FROM ( e + e had ) JK, Steinhauser, Sturm NPB JK, Steinhauser, - PowerPoint PPT Presentation

QUARK MASSES AND α s FROM σ ( e + e − → had ) JK, Steinhauser, Sturm NPB JK, Steinhauser, Teubner PRD l a c i s t c e i s r o y h e h P T e l l B a c i n t Universit¨ at Karlsruhe (TH) r o a AC i P t a t KA u p m o C Forschungsuniversit¨ at • gegr¨ undet 1825 ’ ’ SFB TR9

Main Idea (SVZ) 2

Data 5 5 4.5 4.5 4 4 3.5 3.5 3 3 R(s) R(s) 2.5 2.5 2 2 ▲ BES (2001) ❍ MD-1 1.5 1.5 ▼ CLEO ■ BES (2006) 1 1 pQCD 0.5 0.5 0 0 3.8 4 4.2 4.4 4.6 4.8 2 3 4 5 6 7 8 9 10  s (GeV)  s (GeV) √ √ 5 4.5 4 3.5 3 R(s) 2.5 2 1.5 1 0.5 0 3.65 3.675 3.7 3.725 3.75 3.775 3.8 3.825 3.85 3.875 3.9  s (GeV) √ 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 % ψ ′ PDG (9 %) 2.4 % ψ ′′ PDG (15 %) ψ ′′ region BES 2006 4 % Future improvements: charm region (CLEO) 3% bottom region ?? (CLEO) 4

m Q from SVZ Sum Rules, Moments and Tadpoles Some definitions: Π( q 2 = s + iǫ ) � � R ( s ) = 12 π Im � − q 2 g µν + q µ q ν Π( q 2 ) � � d x e iqx � Tj µ ( x ) j ν (0) � ≡ i with the electromagnetic current j µ 3 Taylor expansion: Π Q ( q 2 ) Q 2 C n z n � ¯ = Q 16 π 2 n ≥ 0 with z = q 2 / (4 m 2 Q ) and m Q = m Q ( µ ) the MS mass. 5

C n up to n = 8 known analytically in order α 2 Coefficients ¯ s [Chetyrkin, JK, Steinhauser, 1996] up to high n ( ∼ 30); VV, AA, PP, SS correlators [Czakon et al., 2006], [Maierh¨ ofer, 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 Pad´ e method or analytically in terms of transcendentals [Schr¨ oder + Vuorinen, Chetyrkin et al., Schr¨ oder + Steinhauser, Laporta, Broadhurst, Kniehl et al.] ¯ C 2 would be desirable! 6

Define the moments � n � � n n ≡ 12 π 2 � � d = 9 1 � M th Π c ( q 2 ) 4 Q 2 ¯ � C n c � d q 2 4 m 2 n ! � c � q 2 =0 d s � M exp = s n +1 R c ( s ) n constraint: M exp = M th n n ➪ m c 7

update compared to NPB619 (2001) • α s = 0 . 1187 ± 0 . 0020 experiment: • Γ e ( J/ψ, ψ ′ ) from BES & CLEO & Babar • ψ (3770) from BES • N 3 LO for n=1 theory: • N 3 LO - estimate for n =2,3,4 • include condensates 12 π 2 Q 2 � α s 1 + α s � � � δ M np c π G 2 ¯ = a n b n n (4 m 2 c ) ( n +2) π • estimate of non-perturbative terms (oscillations, based on Shifman) • careful extrapolation of R uds • careful definition of R c 8

5 4.5 4 3.5 3 R(s) 2.5 2 ▲ BES (2001) ❍ MD-1 1.5 ▼ CLEO ■ BES (2006) 1 pQCD 0.5 0 2 3 4 5 6 7 8 9 10  s (GeV) √ 5 4.5 4 3.5 3 R(s) 2.5 2 1.5 1 0.5 0 4 5  s (GeV) √ 9

Contributions from R = 9 Π M R Γ e δ ( s − M 2 • narrow resonances: R ) α 2 ( s ) • threshold region (2 m D – 4.8 GeV) • perturbative continuum ( E ≥ 4 . 8 GeV) 10

Results ( m c ) C 30 δ ¯ m c (3 GeV) exp np total m c ( m c ) n α s µ n 1 0 . 986 0 . 009 0 . 009 0 . 002 0 . 001 0 . 013 — 1 . 286 2 0 . 979 0 . 006 0 . 014 0 . 005 0 . 000 0 . 016 0 . 006 1 . 280 3 0 . 982 0 . 005 0 . 014 0 . 007 0 . 002 0 . 016 0 . 010 1 . 282 4 1 . 012 0 . 003 0 . 008 0 . 030 0 . 007 0 . 032 0 . 016 1 . 309 n = 1: • m c (3 GeV) = 986 ± 13 MeV • m c ( m c ) = 1286 ± 13 MeV Knowledge of C 30 for n = 2 , 3 !? n other (”experimental”) determinations of M n ? 11

1.3 1.25 1.2 m c (3 GeV) (GeV) 1.15 1.1 1.05 1 0.95 0.9 0.85 0 1 2 3 4 5 n 12

Results ( m b ) C 30 δ ¯ m b (10 GeV) exp total m b ( m b ) n α s µ n 1 3 . 593 0 . 020 0 . 007 0 . 002 0 . 021 — 4 . 149 2 3 . 609 0 . 014 0 . 012 0 . 003 0 . 019 0 . 006 4 . 164 3 3 . 618 0 . 010 0 . 014 0 . 006 0 . 019 0 . 008 4 . 173 4 3 . 631 0 . 008 0 . 015 0 . 021 0 . 027 0 . 012 4 . 185 n = 2: • m b ( m b ) = 4164 ± 25 MeV • m b (10GeV) = 3609 ± 25 MeV • m b ( m t ) = 2703 ± 18 ± 19 MeV • m t /m b = 59 . 8 ± 1 . 3 Knowledge of C 30 for n = 2 , 3 to confirm estimate!? n data above 11GeV? 13

4.3 4.2 4.1 m b (10 GeV) (GeV) 4 3.9 3.8 3.7 3.6 3.5 3.4 0 1 2 3 4 5 n 14

m c (3 GeV) = 0 . 986(13) GeV m b (10 GeV) = 3 . 609(25) GeV m c ( m c ) = 1 . 286(13) GeV m b ( m b ) = 4 . 164(25) GeV (old result: m c ( m c ) = 1 . 304(27)GeV, m b ( m b ) = 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 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 m c (m c ) 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 4.1 4.2 4.3 4.4 4.5 4.6 4.7 m b (m b ) 17

R measurement and α s 3.75 ▼ CLEO (1998) 3.7 ■ CLEO (2007) 3.65 3.6 R(s) 3.55 3.5 3.45 3.4 3.35 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11  s (GeV) √ 18

α s and R basic idea: R exp = R th ( α s , m q ) ➪ α s (weak dependence on variation of m q ) rhad: [Harlander,Steinhauser’02] R th ( s ) : • full quark mass dependence up to O ( α 2 s ) • O ( α 3 s ): ( m 2 q /s ) 0 , ( m 2 q /s ) 1 , ( m 2 q /s ) 2 • . . . • consistent running and decoupling of α s [v. Ritbergen,Larin,Vermaseren’97,Czakon’05] [Chetyrkin,Kniehl,Steinhauser’97] 19

α s and R basic idea: R exp = R th ( α s , m q ) ➪ α s (weak dependence on variation of m q ) rhad: [Harlander,Steinhauser’02] R exp ( s ) ➪ α (4) ( s ) ( n f = 4) s √ s (GeV) α (4) δα sys , cor δα sys , uncor α (4) δα stat ( s ) ( s ) | CLEO s s s s s 10 . 538 0 . 2113 0 . 0026 0 . 0618 0 . 0444 0 . 232 10 . 330 0 . 1280 0 . 0048 0 . 0469 0 . 0445 0 . 142 9 . 996 0 . 1321 0 . 0032 0 . 0516 0 . 0344 0 . 147 9 . 432 0 . 1408 0 . 0039 0 . 0526 0 . 0291 0 . 159 8 . 380 0 . 1868 0 . 0187 0 . 0461 0 . 0195 0 . 218 7 . 380 0 . 1604 0 . 0131 0 . 0404 0 . 0138 0 . 195 6 . 964 0 . 1881 0 . 0221 0 . 0386 0 . 0134 0 . 237 ⇑ massless approx.!!! 20

α s and R basic idea: R exp = R th ( α s , m q ) ➪ α s (weak dependence on variation of m q ) rhad: [Harlander,Steinhauser’02] R exp ( s ) ➪ α (4) ( s ) ( n f = 4) s • Evolve to common scale and combine ➪ α (4) (9 GeV) = 0 . 160 ± 0 . 024 ± 0 . 024 s 21

α s and R basic idea: R exp = R th ( α s , m q ) ➪ α s (weak dependence on variation of m q ) rhad: [Harlander,Steinhauser’02] R exp ( s ) ➪ α (4) ( s ) ( n f = 4) s • Evolve to common scale and combine ➪ α (4) (9 GeV) = 0 . 160 ± 0 . 024 ± 0 . 024 s α (4) (9 GeV) → α (4) ) → α (5) ) → α (5) ( µ dec ( µ dec • ( M Z ) s s s s b b (practically) independent from µ dec (4-loop running and b 3-loop decoupling) RunDec: [Chetyrkin,JK,Steinhauser’00] ➪ α (5) ( M Z ) = 0 . 110 +0 . 010 +0 . 010 − 0 . 011 = 0 . 110 +0 . 014 s [JK,Steinhauser,Teubner’07] − 0 . 012 − 0 . 017 22

α s and R basic idea: R exp = R th ( α s , m q ) ➪ α s (weak dependence on variation of m q ) rhad: [Harlander,Steinhauser’02] R exp ( s ) ➪ α (4) ( s ) ( n f = 4) s • Evolve to common scale and combine ➪ α (4) (9 GeV) = 0 . 160 ± 0 . 024 ± 0 . 024 s α (4) (9 GeV) → α (4) ) → α (5) ) → α (5) ( µ dec ( µ dec • ( M Z ) s s s s b b (practically) independent from µ dec (4-loop running and b 3-loop decoupling) RunDec: [Chetyrkin,JK,Steinhauser’00] ➪ α (5) ( M Z ) = 0 . 110 +0 . 010 +0 . 010 − 0 . 011 = 0 . 110 +0 . 014 s [JK,Steinhauser,Teubner’07] − 0 . 012 − 0 . 017 CLEO analysis: α (5) Z ) | CLEO = 0 . 126 ± 0 . 005 +0 . 015 ( M 2 • s − 0 . 011 massless approximation for R ( s ), no decoupling of α s 23

R : experiment + theory 3.75 ▼ CLEO (1998) 3.7 ■ CLEO (2007) 3.65 3.6 R(s) 3.55 3.5 3.45 3.4 3.35 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11  s (GeV) √ 24

R : experiment + theory 3.75 ▼ CLEO (1998) 3.7 ■ CLEO (2007) 3.65 3.6 R(s) 3.55 3.5 3.45 3.4 3.35 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11  s (GeV) √ 25

α s from R α (5) ( M Z ) = 0 . 110 +0 . 010 +0 . 010 − 0 . 011 = 0 . 110 +0 . 014 • s [JK,Steinhauser,Teubner’07] − 0 . 012 − 0 . 017 • Combine with α (5) ( M Z ) = 0 . 124 +0 . 011 s [JK,Steinhauser’01] − 0 . 014 R measurements between 2 and 10.5 GeV from BES’01, MD-1’96, CLEO’97 α (5) ( M Z ) = 0 . 119 +0 . 009 ➪ s − 0 . 011 Compare: α (5) • ( M Z ) = 0 . 1189 ± 0 . 0010 [Bethke’06] s 26