1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione pep-n...
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αs
from τ decays
and e+e− annihilationPEP-N Workshop 30. April 2001, SLAC Sven Menke, SLAC
• Rτ and the strong coupling− spectral functions− perturbative and non-perturbative QCD− the role of moments− QCD fit results− hypothetical τ decays
• Re+e− and the strong coupling− combination of different experiments− Re+e− including new data up to 4.5 GeV2
− why not use moments too?− preliminary QCD fit results
• Conclusion
The Hadronic Decay Ratio of the τ
Rτ =Γ(τ → ντ hadrons
)Γ (τ → ντ e νe)
• Tree: Rτ = NC (|Vud|2 + |Vus|2)
= 3
• Exp.: Rτ = 3.65
dRτ / ds
s (GeV2)
without QCD
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0 0.5 1 1.5 2 2.5 3
dRτ / ds
s (GeV2)
with QCD
0
0.025
0.05
0.075
0.1
0.125
0.15
0.175
0.2
0.225
0 0.5 1 1.5 2 2.5 3
Rτ and the Spectral FunctionsOptical theorem
Γs
d
dΠIm
hadro
ns
hadrons-W-
W W
~
~
2
Rτ = 12π∫ m2
τ
0ds poly(s) ImΠ(s)
Cauchy’s theorem
Im(s)
Re(s)| | τ
2s = m
Rτ = 6πi∮|s|=m2
τ
ds poly(s) Π(s)
Spectral Functions
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3s (GeV2)
v(s)
OPAL
π π0
3π π0, π 3π0
MC corr.
perturbative QCD (massless)
naïve parton model
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3s (GeV2)
a(s)
OPAL
3π, π 2π0
3π 2π0
MC corr.
perturbative QCD (massless)
naïve parton model
v(s) ∼ ImΠ(1)V a(s) ∼ ImΠ(1)
A
v/a(s)∼[(
1−s
m2τ
)2(1+
2s
m2τ
)]−1∑h
V/A
B(τ→hV/A)
B(τ → e)
(1
N
dN
ds
)V/A
OPAL Collaboration, K. Ackerstaff et al., Eur. Phys. J. C7 (1999) 571.
Theoretical Description of Rτ I
Rτ,V/A =3
2|Vud|2SEW
(1 + δpert + δ
V/Amass + δ
V/A
non-pert
)• perturbative part
1+δpert =1
2πi
∮|s|=m2
τ
ds
s
(1−2
s
m2τ
+2s3
m6τ
−s4
m8τ
)(−s)
dΠ
ds︸ ︷︷ ︸D(s)
CIPT: D(s) ∼1+αs(−s)
π+1.64
α2s (−s)
π2+6.37
α3s (−s)
π3
FOPT: 1+δpert =1+αs(m2
τ )
π+5.20
α2s (m2
τ )
π2+26.4
α3s (m2
τ )
π3
RCPT: D(s) ∼1+
∞∑n=1
κnβn−10
αns (−s)
πn
Theoretical Description of Rτ II
Rτ,V/A =3
2|Vud|2SEW
(1 + δpert + δ
V/Amass + δ
V/A
non-pert
)• power corrections
OPE: δV/A
mass ∼m2
q
m2τ
δV/A
non-pert∼CV/A4
〈O〉D=4
m4τ
+CV/A6
〈O〉D=6
m6τ︸ ︷︷ ︸+CV/A
8
〈O〉D=8
m8τ︸ ︷︷ ︸
〈αsGG〉, . . . δ6V/A δ8
V/A
need several observables with different s dependence
The Definition of Moments of Rτ
Rklτ ,V/A =
∫ m2τ
0ds
(1−
s
m2τ
)k( s
m2τ
)l dRτ,V/A
ds
Cauchy Integral theorem∮ds
sn= 0, for n 6= 1
δD,kl
V/A= 8π2
∑dim O= D
CV/AD 〈O〉
mDτ
D = 2 D = 4 D = 6 D = 8 kl
1
1
0
0
0
0
1
−1
0
0
−3
−3
−1
1
0
−2
−5
3
1
−1
00
10
11
12
13
Comparison of different fits to the τ data (CIPT)
V A V and A
αs(m2τ ) 0.341± 0.017 0.357± 0.019 0.347± 0.012
〈αs GG〉GeV4 0.002± 0.010 −0.011± 0.020 0.001± 0.008
δ6V 0.0259± 0.0041 — 0.0256± 0.0034
δ8V −0.0078± 0.0018 — −0.0080± 0.0013
δ6A — −0.0246± 0.0086 −0.0197± 0.0033
δ8A — 0.0067± 0.0050 0.0041± 0.0019
χ2/d.o.f. 0.07/1 0.06/1 0.63/4
V+A Stat. Br. Syst. Theo. χ2/d.o.f.
αs(m2τ ) 0.348 ±0.002 ±0.009 ±0.002 ±0.019
〈αs GG〉GeV4 −0.003 ±0.007 ±0.007 ±0.006 ±0.005
δ6V+A
0.0012 ±0.0034 ±0.0033 ±0.0029 ±0.00060.16/1
δ8V+A
−0.0010 ±0.0024 ±0.0016 ±0.0015 ±0.0003
Decay Ratio for a hypothetical τ ′
3.25�
3.5�
3.75�
4
1 1.5 2 2.5 3�
s� 0� (GeV2)
�
Rτ'
,V+A
(s0)
OPAL
CIPT
RCPT
FOPT
Unfolded OPAL Data
s� 0� (GeV2)
�
Th
eory
/ D
ata
0.9�
1
1.1
1 1.5 2 2.5 3�
Rτ ′,V+A(s0 = m2
τ ′) ∼
s0∫0
ds
s0
(1−
s
s0
)2[(1 +
2s
s0
)(v(s) + a(s)) + a(0)(s)
]
Running αs
√s (GeV)
α s(√s
)
Rτ (
this
wo
rk) B
jSR
GL
SR
Rτ (
Be,
Bµ,
τ τ)
bb–
th
resh
old
pp
,pp–
→γ+
jets
DIS
(ν� ;
F2
and
F3)
DIS
(µ;
F2)
LG
Tcc
–,b
b– d
ecay
s
Rγ
Jet
rate
s (H
ER
A)
pp–
→b
b−+j
ets
Eve
nt
Sh
apes
(e+ e− )
σ had
(e+ e− )
Eve
nt
Sh
apes
(e+ e− )
Sca
l.Vio
l. (e
+ e− )E
ven
t S
hap
es (
e+ e− )E
ven
t S
hap
es (
e+ e− )
pp–
→W
+jet
sR
l (g
lob
al S
M F
it)
Eve
nt
Sh
apes
(e+ e− )
Eve
nt
Sh
apes
(e+ e− )
Eve
nt
Sh
apes
(e+ e− )
0�
0.1�
0.2�
0.3�
0.4�
0.5�
0.6�
1 10 102
3n
f=
4n
f=
5nf =m m mτ Zb
αs(mZ) exp. theo. evol.
CIPT: 0.1219 ±0.0010 ±0.0017 ±0.0003
FOPT: 0.1191 ±0.0008 ±0.0013 ±0.0003
RCPT: 0.1169 ±0.0007 ±0.0015 ±0.0003
Re+e− and αs
√s in GeV
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8 9 10
Bacci et al.Cosme et al.Mark IPlutoCornell,DORISCrystal BallMD-1 VEPP-4VEPP-2M NDDM2
Rhad
ρ,ω,φ Ψ's Υ's
15 % 15 % 6 %�
3 %�
relative error in �
continuum�
Burkhardt, Pietrzyk '95
Re+e−(s) = 12π ImΠγ(s) = NC
∑f
Q2f
[1+
αs(s)
π+d2
α2s (s)
π2+d3
α3s (s)
π3
]
Averaging of the e+e− data
i i+1 i+2 i+3
Syst.
Stat.
• individual experiments interpolated withtrapezoidal rule
• data points:dk = ck di + (1− ck) di+3, k = i + 1, i + 2
• statistical errors:σk = ck σi + (1− ck)σi+3
• larger than the Gauss errors:σ′k = ck σi ⊕ (1− ck)σi+3
Ratio: rk = σk/σ′k
• Correlation matrix:Vstat
kl = rkrl
(ckclσ
2i + (1− ck)(1− cl)σ2
i+3
)• systematic errors are also interpolated with trapezoidal rule
• correlation is 100 %
• Total Error matrix: V = Vstat + Vsys
• Weighted average over all individual results gives final data points
• Errors scaled with S =√χ2/χ2
68 %if S > 1
Exclusive modes I: 2π, 2π2π0
0�
200
400
600�
800�
1000
1200
1400
0.1�
0.2�
0.3�
0.4�
0.5�
0.6�
0.7�
0.8�
0.9�
1
s (GeV2)
σ(e+ e− →
π+ π− )
(nb)
OLYA
TOF
NA7�
CMD I
CMD II
DM1
DM2
BCF
MEA
0�
5�
10
15
20
25
30�
35�
40
45
50�
0.5�
1 1.5 2 2.5 3�
3.5�
4 4.5
s (GeV2)σ(
e+ e− → π
+ π− π0 π0 ) (n
b)
ND�
M2N
M3N
DM2
OLYA
CMD II
σ(e+e− → π+π−)(s) σ(e+e− → π+π−π0π0)(s)
Exclusive modes II: 4π, 4π2π0
0�
5�
10
15
20
25
30�
35�
40
45
50�
0.5�
1 1.5 2 2.5 3�
3.5�
4 4.5
s (GeV2)
σ(e+ e− →
π+ π− π+ π− )
(nb)
ND�
M3N
MEA
CMD I
DM1
DM2
OLYA
CMD II
0�
2
4
6�
8�
10
12
14
16
18
20
1.5 2 2.5 3�
3.5�
4 4.5 5�
s (GeV2)σ(
e+ e− → π
+ π− π+ π− π0 π0 ) (n
b)
M3N
CMD I
DM2
σ(e+e− → π+π−π+π−)(s) σ(e+e− → π+π−π+π−π0π0)(s)
Exclusive modes III: ωπ0, 6π
0�
2.5
5�
7.5
10
12.5
15
17.5
20
22.5
25
1 1.5 2 2.5 3�
3.5�
4
s (GeV2)
σ(e+ e− →
ωπ0 )
(nb)
ND�
DM2
SND
CMD II
0�
0.5�
1
1.5
2
2.5
3�
3.5�
4
4.5
5�
1.5 2 2.5 3�
3.5�
4 4.5 5�
s (GeV2)σ(
e+ e− → π
+ π− π+ π− π+ π− ) (n
b)
M3N
CMD I
DM1
DM2
σ(e+e− → ωπ0)(s) σ(e+e− → π+π−π+π−π+π−)(s)
Exclusive modes IV: 2ππ0, 2K
0�
2
4
6�
8�
10
12
14
16
18
20
0.5�
1 1.5 2 2.5 3�
3.5�
s (GeV2)
σ(e+ e− →
π+ π− π0 )
(nb)
ND�
M2N
M3N
MEA
DM1
DM2
SND
CMD II
0�
5�
10
15
20
25
30�
35�
40
45
50�
1 1.5 2 2.5 3�
3.5�
4 4.5
s (GeV2)σ(
e+ e− → K
+ K− )
(nb)
MEA
OLYA
BCF
DM1
DM2
σ(e+e− → π+π−π0)(s) σ(e+e− → K+K−)(s)
Exclusive modes V: η2π
0�
0.5�
1
1.5
2
2.5
3�
3.5�
4
4.5
5�
1 1.5 2 2.5 3�
3.5�
4 4.5
s (GeV2)
σ(e+ e− →
ηπ+ π− )
(nb)
ND�
DM2
CMD I
Other exclusive modes without
picture
• 4ππ0
• 3π2π0
• 2π3π0
• K0SK0
L
• 2Kπ0
• 2K2π• K0
SX
σ(e+e− → ηπ+π−)(s)
Re+e− from exclusive modes preliminary
0�
0.5�
1
1.5
2
2.5
3�
3.5�
4
4.5
5�
0.5�
1 1.5 2 2.5 3�
3.5�
4 4.50
�
0.1�
0.2�
0.3�
0.4�
0.5�
0.6�
0.7�
0.8�
0.9�
1
s (GeV2)
R(e
+ e− → h
adro
ns)
Rel
ativ
e E
rrorstat. errors
syst. + stat.PEP-N LOIperturbative QCD (massless)
relative uncertainty
0�
20
40
60�
80�
100
√s (GeV)√s
(G
eV)
0�
0.25�
0.5�
0.75�
1
1.25
1.5
1.75
2
0�
0.25�
0.5�
0.75�
1 1.25 1.5 1.75 2
Re+e−(s) Correlations in %
The Definition of Moments of Re+e−
Rkle+e−(s0) =
∫ s0
0
ds
s0
(1−
s
s0
)k( s
s0
)l
Re+e−(s)
= 6πi∮
|s|=s0
ds
s0
(1−
s
s0
)k( s
s0
)l
Πγ(s)
δD,klγ = 12π2
∑dim O= D
CγD 〈O〉
sD/20
D = 2 D = 4 D = 6 D = 8 kl
1
1
0
0
0
2
3
−1
0
0
1
3
−3
1
0
0
1
−3
3
−1
20
30
31
32
33
inspired by M. Davier and A. Hocker, Phys. Lett. B419 (1998) 419.
Rkl=20,30,31,32,33
e+e− up to 4.5 GeV2 I preliminary
s (GeV2)
Rkl
=20,
30,3
1,32
,33 (e
+ e− → h
adro
ns)
10-2
10-1
0.5�
1 1.5 2 2.5 3�
3.5�
4 4.5
• colored curves denote data andexperimental errors
• black curves show extrapolatedfit results from a fit at s0 = 4 GeV2
with theoretical and experimentalerrors
• theoretical errors (strange quarkmass) dominant
αs(mZ) = 0.118 ± 0.003 ± 0.005
〈αs GG〉 = (0.028 ± 0.006 ± 0.027) GeV4
〈O〉D=6 = −(0.0042± 0.0008± 0.0001) GeV6
〈O〉D=8 = (0.0044 ± 0.0005 ± 0.0006) GeV8
Rkl=20,30,31,32,33
e+e− up to 4.5 GeV2 II preliminary
s (GeV2)
Rkl
=20,
30,3
1,32
,33 (e
+ e− → h
adro
ns)
0�
0.1�
0.2�
0.3�
0.4�
0.5�
0.6�
0.7�
0.8�
0.9�
1
0.5�
1 1.5 2 2.5 3�
3.5�
4 4.5
• black curves show extrapolatedfit results from a fit at s0 = 4 GeV2
with 〈αs GG〉 as free parameter• red curves show extrapolated fit
results from a fit at s0 = 4 GeV2
with ms as free parameter
αs(mZ) = 0.117 ± 0.004 ± 0.002
ms = (0.220 ± 0.036 ± 0.056) GeV
〈O〉D=6 = −(0.0041± 0.0007± 0.0001) GeV6
〈O〉D=8 = (0.0043 ± 0.0005 ± 0.0002) GeV8
Conclusions
• OPE and QCD fits work reliable at low s
• smin0 ≈ 1.5 GeV2 for non-strange τ decays
• smin0 ≈ 1.5− 3 GeV2 for e+e− annihilation (choice of kl)
• experimental error from low s e+e− data on αs(mZ) ≈ 3 %
• expected improvement of experimental errors at PEP-N:
∆αs → 0.93∆αs
∆〈αs GG〉 → 0.72∆〈αs GG〉∆〈O〉D=6 → 0.70∆〈O〉D=6
∆〈O〉D=8 → 0.56∆〈O〉D=8
• sensitivity to non-perturbative QCD parameters could be used toextract ms
fit with ms instead of 〈αs GG〉 as free parameter and〈αs GG〉 = (0.02± 0.01) GeV4
gives ms = (220± 36(26)± 59) MeV