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Register a new userDetermining $alpha_s$ from hadronic $tau$ decay: the pitfalls of truncating the OPE
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We discuss sum-rule determinations of $alpha_s$ from non-strange hadronic $tau$-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. [1-3]. Here, we test this strategy by applying the strategy to the $R$-ratio obtained from $e^+e^-$ data, which extend beyond the $tau$ mass, and, based on the outcome of these tests, we demonstrate the failure of this strategy.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 $alpha_s$, including a representation of duality violations is unavoidable if one wishes to avoid uncontrolled theoretical errors.
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We summarize a comparison of the two strategies which are currently available in the literature for determining the value of $alpha_s(m_tau)$. We will refer to these as the truncated Operator Product Expansion model and the Duality Violation model. After describing the main features of both approaches, we explain why the former fails to pass crucial tests. The latter, on the other hand, passes all the tests known up to date and, therefore, should be currently considered the only reliable method.
We present a new analysis of $alpha_s$ from hadronic $tau$ decays based on the recently revised ALEPH data. The analysis is based on a strategy which we previously applied to the OPAL data. We critically compare our strategy to the one traditionally used and comment on the main differences. Our analysis yields the values $alpha_s(m_tau^2)=0.296pm 0.010$ using fixed-order perturbation theory, and $alpha_s(m_tau^2)=0.310pm 0.014$ using contour-improved perturbation theory. Averaging these values with our previously obtained values from the OPAL data, we find $alpha_s(m_tau^2)=0.303pm 0.009$, respectively, $alpha_s(m_tau^2)=0.319pm 0.012$, as the most reliable results for $alpha_s$ from $tau$ decays currently available.
We determine the strong coupling constant $alpha_s$ from the static QCD potential by matching a theoretical calculation with a lattice QCD computation. We employ a new theoretical formulation based on the operator product expansion, in which renormalons are subtracted from the leading Wilson coefficient. We remove not only the leading renormalon uncertainty of $mathcal{O}(Lambda_{rm QCD})$ but also the first $r$-dependent uncertainty of $mathcal{O}(Lambda_{rm QCD}^3 r^2)$. The theoretical prediction for the potential turns out to be valid at the static color charge distance $Lambda_{rm overline{MS}} r lesssim 0.8$ ($r lesssim 0.4$ fm), which is significantly larger than ordinary perturbation theory. With lattice data down to $Lambda_{rm overline{MS}} r sim 0.09$ ($r sim 0.05$ fm), we perform the matching in a wide region of $r$, which has been difficult in previous determinations of $alpha_s$ from the potential. Our final result is $alpha_s(M_Z^2) = 0.1179^{+0.0015}_{-0.0014}$ with 1.3 % accuracy. The dominant uncertainty comes from higher order corrections to the perturbative prediction and can be straightforwardly reduced by simulating finer lattices.
At small transverse momentum $q_T$, transverse-momentum dependent parton distribution functions (TMDPDFs) arise as genuinely nonperturbative objects that describe Drell-Yan like processes in hadron collisions as well as semi-inclusive deep-inelastic scattering. TMDPDFs naturally depend on the hadron momentum, and the associated evolution is determined by the Collins-Soper equation. For $q_T sim Lambda_mathrm{QCD}$ the corresponding evolution kernel (or anomalous dimension) is nonperturbative and must be determined as an independent ingredient in order to relate TMDPDFs at different scales. We propose a method to extract this kernel using lattice QCD and the Large-Momentum Effective Theory, where the physical TMD correlation involving light-like paths is approximated by a quasi TMDPDF, defined using equal-time correlation functions with a large-momentum hadron state. The kernel is determined from a ratio of quasi TMDPDFs extracted at different hadron momenta.
The path-integral formulation of the hadronic tensor W_{mu u} of deep inelastic scattering is reviewed. It is shown that there are 3 gauge invariant and topologically distinct contributions. The separation of the connected sea partons from those of the disconnected sea can be achieved with a combination of the global fit of the parton distribution function (PDF), the semi-inclusive DIS data on the strange PDF and the lattice calculation of the ratio of the strange to $u/d$ momentum fraction in the disconnected insertion. We shall discuss numerical issues associated with lattice calculation of the hadronic tensor involving a four-point function, such as large hadron momenta and improved maximum entropy method to obtain the spectral density from the hadronic tensor in Euclidean time. We also draw a comparison between the large momentum approach to the parton distribution function (PDF) and the hadronic tensor approach.
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