QUARK MASSES AND αs

FROM σ(e+e− → had)

JK, Steinhauser, Sturm NPB

JK, Steinhauser, Teubner PRD

Universitat Karlsruhe (TH)Forschungsuniversitat • gegrundet 1825 C

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Main Idea (SVZ)

2

Data

▲ BES (2001)❍ MD-1▼ CLEO■ BES (2006)pQCD

√s (GeV)

R(s

)

00.5

11.5

22.5

33.5

44.5

5

2 3 4 5 6 7 8 9 10

√s (GeV)

R(s

)

00.5

11.5

22.5

33.5

44.5

5

3.8 4 4.2 4.4 4.6 4.8

√s (GeV)

R(s

)

00.5

11.5

22.5

33.5

44.5

5

3.65 3.675 3.7 3.725 3.75 3.775 3.8 3.825 3.85 3.875 3.9

pQCD and data agree well in the regions

2 – 3.73 GeV and 5 – 10.52 GeV

3

experiment energy [GeV] date systematic error

BES 2 — 5 2001 4%

MD-1 7.2 — 10.34 1996 4%

CLEO 10.52 1998 2%

PDG J/ψ (7 %) 2.5 %

PDG ψ′ (9 %) 2.4 %

PDG ψ′′ (15 %)

BES ψ′′ region 2006 4%

Future improvements:

charm region (CLEO) 3%

bottom region ?? (CLEO)

4

mQfrom

SVZ Sum Rules, Moments and Tadpoles

Some definitions:

R(s) = 12π Im[

Π(q2 = s+ iǫ)]

(

−q2gµν + qµ qν)

Π(q2) ≡ i∫

dx eiqx〈Tjµ(x)jν(0)〉

with the electromagnetic current jµ

Taylor expansion: ΠQ(q2) = Q2

Q

3

16π2

∑

n≥0

Cn zn

with z = q2/(4m2Q) and mQ = mQ(µ) the MS mass.

5

Coefficients Cn up to n = 8 known analytically in order α2s

[Chetyrkin, JK, Steinhauser, 1996]

up to high n(∼ 30); VV, AA, PP, SS correlators

[Czakon et al., 2006], [Maierhofer, Maier, Marquard, 2007]

➪ reduction to master integrals through Laporta algorithm

[Chetyrkin, JK, Sturm]; confirmed by [Boughezal, Czakon, Schutzmeier]

evaluation of master integrals numerically through difference equations

(30 digits) or Pade method or analytically in terms of transcendentals

[Schroder + Vuorinen, Chetyrkin et al., Schroder + Steinhauser,

Laporta, Broadhurst, Kniehl et al.]

C2 would be desirable!

6

Define the moments

Mthn ≡ 12π2

n!

(

d

dq2

)n

Πc(q2)

∣

∣

∣

∣

∣

∣

q2=0

=9

4Q2c

(

1

4m2c

)n

Cn

Mexpn =

∫

ds

sn+1Rc(s)

constraint:

Mexpn = Mth

n

➪ mc

7

update compared to NPB619 (2001)

experiment: • αs = 0.1187 ± 0.0020

• Γe(J/ψ,ψ′) from BES & CLEO & Babar

• ψ(3770) from BES

theory: • N3LO for n=1

• N3LO - estimate for n =2,3,4

• include condensates

δMnpn =

12π2Q2c

(4m2c )

(n+2)

⟨

αs

πG2⟩

an

(

1 +αs

πbn

)

• estimate of non-perturbative terms

(oscillations, based on Shifman)

• careful extrapolation of Ruds

• careful definition of Rc

8

▲ BES (2001)❍ MD-1▼ CLEO■ BES (2006)pQCD

√s (GeV)R

(s)

00.5

11.5

22.5

33.5

44.5

5

2 3 4 5 6 7 8 9 10

√s (GeV)

R(s

)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4 5

9

Contributions from

• narrow resonances: R =9ΠMRΓeα2(s)

δ(s−M2R)

• threshold region (2mD – 4.8 GeV)

• perturbative continuum (E ≥ 4.8 GeV)

10

Results (mc)

n mc(3 GeV) exp αs µ np total δC30n mc(mc)

1 0.986 0.009 0.009 0.002 0.001 0.013 — 1.2862 0.979 0.006 0.014 0.005 0.000 0.016 0.006 1.2803 0.982 0.005 0.014 0.007 0.002 0.016 0.010 1.2824 1.012 0.003 0.008 0.030 0.007 0.032 0.016 1.309

n = 1:

• mc(3GeV) = 986 ± 13MeV

• mc(mc) = 1286 ± 13MeV

Knowledge of C30n for n = 2,3 !?

other (”experimental”) determinations of Mn ?

11

n

mc(

3 G

eV)

(GeV

)

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

0 1 2 3 4 5

12

Results (mb)

n mb(10 GeV) exp αs µ total δC30n mb(mb)

1 3.593 0.020 0.007 0.002 0.021 — 4.1492 3.609 0.014 0.012 0.003 0.019 0.006 4.1643 3.618 0.010 0.014 0.006 0.019 0.008 4.1734 3.631 0.008 0.015 0.021 0.027 0.012 4.185

n = 2:

• mb(mb) = 4164 ± 25MeV

• mb(10GeV) = 3609 ± 25MeV

• mb(mt) = 2703 ± 18 ± 19MeV

• mt/mb = 59.8 ± 1.3

Knowledge of C30n for n = 2,3 to confirm estimate!?

data above 11GeV?

13

n

mb(

10 G

eV)

(GeV

)

3.4

3.5

3.6

3.7

3.8

3.9

4

4.1

4.2

4.3

0 1 2 3 4 5

14

mc(3 GeV) = 0.986(13) GeV

mc(mc) = 1.286(13) GeV

mb(10 GeV) = 3.609(25) GeV

mb(mb) = 4.164(25) GeV

(old result: mc(mc) = 1.304(27)GeV, mb(mb) = 4.191(51)GeV)

15

Kuehn, Steinhauser, Sturm 07

Buchmueller, Flaecher 05

Hoang, Manohar 05

Hoang, Jamin 04

deDivitiis et al. 03

Rolf, Sint 02

Becirevic, Lubicz, Martinelli 02

Kuehn, Steinhauser 01

PDG 2006

mc(mc)1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6

16

Kuehn, Steinhauser, Sturm 07

Pineda, Signer 06

Della Morte et al. 06

Buchmueller, Flaecher 05

Mc Neile, Michael, Thompson 04

deDivitiis et al. 03

Penin, Steinhauser 02

Pineda 01

Kuehn, Steinhauser 01

Hoang 00

PDG 2006

mb(mb)4.1 4.2 4.3 4.4 4.5 4.6 4.7

17

R measurement and αs

▼ CLEO (1998)

■ CLEO (2007)

√s (GeV)

R(s

)

3.35

3.4

3.45

3.5

3.55

3.6

3.65

3.7

3.75

6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

18

αs and R

basic idea: Rexp = Rth(αs,mq) ➪ αs (weak dependence on variation of mq)

rhad: [Harlander,Steinhauser’02]

Rth(s) :

• full quark mass dependence up to O(α2s)

• O(α3s ): (m2

q/s)0, (m2

q/s)1, (m2

q/s)2

• . . .

• consistent running and decoupling of αs

[v. Ritbergen,Larin,Vermaseren’97,Czakon’05]

[Chetyrkin,Kniehl,Steinhauser’97]

19

αs and R

basic idea: Rexp = Rth(αs,mq) ➪ αs (weak dependence on variation of mq)

rhad: [Harlander,Steinhauser’02]

Rexp(s) ➪ α(4)s (s) (nf = 4)

√s (GeV) α

(4)s (s) δαstat

s δαsys,cors δα

sys,uncors α

(4)s (s)|CLEO

10.538 0.2113 0.0026 0.0618 0.0444 0.23210.330 0.1280 0.0048 0.0469 0.0445 0.1429.996 0.1321 0.0032 0.0516 0.0344 0.1479.432 0.1408 0.0039 0.0526 0.0291 0.1598.380 0.1868 0.0187 0.0461 0.0195 0.2187.380 0.1604 0.0131 0.0404 0.0138 0.1956.964 0.1881 0.0221 0.0386 0.0134 0.237

⇑masslessapprox.!!!

20

αs and R

basic idea: Rexp = Rth(αs,mq) ➪ αs (weak dependence on variation of mq)

rhad: [Harlander,Steinhauser’02]

Rexp(s) ➪ α(4)s (s) (nf = 4)

• Evolve to common scale and combine

➪ α(4)s (9 GeV) = 0.160 ± 0.024 ± 0.024

21

αs and R

basic idea: Rexp = Rth(αs,mq) ➪ αs (weak dependence on variation of mq)

rhad: [Harlander,Steinhauser’02]

Rexp(s) ➪ α(4)s (s) (nf = 4)

• Evolve to common scale and combine

➪ α(4)s (9 GeV) = 0.160 ± 0.024 ± 0.024

• α(4)s (9 GeV) → α

(4)s (µdec

b ) → α(5)s (µdec

b ) → α(5)s (MZ)

(practically) independent from µdecb (4-loop running and

3-loop decoupling) RunDec: [Chetyrkin,JK,Steinhauser’00]

➪ α(5)s (MZ) = 0.110+0.010

−0.012+0.010−0.011 = 0.110+0.014

−0.017 [JK,Steinhauser,Teubner’07]

22

αs and R

basic idea: Rexp = Rth(αs,mq) ➪ αs (weak dependence on variation of mq)

rhad: [Harlander,Steinhauser’02]

Rexp(s) ➪ α(4)s (s) (nf = 4)

• Evolve to common scale and combine

➪ α(4)s (9 GeV) = 0.160 ± 0.024 ± 0.024

• α(4)s (9 GeV) → α

(4)s (µdec

b ) → α(5)s (µdec

b ) → α(5)s (MZ)

(practically) independent from µdecb (4-loop running and

3-loop decoupling) RunDec: [Chetyrkin,JK,Steinhauser’00]

➪ α(5)s (MZ) = 0.110+0.010

−0.012+0.010−0.011 = 0.110+0.014

−0.017 [JK,Steinhauser,Teubner’07]

• CLEO analysis: α(5)s (M2

Z)|CLEO = 0.126 ± 0.005+0.015−0.011

massless approximation for R(s), no decoupling of αs

23

R: experiment + theory

▼ CLEO (1998)

■ CLEO (2007)

√s (GeV)

R(s

)

3.35

3.4

3.45

3.5

3.55

3.6

3.65

3.7

3.75

6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

24

R: experiment + theory

▼ CLEO (1998)

■ CLEO (2007)

√s (GeV)

R(s

)

3.35

3.4

3.45

3.5

3.55

3.6

3.65

3.7

3.75

6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

25

αs from R

• α(5)s (MZ) = 0.110+0.010

−0.012+0.010−0.011 = 0.110+0.014

−0.017 [JK,Steinhauser,Teubner’07]

• Combine with α(5)s (MZ) = 0.124+0.011

−0.014 [JK,Steinhauser’01]

R measurements between 2 and 10.5 GeV from

BES’01, MD-1’96, CLEO’97

➪ α(5)s (MZ) = 0.119+0.009

−0.011

• Compare: α(5)s (MZ) = 0.1189 ± 0.0010 [Bethke’06]

26