QUARK MASSES AND α s FROM σ (e + e − → had) JK, Steinhauser, Sturm NPB JK, Steinhauser, Teubner PRD Universit¨ at Karlsruhe (TH) Forschungsuniversit¨ at • gegr¨ undet 1825 Computational Theoretical Particle Physics SFB TR9 B KA AC ’’
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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SFB TR9
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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