Exploratory calculations of meson resonances andbound states from lattice QCD
Daniel Mohler
Ashburn, VA,April 14, 2015
Collaborators: C. B. Lang, L. Leskovec, S. Prelovsek, R. M. Woloshyn
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 1 / 38
Outline
1 Introduction & Methods
2 The a1(1260) and b1(1230) in πρ and πω scattering
3 P-wave Ds and Bs hadronsD∗s0(2317) and Ds1(2460) as QCD bound statesPositive parity Bs states
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 2 / 38
Outline
1 Introduction & Methods
2 The a1(1260) and b1(1230) in πρ and πω scattering
3 P-wave Ds and Bs hadronsD∗s0(2317) and Ds1(2460) as QCD bound statesPositive parity Bs states
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 3 / 38
Recent progress in lattice QCD
Dynamical simulations with 2+1(+1) quark flavors
Simulations at physical pion masses
Isospin splitting and QCD+QED simulations
Improved heavy quark actions for charm
0
2
4
6
8
10
ΔM
[MeV
]
ΔN
ΔΣ
ΔΞ
ΔD
ΔCG
ΔΞcc
experimentQCD+QEDprediction
BMW 2014 HCH
BMW Collaboration, Borsanyi et al. arXiv:1406.4088
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 4 / 38
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-scatteringSee talk by David Wilson
I will report on exploratory calculations regarding meson resonances andbound states
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 5 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 6 / 38
Euclidean correlators and interpolating fields
On the lattice: Euclidean space correlation functions⟨O2(t)O1(0)
⟩T ∝ ∑
ne−tEn < 0|O2|n >< n|O1|0 >
Use: Interpolating field operator that creates states with correct quantumnumbers.
Example I: Pseudoscalar Mesons with IJPC = 10−+
O(1)π = uγ5d
O(2)π = u
←→D γiγtγ5d
Example II: Nucleon
ON = εabc Γ1 ua(uT
b Γ2 dc−dTb Γ2 uc
)In practice: Many (slightly different) constructions possible!In a QFT they should all be OK; Overlap?
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 7 / 38
Using single hadron interpolators, what does one see?
In practical calculations qq 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;
This 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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 8 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 9 / 38
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:1411.6944Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 10 / 38
Studies within our collaboration (past end present)
ππ scattering and ρ meson widthLang, DM, Prelovsek, Vidmar, PRD 84 054503 (2011)
Kπ scatteringLang, Leskovec, DM, Prelovsek, PRD 86 054508 (2012)Prelovsek, Lang, Leskovec, DM, PRD 88 054508 (2013)
πρ and πω scattering and the a1, b1 resonancesLang, Leskovec, DM, Prelovsek, JHEP 1404 162 (2014)
D mesons including Dπ and D?π with relativistic charm quarksDM, Prelovsek, Woloshyn, PRD 87 034501 (2013)
D∗s0(2317) and Ds1(2460) with qq and D(∗)KDM, Lang, Leskovec, Prelovsek, Woloshyn, PRL 111 222001 (2013)
PRD 90 034510 (2014)
Predicting Bs states with JP = 0+,1+Lang, Prelovsek, DM, Woloshyn, arXiv:1501.01646
Meson scattering and charmoniumPrelovsek & Leskovec PRL 111 192001 (2013)
Prelovsek & Leskovec, Phys.Lett. B727 172 (2013)Prelovsek, Leskovec, DM, arXiv:1310.8127
Prelovsek, Lang, Leskovec, DM, PRD 91 014504 (2015)Lang, Leskovec, DM, Prelovsek arXiv:1503.05363
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 11 / 38
Outline
1 Introduction & Methods
2 The a1(1260) and b1(1230) in πρ and πω scattering
3 P-wave Ds and Bs hadronsD∗s0(2317) and Ds1(2460) as QCD bound statesPositive parity Bs states
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 12 / 38
a1(1260) and b1(1230) as resonances in πρ and πω
scattering
Well established resonances a1(1260) and b1(1230)New a1 resonance: a1(1420)
COMPASS Collaboration, Adolph et al. arXiv:1501.05732
See talks by Paul, Krinner
M = 1414+15−13MeV Γ = 153+8
−23MeVAlternative interpretation: Only one a1 resonance (with mass larger thanthe PDG)
Basdevant, Berger arXiv:1501.04643
See talk by Berger
Our simulation was aimed at qualitative results for the a1(1260)Vast (and maybe unrealistic) simplification: Consider only ρπ and ωπ
elastic scatteringWe do a 2-flavor simulation so no KK∗ for the a1
The ρ is assumed to be stable here
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 13 / 38
Effective energies
4 6 8 10t
0.4
0.6
0.8
1
1.2
1.4
1.6E
eff (t
) a
a1: O
qq, O
ρπ
6 8 10t
a1: O
4 6 8 10t
b1: O
qq,O
ωπ
4 6 8 10t
b1: O
V(0)π(0)
V(1)π(-1)
Combined basis neededClear attractive interaction in the ρπ channelInteraction in the ωπ scattering compatible with no shift→ more statistics needed
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 14 / 38
Resulting phase shift data for the a1
0.3 0.4 0.5 0.6 0.7 0.8
s a2
0
0.2
0.4
0.6
0.8
(p/√
s) c
ot
δ
ρπ threshold
For the a1 the assumption of a stable ρ is a strong assumption, althoughthe effect at our parameters is probably small
Roca, Oset, PRD 85 054507 (2012)
We assume a simple Breit-Wigner shape and neglect channels other thanπρ
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 15 / 38
Resulting resonance masses
resonance a1(1260) b1(1235)quantity mres
a1ga1ρπ aρπ
l=0 mresb1
gb1ωπ
[GeV] [GeV] [fm] [GeV] [GeV]lat 1.435(53)(+0
−109) 1.71(39) 0.62(28) 1.414(36)(+0−83) input
exp 1.230(40) 1.35(30) - 1.2295(32) 0.787(25)
Uncertainties statistical and fit choices only
Higher than physical resonance masses but with large systematics
A rigorous future treatment for the a1 will have to includea1↔ ρπ ↔ 3π and further channelsThe rigorous formalism for that is currently being developed
Hansen, Sharpe, PRD 90 116003 (2014);Polejaeva, Rusetsky, EPJ A48 67 (2012);Briceno, Davoudi, PRD 87 094507 (2013)
No indication for a second low-lying a1 resonance, but might missimportant interpolators
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 16 / 38
Outline
1 Introduction & Methods
2 The a1(1260) and b1(1230) in πρ and πω scattering
3 P-wave Ds and Bs hadronsD∗s0(2317) and Ds1(2460) as QCD bound statesPositive parity Bs states
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 17 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 18 / 38
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−mcc 890.9(3.3)(9.3) 882.0(6.5)(12.6) 882.4(0.3)2MDs
−mcc 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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 19 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 20 / 38
Experimental Ds spectrum
Established states:Ds (JP = 0−) and D∗s (1−)D∗s0(2317) (0+), Ds1(2460) (1+), Ds1(2536) (1+), D∗s2(2573) (2+)
More recent discoveries:D∗s1(2700) seen by BaBar, Belle, LHCb (1−)D∗sJ(2860) seen by BaBarLHCb overlapping 1− and 3− statesD∗sJ(3040) seen by BaBar (1+?,2−?)
j = 12 doublet almost mass-degenerate with non-strange states
Some models suggest a tetraquark/molecular interpretations forcontroversial states
(Most) lattice studies using single hadron (cs) operators get too large orbadly determined masses
Large mπ : D∗s0(2317) below DK threshold;Small mπ : D∗s0(2317) ≈ DK threshold
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 21 / 38
D∗s0(2317) including D meson - Kaon
DM, Lang, Leskovec, Prelovsek, Woloshyn, PRL 111 222001 (2013)
0
100
200
300
400
500
600
700
800
900
M -
M1S
[M
eV]
Ensemble (1)
0
100
200
300
400
500
600
700
800
900Ensemble (2)
qq qq qq + DKqq + DK
Much better quality of the ground state plateau with combined basisD∗s0(2317) as a QCD bound stateSuggests that including multi-hadron levels is vital
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 22 / 38
Possible interpretations
(1) A sub-threshold state stable under the strong interactionWe call this “bound state scenario”This is irrespective of the nature of the stateOne 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 belowthreshold in finite volumeThe additional level would indicate a QCD resonanceOne 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 (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 23 / 38
Using Lüscher’s formula
We can test the plausibility of these scenarios using Lüscher’s formulaand an effective range approximation
M. Lüscher Commun. Math. Phys. 105 (1986) 153; Nucl. Phys. B 354(1991) 531; Nucl. Phys. B 364 (1991) 237.
K−1 = pcotδ (p) =2√πL
Z00(1;q2) ,
≈ 1a0
+12
r0p2 ,
Results for ensembles (1) and (2)
a0 =−0.756±0.025fm r0 = 0.056±0.031fm (1)
a0 =−1.33±0.20fm r0 = 0.27±0.17fm (2)
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 24 / 38
Results for the scattering length a0
DM, Lang, Leskovec, Prelovsek, Woloshyn, PRL 111 222001 (2013)
0 100 200 300 400Mπ[MeV]
-2
-1.5
-1
-0.5
0
Scat
teri
ng le
ngth
[fm
]
Ensemble (1)Ensemble (2)
We compare to the predictions from an indirect calculationLiu et al. PRD 87 014508 (2013).
Our determination robustly leads to negative values.
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 25 / 38
Resulting Ds P-wave spectrum
-200
-100
0
100
200
300
400
500
600
m -
(m
Ds+
3mD
s*)/
4 [
MeV
]
Ensemble (1) mπ = 266 MeV
-200
-100
0
100
200
300
400
500
600
PDGLat: energy levelLat: bound state from phase shift
Ensemble (2) mπ = 156 MeV
Ds D
s D
s0 D
s1 D
s1 D
s2
JP : 0
- 1
- 0
+ 1
+ 1
+ 2
+
Ds D
s D
s0 D
s1 D
s1 D
s2
0- 1
- 0
+ 1
+ 1
+ 2
+
* * * * * *
Remaining differences of the size of discretization uncertainties
Many improvements possible for the Ds states
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 26 / 38
Bs states from experiment
Two p-wave states known from experiment:Bs1(5830) with M = 5828.7(4) MeVB∗s2(5840) with M = 5839.96(20) MeV and Γ = 1.6(5) MeV
Discovered in two body decays into K−B+ at CDF/D0 and also seen byLHCb
Remaining B∗s0 and Bs1 states not measuredLHCb is working on this
Could be seen in electromagnetic transitions, transitions with a single π0
or transitions through a virtual σ with σ → 2π .
Bardeen, Eichten, Hill, PRD 68 054024 (2003)
What can we say?Disclaimer: Will not consider B(∗)
s η , isospin breaking and or s-wave –d-wave mixing
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 27 / 38
Mesons with beauty and the Fermilab method: Expectations
For the Bs meson, noise is expected to fall exponentially with(Mηb +Mηs)/2 < MBs → Results noisier than for Ds
Lepage; also Gregory et al. PRD 83 014506 (2011)
0 3 6 9 12 15 18 21 24 27 30t
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Eff
ecti
ve
mas
ses
- am
Bs ground state
Fitted effective mass (signal)
B noiseFitted effective mass (noise)
Effective masses
m=000
χ2/d.o.f.=18.65/21
χ2/d.o.f.= 5.37/7
Expected heavy-light discretization effects smaller than for charm(unlike onium)Closer to heavy-quark limit: Less mixing for 1+ states
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 28 / 38
Testing our tuning: charm and beauty
Ensemble (1) Ensemble (2) ExperimentmJ/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−mcc 890.9(3.3)(9.3) 882.0(6.5)(12.6) 882.4(0.3)2MDs
−mcc 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)mB∗ −mB - 46.8(7.0)(0.7) 45.78(35)
mBs∗ −mBs - 47.1(1.5)(0.7) 48.7+2.3−2.1
mBs −mB - 81.5(4.1)(1.2) 87.35(23)mY −mηb - 44.2(0.3)(0.6) 62.3(3.2)2mB−mbb - 1190(11)(17) 1182.7(1.0)2mBs
−mbb - 1353(2)(19) 1361.7(3.4)2mBc −mηb −mηc - 169.4(0.4)(2.4) 167.3(4.9)
Errors statistical and scale setting only
Bottom quark slightly to light
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 29 / 38
Previous lattice results
NRQCD b quarks and staggered light quarksStates 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 andexperiment Ds states
Green et al. PRD 69 094505 (2004)
Static-light states on quenched and 2 flavor lattices
Burch et al. PRD 79 014504 (2009)
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 30 / 38
B∗so and Bs1: Results
C. B. Lang, DM, S. Prelovsek, R. M. Woloshyn arXiv:1501.01646
aBK0 =−0.85(10) fm
rBK0 = 0.03(15) fm
MBs0 = 5.711(13)GeV
aB∗K0 =−0.97(16) fm
rB∗K0 = 0.28(15) fm
MBs0 = 5.750(17)GeV
Energy from the difference to the B(∗)K threshold
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 31 / 38
A further sanity check
Discretization errors expected to be smaller than for Ds
5.3
5.4
5.5
5.6
5.7
5.8
5.9
m [
GeV
]
PDGLat: energy level
B*K
B K
Bs B
s
* B
s0
* B
s1 B
s1’ B
s2
JP: 0
- 1
- 0
+ 1
+ 1
+ 2
+
MB′s1= 5.831(9)(6)GeV
MBs2 = 5.853(11)(6)GeV
Uncertainties just statistics and scale setting
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 32 / 38
B∗so and Bs1: Systematic uncertainties
source of uncertainty expected size [MeV]heavy-quark discretization 12
finite volume effects 8unphysical Kaon, isospin & EM 11
b-quark tuning 3dispersion relation 2
spin-average (experiment) 2scale uncertainty 1
3 pt vs. 2 pt linear fit 2total 19
discretiation effects from HQET power counting also considering massmismatches
Oktay, Kronfeld Phys.Rev. D78 014504 (2008)
Finite volume from difference between the energy level and the pole
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 33 / 38
Resulting Bs P-wave spectrum
C. B. Lang, DM, S. Prelovsek, R. M. Woloshyn arXiv:1501.01646
5.3
5.4
5.5
5.6
5.7
5.8
5.9
m [
GeV
]
PDGLat: energy levelLat: bound state from phase shift
Ensemble (2) mπ = 156 MeV
B*K
B K
Bs B
s
* B
s0
* B
s1 B
s1’ B
s2
JP: 0
- 1
- 0
+ 1
+ 1
+ 2
+
Predicted two Bs states with full uncertainty estimateExpecting results from LHCb
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 34 / 38
Comparing to models
0+ 1+
Covariant (U)ChPT 5726(28) 5778(26)NLO UHMChPT 5696(20)(30) 5742(20)(30)LO UChPT 5725(39) 5778(7)LO χ-SU(3) 5643 5690Bardeen, Eichten, Hill 5718(35) 5765(35)rel. quark model 5804 5842rel. quark model 5833 5865rel. quark model 5830 5858HPQCD 2010 5752(16)(5)(25) 5806(15)(5)(25)this work 5713(11)(19) 5750(17)(19)
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 35 / 38
Outline
1 Introduction & Methods
2 The a1(1260) and b1(1230) in πρ and πω scattering
3 P-wave Ds and Bs hadronsD∗s0(2317) and Ds1(2460) as QCD bound statesPositive parity Bs states
4 Conclusions and Outlook
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 36 / 38
Conclusions & Outlook
Studies of QCD resonances and (close-to-threshold) bound states are ofan exploratory nature
This is in contrast to low lying states (full error budgets)Simulations of the a1(1260) and b1(1230)
Simple simulations a1↔ ρπ and b1↔ ωπ possibleMore statistics and partial wave mixing needed for the b1Conceptional progress needed for a1 (three pions rather than ρπ)
Simulations of p-wave heavy-light states are now possibleDs(2317) and Ds(2460) as QCD bound statesD meson states as resonancesPredictions for two Bs states
Further resonancesResults for the ρ resonanceImpressive data for Kπ – Kη coupled channels→ David WilsonCharmonium resonances close to DD, DD∗ and D∗D∗ can be investigated
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 37 / 38
. . .
Thank you!
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 38 / 38
Backup slides
. . .
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 39 / 38
(My) Method of choice: The variational method
Matrix of correlators projected to fixed momentum (will assume 0)
C(t)ij = ∑n
e−tEn 〈0|Oi|n〉⟨
n|O†j |0⟩
Solve the generalized eigenvalue problem:
C(t)~ψ(k) = λ(k)(t)C(t0)~ψ(k)
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. Phys. B339, 222 (1990)
Blossier et al. JHEP 04, 094 (2009)
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 40 / 38
Dπ and D?π scattering
DM, Prelovsek, Woloshyn, PRD 87 034501 (2013)
In the JP = 0+ D?0 channel we extract three levels
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6s
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
(p*/
s1/2 )
cot δ
For the JP = 1+ channel there are two resonances D1(2420) andD1(2430)
0 3 6 9 12 15 18 21t
0.8
1
1.2
1.4
1.6
1.8
2
aE
D*(0)π(0)
D*(-1)π(1)
1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6s
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
(p*/
s1/2 )
cot δ
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 41 / 38
Resonances in Dπ and D?π scattering
DM, Prelovsek, Woloshyn, PRD 87 034501 (2013)
For resonances determine coupling g rather than Γ = g2 p∗s (for s-wave)
Data at mπ = 266MeV on a single volume / lattice spacing
D D* D0* D
1D
1 D2* D
2
-100
0
100
200
300
400
500
600
700
800
900
1000m
- 1
/4 (
mD
+3m
D*)
[M
eV]
lat: naive levellat: resonance
PDG values
J P
: 0 -
1 -
0+
1 +
1 +
2 +
2 -
LHCb 2013BaBar 2010
D∗0(2400) D1(2430)glat [GeV] 2.55±0.21 2.01±0.15gexp [GeV] 1.92±0.14 2.50±0.40
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 42 / 38
Ψ(3770) resonance from Lattice QCDLang, 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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 43 / 38
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)→ DDSpin splitting mχc2(2P)−mχc0(2P) to smallLarge OZI suppressed X(3915)→ ωJ/ψ
Width should be significantly larger than Γχc2(2P)
Investigate DD 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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 44 / 38
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 DD 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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 45 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 46 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 47 / 38
Heavy-light discretization effects in the Fermilab method
0 0.05 0.1
a [fm]
0
10
20
30
40
∆M
[M
eV
]
Mb
ME
M4
w4
discretiation effects from HQET power counting also considering massmismatches
Oktay, Kronfeld Phys.Rev. D78 014504 (2008)
Daniel Mohler (Fermilab) Mesons from lattice QCD Ashburn, VA, April 14, 2015 48 / 38
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) Mesons from lattice QCD Ashburn, VA, April 14, 2015 49 / 38