Bottom hadron spectroscopy from lattice QCD

Stefan Meinel

Department of Physics

Jefferson Lab, October 11, 2010

Some puzzles concerningnon-excited non-exotic heavy hadrons

Ωb: Experiment

) (GeV)b

−ΩM(

5.8 6 6.2 6.4 6.6 6.8 7

Even

ts/(

0.0

4 G

eV

)

0

2

4

6

8

10

12

14

D0 −1

1.3 fbData

Fit

(a)

[D/0, PRL 2008]:MΩb = 6.165(10)(13) GeV

[CDF, PRD 2009]:MΩb = 6.0544(68)(9) GeV

About 6 standard deviations discrepancy

Ωb: Lattice QCD

Figure from [Lewis, arXiv:1010.0889]

See also: Fermilab + staggered [Na and Gottlieb, arXiv:0812.1235]

Our results with NRQCD + DWF at low pion mass will be available soon[Meinel et al., arXiv:0909.3837]

Quarkonium 1S hyperfine splitting: charmonium

Charmonium: M(J/ψ)−M(ηc)

Experiment: 116.6± 1.2 MeV [PDG, JPG 2010]

perturbative QCD (potential NRQCD): ∼ 110+50−30 MeV

[Kniehl et al. PRL 2004]

Quarkonium 1S hyperfine splitting: bottomonium

Bottomonium: M(Υ)−M(ηb)

Experiment: 69.3± 2.8 MeV [BABAR, PRL 2008, 2009,CLEO, PRD 2009]

perturbative QCD (potential NRQCD): 39± 14 MeV [Kniehlet al., PRL 2004]

Perturbation theory should work better in bottomonium than incharmonium. What is going on?

New physics in bottomonium?

Need precise lattice calculation to check perturbative QCD result.

M(Υ)−M(ηb): lattice QCD

M(Υ)−M(ηb) = 54± 12 MeV using Fermilab method[Burch et al., PRD 2010]

M(Υ)−M(ηb) = 61± 14 MeV using NRQCD of order v4

[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 removesthese two sources of error

Mass of the Ωbbb

Baryonic analogue of the Υ.

Reference MΩbbb (GeV)

Ponce, PRD 1979 14.248Hasenfratz et al., PLB 1980 14.30Bjorken 1985 14.76± 0.18Tsuge et al., MPL 1986 13.823Silvestre-Brac, FBS 1996 14.348 . . . 14.398Jia, JHEP 2006 14.37± 0.08Martynenko, PLB 2008 14.569Roberts and Pervin, IJMPA 2008 14.834Bernotas and Simonis, LJP 2009 14.276Zhang and Huang, PLB 2009 13.28± 0.10

1.5 GeV range! Later in this talk: lattice QCD result with12 MeV uncertainty

Heavy quarks on the lattice

Wilson Fermion action

SWF [Ψ,Ψ, U ] = a4

∑x∈aZ4

Ψ(x)

[γµ∇(±)

µ − 1

2a∇(+)

µ ∇(−)µ︸ ︷︷ ︸

removes doublers

+m

]Ψ(x)

with the lattice derivatives

∆+µψ(x) =

1

a[Uµ(x)ψ(x+ µ)− ψ(x)] ,

∆−µψ(x) =1

a[ψ(x)− U−µ(x)ψ(x− µ)] ,

∆±µψ(x) =1

2

[∆+µψ(x) + ∆−µψ(x)

],

where U−µ(x) = U †µ(x− aµ). Can define non-compact gauge fieldAµ through

Uµ(x) = exp [iagAµ(x)] .

Wilson Fermion action: dispersion relation

Energy as a function of momentum:

E(p) = m1 +p2

2m2+O(p4)

For the Wilson quark, at tree level:

m1 = m

(1− 1

2ma+

1

3m2a2 + ...

),

m2 = m

(1− 1

2ma+m2a2 + ...

),

m1

m2= 1− 2

3m2a2 + ...

This indicates large discretization errors (deviations from Lorentzinvariance) when ma not small

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 mπ and mb

1

a.

Thus, a huge number (L/a) of lattice points is needed. Anotherproblem at small a: critical slowing down of topological modes[Luscher, arXiv:1009.5877].

Relativistic b quarks on the lattice

Work at unphysically small m and extrapolate to mb:

introduces systematic errors

Anisotropic lattices with atmb 1 [Klassen, NPB 1998]:

there may still be (asmb)p errors [Harada et al., PRD 2001]

Highly improved actions remove some of the (amb)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

Nonrelativistic b quarks on the lattice

Alternative approach: start with nonrelativistic effective fieldtheory 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

Foldy-Wouthuysen-Tani transformation

Dirac Lagrangian (Minkowski space):

L = Ψ(−m+ iγ0D0 + iγjDj)Ψ

This describes both particles and antiparticles. Projectionoperators for quark / antiquark fields are

1

2(1 + γ0),

1

2(1− γ0)

The term iγjDj couples quarks and antiquarks, as it does notcommute with γ0

→ try to remove this term via field redefinition

Foldy-Wouthuysen-Tani transformation

Ψ = exp

(1

2miγjDj

)Ψ(1),

Ψ = Ψ(1) exp

(1

2miγjDj

)= Ψ(1) exp

(− 1

2miγj←Dj

)results in

L = Ψ(1)(−m+ iγ0D0)Ψ(1) +

∞∑n=1

1

mnΨ(1) O(1)n Ψ(1)

with

O(1)1 = −1

2DjD

j − ig

8[γµ, γν ]Fµν

= −1

2DjD

j − ig

8[γj , γk]Fjk︸ ︷︷ ︸

=OC(1)1

− ig2γj γ0Fj0︸ ︷︷ ︸

=OA(1)1

.

Foldy-Wouthuysen-Tani transformation

Next, remove OA(1)1 by another field redefinition

Ψ(1) = exp

(1

2m2OA(1)1

)Ψ(2),

Ψ(1) = Ψ(2) exp

(1

2m2OA(1)1

)This can be continued to any order in 1/m

Foldy-Wouthuysen-Tani transformation

One obtains

L = Ψ

[−m+ iγ0D0 −

1

2mDjD

j − ig

8m[γj , γk]Fjk

− g

8m2γ0

(Dadj Fj0 −

1

2[γj , γk] Dj , Fk0

)]Ψ

+O(1/m3)

All terms to the given order commute with γ0. The mass term canbe removed via

Ψ → exp(−imx0γ0

)Ψ,

Ψ → Ψ exp(imx0γ0

)

Foldy-Wouthuysen-Tani transformation

Next, write

Ψ =

(ψχ

), Ψ =

(ψ†, −χ†

)and

Ek = F0k, Bj = −1

2εjklFkl

Foldy-Wouthuysen-Tani transformation

One obtains

L = ψ†[iD0 +

D2

2m+

g

2mσ ·B

+g

8m2

((Dad ·E) + iσ · (D×E−E×D)

)]ψ

+ χ†[iD0 −

D2

2m− g

2mσ ·B

+g

8m2

((Dad ·E) + iσ · (D×E−E×D)

)]χ

+ O(1/m3)

Note: these are the tree-level values of the couplings

Power counting: heavy-light hadrons

b

|D0| ∼ |D| ∼ ΛQCD

Then, [Dµ, Dν ] = igFµν implies

|g E| ∼ |g B| ∼ Λ2QCD

Power counting: heavy-light hadrons

b

→ leading-order Lagrangian for heavy quark:

L = ψ† iD0 ψ.

Leads to heavy-quark spin- and flavor symmetry [Shifman,Voloshin, SJNP 1988]. Correction terms are suppressed by powersof (ΛQCD/mb).

Lattice HQET

Continuum Lagrangian (Euclidean):

L = δm ψ†ψ︸︷︷︸dim. 3

+ ψ†D0 ψ︸ ︷︷ ︸dim. 4

Includes all operators of dimension 4 or less that are compatiblewith symmetries → renormalizable!

Lattice action [Eichten, Hill, PLB 1990]:

S =∑x

ψ†(x)[(1 + δm)ψ(x)− U †0(x− 0)ψ(x− 0)

](lattice units with a = 1)

Lattice HQET

Propagator on given gauge field background = Wilson line

Gψ(x, x′) = δx, x′(1 + δm)−(t−t′+1)t−t′−1∏n=0

U †0(x′ + n0).

Treat (ΛQCD/mb) 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.

Power counting: heavy-heavy hadrons

b

b v

|D| ∼ mb v, |D0| ∼ Ekin ∼ mb v2

|g E| ∼ m2b v

3, |g B| ∼ m2b v

4

Power counting: heavy-heavy hadrons

b

b vLeading-order Lagrangian is

L = ψ†[iD0 +

D2

2m

]ψ + χ†

[iD0 −

D2

2m

]χ

Correction terms are suppressed by powers of v2. Forbottomonium, v2 ∼ 0.1.

NRQCD

Continuum Lagrangian (Euclidean):

Lψ = ψ† (D0 +H)ψ

where H contains all terms up to desired order in v2 or(ΛQCD/mb).

Continuum evolution equation for propagator (for fixed backgroundgauge field):

Gψ(t2,x, t′,x′) = T exp

(−∫ t2

t1

(H + ig A0) dt

)Gψ(t1,x, t

′,x′)

Lattice NRQCD

One the lattice, evolution by one time slice is implemented as follows [HPQCD]:

Gψ(t,x, t′,x′) =

(1− δH

2

)(1− H0

2n

)nU†0 (t− 1,x)

×(

1− H0

2n

)n (1− δH

2

)Gψ(t− 1,x, t′,x′)

Here,

H0 = − 1

2mb∆(2)

and δH contains relativistic and Symanzik-improvement corrections (split inH0 and δH for historical/performance reasons).

Need n & 3/(2mb) for numerical stability.

Lattice NRQCD works for both heavy-light and heavy-heavy (andheavy-heavy-heavy!) systems. However, can not take continuum limit - needamb & 1.

Also possible: moving NRQCD [Horgan et al., PRD 2009]

Test of lattice NRQCD: “speed of light”

In relativistic continuum QCD, energies of hadrons satisfy

E2 −M2

p2= 1.

Lattice NRQCD energies are shifted by state-independent constant.Define

c2 ≡[E(p)− E(0) +Mkin,1]2 −M2

kin,1

p2

with

Mkin ≡p2 − [E(p)− E(0)]2

2 [E(p)− E(0)]

Test of lattice NRQCD: “speed of light”

Square of the speed of light, calculated for the ηb(1S) atp = n · 2π/L:

0.99

0.995

1

1.005

1.01

0 1 2 3 4 5 6 7 8 9 10 11 12

c2

n2

L= 24, a≈ 0.11 fm

L= 32, a≈ 0.08 fm

[Meinel, arXiv:1007:3966]

(with Wilson action, results for c2 would be far away from 1)

Bottomonium spectrum

[Meinel, arXiv:1007:3966]

RBC/UKQCD gauge field ensembles

2+1 flavors of domain wall fermions, exact chiral symmetryfor L5 →∞ even at finite a, no doubling problem

better control over operator renormalization and chiralextrapolation, automatic O(a) improvement

Iwasaki gluon action - suppresses residual chiral symmetrybreaking at finite L5

1.8 fm lattices with L = 16, a ≈ 0.11 fm

2.7 fm lattices with L = 24, a ≈ 0.11 fm andL = 32, a ≈ 0.08 fm

lowest pion mass about 300 MeV

[Allton et al., PRD 2007, 2008]

NRQCD action

Includes all terms of order v4 and spin-dependent O(v6) terms[Lepage et al. PRD 1992]

H0 = − 1

2m∆(2),

δH = −c1

(∆(2)

)2

8m3b

+ c2ig

8m2b

(∇ · E− E · ∇

)−c3

g

8m2b

σ ·(∇ × E− E× ∇

)− c4

g

2mbσ · B

+c5a2∆(4)

24mb− c6

a(

∆(2))2

16nm2b

−c7g

8m3b

∆(2), σ · B

−c8

3g

64m4b

∆(2), σ ·

(∇ × E− E× ∇

)−c9

ig2

8m3b

σ · (E× E).

Tree-level: ci = 1. Radiative corrections to spin-dependent couplings not yetknown!

Radial and orbital energy splittings: amb-dependence

Data from L = 32 ensemble with aml = 0.004, order-v4 action:

amb = 1.75 amb = 1.87 amb = 2.05

Υ(2S)−Υ(1S) 0.2422(31) 0.2421(33) 0.2418(31)

2S − 1S 0.2456(32) 0.2454(33) 0.2448(31)

13P −Υ(1S) 0.1901(22) 0.1907(20) 0.1918(19)

13P − 1S 0.1965(22) 0.1969(20) 0.1975(19)

23P − 13P 0.1645(99) 0.1629(94) 0.1592(80)

23P −Υ(1S) 0.353(10) 0.3519(94) 0.3494(82)

23P − 1S 0.359(10) 0.3580(94) 0.3552(82)Υ2(1D)−Υ(1S) 0.3048(39) 0.3051(40) 0.3059(42)

→ Splittings nearly independent of amb

Kinetic mass: amb-dependence

Kinetic mass of of ηb(1S), defined as Mkin ≡p2 − [E(p)− E(0)]2

2 [E(p)− E(0)]

1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10amb

3.9

4.0

4.1

4.2

4.3

4.4

4.5

4.6aM

kin

L = 32, a ≈ 0.08 fm

Fit A · amb + B

Lattice spacing and am(phys.)b

Use Υ(2S)−Υ(1S) splitting to determine a

Determine am(phys.)b such that Mkin(ηb) agrees with

experiment

L3 × T β aml ams a−1 (GeV) am(phys.)b

163 × 32 2.13 0.01 0.04 1.766(52) 2.469(72)163 × 32 2.13 0.02 0.04 1.687(46) 2.604(75)163 × 32 2.13 0.03 0.04 1.651(33) 2.689(56)

243 × 64 2.13 0.005 0.04 1.763(27) 2.487(39)243 × 64 2.13 0.01 0.04 1.732(28) 2.522(42)243 × 64 2.13 0.02 0.04 1.676(42) 2.622(70)243 × 64 2.13 0.03 0.04 1.650(39) 2.691(66)

323 × 64 2.25 0.004 0.03 2.325(32) 1.831(25)323 × 64 2.25 0.006 0.03 2.328(45) 1.829(36)323 × 64 2.25 0.008 0.03 2.285(32) 1.864(27)

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 mπ = 138 MeV

E(mπ, a1) = E(0, a1) +Am2π,

E(mπ, a2) = E(0, a2) +Am2π.

Data from (L = 16, a ≈ 0.11 fm) ensemble extrapolatedindependently (different physical box size)

Radial and orbital energy splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Sp

litti

ng

(GeV

)

Υ(3S)− Υ(1S)

L = 24, a ≈ 0.11 fmL = 32, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Sp

litti

ng

(GeV

)

Υ(3S)− Υ(1S)

L = 16, a ≈ 0.11 fmL = 24, a ≈ 0.11 fmExperiment

Radial and orbital energy splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.42

0.44

0.46

0.48

0.50

0.52

Sp

litti

ng

(GeV

)

13P − 1S

L = 24, a ≈ 0.11 fmL = 32, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.42

0.44

0.46

0.48

0.50

0.52

Sp

litti

ng

(GeV

)

13P − 1S

L = 16, a ≈ 0.11 fmL = 24, a ≈ 0.11 fmExperiment

Radial and orbital energy splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.30

0.35

0.40

0.45

0.50

0.55

Sp

litti

ng

(GeV

)

23P − 13P

L = 24, a ≈ 0.11 fmL = 32, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.30

0.35

0.40

0.45

0.50

0.55

Sp

litti

ng

(GeV

)

23P − 13P

L = 16, a ≈ 0.11 fmL = 24, a ≈ 0.11 fmExperiment

Radial and orbital energy splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.80

Sp

litti

ng

(GeV

)

Υ2(1D)− Υ(1S)

L = 24, a ≈ 0.11 fmL = 32, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.66

0.68

0.70

0.72

0.74

0.76

0.78

0.80

Sp

litti

ng

(GeV

)

Υ2(1D)− Υ(1S)

L = 16, a ≈ 0.11 fmL = 24, a ≈ 0.11 fmExperiment

Radial and orbital energy splittings at mπ = 138 MeV

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6E

(G

eV)

Υ(1S)

Υ(2S)

Υ(3S)

13P–

23P–

Υ2(1D)

ExperimentL= 32, a≈ 0.08 fm

L= 24, a≈ 0.11 fm

L= 16, a≈ 0.11 fm

Spin splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

35

40

45

50

55

60

65

70

75

Sp

litti

ng

(MeV

)

Υ(1S)− ηb(1S)

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

35

40

45

50

55

60

65

70

75

Sp

litti

ng

(MeV

)

Υ(1S)− ηb(1S)

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fmExperiment

1S hyperfine splitting

At leading order: ∝ c24, independent of c3

Spin splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0

10

20

30

40

50

Sp

litti

ng

(MeV

) Υ(2S)− ηb(2S)

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fm

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0

10

20

30

40

50

Sp

litti

ng

(MeV

) Υ(2S)− ηb(2S)

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fm

2S hyperfine splitting

At leading order: ∝ c24, independent of c3

Spin splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

20

25

30

35

40

45

50

55

60

Sp

litti

ng

(MeV

)

1P tensor

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

20

25

30

35

40

45

50

55

60

Sp

litti

ng

(MeV

)

1P tensor

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fmExperiment

1P tensor splitting

−2χb0(1P ) + 3χb1(1P )− χb2(1P )

At leading order: ∝ c24, independent of c3

Spin splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

80

100

120

140

160

180

200

220

240

Sp

litti

ng

(MeV

)

1P spin-orbit

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

80

100

120

140

160

180

200

220

240

Sp

litti

ng

(MeV

)

1P spin-orbit

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fmExperiment

1P spin-orbit splitting

−2χb0(1P )− 3χb1(1P ) + 5χb2(1P )

At leading order: ∝ c3, independent of c4

Spin splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0

2

4

6

8

10

12

Sp

litti

ng

(MeV

) 13P − hb(1P )

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fm

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0

2

4

6

8

10

12

Sp

litti

ng

(MeV

) 13P − hb(1P )

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fm

1P hyperfine splitting

13P − hb(1P )

At leading order: zero

Spin splittings at mπ = 138 MeV

-60

-40

-20

0

20

40∆E

(M

eV)

χb0(1P)

χb1(1P)

χb2(1P)

hb(1P)

Experiment

v action, a≈ 0.08 fm

v action, a≈ 0.11 fm

v action, a≈ 0.08 fm

v action, a≈ 0.11 fm

Spin splittings at mπ = 138 MeV

-80

-60

-40

-20

0

20∆E

(M

eV)

Υ(1S)

ηb(1S)

Υ(2S)

ηb(2S)

Experiment

v action, a≈ 0.08 fm

v action, a≈ 0.11 fm

v action, a≈ 0.08 fm

v action, a≈ 0.11 fm

Effect of v6 terms on spin splittings

S-wave hyperfine and P -wave spin-orbit splitting reduced byabout 20%

P -wave tensor splitting reduced by about 10%

NB: for v4 action, hyperfine and tensor splitting have similarphysics

Radiative corrections to spin splittings

At leading order, hyperfine and tensor splittings are expected to beproportional to c2

4 and independent of c3, so radiative correctionsshould cancel in the ratios

Υ(2S)− ηb(2S)

Υ(1S)− ηb(1S)

and

Υ(1S)− ηb(1S)

1P tensor

Does this also work at order v6 ?

Spin splittings: changing c3 or c4

splitting with c3 6= 1 or c4 6= 1

splitting with all ci = 1

c3 = 0.8 c3 = 1.2 c4 = 0.8 c4 = 1.2

Υ(1S)− ηb(1S) 0.98016(18) 1.02148(19) 0.67151(53) 1.3808(12)Υ(2S)− ηb(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)− ηb(2S)

Υ(1S)− ηb(1S)1.003(89) 1.003(89) 1.02(15) 0.98(10)

Υ(1S)− ηb(1S)

1P tensor0.989(83) 1.013(78) 1.02(10) 0.989(77)

v4 action, a ≈ 0.11 fm

Spin splittings: changing c3 or c4

splitting with c3 6= 1 or c4 6= 1

splitting with all ci = 1

c3 = 0.8 c3 = 1.2 c4 = 0.8 c4 = 1.2

Υ(1S)− ηb(1S) 0.97788(17) 1.02411(20) 0.64656(47) 1.4180(11)Υ(2S)− ηb(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)− ηb(2S)

Υ(1S)− ηb(1S)1.00(13) 1.00(13) 0.97(19) 1.01(14)

Υ(1S)− ηb(1S)

1P tensor0.991(75) 1.018(62) 1.008(95) 1.002(74)

v6 action, a ≈ 0.11 fm

Ratio of hyperfine splittings: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.0

0.2

0.4

0.6

0.8

1.0

Rat

io

Υ(2S)− ηb(2S)

Υ(1S)− ηb(1S)

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fm

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.0

0.2

0.4

0.6

0.8

1.0

Rat

io

Υ(2S)− ηb(2S)

Υ(1S)− ηb(1S)

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fm

Υ(2S)− ηb(2S)

Υ(1S)− ηb(1S)

Ratio of hyperfine and tensor splittings: chiral extrap.

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Rat

io

Υ(1S)− ηb(1S)

1P tensor

v4 action, a ≈ 0.11 fmv4 action, a ≈ 0.08 fmExperiment

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

Rat

io

Υ(1S)− ηb(1S)

1P tensor

v6 action, a ≈ 0.11 fmv6 action, a ≈ 0.08 fmExperiment

Υ(1S)− ηb(1S)

1P tensor

Spin splittings: final results (v6 action, a ≈ 0.08 fm, mπ = 138 MeV)

This work Experiment

Υ(2S)− ηb(2S)

Υ(1S)− ηb(1S)0.403(52)(25) -

Υ(1S)− ηb(1S)

1P tensor1.28(12)(8) 1.467(80)

Υ(2S)− ηb(2S)

1P tensor0.497(87)(32) -

Υ(1S)− ηb(1S) 60.3(5.5)(3.8)(2.1) MeV a 69.3(2.9) MeV

Υ(2S)− ηb(2S)23.5(4.1)(1.5)(0.8) MeV a

-28.0(3.6)(1.7)(1.2) MeV b

13P − hb(1P ) 0.04(93)(20) MeV -

a Using 1P tensor splitting from experimentb Using Υ(1S)− ηb(1S) splitting from experiment

1st error: statistical/fitting, 2nd error: systematic, 3rd error: experimental

Gluon discretization errors still missing, will be included in v2

Ωbbb

[Meinel, arXiv:1008:3154]

Ωbbb correlator

C(Ω)jk αδ(t, t

′,x′) =∑x

εabc εfgh (Cγj)βγ (Cγk)ρσ

×Gafβσ(t,x, t′,x′)Gbgγρ(t,x, t′,x′)Gchαδ(t,x, t

′,x′)

with the NRQCD propagator

G(t,x, t′,x′) =

(Gψ(t,x, t′,x′) 0

0 0

).

For quark smearing, include(1 +

rSnS

∆(2)

)nSat source and/or sink.

Ωbbb correlator

Large (t− t′):

C(Ω)jk → Z2

3/2 e−E3/2 (t−t′) 1

2(1 + γ0)(δjk − 13γjγk)

+ Z21/2 e

−E1/2 (t−t′) 12(1 + γ0)1

3γjγk.

Disentangle J = 32 and J = 1

2 contributions by multiplying withthe projectors

P(3/2)ij = (δij − 1

3γiγj),

P(1/2)ij = 1

3γiγj .

This gives

P(J)ij C

(Ω)jk → Z2

J e−EJ (t−t′) 1

2(1 + γ0)P(J)ik .

Ωbbb correlator: example

Data from RBC/UKQCD ensemble with L = 32, aml = 0.004

10 15 20 25 30 35 40

t

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

C(t

)local− locallocal− smearedsmeared− localsmeared− smeared

Fit includes 7 exponentials and has tmin = 5

Ωbbb correlator: example

Data from RBC/UKQCD ensemble with L = 32, aml = 0.004

10 15 20 25 30 35 40

t

0.48

0.49

0.50

0.51

0.52

0.53

0.54ln

[C(t

)/C

(t+

1)]

local− locallocal− smearedsmeared− localsmeared− smeared

Computing the Ωbbb mass

Energies extracted from fits of two-point functions contain a shiftthat is proportional to the number of heavy quarks in the hadron.

This shift cancels in the energy differences

aEΩbbb −3

2aEΥ

and

aEΩbbb −3

8(aEηb + 3aEΥ)︸ ︷︷ ︸

= 32×(bb spin average)

Ωbbb: dependence on amb

2.3 2.4 2.5 2.6 2.7amb

0.100

0.105

0.110

0.115

0.120

Sp

litti

ng

(lat

tice

un

its)

aEΩbbb− 3

8 (aEηb + 3aEΥ)

aEΩbbb− 3

2aEΥ

1.75 1.80 1.85 1.90 1.95 2.00 2.05amb

0.072

0.074

0.076

0.078

0.080

0.082

0.084

0.086

0.088

Sp

litti

ng

(lat

tice

un

its)

aEΩbbb− 3

8 (aEηb + 3aEΥ)

aEΩbbb− 3

2aEΥ

Ωbbb: chiral extrapolation

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.18

0.19

0.20

0.21

0.22

0.23

0.24

Sp

litti

ng

(GeV

)

EΩbbb− 3

8 (Eηb + 3EΥ)

L = 24, a ≈ 0.11 fmL = 32, a ≈ 0.08 fm

0.0 0.1 0.2 0.3 0.4 0.5

m2π (GeV2)

0.18

0.19

0.20

0.21

0.22

0.23

0.24

Sp

litti

ng

(GeV

)

EΩbbb− 3

8 (Eηb + 3EΥ)

L = 16, a ≈ 0.11 fmL = 24, a ≈ 0.11 fm

Ωbbb: chirally extrapolated/interpolated results

Ensemble type L3 × T mπ (GeV) EΩbbb − 38

(Eηb + 3EΥ) (GeV)

RBC/UKQCD coarse 163 × 32 0.138 0.214(11)RBC/UKQCD coarse 243 × 64 0.138 0.2044(44)RBC/UKQCD fine 323 × 64 0.138 0.1984(29)

MILC coarse 243 × 64 0.460 0.2063(41)RBC/UKQCD coarse 243 × 64 0.460 0.2022(22)

MILC fine 283 × 96 0.416 0.2008(24)RBC/UKQCD fine 323 × 64 0.416 0.1966(24)

MILC ensembles have more accurate gluon action (Luscher-Weisz) but userooted staggered sea quarks. Match R.M.S. pion mass.

Use the following result:

EΩbbb −3

8(Eηb + 3EΥ) = 0.198± 0.003 stat ± 0.011 syst GeV.

EΩbbb− 3

8 (Eηb + 3EΥ): electrostatic correction

ECoulomb = 3(e/3)2

4πε0〈Ωbbb|

1

r|Ωbbb〉+

3

2

(e/3)2

4πε0〈Υ|1

r|Υ〉.

Expectation values from potential models (for Ωbbb from[Silvestre-Brac, FBS 1996]):

〈Υ|1r|Υ〉 = 8.1 fm−1√

〈Υ|r2|Υ〉 = 0.20 fm√〈Ωbbb|r2|Ωbbb〉 = 0.25 fm

Estimate

〈Ωbbb|r−1|Ωbbb〉 = (0.8± 0.4)〈Υ|r−1|Υ〉 = 6.5± 3.2 fm−1

This givesECoulomb = 5.1± 2.5 MeV.

Mass of the Ωbbb: final result

MΩbbb =

[EΩbbb −

3

8(Eηb + 3EΥ)

]LQCD

+ ECoulomb

+3

2

[MΥ

]PDG− 3

8

[ EΥ − Eηb1P tensor

]LQCD

×[1P tensor

]PDG

= 14.371± 0.004 stat ± 0.011 syst ± 0.001 exp GeV.

Mass of the Ωbbb: lattice QCD vs continuum models

Reference MΩbbb (GeV)

Ponce, PRD 1979 14.248Hasenfratz et al., PLB 1980 14.30Bjorken 1985 14.76± 0.18Tsuge et al., MPL 1986 13.823Silvestre-Brac, FBS 1996 14.348 . . . 14.398Jia, JHEP 2006 14.37± 0.08Martynenko, PLB 2008 14.569Roberts and Pervin, IJMPA 2008 14.834Bernotas and Simonis, LJP 2009 14.276Zhang and Huang, PLB 2009 13.28± 0.10

This work 14.371± 0.004 stat ± 0.011 syst ± 0.001 exp

Note: results from Tsuge (1985) and Zhang/Huang (2009) violatebaryon-meson mass inequality

MΩbbb ≥3

2MΥ = 14.1904 GeV

[Adler et al. PRD 1982, Nussinov PRL 1983, Richard PLB 1984]

Outlook

Heavy-light hadrons (with W. Detmold et al.): we arecurrently generating more DWF propagators at a ≈ 0.08 fm.Spectrum results soon. Also: axial couplings

Bottomonium: arXiv:1007:3966v2 will include study of gluondiscretization errors. Currently investigating with latticepotential model

Triply-heavy baryons: possibly include charm quarks, computeexcited states

THANK YOU!

Extra slides

Bottomonium: interpolating operators

fix gauge configurations to Coulomb gauge, use “smearing”function Γ(r), 2× 2-matrix-valued in spinor space

OΓ(p, t) =∑x, x′

χ†(x, t) Γ(x− x′) ψ(x′, t) eip·(x+x′)/2

NB: choice of smearing only affects overlap with states, not their energies

Bottomonium: interpolating operators

Name L S J P C RPC Γ(r)

ηb(nS) 0 0 0 − + A−+1 φnS(r)

Υ(nS) 0 1 1 − − T−−1 φnS(r) σi

hb(nP ) 1 0 1 + − T+−1 φnP (r) ri/r0

χb0(nP ) 1 1 0 + + A++1 φnP (r) (r · σ)/r0

χb1(nP ) 1 1 1 + + T++1 φnP (r) (r× σ)i/r0

χb2(nP ) 1 1 2 + + T++2 φnP (r) (riσj + rjσi)/r0

ηb(nD) 2 0 2 − + T−+2 φnD(r) rirj/r2

0

Υ2(nD) 2 1 2 − − E−− φnD(r) (rirjσk − rjrkσi)/r20

(i 6= j, k 6= j)

State φ(r)

1S exp[−|r|/r0]

2S [1− |r|/(2r0)] exp[−|r|/(2r0)]

3S[1− 2|r|/(3r0) + 2|r|2/(27r2

0)]

exp[−|r|/(3r0)]

1P exp[−|r|/(2r0)]

2P [1− |r|/(6r0)] exp[−|r|/(3r0)]

1D exp[−|r|/(3r0)]

Multi-exponential Bayesian fitting

matrix fits with multiple radial smearing functions (e.g. 1S,2S and 3S) at source and sink

0 5 10 15 20

time

0

5

10

15

20

25

30

corr

elat

or

〈C(Γsk,Γsc,p, t− t′)〉

≈nexp−1∑n=0

An(Γsc)A∗n(Γsk) e−En(t−t′)

Multi-exponential Bayesian fitting

actual fit parameters: ln(E0), A0(Γ), and for n > 0

en ≡ ln(En − En−1),

Bn(Γ) ≡ An(Γ)/A0(Γ)

Bayesian fitting [Lepage et al., NPPS 2002]:χ2 → χ2 + χ2

prior with the Gaussian prior

χ2prior =

∑i

(pi − pi)2

σ2pi

priors for low-lying states: central values from unconstrainedfit at large t, width = 10× error from fit

priors for high-lying states (for L = 24 ensemble, lattice units):

en = −1.4, σen = 1,

Bn(Γ) = 0, σBn(Γ)

= 5

Multi-exponential Bayesian fitting

increase nexp until fit results stabilize

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12

aE

nexp

Υ(1S)

Υ(2S)

Υ(3S)χ

2 /dof

:

45.

8

2.8

3

1.8

3

0.8

6

0.8

0

0.7

9

0.7

9

0.7

9