Analytic Perturbation Theory and Its Applications

A unitary family is a family of unitary operators U(x) acting on a finite dimensional hermitian vector space, depending analytically on a real parameter x. It is monotone if 1 i U (x)U(x) is a positive operator for each x. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator U0, which correspond to the ’commutative’ case U(x) = eU0. Also, for a two-parameter unitary family – for which there is no analytic perturbation theory – we prove an implicit function type theorem for the spectral data under the assumption that the family is monotone in one argument.

We propose a new generalized version of the QCD analytic perturbation theory of Shirkov and Solovtsov for the computation of higher-order corrections in inclusive and exclusive processes. We construct nonpower series expansions for the analytic images of the running coupling and its powers for any fractional (real) power and complete the linear space of these solutions by constructing the index derivative. Using the Laplace transformation in conjunction with dispersion relations, we are able to derive at the one-loop order closed-form expressions for the analytic images in terms of the Lerch function. At the two-loop order we provide approximate analytic images of products of powers of the running coupling and logarithms---typical in higher-order perturbative calculations and when including evolution effects. Moreover, we supply explicit expressions for the two-loop analytic coupling and the analytic images of its powers in terms of one-loop quantities that can strongly simplify two-loop calculations. We also show how to resum powers of the running coupling while maintaining analyticity, a procedure that captures the generic features of Sudakov resummation. The algorithmic rules to obtain analytic-coupling expressions within the proposed fractional analytic perturbation theory from the standard QCD power-series expansion are supplied ready for phenomenological applications and numerical comparisons are given for illustration.

We argue that a technique called analytic perturbation theory leads to a well-defined method for analytically continuing the running coupling constant from the spacelike to the timelike region, which allows us to give a self-consistent definition of the running coupling constant for timelike momentum. The corresponding $\ensuremath{\beta}$ function is proportional to the spectral density, which confirms a hypothesis due to Schwinger.

For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α3/2, α being the fine structure constant. A suitably chosen ground state vector depends analytically on α3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

Entrainment by a rotating magnetic field of a ferrofluid contained in a cylinder is studied on the basis of spin-diffusion theory. The equations for flow velocity and spin velocity, coupled to Maxwell's equations of magnetostatics, are solved in first-harmonic approximation under the assumption that the magnetic field is small compared to the saturation magnetization. The solution leads to a coupled set of nonlinear integral equations, which can be solved numerically by iteration in a recursive scheme by use of the analytic lowest order perturbation theory solution as the initial state. At a critical applied field, the recursive scheme shows bifurcation. At sufficiently high field, the solution with the lower rate of dissipation shows flow in the direction opposite to the rotating applied field.

The Regularized Boussinesq equation: Instability of periodic traveling waves

The phase diagram of the quantum compass ladder model is investigated through numerical density matrix renormalization group based on infinite matrix product state algorithm and analytic effective perturbation theory. For this model we obtain two symmetry-protected topological phases, protected by a symmetry, and a topologically-trivial Z2-symmetry-breaking phase. The symmetry-protected topological phases—labeled by symmetry fractionalization—belong to different topological classes, where the complex-conjugate symmetry uniquely distinguishes them. An important result of this classification is that, as revealed by the nature of the Z2-symmetry-breaking phase, the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. This is in stark contrast to previous studies which require a non-local string order parameter to distinguish the corresponding quantum phase transitions. We numerically examine our results and show that the local-order parameter is related to the magnetization exponent .

In quantum mechanics the temporal decay of certain resonance states is associated with an effective time evolution e−ith(κ) , where h(·) is an analytic family of non-self-adjoint matrices. In general the corresponding resonance states do not decay ex

In the context of the inclusive $\ensuremath{\tau}$ decay, we analyze various forms of perturbative expansions which have appeared as modifications of the original perturbative series. We argue that analytic perturbation theory, which combines renormalization-group invariance and ${Q}^{2}$ analyticity, has significant merits favoring its use to describe the perturbative component of $\ensuremath{\tau}$ decay.

Asymptotic Rayleigh-Schrödinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case. For this purpose we study the spectral properties ofm×m—matrix functionsA(κ) of a complex variable κ which have an asymptotic expansion εA k κ k as τ→0. We show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofA(κ) can be obtained from the set {A κ} by analytic perturbation theory. Special attention is given to the case whereA(κ) is Borel-summable in some sector originating from κ=0 with opening angle >π. Here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofA(κ) which are shown to be holomorphic inS near κ=0 and Borel summable ifA * =A k for allk. We then fit these results into the scheme of Rayleigh-Schrödinger perturbation theory and we give some examples of asymptotic estimates for Schrödinger operators.

Experimental data obtained for the polarized Bjorken sum rule (BSR) Gamma _1^{p-n}(Q^2) are fitted by using predictions derived within a truncated operator product expansion (OPE) approach to QCD. Four QCD versions are considered: perturbative QCD (pQCD) in the {overline{mathrm{MS}}} scheme, Analytic Perturbation Theory (APT), and 2delta and 3delta analytic QCD versions. In contrast to pQCD, these QCD variants do not have Landau singularities at low positive Q^2, which facilitates the fitting procedure significantly. The fitting procedure is applied first to the experimental data of the inelastic part of BSR, and the known elastic contributions are added after the fitting. In general, when 2delta and 3delta QCD coupling is used the fitted curves give the best results, within the Q^2-range of the fit as well as in extended Q^2-intervals. When the fitting procedure is applied to the total BSR, i.e., to the sum of the experimental data and the elastic contribution, the quality of the results deteriorates significantly.

In certain layered electromagnetic media, one can construct a waveguide that supports a harmonic electromagnetic field at a frequency that is embedded in the continuous spectrum. When the structure is perturbed, this embedded eigenvalue moves into the complex plane and becomes a “complex resonance” frequency. The real and imaginary parts of this complex frequency have physical meaning. They lie behind anomalous scattering behaviors known collectively as “Fano resonance”, and people are interested in tuning them to specific values in optical devices. The mathematics involves spectral theory and analytic perturbation theory and is well understood [16], at least on a theoretical level for deterministic (fixed coefficient) media. This dissertation is a study of how random fabrication errors in the waveguide structure affect this kind of resonance. The governing equations are the free harmonic Maxwell equations of electromagnetics, which reduce to a system of ODEs in layered media. The material coefficients (dielectric permittivity ϵ(z) and magnetic permeability μ(z)) are considered to be random variables depending only on the z-variable, with small variance σ(z), and we are interested in how the real and imaginary parts of the complex resonance ω=ω^+iω behave as random variables depending on these coefficients. The first main theorem of this thesis states that, if ϵ(z) and μ(z) are stationary random variables with respect to the variable z, then the variances of ω^ and ωˇ can be computed by considering only events in which ϵ and μ are random variables that are constant in z. The significance of this theorem is that it allows one to compute the mean and variance of ω, to leading order in σ, by means of (deterministic) sensitivity analysis of ω as a function of ϵ and μ. One can consider ω to be a complex-analytic function of ϵ (and μ) if ϵ=ϵ^+iϵˇ, where both parts are Hermitian matrices. The Cauchy-Riemann equations, together with known properties of complex resonances allow one to make interesting and useful conclusions about how the frequency of a resonance and the amount of radiation losses depend on the real and imaginary (lossy) parts of the material coefficients.

We investigate the connection between ghost-free formulations of the RG-invariant QCD perturbation theory in the spacelike and timelike regions. Our basic tool is the “double spectral representation,” similar to the representation for the Adler function, which stems from the first principles of local QFT and relates real functions in the Euclidean and Minkowskian (i.e., timelike) regions. On this base, we establish a simple relation between the approach (known from the early 1980s) of resumming the π2 terms for the invariant coupling function \(\widetilde\alpha (s)\) and QCD observables in the timelike region and the invariant analytic approach (devised a few years ago) leading to the “analyticized” coupling function αan(Q2) and nonpower expansion for observables in the spacelike domain. The function αan(Q2) and the expansion are free of unphysical singularities. The formulated self-consistent scheme, analytic perturbation theory (APT), relates renorm-invariant, effective coupling functions αan(Q2) and \(\widetilde\alpha (s)\), as well as nonpower perturbation expansions for observables in the Euclidean and Minkowskian domains, free of extra singularities and with better convergence in the infrared region. We present a global generalization of the new APT scheme in the case of real QCD, including the domain with various numbers of active quarks. Preliminary estimates indicate that calculations in the framework of the global scheme can produce results quite different from the usual ones for \(\overline \alpha _s \), even in the five-quark region. Numerical examples are given.

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We consider the dynamics generated by an analytically perturbed piecewise linear expanding map on the unit interval. Rigorous analytic perturbation theory for the corresponding transfer operator is used to find the general form of the power series expansion for the topological pressure. This involves integration with respect to the unperturbed Gibbs state which is a highly singular measure supported on the Cantor set; we derive closed-form expressions for the integration of polynomials. Explicit calculations are presented for quadratically perturbed symmetric and nonsymmetric linear systems.