Adler Function Research Articles

Adler function, Bjorken polarized sum rule: confirmation of elements of the {β}-expansion and the diagrams

Different ways exist to obtain the elements of the {β}-expansion for renormgroup invariant quantities. Here we consider independent confirmation within the standard QCD of a number of our results [1] for the values of elements of this expansion for the nonsinglet Adler DA-function, Bjorken polarized sum rules SBjp up to the order N4LO. We suggest an approach to estimate the results of high order QCD calculations using a smaller number of diagrams of the specific type. This type is based on a proposed generalization of Naive NonAbelianization.

Power corrections to the modified QCD perturbative series based on conformal mapping of the Borel plane

Modifications of the QCD perturbative expansions by the subtraction of the dominant infrared renormalon have been proposed recently as attempts to solve the longstanding discrepancy between fixed-order and contour-improved perturbation theory for the hadronic τ decays. In this approach, the modified perturbative series is supplemented by a modified gluon condensate in the operator product expansion of the Adler function. Motivated by these works, we revisited recently a formulation of the QCD perturbation theory, proposed some time ago, which takes into account the renormalons by means of the conformal mapping of the Borel plane. One expects that the modified perturbative series obtained in this framework should be accompanied by modified power-suppressed nonperturbative corrections. However, in the previous studies the focus was on the perturbative series and the question of the possible power corrections has not been considered. In the present paper, we investigate for the first time this problem. Using techniques from the mathematical resurgence theory, we derive the expression of the dominant power correction to the perturbative series obtained by conformal mapping of the Borel plane, and discuss its implications for phenomenological applications. Published by the American Physical Society 2024

Estimating the strong coupling from τ decay using accelerating series convergence

We apply the Euler transformation to accelerate the convergence of the QCD perturbative series with the aim to determine the strong coupling αs in terms of the total τ-decay rate rτ. The variation of the result with the order of the QCD perturbation theory is small and comparable with the uncertainties of rτ. We also present an estimate of a range of the yet unknown 5th and 6th order coefficients k5 and k6 of the Adler function.

Effects of the quark flavour thresholds in the hadronic vacuum polarization contributions to the muon anomalous magnetic moment

The equivalent representations for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in the presence of the quark flavour thresholds are studied. Specifically, the explicit relations between the contributions given by the integration over a finite kinematic interval to expressed in terms of the hadronic vacuum polarization function, Adler function, and the R-ratio of electron–positron annihilation into hadrons are derived. It is shown that the quark flavour thresholds of the hadronic vacuum polarization function generate additional contributions to expressed in terms of the Adler function and the R-ratio and the explicit expressions for such contributions are obtained. The commonly employed dispersion relations, which bind together hadronic vacuum polarization function, Adler function, and R-ratio, are extended to account for the effects due to the quark flavour thresholds. The examined additional contributions due to the heavy quark thresholds to expressed in terms of the R-ratio appear to be quite sizable, that can be of a particular relevance for the data-driven method of assessment of the hadronic part of the muon anomalous magnetic moment.

Representation of the RG-Invariant Quantities in Perturbative QCD through Powers of the Conformal Anomaly

In this work we consider the possibility of representing the perturbative series for renormalization group invariant quantities in QCD in the form of their decomposition in powers of the conformal anomaly {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} in the overline {{text{MS}}} -scheme. We remind that such expansion is possible for the Adler function of the process of {{e}^{ + }}{{e}^{ - }} annihilation into hadrons and the coefficient function of the Bjorken polarized sum rule for the deep-inelastic electron-nucleon scattering, which are both related by the CBK relation. In addition, we study the discussed decomposition for the static quark-antiquark Coulomb-like potential, its relation with the quantity defined by the cusp anomalous dimension and the coefficient function of the Bjorken unpolarized sum rule of neutrino-nucleon scattering. In conclusion we also present the formal results of applying this approach to the non-renormalization invariant ratio between the pole and overline {{text{MS}}} -scheme running mass of heavy quark in QCD and compare them with those already known in the literature. The arguments in favor of the validity of the considered representation in powers of {{beta ({{alpha }_{s}})} mathord{left/ {vphantom {{beta ({{alpha }_{s}})} {{{alpha }_{s}}}}} right. kern-0em} {{{alpha }_{s}}}} for all mentioned renorm-invariant perturbative quantities are discussed.

Resurgence analysis of the Adler function at O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(1/Nf2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {N}_f^2 $$\\end{document})

We compute non-perturbative contributions to the Adler function, the derivative of the vacuum polarization function in gauge theory, using resurgence methods and Borel-summed gauge field propagators. At 2-loop, to order 1/Nf, we construct the full 2-parameter transseries and perform the sum over the non-perturbative sectors. We then introduce a convolution-based method to derive the transseries structure of product series, which can also be used to study higher orders in the expansion in 1/Nf. We compute 3-loop planar diagrams, at order 1/Nf2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {N}_f^2 $$\\end{document}, and for each diagram study the asymptotic behavior and resulting non-perturbative information in the transseries. A structure emerges that, from a resurgence point of view, is quite different from toy models hitherto studied. We study in particular the first and second non-perturbative sectors, their relation to UV and IR renormalons, and how their presence influences the perturbative expansions in neighbouring sectors. Finally, finding that many non-perturbative sectors have asymptotic series, we derive relations among all of them, thus providing an interesting new perspective on the alien lattice for the Adler function.

Resurgent representation of the Adler function in the large- β0 approximation of QCD

Using a full resummation of the Adler function in the large-${\ensuremath{\beta}}_{0}$ approximation of QCD and a mathematical framework of resurgence suitable for the specific properties of the Borel transform in this particular case, we derive a compact resurgent representation of the QCD Adler function, valid in the whole complex momentum plane. The representation is expressed in terms of the inverse Mellin transform of the Borel function and is analytic in the complex momentum plane, except for cuts along the timelike axis and the Landau region of the spacelike axis. It contains a purely nonperturbative term singular at the origin of the coupling plane, depending on a single real arbitrary constant. We compare the resurgent Adler function derived in this work with a previous determination in a similar framework and use its values in the complex plane for a calculation of the hadronic width of the $\ensuremath{\tau}$ lepton in the Standard Model.

The Euclidean Adler function and its interplay with Delta {alpha}_{textrm{QED}}^{textrm{had}} and αs

Three different approaches to precisely describe the Adler function in the Euclidean regime at around 2 GeVs are available: dispersion relations based on the hadronic production data in e+e− annihilation, lattice simulations and perturbative QCD (pQCD). We make a comprehensive study of the perturbative approach, supplemented with the leading power corrections in the operator product expansion. All known contributions are included, with a careful assessment of uncertainties. The pQCD predictions are compared with the Adler functions extracted from Delta {alpha}_{textrm{QED}}^{textrm{had}} (Q2), using both the DHMZ compilation of e+e− data and published lattice results. Taking as input the FLAG value of αs, the pQCD Adler function turns out to be in good agreement with the lattice data, while the dispersive results lie systematically below them. Finally, we explore the sensitivity to αs of the direct comparison between the data-driven, lattice and QCD Euclidean Adler functions. The precision with which the renormalisation group equation can be tested is also evaluated.

Borel–Laplace sum rules with τ decay data, using OPE with improved anomalous dimensions

We perform numerical analysis of double-pinched Borel–Laplace QCD sum rules for the strangeless semihadronic τ-decay data. The D = 0 contribution to the theoretical contour integral in the sum rules is evaluated by the (truncated) Fixed Order perturbation theory method (FO) and by the Principal Value (PV) of the Borel integration. We use for the full Adler function the operator product expansion (OPE) with the terms ∼〈O D 〉 of dimension D = 2n where 2 ≤ n ≤ 5 for the (V+A)-channel, and 2 ≤ n ≤ 7 for the V-channel data. In our previous works Ayala, Cvetič and Teca (2021, Eur. Phys. J. C 81 930), Ayala, Cvetič and Teca (2022, Eur. Phys. J. C 82 362), only the (V+A)-channel data was analysed. In this work, the analysis of a new set of V-channel data is performed as well. Further, a renormalon-motivated construction of the D = 0 part of the Adler function is improved in the u = 3 infrared renormalon sector, by involving the recently known information on the two principal noninteger values of the effective leading-order anomalous dimensions. Additionally, the OPE of the Adler function has now the D = 6 contribution with the principal anomalous dimension (), and terms of higher dimension (with zero anomalous dimension). Cross-checks of the obtained extracted values of α s and of the condensates were performed by reproduction of the (central) experimental values of several double-pinched momenta a (2,n). The averaged final extracted values of the () coupling are: , corresponding to .

Addendum to: The {β}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order Oleft({alpha}_s^4right)

Addendum to: The {β}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order Oleft({alpha}_s^4right)

Addendum: Timelike and spacelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment (2022 J. Phys. G: Nucl. Part. Phys. 49 055001)

This addendum provides results complementary to those obtained in [J. Phys. G 49, 055001 (2022)]. Specifically, an equivalent form of the relation, which binds together the ‘spacelike’ kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment , is obtained. It is shown that the infrared limiting value of the ‘spacelike’ and ‘timelike’ kernel functions, which enter the representations for involving the Adler function and the R-ratio, is identical to the corresponding QED contribution to the muon anomalous magnetic moment of the preceding order in the electromagnetic coupling. The next-to-leading order contributions (which includes the leptonic and hadronic insertions) and (which includes the double hadronic insertion), are studied. The three kernel functions appearing in the representations for , which involve the hadronic vacuum polarization function, Adler function, and the R-ratio, are presented for the cases of the electron and τ-lepton loop insertions.

Nonperturbative Corrections to Adler Function and Hypothesis of Vacuum Dominance

We check the range of values of four-quark condensate, relying on the processing of experimental data on e+e− annihilation into even number of pions (BaBar, CMD-2, OLYA) and extracting from them the values of the nonperturbative corrections. According to the hypothesis of vacuum dominance, the vacuum average of the four-quark condensate is expressed in terms of quark condensate while only vacuum intermediate states contribution is taken into account. However, assuming a possible non-zero effect of non-vacuum intermediate states, we check its influence on the four-quark condensate, or coefficient C6 in operator product expansion. Moreover, we explore how other nonperturbative corrections, C2 (operator with dimension 2), and C4 (which includes gluon condensate), are changed when assuming this effect. It is shown that values of four-quark condensate, or coefficient C6, can vary not more than 15-20%. Other parameters (C2 and C4) take values close to the currently available ones.

Adler function, Bjorken Sum Rule and Crewther-Broadhurst-Kataev relation with generic fermion representations at order [formula omitted

We compute the nonsinglet Adler D-function and the coefficient function for Bjorken polarized sum rules SBjp at order O(αs4) in an extended QCD model with arbitrary number of fermion representations. The Crewther-Broadhurst-Kataev (CBK) relation in this order is confirmed.

The {β}-expansion for Adler function, Bjorken Sum Rule, and the Crewther-Broadhurst-Kataev relation at order O( {alpha}_s^4 )

We derive explicit expressions for the elements of the {β}-expansion for the nonsinglet Adler DA-function and Bjorken polarized sum rules SBjp in the N4LO using recent results by Chetyrkin for these quantities computed within extended QCD including any number of fermion representations. We discuss the properties of the {β}-expansion for DA and SBjp at higher orders which follow from the Crewther [1] and the Broadhurst-Kataev [2] relation.

The QCD Adler Function and the Muon g − 2 Anomaly from Renormalons

We describe the Adler function in Quantum Chromodynamics using a transseries representation within a resurgent framework. The approach is based on a Borel-Ecalle resummation of the infrared renormalons combined with an effective running for the strong coupling. The new approach is flexible enough to give values in agreement with the current Adler function determinations. We then apply our finding to the muon’s anomalous magnetic moment studying the possibility of saturating, solely in terms of the vacuum polarization function, the current discrepancy between the best Standard Model value for the muon’s anomalous magnetic moment and the experimental value obtained by the most recent muon g−2 collaboration. The latter shows that the Adler function’s new representation can also be consistent with recent lattice determinations.

Borel representation of τ hadronic spectral function moments in contour-improved perturbation theory

We show that the Borel representations of $\ensuremath{\tau}$ hadronic spectral function moments based on contour-improved perturbation theory (CIPT) in general differ from those obtained within fixed-order perturbation theory (FOPT) in the presence of IR renormalons in the underlying Adler function. The Borel sums obtained from both types of Borel representations in general differ as well, and the apparently conflicting behavior of the FOPT and CIPT spectral function moment series at intermediate orders, which has been subject to many studies in the past literature, can be understood quantitatively using concrete Borel function models. The difference between the CIPT and FOPT Borel sums, which we call the ``asymptotic separation,'' can be computed analytically for any Borel function model and is proportional to inverse exponential terms in the strong coupling. Even though moments can be designed where the asymptotic separation is strongly suppressed, it is as a matter of principle unavoidable. If the Borel function of the Euclidean Adler function has a sizeable gluon condensate renormalon cut, the asymptotic separation can explain the observed disparity of the CIPT and FOPT spectral function moments at the 5-loop level. The existence of the asymptotic separation implies that the power corrections in the operator product expansion for the spectral function moments in the CIPT expansion approach do not have the commonly assumed analytic standard form.

The overline{mathrm{MS}} -scheme {alpha}_s^5 QCD contributions to the Adler function and Bjorken polarized sum rule in the Crewther-type two-fold {β}-expanded representation

We consider the two-fold expansion in powers of the conformal anomaly and of the strong coupling αs for the non-singlet contributions to Adler D-function and Bjorken polarized sum rule calculated previously in the overline{mathrm{MS}} -scheme at the four-loop level. This representation provides relations between definite terms of different loop orders appearing within the {β}-expansion of these quantities. Supposing the validity of this two-fold representation at the five-loop order and using these relations, we obtain some mathcal{O} ( {alpha}_s^5 ) corrections to the D-function, to the R-ratio of e+e−-annihilation into hadrons and to Bjorken polarized sum rule. These corrections are presented both analytically in the case of the generic simple gauge group and numerically for the SU(3) color group. The arguments in the favor of validity of the two-fold representation are given at least at the four-loop level. Within the {β}-expansion procedure the analytical Riemann ζ4-contributions to the five-loop expressions for the Adler function and Bjorken polarized sum rule are also fixed for the case of the generic simple gauge group.

Using improved operator product expansion in Borel–Laplace sum rules with ALEPH tau decay data, and determination of pQCD coupling

We use improved truncated operator product expansion (OPE) for the Adler function, involving two types of terms with dimension D=6, in the double-pinched Borel–Laplace sum rules and finite energy sum rules for the V + A channel strangeless semihadronic tau decays. The generation of the higher order perturbative QCD terms of the D=0 part of the Adler function is carried out using a renormalon-motivated ansatz incorporating the leading UV renormalon and the first two leading IR renormalons. The trunacted D=0 part of the sum rules is evaluated by two variants of the fixed-order perturbation theory (FO), by principal value of the Borel resummation (PV), and by contour-improved perturbation theory (CI). For the experimental V + A channel spectral function we use the ALEPH tau -decay data. We point out that the truncated FO and PV evaluation methods account correctly for the renormalon structure of the sum rules, while this is not the case for the truncated CI evaluation. We extract the value of the {overline{mathrm{MS}}} coupling alpha _s(m_{tau }^2) = 0.3235^{+0.0138}_{-0.0126} [alpha _s(M_Z^2)=0.1191 pm 0.0016] for the average of the two FO methods and the PV method, which we consider as our main result. If we included in the average also CI extraction, the value would be alpha _s(m_{tau }^2) = 0.3299^{+0.0232}_{-0.0225} [alpha _s(M_Z^2)=0.1199^{+0.0026}_{-0.0028}]. This work is an extension and improvement of our previous work (Ayala et al. in Eur Phys J. C 81(10): 930, 2021, arXiv:2105.00356 [hep-ph]) where we used for the truncated OPE a more naive (and widely used) form and where the extracted values for alpha _s(M_Z^2) were somewhat lower.

Timelike and spacelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment

Abstract The complete set of relations, which mutually express the spacelike and timelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment a μ HVP in terms of each other, is obtained. By making use of the derived relations the explicit expression for the next-to-leading order spacelike kernel function, which enters the representation for a μ HVP involving the hadronic vacuum polarization function, is obtained. The corresponding next-to-leading order spacelike kernel function, which appears in the representation for a μ HVP involving the Adler function, is calculated numerically. The obtained results can be employed in the assessments of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment in the framework of the spacelike methods, such as lattice studies, MUonE project, and others.

Smearing of causality by compositeness divides dispersive approaches into exact ones and precision-limited ones

Scattering off the edge of a composite particle or finite-range interaction can precede that off its center. An effective theory treatment with pointlike particles and contact interactions must find that the scattered experimental wave is slightly advanced, in violation of causality (the fundamental underlying theory being causal). In practice, partial-wave or other projections of multivariate amplitudes exponentially grow with Im(E), so that analyticity is not sufficient to obtain a dispersion relation for them, but only for a slightly modified function (the modified relations additionally connect different J). This can limit the precision of certain dispersive approaches to compositeness based on Cauchy's theorem. Awareness of this may be of interest to some dispersive tests of the Standard Model with hadrons, and to unitarization methods used to extend electroweak effective theories. Interestingly, the Inverse Amplitude Method is safe (as the inverse amplitude has the opposite, convergent behavior allowing contour closure). Generically, one-dimensional sum rules such as for the photon vacuum polarization, form factors or the Adler function are not affected by this uncertainty; nor are fixed-t dispersion relations, cleverly constructed to avoid it and whose consequences are solid.