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Improved alpha_s from Tau Decays(*)

Improved alpha_s from Tau Decays(*). M. Davier, S. Descotes-Genon, A. Hoecker, B. Malaescu , and Z. Zhang. Rencontres de Moriond, QCD and High Energy Interactions, March 2008. (*) arxiv:0803.0979. Outline. Tau Hadronic Spectral Functions Theoretical Framework Tests of Integration Methods

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Improved alpha_s from Tau Decays(*)

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  1. Improved alpha_s from Tau Decays(*) M. Davier, S. Descotes-Genon, A. Hoecker, B. Malaescu, and Z. Zhang Rencontres de Moriond, QCD and High Energy Interactions, March 2008 (*) arxiv:0803.0979

  2. Outline • Tau Hadronic Spectral Functions • Theoretical Framework • Tests of Integration Methods • Impact of Quark-Hadron Duality Violation • Spectral Moments and Fit Results • Test of Asymptotic Freedom • Conclusions

  3. 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:

  4. Currents Separation Separation of V and A components: • 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 are generally not eigenstates of G-parity : - is pure vector - BABAR: fA=0.833±0.024 - rarer modes: fA=0.5±0.5 ALEPH(V+A) BABAR+CVC (V)

  5. Experimental Measurements • From measured leptonic branching ratios: • Vector, Axial-Vector and Strange contributions : (incl. new results from BABAR+Belle)

  6. Of purely nonperturbative origin

  7. Theoretical Prediction of • Problem:ImV/A(J)(s) contains hadronic physics that cannot be predicted in 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

  8. Tau 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

  9. The Perturbative Prediction • Perturbative prediction of Adler function given to N3LO, but how should one • best compute the contour integral A(n)(as) occurring in the prediction of R? Perturbative coefficients of Adler function series, known to n=4 (K4 ≈ 49) P. Baikov, et al., arxiv:0801.1821[hep-ph] • Complex s dependence of as driven by running: RGE -function, known to n=3 In practice, use Taylor development in

  10. Integration Methods • CIPT: at each integration step use Taylor series to compute from the value found at the previous step • FOPT: 6th order Taylor expansion around the physical value and the integration result is also cut at the 6th order • FOPT+: same Taylor expansion with no cut of the integration result • FOPT++: more complete RGE solution and no cut of the integration result • Remarks: • Potential problem for FOPT due to the finite convergence radius of Taylor series • - Avoided by CIPT (use small steps) Im(s) FOPT CIPT Re(s) |s|=s0

  11. Integration Methods: Tests

  12. Integration Methods: Tests Massless perturbative contribution computed for with and estimated by assuming geometric growth. Remaining unknown coefficients were set to zero. • FOPT neglects important contributions to the perturbative series • FOPT uses Taylor expansion in a region where it badly (or does not) converge • It is due to the properties of the kernel that we don’t get higher differences between FOPT and CIPT • CIPT avoids many problems and is to be prefered

  13. 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. Results (contributions to δ(0)): • instantons: < 4.5 · 10-3 • resonances: < 7 · 10-4 Those contributions are within our systematic uncertainties. This problem has also been considered very recently by O. Cata, et al. arxiv: 0803.0246

  14. 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

  15. 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:

  16. 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

  17. Conclusions • Detailed studies of perturbative series: CIPT is to be prefered • Contributions coming from duality violation are within systematic uncertainties • s(m2), extrapolated at MZ scale, is among most precise values of s(MZ2) • s(m2) and s(MZ2) from Z decays provide the most precise test of asymptotic freedom in QCD with an unprecedented precision of 2.4%

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  19. 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:

  20. Spectral Functions:Details

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