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
 Impact of Quark-Hadron Duality Violation
 Spectral Moments and Fit Results
 Test of Asymptotic Freedom
 Conclusions

Tau Hadronic Spectral Functions
R 

    hadrons  

   e  e 



 N
C
V
2
ud
2
 Vus

3
neglecting QCD and
EW corrections
 Hadronic physics factorizes in (vector and axial-vector) Spectral Functions :
1 / a1   V / A    



BR    V  / A  
BR    e  e  
branching fractions
1 dNV / A
NV / A ds
mass spectrum

m2
1  s / m2
 1 s / m 
2
kinematic factor
Fundamental ingredient relating long distance hadrons to short distance quarks (QCD)
•Optical Theorem:
Im 
(1,0)
ud ( s ),V / A
1

v1 a1,0 ( s)
2
2

Currents Separation
R  R ,V  R , A  R ,S
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
 KK modes are generally not eigenstates of G-parity :
- K  K 0 is pure vector
- KK  BABAR: fA=0.833±0.024
- KK  rarer modes: fA=0.5±0.5

ALEPH(V+A)
BABAR+CVC (V)
Experimental Measurements
•From measured leptonic branching ratios:
R 
1  Be  B
Be
1
 uni  1.9726  3.640  0.010
Be
g e  g   g
•Vector, Axial-Vector (S  0) and Strange contributions (S  1):
R  R ,V  R , A  R , S
R ,V  1.783  0.011exp  0.002V / A
R , A  1.695  0.011exp  0.002V / A
R , S  0.1615  0.0040
(incl. new results from BABAR+Belle)
Of purely nonperturbative origin
Theoretical Prediction of R
ds 
s 
R ,U (s0 )  12 SEW 
1  
s
s0 
0 
0
s0


2


s 
(1)
(0)
 1  2  Im U (s  i  )  Im U (s  i  )
s0 


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:
s0
R(s0 )   ds w (s ) Im (s  i  ) 

0
1
ds w (s ) (s )
2i |s|s0
spectral
function
Im(s)
|s| = 
Re(s)
|s| = s0
Potential problems
for OPE
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:
0.0010 (neglected)
R ,U (s0 )  VCKM
2

i , j u ,d ,s
CU ,ij (s0 )

  U(2,mq ) 
SEW  1   (0)  EW


mi (s0 )  m j (s0 )
s0

D  4,6,...
(D )
U



Perturbative contribution
3
4





ds
s
s
s 
 1  2  2        D(s )
2 i 
s 
s0
 s0   s0  
|S|s0 0 
1 
n
where: D(s ) 
K
(

)
a
(  s )

n
s
2
4 n 0
Adler function to avoid
unphysical subtractions:
D(s )  s

(D )
U
d (s )
ds
 ( 1)
D

dim O D
Nonperturbative
contribution
CU (  )
O(  )
D /2
0
s
U
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?
 (0)   K n ( ) A( n ) (as )
n
Perturbative coefficients of
Adler function series, known to
n=4 (K4 ≈ 49)
P. Baikov, et al.,
arxiv:0801.1821[hep-ph]
1
2

 d 1 2e

das
2
  (as )  as  n asn
d ln s
n 0
as (s )  as (s0 )  0as2 (s0 )  ...

 2ei 3  ei 4  asn ( s0ei )

•Complex s dependence of as driven by running:
In practice, use Taylor development in   ln  s s0 
i
RGE -function, known to n=3
Integration Methods




CIPT: at each integration step use Taylor series to compute as ( s)from the
value found at the previous step
FOPT: 6th order Taylor expansion around the physical value as ( s0 )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
Integration Methods: Tests
Integration Methods: Tests
(0)
Massless perturbative contribution  computed for s (m2 )  0.34 with K5,6 and  4
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
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
Spectral Moments

Exploit shape of spectral functions to obtain additional experimental
information:
k

s   s  dR ,U (s0 )
k
R ,U (s0 )   ds  1    
ds
 s0   s0 
0
s0
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:


2
  U(2,mq ,k )   U( D,k ) 
Rk,U (s0 )  VCKM SEW  1   (0,k )  EW
D  4,6,...


with corresponding perturbative and nonperturbative OPE terms

Because of the strong correlations, only four moments are used.

We fit simultaneously
contributions
 s (m2 )
and the leading D=4,6,8 nonperturbative
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:
s (m2 )  0.344  0.005exp  0.007theo

Using 4-loop QCD  -function and 3-loop quark-flavour matching yields:
s (MZ2 )  0.1212  0.0005exp  0.008theo  0.0005evol
Overall comparison
Tau provides:
tau result
- among most precise
s(MZ2) determinations;
- with s(MZ2)Z, the
most precise test of
asymptotic freedom
(1.8-91GeV)
QCD
 s (mZ2 )Z  0.1191  0.0027fit  0.0001trunc
 s (MZ2 )  0.1212  0.0011
 s (MZ2 )   s (mZ2 )Z  0.0021  0.0029
Z result
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%
backup
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. QuarkMass 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:
Dk,U (s0 ) 
Rk,U (s0 )
R ,U (s0 )
Spectral Functions:Details
2
G
2


F
d (  hadrons   ) 
VCKM L H  dPS
4M