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Precise α s 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

Precise α s from Decays(*)

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Precise α s 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. Outline. Motivation: - PowerPoint PPT Presentation

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Page 1: Precise  α s  from    Decays(*)

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

Page 2: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 2

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.)

Page 3: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 3

Experimental Input: Branching Fractions

, , ,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)

(new)

Page 4: Precise  α s  from    Decays(*)

M. Davier alpha_s tau 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

KK0K K

KK

KK

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

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

Page 5: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 5

Of purely nonperturbative originOf purely nonperturbative origin

ALEPH Spectral Functions + BR Updates

Page 6: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 6

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:

0

(1) (0)

2

, 00 0 0

W

0

E( ) 12 1 1 2 ( )Im Im ( )U

s

U U

ds s sR s s i s i

s s sS

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

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

1Im ( )

2ud s V A v a s

• Optical Theorem:

Page 7: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 7

and QCD: The Operator Product Expansion

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

0

2 (2, ) ( ), CKM EW EW

4,6,..

0)

.

(( ) 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:

( )0 / 2

dim 0

( )( ; )D U

U U DO D

OC s

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

Page 8: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 8

The 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

0ln s s

• s dependence of as driven by running:

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

• How to compute the integral in the complex plane?• How to perform a scale transformation?

Re(s)

Im(s)

|s| = s0

|s| =ζ s0

φ

Page 9: Precise  α s  from    Decays(*)

M. Davier alpha_s tau 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)

( )sa s

0( )sa s

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

FOPT

CIPT

Page 10: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 10

RGE Order β0

0

0 00

2

0 0 00

( )( )

1 ( ) l

( ) ( )

n

lnans

s

s

ss

sa s a s

s

a sa s

sa s

s

CIPT FOPT

Page 11: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 11

RGE Order β1,2

0 0

( )( )0 0 11

20 0 0 0 1 0( ) ( )

( ) ( )1ln ln ( )

( ) ( )

ss

s s

a sa ss s an

ss sa s a s

a s a sda ss

a a a s a s s

212

200 2 2 2

ln ln1 1

( ) 1ln ln ln

PDGs

s

a s Os s s

Not exact solution,expansion!

Page 12: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 12

Tests of Integration Methods 0

s D

•FOPT+: the same Taylor expansion for αs(s) as FOPT, with no cut of the integration result

0

3 4

(0)

0 0 0 0| |

6

20

1 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

Page 13: Precise  α s  from    Decays(*)

M. Davier alpha_s tau 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.

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

Page 14: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 14

Integration Methods: Scale dependence

FOPT neglects part of contributions to the perturbative series included in CIPTFOPT neglects part of contributions to the perturbative series included in CIPT FOPT generally does not satisfy the RGEFOPT generally does not satisfy the RGE and uses a Taylor expansion in a uses a Taylor expansion in a

region where it badly convergesregion where it badly converges It is due to the properties of the kernel that we do not get higher differences It is due to the properties of the kernel that we do not get higher differences

between FOPT and CIPTbetween 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:0806.3156)

Conclusions of our tests:

Page 15: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 15

Impact of Quark-Hadron Duality Violation

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

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

Page 16: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 16

Spectral 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 EW4,6,...

( ) P qm kk k k 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

Page 17: Precise  α s  from    Decays(*)

M. Davier alpha_s tau 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:

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 0.0008 0.0005s ZM

Page 18: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 18

Overall 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

Page 19: Precise  α s  from    Decays(*)

M. Davier alpha_s tau 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(m2), extrapolated at MZ scale, is among the

most accurate values of s(MZ2)

s(m2) and s(MZ

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

Page 20: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 20

backup

Page 21: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 21

Tau Hadronic Spectral Functions

2

1 1 22 2

/

/

/ /1 / 1

BR /

BR

1

/ V A

V Ae

V A

e

dN

N

ma V A

s m s mds

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)

(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

22

( )4

FCKM

Gd hadrons V L H dPS

M

Page 22: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 22

Of purely nonperturbative originOf purely nonperturbative origin

Page 23: Precise  α s  from    Decays(*)

M. Davier alpha_s tau Tau08 Workshop Novosibirsk 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 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