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Abstract In a previous article, we have shown that the discrepancy between the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions for τ hadronic spectral function moments, which had affected the precision of α 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 O $$ \mathcal{O} $$ ( α s 5 $$ {\alpha}_s^5 $$ ), we show that using the renormalon-free GC scheme successfully reconciles the results for α s ( m τ 2 $$ {m}_{\tau}^2 $$ ) 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 α s ( m τ 2 $$ {m}_{\tau}^2 $$ ), 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 τ hadronic spectral function moments.

Abstract We propose a simple and easy-to-implement scheme for a renormalon-free gluon condensate (GC) matrix element, which is analogous to implementations of short-distance heavy-quark mass renormalization schemes existing in the literature already for a long time. Because the scheme is based on a perturbative subtraction at the level of the matrix element, with a freely adaptable infrared factorization scale, it can be implemented with little effort for any observable where the GC is relevant. The scheme depends on the renormalon norm of the GC which has to be supplemented independently. We apply the scheme to the fixed-order (FOPT) and contour-improved (CIPT) perturbative expansions of τ hadronic spectral function moments. These expansions exhibit a long-standing discrepancy for moments used in high-precision determinations of the strong coupling in the commonly used GC scheme that is not renormalon-free. We show that the scheme is capable of resolving the FOPT-CIPT discrepancy problem. At the same time, the perturbative behaviour of the moments that previously showed bad convergence properties and for which the non-perturbative corrections from the GC are sizeable, is substantially improved. The new GC scheme may provide a powerful theoretical tool for future phenomenological applications.

Abstract We use rational approximants to study missing higher orders in the massless scalar-current quark correlator. We predict the yet unknown six-loop coefficient of its imaginary part, related to Γ(H → b b ¯ $$ \overline{b} $$ ), to be c 5 = −6900 ± 1400. With this result, the perturbative series becomes almost insensitive to renormalization scale variations and the intrinsic QCD truncation uncertainty is tiny. This confirms the expectation that higher-order loop computations for this quantity will not be required in the foreseeable future, as the uncertainty in Γ(H → b b ¯ $$ \overline{b} $$ ) will remain largely dominated by the Standard Model parameters.

Abstract We calculate the small-momentum expansion of vector, axial-vector, scalar, and pseudo-scalar heavy-quark current correlators in the large-β 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, α 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.

Abstract 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 q 2 and at high q 2, 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) + ν τ 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 q 2. Focusing then on the observables F H and A FB, 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.

Abstract We determine the strong coupling αs(m Z ) from dimensionless ratios of roots of moments of the charm- and bottom-quark vector and charm pseudo-scalar correlators, dubbed R q X , n ≡ M q X , n 1 n / M q X , n + 1 1 n + 1 , $$ {R}_q^{X,n}\equiv \left({M}_q^{X,n}\right)\frac{1}{n}/{\left({M}_q^{X,n+1}\right)}^{\frac{1}{n+1}}, $$ with X = V, P , as well as from the 0-th moment of the charm pseudo-scalar correlator, M c P , 0 . $$ {M}_c^{P,0}. $$ In the quantities we use, the mass dependence is very weak, entering only logarithmically, starting at 𝒪 α s 2 . $$ \left({\alpha}_s^2\right). $$ We carefully study all sources of uncertainties, paying special attention to truncation errors, and making sure that order-by-order convergence is maintained by our choice of renormalization scale variation. In the computation of the experimental uncertainty for the moment ratios, the correlations among individual moments are properly taken into account. Additionally, in the perturbative contributions to experimental vector-current moments, α s (m Z ) is kept as a free parameter such that our extraction of the strong coupling is unbiased and based only on experimental data. The most precise extraction of α s from vector correlators comes from the ratio of the charm-quark moments R c V , 2 $$ {R}_c^{V,2} $$ and reads α s (m Z ) = 0.1168±0.0019, as we have recently discussed in a companion letter. From bottom moments, using the ratio R c V , 2 $$ {R}_c^{V,2} $$ , we find α s (m Z ) = 0.1186±0.0048. Our results from the lattice pseudo-scalar charm correlator agree with the central values of previous determinations, but have larger uncertainties due to our more conservative study of the perturbative error. Averaging the results obtained from various lattice inputs for the n = 0 moment we find α s (m Z ) = 0.1177±0.0020. Combining experimental and lattice information on charm correlators into a single fit we obtain α s (m Z ) = 0.1170±0.0014, which is the main result of this article.

Abstract 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, α 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 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-β 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 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 c 5,1 = 277 ± 51, c 6,1 = 3460±690, and c 7,1 = (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.