Borel Transformation Research Articles

On the stability by convolution product of a resurgent algebra

Various functional spaces take place in Resurgence theory : multiplicative spaces of formal series expansions that one would like to sum; convolutive spaces of analytic functions, the elements of which coming from the former ones by formal Borel transformations; multiplicative spaces of analytic functions deduced from the previous ones by Laplace transformations, thus giving the Borel-Laplace transforms whose asymptotics give back the formal objects one started with. This thesis is devoted to the construction of convolution algebras. Our aim is to provide an original and self-contained proof of the stability under convolution products of the space of endlessly continuable functions, in a way understandable by any youngsearcher in the field. The second part of the thesis concentrates on the convolution space of endlessly continuable functions with simple singularities. We show how the use of the alien derivations bring deep knowlege on the singular structure. We end our work with some problems, still open according to us but of great importance in practice and for which we think that our methods could be applied as well.

Continuum scaling in expansions effective at a large lattice spacing

A new class of truncation schemes of delta expansion on the lattice is studied. We show that the order of expansion in δ which is introduced as the dilation parameter can be taken as large enough and the result gives rise to the Borel transformation with respect to the relevant variable in the lattice models. The explicit simulation of the continuum scaling from the expansion effective at large spacings is investigated in anharmonic oscillators, the d = 2 nonlinear σ model at large and the Gross–Neveu model with Wilson fermions.

A Borel Transform Method for Locating Singularities of Taylor and Fourier Series

Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of P\'olya relates the large d istance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge of a large number of Taylor coefficients we can identify precisely the location of such singularities, as well as their type when they are isolated. There is no risk of getting artefacts with this method, which also gives us access to some of the singularities beyond the convergence disk. The method can also be applied to Fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the Burgers equation. Large precision on scaling exponents (up to twenty accurate digits) can be achieved.

Growth of entire functions of exponential type defined by convolution

Based on the Borel transformation and the Hadamard multiplication theorem on singularities on the convolution of holomorphic functions, results on the growth of entire functions defined by convolution of an entire function of exponential type with a function holomorphic at the origin are obtained.

Pseudo-ɛ-Expansion and the Two-Dimensional Ising Model

Starting from the five-loop renormalization-group expansions for the two-dimensional Euclidean scalar \phi^4 field theory (field-theoretical version of two-dimensional Ising model), pseudo-\epsilon expansions for the Wilson fixed point coordinate g*, critical exponents, and the sextic effective coupling constant g_6 are obtained. Pseudo-\epsilon expansions for g*, inverse susceptibility exponent \gamma, and g_6 are found to possess a remarkable property - higher-order terms in these expansions turn out to be so small that accurate enough numerical estimates can be obtained using simple Pade approximants, i. e. without addressing resummation procedures based upon the Borel transformation.

Critical thermodynamics of a three-dimensional chiral model forN>3

The critical behavior of the three-dimensional $N$-vector chiral model is studied for arbitrary $N$. The known six-loop renormalization-group (RG) expansions are resummed using the Borel transformation combined with the conformal mapping and Pad\'e approximant techniques. Analyzing the fixed point location and the structure of RG flows, it is found that two marginal values of $N$ exist which separate domains of continuous chiral phase transitions $N > N_{c1}$ and $N < N_{c2}$ from the region $N_{c1} > N > N_{c2}$ where such transitions are first-order. Our calculations yield $N_{c1} = 6.4(4)$ and $N_{c2} = 5.7(3)$. For $N > N_{c1}$ the structure of RG flows is identical to that given by the $\epsilon$ and 1/N expansions with the chiral fixed point being a stable node. For $N < N_{c2}$ the chiral fixed point turns out to be a focus having no generic relation to the stable fixed point seen at small $\epsilon$ and large $N$. In this domain, containing the physical values $N = 2$ and $N = 3$, phase trajectories approach the fixed point in a spiral-like manner giving rise to unusual crossover regimes which may imitate varying (scattered) critical exponents seen in numerous physical and computer experiments.

Generalized Borel transform technique in quantum mechanics

We present the Generalized Borel Transform (GBT). This new approach allows one to obtain approximate solutions of Laplace/Mellin transform valid in both, perturbative and non-perturbative regimes. We compare the results provided by the GBT for a solvable model of quantum mechanics with those provided by standard techniques, as the conventional Borel sum, or its modified versions. We found that our approach is very efficient for obtaining both the low and the high energy behavior of the model.

Valence quark distributions in nucleon at low Q2 in QCD

Valence u- and d-quarks distributions in proton are calculated in QCD at low Q 2 and intermediate x, basing on the operator product expansion (OPE). The imaginary part of the virtual photon scattering amplitude on quark current with proton quantum numbers is considered. The initial and final virtualities p 2 1 and p 2 2 of the currents are assumed to be large, negative and different, p 2 1≠ p 2 2. The OPE in p 2 1, p 2 2 up to dimension 6 operators was performed. Double dispersion representations in p 2 1, p 2 2 of the amplitudes in terms of physical states contributions are used. Putting them to be equal to those calculated in QCD, the sum rules for quark distributions are found. The double Borel transformations are applied to the sum rules. Leading order perturbative corrections are accounted. Valence quark distributions are found: u( x) v at 0.15< x<65, d( x) v at 0.25< x<0.55 with an accuracy ∼30% in the middles and ∼50% at the ends of these intervals. The quark distributions obtained are in agreement with those found from the analysis of hard processes data.

On the double series expansion of holomorphic functions

A holomorphic function f in a neighborhood of 0 in C n+1 can be expanded into the double series: f( z)=∑ k=0 ∞∑ l=0 [ k/2] ( z 2) l f k, k−2 l ( z), where f k, k−2 l is a homogeneous harmonic polynomial of degree k−2 l and z 2= z 1 2+⋯+ z n+1 2. We characterized holomorphic functions on the complex Euclidean ball, on the Lie ball or on the dual Lie ball by the growth behavior of homogeneous harmonic polynomials in their double series expansion. In this paper, we consider holomorphic functions and analytic functionals on an N p -ball which lies between the Lie ball and the dual Lie ball, and characterize them by the growth behavior of homogeneous harmonic polynomials. Our results lead a new proof of a known theorem on the Fourier–Borel transformation.

Perturbative approach to the hydrogen atom in a strong magnetic field

The states of hydrogen atom with principal quantum number n <= 3 and zero magnetic quantum number in constant homogeneous magnetic field H are considered. The perturbation theory series is summed with the help of Borel transformation and conformal mapping of the Borel variable. Convergence of approximate energy eigenvalues and their agreement with corresponding existing results are observed for external fields up to n^3 H ~ 5. The possibility of restoring the asymptotic behaviour of energy levels using perturbation theory coefficients is also discussed.

Quark–antiquark potential and generalized Borel transform☆☆Partially supported by CONICET and ANPCyT-Argentina.

The heavy quark potential and particularly the one proposed by Richardson to incorporate both asymptotic freedom and linear confinement is analyzed in terms of a generalized Borel Transform recently proposed. We were able to obtain, in the range of physical interest, an approximate analytical expression for the potential in coordinate space valid even for intermediate distances. The deviation between our approximate potential and the numerical evaluation of the Richardson's one is much smaller than $\Lambda $ of QCD. The $c\overline{c}$ and $b\overline{b}$ quarkonia energy levels agree reasonably well with experimental data for $c$ and $b$ masses in good agreement with the values obtained from experiments.

Check of QCD based on theτ-decay data analysis in the complexq2plane

The thorough analysis of the ALEPH data on hadronic $\ensuremath{\tau}$ decay is performed in the framework of QCD. The perturbative calculations are performed in three- and four-loop approximations. The terms of the operator product expansion (OPE) are accounted for up to the dimension $D=8.$ The value of the QCD coupling constant ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.355\ifmmode\pm\else\textpm\fi{}0.025$ is found from the hadronic branching ratio ${R}_{\ensuremath{\tau}}.$ The $V+A$ and V spectral functions are analyzed using the analytical properties of the polarization operators in the whole complex ${q}^{2}$ plane. Borel sum rules in the complex ${q}^{2}$ plane along the rays, starting from the origin, are used. It is demonstrated that QCD with OPE terms is in agreement with the data for a coupling constant close to the lower error edge ${\ensuremath{\alpha}}_{s}{(m}_{\ensuremath{\tau}}^{2})=0.330.$ The restriction on the value of the gluonic condensate was found to be $〈({\ensuremath{\alpha}}_{s}/\ensuremath{\pi}{)G}^{2}〉=0.006\ifmmode\pm\else\textpm\fi{}0.012 {\mathrm{GeV}}^{2}.$ The analytical perturbative QCD is compared with the data. It is demonstrated to be in strong contradiction with experiment. The restrictions on the renormalon contribution are found. The instanton contributions to the polarization operator are analyzed in various sum rules. In the Borel transformation they appear to be small, but not in the spectral moment sum rules.

Determination of the amplitude of the repulsive-pair potential between particles clothed by end-grafted polymers

We examine the problem of the determination of the repulsive potential between spherical particles clothed by long end-grafted flexible polymers. This potential varying with the distance according to a logarithmic law has a potential amplitude that depends on the number, L, of grafting chains per particle. The purpose of this work is to compute such a potential amplitude. The clothed particles are first regarded as star polymers with small enough diameter and the same number of arms. Then, the amplitude potential is identified to the critical exponent related to the contact probability between cores of these stars, which allows us to find a universal function for the expected potential amplitude depending on L and d-space dimension only. In two-dimensional space, conformal invariance is used to extract the potential amplitude as a function of L. For dimensions greater than 2, the potential amplitude is obtained within the framework of renormalization theory to third order in ɛ=4 − d, where d is the critical dimension of the system. To determine the best three-dimensional expression for the potential amplitude, AL, use is made of the Pade–Borel transformation, which provides a closer form valid for small, intermediate and high values of L. This form of potential amplitude, consistent with the exact scaling asymptotic value of Witten and Pincus [(1986) Macromolecules 19:2509], allows us to find the associated prefactor. The procedure is also extended to interacting stars of different numbers of arms.

Calculation of the continuum contribution to hadronic current correlation functions in a chiral quark model with confinement

A powerful technique for the calculation of hadronic properties is the application of QCD sum rules in a study of correlation functions. The calculation of hadronic current correlation functions may be carried out using the operator-product expansion. The correlator may also be represented by hadronic states, which include one or more discrete levels and a continuum contribution. Comparison of the two methods of calculation of the same correlator allows for the determination of hadronic parameters after one estimates the continuum contribution and specifies the values of the vacuum condensates. In this work we consider two types of contribution to the continuum. The first involves the radially excited states of a meson. These states appear as resonances which ultimately decay into the continuum of multimeson states. In a second process, the multimeson continuum is reached in a direct process, without the excitation of a resonance. For the resonances, we calculate the contribution to the hadronic correlator by calculating the meson decay constants in the region $0&lt;{P}^{2}&lt;6{\mathrm{GeV}}^{2},$ making use of our generalized Nambu--Jona-Lasinio model, which includes a covariant model of confinement. [Our model provides a good fit to the decay constants of the ${a}_{0}(1450)$ and ${K}_{0}^{*}$ (1430) mesons. Our value for the ${a}_{0}(980)$ is about 60% larger than the value obtained by Maltman using QCD sum rules. Also, our values for the pion and kaon decay constants are about 35 and 10% too large, respectively. We have not performed any parameter variations to improve upon these values.] The second contribution to the correlator arises from the direct excitation of multimeson continuum states. For the ${a}_{0}(980)$ and its radial excitations, we calculate both the resonant and direct contributions. In the case of the isovector pseudoscalar states we calculate the resonance contribution and parametrize the direct multimeson contribution so as to reproduce a phenomenological expression of the quark-hadron duality model. In this case the calculated resonance contribution is about 15% of the phenomenological result after a Borel transformation is made. The resonance contribution plus a quark-hadron duality model contribution with ${s}_{0}=1.65{\mathrm{GeV}}^{2}$ yields a good fit to the phenomenological form. In this phenomenological form only the quark-hadron duality model is used with ${s}_{0}=1.53{\mathrm{GeV}}^{2}.$ We also provide values of the resonance contribution to the continuum strength in the case of the strange pseudoscalar states. In the case of the isovector scalar mesons we find significant disagreement between our microscopic calculations and the result of the quark-duality model for the continuum of the hadronic current correlator. Indeed, the contribution of the resonances to the continuum is a least an order of magnitude larger than the values obtained from the quark-hadron duality model. That result may suggest an explanation for the problems encountered by some authors who have studied the isovector scalar mesons using QCD sum rules.

Valence quark distributions in mesons in generalized QCD sum rules

The method for calculation of valence quark distributions at intermediate x is presented. The imaginary part of the virtual photon forward scattering amplitude on the quark current with meson quantum number is considered. The initial and final virtualities ${p}_{1}^{2}$ and ${p}_{2}^{2}$ of the currents are assumed to be large, negative, and different, ${p}_{1}^{2}\ensuremath{\ne}{p}_{2}^{2}.$ The operator product expansion (OPE) in ${p}_{1}^{2}{,p}_{2}^{2}$ up to dimension 6 operators is performed. Double dispersion representations in ${p}_{1}^{2}{,p}_{2}^{2}$ of the amplitude in terms of physical state contributions are used. Setting them equal to those calculated in QCD by OPE, the desired sum rules for quark distributions in mesons are found. The double Borel transformations are applied to the sum rules, destroying nondiagonal transition terms, which deteriorated the accuracy in the previous calculations of quark distributions in the nucleon. Leading-order perturbative corrections are taken into account. Valence quark distributions in the pion and longitudinally and transversally polarized $\ensuremath{\rho}$ mesons are calculated at intermediate $x, 0.2\ensuremath{\lesssim}x\ensuremath{\lesssim}0.7$ and normalization points ${Q}^{2}=2\ensuremath{-}4 {\mathrm{GeV}}^{2}$ with no fitting parameters. The use of the Regge behavior at small x and quark counting rules at large x allows one to find the first and the second moments of valence quark distributions. The obtained quark distributions may be used as an input for evolution equations. In the case of the pion the quark distribution is in agreement with those found from the data on the Drell-Yan process. The quark distributions in transversally and longitudinally polarized $\ensuremath{\rho}$ mesons are essentially different.

Borel summable solutions to one-dimensional Schrödinger equation

It is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be Borel summable to the solutions themselves appear also to be the unique solutions to the one-dimensional (1D) Schrödinger equation having this property. Namely, it is shown in this paper that for the polynomial potentials the Borel function defined by the fundamental solutions can be considered as the canonical one. The latter means that any Borel summable solution can be obtained by the Borel transformation of this unique canonical Borel function multiplied by some ℏ-dependent and Borel summable constant. This justifies the exceptional role the fundamental solutions play in 1D quantum mechanics and completes the relevant semiclassical theory relied on the Borel resummation technique and developed in our other papers.

The Borel Transform and Its Use in the Summation of Asymptotic Expansions

The solution of connection problems on the real line (the x axis) often give asymptotic expansions which are either even or odd. This gives rise to “identically zero” expansions, that is, an asymptotic expansion in which all terms are identically zero at the origin. We show that the Borel transform of these problems have solutions that provide integral representations of the solution. The evaluation of these integrals, as x→0, allows us to compute the exponentially small term that these “identically zero” expansions represent.

N-component Ginzburg-Landau Hamiltonian with cubic anisotropy: A six-loop study

We consider the Ginzburg-Landau Hamiltonian with a cubic-symmetric quartic interaction and compute the renormalization-group functions to six-loop order in $d=3.$ We analyze the stability of the fixed points using a Borel transformation and a conformal mapping that takes into account the singularities of the Borel transform. We find that the cubic fixed point is stable for $N&gt;{N}_{c},$ ${N}_{c}=2.89(4).$ Therefore, the critical properties of cubic ferromagnets are not described by the Heisenberg isotropic Hamiltonian, but instead by the cubic model at the cubic fixed point. For $N=3,$ the critical exponents at the cubic and symmetric fixed points differ very little (less than the precision of our results, which is $\ensuremath{\lesssim}1%$ in the case of $\ensuremath{\gamma}$ and $\ensuremath{\nu}).$ Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly decaying scaling corrections with exponent ${\ensuremath{\omega}}_{2}=0.010(4).$ For $N=2,$ the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent ${\ensuremath{\omega}}_{2}=0.103(8).$ These conclusions are confirmed by a similar analysis of the five-loop $\ensuremath{\epsilon}$ expansion. A constrained analysis, which takes into account that ${N}_{c}=2$ in two dimensions, gives ${N}_{c}=2.87(5).$

Use of the Double Dispersion Relation in QCD Sum Rules with External Fields

In QCD sum rules with external fields, the double dispersion relation is often used to represent the correlation function. In this work, we point out that the double spectral density, when it is determined by successive applications of the Borel transformation, contains spurious terms which should be kept in the subtraction terms in the double dispersion relation. They are zero under the Borel transformation, but, if the dispersion integral is restricted with QCD duality, they contribute to the continuum. for the simple case with zero external momentum, it is shown that subtracting out the spurious terms is equivalent to the QCD sum rules represented by the single dispersion relation.

Improved calculations of quark distributions in hadrons: the case of the pion

The earlier introduced method of calculation of quark distributions in hadrons, based on QCD sum rules, is improved. The imaginary part of the virtual photon forward scattering amplitude on some hadronic current is considered in the case, when initial and final virtualities of the current $p^2_1$ , and $p^2_2$ are different, $p^2_1\ne p^2_2$ . The operator product expansion (OPE) in $p^2_1$ , $p^2_2$ is performed. The sum rule for quark distribution is obtained using double dispersion representation of the amplitude on one side in terms of calculated in QCD OPE and on the other side in terms of physical states contributions. Double Borel transformation in $p^2_1$ , $p^2_2$ is applied to the sum rule, killing background non-diagonal transition terms which deteriorated the accuracy in previous calculations. The case of the valence quark distribution in the pion is considered, which was impossible to treat by the previous method. OPE up to dimension 6 operators is performed and leading order perturbative corrections are accounted. Valence u-quark distribution in $\pi^+$ was found at intermediate x, $0.15 < x < 0.7$ , and normalization point $Q^2=2{\mathrm{\,GeV}}^2$ . These results may be used as input for evolution equations.