Improved alpha_s from Tau Improved alpha_s from Tau Decays(*)Decays(*)

M. Davier, S. Descotes-Genon, A. Hoecker, M. Davier, S. Descotes-Genon, A. Hoecker, B. MalaescuB. Malaescu, and Z. Zhang, and Z. Zhang

Rencontres de Moriond, QCD and High Energy Interactions, March 2008

(*) arxiv:0803.0979

OutlineOutline

Tau Hadronic Spectral FunctionsTau Hadronic Spectral Functions Theoretical FrameworkTheoretical Framework Tests of Integration MethodsTests of Integration Methods Impact of Quark-Hadron Duality ViolationImpact of Quark-Hadron Duality Violation Spectral Moments and Fit ResultsSpectral Moments and Fit Results Test of Asymptotic FreedomTest of Asymptotic Freedom ConclusionsConclusions

Tau Hadronic Spectral FunctionsTau Hadronic Spectral Functions

Hadronic physics factorizes in (vector and axial-vector) Spectral Functions :

2

1 1 22 2

/

/

/ /1 / 1

BR / 1

B

R

/

V A

V Ae

V A

e

dN

N

ma V A

s m s mds

branching fractions mass spectrum kinematic factor

Fundamental ingredient relating long distance hadrons to short distance quarks (QCD)

(1,0)( ), / 1 1,0

1Im ( )

2ud s V A v a s

2 2hadrons

3C ud us

e

R N V Ve

•Optical Theorem:

neglecting QCD and EW corrections

Currents SeparationCurrents Separation

Straightforward for final states with only pions (using G-parity) : Straightforward for final states with only pions (using G-parity) :

- even number of pions ( G = 1 ): vector state- even number of pions ( G = 1 ): vector state

- odd number of pions ( G = -1 ): axial-vector state- odd number of pions ( G = -1 ): axial-vector state modes are generally not eigenstates of G-parity :modes are generally not eigenstates of G-parity :

- is pure vector- is pure vector

- BABAR: f- BABAR: fAA=0.833=0.833±±0.0240.024

- rarer modes: f- rarer modes: fAA=0.5=0.5±±0.50.5

, , ,SV AR RR R

Separation of V and A components:Separation of V and A components:

KK0K K

KK

KK

ALEPH(V+A)ALEPH(V+A)

BABAR+CVC (V)BABAR+CVC (V)

Experimental MeasurementsExperimental Measurements

, , ,V A SR R R R

1 11.9726 3.640 0.010e

unie e

B BR

B B

•From measured leptonic branching ratios:

, exp /

, exp /

,

1.783 0.011 0.002

1.695 0.011 0.002

0.1615 0.0040

V V A

A V A

S

R

R

R

•Vector, Axial-Vector and Strange contributions :( 1)S ( 0)S

eg g g

(incl. new results from BABAR+Belle)

Of purely nonperturbative originOf purely nonperturbative origin

Theoretical Prediction ofTheoretical 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:

R

0

(1) (0)

2

,0 0 0

EW

0

0( ) 12 1 1 2 ( )Im Im ( )U

s

U U

ds s sR s i s i

s s ss S

0

0

0

( ) ( ) Im ( )s

R s ds w s s i

0| |

1 ( ) ( )

2 s s

ds w s si

spectral function

Re(s)

Im(s)

|s| =

|s| = s0

Potential problems for OPE

Tau and QCD: The Operator Product ExpansionTau and QCD: The Operator Product Expansion

Full theoretical ansatz, including nonperturbative operators via the OPE: (in the following: as = s/ )

0

2 (2, )(0) ( ), CKM EW EW

4,6,...

( ) 1 qm DU U U

D

sR V S

0

3 4

0 0 0 0| |

20

2 1 2 2 ( )

1where: ( ) ( ) ( )

4

S s

nn s

n

ds s s si D s

s s s s

D s K a s

Perturbative contribution

( )( )

d sD s s

ds

Adler function to avoid unphysical subtractions:

( )/2

dim 0

( )( 1) ( )D D U

U U DO D

OC

s

Nonperturbative contribution

0.0010 (neglected)

EW correction:

Perturbative quark-mass terms:

0 0

, 0, , , 0

( ) ( )( ) i j

U iji j u d s

m s m sC s

s

The Perturbative PredictionThe Perturbative Prediction

(0) ( ) ( ) ( )nn s

n

K A a

Perturbative coefficients of Adler function series, known to n=4 (K4 ≈ 49)

3 40

11 2 2 ( )

2i i i n i

sd e e e a s e

2

0

( ) ln

nss s n s

n

daa a a

d s

RGE -function, known to n=3

20 0 0( ) ( ) ( ) ...s s sa s a s a s

In practice, use Taylor development in

•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?

0ln s s

•Complex s dependence of as driven by running:

P. Baikov, et al., arxiv:0801.1821[hep-ph]

Integration MethodsIntegration Methods CIPT: at each integration step use Taylor series to compute from the CIPT: at each integration step use Taylor series to compute from the

value found at the previous stepvalue found at the previous step FOPT: 6FOPT: 6thth order Taylor expansion around the physical value and the order Taylor expansion around the physical value and the

integration result is also cut at the 6integration result is also cut at the 6thth order order FOPT+: same Taylor expansion with no cut of the integration resultFOPT+: same Taylor expansion with no cut of the integration result FOPT++: more complete RGE solution and no cut of the integration resultFOPT++: more complete RGE solution and no cut of the integration result

( )sa s

0( )sa s

Remarks:- Potential problem for FOPT due to the finite convergence radius of Taylor series- Avoided by CIPT (use small steps)

Re(s)

Im(s)

|s| = s0

FOPT

CIPT

Integration Methods: TestsIntegration Methods: Tests

Integration Methods: TestsIntegration Methods: Tests

Massless perturbative contribution computed for with and estimated by assuming geometric growth. Remaining unknown coefficients were set to zero.

(0) 2( ) 0.34s m 5,6K 4

FOPT neglects important contributions to the perturbative seriesFOPT neglects important contributions to the perturbative series FOPT uses Taylor expansion in a region where it badly (or does not) FOPT uses Taylor expansion in a region where it badly (or does not)

convergeconverge It is due to the properties of the kernel that we don’t get higher differences It is due to the properties of the kernel that we don’t get higher differences

between FOPT and CIPTbetween FOPT and CIPT CIPT avoids many problems and is to be prefered CIPT avoids many problems and is to be prefered

Impact of Quark-Hadron Duality Impact of Quark-Hadron Duality ViolationViolation

Two models to simulate the contribution of duality violating terms Two models to simulate the contribution of duality violating terms (M.A.Shifman hep-ph/0009131):(M.A.Shifman hep-ph/0009131):

instantons;instantons; resonances.resonances.

This contribution is added to the theoretical computation, and the This contribution is added to the theoretical computation, and the parameters of the models are chosen to match smoothly the V+A parameters of the models are chosen to match smoothly the V+A spectral function, near s=mspectral function, near s=m

22..

Results (contributions to Results (contributions to δδ(0)(0)):): instantons: instantons: < 4.5 · 10< 4.5 · 10-3-3

resonances: resonances: < 7 · 10< 7 · 10-4-4

Those contributions are within our systematic uncertainties.Those contributions are within our systematic uncertainties.

This problem has also been considered very recently by O. Cata, et al. This problem has also been considered very recently by O. Cata, et al. arxiv: 0803.0246 arxiv: 0803.0246

Q-H Duality Violation: OPE only part of the non-perturbative Q-H Duality Violation: OPE only part of the non-perturbative contributions, non-perturbative oscillating terms missed...contributions, non-perturbative oscillating terms missed...

Spectral MomentsSpectral Moments Exploit shape of spectral functions to obtain additional experimental information:

0

, 0, 0

0 00

( )( ) 1

ksUk

U

dR ss sR s ds

s s ds

Le Diberder-Pich, PL B289, 165 (1992)

The region where OPE fails and we have small statistics is suppressed.

2 (2, , )(0, ) ( , ), 0 CKM EW EW

4,6,...

( ) 1 qm kk k D kU U U

D

R s V S

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

2( )s m

Aleph Fit ResultsAleph 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:

2exp theo( ) 0.344 0.005 0.007s m

Using 4-loop QCD -function and 3-loop quark-flavour matching yields:

2exp theo evol( ) 0.1212 0.0005 0.008 0.0005s ZM

Overall comparisonOverall comparisonTau provides:

- among most precise s(MZ

2) determinations;

- with s(MZ2)Z, the most

precise test of asymptotic freedom (1.8-91GeV)

2

2

2 2

( ) 0.1191 0.0027 0.0001

( ) 0.1212 0.0011

( ) ( ) 0.0021 0.0029

s Z Z fit trunc

s Z

s Z s Z Z

m

M

M m

tau result

QCD

Z result

ConclusionsConclusions

Detailed studies of perturbative series: CIPT is Detailed studies of perturbative series: CIPT is to be prefered to be prefered

Contributions coming from duality violation are Contributions coming from duality violation are within systematic uncertaintieswithin systematic uncertainties

s(m2), extrapolated at MZ scale, is among most

precise values of s(MZ2)

s(m2) and s(MZ

2) from Z decays provide the most precise test of asymptotic freedom in QCD with an unprecedented precision of 2.4%

backupbackup

Fit detailsFit 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-Quark-

Mass and Nonperturbative ContributionsMass 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 asqq-bar2 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:

, 0, 0

, 0

( )( )

( )

kUk

UU

R sD s

R s

Spectral Functions:DetailsSpectral Functions:Details

22

( )4

FCKM

Gd hadrons V L H dPS

M