Strong coupling from hadronic τ decays: A critical appraisal

There are some indications from recent determinations of the strong coupling constant alpha_s and the gluon condensate that the Operator Product Expansion may not be accurate enough to describe non-perturbative effects in hadronic tau decays. This breakdown of the Operator Product Expansion is usually referred to as being due to"Duality Violations." With the help of a physically motivated model, we investigate these duality violations. Based on this model, we argue how they may introduce a non-negligible systematic error in the current analysis, which employs finite-energy sum rules with pinched weights. In particular, this systematic effect might affect the precision determination of alpha_s from tau decays. With a view to a possible future application to real data, we present an alternative method for determining the OPE coefficients that might help estimating, and possibly even reducing, this systematic error.

In this review, we discuss why, in the determination of [Formula: see text] from hadronic [Formula: see text] decays, two important assumptions made in most of previous analyses, namely the neglect of higher-dimension condensates and of duality violations (DVs), have introduced uncontrolled systematic errors into this determination. Although the use of pinched weights is usually offered as a justification of these assumptions, we explain why it is not possible to simultaneously suppress these two contributions; particularly since the Operator Product Expansion (OPE) is expected to be an asymptotic, rather than a convergent expansion. There is not only experimental and theoretical evidence for DVs but they also affect the extraction of [Formula: see text].

The earlier introduced method of calculation of quark distributions in hadrons, based on QCD sum rules, is improved. The imaginary part of the virtual photon forward scattering amplitude on some hadronic current is considered in the case, when initial and final virtualities of the current $p^2_1$, and $p^2_2$ are different, $p^2_1\not= p^2_2$. The operator product expansion (OPE) in $p^2_1$, $p^2_2$ is performed. The sum rule for quark distribution is obtained using double dispersion representation of the amplitude on one side in terms of calculated in QCD OPE and on the other side in terms of physical states contributions. Double Borel transformation in $p^2_1$, $p^2_2$ is applied to the sum rule, killing background non-diagonal transition terms, which deteriorated the accuracy in previous calculations. The case of valence quark distribution in pion is considered, which was impossible to treat by the previous method. OPE up to dimension 6 operators is performed and leading order perturbative corrections are accounted. Valence $u$-quark distribution in $\pi^+$ was found at intermediate $x$, $0.15 < x < 0.7$ and normalization point $Q^2=2 GeV^2$. These results may be used as input for evolution equations.

We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in the light-ray operators that appear in the stress-tensor OPE. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in mathcal{N} = 4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).

The large linear superconformal algebra (generated by four spin- currents, seven spin-1 currents, four spin- currents and one spin-2 current) found by Sevrin, Troost and Van Proeyen (and other groups) was realized in the superconformal coset theory previously. The lowest 16 higher spin currents of spins , , and are obtained by starting with the operator product expansions (OPEs) between the four spin- currents from the above large linear superconformal algebra and the lowest higher spin-1 current which is the same as the one in the Wolf space coset theory. These OPEs determine the four higher spin- currents and the next six higher spin-2 currents are obtained from the OPEs between the above four spin- currents associated with the supersymmetry and these four higher spin- currents. The four higher spin- currents can be determined by calculating the OPEs between the above four spin- currents and the higher spin-2 currents. Similarly, the higher spin-3 current is obtained from the OPEs between the four spin- currents and the higher spin- currents. The explicit relations between the above 16 higher spin currents and the corresponding 16 higher spin currents which were found in the extension of large nonlinear superconformal algebra previously are given. By examining the OPEs between the 16 currents from the large linear superconformal algebra and the 16 higher spin currents, the match with the findings of Beccaria, Candu and Gaberdiel is also given. The next 16 higher spin currents of spins , , and occur from the OPEs between the above lowest 16 higher spin currents.

Tau-based finite-energy sum-rule (FESR) analyses often assume that scales s_0\sim m_\tau^2s0∼mτ2 are large enough that (i) integrated duality violations (DVs) can be neglected, and (ii) contributions from non-perturbative OPE condensates of dimension DD scale as (\Lambda_{QCD}/m_\tau )^D(ΛQCD/mτ)D, allowing the OPE series to be truncated at low dimension. The latter assumption is not necessarily valid since the OPE series is not convergent, while the former is open to question given experimental results for the electromagnetic, I=1I=1 vector (VV), I=1I=1 axial vector (AA) and I=1I=1V+AV+A current spectral functions, which show DV oscillations with amplitudes comparable in size to the corresponding \alpha_sαs-dependent perturbative contributions at s\sim2-3s∼2−3 GeV^22. Here, we discuss recently introduced new tools for assessing the numerical relevance of omitted higher-DD OPE contributions. Applying these to the “truncated OPE” strategy used in Refs.[M. Davier et al., Eur. Phys. J. C 74, 2803 (2014); A. Pich and A. Rodríguez-Sánchez, Phys. Rev. D 94, 034027 (2016)] and earlier work by the same authors, we find that this strategy fails to yield reliable results for the strong coupling from hadronic \tauτ decays.

The soft wall model in holographic QCD has Regge trajectories but wrong oper- ator product expansion (OPE) for the two-point vectorial QCD Green function. We modify the dilaton potential to comply with the OPE. We study also the axial two-point function us- ing the same modified dilaton field and an additional scalar fie ld to address chiral symmetry breaking. OPE is recovered adding a boundary term and low energy chiral parameters, Fπ and L10, are well described analytically by the model in terms of Regge spacing and QCD condensates. The model nicely supports and extends previous theoretical analyses advocat- ing Digamma function to study QCD two-point functions in different momentum regions.

In the heavy quark limit of QCD, using the Operator Product Expansion and the non-forward amplitude, as proposed by Nikolai Uraltsev, we formulate sum rules that generalize Bjorken and Uraltsev sum rules. We recover the Uraltsev lower bound for the slope of the Isgur–Wise (IW) function, that we generalize to higher derivatives. We show that these results have a clear interpretation in terms of the Lorentz group, since the IW function is given by an overlap between the initial and final light clouds, related by Lorentz transformations. Both the Lorentz group and the Sum Rules approaches are equivalent. Moreover, we formulate an integral representation of the IW function with a positive measure. Inverting this integral formula, we obtain the measure in terms of the IW function, allowing one to formulate criteria to decide if a given ansatz for the IW function is compatible or not with the sum rule constraints. We compare these theoretical constraints to some forms proposed in the literature.

The QCD sum rule technique is employed to investigate pentaquark states with strangeness $S=+1$ and $I{J}^{\ensuremath{\pi}}=0{\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $1{\frac{1}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $0{\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$, $1{\frac{3}{2}}^{\ifmmode\pm\else\textpm\fi{}}$. Throughout the calculation, emphasis is laid on the establishment of a valid Borel window, which corresponds to a region of the Borel mass, where the operator product expansion converges and the presumed ground state pole dominates the sum rules. Such a Borel window is achieved by constructing the sum rules from the difference of two independent correlators and by calculating the operator product expansion up to dimension 14. Furthermore, we discuss the possibility of the contamination of the sum rules by possible $KN$ scattering states. As a result, we conclude that the $0{\frac{3}{2}}^{+}$ state seems to be the most probable candidate for the experimentally observed ${\ensuremath{\Theta}}^{+}(1540)$, while we also obtain states with $0{\frac{1}{2}}^{\ensuremath{-}}$, $1{\frac{1}{2}}^{\ensuremath{-}}$, $1{\frac{3}{2}}^{+}$ at somewhat higher mass regions.

The diagrammatic approach discussed in the previous chapter is useful to get an order-of-magnitude estimate of the importance of the various contributions to each amplitude, and to find approximate relations among the amplitudes for different decays. However, rigorous computations require a correct treatment of the QCD corrections, and this is best accomplished using the operator product expansion (OPE) already discussed in the context of kaon decays-see Chapter 20. The OPE allows the construction of a low-energy effective Hamiltonian as a sum of local operators multiplied by Wilson coefficients. The effective Hamiltonian is the sum of pieces, each of them of the type in the Wilson coefficients, from the low-energy non-perturbative confinement effects implicitly contained in the operator matrix elements. However, upon change ofµ or of the renormalization scheme, contributions previously included in the matrix elements of some operator may move into the Wilson coefficients or into the matrix elements of other operators.

Dispersive approach to power-behaved contributions in QCD hard processes

This talk reviews the recent progress in the extraction of bound-state characteristics from the operator-product expansion (OPE) for field-theory correlators, which constitutes the basis of the method of QCD sum rules. This progress is mainly related to a deeper understanding of one of the key ingredients of the method of sum rules -the effective continuum threshold. The latter determines to a great extent the numerical values of the bound-state parameters obtained by sum rules. The understanding of properties of the effective continuum threshold allows one to formulate a new algorithm for fixng this quantity for various correlators and in this way gain control over the systematic uncertainties of the bound-state parameters extracted from sum rules. We start with examples from quantum mechanics, where bound-state properties may be calculated independently in two ways: exactly, by solving the Schrdinger equation, and approximately, by the method of dispersive sum rules. Knowing the exact solution is crucial as it allows us to control each step of the sum-rule extraction procedure. On the basis of this analysis, we formulate several improvements of the standard procedures adopted in the method of sum rules in QCD. We then show the impact of these modifications on the extraction of the decay constants of heavy mesons.

We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting (single-trace) terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories.

We verify, both perturbatively and nonperturbatively asymptotically in the ultraviolet (UV), a special case of a low-energy theorem of the NSVZ type in QCD-like theories, recently derived in Phys. Rev. D 95 (2017) 054010, that relates the logarithmic derivative with respect to the gauge coupling, or the logarithmic derivative with respect to the renormalization-group (RG) invariant scale, of an n-point correlator of local operators in one side to an n + 1-point correlator with the insertion of TrF2 at zero momentum in the other side. Our computation involves the operator product expansion (OPE) of the scalar glueball operator, TrF2, in massless QCD, worked out perturbatively in JHEP 12 (2012) 119 — and in its RG-improved form in the present paper — by means of which we extract both the perturbative divergences and the nonperturbative UV asymptotics in both sides. We also discuss the role of the contact terms in the OPE, both finite and divergent, discovered some years ago in JHEP 12 (2012) 119, in relation to the low-energy theorem. Besides, working the other way around by assuming the low-energy theorem for any 2-point correlator of a multiplicatively renormalizable gauge-invariant operator, we compute in a massless QCD-like theory the corresponding perturbative OPE to the order of g2 and nonperturbative asymptotics. The low-energy theorem has a number of applications: to the renormalization in asymptotically free QCD-like theories, both perturbatively and nonperturbatively in the large-N ’t Hooft and Veneziano expansions, and to the way the open/closed string duality may or may not be realized in the would-be solution by canonical string theories for QCD-like theories, both perturbatively and in the ’t Hooft large-N expansion. Our computations will also enter further developments based on the low-energy theorem.

We explain a gluon transversity, transverse-momentum-dependent parton distribution functions (TMDs), and parton distribution functions (PDFs) for spin-1 hadrons. The gluon transversity exists in hadrons with spin more than or equal to one, and it does not exist in the spin-1/2 nucleons. Since there is no direct contribution from the nucleons, it is an appropriate quantity to probe an exotic component in the spin-1 deuteron beyond a simple bound system of the nucleons. We show how the gluon transversity can be measured at hadron accelerator facilities by the Drell-Yan process in addition to lepton-accelerator experiments. Next, possible TMDs are explained for the spin-1 hadrons at the twists 3 and 4 in addition to twist-2 ones by considering tensor polarizations. We found that 30 TMDs exist in the tensor-polarized spin-1 hadron at the twists 3 and 4 in addition to 10 TMDs at the twist 2. There are 3 collinear PDFs at the twists 3 and 4. We also indicate that the corresponding TMD fragmentation functions exist at the twists 3 and 4. Due to the time-reversal invariance in the collinear PDFs, there are new sum rules on the time-reversal odd TMDs. In addition, we obtained a useful twist-2 relation, a sum rule, and relations with multiparton distribution functions by using the operator product expansion and the equation of motion for quarks. These findings are valuable for experimental investigations on polarized deuteron structure functions in 2020's and 2030's at world accelerator facilities.