Logarithmic Branch Research Articles

Asymptotically anti–de Sitter spacetimes and scalar fields with a logarithmic branch

We consider a self-interacting scalar field whose mass saturates the Breitenlohner-Freedman bound, minimally coupled to Einstein gravity with a negative cosmological constant in $D>~3$ dimensions. It is shown that the asymptotic behavior of the metric has a slower fall-off than that of pure gravity with a localized distribution of matter, due to the back-reaction of the scalar field, which has a logarithmic branch decreasing as ${r}^{\ensuremath{-}(D\ensuremath{-}1)/2}\mathrm{ln}r$ for large radius r. We find the asymptotic conditions on the fields which are invariant under the same symmetry group as pure gravity with negative cosmological constant (conformal group in $D\ensuremath{-}1$ dimensions). The generators of the asymptotic symmetries are finite even when the logarithmic branch is considered but acquire, however, a contribution from the scalar field.

String interactions in gravitational wave backgrounds

The non-compact CFT of a class of NS-supported pp-wave backgrounds is solved exactly. The associated tree-level covariant string scattering amplitudes are calculated. The S-matrix elements are well-defined, dual but not analytic as a function of p +. They have poles corresponding to physical intermediate states with p +≠0 and logarithmic branch cuts due to on-shell exchange of spectral-flow images of p +=0 states. When μ→0 a smooth flat space limit is obtained. The μ→∞ limit, unlike the case of RR-supported pp-waves, gives again a flat space theory.

On boundedness of integral means of Blaschke product logarithms

Abstract Using the Fourier series method for the analytic functions, we obtain a result characterizing the behaviour of the integral means of Blaschke product logarithms. Namely, if the zero counting function n(r, B) of the Blaschke product B satisfies the condition where l is a positive function on (0, 1) such that then the q‐integral mean mq (r, log B) = [] is bounded on (0,1), where log B is a branch of the logarithm of B.

Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms

Dedicated to Allan Peterson on the occasion of his 60th birthday. A Floquet theory is presented that generalizes known results for differential equations and for difference equations to the setting of dynamic equations on time scales. Since logarithms of matrices play a key role in Floquet theory, considerable effort is expended in producing case-free exact representations of the principal branch of the matrix logarithm. Such representations were first produced by Putzer as representations of matrix exponentials. Some representations depend on knowledge of the eigenvalues while others depend only on the coefficients of the characteristic polynomial. Logarithms of special forms of matrices are also considered. In particular, it is shown that the square of a nonsingular matrix with real entries has a logarithm with real entries.

Higher-order poles and mass-shell singularities in electron-hydrogen scattering

The forward scattering amplitude for electron-hydrogen scattering is expected to be an analytic function of the energy. However, when calculated within the framework of nonrelativistic quantum mechanics and the Born approximation, the exchange part has double and triple left-hand poles. From the viewpoint of field theory, such higher-order singularities are unexpected: One normally encounters only simple poles and branch points. A simple nonrelativistic model, which interpolates between binding by short-range and long-range forces, is used to show that these singularities are a direct consequence of the fact that the target system is bound by Coulomb forces: If the binding is by short-range forces there is only a simple pole. The double and triple poles emerge from a coalescence of simple poles and logarithmic branch points, in the limit of long-range binding. In the same framework, the direct amplitude has both simple and double poles, regarded as a function of the squared momentum transfer. Now even the simple pole is unexpected because the H atom is neutral. Again, the existence of these singularities is shown to be a consequence of Coulomb binding. The singularities are then studied within the framework of field theory. I show that if the H atom is treated as if it were an elementary particle, described by a field ${\ensuremath{\varphi}}_{\mathrm{H}}$ coupled to electron and proton fields ${\ensuremath{\varphi}}_{e}$ and ${\ensuremath{\varphi}}_{p}$ by a Lagrangian density ${L}_{I}=\ensuremath{-}{G}_{0}({\ensuremath{\varphi}}_{e}{\ensuremath{\varphi}}_{p}{)}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\varphi}}_{\mathrm{H}}+\mathrm{H}.\mathrm{c}.,$ the location of the singularity of the exchange amplitude is immediately obtained from a tree diagram, without the need to carry out any integrations. However, the nature of the singularity found does not agree with the nonrelativistic theory: There is no free lunch. Further analysis indicates that in full-fledged quantum electrodynamics, the vertex function \ensuremath{\Gamma} which describes the virtual decomposition of the H atom into its constituents must itself be singular when all particles are on the mass shell.

On linear independence of values of generalized polylogarithms

The linear independence of the values at certain rational points of generalized polylogarithms, which generate the algebra of all analytic functions with three logarithmic branch points, is established. Hermite-Padé approximations for an Angelesco-Nikishin system defined by a complete binary tree are used.

Summing divergent perturbative series in a strong coupling limit. The Gell-Mann-Low function of the ϕ4 theory

An algorithm is proposed for determining asymptotics of the sum of a perturbative series in the strong coupling limit using given values of the expansion coefficients. Application of the algorithm is illustrated, methods for estimating errors are developed, and an optimization procedure is described. Applied to the ϕ4 theory, the algorithm yields the Gell-Mann-Low function asymptotics of the type β(g)≈7.4g 0.96 for large g. The fact that the exponent is close to unity can be interpreted as a manifestation of the logarithmic branching of the type β(g)∼g (lng)−γ (with γ≈0.14), which is confirmed by independent evidence. In any case, the ϕ4 theory is self-consistent. The procedure of summing perturbative series with arbitrary values of the expansion parameter is discussed.

Three-Gluon Decay Function Using Space-Like Jet Calculus beyond the Leading Order

Three-gluon decay functions in space-like gluon branching are calculated in the next-toleading order of QCD. The calculated results satisfy the crossing symmetry between spacelike branching and time-like branching. Some properties of the decay functions are also examined in the case of soft gluon radiations and the case of branching to a space-like gluon with small momentum fraction. Furthermore, kinematical constraints for the leading logarithmic branching due to interference effects are studied. It is pointed out that the nextto-leading order contributions are not negligible, even if the angular-ordering conditions are imposed on two-body branching processes.

Behavior of logarithmic branch cuts in the self-energy of gluons at finite temperature

We give a simple argument for the cancellation of the $\mathrm{log}(\ensuremath{-}{k}^{2})$ terms $(k$ is the gluon momentum) between the zero-temperature and the temperature-dependent parts of the thermal self-energy.

On the singularity structure of the 2D Ising model susceptibility

Some simplifications of the integrals (2n+1), derived by Wu et al (1976 13 316), that contribute to the zero field susceptibility of the 2D square lattice Ising model are reported. In particular, several alternate expressions for the integrands in (2n+1) are determined which greatly facilitate both the generation of high-temperature series and analytical analysis. One can show that as series, (2n+1) = 22n(s/2)4n(n+1)(1+O(s)) where s is the high-temperature variable sin(2K) with K the conventional normalized inverse temperature. Analysis of the integrals near symmetry points of the integrands shows that (2n+1)(s) is singular on the unit circle at sk = exp(ik) where 2cos(k) = cos(2 k/(2n+1))+cos(2/(2n+1)), -n k, n. The singularities, k = 0 excepted, are logarithmic branch points of order 2n(n+1)-1ln() with = 1-s/sk. There is numerical evidence from series that these van Hove points, in addition to the known points at s = ?1 and ?i, exhaust the singularities on the unit circle. Barring cancellation from extra (unobserved) singularities one can conclude that |s| = 1 is a natural boundary for the susceptibility.

Non-integrability and chaos in classical cosmology

A brief analysis of the dynamics of a Friedmann-Robertson-Walker universe with a conformally coupled, real, self-interacting, massive scalar field, based on the Painlevé theory of differential equations, is presented. Our results complete earlier works done within the framework of dynamical system theory. We conclude that, in general, the system will not be integrable and that the chaos that has been found in a previous work, arises from the presence of movable logarithmic branch points in the solution in the complex plane of time.

Penalty/Barrier Multiplier Methods for Convex Programming Problems

We study a class of methods for solving convex programs, which are based on nonquadratic augmented Lagrangians for which the penalty parameters are functions of the multipliers. This gives rise to Lagrangians which are nonlinear in the multipliers. Each augmented Lagrangian is specified by a choice of a penalty function $\varphi$ and a penalty-updating function $\pi$. The requirements on $\varphi$ are mild and allow for the inclusion of most of the previously suggested augmented Lagrangians. More importantly, a new type of penalty/barrier function (having a logarithmic branch glued to a quadratic branch) is introduced and used to construct an efficient algorithm. Convergence of the algorithms is proved for the case of $\pi$ being a sublinear function of the dual multipliers. The algorithms are tested on large-scale quadratically constrained problems arising in structural optimization.

Weak perturbative Painlevé analysis and descending series solutions

We consider a weak extension of the recently introduced perturbative Painlevé test, whereby as in the weak extension of the standard Painlevé test finite branching is allowed. This extension allows us to capture even more information about the singularity structure of the solutions to nonlinear ordinary (and partial) differential equations, providing a means of uncovering logarithmic branching which might otherwise be missed. Such logarithmic branching might be thought of as providing a greater obstacle to integrability than does finite branching alone. When our new test is passed the resulting expansion is a Puiseux series in ξ = ( x+ x 0), i.e. doubly infinite in rational powers of ξ. The above-mentioned perturbative Painlevé test is included as a special case. However, we are also interested in finding the most efficient means of uncovering such logarithmic branching; we find that for some examples the way to do this is by constructing a descending series solution. For some equations this means that logarithmic branching is exhibited in a particular solution rather than in the general solution. The construction of descending series solutions can be undertaken as a special case of the perturbative Painlevé algorithm. The techniques developed here then allow us to reconsider the analysis of the ordinary differential equation U xx + UU x + kU 3 = 0.

Convergence of psi-series solutions of the Duffing equation and the Lorenz system

The general solutions of both the Duffing equation and the Lorenz system admit meromorphic Laurent series representations about movable poles only for certain values of their respective parameters. In all other parameter regimes, the general solutions contain movable logarithmic branch points. Series expansions about such points are termed psi-series and constitute formal general solutions. In this article, we prove the convergence of the psi-series solutions of the Duffing equation and the Lorenz system, thus establishing that the formal solutions are actual solutions.

QUARK-GLUON VERTEX AND STRUCTURE OF MESONS AND DIQUARKS

Two quark propagators with different analytic structure are employed in Bethe-Salpeter type equations for the pion and scalar diquark form factors. One of the quark propagators has been calculated with the inclusion of a trivial (bare) quark-gluon vertex and, as a consequence, contains a complex conjugate pair of logarithmic branch points. The other quark propagator is obtained using a non-trivial (dressed) vertex ansatz and is entire, with an essential singularity at infinity. The effects of these different quark propagators on the BSE solutions are compared.

Determination of resonance spectra for bound chaotic systems

We consider the computation of the eigenvalues of the evolution operator-the resonance spectrum-by means of the zeros of a zeta function. In particular we address the problems of applying this formalism to bound chaotic systems, caused by e.g. intermittency and non-completeness of the symbolic dynamics. For bound intermittent systems we derive an approximation of the zeta function. With the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces (of the evolution operator) approaching unity as a power law. We also show that the dominant time scale can be much longer than the period of the shortest periodic orbit, as is, for example, the case for the hyperbola billiard. Isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion. Crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function. Semiclassical implications are briefly discussed.

On The Parametric Model of Western Boundary Outflow

A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these values.

Analytic Function with Subordination Class Determined by Rotations

Let A denote the set of all functions analytic in the open unit disk equipped with the topology of uniform convergence on compact subsets of UFor F ∊ A define Let and denote the closed convex hull of s(F) and the extreme points of respectively. Let R denote the class of all F ∊ A such that where Fx(z) = F(xz). Let (Log is the principal branch of logarithm) and A-type set means its translation A + c with c ∊ C An L-domain is a connected open set which is a union of A-type sets. We proved: If Ω is an L-domain and f: U → Ω is a Riemann map, then e f ∊ R. AMS No. 30C80

A perturbative Painlevé approach to nonlinear differential equations

We further improve the Painlevé test so that negative indices (“resonances”) can be treated: we demand single valuedness not only for any pole-like expansion as in the usual Painlevé test, but also for every solution close to it, represented as a perturbation series in a small parameter ε. Order zero is the usual test. Order one, already treated in a preliminary paper, reduces to a (linear) Fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients. Higher orders lead to the analysis of a linear, Fuchsian type inhomogeneous equation. We obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points, arising at every integer index, whether positive or negative, and at every order; those arising at negative indices, including -1, are new, while some conditions may not arise before some high perturbation order. We present several illustrative examples. We discuss the understanding of negative indices, and conclude that they are indistinguishable from positive indices, just as in the Fuchs theory. In particular, negative indices give rise to doubly infinite Laurent series.

Similarity Reductions and Painlevé Properties for the Kupershmidt Equations

Five types of similarity reductions of the Kupershmidt equations which admit a tri-Hamiltonian structure are found by a direct method. Two types of reduction equations which are Painlevé II and IV types are coincident with those obtained by classical Lie approach. Both algebraic and logarithmic branch points for time t can be entered into the solutions of Kupershmidt equations. The integrability of the Kupershmidt equations is re-examined by the singularity analysis using the Weiss-Kruskal approach and the Ablowitz-Ramani-Segur algorithm.