are in better agreement with the exact solution than the solution truncated atM=2. If a δ function is used as an initial condition, the solutions start to oscillate very rapidly with increasingM leading to a δ-function behaviour at the integer points for largeM. If, however, more smooth initial conditions are used, the rapid oscillations die out for increasingM and the solutions converge to interpolations of the exact solution.     

Electromagnetic-field amplification in finite one-dimensional photonic crystals

The electromagnetic-field distribution in a finite one-dimensional photonic crystal is studied using the numerical solution of Maxwell’s equations by the transfer-matrix method. The dependence of the transmission coefficient T on the period d (or the wavelength λ) has the characteristic form with M–1 (M is the number of periods in the structure) maxima with T = 1 in the allowed band of an infinite crystal and zero values in the forbidden band. The field-modulus distribution E(x) in the structure for parameters that correspond to the transmission maxima closest to the boundaries of forbidden bands has maxima at the center of the structure; the value at the maximum considerably exceeds the incident-field strength. For the number of periods M ~ 50, more than an order of magnitude increase in the field amplification is observed. The numerical results are interpreted with an analytic theory constructed by representing the solution in the form of a linear combination of counterpropagating Floquet modes in a periodic structure.     

An Exact Solution of Perfect Fluid in Isotropic Coordinates, Compatible with Relativistic Modeling of Star

We present a spherically symmetric solution of the general relativistic field equations in isotropic coordinates for perfect fluid, compatible with a super dense star modeling. The solution is well behaved for all the values of u lying in the range 0<u≤0.12. Further, we have constructed a super-dense star model with all degree of suitability and by assuming the surface density ρ _b =2×10¹⁴ g/cm³. Corresponding to u=0.12, the resulting well behaved model has maximum mass M=0.912M _Θ with radius R _b ≈11.27 km and Moment of inertia 0.97×10⁴⁵ gm?cm². The good matching of our results for Vela pulsars show the stoutness of our model.     

A variational principle for symplectic connections

We introduce a variational principle for symplectic connections and study the corresponding field equations. For two-dimensional compact symplectic manifolds we determine all solutions of the field equations. For two-dimensional non-compact simply connected symplectic manifolds we give an essentially exhaustive list of solutions of the field equations. Finally we indicate how to construct from solutions of the field equations on (M, ω) solutions of the field equations on the cotangent bundle to M with its standard symplectic structure.     

On the mass operator in a diagonally disordered binary alloy

It is shown that the multiple-occupancy corrections imply identities between skeleton diagrams generating a perturbation calculus for the mass operator M. Self-consistent equations for M_ii, M_i, i+1 are obtained. The corrections to the obtained M begin with terms of order z^-6 where z is the coordination number.     

Neutrino oscillations in the field of a rotating deformed mass

The neutrino oscillations in the field of a rotating deformed mass is investigated. The phase shift is evaluated in the case of weak field limit, slow rotation and small deformation. To this aim the Hartle–Thorne metric is used, which is an approximate solution of the vacuum Einstein equations accurate to second order in the rotation parameter a/M$a/M$ and to first order in the mass quadrupole moment q. Implications on atmospheric, solar and astrophysical neutrinos are discussed.     

What if supersymmetry breaking unifies beyond the GUT scale?

We study models in which soft supersymmetry-breaking parameters of the MSSM become universal at some unification scale, M _in, above the GUT scale, M _GUT. We assume that the scalar masses and gaugino masses have common values, m ₀ and m _1/2 respectively, at M _in. We use the renormalisation-group equations of the minimal supersymmetric SU(5) GUT to evaluate their evolutions down to M _GUT, studying their dependences on the unknown parameters of the SU(5) superpotential. After displaying some generic examples of the evolutions of the soft supersymmetry-breaking parameters, we discuss the effects on physical sparticle masses in some specific examples. We note, for example, that near-degeneracy between the lightest neutralino and the lighter stau is progressively disfavoured as M _in increases. This has the consequence, as we show in (m _1/2,m ₀) planes for several different values of tan?β, that the stau-coannihilation region shrinks as M _in increases, and we delineate the regions of the (M _in,tan?β) plane where it is absent altogether. Moreover, as M _in increases, the focus-point region recedes to larger values of m ₀ for any fixed tan?β and m _1/2. We conclude that the regions of the (m _1/2,m ₀) plane that are commonly favoured in phenomenological analyses tend to disappear at large M _in.     

Spontaneous compactification in generalized Candelas-Weinberg models

In their recent work on the dimensional reduction, Candelas and Weinberg considered a model which is compactified into a direct product space of the Minkowski space (M₄) and an N-dimensional sphere (S_N). In the present paper we investigate generalized models of their type which are compactified into M₄ × S_M × S_N and M₄ × S_M × CP₂. The compactification is caused by the quantum loop effect due to a large number of matter fields. The conditions for the vacuum stability are studied. Numerical computation of the loop effect is undertaken, and it is shown that some of the models of the type M₄ × S_M × S_N admit a stable solution which has finite circumferences of both of the extra spaces and positive coupling constants of the Einstein-Yang-Mills theory in four dimensions.     

Correction to the low-energy scattering of monopoles

Following Manton, and Atiyah and Hitchin we consider approximating solutions to the dynamic Yang-Mills-Higgs equations by motions on the finite-dimensional space M_k of stable k-monopoles. For initial data transverse to M_k the approximate motion will not be geodesic motion but instead will be motion in an effective potential on M_k.     

Relativistic minimal surfaces

We find classical solutions to the equations of motion of an M-dimensional surface moving in a higher-dimensional embedding space-time for arbitrary M. In the case of closed membranes, solutions exist for any topological type (genus).     

General theory of virtual heavy-particle effects in renormalizable field theories (I)

We develop a systematic method of isolating the effects of virtual heavy particles in renormalizable field theories. With a φ⁴-type field-theory model involving two real scalar fields (one with a heavy mass M, and the other light), we show in detail, that up to order $\text{1}\text{M}{}^2$ (but to all orders in renormalized coupling), effects of virtual heavy particles can be completely incorporated into pure light-particle theory via effective local vertices which involve operators of canonical dimension at most six. All the coupling strengths for such effective local interactions are of order $\text{1}\text{M}{}^2$ (the decoupling theorem) and are systematically calculable in renormalized perturbation theory. We also derive a closed set of Callan-Symanzik equations which are satisfied by these coupling strengths. Using these equations, we explicitly sum all the leading logarithms (i.e., log M ～ O(1)) which appear in the perturbative calculations of the effective coupling strengths.     

An explicit derivation of a relation between magnetization M and saturation magnetization Ms

An explicit relationship between a magnetization vector M and its saturation magnetization M_s is derived using the definition of M along with assumptions of the continuum exchange theory. The obtained expression is found to be an extension of the commonly used fixed-length constraint and represents the continuous analog of it for the elementary moment per unit volume. The derivation of this relation is carried out in detail and important potential implications relating to equations for M are also highlighted.     

Calculations of mass and moment of inertia for neutron stars

Masses and moments of inertia for slowly-rotating neutron stars are calculated from the Tolman-Oppenheimer-Volkoff equations and various equations of state for neutron-star matter. We have also obtained pressure and density as a function of the distance from the centre of the star. Generally, two different equations of state are applied for particle densities n > 0.47 fm^?3 and n < 0.47 fm^?3.The maximum mass is, in our calculations for all equations of state except for the unrealistic non-relativistic ideal Fermi gas, given by 1.50 M_⊙ < M < 1.82 M_⊙, which agrees very well with “experimental results”. Corresponding results for the maximum moment of inertia are 9.5 × 10⁴⁴ g · cm² < I < 1.58 × 10⁴⁵ g · cm², which also seem to agree very well with “experimental results”. The radius of the star corresponding to maximum mass and maximum moment of inertia is given by 8.2 km < R < 10.0 km, but a smaller central density ρ_c will give a larger radius.