Borel Transformation Research Articles

BorelT.m: Borel Transformations in QCD Sum Rules with Mathematica

borelT.m: Borel Transformations in QCD Sum Rules with Mathematica

Borel resummation of secular divergences in stochastic inflation

We make use of Borel resummation to extract the exact time dependence from the divergent series found in the context of stochastic inflation. Correlation functions of self-interacting scalar fields in de Sitter spacetime are known to develop secular IR divergences via loops, and the first terms of the divergent series have been consistently computed both with standard techniques for curved spacetime quantum field theory and within the framework of stochastic inflation. We show that Borel resummation can be used to interpret the divergent series and to correctly infer the time evolution of the correlation functions. In practice, we adopt a method called Borel-Padé resummation where we approximate the Borel transformation by a Padé approximant. We also discuss the singularity structures of Borel transformations and mention possible applications to cosmology.

Borel Transform and Scale-Invariant Fractional Derivatives United

The method of Borel transformation for the summation of asymptotic expansions with the power-law asymptotic behavior at infinity is combined with elements of scale-invariant fractional analysis with the goal of calculating the critical amplitudes. The fractional order of specially designed scale-invariant fractional derivatives u is used as a control parameter to be defined uniquely from u-optimization. For resummation of the transformed expansions, we employed the self-similar iterated roots. We also consider a complementary optimization, called b-optimization with the number of iterations b as an alternative fractional control parameter. The method of scale-invariant Fractional Borel Summation consists of three constructive steps. The first step corresponds to u-optimization of the amplitudes with fixed parameter b. When the first step fails, the second step corresponds to b-optimization of the amplitudes with fixed parameter u. However, when the two steps fail, the third step corresponds to the simplified, Borel-light technique. The marginal amplitude should be found by means of the self-similar iterated roots constructed for the transformed series, optimized with either of the two above approaches and corrected with a diagonal Padé approximants. The examples are given when the complementary optimizations,“horses-for-courses” approach outperforms other analytical methods in calculation of critical amplitudes.

Modified Padé–Borel Summation

We revisit the problem of calculating amplitude at infinity for the class of functions with power-law behavior at infinity by means of a resummation procedure based on the truncated series for small variables. Iterative Borel summation is applied by employing Padé approximants of the “odd” and “even” types modified to satisfy the power-law. The odd approximations are conventional and are asymptotically equivalent with an odd number of terms in the truncated series. Even approximants are new, and they are constructed based on the idea of corrected approximants. They are asymptotically equivalent to the even number of terms in truncated series. Odd- and even-modified Padé approximants could be applied with and without a Borel transformation. The four methods are applied to some basic examples from condensed matter physics. We found that modified Padé–Borel summation works well in the case of zero-dimensional field theory with fast-growing coefficients and for similar examples. Remarkably, the methodology of modified Padé–Borel summation appears to be extendible to the instances with slow decay or non-monotonous behavior. In such situations, exemplified by the problem of Bose condensation temperature shift, the results are still very good.

Copy Move Forgery Detection Using Novel Quadsort Moth Flame Light Gradient Boosting Machine

A severe problem in modern information systems is Digital media tampering along with fake information. Even though there is an enhancement in image development, image forgery, either by the photographer or via image manipulations, is also done in parallel. Numerous researches have been concentrated on how to identify such manipulated media or information manually along with automatically; thus conquering the complicated forgery methodologies with effortlessly obtainable technologically enhanced instruments. However, high complexity affects the developed methods. Presently, it is complicated to resolve the issue of the speed-accuracy trade-off. For tackling these challenges, this article put forward a quick and effective Copy-Move Forgery Detection (CMFD) system utilizing a novel Quad-sort Moth Flame (QMF) Light Gradient Boosting Machine (QMF-Light GBM). Utilizing Borel Transform (BT)-based Wiener Filter (BWF) and resizing, the input images are initially pre-processed by eliminating the noise in the proposed system. After that, by utilizing the Orientation Preserving Simple Linear Iterative Clustering (OPSLIC), the pre-processed images, partitioned into a number of grids, are segmented. Next, as of the segmented images, the significant features are extracted along with the feature’s distance is calculated and matched with the input images. Next, utilizing the Union Topological Measure of Pattern Diversity (UTMOPD) method, the false positive matches that took place throughout the matching process are eliminated. After that, utilizing the QMF-Light GBM visualization, the visualization of forged in conjunction with non-forged images is performed. The extensive experiments revealed that concerning detection accuracy, the proposed system could be extremely precise when contrasted to some top-notch approaches.

Analytic Solvability of the H¨ormander Problem and the Borel Transformation of Multiple Laurent Series

In this paper, the initial-boundary value problem of Hormander is formulated in the class of functions representable by Laurent series supported in rational cones. Using the Borel transformation of Laurent series we establish a connection between a differential and a difference problems and prove its global analytic solvability

QCD sum rules with spectral densities solved in inverse problems

We construct QCD sum rules for nonperturbative studies without assuming the quark-hadron duality for the spectral density at low energy on the hadron side. Instead, both resonance and continuum contributions to the spectral density are solved with the operator-product-expansion input on the quark side by treating sum rules as an inverse problem. This new formalism does not involve the continuum threshold, does not require the Borel transformation and stability analysis, and can be extended to extract properties of excited states. Taking the two-current correlator as an example, we demonstrate that the series of $\rho$ resonances can emerge in our formalism, and the decay constants $f_{\rho(770)} (f_{\rho(1450)},f_{\rho(1700)},f_{\rho(1900)})\approx$ 0.22 (0.19, 0.14, 0.14) GeV for the masses $m_{\rho(770)} (m_{\rho(1450)},m_{\rho(1700)},m_{\rho(1900)})\approx$ 0.78 (1.46, 1.70, 1.90) GeV are determined. We also show that the decay width $\Gamma_{\rho(770)}\approx 0.17$ GeV can be obtained by substituting a Breit-Wigner parametrization for the $\rho(770)$ pole on the hadron side. It is observed that quark condensates of dimension-six on the quark side are crucial for establishing those $\rho$ resonances. Handling the conventional sum rules with the duality assumption as an inverse problem, we find that the multiple pole sum rules widely adopted in the literature do not describe the $\rho$ excitations reasonably. The precision of our theoretical outcomes can be improved systematically by including higher-order and higher-power corrections on the quark side. Broad applications of this formalism to abundant low energy QCD observables are expected.

Beautiful mathematics for beauty-full and other multi-heavy hadronic systems

In most non-perturbative methods in hadron physics the calculations are started with a correlation function in terms of some interpolating and transition currents in $ x $-space. For simplicity, the calculations are then transformed to the momentum space by a Fourier transformation. To suppress the contributions of the higher states and continuum; and enhance the ground state contribution, Borel transformation as well as continuum subtraction are applied by the help of quark-hadron duality assumption. In the present study we work out the mathematics required for these processes in the case of light and multi-heavy hadrons. We address a well-known problem in subtraction of the effects of the higher states and continuum and discuss how we find finite results without any divergence by using an appropriate representation of the modified Bessel functions, appearing in the heavy quark propagator, and successive applications of the Borel transformations, which lead to more suppression of the higher states and continuum contributions. The results obtained can be used in determination of the spectroscopic and decay properties of the multi-heavy standard and non-conventional (exotic) systems in many non-perturbative methods, specially the QCD sum rules.

Supersymmetric Solutions and Borel Singularities for N=2 Supersymmetric Chern-Simons Theories.

In supersymmetric (SUSY) field theory, there exist configurations that formally satisfy SUSY conditions but are not on the original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this Letter we discuss that CSS provide important information on the large order behavior of the weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeros) of the Borel transformation of perturbative series whose locations are uniquely determined by actions of the solutions. We demonstrate this for various SUSY observables in 3D N=2 SUSY Chern-Simons matter theories on a sphere. First we construct an infinite number of CSS in the general 3D N=2 SUSY theory with a Lagrangian where an adjoint scalar in the vector multiplet takes a complex value and matter fields are nontrivial. Then we compare their actions with Borel transformations of perturbative expansions by inverse Chern-Simons levels for the observables and see agreement with our conjecture. It turns out that the CSS explain all the Borel singularities for this case.

Borel Transform in the Class W of Quasi-entire Functions

In this article the class W of quasi-entire functions is introduced and the properties of the Borel transform in this class are studied. The class W includes the quasi-entire functions of exponential type whose module is square-integrable on the whole real axis. For such functions the Borel transform formulae can be represented in the form of integrals of Cauchy type from the jumps of functions that are analytic in the whole complex plane outside some cuts in which the jumps are defined. Due to this fact it is possible to obtain Parseval type equalities in the considered class of quasi-entire functions. These results, as a consequence, lead to the known Paley–Wiener results (in particular, the Paley–Wiener theorem). Therefore, they can be considered as a generalization of similar results for the class W of entire functions of exponential type introduced by them. There are different examples of the Borel transform in the class W of quasi-entire functions. In particular, the examples are chosen to show how the obtained results can be generalized in case the module of the quasi-entire function has at most power growth on the real axis. A similar generalization in the class of entire functions of exponential type is known as the Paley–Wiener–Schwartz theorem. In conclusion, there are simple generalizations for the case when the cut which ensures the single-valuedness of the function Borel-associated with the quasi-entire function of the class W does not necessarily coincide with the real axis.

Operational Versus Umbral Methods and the Borel Transform

Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well.

Differential Equations with Infinitely Many Derivatives and the Borel Transform

Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Gorka et al., Class Quantum Gravity 29:065017, 2012; Gorka et al., Ann Henri Poincare 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory.

Building Meromorphic Solutions ofq-Difference Equations Using a Borel–Laplace Summation

After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is put in the Birkhoff-Guenther normal form, we prove that the solutions we compute are exactly the same as the one constructed by Ramis, Sauloy and Zhang.

DETERMINATION OF THE STRONG COUPLING FROM HADRONIC TAU DECAYS USING RENORMALIZATION GROUP SUMMED PERTURBATION THEORY

We determine the strong coupling constant αs from the τ hadronic width using a renormalization group summed (RGS) expansion of the QCD Adler function. The main theoretical uncertainty in the extraction of αs is due to the manner in which renormalization group invariance is implemented, and the as yet uncalculated higher order terms in the QCD perturbative series. We show that new expansion exhibits good renormalization group improvement and the behavior of the series is similar to that of the standard CIPT expansion. The value of the strong coupling in [Formula: see text] scheme obtained with the RGS expansion is [Formula: see text]. The convergence properties of the new expansion can be improved by Borel transformation and analytic continuation in the Borel plane. This is discussed elsewhere in these issues.

A Connection Formula of the Hahn-Exton q-Bessel Function

We show a connection formula of the Hahn-Exton $q$-Bessel function around the origin and the infinity. We introduce the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit $p\to 1^-$ of the connection formula.

Domination conditions and spectral multiplicity of operators

We study the spectral multiplicity for the direct sum A⊕B of operators A and B on the Banach spaces X and Y. Under some domination conditions ‖P(B)‖≦C‖P(A)‖, in particular, ‖Bn‖≦C‖An‖, n≧0, we prove the addition formulas μ(A⊕B)=μ(A)+μ(B) for spectral multiplicities. We give valuable new applications of the main result of the author’s paper [12]. We also use the so-called Borel transformation and generalized Duhamel product in calculating the spectral multiplicity of a direct sum of the form T⊕A, where T is a weighted shift operator on the Wiener algebra \(W(\mathbb{D})\).

RτAND HIGGS DECAY WIDTH IN CORGI APPROACH, BASED ON THE LEADING-b APPROXIMATION

We consider Adler D-function for the vector and scalar correlators with recent QCD analysis at N4LO approximation. The portion of perturbative coefficients of these observable containing the leading power of b, the first beta-function coefficient, is resummed to all-orders. The factorial behavior of coefficients allows us to use the Borel transformation to reveal the existent singularities as renormalons. CORGI approach is employed to avoid the renormalization dependence of the observable. In addition to the CORGI approach, the calculations of the standard perturbative QCD approach which uses the [Formula: see text] scheme with a physical choice of renormalization scale, are also presented. A comparison between the results of the standard and CORGI approaches for Rτand Higgs decay width is done. The comparison anticipates a better result for the CORGI approach. As an adjunct to these studies, the difference between the all order and fixed order result at N4LO approximation is used to estimate the uncertainty in [Formula: see text] extracted from Rτmeasurements. We find at N4LO approximation a smaller uncertainty with respect to the previous result which has been done at N4LO approximation.

Strong coupling asymptotics of the β-function in [formula omitted] theory and QED

The well-known algorithm for summing divergent series is based on the Borel transformation in combination with the conformal mapping. A modification of this algorithm allows one to determine a strong coupling asymptotics of the sum of the series through the values of the expansion coefficients. An application of the algorithm to the β-function of φ 4 theory leads to the asymptotics β ( g ) = β ∞ g α at g → ∞ , where α ≈ 1 for space dimensions d = 2 , 3 , 4 . The natural hypothesis arises, that the asymptotic behavior is β ( g ) ∼ g for all d. Consideration of the “toy” zero-dimensional model confirms the hypothesis and reveals the origin of this result: it is related to a zero of a certain functional integral. A generalization of this mechanism to the arbitrary space dimensionality leads to the linear asymptotics of β ( g ) for all d. The same idea can be applied to QED and gives the asymptotics β ( g ) = g , where g is the running fine structure constant. A relation to the “zero charge” problem is discussed.

Summation of divergent series: Order-dependent mapping

Summation methods play a very important role in quantum field theory because all perturbation series are divergent and the expansion parameter is not always small. A number of methods have been tried in this context, most notably Padé approximants, Borel–Padé summation, Borel transformation with mapping, which we briefly describe and one on which we concentrate here, Order-Dependent Mapping (ODM). We recall the basis of the method, for a class of series we give intuitive arguments to explain its convergence and illustrate its properties by several simple examples. Since the method was proposed, some rigorous convergence proofs were given. The method has also found a number of applications and we shall list a few.

On the Local Borel Transform of Perturbation Theory

We prove existence of the local Borel transform for the perturbative series of massive $${\varphi_4^4}$$ -theory. As compared to previous proofs in the literature, the present bounds are much sharper as regards the dependence on external momenta, they are explicit in the number of external legs, and they are obtained quite simply through a judiciously chosen induction hypothesis applied to the Wegner-Wilson-Polchinski flow equations. We pay attention not to generate an astronomically large numerical constant for the inverse radius of convergence of the Borel transform.