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