Fractional Analytic Perturbation Theory Research Articles

Improved calculation of the γ*γ → π process at low Q2 using LCSR’s and renormalization-group summation

We study two versions of lightcone sum rules to calculate the γ*γ → π0 transition form factor (TFF) within QCD. While the standard version is based on fixed-order perturbation theory by means of a power-series expansion in the strong coupling, the new method incorporates radiative corrections by renormalization-group summation and generates an expansion within a generalized fractional analytic perturbation theory involving only analytic couplings. Using this scheme, we determine the relative nonperturbative parameters and the first two Gegenbauer coefficients of the pion distribution amplitude (DA) to obtain TFF predictions in good agreement with the preliminary BESIII data, while the best-fit pion DA satisfies the most recent lattice constraints on the second moment of the pion DA at the three-loop level.

Form factor π0γ*γ in lightcone sum rules combined with renormalization-group summation vs experimental data.

We consider the lightcone sum-rule (LCSR) description of the pionphoton transition form factor in combination with the renormalization group of QCD. The emerging scheme represents a certain version of Fractional Analytic Perturbation Theory and significantly extends the applicability domain of perturbation theory towards lower momenta Q2 ≲ 1 GeV2. We show that the predictions calculated herewith agree very well with the released preliminary data of the BESIII experiment, which have very small errors just in this region, while the agreement with other data at higher Q2 is compatible with the LCSR predictions obtained recently by one of us using fixed-order perturbation theory.

Calculation of the pion-photon transition form factor using dispersion relations and renormalization-group summation

We consider the lightcone sum-rule description of the pion-photon transition form factor, based on dispersion relations, in combination with the renormalization group of QCD, in terms of the formal solution of the Efremov-Radyushkin-Brodsky-Lepage evolution equation, and show that the emerging scheme amounts to a certain version of Fractional Analytic Perturbation Theory (FAPT). In order to ensure the correct asymptotic behavior of the considered physical quantity, this modified FAPT version has to be supplemented by process-specific boundary conditions---in contrast to the standard one. However, it provides the advantage of significantly improving the inclusion of radiative corrections in the low-momentum regime of QCD perturbation theory using renormalization-group summation.

Non-singlet Q2-evolution by inverse Mellin transform in the analytic perturbation theory

We discuss the application of the analytic approach called the fractional Analytic Perturbation Theory (APT) to the QCD analysis of the non-singlet structure function xF3(x,Q2). The inverse Mellin transform method applyed for the fit of experimental data and for estimation of the Jacobi polynomial method accuracy in extraction of values of the scale parameter ΛQCD and the form of the xF3 structure function. Our estimates give the accuracy of the Jacobi polynomials method for the x-shape of the structure function about 10%, and accuracy for the scale parameter ΛQCD 4%.

AnQCD: Fortran programs for couplings at complex momenta in various analytic QCD models

We provide three Fortran programs which evaluate the QCD analytic (holomorphic) couplings Aν(Q2) for complex or real squared momenta Q2. These couplings are holomorphic analogs of the powers a(Q2)ν of the underlying perturbative QCD (pQCD) coupling a(Q2)≡αs(Q2)/π, in three analytic QCD models (anQCD): Fractional Analytic Perturbation Theory (FAPT), Two-delta analytic QCD (2δanQCD), and Massive Perturbation Theory (MPT). The index ν can be noninteger. The provided programs do basically the same job as the Mathematica package anQCD.m published by us previously (Ayala and Cvetič, 2015), but are now written in Fortran. Program summaryProgram title: AanQCDextCatalogue identifier: AEYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEYK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 12105No. of bytes in distributed program, including test data, etc.: 98822Distribution format: tar.gzProgramming language: Fortran.Computer: Any work-station or PC where Fortran 95/2003/2008 (gfortran) is running.Operating system: Operating system Linux (Ubuntu and Scientific Linux), Windows (in all cases using gfortran).Classification: 11.1, 11.5.Nature of problem: Calculation of values of the running analytic couplings Aν(Q2;Nf) for general complex squared momenta Q2≡−q2, in three analytic QCD models, where Aν(Q2;Nf) is the analytic (holomorphic) analog of the power (αs(Q2;Nf)/π)ν. Here, Aν(Q2;Nf) is a holomorphic function in the Q2 complex plane, with the exception of the negative semiaxis (−∞,−Mthr2], reflecting the analyticity properties of the spacelike renormalization invariant quantities D(Q2) in QCD. In contrast, the perturbative QCD power (αs(Q2;Nf)/π)ν has singularities even outside the negative semiaxis (Landau ghosts). The three considered models are: Analytic Perturbation theory (APT); Two-delta analytic QCD (2δanQCD); Massive Perturbation Theory (MPT). We refer to Ref. [1] for more details and literature.Solution method: The Fortran programs for FAPT and 2δanQCD models contain routines and functions needed to perform two-dimensional numerical integrations involving the spectral function, in order to evaluate Aν(Q2) couplings. In MPT model, one-dimensional numerical integration involving A1(Q2) is sufficient to evaluate any Aν(Q2) coupling.Restrictions: For unphysical choices of the input parameters the results are meaningless. When Q2 is close to the cut region of the couplings (Q2 real negative), the calculations can take more time and can have less precision.Running time: For evaluation of a set of about 10 related couplings, the times vary in the range t∼101–102 s. MPT requires less time, t∼1–101 s.

How to perform a QCD analysis of DIS in analytic perturbation theory

We apply (Fractional) Analytic Perturbation Theory (FAPT) to the QCD analysis of the nonsinglet nucleon structure function $F_2(x,Q^2)$ in deep inelastic scattering up to the next leading order and compare the results with ones obtained within the standard perturbation QCD. Based on a popular parameterization of the corresponding parton distribution we perform the analysis within the Jacobi Polynomial formalism and under the control of the numerical inverse Mellin transform. To reveal the main features of the FAPT two-loop approach, we consider a wide range of momentum transfer from high $Q^2\sim 100 {\rm GeV}^2$ to low $Q^2\sim 0.3 {\rm GeV}^2$ where the approach still works.

AnQCD: A Mathematica package for calculations in general analytic QCD models

We provide a Mathematica package that evaluates the QCD analytic couplings (in the complex domain) Aν(Q2), which are analytic analogs of the powers a(Q2)ν of the underlying perturbative QCD (pQCD) coupling a(Q2)≡αs(Q2)/π, in three analytic QCD models (anQCD): Fractional Analytic Perturbation Theory (FAPT), Two-delta analytic QCD (2δanQCD), and Massive Perturbation Theory (MPT). The analytic (holomorphic) running couplings Aν(Q2), in contrast to the corresponding pQCD expressions a(Q2)ν, reflect correctly the analytic properties of the spacelike observables D(Q2) in the complex Q2 plane as dictated by the general principles of quantum field theory. They are thus more suited for evaluations of such physical quantities, especially at low momenta |Q2|∼1GeV2. Program summaryProgram title:anQCDCatalogue identifier: AEVK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEVK_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1678No. of bytes in distributed program, including test data, etc.: 46323Distribution format: tar.gzProgramming language: Mathematica versions 9.0.1 and 10.0.1.Computer: Any work-station or PC where Mathematica is running.Operating system: Linux and Mac OS X, software Mathematica 9.0.1 and Mathematica 10.0.1.Classification: 11.1, 11.5.Nature of problem:Evaluation of the values for analytic couplings Aν(Q2;Nf) in analytic QCD [the analytic analog of the power (αs(Q2;Nf)/π)ν] based on the dispersion relation; Aν represents a physical (holomorphic) function in the plane of complex squared momenta −q2≡Q2. In anQCD.m we collect the formulas for three different analytic models depending on the energy scale, Q2, number of flavors Nf, the QCD scale Λ¯Nf, and the (nonpower) index ν. The considered models are: Analytic Perturbation theory (APT), Two-delta analytic QCD (2δanQCD) and Massive Perturbation Theory (MPT).Solution method:anQCD uses Mathematica functions to perform numerical integration of spectral function for each analytic model, in order to obtain the corresponding analytic images Aν(Q2) via dispersion relation.Restrictions:It could be that for an unphysical choice of the input parameters the results are meaningless.Running time:For all operations the running time does not exceed a few seconds.

On applications of Mathematica Package "FAPT" in QCD

We consider computational problems in the framework of nonpower Analityc Perturbation Theory and Fractional Analytic Perturbation Theory that are the generalization of the standard QCD perturbation theory. The singularity-free, finite couplings appear in these approaches as analytic images of the standard QCD coupling powers αvs(Q2) in the Euclidean and Minkowski domains, respectively. We provide a package "FAPT" based on the system Mathematica for QCD calculations of the images up to N3LO of renormalization group evolution. Application of these approaches to Bjorken sum rule analysis and Q2-evolution of higher twist μp−n4 is considered.

Gluon propagator in fractional analytic perturbation theory

We consider the gluon propagator in the Landau gauge at low spacelike momenta and with the dressing function $Z({Q}^{2})$ at the two-loop order. We incorporate the nonperturbative effects by making the (noninteger) powers of the QCD coupling in the dressing function $Z({Q}^{2})$ analytic (holomorphic) via the fractional analytic perturbation theory (FAPT) model, and simultaneously introducing the gluon dynamical mass in the propagator as motivated by the previous analyses of the Dyson-Schwinger equations. The obtained propagator has behavior compatible with the unquenched lattice data (${N}_{f}=2+1$) at low spacelike momenta $0.4\text{ }\text{ }\mathrm{GeV}<Q\ensuremath{\lesssim}10\text{ }\text{ }\mathrm{GeV}$. We conclude that the removal of the unphysical Landau singularities of the powers of the coupling via the (F)APT prescription, in conjunction with the introduction of the dynamical mass $M\ensuremath{\approx}0.62\text{ }\text{ }\mathrm{GeV}$ of the gluon, leads to an acceptable behavior of the propagator in the infrared regime.

Taming Landau singularities in QCD perturbation theory: The analytic approach 2.0

The aim of this topical article is to outline the fundamental ideas underlying the recently developed Fractional Analytic Perturbation Theory (FAPT) of QCD and present its main calculational tools together with key applications. For this, it is first necessary to review previous methods to apply QCD perturbation theory at low spacelike momentum scales, where the influence of the Landau singularities becomes inevitable. Several concepts are considered and their limitations are pointed out. The usefulness of FAPT is discussed in terms of two characteristic hadronic quantities: the perturbatively calculable part of the pion's electromagnetic form factor in the spacelike region and the Higgs-boson decay into a $b\bar b$ pair in the timelike region. In the first case, the focus is on the optimization of the prediction with respect to the choice of the renormalization scheme and the dependence on the renormalization and the factorization scales. The second case serves to show that the application of FAPT to this reaction reaches already at the four-loop level an accuracy of the order of 1%, avoiding difficulties inherent in the standard perturbative expansion. The obtained results are compared with estimates from fixed-order and contour-improved QCD perturbation theory. Using the brand-new Higgs mass value of about 125 GeV, measured at the Large Hadron Collider (CERN), a prediction for $\Gamma_{H\to b\bar{b}}=2.4 \pm 0.15 {\rm MeV}$ is extracted.

Calculation of binding energies and masses of quarkonia in analytic QCD models

We extract MSbar quark masses mqbar (q=b,c) from the evaluation of the masses of quarkonia Upsilon(1S) and J/\psi(1S), performed in two analytic QCD models, and in perturbative QCD in two renormalization schemes. In analytic QCD the running coupling has no unphysical singularities in the low-momentum regime. We apply the analytic model of Shirkov et al.[Analytic Perturbation Theory (APT)], extended by Bakulev et al.[Fractional Analytic Perturbation Theory (FAPT)], and the two-delta analytic model (2delanQCD). The latter, in contrast to (F)APT, at higher energies basically coincides with the perturbative QCD (in the same scheme). We use the renormalon-free mass mqbar as input. The separation of the soft and ultrasoft parts of the binding energy E is performed by the requirement of the cancellation of the leading infrared renormalon. The analysis in the 2delanQCD model indicates that the low-momentum ultrasoft regime is important for the extraction of the masses mqbar, especially mbbar. The 2delanQCD model gives us clues on how to estimate the influence of the ultrasoft sector on mqbar in general. These effects lead to relatively large values mbbar(mbbar) =approx (4.35 +- 0.08) GeV in the 2delanQCD model, which, however, are compatible with recent lattice calculations. In perturbative QCD in MSbar scheme these effects are even stronger and give larger uncertainties in mbbar. The (F)APT model gives small ultrasoft effects and the extracted values of mbbar agree with those in most of the literature: mbbar(mbbar) =approx 4.2 GeV. The extracted values of mcbar(mcbar) in all four models are about 1.26-1.27 GeV and agree well with those in the literature.

FAPT: A Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory

We provide here all the procedures in Mathematica which are needed for the computation of the analytic images of the strong coupling constant powers in Minkowski (Āν(s;nf) and Aνglob(s)) and Euclidean (Āν(Q2;nf) and Aνglob(Q2)) domains at arbitrary energy scales (s and Q2, correspondingly) for both schemes — with fixed number of active flavours nf=3,4,5,6 and the global one with taking into account all heavy-quark thresholds. These singularity-free couplings are inevitable elements of Analytic Perturbation Theory (APT) in QCD, proposed in [10,69,70], and its generalization — Fractional APT, suggested in [42,46,43], needed to apply the APT imperative for renormalization-group improved hadronic observables. Program summaryProgram title: FAPTCatalogue identifier: AENJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1985No. of bytes in distributed program, including test data, etc.: 1895776Distribution format: tar.gzProgramming language: Mathematica.Computer: Any work-station or PC where Mathematica is running.Operating system: Windows XP, Mathematica (versions 5 and 7).Classification: 11.5.Nature of problem:The values of analytic images Āν(Q2) and Āν(s) of the QCD running coupling powers αsν(Q2) in Euclidean and Minkowski regions, correspondingly, are determined through the spectral representation in the QCD Analytic Perturbation Theory (APT). In the program FAPT we collect all relevant formulas and various procedures which allow for a convenient evaluation of Āν(Q2) and Āν(s) using numerical integrations of the relevant spectral densities.Solution method:FAPT uses Mathematica functions to calculate different spectral densities and then performs numerical integration of these spectral integrals to obtain analytic images of different objects.Restrictions:It could be that for an unphysical choice of the input parameters the results are without any meaning.Running time:For all operations the run time does not exceed a few seconds. Usually numerical integration is not fast, so that we advise the use of arrays of precalculated data and then to apply the routine Interpolate(as shown in supplied example of the program usage, namely in the notebook FAPT_Interp.nb).

Resummation Approach in QCD Analytic Perturbation Theory

We discuss the resummation approach in QCD Analytic Perturbation Theory (APT). We start with a simple example of asymptotic power series for a zero-dimensional analog of the scalar $g\,\phi^4$ model. Then we give a short historic preamble of APT and show that renormgroup improvement of the QCD perturbation theory dictates to use the Fractional APT (FAPT). After that we discuss the (F)APT resummation of nonpower series and provide the one-, two-, and three-loop resummation recipes. We show the results of applications of these recipes to the estimation of the Adler function $D(Q^2)$ in the $N_f=4$ region of $Q^2$ and of the Higgs-boson-decay width $\Gamma_{H\to b\bar{b}}(m_H^2)$ for $M_H=100-180$ GeV$^2$.

Resummation in QCD fractional analytic perturbation theory

We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov and Solovtsov, to fractional powers of coupling -- Fractional APT (FAPT). The basic aspects of FAPT is shortly summarized. We describe how to treat heavy-quark thresholds in FAPT and then show how to resum perturbative series in both the one-loop APT and FAPT. As an application we consider FAPT description of the Higgs boson decay $H^0\to b\bar{b}$. The main conclusion is: To achieve an accuracy of the order of 1% it is enough to take into account up to the third correction.

Higher-order QCD perturbation theory in different schemes: from FOPT to CIPT to FAPT

Results on the resummation of non-power-series expansions of the Adler function of a scalar, $D_S$, and a vector, $D_V$, correlator are presented within fractional analytic perturbation theory (FAPT). The first observable can be used to determine the decay width $\Gamma_{H\to b\bar{b}}$ of a scalar Higgs boson to a bottom-antibottom pair, while the second one is relevant for the $e^+e^-$ annihilation cross section. The obtained estimates are compared with those from fixed-order (FOPT) and contour-improved perturbation theory (CIPT), working out the differences. We prove that although FAPT and CIPT are conceptually different, they yield identical results. The convergence properties of these expansions are discussed and predictions are extracted for the resummed series of $R_S$ and $D_V$ using one- and two-loop coupling running, and making use of appropriate generating functions for the coefficients of the perturbative series.

Pion form factor in QCD sum rules, local duality approach, and [formula omitted] fractional analytic perturbation theory

Using the results on the electromagnetic pion Form Factor (FF) obtained in the $O(\alpha_s)$ QCD sum rules with non-local condensates \cite{BPS09} we determine the effective continuum threshold for the local duality approach. Then we apply it to construct the $O(\alpha_s^2)$ estimation of the pion FF in the framework of the fractional analytic perturbation theory.

PION FORM FACTOR IN QCD SUM RULES WITH NONLOCAL CONDENSATES AND IN THE LOCAL-DUALITY APPROACH

We discuss the QCD sum-rule approach for the spacelike electromagnetic pion form factor in the O(αs) approximation. We show that the nonlocality of the condensates is a key point to include nonperturbative contributions to the pion form factor. We compare our results with the Local-Duality predictions and show that the continuum threshold s0(Q2) parameter is highly underestimated in the Local-Duality approach at Q2 ≳ 2 GeV 2. Using our fit for this parameter, [Formula: see text] and applying the fractional analytic perturbation theory, we estimate with an accuracy of the order of 1% the [Formula: see text] contribution to the pion's form factor.

Global fractional analytic perturbation theory in QCD with selected applications

We give the generalization of Fractional Analytic Perturbation Theory (FAPT) for QCD observables, recently developed both for the Euclidean and Minkowski regions of squared momentum transfer q^2, which takes into account heavy-quark thresholds. The original analytic approach to QCD, initiated by Jones, Solovtsov and Shirkov, is shortly summarized. We also shortly consider the basic aspects of FAPT and then concentrate on the accounting for the heavy-quark thresholds problem and the construction of global version of FAPT. We discuss what one should use as an analytic coupling in the timelike region q^2=s>0 for the e^{+}e^{-}-annihilation and the pion form factor, and consider applications to phenomenologically relevant processes (the factorizable part of the pion form factor and the Higgs boson decay into a b\bar{b} pair), as well as to the summation of perturbative series.

Erratum: Fractional analytic perturbation theory in Minkowski space and application to Higgs boson decay into abb¯pair [Phys. Rev. D75, 056005 (2007)

We work out and discuss the Minkowski version of Fractional Analytic Perturbation Theory (FAPT) for QCD observables, recently developed and presented by us for the Euclidean region. The original analytic approach to QCD, initiated by Shirkov and Solovtsov, is summarized and relations to other proposals to achieve an analytic strong coupling are pointed out. The developed framework is applied to the Higgs boson decay into a b\bar{b} pair, using recent results for the massless correlator of two quark scalar currents in the \bar{MS} scheme. We present calculations for the decay width within MFAPT including those non-power-series contributions that correspond to the O(\alpha_{s}^{3})-terms, taking also into account evolution effects of the running coupling and the b-quark-mass renormalization. Comparisons with previous results within standard QCD perturbation theory are performed and the differences are pointed out. The interplay between effects originating from the analyticity requirement and the analytic continuation from the spacelike to the timelike region and those due to the evolution of the heavy-quark mass is addressed, highlighting the differences to the conventional QCD perturbation theory.

Fractional analytic perturbation theory in Minkowski space and applicationto Higgs boson decay into abb¯pair

We work out and discuss the Minkowski version of Fractional Analytic Perturbation Theory (MFAPT) for QCD observables, recently developed and presented by us for the Euclidean region. The original analytic approach to QCD, initiated by Shirkov and Solovtsov, is summarized and relations to other proposals to achieve an analytic strong coupling are pointed out. The developed framework is applied to the Higgs boson decay into a bb pair, using recent results for the massless correlator of two quark scalar currents in the MS scheme. We present calculations for the decay width within MFAPT including those non-power-series contributions that correspond to the O � 3 � - terms, taking also into account evolution effects of the running coupling and the b-quark-mass renormalization. Comparisons with previous results within standard QCD perturbation theory are performed and the differences are pointed out. The interplay between effects originating from the analyticity requirement and the analytic continuation from the spacelike to the timelike region and those due to the evolution of the heavy-quark mass is addressed, highlighting the differences from the conventional QCD perturbation theory.