Borel Method Research Articles

Iterative Borel Summation with Self-Similar Iterated Roots

Borel summation is applied iteratively in conjunction with self-similar iterated roots. In general form, the iterative Borel summation is presented in the form of a multi-dimensional integral. It can be developed only numerically and is rarely used. Such a technique is developed in the current paper analytically and is shown to be more powerful than the original Borel summation. The self-similar nature of roots and their asymptotic scale invariance allow us to find critical indices and amplitudes directly and explicitly. The locations of poles remain the same with the uncontrolled self-similar Borel summation. The number of steps employed in the course of iterations is used as a continuous control parameter. To introduce control into the discrete version of the iterative Borel summation, instead of the exponential function, we use a stretched (compacted) exponential function. For the poles, considering inverse quantities is prescribed. The simplest scheme of the iterative Borel method, based on averaging over the one-step and two-step Borel iterations, works well when lower and upper bounds are established by making those steps. In the situations when only a one-sided bound is found, the iterative Borel summation with the number of iterations employed as the control works best by extrapolating beyond the bound. Several key examples from condensed matter physics are considered. Iterative application of Borel summation leads to an improvement compared with a conventional, single-step application of the Borel summation.

Phenomenology of combined resummation for Higgs and Drell-Yan

We study the phenomenological impact of a recently suggested formalism for the combination of threshold and a so-called threshold-improved transverse momentum resummation, by using it to improve the fixed-order results. This formalism allows for a systematic improvement of the transverse momentum resummation that is valid in the entire range of pT by the inclusion of the threshold contribution. We use the Borel method as a suitable prescription for defining the inverse Mellin and Fourier transforms in the context of combined resummed expression. The study is applied to two QCD processes, namely the Higgs boson produced via gluon fusion and Z boson production via the Drell-Yan mechanism. We compare our results to the standard transverse momentum resummation, as well as to the fixed-order results. We find that the threshold-improved transverse momentum resummation leads to faster perturbative convergence at small pT while the inclusion of threshold resummation improves the agreement with fixed-order calculations at medium and large pT. These effects are more pronounced in the case of Higgs which is known to have slower perturbative convergence.

Analytic Continuation Methods for Multivalued Functions of One Variable and Their Application to the Solution of Algebraic Equations

The paper discusses several methods of analytic continuation of a multivalued function of one variable given on a part of its Riemann surface in the form of a Puiseux series generated by the power function z = w1/ρ, where ρ 1/2 and ρ ≠ 1. We present a many-sheeted variant of G.Polya’s theorem describing the relation between the indicator and conjugate diagrams for entire functions of exponential type. The description is based on V. Bernstein’s construction for the many-sheeted indicator diagram of an entire function of order ρ ≠ 1 and normal type. The summation domain of the “proper” Puiseux series (the many-sheeted “Borel polygon”) is found with the use of a generalization of the Borel method. This result seems to be new even in the case of a power series. The theory is applied to describe the domains of analytic continuation of Puiseux series representing the inverses of rational functions. As a consequence, a new approach to the solution of algebraic equations is found.

Korovkin type approximation theorems via power series method

In this paper we consider power series method which is also member of the class of all continuous summability methods. The power series method includes Abel method as well as Borel method. We investigate, using the power series method, Korovkin type approximation theorems for the sequence of positive linear operators defined on C[a, b] and \(L_{q}[a,b]\), \(1\le q<\infty \), respectively. We also study some quantitative estimates for \(L_{q}\) approximation and give the rate of convergence of these operators.

Renormalization-group description of nonequilibrium critical short-time relaxation processes: A three-loop approximation

The influence of nonequilibrium initial values of the order parameter on its evolution at a critical point is described using a renormalization group approach of the field theory. The dynamic critical exponent $\theta'$ of the short time evolution of a system with an $n$-component order parameter is calculated within a dynamical dissipative model using the method of $\varepsilon$-expansion in a three-loop approximation. Numerical values of $\theta'$ for three-dimensional systems are determined using the Pad\'{e}-Borel method for the summation of asymptotic series.

A one-sided Tauberian theorem for the Borel summability method

We establish a quantitative version of Vijayaraghavan's classical result and use it to give a short proof of the known theorem that a real sequence ( s n ) which is summable by the Borel method, and which satisfies the one-sided Tauberian condition that n (s n−s n−1) is bounded below must be convergent.

Resummation of the divergent perturbation series for a hydrogen atom in an electric field

We consider the resummation of the perturbation series describing the energy displacement of a hydrogenic bound state in an electric field (known as the Stark effect or the LoSurdo-Stark effect), which constitutes a divergent formal power series in the electric field strength. The perturbation series exhibits a rich singularity structure in the Borel plane. Resummation methods are presented which appear to lead to consistent results even in problematic cases where isolated singularities or branch cuts are present on the positive and negative real axis in the Borel plane. Two resummation prescriptions are compared: (i) a variant of the Borel-Pade resummation method, with an additional improvement due to utilization of the leading renormalon poles (for a comprehensive discussion of renormalons see [M. Beneke, Phys. Rep. vol. 317, p. 1 (1999)]), and (ii) a contour-improved combination of the Borel method with an analytic continuation by conformal mapping, and Pade approximations in the conformal variable. The singularity structure in the case of the LoSurdo-Stark effect in the complex Borel plane is shown to be similar to (divergent) perturbative expansions in quantum chromodynamics.

Realization of compact Lie algebras in Kahler manifolds

The Berezin quantization on a simply connected homogeneous Kahler manifold, which is considered as a phase space for a dynamical system, enables a description of the quantal system in a (finite-dimensional) Hilbert space of holomorphic functions corresponding to generalized coherent states. The Lie algebra associated with the manifold symmetry group is given in terms of first-order differential operators. In the classical theory, the Lie algebra is represented by the momentum maps which are functions on the manifold, and the Lie product is the Poisson bracket given by the Kahler structure. The Kahler potentials are constructed for the manifolds related to all compact semi-simple Lie groups. The complex coordinates are introduced by means of the Borel method. The Kahler structure is obtained explicitly for any unitary group representation. The cocycle functions for the Lie algebra and the Killing vector fields on the manifold are also obtained.

A General Tauberian Condition that Implies Euler Summability

AbstractLet V be any summability method (whether linear or conservative or not), 0 &lt; p &lt; 1 and s a real or complex sequence. Let Ep denote the matrix of the Euler method. A theorem is proved, giving a condition under which the V-summability of Eps will imply the Ep-summability of s. This extends, in generalized form, an earlier result of N. H. Bingham who considered the case where s is a real sequence and V = B (Borel's method). It is also proved that even for real sequences, the condition given in the theorem cannot be replaced by the condition used by Bingham.

A Tauberian Theorem for the General Euler-Borel Summability Method

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m &gt; n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

Convexity theorems for the circle methods of summability

A well-known theorem due to Littlewood (1911) and Andersen (1921) tells us that the family of summability methods consisting of the Abel method and all Cesàro methods C α , α > −1, is convex. In this paper we prove that this theorem remains true if we replace the Abel method and the Cesàro methods by the Borel method and the Euler-Knopp methods of positive order, respectively. Furthermore, we give some variants of this result in case of other circle methods of summability.

Summation of the perturbation expansion and the QCD corrections to tau decay.

The problem of summation of the perturbation expansion in QCD is discussed. It is argued that this expansion is summable by a modified Borel method. The QCD corrections to the semileptonic width of a heavy lepton are considered as an example. The possible differences between the predictions from the resummed expansion and the truncated expansion are illustrated. It is shown that the use of a Borel-type approximant may eliminate certain discrepancy between the theoretical prediction for the $\ensuremath{\tau}$ lepton and experiment.

Novel summability methods generalizing the Borel method

The paper deals with moment constant summability methods. A method is constructed which provides an analytic continuation of a function regular at the origin onto its Mittag-Leffler (principal) star. In this sense the method is optimal in contrast to the Borel one. In the next a one parameter family of such methods is introduced. The methods may be useful both in field theory and in statistical physics. Applications to the Nevanlinna theorem, the Rayleigh-Schro-dinger perturbation theory and the dispersion-like integral are given. The proofs of theorems can be easily adapted to the study of the Mellin transform of some entire functions and a simpler proof of asymptotic properties of the gamma function can be obtained.

A Tauberian theorem for the summation of double series by Borel's method

A Tauberian theorem for the summation of double series by Borel's method

Perturbative solution for the generalised anharmonic oscillators

Serious difficulties have been encountered in the past when attempting the straightforward application of the Borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the Borel transform series which is usually performed by means of conformal mapping techniques. The author shows that this continuation may be accomplished by combining the Borel method with the first confluent form of the epsilon algorithm of Wynn (1960, 1968) for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators, two different approaches are presented, both of which lead to accurate results.

Borel summability of divergent Born series

The Borel summability method is applied to summing diverging Born series obtained from the Lippmann-Schwinger equation for the two-bodyT-matrix with the1S0 Reid potential. It is found, in practice, that appreciable numerical difficulties prevent its direct application. However, a combination of the first confluent form of the e-algorithm of Wynn with the Borel method is shown to yield a rapidly converging solution of the scattering equations of comparable accuracy to other standard methods.

Application of the Borel Method of Summation to a Quantum Mechanical Model Problem

The Borel method of summation is applied to Rayleigh-Schrödinger expansions with non-zero radius of convergence. The region of Borel summability in the complex plane of the interaction parameter is determined, and the effect of intruder states is discussed. For intruder states which are strongly coupled to the ground state the perturbation series can be Borel summable much further out than the radius of convergence. The application of Padé approximants to the Borel series is compared to the use of Padé approximants directly on the original series. Numerical applications are made to a simple model consisting of a 2 × 2 matrix Hamiltonian.

Resonances in Stark effect and perturbation theory

It is proved that the action of a weak electric field shifts the eigenvalues of the Hydrogen atom into resonances of the Stark effect, uniquely determined by the perturbation series through the Borel method. This is obtained by combining the Balslev-Combes technique of analytic dilatations with Simon's results on anharmonic oscillators.

Steinhaus Type Theorems for Summability Matrices

Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the $(C,1)$ method. Also, it is proved that no coregular matrix can almost sum all bounded sequences—a generalization of Steinhaus’ theorem.

Steinhaus type theorems for summability matrices

Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the ( C , 1 ) (C,1) method. Also, it is proved that no coregular matrix can almost sum all bounded sequences—a generalization of Steinhaus’ theorem.