Precise α s from  Decays(*)

1. Precise αs from  Decays(*) M. Davier, S. Descotes-Genon, A. Hoecker, B. Malaescu, and Z. Zhang Tau08 Workshop Novosibirsk, Sept. 22-25 2008 (*) arxiv:0803.0979; published in EPJ C Rev. Mod. Phys. 78 (2006) 1043

2. Outline • Motivation: • final ALEPH  spectral functions with improvements of branching ratios • (BABAR and Belle) • Better knowledge of perturbative series (P. Baikov, et al.) • 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 t spectral functions (ALEPH, OPAL) and phenomenological framework (Braaten-Narison-Pich, LeDiberder,…) Tau08 Workshop Novosibirsk

3. Experimental Input: Branching Fractions • From measured leptonic branching ratios: • Vector, Axial-Vector and Strange contributions : (new) (incl. new results from BABAR+Belle) Tau08 Workshop Novosibirsk

4. 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  fA=0.833±0.024 - modes: fA=0.5±0.5 ALEPH(V+A) BABAR+CVC (V) Tau08 Workshop Novosibirsk

5. ALEPH Spectral Functions + BR Updates Tau08 Workshop Novosibirsk Of purely nonperturbative origin

6. Theoretical Prediction of R • Optical Theorem: • 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: Potential problems for OPE spectral function Im(s) |s|= Re(s) |s|=s0 Tau08 Workshop Novosibirsk

7.  and QCD: The Operator Product Expansion Perturbative quark-mass terms: • Full theoretical ansatz, including nonperturbative operators via the OPE: (in the following: as = s/) EW correction: Perturbative contribution Adler function to avoid unphysical subtractions: Nonperturbative contribution Tau08 Workshop Novosibirsk

8. The Perturbative Prediction • Perturbative prediction of Adler function given to N3LO Perturbative coefficients of Adler function series, known to n=4 (K4 ≈ 49) P. Baikov, et al., arxiv:0801.1821[hep-ph] Im(s) • s dependence of as driven by running: RGE -function, known to n=3 Re(s) In practice, use Taylor development in |s|=ζs0 φ |s|=s0 • How to compute the integral in the complex plane? • How to perform a scale transformation? Tau08 Workshop Novosibirsk

9. 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 as(s0) Im(s) CIPT Re(s) FOPT |s|=s0 • 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! Tau08 Workshop Novosibirsk

10. RGE Order β0 CIPT FOPT Tau08 Workshop Novosibirsk

11. RGE Order β1,2 Not exact solution, expansion! Tau08 Workshop Novosibirsk

12. Tests of Integration Methods • FOPT+: the same Taylor expansion for αs(s) as FOPT, with no cut of the integration result Tau08 Workshop Novosibirsk

13. 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. Tau08 Workshop Novosibirsk

14. Integration Methods: Scale dependence Conclusions of our tests: • FOPT neglects part of contributions to the perturbative series included in CIPT • 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 • 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:0806.3156) Tau08 Workshop Novosibirsk

15. Impact of Quark-Hadron Duality Violation Q-H Duality Violation: OPE only part of the non-perturbative contributions, non-perturbative oscillating terms missed... Two models to simulate the contribution of duality violating terms (M.A.Shifman hep-ph/0009131): • 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 · 10-3 • resonances: < 7 · 10-4 Those contributions are within our systematic uncertainties. This problem has also been considered by O. Cata, et al. arxiv: 0803.0246 Tau08 Workshop Novosibirsk

16. 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. • Theory prediction very similar to R: with corresponding perturbative and nonperturbative OPE terms • Because of the strong correlations, only four moments are used. • We fit simultaneously and the leading D=4,6,8 nonperturbative contributions Tau08 Workshop Novosibirsk

17. Aleph Fit Results • The combined fit of R and spectral moments (k=1, =0,1,2,3) gives (at s0=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: Tau08 Workshop Novosibirsk

18. Tau provides: - among most precise s(MZ2) determinations; - with s(MZ2)Z,the most precise test of asymptotic freedom (1.8-91GeV) Overall comparison tau result QCD Z result Tau08 Workshop Novosibirsk

19. 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 MZ scale, is among the most accurate values of s(MZ2) • s(m2) and s(MZ2) from Z decays provide the most precise test of asymptotic freedom in QCD with a precision of 2.4% Tau08 Workshop Novosibirsk

20. backup Tau08 Workshop Novosibirsk

21. Tau Hadronic Spectral Functions neglecting QCD and EW corrections • Hadronic physics factorizes in(vector and axial-vector) Spectral Functions : branching fractionsmass spectrum kinematic factor Fundamental ingredient relating long distance hadrons to short distance quarks (QCD) • Optical Theorem: Tau08 Workshop Novosibirsk

22. Tau08 Workshop Novosibirsk Of purely nonperturbative origin

23. 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 mq2/s0, which is negligible for q=u,d • D=4: dominant contributions from gluon- and quark-field condensations (gluon condensate asGG 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 as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 O8 determined from data • For practical reasons it is convenient to normalize the spectral moments: Tau08 Workshop Novosibirsk