Bottom hadron spectroscopy from lattice QCD

Transcription

1 Bottom hadron spectroscopy from lattice QCD Stefan Meinel Department of Physics Jefferson Lab, October 11, 2010

2 Some puzzles concerning non-excited non-exotic heavy hadrons

3 Ω b : Experiment Events/(0.04 GeV) D0 1.3 fb 1 Data Fit (a) M(Ω 6.6 b ) 6.8 (GeV) 7 [D/0, PRL 2008]: M Ωb = 6.165(10)(13) GeV [CDF, PRD 2009]: M Ωb = (68)(9) GeV About 6 standard deviations discrepancy

4 Ω b : Lattice QCD mass [ GeV/c 2 ] CDF experiment Lewis,Woloshyn PRD,2009 Burch,Hagan,Lang,Limmer,Schafer PRD,2009 Detmold,Lin,Wingate NPB,2009 Lin,Cohen,Mathur,Orginos PRD,2009 Λ b Ξ b Σ b Ξ' b Ω b Σ * b Ξ * b Ω * b Figure from [Lewis, arxiv: ] See also: Fermilab + staggered [Na and Gottlieb, arxiv: ] Our results with NRQCD + DWF at low pion mass will be available soon [Meinel et al., arxiv: ]

5 Quarkonium 1S hyperfine splitting: charmonium Charmonium: M(J/ψ) M(η c ) Experiment: ± 1.2 MeV [PDG, JPG 2010] perturbative QCD (potential NRQCD): MeV [Kniehl et al. PRL 2004]

6 Quarkonium 1S hyperfine splitting: bottomonium Bottomonium: M(Υ) M(η b ) Experiment: 69.3 ± 2.8 MeV [BABAR, PRL 2008, 2009, CLEO, PRD 2009] perturbative QCD (potential NRQCD): 39 ± 14 MeV [Kniehl et al., PRL 2004] Perturbation theory should work better in bottomonium than in charmonium. What is going on?

7 New physics in bottomonium? Need precise lattice calculation to check perturbative QCD result.

8 M(Υ) M(η b ): lattice QCD M(Υ) M(η b ) = 54 ± 12 MeV using Fermilab method [Burch et al., PRD 2010] M(Υ) M(η b ) = 61 ± 14 MeV using NRQCD of order v 4 [Gray et al., PRD 2005] Dominant errors on NRQCD result: relativistic (10%) and radiative (20%) Later in this talk: a new NRQCD calculation that largely removes these two sources of error

9 Mass of the Ω bbb Baryonic analogue of the Υ. Reference M Ωbbb (GeV) Ponce, PRD Hasenfratz et al., PLB Bjorken ± 0.18 Tsuge et al., MPL Silvestre-Brac, FBS Jia, JHEP ± 0.08 Martynenko, PLB Roberts and Pervin, IJMPA Bernotas and Simonis, LJP Zhang and Huang, PLB ± GeV range! Later in this talk: lattice QCD result with 12 MeV uncertainty

10 Heavy quarks on the lattice

11 Wilson Fermion action S W F [Ψ, Ψ, U] = a 4 x az 4 Ψ(x) with the lattice derivatives [ γ µ (±) µ 1 2 a (+) µ ( ) µ +m Ψ(x) }{{} removes doublers + µ ψ(x) = 1 a [U µ(x)ψ(x + ˆµ) ψ(x)], µ ψ(x) = 1 a [ψ(x) U µ(x)ψ(x ˆµ)], ± µ ψ(x) = 1 2 [ + µ ψ(x) + µ ψ(x) ], ] where U µ (x) = U µ(x aˆµ). Can define non-compact gauge field A µ through U µ (x) = exp [iaga µ (x)].

12 Wilson Fermion action: dispersion relation Energy as a function of momentum: E(p) = m 1 + p2 2m 2 + O(p 4 ) For the Wilson quark, at tree level: ( m 1 = m ma + 1 ) 3 m2 a , ( m 2 = m 1 1 ) 2 ma + m2 a , m 1 m 2 = m2 a This indicates large discretization errors (deviations from Lorentz invariance) when ma not small

13 Heavy quarks on the lattice Compton wavelength vs lattice spacing: λ = 2π m a For precise lattice calculations in b physics using relativistic action, would need simultaneously 1 L m π and m b 1 a. Thus, a huge number (L/a) of lattice points is needed. Another problem at small a: critical slowing down of topological modes [Lüscher, arxiv: ].

14 Relativistic b quarks on the lattice Work at unphysically small m and extrapolate to m b : introduces systematic errors Anisotropic lattices with a t m b 1 [Klassen, NPB 1998]: there may still be (a s m b ) p errors [Harada et al., PRD 2001] Highly improved actions remove some of the (am b ) p errors: with HISQ [Follana et al., PRD 2007] still need a < 0.03 fm. Critical slowing down? Fermilab method [El-Khadra et al., PRD 1997]: difficult parameter tuning, if incomplete still large errors

15 Nonrelativistic b quarks on the lattice Alternative approach: start with nonrelativistic effective field theory in the continuum, then discretize Lattice NRQCD [Lepage, PRD 1991, 1992]: can not take continuum limit Lattice HQET [Eichten, Hill, PLB 1990]: only for heavy-light hadrons

16 Foldy-Wouthuysen-Tani transformation Dirac Lagrangian (Minkowski space): L = Ψ( m + iˆγ 0 D 0 + iˆγ j D j )Ψ This describes both particles and antiparticles. Projection operators for quark / antiquark fields are 1 2 (1 + ˆγ0 ), 1 2 (1 ˆγ0 ) The term iˆγ j D j couples quarks and antiquarks, as it does not commute with ˆγ 0 try to remove this term via field redefinition

17 Foldy-Wouthuysen-Tani transformation ( ) 1 Ψ = exp 2m iˆγj D j Ψ (1), ( ) ( 1 Ψ = Ψ (1) exp 2m iˆγj D j = Ψ (1) exp 1 2m iˆγj results in ) D j with L = Ψ (1) ( m + iˆγ 0 D 0 )Ψ (1) + n=1 1 m n Ψ (1) O (1)n Ψ (1) O (1)1 = 1 2 D jd j ig 8 [ˆγµ, ˆγ ν ]F µν = 1 2 D jd j ig 8 [ˆγj, ˆγ k ]F jk }{{} =O C (1)1 ig 2 ˆγjˆγ 0 F j0. }{{} =O(1)1 A

18 Foldy-Wouthuysen-Tani transformation Next, remove O A (1)1 by another field redefinition ( ) 1 Ψ (1) = exp 2m 2 OA (1)1 Ψ (2), ( ) 1 Ψ (1) = Ψ (2) exp 2m 2 OA (1)1 This can be continued to any order in 1/m

19 Foldy-Wouthuysen-Tani transformation One obtains [ L = Ψ m + iˆγ 0 D 0 1 2m D jd j ig 8m [ˆγj, ˆγ k ]F jk g ( 8m 2 ˆγ0 Dj ad F j0 1 )] 2 [ˆγj, ˆγ k ] {D j, F k0 } +O(1/m 3 ) All terms to the given order commute with ˆγ 0. The mass term can be removed via Ψ Ψ exp ( imx 0ˆγ 0) Ψ, Ψ Ψ exp ( imx 0ˆγ 0)

20 Foldy-Wouthuysen-Tani transformation Next, write and Ψ = ( ψ χ ), Ψ = ( ψ, χ ) E k = F 0k, B j = 1 2 ɛ jklf kl

21 Foldy-Wouthuysen-Tani transformation One obtains [ L = ψ id 0 + D2 2m + g 2m σ B + g 8m 2 ( (D ad E) + iσ (D E E D)) ] ψ + χ [ id 0 D2 2m + O(1/m 3 ) g 2m σ B + g 8m 2 ( (D ad E) + iσ (D E E D)) ] χ Note: these are the tree-level values of the couplings

22 Power counting: heavy-light hadrons b Then, [D µ, D ν ] = igf µν implies D 0 D Λ QCD g E g B Λ 2 QCD

23 Power counting: heavy-light hadrons b leading-order Lagrangian for heavy quark: L = ψ id 0 ψ. Leads to heavy-quark spin- and flavor symmetry [Shifman, Voloshin, SJNP 1988]. Correction terms are suppressed by powers of (Λ QCD /m b ).

24 Lattice HQET Continuum Lagrangian (Euclidean): L = δm ψ ψ }{{} dim. 3 + ψ D 0 ψ }{{} dim. 4 Includes all operators of dimension 4 or less that are compatible with symmetries renormalizable! Lattice action [Eichten, Hill, PLB 1990]: S = x (lattice units with a = 1) ψ (x) [ (1 + δm)ψ(x) U 0 (x ˆ0)ψ(x ˆ0) ]

25 Lattice HQET Propagator on given gauge field background = Wilson line t t 1 G ψ (x, x ) = δ x, x (1 + δm) (t t +1) U 0 (x + nˆ0). n=0 Treat (Λ QCD /m b ) corrections as insertions in correlation functions. When renormalized nonperturbatively [Maiani et al. NPB 1992], theory remains renormalizable and continuum limit is possible [ALPHA Collaboration]. Works only for heavy-light hadrons.

26 Power counting: heavy-heavy hadrons b b v D m b v, D 0 E kin m b v 2 g E m 2 b v3, g B m 2 b v4

27 Power counting: heavy-heavy hadrons b b Leading-order Lagrangian is [ ] ] L = ψ id 0 + D2 ψ + χ [id 0 D2 χ 2m 2m Correction terms are suppressed by powers of v 2. For bottomonium, v v

28 NRQCD Continuum Lagrangian (Euclidean): L ψ = ψ (D 0 + H) ψ where H contains all terms up to desired order in v 2 or (Λ QCD /m b ). Continuum evolution equation for propagator (for fixed background gauge field): ( G ψ (t 2, x, t, x ) = T exp t2 t 1 ) (H + ig A 0 ) dt G ψ (t 1, x, t, x )

29 Lattice NRQCD One the lattice, evolution by one time slice is implemented as follows [HPQCD]: ( G ψ (t, x, t, x ) = 1 δh ) ( ) n 1 H0 U 0 (t 1, x) 2 2n ( ) n ( 1 H0 1 δh ) G ψ (t 1, x, t, x ) 2n 2 Here, H 0 = 1 2m b (2) and δh contains relativistic and Symanzik-improvement corrections (split in H 0 and δh for historical/performance reasons). Need n 3/(2m b ) for numerical stability. Lattice NRQCD works for both heavy-light and heavy-heavy (and heavy-heavy-heavy!) systems. However, can not take continuum limit - need am b 1. Also possible: moving NRQCD [Horgan et al., PRD 2009]

30 Test of lattice NRQCD: speed of light In relativistic continuum QCD, energies of hadrons satisfy E 2 M 2 p 2 = 1. Lattice NRQCD energies are shifted by state-independent constant. Define with c 2 [E(p) E(0) + M kin,1] 2 M 2 kin,1 p 2 M kin p2 [E(p) E(0)] 2 2 [E(p) E(0)]

31 Test of lattice NRQCD: speed of light Square of the speed of light, calculated for the η b (1S) at p = n 2π/L: L = 24, a 0.11 fm L = 32, a 0.08 fm c n 2 [Meinel, arxiv:1007:3966] (with Wilson action, results for c 2 would be far away from 1)

32 Bottomonium spectrum [Meinel, arxiv:1007:3966]

33 RBC/UKQCD gauge field ensembles 2+1 flavors of domain wall fermions, exact chiral symmetry for L 5 even at finite a, no doubling problem better control over operator renormalization and chiral extrapolation, automatic O(a) improvement Iwasaki gluon action - suppresses residual chiral symmetry breaking at finite L fm lattices with L = 16, a 0.11 fm 2.7 fm lattices with L = 24, a 0.11 fm and L = 32, a 0.08 fm lowest pion mass about 300 MeV [Allton et al., PRD 2007, 2008]

34 NRQCD action Includes all terms of order v 4 and spin-dependent O(v 6 ) terms [Lepage et al. PRD 1992] H 0 = 1 2m (2), ) 2 ( (2) δh = c 1 c 3 8m 3 b g 8m 2 b a 2 (4) +c 5 24m b g c 7 8m 3 b ig ) + c 2 ( 8m Ẽ Ẽ 2 b ) σ ( Ẽ Ẽ ) 2 a ( (2) c 6 16n m 2 b { (2), σ B ig 2 c 9 σ (Ẽ 8m Ẽ). 3 b } 3g c 8 64m 4 b c 4 g 2m b σ B { ( (2), σ Ẽ Ẽ )} Tree-level: c i = 1. Radiative corrections to spin-dependent couplings not yet known!

35 Radial and orbital energy splittings: am b -dependence Data from L = 32 ensemble with am l = 0.004, order-v 4 action: am b = 1.75 am b = 1.87 am b = 2.05 Υ(2S) Υ(1S) (31) (33) (31) 2S 1S (32) (33) (31) 1 3 P Υ(1S) (22) (20) (19) 1 3 P 1S (22) (20) (19) 2 3 P 1 3 P (99) (94) (80) 2 3 P Υ(1S) 0.353(10) (94) (82) 2 3 P 1S 0.359(10) (94) (82) Υ 2(1D) Υ(1S) (39) (40) (42) Splittings nearly independent of am b

36 Kinetic mass: am b -dependence Kinetic mass of of η b (1S), defined as M kin p2 [E(p) E(0)] 2 amkin L = 32, a 0.08 fm Fit A am b + B 2 [E(p) E(0)] am b

37 Lattice spacing and am (phys.) b Use Υ(2S) Υ(1S) splitting to determine a Determine am (phys.) b such that M kin (η b ) agrees with experiment L 3 T β am l am s a 1 (GeV) am (phys.) b (52) 2.469(72) (46) 2.604(75) (33) 2.689(56) (27) 2.487(39) (28) 2.522(42) (42) 2.622(70) (39) 2.691(66) (32) 1.831(25) (45) 1.829(36) (32) 1.864(27)

38 Chiral extrapolation Interpolate spin splittings to am (phys.) b for each ensemble Convert to physical units on each ensemble Simultaneously extrapolate data from (L = 32, a 0.08 fm) and (L = 24, a 0.11 fm) to m π = 138 MeV E(m π, a 1 ) = E(0, a 1 ) + A m 2 π, E(m π, a 2 ) = E(0, a 2 ) + A m 2 π. Data from (L = 16, a 0.11 fm) ensemble extrapolated independently (different physical box size)

39 Radial and orbital energy splittings: chiral extrapolation L = 24, a 0.11 fm L = 32, a 0.08 fm Experiment L = 16, a 0.11 fm L = 24, a 0.11 fm Experiment Splitting (GeV) Υ(3S) Υ(1S) Splitting (GeV) Υ(3S) Υ(1S) m 2 π (GeV 2 ) m 2 π (GeV 2 )

40 Radial and orbital energy splittings: chiral extrapolation L = 24, a 0.11 fm L = 32, a 0.08 fm Experiment L = 16, a 0.11 fm L = 24, a 0.11 fm Experiment Splitting (GeV) P 1S Splitting (GeV) P 1S m 2 π (GeV 2 ) m 2 π (GeV 2 )

41 Radial and orbital energy splittings: chiral extrapolation 0.55 L = 24, a 0.11 fm L = 32, a 0.08 fm Experiment 0.55 L = 16, a 0.11 fm L = 24, a 0.11 fm Experiment Splitting (GeV) P 1 3 P Splitting (GeV) P 1 3 P m 2 π (GeV 2 ) m 2 π (GeV 2 )

42 Radial and orbital energy splittings: chiral extrapolation Splitting (GeV) L = 24, a 0.11 fm L = 32, a 0.08 fm Experiment Υ 2 (1D) Υ(1S) Splitting (GeV) L = 16, a 0.11 fm L = 24, a 0.11 fm Experiment Υ 2 (1D) Υ(1S) m 2 π (GeV 2 ) m 2 π (GeV 2 )

43 Radial and orbital energy splittings at m π = 138 MeV Υ(3S) 2 3 P Υ 2 (1D) E (GeV) Υ(2S) 1 3 P Υ(1S) Experiment L = 32, a 0.08 fm L = 24, a 0.11 fm L = 16, a 0.11 fm 9.2

44 Spin splittings: chiral extrapolation Splitting (MeV) m 2 π (GeV 2 ) v 4 action, a 0.11 fm v 4 action, a 0.08 fm Experiment Υ(1S) η b (1S) Splitting (MeV) m 2 π (GeV 2 ) v 6 action, a 0.11 fm v 6 action, a 0.08 fm Experiment Υ(1S) η b (1S) 1S hyperfine splitting At leading order: c 2 4, independent of c 3

45 Spin splittings: chiral extrapolation Splitting (MeV) v 4 action, a 0.11 fm v 4 action, a 0.08 fm Υ(2S) η b (2S) Splitting (MeV) v 6 action, a 0.11 fm v 6 action, a 0.08 fm Υ(2S) η b (2S) m 2 π (GeV 2 ) m 2 π (GeV 2 ) 2S hyperfine splitting At leading order: c 2 4, independent of c 3

46 Spin splittings: chiral extrapolation Splitting (MeV) m 2 π (GeV 2 ) v 4 action, a 0.11 fm v 4 action, a 0.08 fm Experiment 1P tensor Splitting (MeV) P tensor splitting m 2 π (GeV 2 ) v 6 action, a 0.11 fm v 6 action, a 0.08 fm Experiment 1P tensor 2χ b0 (1P ) + 3χ b1 (1P ) χ b2 (1P ) At leading order: c 2 4, independent of c 3

47 Spin splittings: chiral extrapolation Splitting (MeV) m 2 π (GeV 2 ) v 4 action, a 0.11 fm v 4 action, a 0.08 fm Experiment 1P spin-orbit Splitting (MeV) m 2 π (GeV 2 ) v 6 action, a 0.11 fm v 6 action, a 0.08 fm Experiment 1P spin-orbit 1P spin-orbit splitting 2χ b0 (1P ) 3χ b1 (1P ) + 5χ b2 (1P ) At leading order: c 3, independent of c 4

48 Spin splittings: chiral extrapolation Splitting (MeV) v 4 action, a 0.11 fm v 4 action, a 0.08 fm 1 3 P h b (1P ) Splitting (MeV) v 4 action, a 0.11 fm v 4 action, a 0.08 fm 1 3 P h b (1P ) m 2 π (GeV 2 ) 1P hyperfine splitting m 2 π (GeV 2 ) At leading order: zero 1 3 P h b (1P )

49 Spin splittings at m π = 138 MeV 40 E (MeV) χ b2 (1P) χ b1 (1P) χ b0 (1P) h b (1P) Experiment v 4 action,a 0.08 fm v 4 action,a 0.11 fm v 6 action,a 0.08 fm v 6 action,a 0.11 fm

50 Spin splittings at m π = 138 MeV 20 0 Υ(1S) Υ(2S) E (MeV) η b (1S) η b (2S) Experiment v 4 action,a 0.08 fm v 4 action,a 0.11 fm v 6 action,a 0.08 fm v 6 action,a 0.11 fm -80

51 Effect of v 6 terms on spin splittings S-wave hyperfine and P -wave spin-orbit splitting reduced by about 20% P -wave tensor splitting reduced by about 10% NB: for v 4 action, hyperfine and tensor splitting have similar physics

52 Radiative corrections to spin splittings At leading order, hyperfine and tensor splittings are expected to be proportional to c 2 4 and independent of c 3, so radiative corrections should cancel in the ratios and Υ(2S) η b (2S) Υ(1S) η b (1S) Υ(1S) η b (1S) 1P tensor Does this also work at order v 6?

53 Spin splittings: changing c 3 or c 4 splitting with c 3 1 or c 4 1 splitting with all c i = 1 c 3 = 0.8 c 3 = 1.2 c 4 = 0.8 c 4 = 1.2 Υ(1S) η b (1S) (18) (19) (53) (12) Υ(2S) η b (2S) 0.983(87) 1.025(91) 0.68(10) 1.35(14) 1P tensor 0.991(84) 1.008(76) 0.658(67) 1.40(11) 1P spin orbit 0.871(29) 1.129(31) 0.936(32) 1.059(39) Υ(2S) η b (2S) Υ(1S) η b (1S) Υ(1S) η b (1S) 1P tensor 1.003(89) 1.003(89) 1.02(15) 0.98(10) 0.989(83) 1.013(78) 1.02(10) 0.989(77) v 4 action, a 0.11 fm

54 Spin splittings: changing c 3 or c 4 splitting with c 3 1 or c 4 1 splitting with all c i = 1 c 3 = 0.8 c 3 = 1.2 c 4 = 0.8 c 4 = 1.2 Υ(1S) η b (1S) (17) (20) (47) (11) Υ(2S) η b (2S) 0.98(13) 1.03(13) 0.63(12) 1.44(19) 1P tensor 0.987(71) 1.006(62) 0.641(59) 1.41(11) 1P spin orbit 0.845(28) 1.154(32) 0.920(29) 1.077(40) Υ(2S) η b (2S) Υ(1S) η b (1S) Υ(1S) η b (1S) 1P tensor 1.00(13) 1.00(13) 0.97(19) 1.01(14) 0.991(75) 1.018(62) 1.008(95) 1.002(74) v 6 action, a 0.11 fm

55 Ratio of hyperfine splittings: chiral extrapolation Υ(2S) η b (2S) Υ(1S) η b (1S) v 4 action, a 0.11 fm v 4 action, a 0.08 fm Υ(2S) η b (2S) Υ(1S) η b (1S) v 6 action, a 0.11 fm v 6 action, a 0.08 fm Ratio Ratio m 2 π (GeV 2 ) m 2 π (GeV 2 ) Υ(2S) η b (2S) Υ(1S) η b (1S)

56 Ratio of hyperfine and tensor splittings: chiral extrap Υ(1S) η b (1S) 1P tensor v 4 action, a 0.11 fm v 4 action, a 0.08 fm Experiment Υ(1S) η b (1S) 1P tensor v 6 action, a 0.11 fm v 6 action, a 0.08 fm Experiment Ratio Ratio m 2 π (GeV 2 ) m 2 π (GeV 2 ) Υ(1S) η b (1S) 1P tensor

57 Spin splittings: final results (v 6 action, a 0.08 fm, m π = 138 MeV) Υ(2S) η b (2S) Υ(1S) η b (1S) This work 0.403(52)(25) - Experiment Υ(1S) η b (1S) 1P tensor 1.28(12)(8) 1.467(80) Υ(2S) η b (2S) 1P tensor 0.497(87)(32) - Υ(1S) η b (1S) 60.3(5.5)(3.8)(2.1) MeV a 69.3(2.9) MeV Υ(2S) η b (2S) 23.5(4.1)(1.5)(0.8) MeV a (3.6)(1.7)(1.2) MeV b 1 3 P h b (1P ) 0.04(93)(20) MeV - a Using 1P tensor splitting from experiment b Using Υ(1S) η b (1S) splitting from experiment 1st error: statistical/fitting, 2nd error: systematic, 3rd error: experimental Gluon discretization errors still missing, will be included in v2

58 Ω bbb [Meinel, arxiv:1008:3154]

59 Ω bbb correlator C (Ω) jk αδ (t, t, x ) = x ɛ abc ɛ fgh (Cγ j ) βγ (Cγ k ) ρσ with the NRQCD propagator G af βσ (t, x, t, x ) G bg γρ(t, x, t, x ) G ch αδ (t, x, t, x ) G(t, x, t, x ) = ( Gψ (t, x, t, x ) For quark smearing, include ( 1 + r ) ns S (2) n S at source and/or sink. ).

60 Ω bbb correlator Large (t t ): C (Ω) jk Z 2 3/2 e E 3/2 (t t ) 1 2 (1 + γ 0)(δ jk 1 3 γ jγ k ) + Z 2 1/2 e E 1/2 (t t ) 1 2 (1 + γ 0) 1 3 γ jγ k. Disentangle J = 3 2 and J = 1 2 the projectors contributions by multiplying with P (3/2) ij = (δ ij 1 3 γ iγ j ), P (1/2) ij = 1 3 γ iγ j. This gives P (J) ij C (Ω) jk Z2 J e E J (t t ) 1 2 (1 + γ 0)P (J) ik.

61 Ω bbb correlator: example Data from RBC/UKQCD ensemble with L = 32, am l = local local local smeared smeared local smeared smeared C(t) t Fit includes 7 exponentials and has t min = 5

62 Ω bbb correlator: example Data from RBC/UKQCD ensemble with L = 32, am l = ln[c(t)/c(t + 1)] local local local smeared smeared local smeared smeared t

63 Computing the Ω bbb mass Energies extracted from fits of two-point functions contain a shift that is proportional to the number of heavy quarks in the hadron. This shift cancels in the energy differences ae Ωbbb 3 2 ae Υ and ae Ωbbb 3 8 (ae η b + 3aE Υ ) }{{} = 3 2 (b b spin average)

64 Ω bbb : dependence on am b Splitting (lattice units) ae Ωbbb 3 8 (ae ηb + 3aE Υ) ae Ωbbb 3 2 ae Υ Splitting (lattice units) ae Ωbbb 3 8 (ae ηb + 3aE Υ) ae Ωbbb 3 2 ae Υ am b am b

65 Ω bbb : chiral extrapolation L = 24, a 0.11 fm L = 32, a 0.08 fm L = 16, a 0.11 fm L = 24, a 0.11 fm Splitting (GeV) E Ωbbb 3 8 (E η b + 3E Υ ) Splitting (GeV) E Ωbbb 3 8 (E η b + 3E Υ ) m 2 π (GeV 2 ) m 2 π (GeV 2 )

66 Ω bbb : chirally extrapolated/interpolated results Ensemble type L 3 T m π (GeV) E Ωbbb 3 8 (Eη b + 3EΥ) (GeV) RBC/UKQCD coarse (11) RBC/UKQCD coarse (44) RBC/UKQCD fine (29) MILC coarse (41) RBC/UKQCD coarse (22) MILC fine (24) RBC/UKQCD fine (24) MILC ensembles have more accurate gluon action (Lüscher-Weisz) but use rooted staggered sea quarks. Match R.M.S. pion mass. Use the following result: E Ωbbb 3 8 (E η b + 3E Υ ) = ± stat ± syst GeV.

67 E Ωbbb 3 8 (E η b + 3E Υ ): electrostatic correction E Coulomb = 3 (e/3)2 Ω bbb 1 4πɛ 0 r Ω bbb + 3 (e/3) 2 Υ 1 2 4πɛ 0 r Υ. Expectation values from potential models (for Ω bbb from [Silvestre-Brac, FBS 1996]): Υ 1 Υ = 8.1 fm 1 r Υ r 2 Υ = 0.20 fm Ωbbb r 2 Ω bbb = 0.25 fm Estimate Ω bbb r 1 Ω bbb = (0.8 ± 0.4) Υ r 1 Υ = 6.5 ± 3.2 fm 1 This gives E Coulomb = 5.1 ± 2.5 MeV.

68 Mass of the Ω bbb : final result M Ωbbb = [ E Ωbbb 3 ] 8 (E η b + 3E Υ ) [M Υ ] PDG 3 8 LQCD + E Coulomb [ EΥ E [ ] ηb 1P tensor ]LQCD 1P tensor PDG = ± stat ± syst ± exp GeV.

69 Mass of the Ω bbb : lattice QCD vs continuum models Reference M Ωbbb (GeV) Ponce, PRD Hasenfratz et al., PLB Bjorken ± 0.18 Tsuge et al., MPL Silvestre-Brac, FBS Jia, JHEP ± 0.08 Martynenko, PLB Roberts and Pervin, IJMPA Bernotas and Simonis, LJP Zhang and Huang, PLB ± 0.10 This work ± stat ± syst ± exp Note: results from Tsuge (1985) and Zhang/Huang (2009) violate baryon-meson mass inequality M Ωbbb 3 2 MΥ = GeV [Adler et al. PRD 1982, Nussinov PRL 1983, Richard PLB 1984]

70 Outlook Heavy-light hadrons (with W. Detmold et al.): we are currently generating more DWF propagators at a 0.08 fm. Spectrum results soon. Also: axial couplings Bottomonium: arxiv:1007:3966v2 will include study of gluon discretization errors. Currently investigating with lattice potential model Triply-heavy baryons: possibly include charm quarks, compute excited states THANK YOU!

71 Extra slides

72 Bottomonium: interpolating operators fix gauge configurations to Coulomb gauge, use smearing function Γ(r), 2 2-matrix-valued in spinor space O Γ (p, t) = x, x χ (x, t) Γ(x x ) ψ(x, t) e ip (x+x )/2 NB: choice of smearing only affects overlap with states, not their energies

73 Bottomonium: interpolating operators Name L S J P C R P C Γ(r) η b (ns) A + 1 φ ns(r) Υ(nS) T 1 φ ns(r) σ i h b (np ) T + 1 φ np (r) r i /r 0 χ b0 (np ) A ++ 1 φ np (r) (r σ)/r 0 χ b1 (np ) T ++ 1 φ np (r) (r σ) i /r 0 χ b2 (np ) T ++ 2 φ np (r) (r i σ j + r j σ i )/r 0 η b (nd) T + 2 φ nd(r) r i r j /r0 2 Υ 2(nD) E φ nd(r) (r i r j σ k r j r k σ i )/r0 2 State φ(r) 1S exp[ r /r 0] 2S [1 r /(2r 0)] exp[ r /(2r [ 0)] 3S 1 2 r /(3r0) + 2 r 2 /(27r0) ] 2 exp[ r /(3r 0)] 1P exp[ r /(2r 0)] 2P [1 r /(6r 0)] exp[ r /(3r 0)] 1D exp[ r /(3r 0)] (i j, k j)

74 Multi-exponential Bayesian fitting matrix fits with multiple radial smearing functions (e.g. 1S, 2S and 3S) at source and sink 30 correlator C(Γ sk, Γ sc, p, t t ) n exp 1 n=0 A n (Γ sc ) A n(γ sk ) e En(t t ) time

75 Multi-exponential Bayesian fitting actual fit parameters: ln(e 0 ), A 0 (Γ), and for n > 0 e n ln(e n E n 1 ), B n (Γ) A n (Γ)/A 0 (Γ) Bayesian fitting [Lepage et al., NPPS 2002]: χ 2 χ 2 + χ 2 prior with the Gaussian prior χ 2 prior = i σ 2 p i (p i p i ) 2 priors for low-lying states: central values from unconstrained fit at large t, width = 10 error from fit priors for high-lying states (for L = 24 ensemble, lattice units): ẽ n = 1.4, σẽn = 1, B n (Γ) = 0, σ Bn(Γ) = 5

76 Multi-exponential Bayesian fitting increase n exp until fit results stabilize Υ(3S) ae χ 2 /dof: Υ(2S) Υ(1S) n exp