Adler Function Research Articles

Renormalon subtraction in OPE using Fourier transform: formulation and application to various observables

Properly separating and subtracting renormalons in the framework of the op- erator product expansion (OPE) is a way to realize high precision computation of QCD effects in high energy physics. We propose a new method (FTRS method), which enables to subtract multiple renormalons simultaneously from a general observable. It utilizes a property of Fourier transform, and the leading Wilson coefficient is written in a one-parameter integral form whose integrand has suppressed (or vanishing) renormalons. The renormalon subtraction scheme coincides with the usual principal-value prescription at large orders. We perform test analyses and subtract the mathcal{O}left({Lambda}_{mathrm{QCD}}^4right) renormalon from the Adler function, the mathcal{O}left({Lambda}_{mathrm{QCD}}^2right) renormalon from the B → Xul overline{nu} decay width, and the mathcal{O} (ΛQCD) and mathcal{O}left({Lambda}_{mathrm{QCD}}^2right) renormalons from the B, D meson masses. The analyses show good consistency with theoretical expectations, such as improved convergence and scale dependence. In particular we obtain overline{Lambda} FTRS = 0.495 ± 0.053 GeV and ( {mu}_{pi}^2 )FTRS = −0.12 ± 0.23 GeV2 for the non-perturbative parameters of HQET. We explain the formulation and analyses in detail.

Determination of αs value from tau decays with a renormalon-motivated approach

We apply Borel-Laplace sum rules to the data of the semihadronic tau decay rate. For the higher order terms of the Adler function in the leadingtwist (D = 0) contribution we use a renormalon-motivated model, where the correct leading anomalous dimensions are taken into account in the IR u = 3 (and u = 2) renormalon contributions. In the evaluation of D = 0 contribution of the sum rules we apply two methods: (a) fixed order perturbation theory (FO) and (b) Borel resummation of the singular part with the Principal Value prescription (PV). We use as data the ALEPH data for the (V+A)-channel, and a combined set of data for the V-channel. In the D = 6 OPE term of the Adler function we account for the leading nonzero (and noninteger) anomalous dimension. In the OPE for the Adler function we include terms with dimension up to D = 10 for the (V+A)-channel, and up to D = 14 for the V-channel. In such cases, the extracted values of the coupling αs and of the condensates show a reasonably good convergence under the increase of OPE terms. In order to suppress the quark-hadron duality violations, our sum rules are doubly-pinched in the Minkowskian point. We obtain the averaged extracted values of the coupling αs(m2τ) = 0.3169+0.0070-0.0096, corresponding to αs(M2Z) = 0.1183+0.0009-0.0012.

Reconciling the FOPT and CIPT Predictions for τ Hadronic Spectral Function Moments

Recently it has been clarified by Hoang and Regner that the longstanding discrepancy between the CIPT and FOPT expansion approaches in αs determinations from the τ hadronic spectral function moments has been caused by an inconsistency of CIPT with the standard OPE approach. This inconsistency arises in the presence of IR renormalons in the underlying Adler function and is numerically dominated by the dimension-4 gluon condensate renormalon. In this talk we report on an approach to reconcile the CIPT based on a perturbative definition of a renormalon-free and scale-invariant gluon condensate scheme, called RF GC scheme. The scheme implies perturbative subtractions which eliminate the CIPT inconsistency for all practical applications of the τ hadronic spectral function moments. The scheme depends on the gluon condensate renormalon norm Ng as an independent input and on an IR subtraction scale R. We discuss three different approaches to determine Ng which yield consistent results and we apply the RF GC scheme in two full-fledged phenomenological αs determinations based on the truncated OPE and the duality violation model approach. In the RF GC scheme the long-standing CIPT-FOPT discrepancy problem is gone and the CIPT and FOPT αs determinations can be consistently combined.

Determination of perturbative QCD coupling from ALEPH tau decay data using pinched Borel\u2013Laplace and Finite Energy Sum Rules

We present a determination of the perturbative QCD (pQCD) coupling using the V+A channel ALEPH tau -decay data. The determination involves the double-pinched Borel–Laplace Sum Rules and Finite Energy Sum Rules. The theoretical basis is the Operator Product Expansion (OPE) of the V+A channel Adler function in which the higher order terms of the leading-twist part originate from a model based on the known structure of the leading renormalons of this quantity. The applied evaluation methods are contour-improved perturbation theory (CIPT), fixed-order perturbation theory (FOPT), and Principal Value of the Borel resummation (PV). All the methods involve truncations in the order of the coupling. In contrast to the truncated CIPT method, the truncated FOPT and PV methods account correctly for the suppression of various renormalon contributions of the Adler function in the mentioned sum rules. The extracted value of the {overline{mathrm{MS}}} coupling is alpha _s(m_{tau }^2) = 0.3116 pm 0.0073 [alpha _s(M_Z^2)=0.1176 pm 0.0010] for the average of the FOPT and PV methods, which we regard as our main result. On the other hand, if we include in the average also the CIPT method, the resulting values are significantly higher, alpha _s(m_{tau }^2) = 0.3194 pm 0.0167 [alpha _s(M_Z^2)=0.1186 pm 0.0021].

Higher-order behaviour of two-point current correlators

Estimates of higher-order contributions for perturbative series in QCD, in view of their asymptotic nature, are delicate, though indispensable for a reliable error assessment in phenomenological applications. In this work, the Adler function and the scalar correlator are investigated, and models for Borel transforms of their perturbative series are constructed, which respect general constraints from the operator product expansion and the renormalisation group. As a novel ingredient, the QCD coupling is employed in the so-called $C$-scheme, which has certain advantages. For the Adler function, previous results obtained directly in the $\overline{\rm MS}$ scheme are supported. Corresponding results for the scalar correlation function are new. It turns out that the substantially larger perturbative corrections for the scalar correlator in $\overline{\rm MS}$ are dominantly due to this scheme choice, and can be largely reduced through more appropriate renormalisation schemes, which are easy to realise in the $C$-scheme.

Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders * *Supported by the Chongqing Graduate Research and Innovation Foundation (ydstd1912, CYB21045), the National Natural Science Foundation of China (11625520, 12047564) and the Fundamental Research Funds for the Central Universities (2020CQJQY-Z003)

In this study, we investigate in detail the generalized Crewther Relation (GCR) between the Adler function (D) and the Gross-Llewellyn Smith sum rules coefficient ( ) using the newly proposed single-scale approach of the principle of maximum conformality (PMC). The resultant GCR is scheme-independent, with the residual scale dependence due to unknown higher-order terms highly suppressed. Thus, a precise test of QCD theory without renormalization schemes and scale ambiguities can be achieved by comparing with data. Moreover, a demonstration of the scheme independence of the commensurate scale relation up to all orders is presented. Additionally, for the first time, the Pade approximation approach has been adopted for estimating the unknown -loop contributions from the known four-loop perturbative series.

Conformal mappings in perturbative QCD

We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modified expansions have a tamed behaviour at large orders and may even converge under some conditions. On the other hand, the expansion functions exhibit nonperturbative features similar to those of the expanded function. Using these properties, it was suggested recently that the expansions based on the conformal mapping of the Borel plane may provide an alternative to the standard OPE. We briefly review the arguments in favour of this conjecture and discuss the application of the method to the Adler function for massless quarks and the static quark self-energy calculated in lattice QCD.

Resurgence of the QCD Adler function

We study the QCD Adler function in the energy region ≈0.7−2.5 GeV, in which the non-perturbative effects become dominant. Our analysis is a renormalon-based evaluation using transseries within the resurgence of the Renormalization-Group-Equation and does not require the Operator-Product-Expansion.

Lattice-motivated QCD coupling and hadronic contribution to muon g − 2

We present an updated version of a QCD coupling which fulfills various physically motivated conditions: at high momenta it practically coincides with the perturbative QCD coupling; at intermediate momenta it reproduces correctly the physics of the semihadronic tau decay; and at very low momenta it is suppressed as suggested by large-volume lattice calculations. An earlier presented analysis is updated here in the sense that the Adler function, in the regime |Q 2|≲ 1 GeV2, is evaluated by a renormalon-motivated resummation method. This Adler function is then used here in the evaluation of the quantities related with the semihadronic (strangeless) τ-decay spectral functions, including Borel–Laplace sum rules in the (V + A)-channel. The analysis is then extended to the evaluation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, , where we include in the Adler function the V-channel higher-twist OPE terms which are regulated in the infrared (IR) by mass parameters which are expected to be ≲1 GeV. The correct value of can be reproduced with the mentioned IR-regulating mass parameters if the value of the condensate is positive (and thus the gluon condensate value is positive). This restriction and the requirement of the acceptable quality of the fits to the various mentioned sum rules then lead us to the restriction .

Conformal mapping of the Borel plane: Going beyond perturbative QCD

The power corrections in the Operator Product Expansion (OPE) of QCD correlators can be viewed mathematically as an illustration of the transseries concept, which allows to recover a function from its asymptotic divergent expansion. Alternatively, starting from the divergent behavior of the perturbative QCD encoded in the singularities in the Borel plane, a modified expansion can be defined by means of the conformal mapping of this plane. A comparison of the two approaches concerning their ability to recover nonperturbative properties of the true correlator was not explored up to now. In the present paper, we make a first attempt to investigate this problem. We use for illustration the Adler function and observables expressed as integrals of this function along contours in the complex energy plane. We show that the expansions based on the conformal mapping of the Borel plane go beyond finite-order perturbation theory, containing an infinite number of terms when reexpanded in powers of the coupling. Moreover, the expansion functions exhibit nonperturbative features of the true function, while the expansions have a tamed behavior at large orders and are expected even to be convergent. Using these properties, we argue that there are no mathematical reasons for supplementing the expansions based on the conformal mapping of the Borel plane by additional arbitrary power corrections. Therefore, we make the conjecture that they provide an alternative to the standard OPE in approximating the QCD correlator. This conjecture allows to slightly improve the accuracy of the strong coupling extracted from the hadronic $\tau$ decay width. Using the optimal expansions based on conformal mapping and the contour-improved prescription of renormalization-group resummation, we obtain $\alpha_s(m_\tau^2)=0.314 \pm 0.006$, which implies $\alpha_s(m_Z^2)=0.1179 \pm 0.0008$.

Infrared-suppressed QCD coupling and the hadronic contribution to muon g-2

A variant of QCD with the coupling suppressed in the infrared (IR) regime, as suggested by large-volume lattice calculations of the Landau gauge gluon and ghost dressing functions, is considered. The coupling is further restricted by the condition of approximate coincidence with perturbative QCD in the high momentum regime, and by the τ-lepton semihadronic decay rate in the intermediate momentum regime, the rate which is evaluated by a renormalon-motivated resummation method. The obtained coupling turns out to be free of Landau singularities. The D = 4, 6 condensate values of the Adler function are then extracted by application of the Borel sum rules to the OPAL and ALEPH (V + A)-channel data of τ-decay, and the corresponding V-channel condensate values are deduced as well. We then show that the correct value of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, , is reproduced by regularizing the D = 4, 6 OPE terms in the V-channel Adler function with IR-regularization masses , suggesting the internal consistency of the presented QCD framework.

Higher-order QCD corrections to hadronic τ decays from Padé approximants

The mathematical method of Padé approximants is put forward to reconstruct the Borel transformed series of the Adler function and extract higher-order corrections in a model-independent way. The use of the D-log Padé approximants provides valuable information to its analytic structure and yields, together with the standard Padé approximants, reliable model-independent predictions of the higher-order coefficients, as for example, c5,1=277±51. Such reconstruction strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

Higher-order perturbative coefficients in QCD from series acceleration by conformal mappings

The present calculations in perturbative QCD reach the order ${\ensuremath{\alpha}}_{s}^{4}$ for several correlators calculated to five loops, and the huge computational difficulties make unlikely the full six-loop calculation in the near future. This situation has practical consequences; in particular the treatment of the higher orders of the perturbation series for the current-current correlator of light quarks is one of the main sources of errors in the extraction of the strong coupling from hadronic $\ensuremath{\tau}$ decays. Several approximate estimates of the next coefficients of the corresponding Adler function have been proposed, using various arguments. In the present paper we exploit the analytic structure of the Adler function in the Borel plane, which allows the definition of an improved perturbative expansion in powers of a conformal variable which maps the cut Borel plane onto the unit disk. The new expansions converge in a larger domain of the Borel plane and, when reexpanded in powers of the strong coupling, yield definite values for the higher perturbative coefficients. We apply the method to the Adler function in the $\overline{\mathrm{MS}}$ scheme and to a suitable weighted integral of this function in the complex $s$ plane, chosen such as to avoid model-dependent assumptions on analyticity. Our results ${c}_{5,1}=287\ifmmode\pm\else\textpm\fi{}40$, ${c}_{6,1}=2948\ifmmode\pm\else\textpm\fi{}208$ and ${c}_{7,1}=(1.89\ifmmode\pm\else\textpm\fi{}0.75)\ifmmode\times\else\texttimes\fi{}{10}^{4}$, for the six-, seven- and eight-loop coefficients, respectively, agree with a recent determination from Pad\'e approximants applied to the perturbative expansion of the hadronic $\ensuremath{\tau}$ decay rate.

Non-perturbative quantum chromodynamics and short strings in electron-positron pair annihilation to hadrons

We develop a new mathematical method to study the non-perturbative quantum chromodynamics effects in low-energy processes using ordinary and analytic perturbation theory (PT and APT). We apply this tool to extract the lowest dimensional condensates from the high-precision fits of the data on e+e–-annihilation to pion channels:  e+e− → π+π−, e+e− → 2π+2π−, e+e− → π+π−2π0, e+e− → 3π+3π− and e+e− → 2π+2π−2π0. The special attention is devoted to the existence of the contribution of dimension 2 operator. Our consideration is based on the dependence of Adler function corresponding to e+e−-annihilation into pions on both Q and Borel transform parameter M2. It is shown that within the framework of the proposed method the C2 coefficient of the dimension 2 operator is negative and is closer to zero for analytic perturbation theory, which may therefore be partially responsible for the effect of short strings.

Renormalon-motivated evaluation of QCD observables

A method of evaluation of spacelike QCD observables ${\cal D}(Q^2)$ is developed, motivated by the renormalon structure of these quantities. A related auxiliary quantity ${\widetilde {\cal D}}(Q^2)$ is introduced, which is renomalization scale independent only at the one-loop level, and a large-$\beta_0$-type renormalon motivated ansatz is made for the Borel transform of this quantity. This leads to a `dressed' Borel transform of the considered observable ${\cal D}(Q^2)$. From there, a Neubert-type distribution is obtained for the observable. The described method is then applied to the massless Adler function and the related decay ratio of the $\tau$ lepton semihadronic decays. Comparisons are then made with an evaluation method at higher truncated orders, developed in our earlier works, which is a renormalization scale invariant extension of the diagonal Pad\'e approximants.

Renormalization-scheme variation of a QCD perturbation expansion with tamed large-order behavior

The renormalization-scheme and scale dependence of the truncated QCD perturbative expansions is one of the main sources of theoretical error of the standard model predictions, especially at intermediate energies. Recently, a class of renormalization schemes, parametrized by a single real number $C$, has been defined and investigated in the frame of the standard perturbation expansions in powers of the coupling. In the present paper we investigate the $C$-scheme variation of a Borel-improved QCD perturbation series, which implements information about the large-order divergent character of perturbation theory by means of an optimal conformal mapping of the Borel plane. In the new expansions, the powers of the strong coupling are replaced by a set of expansion functions with properties which resemble those of the expanded correlators, having in particular a singular behavior at the origin of the complex coupling plane. On the other hand, the new expansions have a tamed increase at high orders, as demonstrated by previous studies in the $\overline{\mathrm{MS}}$ renormalization scheme. Using as examples the Adler function and the hadronic decay width of the $\ensuremath{\tau}$ lepton, we investigate the properties of the Borel-improved expansions in the $C$-scheme, in comparison with the standard expansions in the $C$-scheme and the expansions in $\overline{\mathrm{MS}}$. The variation with the renormalization scale and the prescription for the choice of an optimal value of the parameter $C$ are discussed. The good large-order behavior of the Borel-improved expansions is proved also in the $C$-scheme, which is a further argument in favor of using them in applications of perturbative QCD at intermediate energies.

Higher-order QCD corrections to hadronic \u03c4 decays from Pad\xe9 approximants

Perturbative QCD corrections to hadronic τ decays and e+e− annihilation into hadrons below charm are obtained from the Adler function, which at present is known in the chiral limit to five-loop accuracy. Extractions of the strong coupling, αs, from these processes suffer from an ambiguity related to the treatment of unknown higher orders in the perturbative series. In this work, we exploit the method of Padé approximants and its convergence theorems to extract information about higher-order corrections to the Adler function in a systematic way. First, the method is tested in the large-β0 limit of QCD, where the perturbative series is known to all orders. We devise strategies to accelerate the convergence of the method employing renormalization scheme variations and the so-called D-log Padé approximants. The success of these strategies can be understood in terms of the analytic structure of the series in the Borel plane. We then apply the method to full QCD and obtain reliable model-independent predictions for the higher-order coefficients of the Adler function. For the six-, seven-, and eight-loop coefficients we find c5,1 = 277 ± 51, c6,1 = 3460±690, and c7,1 = (2.02±0.72)×104, respectively, with errors to be understood as lower and upper bounds. Our model-independent reconstruction of the perturbative QCD corrections to the τ hadronic width strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

Novel demonstration of the renormalization group invariance of the fixed-order predictions using the principle of maximum conformality and the C -scheme coupling

As a basic requirement of the renormalization group invariance, any physical observable must be independent of the choice of both the renormalization scheme and the initial renormalization scale. In this paper, we show that by using the newly suggested $C$-scheme coupling, one can obtain a demonstration that the {\it Principle of Maximum Conformality} prediction is scheme-independent to all-orders for any renormalization schemes, thus satisfying all of the conditions of the renormalization group invariance. We illustrate these features for the non-singlet Adler function and for $\tau $ decay to $\nu +$ hadrons at the four-loop level.

Holographic CBK relation

The Crewther–Broadhurst–Kataev (CBK) relation connects the Bjorken function for deep-inelastic sum rules (or the Gross–Llewellyn Smith function) with the Adler function for electron–positron annihilation in QCD; it has been checked to hold up to four loops in perturbation theory. Here we study non-perturbative terms in the CBK relation using a holographic dual theory that is believed to capture properties of QCD. We show that for the large invariant momenta the perturbative CBK relation is exactly satisfied. For the small momenta non-perturbative corrections enter the relation and we calculate their significant effects. We also give an exact holographic expression for the Bjorken function, as well as for the entire three-point axial-vector–vector correlation function, and check their consistency in the conformal limit.

Absence of even-integer ζ-function values in Euclidean physical quantities in QCD

At order αs4 in perturbative quantum chromodynamics, even-integer ζ-function values are present in Euclidean physical correlation functions like the scalar quark correlation function or the scalar gluonium correlator. We demonstrate that these contributions cancel when the perturbative expansion is expressed in terms of the so-called C-scheme coupling αˆs which has recently been introduced in Ref. [1]. It is furthermore conjectured that a ζ4 term should arise in the Adler function at order αs5 in the MS‾-scheme, and that this term is expected to disappear in the C-scheme as well.