Precise α s from  Decays(*) M. Davier, S. Descotes-Genon, A. Hoecker, B. Malaescu, and Z. Zhang Tau08 Workshop Novosibirsk, Sept (*) arxiv: ; published in EPJ C Rev. Mod. Phys. 78 (2006) 1043

M. Davier alpha_s tauTau08 Workshop Novosibirsk2 Outline Theoretical Framework Tests of Integration Methods Impact of Quark-Hadron Duality Violation Spectral Moments and Fit Results Test of Asymptotic Freedom Conclusions Work started 10 years ago with availability of  spectral functions (ALEPH, OPAL) and phenomenological framework (Braaten- Narison-Pich, LeDiberder,…) Motivation: final ALEPH  spectral functions with improvements of branching ratios (BABAR and Belle) Better knowledge of perturbative series (P. Baikov, et al.)

M. Davier alpha_s tauTau08 Workshop Novosibirsk3 Experimental Input: Branching Fractions From measured leptonic branching ratios: Vector, Axial-Vector and Strange contributions : (incl. new results from BABAR+Belle) (new)

M. Davier alpha_s tauTau08 Workshop Novosibirsk4 Separation of V/A Currents Straightforward for final states with only pions (using G-parity) : - even number of pions ( G = 1 ): vector state - odd number of pions ( G = -1 ): axial-vector state modes with are generally not eigenstates of G-parity : - is pure vector -ISR BaBaR results on ee  interferences in Dalitz plot (K*)  separation of I=0 and I=1 cross sections using CVC  f A =0.833± modes: f A =0.5±0.5 ALEPH(V+A) BABAR+CVC (V)

M. Davier alpha_s tauTau08 Workshop Novosibirsk5 Of purely nonperturbative origin ALEPH Spectral Functions + BR Updates

M. Davier alpha_s tauTau08 Workshop Novosibirsk6 Theoretical Prediction of R  Problem: Im  V/A (J) (s) contains hadronic physics that cannot be predicted by QCD in this region of the real axis However, owing to the analyticity of  (s), one can use Cauchy’s theorem: spectral function Re(s) Im(s) |s| = |s| =  |s| = s0|s| = s0 Potential problems for OPE Optical Theorem:

M. Davier alpha_s tauTau08 Workshop Novosibirsk7  and QCD: The Operator Product Expansion Full theoretical ansatz, including nonperturbative operators via the OPE: (in the following: a s =  s /  ) Perturbative contribution Adler function to avoid unphysical subtractions: Nonperturbative contribution EW correction: Perturbative quark-mass terms:

M. Davier alpha_s tauTau08 Workshop Novosibirsk8 The Perturbative Prediction Perturbative coefficients of Adler function series, known to n=4 (K 4 ≈ 49) RGE  -function, known to n=3 In practice, use Taylor development in Perturbative prediction of Adler function given to N 3 LO s dependence of a s driven by running: P. Baikov, et al., arxiv: [hep-ph] How to compute the integral in the complex plane? How to perform a scale transformation? Re(s) Im(s) |s| = s0|s| = s0 |s| =ζ s0|s| =ζ s0 φ

M. Davier alpha_s tauTau08 Workshop Novosibirsk9 Numerical Methods CIPT: at each step use Taylor series to compute from the value found at the previous step (LeDiberder-Pich, Pivovarov) FOPT: fixed order Taylor expansion around the physical value and the integration result is cut at the same order in a s (s 0 ) Remarks: - Potential problem for FOPT due to the use of Taylor series for large | η | - Avoided by CIPT (use small steps) - Analytically, the FOPT result can also be obtained by making small steps, with a fixed order cut of the result at each step, BUT the RGE is modified at each step! Re(s) Im(s) |s| = s0|s| = s0 FOPT CIPT

M. Davier alpha_s tauTau08 Workshop Novosibirsk10 RGE Order β 0 CIPT FOPT

M. Davier alpha_s tauTau08 Workshop Novosibirsk11 RGE Order β 1,2 Not exact solution, expansion!

M. Davier alpha_s tauTau08 Workshop Novosibirsk12 Tests of Integration Methods FOPT+: the same Taylor expansion for α s (s) as FOPT, with no cut of the integration result

M. Davier alpha_s tauTau08 Workshop Novosibirsk13 Numerical Tests of Integration Methods: δ (0) Massless perturbative contribution computed for with and estimated by assuming geometric growth. Higher unknown coefficients were set to zero.

M. Davier alpha_s tauTau08 Workshop Novosibirsk14 Integration Methods: Scale dependence FOPT neglects part of contributions to the perturbative series included in CIPT FOPT neglects part of contributions to the perturbative series included in CIPT FOPT generally does not satisfy the RGEuses a Taylor expansion in a region where it badly converges FOPT generally does not satisfy the RGE and uses a Taylor expansion in a region where it badly converges It is due to the properties of the kernel that we do not get higher differences between FOPT and CIPT It is due to the properties of the kernel that we do not get higher differences between FOPT and CIPT CIPT avoids these problems and is to be preferred * CIPT avoids these problems and is to be preferred * *See however a recent study of the role of higher order contributions (with different approach and conclusions): M.Beneke and M. Jamin (arXiv: ) Conclusions of our tests:

M. Davier alpha_s tauTau08 Workshop Novosibirsk15 Impact of Quark-Hadron Duality Violation Two models to simulate the contribution of duality violating terms (M.A.Shifman hep-ph/ ): instantons; resonances. This contribution is added to the theoretical computation, and the parameters of the models are chosen to match smoothly the V+A spectral function, near s=m  2. Contributions to δ (0) (~0.2): instantons: < 4.5 · resonances: < 7 · Those contributions are within our systematic uncertainties. This problem has also been considered by O. Cata, et al. arxiv: Q-H Duality Violation: OPE only part of the non-perturbative contributions, non-perturbative oscillating terms missed...

M. Davier alpha_s tauTau08 Workshop Novosibirsk16 Spectral Moments Exploit shape of spectral functions to obtain additional experimental information: Le Diberder-Pich, PL B289, 165 (1992) The region where OPE fails and we have small statistics is suppressed. with corresponding perturbative and nonperturbative OPE terms Theory prediction very similar to R  : Because of the strong correlations, only four moments are used. We fit simultaneously and the leading D=4,6,8 nonperturbative contributions

M. Davier alpha_s tauTau08 Workshop Novosibirsk17 Aleph Fit Results The combined fit of R  and spectral moments (k=1, =0,1,2,3) gives (at s 0 =m  2 ): Theory framework: tests  CIPT method preferred, no CIPT-vs.-FOPT syst. The fit to the V+A data yields: Using 4-loop QCD  -function and 3-loop quark-flavour matching yields:

M. Davier alpha_s tauTau08 Workshop Novosibirsk18 Overall comparison Tau provides: - among most precise  s (M Z 2 ) determinations; - with  s (M Z 2 ) Z, the most precise test of asymptotic freedom (1.8-91GeV) tau result QCD Z result

M. Davier alpha_s tauTau08 Workshop Novosibirsk19 Conclusions Detailed studies of perturbative series: CIPT is to be preferred Contributions coming from duality violation are within systematic uncertainties  s (m  2 ), extrapolated at M Z scale, is among the most accurate values of  s (M Z 2 )  s (m  2 ) and  s (M Z 2 ) from Z decays provide the most precise test of asymptotic freedom in QCD with a precision of 2.4%

M. Davier alpha_s tauTau08 Workshop Novosibirsk20 backup

M. Davier alpha_s tauTau08 Workshop Novosibirsk21 Tau Hadronic Spectral Functions  Hadronic physics factorizes in (vector and axial-vector) Spectral Functions : branching fractions mass spectrum kinematic factor Fundamental ingredient relating long distance hadrons to short distance quarks (QCD) Optical Theorem: neglecting QCD and EW corrections

M. Davier alpha_s tauTau08 Workshop Novosibirsk22 Of purely nonperturbative origin

M. Davier alpha_s tauTau08 Workshop Novosibirsk23 Fit details Although  (0) is the main contribution, and the one that provides the sensitivity to  s, we must not forget the other terms in the OPE (i.e. Quark- Mass and Nonperturbative Contributions): D=2 (mass dimension): quark-mass terms are  m q 2 /s 0, which is negligible for q=u,d D=4: dominant contributions from gluon- and quark-field condensations (gluon condensate  a s GG  is determined from data) D=6: dominated by large number of four-quark dynamical operators that  assuming factorization (vacuum saturation)  can be reduced to an effective scale-independent operator  a s  qq -bar  2 that is determined from data D=8: structure of quark-quark, quark-gluon and four-gluon condensates absorbed in single phenomenological operator  O 8  determined from data For practical reasons it is convenient to normalize the spectral moments: