1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N...

22
α s from τ decays and e + e - annihilation PEP-N Workshop 30. April 2001, SLAC Sven Menke, SLAC R τ and the strong coupling - spectral functions - perturbative and non-perturbative QCD - the role of moments - QCD fit results - hypothetical τ decays R e + e - and the strong coupling - combination of different experiments - R e + e - including new data up to 4.5 GeV 2 - why not use moments too? - preliminary QCD fit results Conclusion

Transcript of 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N...

Page 1: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 2: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 3: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 4: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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.

Page 5: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

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

Page 6: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 7: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 8: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 9: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

]

Page 10: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

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

Page 11: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

]

Page 12: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 13: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 14: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 15: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 16: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 17: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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)

Page 18: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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 %

Page 19: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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.

Page 20: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 21: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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

Page 22: 1s from 1 decays and 1e+e- annihilationmenke/pepn01.pdf · from˝ decayss and e+annihilatione PEP-N Workshop30. April 2001, SLACSven Menke, SLAC R ˝and the strong couplingspectral

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