Transcript
Bottom hadron spectroscopy from lattice QCD Stefan Meinel Department of Physics
Jefferson Lab, October 11, 2010
Some puzzles concerning non-excited non-exotic heavy hadrons
Events/(0.04 GeV)
Ωb : Experiment
14
(a)
D0 1.3 fb −1
Data Fit
12 10 8 6 4 2 0
5.8
6
6.2
6.4
6.6
−
6.8
7
M(Ωb) (GeV)
[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 [Kniehl et al., PRL 2004] Perturbation theory should work better in bottomonium than in charmonium. 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 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
Mass of the Ωbbb Baryonic analogue of the Υ. Reference Ponce, PRD 1979 Hasenfratz et al., PLB 1980 Bjorken 1985 Tsuge et al., MPL 1986 Silvestre-Brac, FBS 1996 Jia, JHEP 2006 Martynenko, PLB 2008 Roberts and Pervin, IJMPA 2008 Bernotas and Simonis, LJP 2009 Zhang and Huang, PLB 2009
MΩbbb (GeV) 14.248 14.30 14.76 ± 0.18 13.823 14.348 . . . 14.398 14.37 ± 0.08 14.569 14.834 14.276 13.28 ± 0.10
1.5 GeV range! Later in this talk: lattice QCD result with 12 MeV uncertainty
Heavy quarks on the lattice
Wilson Fermion action "
SFW [Ψ, Ψ, U ] = a4
X
x∈aZ
# 1 (−) Ψ(x) γµ ∇(±) a∇(+) +m Ψ(x) µ − µ ∇µ 2 4 {z } | removes doublers
with the lattice derivatives ∆+ µ ψ(x) = ∆− µ ψ(x) = ∆± µ ψ(x) =
1 [Uµ (x)ψ(x + µ ˆ) − ψ(x)] , a 1 [ψ(x) − U−µ (x)ψ(x − µ ˆ)] , a 1 + ∆µ ψ(x) + ∆− µ ψ(x) , 2
where U−µ (x) = Uµ† (x − aˆ µ). Can define non-compact gauge field Aµ through Uµ (x) = exp [iagAµ (x)] .
Wilson Fermion action: dispersion relation Energy as a function of momentum: E(p) = m1 +
p2 + O(p4 ) 2m2
For the Wilson quark, at tree level: 1 1 2 2 m1 = m 1 − ma + m a + ... , 2 3 1 m2 = m 1 − ma + m2 a2 + ... , 2 m1 2 = 1 − m2 a2 + ... m2 3 This indicates large discretization errors (deviations from Lorentz invariance) 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 1 mπ and mb . L a Thus, a huge number (L/a) of lattice points is needed. Another problem at small a: critical slowing down of topological modes [L¨ uscher, arXiv:1009.5877].
Relativistic b quarks on the lattice Work at unphysically small m and extrapolate to mb : introduces systematic errors Anisotropic lattices with at mb 1 [Klassen, NPB 1998]:
there may still be (as mb )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 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
Foldy-Wouthuysen-Tani transformation
Dirac Lagrangian (Minkowski space): L = Ψ(−m + iˆ γ 0 D0 + iˆ γ j Dj )Ψ This describes both particles and antiparticles. Projection operators for quark / antiquark fields are 1 (1 + γˆ 0 ), 2
1 (1 − γˆ 0 ) 2
The term iˆ γ j Dj couples quarks and antiquarks, as it does not commute with γˆ 0 → try to remove this term via field redefinition
Foldy-Wouthuysen-Tani transformation
1 j Ψ = exp iˆ γ Dj Ψ(1) , 2m ← 1 1 Ψ = Ψ(1) exp iˆ γ j Dj = Ψ(1) exp − iˆ γ j Dj 2m 2m results in ∞ X 1 L = Ψ(1) (−m + iˆ γ D0 )Ψ(1) + Ψ O Ψ mn (1) (1)n (1) 0
n=1
with 1 ig µ ν O(1)1 = − Dj Dj − [ˆ γ , γˆ ]Fµν 2 8 1 ig j k ig = − Dj Dj − [ˆ γ , γˆ ]Fjk − γˆ j γˆ 0 Fj0 . | 2 {z8 } | 2 {z } C =O(1)1
A =O(1)1
Foldy-Wouthuysen-Tani transformation
A Next, remove O(1)1 by another field redefinition
Ψ(1) Ψ(1)
1 A = exp O Ψ(2) , 2m2 (1)1 1 A = Ψ(2) exp O 2m2 (1)1
This can be continued to any order in 1/m
Foldy-Wouthuysen-Tani transformation One obtains 1 ig j k ˜ L = Ψ −m + iˆ γ 0 D0 − Dj D j − [ˆ γ , γˆ ]Fjk 2m 8m g 0 1 j k ad ˜ − 2 γˆ Dj Fj0 − [ˆ γ , γˆ ] {Dj , Fk0 } Ψ 8m 2 +O(1/m3 ) All terms to the given order commute with γˆ 0 . The mass term can be removed via ˜ → exp −imx0 γˆ 0 Ψ, ˜ Ψ ˜ → Ψ ˜ exp imx0 γˆ 0 Ψ
Foldy-Wouthuysen-Tani transformation
Next, write ˜ = Ψ
ψ χ
,
˜ = ψ†, Ψ
−χ†
and Ek = F0k ,
1 Bj = − jkl Fkl 2
Foldy-Wouthuysen-Tani transformation One obtains D2 g L = ψ iD0 + + σ·B 2m 2m g ad + 2 (D ·E) + iσ · (D × E − E × D) ψ 8m D2 g + χ† iD0 − − σ·B 2m 2m g ad + 2 (D ·E) + iσ · (D × E − E × D) χ 8m †
+ 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 powers of (ΛQCD /mb ).
Lattice HQET
Continuum Lagrangian (Euclidean): L = δm ψ † ψ + ψ † D0 ψ |{z} | {z } dim. 3
dim. 4
Includes all operators of dimension 4 or less that are compatible with symmetries → renormalizable! Lattice action [Eichten, Hill, PLB 1990]: S=
X x
ψ † (x) (1 + δm)ψ(x) − U0† (x − ˆ 0)ψ(x − ˆ0)
(lattice units with a = 1)
Lattice HQET
Propagator on given gauge field background = Wilson line 0
−(t−t0 +1)
Gψ (x, x ) = δx, x0 (1 + δm)
0 t−t Y−1
U0† (x0 + nˆ0).
n=0
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
v
b
|D| ∼ mb v,
|D0 | ∼ Ekin ∼ mb v 2
|g E| ∼ m2b v 3 ,
|g B| ∼ m2b v 4
Power counting: heavy-heavy hadrons
b
b
v
Leading-order Lagrangian is D2 D2 † † L = ψ iD0 + ψ + χ iD0 − χ 2m 2m Correction terms are suppressed by powers of v 2 . For bottomonium, v 2 ∼ 0.1.
NRQCD
Continuum Lagrangian (Euclidean): Lψ = ψ † (D0 + H) ψ where H contains all terms up to desired order in v 2 or (ΛQCD /mb ). Continuum evolution equation for propagator (for fixed background gauge field): Z t2 Gψ (t2 , x, t0 , x0 ) = T exp − (H + ig A0 ) dt Gψ (t1 , x, t0 , x0 ) t1
Lattice NRQCD One the lattice, evolution by one time slice is implemented as follows [HPQCD]: n H0 δH 1− Gψ (t, x, t0 , x0 ) = 1− U0† (t − 1, x) 2 2n n H0 δH × 1− 1− Gψ (t − 1, x, t0 , x0 ) 2n 2 Here,
1 ∆(2) 2mb and δH contains relativistic and Symanzik-improvement corrections (split in H0 and δH for historical/performance reasons). H0 = −
Need n & 3/(2mb ) 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 amb & 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 − M 2 = 1. p2 Lattice NRQCD energies are shifted by state-independent constant. Define 2
c ≡
2 [E(p) − E(0) + Mkin,1 ]2 − Mkin,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) at p = n · 2π/L: 1.01 L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm
c2
1.005
1
0.995
0.99
0
1
2
3
4
5
6
7
8
9 10 11 12
2
n
[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 symmetry for L5 → ∞ 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 L5 1.8 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]
NRQCD action Includes all terms of order v 4 and spin-dependent O(v 6 ) terms [Lepage et al. PRD 1992] H0
=
δH
=
1 (2) ∆ , 2m 2 ∆(2) ig e −E e ·∇ ∇·E + c2 −c1 3 2 8mb 8mb g e −E e ×∇ e e ×E e − c4 g σ · B −c3 σ· ∇ 2 8mb 2mb 2 (2) 2 (4) a ∆ a ∆ +c5 − c6 24mb 16n m2b n o n o g e −c8 3g e −E e ×∇ e ×E e −c7 ∆(2) , σ · B ∆(2) , σ · ∇ 3 4 8mb 64mb −
−c9
ig 2 e × E). e σ · (E 8m3b
Tree-level: ci = 1. Radiative corrections to spin-dependent couplings not yet known!
Radial and orbital energy splittings: amb -dependence
Data from L = 32 ensemble with aml = 0.004, order-v 4 action: Υ(2S) − Υ(1S) 2S − 1S 13 P − Υ(1S) 13 P − 1S 23 P − 13 P 23 P − Υ(1S) 23 P − 1S Υ2 (1D) − Υ(1S)
amb = 1.75 0.2422(31) 0.2456(32) 0.1901(22) 0.1965(22) 0.1645(99) 0.353(10) 0.359(10) 0.3048(39)
amb = 1.87 0.2421(33) 0.2454(33) 0.1907(20) 0.1969(20) 0.1629(94) 0.3519(94) 0.3580(94) 0.3051(40)
→ Splittings nearly independent of amb
amb = 2.05 0.2418(31) 0.2448(31) 0.1918(19) 0.1975(19) 0.1592(80) 0.3494(82) 0.3552(82) 0.3059(42)
Kinetic mass: amb -dependence Kinetic mass of of ηb (1S), defined as Mkin 4.6
p2 − [E(p) − E(0)]2 ≡ 2 [E(p) − E(0)]
4.5 4.4 aMkin
4.3 4.2 4.1 4.0 3.9
L = 32, a ≈ 0.08 fm Fit A · amb + B
1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 amb
(phys.)
Lattice spacing and amb
Use Υ(2S) − Υ(1S) splitting to determine a (phys.)
Determine amb experiment
such that Mkin (ηb ) agrees with
(phys.)
L3 × T 163 × 32 163 × 32 163 × 32
β 2.13 2.13 2.13
aml 0.01 0.02 0.03
ams 0.04 0.04 0.04
a−1 (GeV) 1.766(52) 1.687(46) 1.651(33)
amb 2.469(72) 2.604(75) 2.689(56)
243 × 64 243 × 64 243 × 64 243 × 64
2.13 2.13 2.13 2.13
0.005 0.01 0.02 0.03
0.04 0.04 0.04 0.04
1.763(27) 1.732(28) 1.676(42) 1.650(39)
2.487(39) 2.522(42) 2.622(70) 2.691(66)
323 × 64 323 × 64 323 × 64
2.25 2.25 2.25
0.004 0.006 0.008
0.03 0.03 0.03
2.325(32) 2.328(45) 2.285(32)
1.831(25) 1.829(36) 1.864(27)
Chiral extrapolation
(phys.)
Interpolate spin splittings to amb
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 ) + A m2π , E(mπ , a2 ) = E(0, a2 ) + A m2π . Data from (L = 16, a ≈ 0.11 fm) ensemble extrapolated independently (different physical box size)
Radial and orbital energy splittings: chiral extrapolation
1.3
1.3 L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm Experiment
1.1
Υ(3S) − Υ(1S)
1.0 0.9 0.8 0.7
L = 16, a ≈ 0.11 fm L = 24, a ≈ 0.11 fm Experiment
1.2 Splitting (GeV)
Splitting (GeV)
1.2
1.1
Υ(3S) − Υ(1S)
1.0 0.9 0.8
0.0
0.1
0.2 m2π
0.3 2
(GeV )
0.4
0.5
0.7
0.0
0.1
0.2
0.3
m2π (GeV2)
0.4
0.5
Radial and orbital energy splittings: chiral extrapolation
0.52
0.52
Splitting (GeV)
0.50
13 P
0.48
− 1S
0.46 0.44 0.42
Splitting (GeV)
L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm Experiment
0.50
L = 16, a ≈ 0.11 fm L = 24, a ≈ 0.11 fm Experiment
0.48
13P − 1S
0.46 0.44
0.0
0.1
0.2 m2π
0.3 2
(GeV )
0.4
0.5
0.42
0.0
0.1
0.2
0.3
m2π (GeV2)
0.4
0.5
Radial and orbital energy splittings: chiral extrapolation
L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm Experiment
0.50
23 P − 13 P
0.45 0.40 0.35 0.30
L = 16, a ≈ 0.11 fm L = 24, a ≈ 0.11 fm Experiment
0.55 Splitting (GeV)
Splitting (GeV)
0.55
0.50
23 P − 13 P
0.45 0.40 0.35
0.0
0.1
0.2 m2π
0.3 2
(GeV )
0.4
0.5
0.30
0.0
0.1
0.2
0.3
m2π (GeV2)
0.4
0.5
Radial and orbital energy splittings: chiral extrapolation
0.80
0.80 L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm Experiment
0.76
Υ2(1D) − Υ(1S)
0.74 0.72 0.70 0.68 0.66
L = 16, a ≈ 0.11 fm L = 24, a ≈ 0.11 fm Experiment
0.78 Splitting (GeV)
Splitting (GeV)
0.78
0.76
Υ2(1D) − Υ(1S)
0.74 0.72 0.70 0.68
0.0
0.1
0.2 m2π
0.3 2
(GeV )
0.4
0.5
0.66
0.0
0.1
0.2
0.3
m2π (GeV2)
0.4
0.5
Radial and orbital energy splittings at mπ = 138 MeV 10.6 10.4
Υ(3S)
E (GeV)
10.2 10
3P 2–
Υ2(1D) Υ(2S)
3P 1–
9.8 Experiment
9.6 9.4 9.2
Υ(1S)
L = 32, a ≈ 0.08 fm L = 24, a ≈ 0.11 fm L = 16, a ≈ 0.11 fm
Spin splittings: chiral extrapolation
75
75
60
Υ(1S) − ηb(1S)
Splitting (MeV)
55 50
65
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm Experiment
60
Υ(1S) − ηb(1S)
70 Splitting (MeV)
65
v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm Experiment
70
55 50
45
45
40
40
35
0.0
0.1
0.2 m2π
0.3
0.4
0.5
35
0.0
0.1
2
0.2
0.3
m2π (GeV2)
(GeV )
1S hyperfine splitting At leading order: ∝ c24 , independent of c3
0.4
0.5
Spin splittings: chiral extrapolation
50
50 v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm
Υ(2S) − ηb(2S) 30 20 10 0
40 Splitting (MeV)
Splitting (MeV)
40
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm
Υ(2S) − ηb(2S) 30 20 10
0.0
0.1
0.2 m2π
0.3
0.4
0.5
0
0.0
0.1
2
0.2
0.3
m2π (GeV2)
(GeV )
2S hyperfine splitting At leading order: ∝ c24 , independent of c3
0.4
0.5
Spin splittings: chiral extrapolation 60
60 v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm Experiment
50
1P tensor
45 40 35
50
40 35 30
25
25 0.0
0.1
0.2 m2π
0.3
0.4
0.5
1P tensor
45
30
20
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm Experiment
55 Splitting (MeV)
Splitting (MeV)
55
20
0.0
2
0.1
0.2
0.3
m2π (GeV2)
(GeV )
1P tensor splitting −2χb0 (1P ) + 3χb1 (1P ) − χb2 (1P ) At leading order: ∝ c24 , independent of c3
0.4
0.5
Spin splittings: chiral extrapolation 240
240
180
1P spin-orbit
Splitting (MeV)
160 140
200
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm Experiment
180
1P spin-orbit
220 Splitting (MeV)
200
v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm Experiment
220
160 140
120
120
100
100
80
0.0
0.1
0.2 m2π
0.3
0.4
0.5
80
0.0
0.1
2
0.2
0.3
m2π (GeV2)
(GeV )
1P spin-orbit splitting −2χb0 (1P ) − 3χb1 (1P ) + 5χb2 (1P ) At leading order: ∝ c3 , independent of c4
0.4
0.5
Spin splittings: chiral extrapolation
v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm
Splitting (MeV)
10
13P − hb(1P )
8 6 4
10
6 4 2
0
0 0.1
0.2
0.3
0.4
0.5
13P − hb(1P )
8
2
0.0
v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm
12 Splitting (MeV)
12
0.0
0.1
m2π (GeV2)
0.3
m2π (GeV2)
1P hyperfine splitting 13 P − hb (1P ) At leading order: zero
0.2
0.4
0.5
Spin splittings at mπ = 138 MeV 40
20
∆E
(MeV)
χb2(1P) hb(1P)
0 χb1(1P) -20
Experiment
-40
-60
χb0(1P)
v v v v
action, action, action, action,
a ≈ 0.08 a ≈ 0.11 a ≈ 0.08 a ≈ 0.11
fm fm fm fm
Spin splittings at mπ = 138 MeV 20
∆E
(MeV)
0
Υ(1S)
Υ(2S)
-20
ηb(2S)
-40
Experiment
-60 ηb(1S) -80
v v v v
action, action, action, action,
a ≈ 0.08 a ≈ 0.11 a ≈ 0.08 a ≈ 0.11
fm fm fm fm
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
Radiative corrections to spin splittings
At leading order, hyperfine and tensor splittings are expected to be proportional to c24 and independent of c3 , so radiative corrections should cancel in the ratios
and
Υ(2S) − ηb (2S) Υ(1S) − ηb (1S) Υ(1S) − ηb (1S) 1P tensor
Does this also work at order v 6 ?
Spin splittings: changing c3 or c4
splitting with c3 6= 1 or c4 6= 1 splitting with all ci = 1 c3 = 0.8 0.98016(18) 0.983(87) 0.991(84) 0.871(29)
c3 = 1.2 1.02148(19) 1.025(91) 1.008(76) 1.129(31)
c4 = 0.8 0.67151(53) 0.68(10) 0.658(67) 0.936(32)
c4 = 1.2 1.3808(12) 1.35(14) 1.40(11) 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 tensor
0.989(83)
1.013(78)
1.02(10)
0.989(77)
Υ(1S) − ηb (1S) Υ(2S) − ηb (2S) 1P tensor 1P spin − orbit
v 4 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 0.97788(17) 0.98(13) 0.987(71) 0.845(28)
c3 = 1.2 1.02411(20) 1.03(13) 1.006(62) 1.154(32)
c4 = 0.8 0.64656(47) 0.63(12) 0.641(59) 0.920(29)
c4 = 1.2 1.4180(11) 1.44(19) 1.41(11) 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 tensor
0.991(75)
1.018(62)
1.008(95)
1.002(74)
Υ(1S) − ηb (1S) Υ(2S) − ηb (2S) 1P tensor 1P spin − orbit
v 6 action, a ≈ 0.11 fm
Ratio of hyperfine splittings: chiral extrapolation
1.0 v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm
0.6 0.4 0.2 0.0
0.8
Ratio
Ratio
0.8
1.0 Υ(2S) − ηb(2S) Υ(1S) − ηb(1S)
Υ(2S) − ηb(2S) Υ(1S) − ηb(1S)
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm
0.6 0.4 0.2
0.0
0.1
0.2
0.3
0.4
0.5
0.0
0.0
m2π (GeV2)
0.1
0.2
0.3
m2π (GeV2)
Υ(2S) − ηb (2S) Υ(1S) − ηb (1S)
0.4
0.5
Ratio of hyperfine and tensor splittings: chiral extrap.
2.4 2.2
Υ(1S) − ηb(1S) 1P tensor
2.4
v 4 action, a ≈ 0.11 fm v 4 action, a ≈ 0.08 fm Experiment
2.2
1.8 1.6
1.8 1.6
1.4
1.4
1.2
1.2
1.0
v 6 action, a ≈ 0.11 fm v 6 action, a ≈ 0.08 fm Experiment
2.0 Ratio
Ratio
2.0
Υ(1S) − ηb(1S) 1P tensor
0.0
0.1
0.2
0.3
0.4
0.5
1.0
0.0
m2π (GeV2)
0.1
0.2
0.3
m2π (GeV2)
Υ(1S) − ηb (1S) 1P tensor
0.4
0.5
Spin splittings: final results
(v 6 action, a ≈ 0.08 fm, mπ = 138 MeV)
This work Υ(2S) − ηb (2S) Υ(1S) − ηb (1S) Υ(1S) − ηb (1S) 1P tensor Υ(2S) − ηb (2S) 1P tensor Υ(1S) − ηb (1S)
a b
Experiment
0.403(52)(25)
-
1.28(12)(8)
1.467(80)
0.497(87)(32)
-
60.3(5.5)(3.8)(2.1) MeV
a
23.5(4.1)(1.5)(0.8) MeV 28.0(3.6)(1.7)(1.2) MeV
a
Υ(2S) − ηb (2S) 13 P − hb (1P )
0.04(93)(20) MeV
b
69.3(2.9) MeV -
Using 1P tensor splitting from experiment 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
(Ω)
Cjk αδ (t, t0 , x0 ) =
X
abc f gh (Cγj )βγ (Cγk )ρσ
x 0 0 bg 0 0 ch 0 0 × Gaf βσ (t, x, t , x ) Gγρ (t, x, t , x ) Gαδ (t, x, t , x )
with the NRQCD propagator 0
0
G(t, x, t , x ) =
Gψ (t, x, t0 , x0 ) 0 0 0
For quark smearing, include rS (2) nS ∆ 1+ nS at source and/or sink.
.
Ωbbb correlator Large (t − t0 ): (Ω)
Cjk
→
Disentangle J = the projectors
0
2 Z3/2 e−E3/2 (t−t ) 12 (1 + γ0 )(δjk − 31 γj γk ) 0
2 + Z1/2 e−E1/2 (t−t ) 12 (1 + γ0 ) 13 γj γk . 3 2
and J = (3/2)
Pij
(1/2)
Pij
1 2
contributions by multiplying with
= (δij − 13 γi γj ), =
1 3 γi γj .
This gives (J)
(Ω)
0
(J)
Pij Cjk → ZJ2 e−EJ (t−t ) 12 (1 + γ0 )Pik .
Ωbbb correlator: example Data from RBC/UKQCD ensemble with L = 32, aml = 0.004 100 local − local local − smeared smeared − local smeared − smeared
10−2 10−4 C(t)
10−6 10−8 10−10 10−12 10−14 10−16
10
15
20
25 t
30
Fit includes 7 exponentials and has tmin = 5
35
40
Ωbbb correlator: example Data from RBC/UKQCD ensemble with L = 32, aml = 0.004 0.54 local − local local − smeared smeared − local smeared − smeared
ln[C(t)/C(t + 1)]
0.53 0.52 0.51 0.50 0.49 0.48
10
15
20
25 t
30
35
40
Computing the Ωbbb 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 3 aEΩbbb − aEΥ 2 and aEΩbbb −
3 (aEηb + 3aEΥ ) |8 {z }
= 32 ×(b¯b spin average)
Ωbbb : dependence on amb
0.088
0.120
aEΩbbb − 38 (aEηb + 3aEΥ)
0.110
aEΩbbb −
3 2 aEΥ
0.105
Splitting (lattice units)
Splitting (lattice units)
0.086 0.115
0.084 0.082 0.080 0.078
aEΩbbb − 38 (aEηb + 3aEΥ) aEΩbbb − 32 aEΥ
0.076 0.074
0.100
2.3
2.4
2.5 amb
2.6
2.7
0.072
1.75 1.80 1.85 1.90 1.95 2.00 2.05 amb
Ωbbb : chiral extrapolation
0.24
0.24 L = 24, a ≈ 0.11 fm L = 32, a ≈ 0.08 fm
0.22
EΩbbb − 38 (Eηb + 3EΥ)
0.21 0.20 0.19 0.18
L = 16, a ≈ 0.11 fm L = 24, a ≈ 0.11 fm
0.23 Splitting (GeV)
Splitting (GeV)
0.23
0.22
EΩbbb − 38 (Eηb + 3EΥ)
0.21 0.20 0.19
0.0
0.1
0.2
0.3 2
m2π (GeV )
0.4
0.5
0.18
0.0
0.1
0.2
0.3
m2π (GeV2)
0.4
0.5
Ωbbb : chirally extrapolated/interpolated results Ensemble type RBC/UKQCD coarse RBC/UKQCD coarse RBC/UKQCD fine
L3 × T 163 × 32 243 × 64 323 × 64
mπ (GeV) 0.138 0.138 0.138
MILC coarse RBC/UKQCD coarse
243 × 64 243 × 64
0.460 0.460
0.2063(41) 0.2022(22)
MILC fine RBC/UKQCD fine
283 × 96 323 × 64
0.416 0.416
0.2008(24) 0.1966(24)
EΩbbb −
3 8
(Eηb + 3EΥ ) (GeV) 0.214(11) 0.2044(44) 0.1984(29)
MILC ensembles have more accurate gluon action (L¨ uscher-Weisz) but use rooted staggered sea quarks. Match R.M.S. pion mass.
Use the following result: EΩbbb −
3 (Eηb + 3EΥ ) = 0.198 ± 0.003 stat ± 0.011 syst GeV. 8
EΩbbb − 83 (Eηb + 3EΥ ): electrostatic correction ECoulomb = 3
(e/3)2 1 3 (e/3)2 1 hΩbbb | |Ωbbb i + hΥ| |Υi. 4π0 r 2 4π0 r
Expectation values from potential models (for Ωbbb from [Silvestre-Brac, FBS 1996]): 1 hΥ| |Υi = 8.1 fm−1 r p hΥ|r2 |Υi = 0.20 fm
p hΩbbb |r2 |Ωbbb i = 0.25 fm Estimate
hΩbbb |r−1 |Ωbbb i = (0.8 ± 0.4)hΥ|r−1 |Υi = 6.5 ± 3.2 fm−1 This gives ECoulomb = 5.1 ± 2.5 MeV.
Mass of the Ωbbb : final result
MΩbbb
3 + ECoulomb = EΩbbb − (Eηb + 3EΥ ) 8 LQCD i h i 3h 3 h EΥ − Eηb i + MΥ − × 1P tensor 2 8 1P tensor LQCD PDG PDG
= 14.371 ± 0.004 stat ± 0.011 syst ± 0.001 exp GeV.
Mass of the Ωbbb : lattice QCD vs continuum models Reference Ponce, PRD 1979 Hasenfratz et al., PLB 1980 Bjorken 1985 Tsuge et al., MPL 1986 Silvestre-Brac, FBS 1996 Jia, JHEP 2006 Martynenko, PLB 2008 Roberts and Pervin, IJMPA 2008 Bernotas and Simonis, LJP 2009 Zhang and Huang, PLB 2009
MΩbbb (GeV) 14.248 14.30 14.76 ± 0.18 13.823 14.348 . . . 14.398 14.37 ± 0.08 14.569 14.834 14.276 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) violate baryon-meson mass inequality 3 MΥ = 14.1904 GeV 2 [Adler et al. PRD 1982, Nussinov PRL 1983, Richard PLB 1984] MΩbbb ≥
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!
Extra slides
Bottomonium: interpolating operators fix gauge configurations to Coulomb gauge, use “smearing” function Γ(r), 2 × 2-matrix-valued in spinor space X 0 χ† (x, t) Γ(x − x0 ) ψ(x0 , t) eip·(x+x )/2 OΓ (p, t) = x, x0
NB: choice of smearing only affects overlap with states, not their energies
Bottomonium: interpolating operators Name ηb (nS) Υ(nS) hb (nP ) χb0 (nP ) χb1 (nP ) χb2 (nP ) ηb (nD) Υ2 (nD)
L 0 0 1 1 1 1 2 2
S 0 1 0 1 1 1 0 1
J 0 1 1 0 1 2 2 2
P − − + + + + − −
C + − − + + + + −
RP C A−+ 1 T1−− T1+− A++ 1 T1++ T2++ T2−+ E −−
Γ(r) φnS (r) φnS (r) σ i φnP (r) ri /r0 φnP (r) (r · σ)/r0 φnP (r) (r × σ)i /r0 φnP (r) (ri σ j + rj σ i )/r0 φnD (r) ri rj /r02 φnD (r) (ri rj σ k − rj rk σ i )/r02 (i 6= j, k 6= j)
State 1S 2S 3S 1P 2P 1D
φ(r) exp[−|r|/r0 ] [1 − |r|/(2r0 )] exp[−|r|/(2r0 )] 1 − 2|r|/(3r0 ) + 2|r|2 /(27r02 ) exp[−|r|/(3r0 )] exp[−|r|/(2r0 )] [1 − |r|/(6r0 )] exp[−|r|/(3r0 )] exp[−|r|/(3r0 )]
Multi-exponential Bayesian fitting matrix fits with multiple radial smearing functions (e.g. 1S, 2S and 3S) at source and sink 30
hC(Γsk , Γsc , p, t − t0 )i
25
nexp −1
correlator
20
≈
15
X
0
An (Γsc ) A∗n (Γsk ) e−En (t−t )
n=0
10
5
0 0
5
10
time
15
20
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 + χ2prior with the Gaussian prior χ2prior =
X (pi − p˜i )2 i
σp2˜i
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): een = −1.4, σeen = 1, en (Γ) = 0, σ e B =5 Bn (Γ)
Multi-exponential Bayesian fitting
increase nexp until fit results stabilize 1
Υ(3S)
0.6
Υ(2S)
aE
0.8
0.4
0
2
4
6
nexp
8
0.79
0.79
0.79
0.80
0.86
1.83
2.83
45.8
χ2/dof:
Υ(1S) 0.2
10
12
