The physics of hadronic tau decays

Michel Davier, Andreas Höcker, and Zhiqing Zhang

Rev. Mod. Phys. 78, 1043 – Published 4 October 2006

Abstract

Hadronic $τ$ decays provide a clean laboratory for the precise study of quantum chromodynamics (QCD). Observables based on the spectral functions of hadronic $τ$ decays can be related to QCD quark-level calculations to determine fundamental quantities like the strong-coupling constant, parameters of the chiral Lagrangian $∣ V_{u s} ∣$ , the mass of the strange quark, and to simultaneously test the concept of quark-hadron duality. Using the best available measurements and a revisited analysis of the theoretical framework, the value $α_{s} (m_{τ}^{2}) = 0.345 \pm {0.004}_{\exp} \pm {0.009}_{th}$ is obtained. Taken together with the determination of $α_{s} (M_{Z}^{2})$ from the global electroweak fit, this result leads to the most accurate test of asymptotic freedom: the value of the logarithmic slope of $α_{s}^{- 1} (s)$ is found to agree with QCD at a precision of 4%. The $τ$ spectral functions can also be used to determine hadronic quantities that, due to the nonperturbative nature of long-distance QCD, cannot be computed from first principles. An example for this is the contribution from hadronic vacuum polarization to loop-dominated processes like the anomalous magnetic moment of the muon. This article reviews the measurements of nonstrange and strange $τ$ spectral functions and their phenomenological applications.

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DOI:https://doi.org/10.1103/RevModPhys.78.1043

Authors & Affiliations

Michel Davier^*, Andreas Höcker^†, and Zhiqing Zhang^‡

Laboratoire de l’Accélérateur Linéaire, IN2P3/CNRS and Université de Paris-Sud 11, BP34, F-91898 Orsay, France

^*Electronic address: davier@lal.in2p3.fr
^†Present address: CERN, PH Division, CH-1211 Geneva 23, Switzerland. Electronic address: hoecker@lal.in2p3.fr
^‡Electronic address: zhangzq@lal.in2p3.fr

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Vol. 78, Iss. 4 — October - December 2006

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Figure 1
(Color online) Upper: The spectral function (see Sec. 4 for the definition) for the decay $τ^{-} \to ω π^{-} ν_{τ}$ vs the hadronic mass from the CLEO analysis (166). The curves are results of fits to various combinations of $ρ$ , $ρ (1450)$ , and $ρ (1700)$ resonance contributions. Lower: The angular distribution in the $ω π$ center-of-mass frame compared to predictions for different $(l)$ partial waves. Second-class currents would reveal themselves as $l = 0, 2$ waves.Reuse & Permissions
Figure 2
(Color online) Measurements of the branching fraction for $τ^{-} \to e^{-} {\bar{ν}}_{e} ν_{τ}$ (upper) and $τ^{-} \to μ^{-} {\bar{ν}}_{μ} ν_{τ}$ (lower). References for experiments are 296; 25; 31; 10; 12; 3, 4. The world averages are indicated by the shaded vertical bands.Reuse & Permissions
Figure 3
(Color online) Comparison of results on the main branching fractions for $τ$ hadronic modes (105; 24; 287; 36; 184; 47; 31; 14, 17; 40; 296). World averages are indicated by the shaded vertical bands.Reuse & Permissions
Figure 4
(Color online) Left-hand plots: The inclusive vector (upper) and axial-vector (lower) spectral functions as measured by 296). Shaded areas indicate the contributing exclusive $τ$ decay channels. Curves show predictions from the parton model (dotted) and from massless perturbative QCD using $α_{s} (M_{Z}^{2}) = 0.120$ (solid). Right-hand-plots: Comparison of the inclusive vector (upper) and axial-vector (lower) spectral functions obtained by ALEPH and OPAL (16).Reuse & Permissions
Figure 5
(Color online) Inclusive vector plus axial-vector (upper) and vector minus axial-vector spectral function (lower) as measured by 296 (dots with errors bars) and 16 (shaded 1 standard deviation errors). Lines show predictions from the parton model (dotted) and from massless perturbative QCD using $α_{s} (M_{Z}^{2}) = 0.120$ (solid). They cancel to all orders in the difference.Reuse & Permissions
Figure 6
(Color online) Inclusive hadronic vector plus axial-vector spectral function from $τ$ decays into $S = \pm 1$ final states with its different contributions as indicated from 55 and 5. The kaon pole is not included in the ALEPH plot. The parton model prediction is given by the dashed line.Reuse & Permissions
Figure 7
(Color online) Upper: Relative comparison of the $π^{+} π^{-}$ spectral functions extracted from $τ$ data for the different experiments, expressed as a ratio to the average $τ$ spectral function. Lower: The $ρ$ region. For CLEO only statistical errors are shown.Reuse & Permissions
Figure 8
(Color online) Comparison of $π^{+} π^{-}$ spectral functions from isospin-breaking corrected $τ$ data (world average) expressed as $e^{+} e^{-}$ cross sections and $e^{+} e^{-}$ annihilation. The band indicates the combined $τ$ and $e^{+} e^{-}$ result within $1 σ$ errors. It is given for illustration purpose only. A compendium of references for the $e^{+} e^{-}$ data has been given by 144, 145).Reuse & Permissions
Figure 9
(Color online) Relative comparison of the isospin-breaking corrected $τ$ data (world average) and $π^{+} π^{-}$ spectral functions from $e^{+} e^{-}$ annihilation, expressed as a ratio to the $τ$ spectral function. The shaded band gives the uncertainty on the $τ$ spectral function. The lower plot zooms into the $ρ (770)$ peak region where the data density is high. The $ρ - ω$ mixing has been phenomenologically corrected using a fit to the CMD-2 data. The larger mixing observed in the SND data, expressed as a 40% increase in the value for the $ω \to π^{+} π^{-}$ branching fraction (13), is responsible for the residual effect after correction seen in the lower plot.Reuse & Permissions
Figure 10
(Color online) Comparison of the $2 π^{+} 2 π^{-}$ (upper) and $π^{+} π^{-} 2 π^{0}$ (lower) spectral functions from $e^{+} e^{-}$ and isospin-breaking corrected $τ$ data (ALEPH), expressed as $e^{+} e^{-}$ cross sections. References for the $e^{+} e^{-}$ data have been given by 144, 145. The BABAR results are taken from 39.Reuse & Permissions
Figure 11
(Color online) Measured branching fractions for $τ^{-} \to π^{-} π^{0} ν_{τ}$ (36; 15; 296) compared to the predictions from $e^{+} e^{-} \to π^{+} π^{-}$ spectral functions, applying the isospin-breaking correction factors discussed in Sec. 5A. For $e^{+} e^{-}$ results we have used only data from indicated experiments where they are available ( $0.61 - 0.96 GeV$ for CMD-2, $0.60 - 0.97 GeV$ for KLOE, and $0.39 - 0.97 GeV$ for SND), and all data combined in the remaining energy domains below $m_{τ}$ . Since the compatibility between $e^{+} e^{-}$ data is marginal (cf. Fig. 9), we have refrained from taking their average.Reuse & Permissions
Figure 12
(Color online) Phenomenological fit to the ALEPH $π^{+} π^{0}$ spectral function (upper) and to $e^{+} e^{-}$ annihilation data (lower) using the Gounaris-Sakurai parametrization. From 296.Reuse & Permissions
Figure 13
(Color online) Representative diagrams contributing to $a_{μ}$ . Lowest-order diagram (a) and first-order QED correction (b); lowest-order hadronic contribution (c) and hadronic light-by-light scattering (d); weak interaction diagrams (e), (f); possible contributions from lowest-order supersymmetry (g), (h).Reuse & Permissions
Figure 14
(Color online) Fit of the pion form factor from $4 m_{π}^{2}$ to $0.35 {GeV}^{2}$ using a third-order expansion with constraints at $s = 0$ and measured pion rms charge radius from spacelike data (29). The result of the fit is integrated only up to $0.25 {GeV}^{2}$ . The $e^{+} e^{-}$ data do not yet include the recent SND measurement (13).Reuse & Permissions
Figure 15
(Color online) Compilation of data contributing to $a_{μ}^{had, LO}$ . Shown is the total hadronic over muonic cross-section ratio $R$ . The shaded band below $2 GeV$ represents the sum of exclusively measured channels, with the exception of contributions from narrow resonances that are given as dashed lines. All data points shown correspond to inclusive measurements. The cross-hatched band gives the prediction from (essentially) perturbative QCD (see text).Reuse & Permissions
Figure 16
(Color online) Comparison of the result (71) (209) labeled DEHZ 04 with the BNL measurement (66). Also given are the previous estimate (145), where the triangle with the dotted error bar indicates the $τ$ -based result, as well as estimates from 202; 221; 155, not yet including the KLOE data.Reuse & Permissions
Figure 17
Integration contour for the right-hand side of Eq. (83).Reuse & Permissions
Figure 18
(Color online) Comparison between the iteratively evolved $α_{s} (m_{τ}^{2} e^{i φ})$ (exact solution) and fourth-order Taylor expansion (81) on the circle $φ = - π, \dots, π$ (260). This example uses $α_{s} (m_{τ}^{2}) = 0.35$ .Reuse & Permissions
Figure 19
(Color online) Evolution of $a_{τ} (s_{0})$ for different perturbative methods, and for variations of the unknown higher-order perturbative coefficients $K_{4}$ (dashed) and $K_{5}$ (dotted). All methods are normalized to $a_{τ} (m_{τ}^{2}) = 0.22$ . The inseted frames give the $α_{s} (m_{τ}^{2})$ values derived from this $a_{τ}$ value in the respective methods. The error bars show uncertainties, added in quadrature, from the $K_{4}$ and $K_{5}$ variations quoted on the plots.Reuse & Permissions
Figure 20
Multifermion loop insertion (renormalons) into a fermion-antifermion vacuum-polarization diagram.Reuse & Permissions
Figure 21
Ultraviolet (UV) and infrared (IR) renormalons in the Borel plane of the Adler function.Reuse & Permissions
Figure 22
(Color online) Upper: The ratio $R_{τ, V + A}$ versus the square “ $τ$ mass” $s_{0}$ . Curves are plotted as error bands to emphasize their strong point-to-point correlations in $s_{0}$ . Also shown is the theoretical prediction using CIPT and results for $R_{τ, V + A}$ and nonperturbative terms from Table . Lower: The running of $α_{s} (s_{0})$ obtained from the fit of the theoretical prediction to $R_{τ, V + A} (s_{0})$ using CIPT. The shaded band shows data including only experimental errors. The curve gives the four-loop RGE evolution for three flavors.Reuse & Permissions
Figure 23
(Color online) Top: The evolution of $α_{s} (m_{τ}^{2})$ , Eq. (139), to higher scales $μ$ using the four-loop RGE and the three-loop matching conditions applied at heavy quark-pair thresholds (hence the discontinuities at $2 m_{c}$ and $2 m_{b}$ ). The evolution is compared with other independent measurements (see text) covering scales varying over more than two orders magnitude. Experimental values are discussed in the text; they are mostly taken from 72. Bottom: The corresponding extrapolated $α_{s}$ values at $M_{Z}$ . The shaded band displays the $τ$ decay result within errors.Reuse & Permissions
Figure 24
Integrand of Eq. (155) with $(k = 0, ℓ = 0)$ for the difference of the Cabibbo-corrected nonstrange and strange invariant mass spectra from 55. The contribution from massless perturbative QCD (pQCD) vanishes. To guide the eye, the solid line interpolates between the bins of constant $0.1 {GeV}^{2}$ width.Reuse & Permissions
Figure 25
(Color online) The observables $δ R_{τ}^{k 0} (s_{0})$ as a function of the $τ$ mass squared $s_{0}$ confronted with the QCD predictions (left plots), and the derived $m_{s} (s_{0})$ compared with the QCD RGE running (right plots). The bands correspond to the experimental and theoretical uncertainties (left plots), and experimental errors only (right plots). By construction, data and theory agree at $s_{0} = m_{τ}^{2}$ . From 111.Reuse & Permissions
Figure 26
Determinations of the $\bar{MS}$ mass ${\bar{m}}_{s} (4 {GeV}^{2})$ based on $τ$ spectral functions and on other approaches: $K^{*}$ and $e^{+} e^{-}$ sum rules, lattice calculations.Reuse & Permissions
Figure 27
(Color online) The values of $∣ V_{u s} ∣$ and $m_{s} (m_{τ}^{2})$ derived from the experimental strange and nonstrange moments obtained from the ALEPH spectral functions normalized to world average branching ratios. The various curves correspond to the different $k, ℓ = 0$ moments with $k$ from 0 to 4. The width of the bands indicates the present experimental accuracy. The longitudinal components are subtracted using $K$ and $π$ branching fractions together with $τ − μ$ universality, and the phenomenological contributions estimated by 176. Data points indicate the $∣ V_{u s} ∣$ and $m_{s} (m_{τ}^{2})$ results from outside $τ$ physics. The two $∣ V_{u s} ∣$ values shown correspond to the average of the $K_{ℓ 3}$ and $K_{μ 2}$ results (full circle), and to the $\sqrt{1 - {∣ V_{u d} ∣}^{2}}$ unitarity-based number (open circle) (215).Reuse & Permissions
Figure 28
(Color online) The 68% C.L. contours in the $∣ V_{u s} ∣ − vs − m_{s} (m_{τ}^{2})$ plane from the fit of different pairs of moments derived from the experimental strange and nonstrange ALEPH spectral functions normalized to world average branching ratios. The longitudinal components are subtracted using the $τ − μ$ universality-improved $K$ and $π$ branching fractions, and phenomenological contributions estimated by 176. The results exhibit a systematic trend with the orders of the pair chosen. Data points show the best present knowledge of $∣ V_{u s} ∣$ (the open circle determined using unitarity) and $m_{s} (m_{τ}^{2})$ from outside $τ$ physics. The two $∣ V_{u s} ∣$ values shown correspond to the average of the $K_{ℓ 3}$ and $K_{μ 2}$ results (full circle), and to the $\sqrt{1 - {∣ V_{u d} ∣}^{2}}$ unitarity-based number (open circle) (215).Reuse & Permissions
Figure 29
(Color online) The first and second Weinberg sum rules (upper left and right), and the DMO and DGLMY sum rules (lower left and right) vs the upper integration cutoff $(s_{0})$ . The horizontal bands indicate chiral predictions within errors, assuming full saturation of the integrals.Reuse & Permissions
Figure 30
(Color online) The Borel sum rule (194) (upper) and the real part of the complex Borel sum rule (196) for $ϕ = 9 π ∕ 10$ (lower), at $M = 1 GeV$ , and as a function of the integral cutoff $s_{0}$ . The chiral prediction shown in the lower plot does not include OPE power corrections.Reuse & Permissions
Figure 31
Real part (196) of the Borel transformation for $ϕ = 9 π ∕ 10$ and its imaginary part (196) for $ϕ = 4 π ∕ 5$ as a function of the modulus $M^{2}$ of the Borel parameter. The leading order $D = 12$ operator $a_{12}$ vanishes in these sum rules. The shaded band shows the ALEPH data (50, 53), and lines display the leading-order operator series, i.e., neglecting all logarithmic terms. Numbers indicate the highest dimension considered in the operator sum. From 322.Reuse & Permissions
Figure 32
Integration contour around the circles at $s = m_{τ}^{2}$ and $s = s_{th}$ .Reuse & Permissions

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