QCD to XYZ
From quarks and gluons to exotic hadrons
Daniel Mohler
Graz,
April 27, 2016
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
1 / 35
Quantum Chromodynamics (QCD) and hadrons
QCD: Theory that describes the strong interaction between quarks and gluons
within the Standard Model of Particle Physics
Asymptotic Freedom: Theory perturbative at
high energies/short distances
→ Perturbative calculations for high-energy
physics
Confinement: Color-charged particles do not exist individually but only
confined into composite objects called hadrons.
Textbook classification: Quark-antiquark mesons and three-quark baryons
→
c
Arpad
Horvath
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
2 / 35
Exotic mesons in the heavy-quark spectrum
Surprise results from the Belle and BaBar experiments
(built to investigate the difference between matter and antimatter):
D∗s0 (2317):
Y(4260):
PRL 91 262001 (2003)
PRL 90 242001 (2003)
PRL 95 142001 (2005)
Events / 20 MeV/c2
X(3872):
40
104
103
30
102
10
1 3.6 3.8
20
4
4.2 4.4 4.6 4.8
5
10
0
3.8
4
4.2
4.4
4.6
4.8
5
m(π+π-J/ψ) (GeV/c2)
Highest-cited Belle paper
Daniel Mohler (HIM)
Third-most cited BaBar
paper
From quarks and gluons to exotic hadrons
Fourth-most cited BaBar
paper
Graz, April 27, 2016
3 / 35
10 years later: Many more surprises
Data
Global fit
Three-body PS (global fit)
±1σ uncertainty band
Event-mixing (J/ ψ, φ, K+)
Event-mixing (J/ ψ, φ K+ )
1D fit
250
200
150
100
50
0
1.1
1.2
1.3
1.4
Events / 0.01 GeV/c2
N(B+) / 20 MeV
300
CMS, s = 7 TeV, L=5.2 fb-1
Data
100
Total fit
Background fit
80
PHSP MC
Sideband
60
40
Z(4430)± : Belle, LHCb
Candidates / ( 0.2 GeV2 )
Zc (3900)± : BESIII,
Belle, data from Cleo
Y(4140): CDF, CMS
1000
500
20
0
0
3.7
3.8
3.9
8000
6000
120
Events/(15 MeV)
10000
20
100
80
60
22
m2ψ'π− [GeV2]
Pc (4450), Pc (4380):
LHCb
Zc (4020)± : BESIII
Events/ ( 0.005GeV/c2 )
Events / 10 MeV/c2
Belle
(a)
18
4.0
Mmax(π±J/ψ) (GeV/c2)
800
LHCb
(b)
700
600
500
400
4000
40
2000
20
0
0
-2000
10.4
16
1.5
∆m [GeV]
Zb (10610)+ , Zb (10650)+ :
12000
LHCb
300
10.5
10.6
200
3.7
3.8
3.9
4.0
4.1
4.2
Mπ±hc(GeV/c2)
Mmiss(π), GeV/c2
Daniel Mohler (HIM)
100
0
4
4.2
4.4
4.6
4.8
5
mJ / ψp [GeV]
10.7
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
4 / 35
Exotic hadron spectroscopy: experiment ↔ theory
Vigorous and varied experiment program(current and planned)
Heavy mesons: LHCb, BESIII, PANDA, BelleII, . . .
Light mesons: GlueX, MesonEx, COMPASS, . . .
Baryons: CLAS12, ELSA, E45@JPARC, MAMI . . .
Should be accompanied by an equally vigorous theory effort
Current theory understanding of states with exotic properties relies on
models rather than first-principle calculations.
Molecules, Tetraquarks, Hybrid mesons, etc.
Many models – and none match all observations
The research I pursue will aid the understanding of exotic heavy mesons
directly from QCD
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
5 / 35
Exotic hadron spectroscopy: experiment ↔ theory
Vigorous and varied experiment program(current and planned)
Heavy mesons: LHCb, BESIII, PANDA, BelleII, . . .
Light mesons: GlueX, MesonEx, COMPASS, . . .
Baryons: CLAS12, ELSA, E45@JPARC, MAMI . . .
Should be accompanied by an equally vigorous theory effort
Current theory understanding of states with exotic properties relies on
models rather than first-principle calculations.
Molecules, Tetraquarks, Hybrid mesons, etc.
Many models – and none match all observations
The research I pursue will aid the understanding of exotic heavy mesons
directly from QCD
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
5 / 35
Exotic hadron spectroscopy: experiment ↔ theory
Vigorous and varied experiment program(current and planned)
Heavy mesons: LHCb, BESIII, PANDA, BelleII, . . .
Light mesons: GlueX, MesonEx, COMPASS, . . .
Baryons: CLAS12, ELSA, E45@JPARC, MAMI . . .
Should be accompanied by an equally vigorous theory effort
Current theory understanding of states with exotic properties relies on
models rather than first-principle calculations.
Molecules, Tetraquarks, Hybrid mesons, etc.
Many models – and none match all observations
The research I pursue will aid the understanding of exotic heavy mesons
directly from QCD
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
5 / 35
Outline
1
Introduction and Motivation
Hadrons - the bound states of quarks and gluons
Quantum Chromodynamics (QCD) and the Lattice
2
Modern lattice hadron spectroscopy
Spectroscopy and properties of bound states
What about resonances/ threshold states?
The simplest resonance: The ρ meson
Ψ(3770) - a heavier brother of the ρ
3
Towards exotic hadrons
D∗s0 (2317) and Ds1 (2460) and their b-quark cousins
χ0c0 and X/Y(3915)
4
Summary and future research directions
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
6 / 35
Lattice Quantum Chromodynamics: What do we calculate?
Regularization of QCD by a 4-d Euclidean space-time
lattice. (Kenneth Wilson 1974)
Provides a calculational method for QCD
Euclidean correlator of two Hilbert-space operators Ô1 and Ô2 .
D
E X
Ô2 (t)Ô1 (0) =
e−t∆En h0|Ô2 |nihn|Ô1 |0i
n
1
=
Z
Z
D[ψ, ψ̄, U]e−SE O2 [ψ, ψ̄, U]O1 [ψ, ψ̄, U]
Last line is a path integral over the Euclidean action SE,QCD [ψ, ψ̄, U];
(a sum over quantum fluctuations)
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
7 / 35
Lattice QCD: What do we calculate?
Fermion integral can be done explicitly
Rest can be evaluated with Monte Carlo simulations using methods well
established in statistical physics
gluons: Uµ
}
a
quarks: ψ
Λ
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
8 / 35
Why Lattice Quantum Chromodynamics?
Understand the theory of the strong interaction at low energies
Confinement and hadron properties
Chiral Symmetry breaking
(QCD dynamics responsible for > 98% of the nucleon mass)
Lattice QCD as a tool to understand strong-interaction contributions
precision flavor physics
muon physics (g − 2, . . . )
neutrino physics
dark matter searches
Higgs boson decays (precision quark masses)
QCD for hadronic (and nuclear) physics
Understand hadronic degrees of freedom (and how they arise)
Understand connection to nuclei and their properties
Lattice QCD is a non-perturbative, systematically improvable method leading
to quantifiable uncertainties
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
9 / 35
Stable hadron states: A lattice success story
Light mesons and baryons
Heavy mesons
12
MESON MASS (GeV/c2)
10
’
db
db
expt
fix params
postdcns
predcns
rb2
rb1(2P)
rb0
rb2
rb1(1P)
r
[(1D)
b0
8
Bc’
Bc
6
4
Bs
B
d ’c
dc
0
s’
J/s
hc
*’
B c
*
Bc
B*s
B*
*
B
c0
rc2
rc1
rc0
Ds
D
2
Example from BMW
Dürr et al. Science 322 (2008)
hb(2P)
[’’
[’
[ hb(1P)
d
/
K
Example from HPQCD
Dowdall et al. PRD 86 094510 (2012)
Hadrons stable under QCD: full control of systematic uncertainties
Routinely done for a wide variety of observables
(for example for flavor physics)
Goal: Extend this success to hadron resonances
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
10 / 35
Two kinds of progress...
Precision results:
=
= +
= + +
/
1.14
1.18
1.22
1.26
Exploratory studies:
our estimate for = + +
ETM 13F
HPQCD 13A
MILC 13A
MILC 11 (stat. err. only)
ETM 10E (stat. err. only)
our estimate for = +
RBC/UKQCD 12
Laiho 11
MILC 10
JLQCD/TWQCD 10
RBC/UKQCD 10A
PACS-CS 09
BMW 10
JLQCD/TWQCD 09A (stat. err. only)
MILC 09A
MILC 09
Aubin 08
PACS-CS 08, 08A
RBC/UKQCD 08
HPQCD/UKQCD 07
NPLQCD 06
MILC 04
our estimate for =
ALPHA 13
BGR 11
ETM 10D (stat. err. only)
ETM 09
QCDSF/UKQCD 07
Example: FLAG review
See http://itpwiki.unibe.ch/flag/
600
800
1000
1200
1400
1600
-100
-300
-500
Example: πK-ηK-scattering
Dudek et al.
PRL 113 182001 (2014)
I will report on exploratory calculations with regard to heavy meson
resonances and bound states
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
11 / 35
Observables: Examples for correlation functions
Need: Interpolating field operator that creates states with correct quantum
numbers.
Example I: Pseudoscalar Mesons with IJ PC = 10−+
O(1)
π = ūγ5 d
←
→
O(2)
π = ū D γi γt γ5 d
Can obtain mass from 2-point correlator with Oπ and Ōπ
Example II: Nucleon
ON = abc Γ1 ua uTb Γ2 dc − dbT Γ2 uc
In practice: Many (slightly different) constructions possible!
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
12 / 35
(My) Method of choice: The variational method
Matrix of correlators projected to fixed momentum (will assume 0)
D
E
X
C(t)ij =
e−tEn h0|Oi |ni n|O†j |0
n
Solve the generalized eigenvalue problem:
~ (k) = λ(k) (t)C(t0 )ψ
~ (k)
C(t)ψ
λ(k) (t) ∝ e−tEk 1 + O e−t∆Ek
At large time separation: only a single state in each eigenvalue.
Eigenvectors can serve as a fingerprint.
Michael Nucl.
Phys.
B259, 58 (1985)
Lüscher and Wolff Nucl.
Blossier et al.
Daniel Mohler (HIM)
Phys.
B339, 222 (1990)
JHEP 04, 094 (2009)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
13 / 35
Technicalities: The “Distillation” method
Peardon et al. PRD 80, 054506 (2009)
Morningstar et al. PRD 83, 114505 (2011)
Idea: Construct separable quark smearing operator using low modes of
the 3D lattice Laplacian
Spectral decomposition for an N × N matrix:
f (A) =
N
X
f (λ(k) ) v(k) v(k)† .
k=1
With
f (∇2 )
=
qs ≡
Θ(σs2
N
X
+
∇2 )
(Laplacian-Heaviside (LapH) smearing):
Θ(σs2 + λ(k) )v(k) v(k)† q =
k=1
Nv
X
v(k) v(k)† q .
k=1
Advantages: momentum projection at source; large interpolator freedom,
small storage
Disadvantages: expensive; unfavorable volume scaling
Stochastic approach (mostly) eliminates bad volume scaling
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
14 / 35
Using single hadron interpolators, what do we see?
In practical calculations q̄q and qqq interpolators couple very weakly to
multi-hadron states
McNeile & Michael, Phys. Lett. B 556, 177 (2003); Engel, DM 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 string breaking studies
Pennanen & Michael hep-lat/0001015;Bernard et al.
PRD 64 074509 2001;
This necessitates the inclusion of hadron-hadron interpolators
We know: Energy levels 6= resonance masses
Naïve expectation: Correct up to O(ΓR (mπ ))
Was good enough for heavy pion masses where one would deal with
bound states or very narrow resonances.
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
15 / 35
An example: Different rho momentum frames
without ππ
with ππ
interpolator set:
En a
11
P=(0,0,0)
0.6
0.4
1
En a
qq
0.8
1
2
3
4
5
6
8
0.8
0.6
P=(0,0,1)
0.4
1
En a
7
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
, O6
6: O1,2,3,4,5
0.8
7: O1,2,3,4
0.6
P=(1,1,0)
0.4
1
2
3
4
5
6
7
8: O1,2,3
8
interpolator set
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
16 / 35
Scattering in finite volume: 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
(2)
−−→
En (L)
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 LATTICE plenaries by Raúl A. Briceño arXiv:1411.6944
and Max Hansen arXiv:1511.04737
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
17 / 35
A look at the Particle Data Group booklet
I will discuss the following examples:
Light hadrons
Heavy-light hadrons
Charmonium
+ b-quark analogues
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
18 / 35
Studies within our collaboration (look at the past)
ππ scattering and ρ meson width
Lang, DM, Prelovsek, Vidmar, PRD 84 054503 (2011)
Kπ scattering
Lang, Leskovec, DM, Prelovsek, PRD 86 054508 (2012)
Prelovsek, Lang, Leskovec, DM, PRD 88 054508 (2013)
πρ and πω scattering and the a1 , b1 resonances
Lang, Leskovec, DM, Prelovsek, JHEP 1404 162 (2014)
D mesons including Dπ and D? π with relativistic charm quarks
DM, Prelovsek, Woloshyn, PRD 87 034501 (2013)
D∗s0 (2317) and Ds1 (2460) with q̄q and D(∗) K
DM, Lang, Leskovec, Prelovsek, Woloshyn, PRL 111 222001 (2013)
PRD 90 034510 (2014)
Predicting Bs states with J P = 0+ , 1+
Lang, Prelovsek, DM, Woloshyn, Phys. Lett.B750 17-21 (2015)
Heavy meson scattering and charmonium
Prelovsek & Leskovec PRL 111 192001
Prelovsek & Leskovec, Phys.Lett. B727 172
Prelovsek, Lang, Leskovec, DM, PRD 91 014504
Lang, Leskovec, DM, Prelovsek, JHEP 1509 089
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
(2013)
(2013)
(2015)
(2015)
Graz, April 27, 2016
19 / 35
Technicalities: Lattices used
ID
(1)
(2)
NL3 × NT
163 × 32
323 × 64
Nf
2
2+1
a[fm]
0.1239(13)
0.0907(13)
L[fm]
1.98
2.90
#configs
280/279
196
mπ [MeV]
266(3)(3)
156(7)(2)
mK [MeV]
552(2)(6)
504(1)(7)
Ensemble (1) has 2 flavors of nHYP-smeared quarks
Gauge 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 distillation
On the larger volume we use stochastic distillation
Peardon et al. PRD 80, 054506 (2009);
Morningstar et al. PRD 83, 114505 (2011)
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
20 / 35
The ρ resonance - a benchmark calculation
From Lang, DM, Prelovsek, Vidmar, PRD 84 054503 (2011); erratum ibid;
δ1
150
100
50
gρππ = 5.61(12); mρ = 0.4846(37)
lattice data
0
0.1
0.2
0.15
0.25
s
0.3
0.35
0.4
We extract gρππ rather than Γ
Γ(s) =
p? 3 g2ρππ
s 6π
Results for mπ = 266(3)(3)MeV
gρππ = 5.61(12)
Daniel Mohler (HIM)
mρ = 772(6)(8) MeV
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
21 / 35
The ρ resonance - comparing results for the coupling
(phys)
gρππ
≈ 5.97
mρ = 775.11(34) MeV
7.5
PACS-CS (2011)
Lang et al. (2011)
ETMC (2011)
physical value
7
gρππ
6.5
6
5.5
5
4.5
4
0
100
200
300
Mπ/MeV
400
500
Caution: To date no simulation with full control of systematics
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
22 / 35
The ρ resonance - comparing results for the coupling
(phys)
gρππ
≈ 5.97
mρ = 775.11(34) MeV
7.5
PACS-CS (2011)
Lang et al. (2011)
ETMC (2011)
GWU (2012)
HSC (2012)
GWU (2015)
HSC (2015)
Bulava et al. (2015)
Bali et al. (2015)
physical value
7
gρππ
6.5
6
5.5
5
4.5
4
0
100
200
300
Mπ/MeV
400
500
Caution: To date no simulation with full control of systematics
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
22 / 35
ρ resonance: Another look at an incomplete basis
Wilson et al.
PRD 92 094502 (2015)
180
150
120
90
60
30
0
0.08
0.10
0.12
0.14
0.16
At first sight spectrum seems well determined
Energy levels cluster close to resonance energy
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
23 / 35
Ψ(3770) resonance
Lang, Leskovec, DM, Prelovsek, JHEP 1509 089 (2015)
3.9
mπ = 266 MeV
mπ=156 MeV
exp.
3.9
ψ(3770)
3.8
3.7
3.8
3.7
ψ(2S)
E [GeV]
3.6
3.6
_
D(0)D(0)
3.5
mD++mD- 3.5
2mD0
3.4
3.4
3.3
3.3
3.2
3.2
J/ψ
3.1
3.1
fit (i)
Ensemble(1)
Ensemble(2)
Experiment
fit (ii)
fit (i)
fit (ii)
Mass [MeV]
3784(7)(8)
3786(56)(10)
3773.15(33)
gΨ(3770)DD
13.2 (1.2)
24(19)
18.7(1.4)
First resonance determination of a charmonium state
Proof of principle - many improvements possible
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
24 / 35
Exotic Ds (charm-strange) and Bs (bottom-strange)
candidates
Established s and p-wave states:
Ds (J P = 0− ) and D∗s (1− )
∗
Ds0 (2317) (0+ ), Ds1 (2460) (1+ ),
Ds1 (2536) (1+ ), D∗s2 (2573) (2+ )
Peculiarity: Mcs̄ ≈ Mcd̄
→
Bs (J P = 0− ) and B∗s (1− )
Bs1 (5830) (1+ ), B∗s2 (5840) (2+ )
exotic structure? (tetraquark, molecule)
Traditional lattice studies (using single hadron operators) tend get too
large or badly determined masses
Observed Bs p-wave states from two body decays into K − B+
(CDF/D0 and LHCb)
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
25 / 35
Exotic Ds (charm-strange) and Bs (bottom-strange)
candidates
Established s and p-wave states:
Ds (J P = 0− ) and D∗s (1− )
∗
Ds0 (2317) (0+ ), Ds1 (2460) (1+ ),
Ds1 (2536) (1+ ), D∗s2 (2573) (2+ )
Peculiarity: Mcs̄ ≈ Mcd̄
→
Bs (J P = 0− ) and B∗s (1− )
Bs1 (5830) (1+ ), B∗s2 (5840) (2+ )
exotic structure? (tetraquark, molecule)
Traditional lattice studies (using single hadron operators) tend get too
large or badly determined masses
Observed Bs p-wave states from two body decays into K − B+
(CDF/D0 and LHCb)
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
25 / 35
Exotic Ds (charm-strange) and Bs (bottom-strange)
candidates
Established s and p-wave states:
Ds (J P = 0− ) and D∗s (1− )
∗
Ds0 (2317) (0+ ), Ds1 (2460) (1+ ),
Ds1 (2536) (1+ ), D∗s2 (2573) (2+ )
Peculiarity: Mcs̄ ≈ Mcd̄
→
Bs (J P = 0− ) and B∗s (1− )
Bs1 (5830) (1+ ), B∗s2 (5840) (2+ )
exotic structure? (tetraquark, molecule)
Traditional lattice studies (using single hadron operators) tend get too
large or badly determined masses
Observed Bs p-wave states from two body decays into K − B+
(CDF/D0 and LHCb)
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
25 / 35
Discretization effects for charm and beauty
−1
NRQCD for quarkonia
relative error
relative error
−1
from w4/6
10
−2
−2
10
10
−1
−1
−1
10
10
−2
−2
10
10
−1
−2
10
−1
10
10
10
3
3
−2
10
−1
−1
10
i
2
10
i
2
from 1/4mE
from 1/2mB
−2
10
HQET for heavy-light
10
from 1/8m4
from (w4 + w4′ )/4
from 1/2mB
−2
10
10
from 1/8m4
−1
−1
10
−2
−2
10
10
−3
−3
10 0.01
0.1
a (fm)
0.01
0.1
a (fm)
10
from w4/6
relative error
from wB /4
NRQCD for quarkonia
relative error
−2
−2
10
10
−3
from (w4 + w4′ )/4
HQET for heavy-light
from 1/4mE
from wB /4
−1
10
−3
10 0.01
0.1
a (fm)
0.01
0.1
10
a (fm)
From Oktay, Kronfeld, PRD 78 014504 (2008)
We still expect sizable discretization effects for bottomonium and
charm-light states
Some discretization effects remain sizable for at < as ≈ 0.1fm
Modified dispersion relation makes moving frames not straight-forward
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
26 / 35
Testing our tuning: charm and beauty
mJ/Ψ − mηc
mD∗s − mDs
mD∗ − mD
2mD − mc̄c
2MDs − mc̄c
m Ds − m D
mB∗ − mB
mBs∗ − mBs
mBs − mB
mY − mηb
2mB − mb̄b
2mBs − mb̄b
2mBc − mηb − mηc
Ensemble (1)
107.9(0.3)(1.1)
120.4(0.6)(1.3)
129.4(1.8)(1.4)
890.9(3.3)(9.3)
1065.5(1.4)(11.2)
96.6(0.9)(1.0)
-
Ensemble (2)
107.1(0.2)(1.5)
142.1(0.7)(2.0)
148.4(5.2)(2.1)
882.0(6.5)(12.6)
1060.7(1.1)(15.2)
94.0(4.6)(1.3)
46.8(7.0)(0.7)
47.1(1.5)(0.7)
81.5(4.1)(1.2)
44.2(0.3)(0.6)
1190(11)(17)
1353(2)(19)
169.4(0.4)(2.4)
Experiment
113.2(0.7)
143.8(0.4)
140.66(10)
882.4(0.3)
1084.8(0.6)
98.87(29)
45.78(35)
+2.3
48.7−2.1
87.35(23)
62.3(3.2)
1182.7(1.0)
1361.7(3.4)
167.3(4.9)
Errors statistical and scale setting only
Bottom quark slightly too light
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
27 / 35
D∗s0 (2317): D-meson – Kaon s-wave scattering
M. Lüscher Commun. Math. Phys. 105 (1986) 153;
Nucl. Phys. B 354 (1991) 531; Nucl. Phys. B 364 (1991) 237.
2
p cot δ(p) = √ Z00 (1; q2 )
πL
1
1
≈
+ r0 p2
a0 2
Mohler et al. PRL 111 222001 (2013)
Lang, DM et al. PRD 90 034510 (2014)
Results for ensembles (1) and (2)
p cot δ [GeV]
0
a0 = −0.756 ± 0.025fm
-0.2
(1)
r0 = −0.056 ± 0.031fm
-0.4
12
-0.6
a0 = −1.33 ± 0.20fm
-0.8
(2)
r0 = 0.27 ± 0.17fm
-1
-0.1
0
0.1
2
0.2
0.3
0.4
0.5
2
p [GeV ]
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
28 / 35
D∗s0 (2317): D-meson – Kaon s-wave scattering
M. Lüscher Commun. Math. Phys. 105 (1986) 153;
Nucl. Phys. B 354 (1991) 531; Nucl. Phys. B 364 (1991) 237.
2
p cot δ(p) = √ Z00 (1; q2 )
πL
1
1
≈
+ r0 p2
a0 2
Mohler et al. PRL 111 222001 (2013)
Lang, DM et al. PRD 90 034510 (2014)
Results for ensembles (1) and (2)
p cot δ [GeV]
0
a0 = −0.756 ± 0.025fm
-0.2
(1)
r0 = −0.056 ± 0.031fm
-0.4
12
-0.6
a0 = −1.33 ± 0.20fm
-0.8
(2)
r0 = 0.27 ± 0.17fm
-1
-0.1
0
0.1
2
0.2
0.3
0.4
0.5
2
p [GeV ]
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
28 / 35
B∗s0 and Bs1 : Results
aBK
0 = −0.85(10) fm
B∗s0
r0BK = 0.03(15) fm
MB∗s0 = 5.711(13) GeV
aB0
Bs1
∗K
∗
r0B K
= −0.97(16) fm
= 0.28(15) fm
MBs1 = 5.750(17) GeV
Energy from the difference to the B(∗) K threshold
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
29 / 35
Spectrum results
Mohler et al. PRL 111 222001 (2013)
Lang, DM et al. PRD 90 034510 (2014)
Lang, Mohler, Prelovsek, Woloshyn PLB 750 17 (2015)
m - (mDs+3mDs*)/4 [MeV]
Ensemble (1)
Ensemble (2)
600
600
500
500
400
400
300
300
200
200
100
100
0
PDG
Lat: energy level
Lat: bound state
from phase shift
0
-100
-200
Ds
P
J :
0
-
*
Ds
-
1
*
Ds0
+
0
Ds1
1
+
Ds1
1
+
*
Ds2
-100
Ds
+
2
0
-
*
Ds
1
-
*
Ds0
+
0
Ds1
1
+
Ds1
1
+
*
Ds2
2
+
Discretization uncertainties
sizeable for charm
-200
Full uncertainty estimate only for
magenta Bs states
Prediction of exotic states from
Lattice QCD!
Many improvements possible for
the Ds states
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
30 / 35
Spectrum results
Mohler et al. PRL 111 222001 (2013)
Lang, DM et al. PRD 90 034510 (2014)
Lang, Mohler, Prelovsek, Woloshyn PLB 750 17 (2015)
Ensemble (1)
Ensemble (2) mπ = 156 MeV
Ensemble (2)
600
600
5.9
500
500
5.8
400
400
300
300
200
200
100
100
-100
-200
P
J :
*
*
Ds
Ds
Ds0
-
-
+
0
1
0
Ds1
1
+
Ds1
1
+
*
Ds2
5.7
5.6
5.5
PDG
Lat: energy level
Lat: bound state
from phase shift
5.4
-100
*
Ds
Ds
-
-
+
2
BK
0
PDG
Lat: energy level
Lat: bound state
from phase shift
0
m [GeV]
m - (mDs+3mDs*)/4 [MeV]
*
BK
0
1
*
Ds0
+
0
*
Ds1
Ds1
Ds2
+
+
+
1
1
2
-200
5.3
Bs
P
J :
-
0
Bs
*
-
1
Bs0
*
+
0
Bs1
+
1
Bs1’
1
+
Bs2
+
2
Discretization uncertainties
sizeable for charm
Full uncertainty estimate only for
magenta Bs states
Many improvements possible for
the Ds states
Prediction of exotic states from
Lattice QCD!
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
30 / 35
Comparing to models
Covariant (U)ChPT
NLO UHMChPT
LO UChPT
LO χ-SU(3)
Bardeen, Eichten, Hill
rel. quark model
rel. quark model
rel. quark model
HPQCD 2010
this work
Daniel Mohler (HIM)
0+
5726(28)
5696(20)(30)
5725(39)
5643
5718(35)
5804
5833
5830
5752(16)(5)(25)
5713(11)(19)
From quarks and gluons to exotic hadrons
1+
5778(26)
5742(20)(30)
5778(7)
5690
5765(35)
5842
5865
5858
5806(15)(5)(25)
5750(17)(19)
Graz, April 27, 2016
31 / 35
χ0c0 and X/Y(3915)
PDG interprets X(3915) as a regular charmonium (χ0c0 )
Some 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) → D̄D
Spin splitting mχc2 (2P) − mχc0 (2P) too small
Large OZI suppressed X(3915) → ωJ/ψ
Width should be significantly larger than Γχc2 (2P)
Zhou et al. (PRL 115 2, 022001 (2015)) argue that what is dubbed
X(3915) is the spin 2 state already known and suggests that a broader
state is hiding in the experiment data.
Investigate D̄D scattering in S-wave on the lattice!
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
32 / 35
χ0c0 and X/Y(3915)
PDG interprets X(3915) as a regular charmonium (χ0c0 )
Some 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) → D̄D
Spin splitting mχc2 (2P) − mχc0 (2P) too small
Large OZI suppressed X(3915) → ωJ/ψ
Width should be significantly larger than Γχc2 (2P)
Zhou et al. (PRL 115 2, 022001 (2015)) argue that what is dubbed
X(3915) is the spin 2 state already known and suggests that a broader
state is hiding in the experiment data.
Investigate D̄D scattering in S-wave on the lattice!
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
32 / 35
χ0c0 : Exploratory lattice calculation
Lang, Leskovec, DM, Prelovsek, JHEP 1509 089 (2015)
(c)
(b)
1.00
0.80
0.80
0.60
0.60
0.40
0.40
0.20
0.20
0.00
0.00
-0.20
-0.20
-0.40
p cotδ/√s
p cotδ/√s
(a)
1.00
-0.40
-0.6
-0.4
-0.2
0.0
2
0.2
0.4
-0.6
-0.4
2
-0.2
2
p [GeV ]
0.0
0.2
0.4
-0.6
-0.4
2
p [GeV ]
-0.2
2
0.0
0.2
0.4
2
p [GeV ]
Assumes only D̄D is relevant
Lattice data suggests a fairly narrow resonance with
3.9GeV < M < 4.0GeV and Γ < 100MeV
Future experiment and lattice QCD results needed to clarify the situation
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
33 / 35
Emerging understanding from QCD
Ds D*D
* D D D*
s s0 s1 s1 s2
1000
900
800
700
_
m-m_cc [MeV]
600
500
400
300
200
100
0
-100
-200
800
_
m - mDs [MeV]
1100
600
500
_
m-m D [MeV]
400
600
300
400
200
100
0
200
lat: naive level
res. / bound state
0
-100
ηc Ψ h χc0 χc1 χc2 ηc2 Ψ Ψ h χc3
c
2
3 c3
D D*
D0*D1 D1D2*D2
Comprehensive studies where feasible
spectrum around lowest few thresholds
radiative transitions
Extend exploratory results to higher masses and search for manifestly
exotic states
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
34 / 35
...
Thank you!
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
35 / 35
...
Backup Slides
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
36 / 35
First examples of emerging understanding from QCD
lattice (mπ~266 MeV)
600
500
400
400
300
300
200
200
100
100
-100
-200
Ds
P
J :
0
-
*
Ds
-
1
*
Ds0
+
0
Ds1
1
+
Ds1
1
+
1100
D(1)D*(-1)
1000
900
J/ψ(0)ω(0)
D(0)D*(0)
X(3872)
800
X(3872)
700
600
0
PDG
Lat: energy level
Lat: bound state
from phase shift
0
Exp
*
Ds2
Ds
+
2
0
-
*
*
Ds
1
Ds0
-
Ds1
+
0
1
+
Ds1
1
+
*
O: cc
+
Mohler et al.
PRL 111 222001
Ensemble (2) m = 156 MeV
π
500
400
-200
Ds2
2
χc1(1P)
χc1(1P)
-100
O: cc
DD*
J/ψ ω
c
Ensemble (2) mπ = 156 MeV
500
m - 1/4 (mη +3 mJ/ψ) [MeV]
m - (mDs+3mDs*)/4 [MeV]
Ensemble (1) mπ = 266 MeV
600
pole
L→∞
Prelovsek, Leskovec, PRL 111
5.9
192001 (2013)
*
BK
5.8
BK
5.7
1200
E - E(1S) MeV
m [GeV]
1400
5.6
5.5
PDG
Lat: energy level
Lat: bound state
from phase shift
5.4
5.3
Bs
P
J :
-
0
Bs
*
-
1
Bs0
*
+
0
Bs1
+
1
Bs1’
1
+
*
1000
D(-1)D (1)
D(0)D*(0)
800
600
400
cc (I=0)
*
cc + DD (I=0)
*
DD (I=0)
Bs2
Lee, DM et al.
+
2
arXiv:1411.1389
Lang, DM et al. PLB 750 17
Examples of bound states with a large four-quark component
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
37 / 35
Testing our tuning: charm and light
mπ
mK
mφ
mηs
mJ/Ψ − mηc
mD∗s − mDs
mD∗ − mD
2mD − mc̄c
2MDs − mc̄c
mDs − mD
Ensemble (1)
266(3)(3)
552(1)(6)
1015.8(1.8)(10.7)
732.3(0.9)(7.7)
107.9(0.3)(1.1)
120.4(0.6)(1.3)
129.4(1.8)(1.4)
890.9(3.3)(9.3)
1065.5(1.4)(11.2)
96.6(0.9)(1.0)
Ensemble (2)
156(7)(2)
504(1)(7)
1018.4(2.8)(14.6)
692.9(0.5)(9.9)
107.1(0.2)(1.5)
142.1(0.7)(2.0)
148.4(5.2)(2.1)
882.0(6.5)(12.6)
1060.7(1.1)(15.2)
94.0(4.6)(1.3)
Experiment
139.5702(4)
493.677(16)
1019.455(20)
688.5(2.2)*
113.2(0.7)
143.8(0.4)
140.66(10)
882.4(0.3)
1084.8(0.6)
98.87(29)
A single ensemble: Discrepancies due to discretization and unphysical
light-quark masses expected
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
38 / 35
D∗s0 (2317) including D meson - Kaon
DM, Lang, Leskovec, Prelovsek, Woloshyn, PRL 111 222001 (2013)
M - M1S [MeV]
Ensemble (1)
Ensemble (2)
900
900
800
800
700
700
600
600
500
500
400
400
300
300
200
200
100
100
0
0
qq
qq + DK
qq
qq + DK
Much better quality of the ground state plateau with combined basis
D∗s0 (2317) as a QCD bound state
Suggests that including multi-hadron levels is vital
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
39 / 35
Previous lattice results
NRQCD b quarks and staggered light quarks
States predicted slightly below the B(∗) K thresholds:
MB∗s0 = 5752(16)(5)(25)
MBs1 = 5806(15)(5)(25)
Gregory et al.
PRD 83 014506 (2011)
Static-light mesons with the transition amplitude method
McNeile, Michael, Thompson, PRD 70 054501 (2004)
Static-light mesons plus interpolation between static light states and
experiment Ds states
Green et al.
PRD 69 094505 (2004)
Static-light states on quenched and 2 flavor lattices
Burch et al.
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
PRD 79 014504 (2009)
Graz, April 27, 2016
40 / 35
Possible interpretations
(1) A sub-threshold state stable under the strong interaction
We call this “bound state scenario”
This is irrespective of the nature of the state
One expects a negative scattering length in this case
See Sasaki and Yamazaki, PRD 74 114507 (2006) for details.
(2) A resonance in a channel with attractive interaction
The lowest state corresponds to the scattering level shifted below
threshold in finite volume
The additional level would indicate a QCD resonance
One expects a positive scattering length in this case
This is the situation for the D∗0 (2400)
DM, Prelovsek, Woloshyn, PRD 87 034501 (2013).
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
41 / 35
B∗so and Bs1 : Systematic uncertainties
source of uncertainty
heavy-quark discretization
finite volume effects
unphysical Kaon, isospin & EM
b-quark tuning
dispersion relation
spin-average (experiment)
scale uncertainty
3 pt vs. 2 pt linear fit
total
expected size [MeV]
12
8
11
3
2
2
1
2
19
discretiation effects from HQET power counting also considering mass
mismatches
Oktay, Kronfeld Phys.Rev.
D78 014504 (2008)
Finite volume from difference between the energy level and the pole
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
42 / 35
Search for Zc+ with I G J PC = 1+ 1+−
D(2) D*(-2)
D*(1) D*(-1)
J/ψ(2) π(−2)
ψ3 π
D(1) D*(-1)
ψ1D π
D* D*
ηc(1)ρ(−1)
ψ2S π
D D*
j/ψ(1) π(-1)
ηc ρ
J/ψ π
4.6
4.4
E[GeV]
4.2
4
3.8
3.6
3.4
3.2
Exp.
Lattice
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 (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
43 / 35
Exotic meson physics for P̄ANDA, BelleII, BESIII, LHCb
Simulate a large basis of quark-antiquark (regular and hybrid), meson-meson
and tetraquark operators with a variety of quantum numbers
Study the spectrum and extract bound states and resonances
Study quark mass dependence to confirm/falsify model expectations
Study operator overlaps to learn about structure
Study (radiative) transition amplitudes to learn about structure
Promising examples (similar for heavy-light states):
X(3872)
Establish relation between the observed candidate and the X(3872)
Study charm-quark variation and compare to models/ EFT
Study radiative transitions of the candidate state
χ0c0 /X(3915)
Study DD̄∗ and J/ψω scattering and establish resonances < 4GeV
Charged charmonium-like Zc states
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
44 / 35
Modern methods
Use Lüscher’s method to access scattering phase shifts/ inelasticities
→ bound state and resonance poles
State of the art propagator calculations: distillation method
Handles all smeared timeslice-to-timeslice correlators
Allows for storing the quark propagators
Highly flexible for large synergy between different projects
Provides flexibility to optionally address light quark exotics, high spin
states and baryons
Improved heavy quark action (either Fermilab approach or highly
improved actions)
→ small and well understood discretization effects
Methods are mostly established but the combination of methods is unique.
A next generation resonance project will profit from lattice gauge fields
made available by various lattice collaborations (MILC, CLS, . . . )
Daniel Mohler (HIM)
From quarks and gluons to exotic hadrons
Graz, April 27, 2016
45 / 35