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Bottom hadron spectroscopy from lattice QCD
Stefan Meinel
Department of Physics
Jefferson Lab, October 11, 2010
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Some puzzles concerningnon-excited non-exotic heavy hadrons
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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
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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]
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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]
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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?
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New physics in bottomonium?
Need precise lattice calculation to check perturbative QCD result.
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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
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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
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Heavy quarks on the lattice
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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)] .
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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
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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].
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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
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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
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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
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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
.
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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
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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
)
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Foldy-Wouthuysen-Tani transformation
Next, write
Ψ =
(ψχ
), Ψ =
(ψ†, −χ†
)and
Ek = F0k, Bj = −1
2εjklFkl
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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
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Power counting: heavy-light hadrons
b
|D0| ∼ |D| ∼ ΛQCD
Then, [Dµ, Dν ] = igFµν implies
|g E| ∼ |g B| ∼ Λ2QCD
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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).
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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)
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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.
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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
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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.
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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′)
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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]
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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)]
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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)
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Bottomonium spectrum
[Meinel, arXiv:1007:3966]
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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]
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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!
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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
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ?
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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
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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
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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)
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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
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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
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Ωbbb
[Meinel, arXiv:1008:3154]
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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.
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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 .
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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
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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
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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)
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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
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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
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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.
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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.
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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.
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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]
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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!
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Extra slides
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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
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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)]
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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′)
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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
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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