Search

We describe how recent determinations of exclusive-mode contributions to $a_\mu^{\rm LO,HVP}$, the leading-order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon, can be used to provide, up to small electromagnetic (EM) corrections accessible from the lattice, a data-based dispersive determination of $a_\mu^{\rm lqc;\, IL}$, the isospin-limit, light-quark-connected contribution to $a_\mu^{\rm LO,HVP}$. Such a determination is of interest in view of the existence of a number of lattice results for this quantity, emerging evidence for a tension between lattice and dispersive determinations of $a_\mu^{\rm LO,HVP}$, and the desire to clarify the source of this tension. Taking as input for the small EM correction that must be applied to the purely data-driven dispersive determination the result $-0.93(58)\times 10^{-10}$ obtained in a recent BMW lattice study, we find $a_\mu^{\rm lqc;\, IL}$ to be $635.0(2.7)\times 10^{-10}$ if the results of Keshavarzi, Nomura and Teubner are used for the exclusive-mode contributions and $638.4(4.1)\times 10^{-10}$ if instead those of Davier, H\"ocker, Malaescu and Zhang are used., Comment: 19 pages, 1 figure

We employ Pad\'e approximants in the study of the analytic structure of the four-dimensional $SU(2)$ Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, serve as fitting functions for the lattice data. We carefully propagate the uncertainties due to the fitting procedure, taking into account all possible correlations. For the gluon-propagator data, we confirm the presence of a pair of complex poles at $p_{\rm pole}^2 = \left[(-0.37 \,\pm\, 0.05_{\rm stat}\,\pm\, 0.08_{\rm sys}) \pm i\,(0.66\, \pm\, 0.03_{\rm stat}\, \pm\, 0.02_{\rm sys})\right]\, \mathrm{GeV}^2$, where the first error is statistical and the second systematic. The existence of this pair of complex poles, already hinted upon in previous works, is thus put onto a firmer basis, thanks to the model independence and to the careful error propagation of our analysis. For the ghost propagator, the Pad\'es indicate the existence of a single pole at $p^2 = 0$, as expected. In this case, our results also show evidence of a branch cut along the negative real axis of $p^2$. This is corroborated with another type of approximant, the D-Log Pad\'es, which are better suited to studying functions with a branch cut and are applied here for the first time in this context. Due to particular features and limited statistics of the gluon-propagator data, our analysis is inconclusive regarding the presence of a branch cut in the gluon case., Comment: 36 pages, 12 figures. Small changes in the text, references added, results unchanged. Accepted for publication in JHEP

Padé approximants are employed in order to study the analytic structure of the four-dimensional SU(2) Landau-gauge gluon and ghost propagators in the infrared regime. The approximants, which are model independent, are used as fitting functions to lattice data for the propagators, carefully propagating uncertainties due to the fit procedure and taking into account all possible correlations. Applying this procedure systematically to the gluon-propagator data, we observe the presence of a pair of complex poles at $p^2_{\mathrm{pole}} = (-0.37 \pm 0.05_{\mathrm{stat}} \pm 0.08_{\mathrm{sys}}) \pm \, i\, (0.66 \pm 0.03_{\mathrm{stat}} \pm 0.02_{\mathrm{sys}}) \, \mathrm{GeV}^2$, where ``stat'' represents the statistical error and ``sys'' the systematic one. We also find a zero on the negative real axis of $p^2$, at $p^2_{\mathrm{zero}} = (-2.9 \pm 0.4_{\mathrm{stat}} \pm 0.9_{\mathrm{sys}}) \, \mathrm{GeV}^2$. We thus note that our procedure -- which is based on a model-independent approach and includes careful error propagation -- confirms the presence of a pair of complex poles in the gluon propagator, in agreement with previous works. For the ghost propagator, the Padés indicate the existence of the single pole at $p^2 = 0$, as expected. We also find evidence of a branch cut on the negative real axis. Through the use of the so-called D-Log Padé method, which is designed to approximate functions with cuts, we corroborate the existence of this cut for the ghost propagator., 10 pages, 3 figures. Contribution to the Proceedings of Lattice 2022, Bonn, Germany, 8th-13th August, 2022

In a recent work by some of us it was shown that the long-standing discrepancy between the QCD perturbation series for the inclusive hadronic tau decay rate computed in the CIPT and FOPT expansion approaches can be understood from the fact that CIPT has an infrared (IR) sensitivity that it not compatible with the standard form of the operator production expansion (OPE). For concrete IR renormalon models for the QCD Adler function the resulting CIPT-FOPT discrepancy, the asymptotic separation, can be calculated analytically from the Borel representation of the CIPT series expansion. If the known perturbative corrections for the QCD Adler function at the 5-loop level already have a sizeable contribution from the asymptotic behavior related to the gluon condensate (GC) renormalon, the asymptotic separation is dominated by that renormalon. This implies that the CIPT expansion can be reconciled with FOPT, when a renormalon-free scheme for the GC is adopted. In this talk we discuss such a renormalon-free scheme for the GC, which involves perturbative subtractions in analogy to using short-distance quark mass schemes instead of the pole mass. Using a concrete realistic high-order Borel model for the Adler function consistent with the known corrections up to 5 loops and containing a sizeable GC renormalon contribution, we show that the CIPT-FOPT discrepancy can be avoided when switching to the renormalon-free GC scheme. At the same time, the perturbative convergence of $\tau$ hadronic spectral funtion moments strongly sensitive to the GC OPE corrections is considerably improved. We show that these improvements may lead to higher precision for strong coupling determinations., Comment: 9 pages, 4 figures, proceedings contribution to Loops and Legs in Quantum Field Theory (LL2022)

In a previous article, we have shown that the discrepancy between the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for $\tau$ hadronic spectral function moments, which had affected the precision of $\alpha_s$ determinations for many years, may be reconciled by employing a renormalon-free (RF) scheme for the gluon condensate (GC) matrix element. In addition, the perturbative convergence of spectral function moments with a sizeable GC correction can be improved. The RF GC scheme depends on an IR factorization scale $R$ and the normalization $N_g$ of the GC renormalon. In the present work, we use three different methods to determine $N_g$, yielding a result with an uncertainty of $40\%$. Following two recent state-of-the-art strong coupling determination analyses at ${\cal O}(\alpha_s^5)$, we show that using the renormalon-free GC scheme successfully reconciles the results for $\alpha_s({m_\tau^2})$ based on CIPT and FOPT. The uncertainties due to variations of $R$ and the uncertainty of $N_g$ only lead to a small or moderate increase of the final uncertainty of $\alpha_s(m_\tau^2)$, and affect mainly the CIPT expansion method. The FOPT and CIPT results obtained in the RF GC scheme may be consistently averaged. The RF GC scheme thus constitutes a powerful new ingredient for future analyses of $\tau$ hadronic spectral function moments., Comment: 43 pages, 5 figures; presentation in section 3.2 improved, results unchanged; version published in the Journal of High Energy Physics

We point out that the sum of the strange-quark-connected and full, three-flavor quark-disconnected contributions to the leading-order hadronic vacuum polarization contribution, $a_\mu^{\mathrm{LO,HVP}}$, to the anomalous magnetic moment of the muon, is a physical observable, and provide a data-based determination of this quantity in the isospin limit. The result, $40.1(1.5)\times 10^{-10}$, or $38.7(2.0)\times 10^{-10}$, depending on which data compilation is used, serves as a target of comparison for lattice calculations of the same isospin-limit combination. Subtracting from this result the average of lattice determinations of the strange-quark-connected contribution, one obtains also an alternate determination of the isospin-limit three-flavor disconnected contribution to $a_\mu^{\mathrm{LO,HVP}}$. The result of this determination, $-13.3(1.5)\times 10^{-10}$, or $-14.6(2.0)\times 10^{-10}$, depending on which data compilation is used, agrees well and is competitive with the most precise current lattice determination., Comment: 32 pages, 1 figure

The mathematical method of Pade 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 Pade approximants provides valuable information to its analytic structure and yields, together with the standard Pade approximants, reliable model-independent predictions of the higher-order coefficients, as for example, c 5 , 1 = 277 ± 51 . Such reconstruction strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting.

We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-$\beta_0$ limit of QCD, extending the analysis of Grozin and Sturm beyond the vector current. Our results are used to study the higher-order behaviour of dimensionless ratios of vector and pseudo-scalar moments used for the precise extraction of the strong coupling, $\alpha_s$, from relativistic quarkonium sum rules and lattice data, respectively. We show that these ratios benefit from a partial cancellation of the leading renormalon singularities. Our results can guide the design of combinations of moments with improved perturbative behaviour., Comment: 27 pages, 7 figures. References added. Version to appear in JHEP

In this paper we consider the decay D+ → π+ℓ+ℓ−, addressing in particular the resonance contributions as well as the relatively large contributions from the weak annihilation diagrams. For the weak annihilation diagrams we include known results from QCD factorisation at low q2 and at high q2, adapting the existing calculation for B decays in the Operator Product Expansion. The hadronic resonance contributions are obtained through a dispersion relation, modelling the spectral functions as towers of Regge-like resonances in each channel, as suggested by Shifman, imposing the partonic behaviour in the deep Euclidean. The parameters of the model are extracted using e+e− → (hadrons) and τ → (hadrons) + ντ data as well as the branching ratios for the resonant decays D+ → π+R(R → ℓ+ℓ−), with R = ρ, ω, and ϕ. We perform a thorough error analysis, and present our results for the Standard Model differential branching ratio as a function of q2. Focusing then on the observables FH and AFB, we consider the sensitivity of this channel to effects of physics beyond the Standard Model, both in a model independent way and for the case of leptoquarks.

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., Comment: 38 pages, 8 figures, published version

After discussing difficulties in determining αs from tau decay due to the existence of Duality Violations and the associated asymptotic nature of the OPE, we describe a new determination based on an improved vector isovector spectral function, now based solely on experimental input, obtained by (i) combining ALEPH and OPAL results for 2π + 4π and (ii) replacing K−K0 and higher-multiplicity exclusive-mode contributions, both previously estimated using Monte Carlo, with new experimental BaBar results for K−K0 and results implied by e+e− cross sections and CVC for the higher-multiplicity modes. We find αs(mτ) = 0.3077±0.0075, which corresponds to αs(mZ) = 0.1171±0.0010. Finally, we comment on some of the shortcomings in the criticism of our approach by Pich and Rodriguez-Sanchez.

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.

We present our recent results on the extraction of the strong coupling, $\alpha_s$, from a new compilation of the hadronic $R$-ratio obtained from the available data for $e^+e^- \to {\rm hadrons}$. The determination uses all data up to $s=4$~GeV$^2$, where $s$ is the square of the center-of-mass energy. We obtain competitive values with only a small contamination from non-perturbative effects and with smaller theoretical uncertainties than the extractions from hadronic $\tau$ decays.

Precise extractions of $\alpha_s$ from $\tau\to {\rm (hadrons)}+\nu_\tau$ and from $e^+e^-\to {\rm (hadrons)}$ below the charm threshold rely on finite energy sum rules (FESRs) where the experimental side is given by integrated spectral function moments. Here we study the renormalons that appear in the Borel transform of polynomial moments in the large-$\beta_0$ limit and in full QCD. In large-$\beta_0$, we establish a direct connection between the renormalons and the perturbative behaviour of moments often employed in the literature. The leading IR singularity is particularly prominent and is behind the fate of moments whose perturbative series are unstable, while those with good perturbative behaviour benefit from partial cancellations of renormalon singularities. The conclusions can be extended to QCD through a convenient scheme transformation to the $C$-scheme together with the use of a modified Borel transform which make the results particularly simple; the leading IR singularity becomes a simple pole, as in large-$\beta_0$. Finally, for the moments that display good perturbative behaviour, we discuss an optimized truncation based on renormalisation scheme (or scale) variation. Our results allow for a deeper understanding of the perturbative behaviour of integrated spectral function moments and provide theoretical support for low-$Q^2$ $\alpha_s$ determinations., Comment: 26 pages, 5 figures

The strong coupling, αs, governs perturbative Quantum Chromodynamics (QCD) and is one of the free parameters of the Standard Model. We introduce a new method that allows a precise extraction of αs(mZ) from dimensionless ratios of roots of moments of the charm-quark vector correlator. The ratios we use in our analysis have a rather weak logarithmic quark-mass dependence, starting at O(αs2), and can be obtained from experimental data with good precision, since they benefit from positive correlations among the individual experimentally determined moments. We perform a careful and conservative error analysis with special emphasis on uncertainties related to the truncation of perturbation theory, treating the renormalization scales such as to ensure order-by-order convergence. Our final result, with expressions at O(αs3), is αs(mZ)=0.1168±0.0019.

Perturbative QCD corrections to hadronic $\tau$ 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, $\alpha_s$, 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 Pad\'e 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-$\beta_0$ 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 Pad\'e 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 $c_{5,1} = 277\pm 51$, $c_{6,1}=3460\pm 690$, and $c_{7,1}=(2.02\pm0.72)\times 10^4$, respectively, with errors to be understood as lower and upper bounds. Our model-independent reconstruction of the perturbative QCD corrections to the $\tau$ hadronic width strongly favours the use of fixed-order perturbation theory (FOPT) for the renormalization-scale setting., Comment: 40 pages, 8 figures. Five references added. Minor changes. To appear in Journal of High Energy Physics

The operator product expansion (OPE), truncated in dimension, is employed in many contexts. An example is the extraction of the strong coupling, ${\ensuremath{\alpha}}_{s}$, from hadronic $\ensuremath{\tau}$-decay data, using a variety of analysis methods based on finite-energy sum rules. Here, we reconsider a long-used method, which parametrizes nonperturbative contributions to the $I=1$ vector and axial vacuum polarizations with the OPE, setting several higher-dimension coefficients to zero in order to implement the method in practice. The assumption that doing this has a negligible effect on the value of ${\ensuremath{\alpha}}_{s}$ is tantamount to the assumption that the low-dimension part of the OPE converges rapidly with increasing dimension near the $\ensuremath{\tau}$ mass. Were this assumption valid, it would certainly have to be valid at energies above the $\ensuremath{\tau}$ mass as well. It follows that the method can be tested using data obtained from ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\text{hadrons}$, as they are not limited by the kinematic constraints of $\ensuremath{\tau}$ decays. We carry out such an investigation using a recent high-precision compilation for the $R$ ratio, arguing that it provides insights into the validity of the strategy, even if it probes a different, though related, channel. We find that ${e}^{+}{e}^{\ensuremath{-}}$-based tests call into question the implied assumption of rapid convergence of the low-dimension part of the OPE around the $\ensuremath{\tau}$ mass and thus underscore the need to restrict finite-energy sum-rule analyses to observables which receive only contributions from lower-order terms in the OPE.

The QCD coupling, $\alpha_s$, is not a physical observable since it depends on conventions related to the renormalization procedure. Here we discuss a redefinition of the coupling where changes of scheme are parametrised by a single parameter $C$. The new coupling is denoted $\hat \alpha_s$ and its running is scheme independent. Moreover, scheme variations become completely analogous to renormalization scale variations. We discuss how the coupling $\hat \alpha_s$ can be used in order to optimize predictions for the inclusive hadronic decays of the tau lepton. Preliminary investigations of the $C$-scheme in the presence of higher-order terms of the perturbative series are discussed here for the first time., Comment: 4 pages, 3 figures. Contribution to "The 14th International Workshop on Tau Lepton Physics", 19--23 September 2016, IHEP, Beijing, China

While the idea of using the operator product expansion (OPE) to extract the strong coupling from hadronic τ decay data is not new, there is an ongoing controversy over how to include quark-hadron “duality violations” ( i.e ., resonance effects) which are not described by the OPE. One approach attempts to suppress duality violations enough that they might become negligible, but pays the price of an uncontrolled OPE truncation. We critically examine a recent analysis using this approach and show that it fails to properly account for non-perturbative effects, making the resulting determination of the strong coupling unreliable. In a different approach duality violations are taken into account with a model, avoiding the OPE truncation. This second approach provides a self-consistent determination of the strong coupling from τ decays.

We review how the current precision attained in the extraction of αs(mτ)from hadronic τ decays requires the inclusion of Duality Violations (DVs) in the analysis, even though these decays are largely dominated by perturbation theory. A weighted average using the OPAL and ALEPH experimental data yields αs(mZ) =0.1165±0.0012 andαs(mZ) =0.1185±0.0015 in fixed-order and contour-improved perturbation theory, respectively.

We present a brief summary of our recent determination of $αs$ from $e+e$ −→hadrons, in the region $3≤s≤4$ GeV2, with $s$ the square of the center-of-massenergy

We discuss sum-rule determinations of \alpha_sαs from non-strange hadronic \tauτ-decay data. We investigate, in particular, the reliability of the assumptions underlying the “truncated OPE strategy,” which specifies a certain treatment of non-perturbative contributions, and which was employed in Refs. . Here, we test this strategy by applying the strategy to the RR-ratio obtained from e^+e^-e+e− data, which extend beyond the \tauτ mass, and demonstrate that the assumptions underlying this strategy are not, in general, valid. We then present a brief overview of new results on the form of duality-violating non-perturbative contributions, which are conspicuously present in the experimentally determined spectral functions. As we show, with the current precision claimed for the extraction of \alpha_sαs, including a representation of duality violations is unavoidable if one wishes to avoid uncontrolled theoretical errors.

One of the main sources of theoretical uncertainty in the extraction of the strong coupling from hadronic tau decays stems from the renormalization group improvement of the series. Perturbative series in QCD are divergent but are (most likely) asymptotic expansions. One needs knowledge about higher orders to be able to choose the optimal renormalization-scale setting procedure. Here, we discuss the use of Padé approximants as a model-independent and robust method to extract information about the higher-order terms. We show that in hadronic \tauτ decays the fixed-order expansion, known as fixed-order perturbation theory (FOPT), is the most reliable mainstream method to set the scale. This fully corroborates previous conclusions based on the available knowledge about the leading renormalon singularities of the perturbative series.

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.

International audience

Parametrizing beyond Standard Model physics by the S U ( 3 ) × S U ( 2 ) L × U ( 1 ) Y dimension-six effective lagrangian, we study the impact of anomalous Higgs couplings in angular asymmetries of the crossing symmetric processes H → Z l + l − and e + e − → H Z . In the light of present bounds on d = 6 couplings, we show that some asymmetries can reveal BSM effects that would otherwise be hidden in other observables. The d = 6 H Z γ couplings as well as (to a lesser extent) H Z l l contact interactions can generate asymmetries at the several percent level, albeit having less significant effects on the di-lepton invariant mass spectrum of the decay H → Z l + l − . The higher di-lepton invariant mass probed in e + e − → H Z can lead to complementary anomalous coupling searches at e + e − colliders.

In the extraction of α s 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 α s . In order to improve the strategy used in α s 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 α s 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.

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.

We investigate the origin of the quark-hadron duality-violating terms in the expansion of the QCD two-point vector correlation function at large energies in the complex $q^2$ plane. Starting from the dispersive representation for the associated polarization, the analytic continuation of the operator product expansion from the Euclidean to the Minkowski region is performed by means of a generalized Borel-Laplace transform, borrowing techniques from hyperasymptotics. We establish a connection between singularities in the Borel plane and quark-hadron duality violating contributions. Starting with the assumption that for QCD at $N_c=\infty$ the spectrum approaches a Regge trajectory at large energy, we obtain an expression for quark-hadron duality violations at large, but finite $N_c$., Comment: 38 pages, 19 figures

We introduce parametrizations of hadronic three-body $B$ and $D$ weak decay amplitudes that can be readily implemented in experimental analyses and are a sound alternative to the simplistic and widely used sum of Breit-Wigner type amplitudes, also known as the isobar model. These parametrizations can be particularly useful in the interpretation of $CP$ asymmetries in the Dalitz plots. They are derived from previous calculations based on a quasi-two-body factorization approach in which two-body hadronic final-state interactions are fully taken into account in terms of unitary $S$- and $P$-wave $\ensuremath{\pi}\ensuremath{\pi}$, $\ensuremath{\pi}K$, and $K\overline{K}$ form factors. These form factors can be determined rigorously, fulfilling fundamental properties of quantum field-theory amplitudes such as analyticity and unitarity, and are in agreement with the low-energy behavior predicted by effective theories of QCD. They are derived from sets of coupled-channel equations using $T$-matrix elements constrained by experimental meson-meson phase shifts and inelasticities, chiral symmetry, and asymptotic QCD. We provide explicit amplitude expressions for the decays ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, $B\ensuremath{\rightarrow}K{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}{K}^{\ifmmode\pm\else\textpm\fi{}}$, ${D}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$, ${D}^{+}\ensuremath{\rightarrow}{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$, and ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, for which we have shown in previous studies that this approach is phenomenologically successful; in addition, we provide expressions for the ${D}^{0}\ensuremath{\rightarrow}{K}_{S}^{0}{K}^{+}{K}^{\ensuremath{-}}$ decay. Other three-body hadronic channels can be parametrized likewise.

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.

The presence of terms that violate quark-hadron duality in the expansion of QCD Green's functions is a generally accepted fact. Recently, a new approach was proposed for the study of duality violations (DVs), which exploits the existence of a rigorous lower bound on the functional distance, measured in a certain norm, between a "true" correlator and its approximant calculated theoretically along a contour in the complex energy plane. In the present paper we pursue the investigation of functional-analysis based tests towards their application to real spectral function data. We derive a closed analytic expression for the minimal functional distance based on the general weighted $L^2$ norm and discuss its relation with the distance measured in $L^\infty$ norm. Using fake data sets obtained from a realistic toy model in which we allow for covariances inspired from the publicly available ALEPH spectral functions, we obtain by Monte Carlo simulations the statistical distribution of the strength parameter that measures the magnitude of the DV term added to the usual operator product expansion (OPE). The results show that, if the region with large errors near the end-point of the spectrum in $\tau$ decays is excluded, the functional-analysis based tests using either $L^2$ or $L^\infty$ norms are able to detect, in a statistically significant way, the presence of DVs in realistic spectral function pseudodata., Comment: 24 pages, 5 figures. Reference added. Accepted for publication in Physical Review D

Several different analysis methods have been developed to determine the strong coupling via finite-energy sum-rule analyses of hadronic $\tau$ decay data. While most methods agree on the existence of the well-known ambiguity in the choice of a resummation scheme due to the slow convergence of QCD perturbation theory at the $\tau$ mass, there is an ongoing controversy over how to deal properly with non-perturbative effects. These are small, but not negligible, and include quark-hadron "duality violations" (i.e., resonance effects) which are not described by the operator product expansion (OPE). In one approach, an attempt is made to suppress duality violations enough that they might become negligible. The number of OPE parameters to be fit, however, then exceeds the number of available sum rules, necessitating an uncontrolled OPE truncation, in which a number of higher-dimension OPE contributions in general present in QCD are set to zero by hand. In the second approach, truncation of the OPE is avoided by construction, and duality violations are taken into account explicitly, using a physically motivated model. In this article, we provide a critical appraisal of a recent analysis employing the first approach and demonstrate that it fails to properly account for non-perturbative effects, making the resulting determination of the strong coupling unreliable. The second approach, in contrast, passes all self-consistency tests, and provides a competitive determination of the strong coupling from $\tau$ decays., Comment: 30 pages, 10 figures, layout issue in subsection header fixed

We review our recent determination of the strong coupling α s from the OPAL data for non-strange hadronic tau decays. We find that α s ( m τ 2 ) = 0.325 ± 0.018 using fixed-order perturbation theory, and α s ( m τ 2 ) = 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.

The Quantum Chromodynamics (QCD) coupling $\alpha_s$ is a central parameter in the Standard Model of particle physics. However, it depends on theoretical conventions related to renormalisation and hence is not an observable quantity. In order to capture this dependence in a transparent way, a novel definition of the QCD coupling, denoted by $\hat a$, is introduced, whose running is explicitly renormalisation scheme invariant. The remaining renormalisation scheme dependence is related to transformations of the QCD scale $\Lambda$, and can be parametrised by a single parameter $C$. Hence, we call $\hat a$ the $C$-scheme coupling. The dependence on $C$ can be exploited to study and improve perturbative predictions of physical observables. This is demonstrated for the QCD Adler function and hadronic decays of the $\tau$ lepton., Comment: 10 pages, 6 figures, proceedings of XIIth Quark Confinement and the Hadron Spectrum, 28 August - 4 September (2016), Thessaloniki (Greece)

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.

Recently, we extracted the strong coupling, $\alpha_s(m_\tau^2)$, from the revised ALEPH data for non-strange hadronic tau decays. Our analysis is based on a method previously used for the determination of the strong coupling from OPAL data. In our strategy, we employ different moments of the spectral functions both with and without pinching, including Duality Violations, in order to obtain fully self-consistent analyses that do not rely on untested assumptions (such as the smallness of higher dimension contributions in the OPE). Here we discuss the $\alpha_s$ values obtained from the ALEPH and the OPAL data, the robustness of the analysis, as well as non-perturbative contributions from DVs and the OPE. We show that, although the $\alpha_s$ determination is sound, non-perturbative effects limit the accuracy with which one can extract the strong coupling from tau decay data. Finally, we discuss the compatibility of the data sets and the possibility of a combined analysis., Comment: Invited contribution to the workshop "Determination of the Fundamental Parameters in QCD" (MITP/Mainz, 7-11 March 2016). 14 pages 5 figures