charmonium spectroscopy from lattice qcd · daniel mohler (fermilab) charmonium spectroscopy...
TRANSCRIPT
Charmonium spectroscopy from Lattice QCD
Daniel Mohler
Baltimore,April 9, 2015
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 1 / 32
Outline
1 Introduction
2 Charmonia from single-hadron operatorsLow lying charmoniaNew results on the 1S hyperfine splittingNew results on charmonium 1P states and 2S statesExited state energy levels from q̄qGluonic excitations
3 Charmonia including meson-meson statesThe ψ(2S) and Ψ(3770)χ ′c0 and X/Y(3915)Lattice results for the X(3872)Searches for Zc and Y(4140)
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 2 / 32
Outline
1 Introduction
2 Charmonia from single-hadron operatorsLow lying charmoniaNew results on the 1S hyperfine splittingNew results on charmonium 1P states and 2S statesExited state energy levels from q̄qGluonic excitations
3 Charmonia including meson-meson statesThe ψ(2S) and Ψ(3770)χ ′c0 and X/Y(3915)Lattice results for the X(3872)Searches for Zc and Y(4140)
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 3 / 32
Two kinds of progress...
Precision results:
1.14 1.18 1.22 1.26
=+
+=
+=
QCDSF/UKQCD 07 ETM 09 ETM 10D (stat. err. only) BGR 11 ALPHA 13
our estimate for =
MILC 04 NPLQCD 06 HPQCD/UKQCD 07 RBC/UKQCD 08 PACS-CS 08, 08A Aubin 08 MILC 09 MILC 09A JLQCD/TWQCD 09A (stat. err. only) BMW 10 PACS-CS 09 RBC/UKQCD 10A JLQCD/TWQCD 10 MILC 10 Laiho 11 RBC/UKQCD 12
our estimate for = +
ETM 10E (stat. err. only) MILC 11 (stat. err. only) MILC 13A HPQCD 13A ETM 13F
our estimate for = + +
/
Example: FLAG reviewSee http://itpwiki.unibe.ch/flag/
Exploratory studies:
-500
-300
-100
600 800 1000 1200 1400 1600
Example: πK-ηK-scatteringDudek et al. PRL 113 182001 (2014)
See talk by David Wilson Fri
I will report on both kind of progress with regard to charmonium
There will be preliminary data - use with caution
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 4 / 32
Outline
1 Introduction
2 Charmonia from single-hadron operatorsLow lying charmoniaNew results on the 1S hyperfine splittingNew results on charmonium 1P states and 2S statesExited state energy levels from q̄qGluonic excitations
3 Charmonia including meson-meson statesThe ψ(2S) and Ψ(3770)χ ′c0 and X/Y(3915)Lattice results for the X(3872)Searches for Zc and Y(4140)
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 5 / 32
Low-lying charmonium: A precision benchmark
Well understood from potential modelsWell determined in experimentWell separated from open-charm threshold(s)Spin-dependent mass splittings extremely sensitive to the charm-quarkmass and heavy-quark discretization
meson mass width
ηc 2983.7(7) 32.0(9) MeV
J/Ψ 3096.916(11) 92.9(2.8) keV
χc0 3414.75(31) 10.3(6) MeV
χc1 3510.66(3) 0.86(5) MeV
χc2 3556.20(9) 1.97(11) MeV
hc 3525.38(11) 0.7(4) MeV
ηc(2S) 3639.4(1.3) 11.3(+3.2)(−2.9) MeV
Ψ(2S) 3686.109(+12)(−14) 299(8) keV
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 6 / 32
Preliminary results from Fermilab-MILC
Fermilab Lattice and MILC collaborations - to be published
We use the Fermilab method for the charm quarkEl-Khadra et al., PRD 55, 3933 (1997)
mc tuned by demanding the Ds meson kinetic mass to be physical
We quote splittings among charmonium states
5 lattice spacings with two different light sea-quark masses→ Controlled extrapolation to the chiral-continuum limit
2+1 flavor simulation with high statistics
Follows previous effortsT. Burch et al. PRD 81 034508, 2010
Charm annihilation contributions are omitted
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 7 / 32
Fermilab-MILC results for the 1S hyperfine splitting
0 0.05 0.1 0.15 0.2a [fm]
0.16
0.165
0.17
0.175
0.18
0.185
0.19
0.195
∆MH
F r
1
lattice data 0.2 mh
lattice data 0.1 mh
fit results at lattice parameters
leading + subleading shapes
χ2
aug/d.o.f.
aug = 2.84/7
PRELIMINARY
Curves for physical (black), 0.1ms, and 0.2ms light-quark massesThis corresponds to MJ/Ψ−Mηc = 118.1(2.1)
(−1.5−4.0
)Errors include statistics, chiral and continuum extrapolationsContribution from disconnected diagrams expected
(−1.5−4.0
)Levkova and DeTar, PRD 83 074504, 2011
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 8 / 32
Results from the χQCD Collaboration
χQCD Collaboration, Yang et al. arXiv:1410.3343
0.6
0.8
1
1.2
1.4
1.6
r0(fm) mc ms MJ/ψ-Mηc
Mχc0
Mχc1
MhcfDs
Ra
tio
1.09(3)0.095(5)
0.1166(12)3.41475(31)
3.51066(7)3.52541(16)
0.2575(61)
0.458(14)1.111(24)
0.103(9)0.1120(61)
3.408(47)3.496(49)
3.516(29)0.256(5)
Exp (PDG)this work (MDs
, MDs* and MJ/ψ as Input)
Results from Overlap fermions on 2+1 flavor domain wall gaugeconfigurationsUncertainty do not include charm-quark annihilation effects
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 9 / 32
Preliminary results from the HPQCD Collaboration
HPQCD, Galloway et al. PoS LATTICE2014 (2014) 092
0.0 0.2 0.4 0.6 0.8
(amc)2
100
105
110
115
120
MJ/Ψ−Mηc
/M
eV
HISQ on HISQ 2+1+1 ml/ms = 1/5
HISQ on HISQ 2+1+1 ml/ms = 1/10
HISQ on HISQ 2+1+1 ml/ms = phys
expt (PDG)
Uses MILC 2+1+1 flavor HISQ (Highly Improved Staggered Quarks)ensembles
Systematics on 1S hyperfine dominated by estimate of annihilationeffects
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 10 / 32
A comparison of hyperfine splittings
105 110 115 120 125 130 135 140M
J/ψ-Mηc[MeV]
PDG
FNAL-MILC ’09
HPQCD ’12
ETMC ’12
Briceno et al ’12
χQCD ’14
FNAL-MILC preliminary
HPQCD preliminary
All results at physical quark masses and in the continuum limitLattice numbers exclude (now dominant?) annihilation effectsEstimate from data expects a shift of -1.5..-4.5 MeV
Levkova and DeTar, PRD 83 074504, 2011
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 11 / 32
Fermilab-MILC 1P-1S and 1P hyperfine splittings
0 0.05 0.1a [fm]
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
(M1P
- M
1S)
r 1
lattice data 0.2mh
lattice data 0.1mh
fit result at lattice parameters
leading + subleading shapes
χ2
aug/d.o.f.
aug = 9.79/7
PRELIMINARY0 0.05 0.1
a[fm]-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
(M1P
-Mh c)r
1
leading + subleading shapes
χ2
aug/d.o.f.
aug = 4.28/7
PRELIMINARY
Not yet included: Scale-setting uncertainty
Mass difference This analysis [MeV] Experiment [MeV]
1P1S 457.3±3.6 457.5±0.31P hyperfine −6.2±4.1 −0.10±0.22
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 12 / 32
Fermilab-MILC P-wave spin-orbit and tensor splittings
∆MSpin−Orbit = (5Mχc2 −3Mχc1 −2Mχc0 )/9
0 0.05 0.1 0.15 0.2a [fm]
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
∆MSO
r1
lattice data 0.2 mh
lattice data 0.1 mh
fit result at lattice parameters
leading + subleading shapes
χ2
aug/d.o.f.
aug = 5.79/7
PRELIMINARY
∆MTensor = (3Mχc1 −Mχc2 −2Mχc0 )/9
0 0.05 0.1 0.15 0.2a [fm]
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
∆MT
e r1
lattice data 0.2 mh
lattice data 0.1 mh
fit result at lattice parameters
leading + subleading shapes
χ2
aug/d.o.f.
aug = 9.07/7
PRELIMINARY
Mass difference This analysis [MeV] Experiment [MeV]
1P spin-orbit 49.5±2.5 46.6±0.11P tensor 17.3±2.9 16.25±0.07
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 13 / 32
Preliminary results from the HPQCD Collaboration
HPQCD, Galloway et al. PoS LATTICE2014 (2014) 092
0−+ 1−− 1+− 0++ 1++
JPC
3.0
3.2
3.4
3.6
3.8
Mass
/G
eV χc0
χc1
ηc
η′c
hc
J/Ψ
Ψ′
expt (PDG)
m`/ms = 1/5
m`/ms = 1/10
m`/ms = phys
0.000 0.005 0.010 0.015 0.020 0.025
a2/fm2
550
600
650
700
750
Ch
arm
on
ium
spin
-avg.
2S
-1S
splitt
ing
/M
eV HISQ on HISQ 2+1+1 ml/ms = 1/5
HISQ on HISQ 2+1+1 ml/ms = 1/10
HISQ on HISQ 2+1+1 ml/ms = phys
expt (PDG)
Uses MILC 2+1+1 flavor HISQ (Highly Improved Staggered Quarks)ensembles
Shaded region shows the systematic uncertainty
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 14 / 32
Exited state energy levels from q̄q operators
HSC, L. Liu et al. JHEP 1207 126 (2012)
DDDD
DsDsDsDs
0-+0-+ 1--1-- 2-+2-+ 2--2-- 3--3-- 4-+4-+ 4--4-- 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 3++3++ 4++4++ 1-+1-+ 0+-0+- 2+-2+-0
500
1000
1500
M-
MΗc
HMeV
L
A large number of energy levels including higher spin states can beidentified
Mesons with spin-exotic quantum numbers!
Relation of energy levels to resonances not straight forward
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 15 / 32
Gluonic excitations
HSC, L. Liu et al. JHEP 1207 126 (2012)
DDDD
DsDsDsDs
0-+0-+ 1--1-- 2-+2-+ 1-+1-+ 0++0++ 1+-1+- 1++1++ 2++2++ 3+-3+- 0+-0+- 2+-2+-0
500
1000
1500M
-M
Ηc
HMeV
L
Hybrid mesons with both regular and spin-exotic quantum numbers
Relation of energy levels to resonances not straight forward
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 16 / 32
Outline
1 Introduction
2 Charmonia from single-hadron operatorsLow lying charmoniaNew results on the 1S hyperfine splittingNew results on charmonium 1P states and 2S statesExited state energy levels from q̄qGluonic excitations
3 Charmonia including meson-meson statesThe ψ(2S) and Ψ(3770)χ ′c0 and X/Y(3915)Lattice results for the X(3872)Searches for Zc and Y(4140)
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 17 / 32
Single hadron interpolators: what do we see?
In practical calculations q̄q and qqq interpolators couple very weakly tomulti-hadron states
McNeile & Michael, Phys. Lett. B 556, 177 (2003); Engel et al. PRD 82, 034505 (2010);Bulava et al. PRD 82, 014507(2010); Dudek et al. PRD 82, 034508(2010);
This is not unlike observations in QCD string-breaking studiesPennanen & Michael hep-lat/0001015;Bernard et al. PRD 64 074509 2001;
Necessitates the inclusion of hadron-hadron interpolators
We know: Energy levels 6= resonance massesNaïve expectation: Correct up to O(ΓR(mπ))
Was good enough for heavy pion masses where one would deal withbound states or very narrow resonances.
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 18 / 32
The Lüscher method
M. Lüscher Commun. Math. Phys. 105 (1986) 153; Nucl. Phys. B 354 (1991)531; Nucl. Phys. B 364 (1991) 237.
E = E(p1)+E(p2) E = E(p1)+E(p2)+∆E
En(L)(2)−→ δl
(3)−→ mR; ΓR or coupling g
(1) Extract energy levels En(L) in a finite box(2) Lüscher formula→ phase shift of the continuum scattering amplitude(3) Extract resonance parameters (similar to experiment)
2-hadron scattering and transitions well understood;progress for 3 (or more) hadrons but difficult
See LATTICE2014 plenary by Raúl A. Briceño, arXiv:1411.6944
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 19 / 32
Technicalities: Lattices used
ID N3L×NT Nf a[fm] L[fm] #configs mπ [MeV] mK[MeV]
(1) 163×32 2 0.1239(13) 1.98 280/279 266(3)(3) 552(2)(6)(2) 323×64 2+1 0.0907(13) 2.90 196 156(7)(2) 504(1)(7)
Ensemble (1) has 2 flavors of nHYP-smeared quarksGauge ensemble from Hasenfratz et al. PRD 78 054511 (2008)
Hasenfratz et al. PRD 78 014515 (2008)
Ensemble (2) has 2+1 flavors of Wilson-Clover quarks
PACS-CS, Aoki et al. PRD 79 034503 (2009)
On the small volume we use distillationOn the larger volume we use stochastic distillation
Peardon et al. PRD 80, 054506 (2009);
Morningstar et al. PRD 83, 114505 (2011)
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 20 / 32
Ψ(3770) resonance from Lattice QCD
Lang, Leskovec, DM, Prelovsek, arXiv:1503.05363
-0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
p2[GeV
2]
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
p3 c
otδ
/√s [G
eV
2]
BW fit (i)fit (ii)
ensemble (1)
-0.20 -0.10 0.00 0.10 0.20 0.30 0.40 0.50
p2[GeV
2]
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
p3 c
otδ
/√s [G
eV
2]
BW fit (i)fit (ii)
ensemble (2)
Proof of principle - many improvements possible
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 21 / 32
Ψ(3770) resonance: ResultsLang, Leskovec, DM, Prelovsek, arXiv:1503.05363
fit (i) fit (ii)
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
E [
GeV
]
fit (i) fit (ii)
D(0)D_
(0)
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
mD
++mD
-
2mD
0
mπ = 266 MeV mπ=156 MeV exp.
J/ψ
ψ(2S)
ψ(3770)
Mass [MeV] gΨ(3770)DDEnsemble(1) 3784(7)(8) 13.2 (1.2)Ensemble(2) 3786(56)(10) 24(19)Experiment 3773.15(33) 18.7(1.4)
First resonance determination of a charmonium stateProof of principle - many improvements possible
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 22 / 32
χ ′c0 and X/Y(3915)
PDG is identifying the X(3915) with the χ ′c0
Based on BaBar determination of its quantum numbersSome of the reasons to doubt this assignment:
Guo, Meissner Phys. Rev. D86, 091501 (2012)
Olsen, arxiv 1410.6534
No evidence for fall-apart mode X(3915)→ D̄DSpin splitting mχc2(2P)−mχc0(2P) to smallLarge OZI suppressed X(3915)→ ωJ/ψ
Width should be significantly larger than Γχc2(2P)
Investigate D̄D scattering in S-wave on the lattice!Candidates:
m = 3837.6±11.5 MeV, Γ = 221±19 MeV Guo&Meissner
m = 3878±48 MeV, Γ = 347+316−143
MeV Olsen
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 23 / 32
χ ′c0: Exploratory lattice calculation
Lang, Leskovec, DM, Prelovsek, arXiv:1503.05363
-0.6 -0.4 -0.2 0.0 0.2 0.4
p2[GeV
2]
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
p co
tδ/√
s
(a)
-0.6 -0.4 -0.2 0.0 0.2 0.4
p2[GeV
2]
(b)
-0.6 -0.4 -0.2 0.0 0.2 0.4
p2[GeV
2]
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
p co
tδ/√
s
(c)
Assumes only D̄D is relevant
Lattice data suggests a fairly narrow resonance with3.9GeV < M < 4.0GeV and Γ < 100MeV
Future experiment and lattice QCD results needed to clarify the situation
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 24 / 32
An X(3872) candidate from Lattice QCD
lattice (mπ~266 MeV)
400
500
600
700
800
900
1000
1100
m -
1/4
(m
η c+3
mJ/
ψ)
[M
eV]
Exp
D(0)D*(0)
J/ψ(0)ω(0)
D(1)D*(-1)
χc1
(1P)
X(3872)
χc1
(1P)
X(3872)
O: cc O: cc DD* J/ψ ω
poleL→∞
Prelovsek, Leskovec, PRL 111
192001 (2013)
400
600
800
1000
1200
1400
E -
E(1
S) M
eV
D(0)D*(0)
D(-1)D*(1)
cc (I=0) cc + DD* (I=0) DD* (I=0)
Lee, DeTar, DM, Na,
arXiv:1411.1389
Neglects charm annihilation and J/ψω
Seen only when q̄q and D̄∗D are used
The two simulations have vastly different systematics (yet results aresimilar)
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 25 / 32
An X(3872) candidate from Lattice QCD II
Padmanath, Lang, Prelovsek, arXiv:1503.03257
3.45
3.6
3.75
3.9
4.05
4.2
4.35
4.5
En [
GeV
]
Lat. - OMM17 Lat. - O
MM17 - O
-c c
D(0) -D*(0)
J/Ψ(0) ω(0)
D(1) -D* (-1)
J/Ψ(1) ω(-1)
ηc(1) σ(-1)
-30
-20
-10
0
10
Exp. Lat. Lat.-O4q [31] [32]
mX(3872)−mD−m-D*
770
790
810
830
850mX(3872)−ms.a.
Without q̄q interpolatorssignal vanishes
Simulations stillunphysical in many ways
Discretization and finitevolume effects sizable!
Makes interpretation aspure molecule or puretetraquark unlikely
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 26 / 32
Search for Z+c with IGJPC = 1+1+−
Lattice
D(2) D*(-2)D*(1) D*(-1)J/ψ(2) π(−2)ψ3 πD(1) D*(-1)ψ
1Dπ
D* D*η
c(1)ρ(−1)
ψ2S
πD D*j/ψ(1) π(-1)η
c ρ
J/ψ π
Exp.
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
E[G
eV]
Prelovsek, Lang, Leskovec, DM, Phys.Rev. D91 014504 (2015)
Simple level counting approach
We find 13 two meson states as expected
We find no extra energy level that could point to a Zc candidate
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 27 / 32
Search for Y(4140) with JPC = 1++
Padmanath, Lang, Prelovsek, arXiv:1503.03257
3.45
3.6
3.75
3.9
4.05
4.2
4.35
4.5
En [
GeV
]
Expt. Lat. Lat. − O4q
Ds(0) -Ds*(0)
J/Ψ(0) φ(0)
Ds(1) -Ds* (-1)
J/Ψ(1) φ(-1)
Considered only J/ψφ and DsD̄∗sNo additional energy level seen
Further simulations necessary
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 28 / 32
Outline
1 Introduction
2 Charmonia from single-hadron operatorsLow lying charmoniaNew results on the 1S hyperfine splittingNew results on charmonium 1P states and 2S statesExited state energy levels from q̄qGluonic excitations
3 Charmonia including meson-meson statesThe ψ(2S) and Ψ(3770)χ ′c0 and X/Y(3915)Lattice results for the X(3872)Searches for Zc and Y(4140)
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 29 / 32
Snapshot of spectrum calculations at mπ = 266MeV
ηc Ψ h
cχ
c0χ
c1χ
c2η
c2 Ψ2
Ψ3
hc3
χc3
-100
0
100
200
300
400
500
600
700
800
900
1000
1100m
-m_ c_ c [M
eV]
Ds
Ds*D
s0* D
s1D
s1D
s2*
-200-1000100200300400500600
m -
m_ Ds [
MeV
]
D D* D0*D
1D
1D2*D
2
0
200
400
600
800
m-m_
D [
MeV
]
lat: naive levelres. / bound state
First principles determination of resonances and bound states just startingProgram will have to be done with a number of lattice spacings/volumes
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 30 / 32
Conclusions & Outlook
Most current charmonium and charmed meson results are of anexploratory nature
Low-lying charmonium states are under good control (full error budgets)
New ideas might be needed for charm-annihilation contributions
Progress with regard to hybrids and higher spin states
Bound states and resonances close to DD̄, DD̄∗ and D∗D̄∗ can beinvestigated
No signal yet for manifestly exotic four quark states
For states well established on the lattice: calculation of radiativetransitions, etc. possible (and will be done)
A lot of progress but still much to do for excited charmonium
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 31 / 32
. . .
Thank you!
... also to my collaborators Carleton DeTar, Andreas Kronfeld,Christian Lang, Song-Haeng Lee, Luka Leskovec, Ludmila Levkova,
Sasa Prelovsek and Jim Simone
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 32 / 32
Backup slides
. . .
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 33 / 32
Heavy quarks using the Fermilab method
El-Khadra et al., PRD 55,3933
We tune κ for the spin averaged kinetic mass (Mηc +3MJ/Ψ)/4 toassume its physical valueGeneral form for the dispersion relation
Bernard et al. PRD83:034503,2011
E(p) = M1 +p2
2M2− a3W4
6 ∑i
p4i −
(p2)2
8M34+ . . .
We compare results from three different fit strategiesEnergy splittings are expected to be close to physicalFor MeV values of masses
M = ∆M+Msa,phys
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 34 / 32
Technicalities: Lattices used
ID N3L×NT Nf a[fm] L[fm] #configs mπ [MeV] mK[MeV]
(1) 163×32 2 0.1239(13) 1.98 280/279 266(3)(3) 552(2)(6)(2) 323×64 2+1 0.0907(13) 2.90 196 156(7)(2) 504(1)(7)
Ensemble (1) has 2 flavors of nHYP-smeared quarksGauge ensemble from Hasenfratz et al. PRD 78 054511 (2008)
Hasenfratz et al. PRD 78 014515 (2008)
Ensemble (2) has 2+1 flavors of Wilson-Clover quarks
PACS-CS, Aoki et al. PRD 79 034503 (2009)
On the small volume we use distillationOn the larger volume we use stochastic distillation
Peardon et al. PRD 80, 054506 (2009);
Morningstar et al. PRD 83, 114505 (2011)
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 35 / 32
Testing our tuning: charm and light
Ensemble (1) Ensemble (2) Experimentmπ 266(3)(3) 156(7)(2) 139.5702(4)mK 552(1)(6) 504(1)(7) 493.677(16)mφ 1015.8(1.8)(10.7) 1018.4(2.8)(14.6) 1019.455(20)mηs 732.3(0.9)(7.7) 692.9(0.5)(9.9) 688.5(2.2)*
mJ/Ψ−mηc 107.9(0.3)(1.1) 107.1(0.2)(1.5) 113.2(0.7)mD∗s −mDs 120.4(0.6)(1.3) 142.1(0.7)(2.0) 143.8(0.4)mD∗−mD 129.4(1.8)(1.4) 148.4(5.2)(2.1) 140.66(10)2mD−mc̄c 890.9(3.3)(9.3) 882.0(6.5)(12.6) 882.4(0.3)2MDs
−mc̄c 1065.5(1.4)(11.2) 1060.7(1.1)(15.2) 1084.8(0.6)mDs−mD 96.6(0.9)(1.0) 94.0(4.6)(1.3) 98.87(29)
A single ensemble: Discrepancies due to discretization and unphysicallight-quark masses expected
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 36 / 32
Low-lying charmonium spectrum
ηc Ψ h
cχ
c0χ
c1χ
c2η
c2 Ψ2
Ψ3
hc3
χc3
-1000
100200300400500600700800900
10001100
Ene
rgy
diff
eren
ce [
MeV
]
JPC
: 0- +
1- -
1+ -
0+ +
1+ +
2+ +
2- +
2 - -
3- -
3+ -
3++
DM, S. Prelovsek, R. M. Woloshyn, PRD 87 034501 (2013);
Serves as further confirmation of our heavy-quark approachData from 1 ensemble; Errors statistical + scale setting
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 37 / 32
Contractions
t f tiqqqq cs A1
t f ti
D
K
qqc
sC1
t f ti
D
K
c
sB1
t f ti
DD
KK
c
s
D1
t f ti
DD
KK
c
s
D2
Handled efficiently within the distillation method
Peardon et al. PRD 80, 054506 (2009)Morningstar et al. PRD 83, 114505 (2011)
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 38 / 32
An example: Different rho momentum frames
Lang, DM, Prelovsek, Vidmar, PRD 84 054503 (2011) & erratum ibid
0.4
0.6
0.8
1
En a
1 2 3 4 5 6 7 8
1
0.4
0.6
0.8
1
En a
1 2 3 4 5 6 7 8interpolator set
0.4
0.6
0.8
1
En a
interpolator set:
qq ππ
1: O1,2,3,4,5
, O6
2: O1,2,3,4
, O6
3: O1,2,3
, O6
4: O2,3,4,5
, O6
5: O1 , O
6
6: O1,2,3,4,5
7: O1,2,3,4
8: O1,2,3
P=(1,1,0)
P=(0,0,1)
P=(0,0,0)
with ππ without ππ
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 39 / 32
2S states
Fermilab/MILC, DeTar et al. arXiv:1410.3343
Simple analysis of 2S states lead to splittings much larger than inexperiment
Strong dependence on fit range and very noisy data
Daniel Mohler (Fermilab) Charmonium spectroscopy Baltimore, April 9, 2015 40 / 32