Contour-improved Perturbation Theory Research Articles

Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays

Mathematical aspects of the asymptotic expansion in contour improved perturbation theory for hadronic tau decays

Difference between fixed-order and contour-improved perturbation theory

Difference between fixed-order and contour-improved perturbation theory

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.

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.

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].

Strong coupling from an improved τ vector isovector spectral function

We combine ALEPH and OPAL results for the spectral distributions measured in $\tau\to\pi^-\pi^0\nu_\tau$, $\tau\to 2\pi^-\pi^+\pi^0\nu_\tau$ and $\tau\to\pi^-3\pi^0\nu_\tau$ decays with (i) recent BaBar results for the analogous $\tau\to K^- K^0\nu_\tau$ distribution and (ii) estimates of the contributions from other hadronic $\tau$-decay modes obtained using CVC and electroproduction data, to obtain a new and more precise non-strange, inclusive vector, isovector spectral function. The BaBar $K^- K^0$ and CVC/electroproduction results provide us with alternate, entirely data-based input for the contributions of all exclusive modes for which ALEPH and OPAL employed Monte-Carlo-based estimates. We use the resulting spectral function to determine $\alpha_s(m_\tau)$, the strong coupling at the $\tau$ mass scale, employing finite energy sum rules. Using the fixed-order perturbation theory (FOPT) prescription, we find $\alpha_s(m_\tau)=0.3077\pm 0.0075$, which corresponds to the five-flavor result $\alpha_s(M_Z)=0.1171\pm 0.0010$ at the $Z$ mass. While we also provide an estimate using contour-improved perturbation theory (CIPT), we point out that the FOPT prescription is to be preferred for comparison with other $\alpha_s$ determinations employing the $\overline{{\rm MS}}$ scheme, especially given the inconsistency between CIPT and the standard operator product expansion recently pointed out in the literature. Additional experimental input on the dominant $2\pi$ and $4\pi$ modes would allow for further improvements to the current analysis.

Precision Determination of the Light Quark Masses from QCD Finite Energy Sum Rules

The light quark masses are determined from a QCD finite energy sum rule, using the pseudoscalar correlator to six-loop order in perturbative QCD, with the leading vacuum condensates and higher order quark mass corrections included. Both the fixed order perturbation theory (FOPT) method and contour improved perturbation theory (CIPT) method are explored. The results in the latter framework exhibit good convergence and stability in the window s0 = 3.0 5.0 GeV2 for the strange quark and s0 = 1.5 4.0 GeV2 for the up and down quarks; where s0 is the radius of the integration contour in the complex s-plane. The results are: , and the sum is . These proceedings critically explore how the current results - computed precisely in a modern computer language, Mathematica compare to past determinations in literature.

The strong coupling from $e^+e^-\to$~hadrons

We use a new compilation of the hadronic RR-ratio from available data for the process e^+e^-\toe+e−→ hadrons below the charm mass to determine the strong coupling \alpha_sαs, using finite-energy sum rules. Quoting our results at the \tauτ mass to facilitate comparison to the results obtained from similar analyses of hadronic \tauτ-decay data, we find \alpha_s(m_\tau^2)=0.298\pm 0.016\pm 0.006αs(mτ2)=0.298±0.016±0.006 in fixed-order perturbation theory, and \alpha_s(m_\tau^2)=0.304\pm 0.018\pm 0.006αs(mτ2)=0.304±0.018±0.006 in contour-improved perturbation theory, where the first error is statistical, and the second error combines various systematic effects. These values are in good agreement with a recent determination from the OPAL and ALEPH data for hadronic \tauτ decays. We briefly compare the R(s)R(s)-based analysis with the \tauτ-based analysis.

Up- and down-quark masses from QCD sum rules

The QCD up- and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are: {overline{m}}_uleft(2 mathrm{GeV}right)=left(2.6pm 0.4right) MeV, {overline{m}}_dleft(2 mathrm{GeV}right)=left(5.3pm 0.4right) MeV, and the sum {overline{m}}_{ud}equiv left({overline{m}}_u+{overline{m}}_dright)/2 , is {overline{m}}_{ud}left(2 mathrm{GeV}right)=left(3.9pm 0.3right) MeV.

Strong coupling from e+e−→hadrons below charm

We use a new compilation of the hadronic $R$-ratio from available data for the process $e^+e^-\to\mbox{hadrons}$ to determine the strong coupling, $\alpha_s$. We make use of all data for the $R$-ratio from threshold to a center-of-mass energy of 2 GeV by employing finite-energy sum rules. Data above 2 GeV, for which at present far fewer high-precision experimental data are available, do not provide much additional constraint but are fully consistent with the values for $\alpha_s$ we obtain. Quoting our results at the $\tau$ mass to facilitate comparison to the results obtained from analogous analyses of hadronic $\tau$-decay data, we find $\alpha_s(m_\tau^2)=0.298\pm 0.016\pm 0.006$ in fixed-order perturbation theory, and $\alpha_s(m_\tau^2)=0.304\pm 0.018\pm 0.006$ in contour-improved perturbation theory, where the first error is statistical, and the second error reflects our estimate of various systematic effects. These values are in good agreement with a recent determination from the OPAL and ALEPH data for hadronic $\tau$ decays.

Tau hadronic spectral function moments: perturbative expansion and αs extractions

In the extraction of αs from hadronic τ decays different moments of the spectral functions have been used. Furthermore, the two mainstream renormalization group improvement (RGI) frameworks, namely Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT), lead to conflicting values of αs. In order to improve the strategy used in αs determinations, we have performed a systematic study of the perturbative behaviour of these spectral moments in the context of FOPT and CIPT. Higher order coefficients of the perturbative series, yet unknown, were modelled using available knowledge of the renormalon content of the QCD Adler function. We conclude that within these RGI frameworks some of the moments often employed in αs extractions should be avoided due to their poor perturbative behaviour. Finally, under reasonable assumptions about higher orders, we conclude that FOPT is the preferred method to perform the renormalization group improvement of the perturbative series.

αs from the updated ALEPH data for hadronic τ decays

We extract the strong coupling αs(mτ2) from the recently updated ALEPH non-strange spectral functions obtained from hadronic τ decays. We apply a self-consistent analysis method, first tested in the analysis of OPAL data, to extract αs(mτ2) and non-perturbative contributions. The analysis yields αsFO(mτ2)=0.296±0.010, using Fixed Order Perturbation Theory (FOPT), and αsCI(mτ2)=0.310±0.014, using Contour Improved Perturbation Theory (CIPT). The weighted average of these results with those previously obtained from OPAL data give αsFO(mτ2)=0.303±0.009 and αsCI(mτ2)=0.319±0.012, which gives, after evolution to the Z boson mass scale, αsFO(mZ2)=0.1165±0.0012 and αsCI(mZ2)=0.1185±0.0015, respectively. We observe that non-perturbative effects limit the accuracy with which αs can be extracted from τ decay data.

Formula omitted] from the (revised) ALEPH data for τ decay

We present a new analysis of αs from hadronic τ decays based on the recently revised ALEPH data, employing a strategy previously applied to the OPAL data. We examine critically the approach traditionally used, comparing it to our strategy, and commenting on the main differences between the two. Our analysis yields the values αs(mτ2)=0.296±0.010 using fixed-order perturbation theory, and αs(mτ2)=0.310±0.014 using contour-improved perturbation theory. Averaging these values with our previously obtained values from the OPAL data, we find αs(mτ2)=0.303±0.009, respectively, αs(mτ2)=0.319±0.012, as the most reliable results for αs from τ decays currently available.

Strong coupling from the revised ALEPH data for hadronicτdecays

We apply an analysis method previously developed for the extraction of the strong coupling from the OPAL data to the recently revised ALEPH data for non-strange hadronic $\tau$ decays. Our analysis yields the values $\alpha_s(m_\tau^2)=0.296\pm 0.010$ using fixed-order perturbation theory, and $\alpha_s(m_\tau^2)=0.310\pm 0.014$ using contour-improved perturbation theory. Averaging these values with our previously obtained values from the OPAL data, we find $\alpha_s(m_\tau^2)=0.303\pm 0.009$, respectively, $\alpha_s(m_\tau^2)=0.319\pm 0.012$. We present a critique of the analysis method employed previously, for example in analyses by the ALEPH and OPAL collaborations, and compare it with our own approach. Our conclusion is that non-perturbative effects limit the accuracy with which the strong coupling, an inherently perturbative quantity, can be extracted at energies as low as the $\tau$ mass. Our results further indicate that systematic errors on the determination of the strong coupling from analyses of hadronic $\tau$-decay data have been underestimated in much of the existing literature.

The strong coupling from tau decays without prejudice

We review our recent determination of the strong coupling αs from the OPAL data for non-strange hadronic tau decays. We find that αs(mτ2)=0.325±0.018 using fixed-order perturbation theory, and αs(mτ2)=0.347±0.025 using contour-improved perturbation theory. At present, these values supersede any earlier determinations of the strong coupling from hadronic tau decays, including those from ALEPH data.

PERTURBATIVE EXPANSION OF τ HADRONIC SPECTRAL FUNCTION MOMENTS

In the extraction of αs from hadronic τ decay data several moments of the spectral functions have been employed. Furthermore, different renormalization group improvement (RGI) frameworks have been advocated, leading to conflicting values of αs. Recently, we performed a systematic study of the perturbative behavior of these moments in the context of the two main-stream RGI frameworks: Fixed Order Perturbation Theory (FOPT) and Contour Improved Perturbation Theory (CIPT). The yet unknown higher order coefficients of the perturbative series were modelled using the available knowledge of the renormalon singularities of the QCD Adler function. We were able to show that within these RGI frameworks some of the commonly employed moments should be avoided due to their poor perturbative behavior. Furthermore, under reasonable assumptions about the higher order behavior of the perturbative series FOPT provides the preferred RGI framework.

Updated determination of [formula omitted] from hadronic τ decays

Recently we introduced a new framework to extract αs from the non-strange hadronic τ-decay spectral functions taking into account duality violations and treating nonperturbative effects self-consistently. Using an updated version of the OPAL 1998 data, we obtain αsFO(mτ2)=0.325±0.018 and αsCl(mτ2)=0.347±0.025 in fixed-order and contour-improved perturbation theory, respectively. The results from our multi-parameter nonlinear fits were corroborated by a Markov-Chain Monte Carlo investigation of the posterior probability distributions of the model parameters.

Updated determination ofαsfromτdecays

Employing our previous framework to treat non-perturbative effects self-consistently, including duality violations, we update the determination of the strong coupling, \alpha_s, using a modified version of the 1998 OPAL data, updated to reflect current values of exclusive mode hadronic \tau decay branching fractions. Our best n_f=3 values from the updated OPAL data are \alpha_s(m^2_\tau)=0.325+-0.018 and \alpha_s(m^2_\tau)=0.347+-0.025 in fixed-order and contour-improved perturbation theory, respectively. To account for non-perturbative effects, non-linear, multi-parameter fits are necessary. We have, therefore, investigated the posterior probability distribution of the model parameters underlying our fits in more detail. We find that OPAL data alone provide only weak constraints on some of the parameters needed to model duality violations, especially in the case of fits involving axial vector channel data, making additional prior assumptions on the expected size of these parameters necessary at present. We provide evidence that this situation could be greatly improved if hadronic spectral functions based on the high-statistics BaBar and Belle data were to be made available.

Determination ofαs(Mτ2)from improved fixed order perturbation theory

We revisit the extraction of ${\ensuremath{\alpha}}_{s}({M}_{\ensuremath{\tau}}^{2})$ from the QCD perturbative corrections to the hadronic $\ensuremath{\tau}$ branching ratio, using an improved fixed-order perturbation theory based on the explicit summation of all renormalization-group accessible logarithms, proposed some time ago in the literature. In this approach, the powers of the coupling in the expansion of the QCD Adler function are multiplied by a set of functions ${D}_{n}$, which depend themselves on the coupling and can be written in a closed form by iteratively solving a sequence of differential equations. We find that the new expansion has an improved behavior in the complex energy plane compared to that of the standard fixed-order perturbation theory (FOPT), and is similar but not identical to the contour-improved perturbation theory (CIPT). With five terms in the perturbative expansion we obtain in the $\overline{\mathrm{MS}}$ scheme ${\ensuremath{\alpha}}_{s}({M}_{\ensuremath{\tau}}^{2})=0.338\ifmmode\pm\else\textpm\fi{}0.010$, using as input a precise value for the perturbative contribution to the hadronic width of the $\ensuremath{\tau}$ lepton reported recently in the literature.

From τ decays revisited

Being a determination at low energies, the analysis of hadronic tau decay data provides a rather precise determination of the strong coupling alpha_s after evolving the result to M_Z. At such a level of precision, even small non-perturbative effects become relevant for the central value and error. While those effects had been taken into account in the framework of the operator product expansion, contributions going beyond it, so-called duality violations, have previously been neglected. The following investigation fills this gap through a finite-energy sum rule analysis of tau decay spectra from the OPAL experiment, including duality violations and performing a consistent fit of all appearing QCD parameters. The resulting values for alpha_s(M_tau) are 0.307(19) in fixed-order perturbation theory and 0.322(26) in contour-improved perturbation theory, which translates to the n_f=5 values 0.1169(25) and 0.1187(32) at M_Z, respectively.