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:

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 fractions mass spectrum kinematic factor
Fundamental ingredient relating long distance hadrons to short distance quarks (QCD)
Optical Theorem:

4. Separation of V and A components:
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
Full theoretical ansatz, including nonperturbative operators via the OPE: (in the following: as = s/ )
Perturbative quark-mass terms:
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: [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/ ):
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:

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. Overall comparison Tau provides:
- among most precise s(MZ2) determinations;
- with s(MZ2)Z, the most precise test of asymptotic freedom (1.8-91GeV)
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%

18. backup

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