Precise quark masses and strong coupling constant from heavy quark current correlators

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1 Precise quark masses ad strog couplig costat from heavy quark curret correlators Christia Sturm Physics Departmet Brookhave Natioal Laboratory High Eergy Theory Group Upto, New York I. Itroductio & Motivatio II. Methods & Calculatio III. Aalysis & Results IV. Summary & Coclusio Precise quark masses ad strog couplig costat from heavy quark curret correlators 0/17

2 Itroductio Motivatio Precise determiatio of quark masses ad strog couplig: Quark masses/strog couplig are fudametal parameters of the Stadard Model Quark masses play a importat role i Higgs physics: e.g. decay: Γ(H b b) = G f M h 4 2π m2 b (1 + O(α s) +... ), Γ(H c c) Flavor physics Test ruig of strog couplig... Compariso with other methods Here: 2 methods: Heavy quark correlators i combiatio with R-ratio & Lattice simulatios s. talk by Eduardo Follaa Precise quark masses ad strog couplig costat from heavy quark curret correlators 1/17

3 Method I: R-ratio Experimet R(s) = σ(e+ e Hadros) σ(e + e µ + µ ) R(s) pqcd BES (2001) J/ψ ψ, MD-1 CLEO BES (2006) s (GeV) Precise quark masses ad strog couplig costat from heavy quark curret correlators 2/17

4 Method I Relatio: Experimet Theory Heavy quark correlator Π µν (q, j) = i dx e iqx 0 T j µ (x)j ν (0) 0 Here: j µ (x) electromagetic heavy quark vector curret Π µν (q, j)= q q =( g µν + q µ q ν /q 2 )Π(q 2 ) R(s) = σ(e+ e Hadros) = 12πIm [ Π(q 2 = s + iε) ] σ(e + e µ + µ ) dπ e + e q Q 2 = 2 Im ( ) With the help of dispersio-relatios: Π(q 2 ) = Π(q 2 = 0) + q2 R(s) 12π 2 ds s(s q 2 ) Precise quark masses ad strog couplig costat from heavy quark curret correlators 3/17

5 Method I Relatio: Theory Experimet Exp. momets are related to derivatives of Π(q 2 ) at q 2 = 0: ( ) 12π d! dq 2 Π(q ) 2 ds = M exp = q2 =0 s +1 Rexp (s) I terms of expasio coefficiets: Π(q 2 ) = 3Q2 ( ) f 16π 2 C v q 2 Q 4m 2, f : charge of quark m = m(µ) : MS mass C v ca be calculated perturbatively First ad higher derivatives of Π(q 2 ) allow direct determiatio of the MS charm- ad bottom-quark mass: ( ) 1/(2) 9 Theory m(µ) = 1 2 Q 2 f 4 C v M exp Experimet c-quarks: Novikov et al. 78; b-quarks: Reiders et al. 85 C v deped o the quark mass through log(m(µ) 2 /µ 2 ) Precise quark masses ad strog couplig costat from heavy quark curret correlators 4/17

6 Method II: Data from Lattice I. Alliso, E. Dalgic, C.T.H. Davies, E. Follaa, R.R. Horga, K. Horbostel, G.P. Lepage, C. McNeile, J. Shigemitsu, H. Trottier, R.M. Woloshy(HPQCD), K.G. Chetyrki, J.H. Küh, M. Steihauser, C.S. Idea: Replace momets obtaied from R-ratio by computatio of correlator through lattice simulatios Allows to substitute electromagetic curret by pseudoscalar operator r 2k+2 = (C p k/c p,(0) k ) 1 2k 2, k = 2, 3,... C p k: expasio coeff. of pseudoscalar correlator Quark mass: m c (µ) = mexp η c r pqcd 2k+2 Pert. theory 2 R LQCD 2k+2 Lattice Sim. s. talk E. Follaa Strog couplig: r 4 (α s, µ/m c ) = R LQCD 4 + ratios of momets, weak depedece o quark mass Precise quark masses ad strog couplig costat from heavy quark curret correlators 5/17

7 Pert. calculatio of expasio coefficiets Theory Expasio diagrammatically: + q 2 ( +... Oe-scale multi-loop itegrals i pqcd Sample diagrams )... 3-loop(order α 2 s) coefficiets C up to =8Chetyrki,Küh,Steihauser 96 up to higher momets 30 Czako et al. 06; Maierhöfer, Maier, Marquard 07 for correlators VV, AA, PP, SS Precise quark masses ad strog couplig costat from heavy quark curret correlators 6/17

8 Pert. calculatio of expasio coefficiets Theory 4-loop (order α 3 s) coefficiets: Use Itegratio-by-parts idetities K.G. Chetyrki, F.V. Tkachov i combiatio with Laporta s Algorithm S. Laporta, E. Remiddi to map all itegrals o small set of master itegrals Requires solutio of liear system of equatios several millio of equatios; several GB of solutios Computatio of 13 master itegrals umerically through differece equatios(30 digits) or Padé method or Y.Schröder, A.Vuorie; Chetyrki et al., aalytically Schröder, Steihauser; Laporta, Broadhurst, Kiehl et al. Vector case: - first momets C 0, C 1 K. G. Chetyrki, J. H. Küh, C.S. 06; R. Boughezal, M. Czako, T. Schutzmeier 06 - secod momet C 2 A. Maier, P. Maierhöfer, P. Marquard 08 Pseudoscalar case: - first momets C 0, C 1, C 2 I. Alliso, E. Dalgic, C.T.H. Davies, E. Follaa, R.R. Horga, K. Horbostel, G.P. Lepage, C. McNeile, J. Shigemitsu, H. Trottier, R.M. Woloshy, K.G. Chetyrki, J.H. Küh, M. Steihauser, C.S third momet C 3 A. Maier, P. Maierhöfer, P. Marquard 08 Precise quark masses ad strog couplig costat from heavy quark curret correlators 7/17

9 Result C = C (0) ( αs ) ( π ) 2 ( ( αs π ( αs C (10) + C (11) l mc ) ) C (20) + C (21) l mc + C (22) lm 2 c ) 3 ( ) C (30) + C (31) l mc + C (32) lm 2 c + C (33) lm 3 c π +...,with l mc = log(mc/µ 2 2 ) Pseudoscalar case( f = 4): 1-loop 2-loop 3-loop 4-loop C (0) C (10) C (11) C (20) C (21) C (22) C (30) C (31) C (32) C (33) Result also available completely aalytically Precise quark masses ad strog couplig costat from heavy quark curret correlators 8/17

10 Result C = C (0) ( αs ) ( π ) 2 ( ( αs π ( αs C (10) + C (11) l mc ) ) C (20) + C (21) l mc + C (22) lm 2 c ) 3 ( ) C (30) + C (31) l mc + C (32) lm 2 c + C (33) lm 3 c π +...,with l mc = log(mc/µ 2 2 ) Vector case( f = 4): 1-loop 2-loop 3-loop 4-loop C (0) C (10) C (11) C (20) C (21) C (22) C (30) C (31) C (32) C (33) Result also available completely aalytically Precise quark masses ad strog couplig costat from heavy quark curret correlators 8/17

11 Aalysis Extractio of the exp. momets from R(s) (charm quark case) J.H. Küh, M. Steihauser, C.S. Determie: M exp For charm quarks: M res = ds R exp (s) = M res s +1 + M thr + M cot : Cotais: J/Ψ, Ψ(2S) treated i arrow width approximatio 2 α α(s) δ(s M 2 R ) R res (s) = 9πM RΓ ee α 2 J/Ψ Ψ(2S) M Ψ (GeV) (11) (34) Γ ee(kev) 5.55(14) 2.48(6) (α/α(m Ψ )) M thr : BES data ( s 3.73 GeV) subtract backgroud from R uds, u,d,s c R from data below 3.73 GeV, s-depedece from theory Precise quark masses ad strog couplig costat from heavy quark curret correlators 9/17

12 Aalysis Extractio of the exp. momets from R(s) (charm quark case) J.H. Küh, M. Steihauser, C.S. M cot : pqcd above s 4.8 GeV, o data, M exp : M res R(s) with full quark mass depedece rhad: R. Harlader, M. Steihauser ( 1) 10 ( 1) 10 ( 1) 10 ( 1) 10 ( 1) (25) (15) (11) (31) (2) (25) (8) (3) (27) (0) (26) (5) (1) (27) (14) (27) (3) (0) (27) (54) δm p M thresh M cot M exp M p = 12π2 Q 2 c (4m 2 c) (+2) αs π G2 a ( 1 + α s π b ) D.J. Broadhurst, P.A. Baikov, V.A. Ilyi, J. Fleischer, O.V. Tarasov, V.A. Smirov Precise quark masses ad strog couplig costat from heavy quark curret correlators 10/17

13 Aalysis Determiatio of the charm quark mass from R(s) J.H. Küh, M. Steihauser, C.S. ( ) M th + M p = M exp with M th = Q2 q C 4mc 2 ( ) 1/(2) 9 m(µ) = 1 2 Q 2 f 4 M exp C M p µ = (3 ± 1) GeV α s (M Z ) = ± m c(3 GeV) exp α s µ p total (30) δ C m c(m c) C (30) 3 5.2, 6.0 C (30) =2: old m c(3 GeV) = GeV; estimated 6.0 C (30) 2 7.0; ew: m c(3 GeV) = GeV; C (30) 2 = A. Maier, P. Maierhöfer, P. Marquard Precise quark masses ad strog couplig costat from heavy quark curret correlators 11/17

14 Aalysis Determiatio of the charm quark mass from R(s) J.H. Küh, M. Steihauser, C.S. Charm-quarks m c (3 GeV) (GeV) m c (3 GeV) = 0.986(13) GeV Precise quark masses ad strog couplig costat from heavy quark curret correlators 12/17

15 Aalysis Extractio of the exp. momets from R(s) (bottom quark case) J.H. Küh, M. Steihauser, C.S. M th : aalog to charm case, oly f = 5 M p : egligible M res M thr. M cot : Υ(1S), Υ(2S), Υ(3S), Υ(4S) : CLEO data up to GeV : pqcd above GeV M exp : 10 (2+1) 10 (2+1) 10 (2+1) 10 (2+1) (23) 0.296(32) 2.911(18) 4.601(43) (23) 0.249(27) 1.173(11) 2.881(37) (24) 0.209(22) 0.624(7) 2.370(34) (25) 0.175(19) 0.372(5) 2.178(32) M res M thresh M cot M exp Precise quark masses ad strog couplig costat from heavy quark curret correlators 13/17

16 Aalysis Determiatio of the bottom quark mass from R(s) J.H. Küh, M. Steihauser, C.S. µ = (10 ± 5) GeV; α s(m Z ) = ± m b (10 GeV) exp α s µ total (30) δ C m b (m b ) C (30) 3 5.2, 6.0 C (30) =2: old m b (10 GeV) = 3.609(25) GeV; estimated 6.0 C (30) 2 7.0; ew: m b (10 GeV) = 3.607(19) GeV; C (30) 2 = A. Maier, P. Maierhöfer, P. Marquard Precise quark masses ad strog couplig costat from heavy quark curret correlators 14/17

17 Method II Charm mass & strog couplig costat Extract momets R 2k+2 by lattice simulatio of pseudoscalar correlator I. Alliso, E. Dalgic, C.T.H. Davies, E. Follaa, R.R. Horga, K. Horbostel, G.P. Lepage, C. McNeile, J. Shigemitsu, H. Trottier, R.M. Woloshy(HPQCD), K.G. Chetyrki, J.H. Küh, M. Steihauser, C.S. s. talk by Eduardo Follaa Results: m c (3 GeV) = 0.986(10) GeV lattice + pqcd ( compared to m c(3 GeV) = 0.986(13) GeV) e + e + pqcd ᾱ s (M z ) = (12) lattice + pqcd Also VV ad AA correlator ca be employed through reduced momets R (j) 2k+2 = 2mc(µ) m j C j k C j,(0) k C j,(0) k 1 C j k 1 Compariso: R v 2k+2 Rexp 2k k Experimetal ad simulatio results agree withi 2% v exp R 2k+2 /R 2k Precise quark masses ad strog couplig costat from heavy quark curret correlators 15/17 1

18 2 2 2 Compariso with other methods charm-quarks bottom-quarks HPQCD + Karlsruhe/BNL 08 lattice + pqcd Kueh, Steihauser, Sturm 07 low-momet sum rules, NNNLO Buchmueller, Flaecher 05 B decays α s β 0 Hoag, Maohar 05 B decays α s β 0 Hoag, Jami 04 NNLO momets dedivitiis et al. 03 lattice queched Rolf, Sit 02 lattice (ALPHA) queched Becirevic, Lubicz, Martielli 02 lattice queched Kueh, Steihauser 01 low-momet sum-rules, NNLO QWG 2004 PDG m c(3 GeV) (GeV) Kueh, Steihauser, Sturm 07 low-momet sum rules, NNNLO Pieda, Siger 06 Y sum rules, NNLL (ot complete) Della Morte et al. 06 lattice (ALPHA) queched Buchmueller, Flaecher 05 B decays α s β 0 McNeile, Michael, Thompso 04 lattice (UKQCD) dedivitiis et al. 03 lattice queched Pei, Steihauser 02 Y(1S), NNNLO Pieda 01 Y(1S), NNLO Kueh, Steihauser 01 low-momet sum rules, NNLO Hoag 00 Y sum rules, NNLO QWG 2004 PDG m b (m ) (GeV) b Precise quark masses ad strog couplig costat from heavy quark curret correlators 16/17

19 Summary & Coclusio Heavy quark curret correlators ca be used to perform a precise quark mass determiatio i combiatio with experimetally measured R-ratio ad with lattice simulatios Extractio of charm- ad bottom-quark masses from R-ratio icludig NNNLO results i pqcd Charm-quark mass ad strog couplig from lattice simulatios icludig NNNLO results i pqcd Quark masses & strog couplig: Charm-mass: m c (3 GeV) = 0.986(13) GeV e + e + pqcd m c (3 GeV) = 0.986(10) GeV lattice + pqcd Bottom-mass: m b (10 GeV) = 3.607(19) GeV e + e + pqcd + C (30) 2 strog couplig: ᾱ s (M z ) = (12) lattice + pqcd Precise quark masses ad strog couplig costat from heavy quark curret correlators 17/17

Heavy Quark Masses. Matthias Steinhauser TTP, University of Karlsruhe. May 2008, CAQCD08, University of Minnesota

Heavy Quark Masses. Matthias Steinhauser TTP, University of Karlsruhe. May 2008, CAQCD08, University of Minnesota Heavy Quark Masses Matthias Steihauser TTP, Uiversity of Karlsruhe May 2008, CAQCD08, Uiversity of Miesota i collaboratio with Kostja Chetyrki, Has Küh, Christia Sturm ad (i part with) the HPQCD Collaboratio: