Bound states on the lattice

Transcription

1 Bound states on the lattice Daniel Mohler Admont, February, 2017 Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

2 A simplistic take on fermion discretizations family staggered Wilson-like Ginsparg-Wilson fermions such as ASQTAD Wilson Overlap HISQ Wilson-Clover Domain Wall Twisted Mass Chirally Improved Cost $ $$ $$$ Chiral Symmetry Issues remnant symmetry Lattice version: Dγ 5 + γ 5D = adγ 5D Taste multiplets light quarks are tough rooting procedure Suitable choice may depend on the project Collaborations pick an action (and stick to it) As always: the full story is messy Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

3 Scale setting and physical quark masses Compare: Discussion by Christian Fischer The QCD action contains parameters m l (,m s,m c, m b ) and β = 6 g 2 Example: for 2+1 flavors we need to fix 3 parameters to obtain physical results Physical results for (all) observables only in the infinite volume and continuum limits! Comparisons away from the physical point can be misleading m = 0 chiral limit constant physics constant a continuum limit m = β = 0 ( g= ) quenched limit β = ( g= 0) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

4 Scale setting: What is a good choice? Ideally the quantity used... can be obtained without much computation is well defined in the finite volume has a small statistical uncertainty is well determined from experiment has no strong dependence on hard-to-control systematics depends strongly on only a single parameter is not a valuable output from QCD Good quantities: Masses of hadrons stable under QCD m π, m K, m Ω, Charmonium masses Light meson decay constants f π, f K Quite often an intermediary scale is used: Sommer parameter r 0, Wilson-flow variables t 0, w 0 Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

5 The landscape of lattice simulations L[fm] % 0.3% 1% ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 (2) CLS '10(2) M π [MeV] Plots from Christian Hoelbling Acta Phys.Polon. B45 no.12, 2143, 2014 Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

6 The landscape of lattice simulations M π [MeV] ETMC '09 (2) ETMC '10 (2+1+1) MILC '10 MILC '12 QCDSF '10 (2) QCDSF-UKQCD '10 BMWc '10 BMWc'08 PACS-CS '09 RBC/UKQCD '10 JLQCD/TWQCD '09 HSC '08 BGR '10 CLS '10 (2) a[fm] Plots from Christian Hoelbling Acta Phys.Polon. B45 no.12, 2143, 2014 Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

7 Lattice simulations: Specific examples N f = Highly Improved Staggered Quarks (HISQ) completed in progress physical point a (fm) M π =135 MeV M π [MeV] Plot from the MILC collaboration (private communication) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

8 Lattice simulations: Specific examples N f = CLS simulations (Wilson) J500 N300 H200 N202 B450 U103 H J501 N302 N203 S400 U102 H102 mπ[mev] J303 S201 N200 N401 U101 H105 N D200 S100 C101 D D150 physical D a 2 [fm 2 ] Plot from Bali et. al. arxiv: Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

9 A practical example of a combined extrapolation Charmonium splittings (see first Lecture by Christian Fischer) In practice one usually performs a combined extrapolation A more complicated example from charmonium splittings M = M 0 + c 1 (2x l + x h ) + c 2 f 1 (a) + c 3 f 2 (a) +... x l = m ud,sea m ud,phys m s,phys x h = m s,sea m s,phys m s,phys In this case: Mistuning of sea-quark masses differing from ensemble to ensemble An special heavy-quark action that complicates the continuum extrapolation Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

10 Chiral-continuum extrapolation for low-lying charmonia M HF = M n 3 L M n 1 L lattice data fit results at lattice parameters M HF r χ 2 aug /d.o.f. aug = 2.82/ a [fm] Preliminary data from the Fermilab-MILC collaboration - to be published Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

11 Chiral-continuum extrapolation for low-lying charmonia M 1P 1S = M 1P M 1S, M 1P = (M χc0 + 3M χc1 + 5M χc2 )/9, M 1S = (M ηc + 3M J/ψ )/ lattice data fit result at lattice parameters (M 1P - M 1S ) r χ 2 aug /d.o.f. aug = 5.83/ a [fm] Preliminary data from the Fermilab-MILC collaboration Daniel Mohler (HIM) - to Bound be statespublished on the lattice Admont, February, / 55

12 Outline of Lecture 3 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

13 Stable hadron states: A lattice success story Light mesons and baryons b b Heavy mesons h b (2P) h b (1P) b2 b1 (2P) b0 b2 b0 b1 (1P) expt fix params postdcns predcns (1D) MESON MASS (GeV/c 2 ) c c c2 h c1 c J/ c0 B c B c B s B B * B * c c B * s B * B * c0 2 D s D Example from BMW Dürr et al. Science 322 (2008) 0 Example from HPQCD Dowdall et al. PRD (2012) Hadrons stable under QCD: full control of systematic uncertainties Routinely done for a wide variety of observables (for example for flavor physics) Future goal: Extend this success to hadron resonances Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55 K

14 Precision flavor physics: The FLAG report Example for fully systematic results: Aims to answer: What is currently the best lattice value for a particular quantity? Uses symbols derived from rigorous quality criteria and covers precision results ±/ ± ±/ ± = + + FLAG average for = + + ETM 14E FNAL/MILC 14A ETM 13F HPQCD 13A MILC 13A MILC 11 (stat. err. only) ETM 10E (stat. err. only) = + + FLAG average for = + + ETM 14E FNAL/MILC 14A ETM 13F HPQCD 13A MILC 13A MILC 11 (stat. err. only) ETM 10E (stat. err. only) = + FLAG average for = + RBC/UKQCD 14B 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 = + FLAG average for = + RBC/UKQCD 14B 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 = FLAG average for = ETM 14D (stat. err. only) ALPHA 13A BGR 11 ETM 10D (stat. err. only) ETM 09 QCDSF/UKQCD 07 = FLAG average for = ETM 14D (stat. err. only) ALPHA 13A BGR 11 ETM 10D (stat. err. only) ETM 09 QCDSF/UKQCD 07 Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

15 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

16 A lesson from the ρ resonance Plot from Lang, DM, Prelovsek, Vidmar, PRD (2011); E n a E n a E n a with ππ without ππ P=(0,0,0) P=(0,0,1) P=(1,1,0) interpolator set interpolator set: qq ππ 1: O 1,2,3,4,5, O 6 2: O 1,2,3,4, O 6 3: O 1,2,3, O 6 4: O 2,3,4,5, O 6 5: O 1, O 6 6: O 1,2,3,4,5 7: O 1,2,3,4 8: O 1,2,3 A diverse interpolator basis is vital to determine the true spectrum! Beware: Effective energies may seem to reach a plateau (with good fit χ 2 ) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

17 A lesson from the ρ resonance Plot Wilson et al. PRD (2015) A diverse interpolator basis is vital to determine the true spectrum! Beware: Effective energies may seem to reach a plateau (with good fit χ 2 ) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

18 A lesson from the ρ resonance Data from Mohler et al. PRL (2013) Effective masses - am1.1 ground state 2x2 basis excited state 2x2 basis just interpolator 1 of t A diverse interpolator basis is vital to determine the true spectrum! Beware: Effective energies may seem to reach a plateau (with good fit χ 2 ) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

19 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

20 Recent progress: Singly charmed baryon states Recent years have seen a number of studies Studies by Briceno et. al, ETMC, and Brown et al. feature a full chiral-continuum extrapolation Different methods agree quite well Plot from Perez-Rubio, Collins, Bali, PRD (2015) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

21 Recent progress: Singly charmed baryon states Doubly charmed baryons are predictions Only one candidate seen in the SELEX experiment (not confirmed by other experiments) Different methods agree quite well Plot from Perez-Rubio, Collins, Bali, PRD (2015) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

22 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

23 A reminder about the lightest nonet of scalar mesons mass I=0,1 I=1/2 I=0 Observed scalars (below 1 GeV)?? κ(800) σ(600) I=0,1 a0(980) f0(980) I=1/2 I=0 mass qq nonet (vector meson case) ss uu dd us φ ud K * ρ,ω 1 1/2 0 1/2 1 I 3 1 1/2 0 1/2 1 I 3 Tetraquark picture naturally leads to the correct pattern See also previous lectures at this school In a quantum field theory: Mixing simple models may be misleading Goal:Determine poles in the complex plane using the Lüscher method, including both meson-meson (4 quark) and quark-antiquark interpolators Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

24 A reminder about the lightest nonet of scalar mesons mass I=0,1 I=1/2 I=0 mass Observed scalars (below 1 GeV) I=0,1 a0(980) f0(980)? I=1/2? κ(800) I=0 σ(600) Tetraquark nonet ussu dssd udud udds ussd 1 1/2 0 1/2 1 I 3 1 1/2 0 1/2 1 I 3 Tetraquark picture naturally leads to the correct pattern See also previous lectures at this school In a quantum field theory: Mixing simple models may be misleading Goal:Determine poles in the complex plane using the Lüscher method, including both meson-meson (4 quark) and quark-antiquark interpolators Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

25 From the ρ to coupled channels For the elastic case: One phase-shift point for each energy level For two coupled channels: Have two phase-shifts and one inelasticity η Would need 3 data points at the same E cm Instead try out various parameterizations of the energy dependence of the t-matrix and test if they can describe the data. Given enough lattice data points this approach should work Notice the similarity to the analysis of experiment data! Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

26 Coupled channel πk ηk scattering Wilson, Dudek, Edwards, Thomas PRD (2015) and PRL (2014) First coupled channel study in Lattice QCD Channels are mostly decoupled. Finds poles related to the f 0 (500) or κ and K 0 (1430), K (892), K 2 (1430) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

27 Coupled channel πk ηk scattering Wilson, Dudek, Edwards, Thomas PRD (2015) and PRL (2014) First coupled channel study in Lattice QCD Channels are mostly decoupled. Finds poles related to the f 0 (500) or κ and K 0 (1430), K (892), K 2 (1430) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

28 The a 0 resonance in πη KK scattering J. Dudek et al. PRD (2016) Channels are tightly coupled Prominent cusp-like structure in πη πη close to KK threshold Rhs plot illustrates systematics from different parameterizations Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

29 Isoscalar ππ scattering and the σ meson resonance. J. Dudek et al. arxiv: Lowering the pion mass phase shift approaches experiment Analysis of lattice results is similar to experimental data analysis Phase-shifts alone will not settle controversy over nature Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

30 Isoscalar ππ scattering and the σ meson resonance. J. Dudek et al. arxiv: Lowering the pion mass phase shift approaches experiment Analysis of lattice results is similar to experimental data analysis Phase-shifts alone will not settle controversy over nature Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

31 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

32 Hybrid mesons and baryons Dudek PRD (2011); Caveat: No multi-hadron operators Provides a qualitative picture (number and energy scale of hybrid states) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

33 Hybrid mesons and baryons Dudek & Edwards PRD (2012) Caveat: No multi-hadron operators Provides a qualitative picture (number and energy scale of hybrid states) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

34 Hybrid candidates in the heavy-quark spectrum Cheung et al. arxiv: Similar results also for charm-light mesons Caveat: No multi-hadron operators Provides a qualitative picture (number and energy scale of hybrid states) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

35 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

36 Exotic D s and B s candidates Established s and p-wave states: D s (J P = 0 ) and D s (1 ) D s0 (2317) (0+ ), D s1 (2460) (1 + ), D s1 (2536) (1 + ), D s2 (2573) (2+ ) B s (J P = 0 ) and B s (1 ) B s1 (5830) (1 + ), B s2 (5840) (2+ ) Peculiarity: M c s M c d exotic structure? (tetraquark, molecule) Traditional lattice studies (using single hadron operators) tend get too large or badly determined masses Lecture by Guo: States of interest correspond to j = 1 2 multiplet decaying in S-wave B s cousins of the D s0 (2317) and D s1(2460) not (yet) seen in experiment LHCb should be able to see these Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

37 Exotic D s and B s candidates Established s and p-wave states: D s (J P = 0 ) and D s (1 ) D s0 (2317) (0+ ), D s1 (2460) (1 + ), D s1 (2536) (1 + ), D s2 (2573) (2+ ) B s (J P = 0 ) and B s (1 )? B s1 (5830) (1 + ), B s2 (5840) (2+ ) Peculiarity: M c s M c d exotic structure? (tetraquark, molecule) Traditional lattice studies (using single hadron operators) tend get too large or badly determined masses Lecture by Guo: States of interest correspond to j = 1 2 multiplet decaying in S-wave B s cousins of the D s0 (2317) and D s1(2460) not (yet) seen in experiment LHCb should be able to see these Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

38 D s0 (2317) including D meson - Kaon DM, Lang, Leskovec, Prelovsek, Woloshyn, PRL (2013) 900 Ensemble (1) Ensemble (2) M - M 1S [MeV] qq qq + DK qq qq + DK 0 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) Bound states on the lattice Admont, February, / 55

39 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 (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 (2013). Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

40 Using Lüscher s formula We can test the plausibility of these scenarios using Lüscher s formula and 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 = p cot δ(p) = 2 πl Z 00 (1; q 2 ), 1 a r 0p 2, Results for ensembles (1) and (2) a 0 = ± 0.025fm r 0 = ± 0.031fm (1) a 0 = 1.33 ± 0.20fm r 0 = 0.27 ± 0.17fm (2) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

41 Results: D( ) K scattering C. B. Lang, DM, S. Prelovsek, R. M. Woloshyn, PRD (2014) Sara Collins, Gunnar Bali - private communication Ensemble (1) mπ = 266 MeV Ensemble (2) mπ = 156 MeV PDG Lat: energy level Lat: bound state from phase shift Ds P J : 0 - * Ds - 1 * Ds0 + 0 Ds1 1 + Ds1 1 + * Ds2 + 2 MeV (MD + MK ) MDs0 m - (mds+3mds*)/4 [MeV] Expt Ds 0 - * Ds 1 - * Ds0 + 0 Ds1 1 + Ds1 1 + * -200 Ds2 2 Lang et al Lang et al a = a = Lmπ fm, fm, Mπ Mπ Mπ Mπ = 160 = 290 = 156 = 266 MeV MeV MeV MeV Discretization uncertainties sizeable for charm Many improvements possible for the Ds states Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

42 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

43 Testing our tuning: charm and beauty Ensemble (1) Ensemble (2) Experiment m J/Ψ m ηc 107.9(0.3)(1.1) 107.1(0.2)(1.5) 113.2(0.7) m D s m Ds 120.4(0.6)(1.3) 142.1(0.7)(2.0) 143.8(0.4) m D m D 129.4(1.8)(1.4) 148.4(5.2)(2.1) (10) 2m D m cc 890.9(3.3)(9.3) 882.0(6.5)(12.6) 882.4(0.3) 2M Ds m cc (1.4)(11.2) (1.1)(15.2) (0.6) m Ds m D 96.6(0.9)(1.0) 94.0(4.6)(1.3) 98.87(29) m B m B (7.0)(0.7) 45.78(35) m Bs m Bs (1.5)(0.7) m Bs m B (4.1)(1.2) 87.35(23) m Y m ηb (0.3)(0.6) 62.3(3.2) 2m B m bb (11)(17) (1.0) 2m Bs m bb (2)(19) (3.4) 2m Bc m ηb m ηc (0.4)(2.4) 167.3(4.9) Errors statistical and scale setting only Bottom quark slightly too light However: Large discretization effects in dispersion relation! Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

44 B so and B s1 : Results a BK 0 = 0.85(10) fm r BK 0 = 0.03(15) fm M Bs0 = 5.711(13) GeV a B K 0 = 0.97(16) fm r B K 0 = 0.28(15) fm M Bs0 = 5.750(17) GeV Energy from the difference to the B ( ) K threshold Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

45 A further sanity check Discretization errors expected to be smaller than for D s Closer to the heavy-quark limit B * K B K m [GeV] M B s1 = 5.831(9)(6) GeV M Bs2 = 5.853(11)(6) GeV 5.4 PDG Lat: energy level 5.3 B s * B s * B s0 B s1 B s1 B s2 J P : Uncertainties just statistics and scale setting Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

46 B so and B s1 : Systematic uncertainties source of uncertainty expected size [MeV] heavy-quark discretization 12 finite volume effects 8 unphysical Kaon, isospin & EM 11 b-quark tuning 3 dispersion relation 2 spin-average (experiment) 2 scale uncertainty 1 3 pt vs. 2 pt linear fit 2 total 19 discretization effects from HQET power counting also considering mass mismatches Oktay, Kronfeld Phys.Rev. D (2008) Finite volume from difference between the energy level and the pole Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

47 Spectrum results Lang, DM, Prelovsek, Woloshyn PLB (2015) Ensemble (2) m π = 156 MeV B * K B K m [GeV] PDG 5.4 Lat: energy level Lat: bound state from phase shift 5.3 B s * B s * B s0 B s1 B s1 B s2 J P : Full uncertainty estimate only for magenta B s states Prediction of exotic states from Lattice QCD! Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

48 Comparing to models 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) Bardeen, Eichten, Hill 5718(35) 5765(35) rel. quark model rel. quark model rel. quark model HPQCD (16)(5)(25) 5806(15)(5)(25) this work 5713(11)(19) 5750(17)(19) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

49 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

50 An X(3872) candidate from Lattice QCD lattice (m π ~266 MeV) Exp D(1)D*(-1) J/ψ(0)ω(0) D(0)D*(0) O: cc O: cc DD* J/ψ ω X(3872) χ c1 (1P) pole L X(3872) χ c1 (1P) m - 1/4 (m ηc +3 m J/ψ ) [MeV] Prelovsek, Leskovec, PRL (2013) Neglects charm annihilation and J/ψω Seen only when qq and D D are used E - E(1S) MeV D(-1)D * (1) D(0)D * (0) cc (I=0) cc + DD * (I=0) DD * (I=0) Lee, DeTar, DM, Na, arxiv: The two simulations have vastly different systematics (yet results are similar) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

51 An X(3872) candidate from Lattice QCD II Padmanath, Lang, Prelovsek, PRD (2015) E n [GeV] Lat. - O MM 17 Lat. - O MM 17 - O- c c J/Ψ(1) ω(-1) D(1) - D* (-1) J/Ψ(0) - ω(0) D(0) D*(0) η c (1) σ(-1) Without qq interpolators signal vanishes Simulations still unphysical in many ways Discretization and finite volume effects sizable! m X(3872) m s.a. m X(3872) m D m- D* Exp. Lat. Lat.-O 4q [31] [32] Makes interpretation as pure molecule or pure tetraquark unlikely Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

52 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

53 2 2 ) ) Charged charmonium-like Z c states: Selected studies 1 Search for a Z c + state from Lattice QCD Prelovsek, Lang, Leskovec, DM, Phys.Rev. D (2015) 2 Z c (3900) with the HALQCD method Ikeda et al. PRL (2016) Both of these studies are purely qualitative 2 Events / 0.01 GeV/c Z c (3900) ± : BESIII, Belle, data from Cleo Data Total fit M max (π ± J/ψ) (GeV/c ) Background fit PHSP MC Sideband Events/ ( 0.005GeV/c Z c (4020) ± : BESIII M π ± (GeV/c ) h c Candidates / ( 0.2 GeV Z(4430) ± : Belle, LHCb LHCb m ψ'π [GeV ] Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

54 Search for a Z + c state from Lattice QCD Prelovsek, Lang, Leskovec, DM, Phys.Rev. D (2015) Search for a Z c + in the I G J PC = channel Aim at simulating all meson-meson states below 4.3GeV Caveat: Neglects 3-particle states Include tetraquark interpolators of type 3 c 3 c Count energy levels and identify them according to their overlaps Hope: See an extra level, as would be expected for a (narrow) resonance More rigorous approach (a la Lüscher) quite challenging Coupled channel system with many channels Small shifts in finite volume and (largish) discretization effects Thresholds should be close to physical Suitable ensembles are (probably) not available at the moment. Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

55 A look at the spectrum of scattering states Expect level close to non-interacting scattering states J/Ψπ η cρ JΨ(1)π( 1) DD Ψ 2Sπ D D Ψ 3770π D(1)D ( 1) Ψ 3π JΨ(2)π( 2) D (1)D ( 1) D(2)D ( 2) E[GeV] Lattice ψ 3 π D(1) D*(-1) ψ(3770) π D* D* ψ(2s) π D D* j/ψ(1) π(-1) η c ρ J/ψ π Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

56 Search for Z + c with I G J PC = Prelovsek, Lang, Leskovec, DM, Phys.Rev. D (2015) E[GeV] D(2) D*(-2) D*(1) D*(-1) J/ψ(2) π( 2) ψ 3 π D(1) D*(-1) ψ 1D π D* D* η c (1)ρ( 1) ψ 2S π D D* j/ψ(1) π(-1) η c ρ J/ψ π Exp. Lattice Simple level counting approach We find 13 two meson states as expected We find no extra energy level that could point to a Z c candidate Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

57 Z c (3900) with the HALQCD method I Ikeda et al. PRL (2016) Coupled-channel scattering J/Ψπ, η c ρ, DD, I G (J PC ) = Uses 2+1 flavor gauge configurations with a = 0.907(13) and m π = 410, 570, 700 HALQCD method Ishii et al. PLB 712, 437 (2012) Calculate a potential as a function of distance r Solve Schrödinger equation with given V(r) and determine scattering phase shifts Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

58 Z c (3900) with the HALQCD method II Ikeda et al. PRL (2016) Im[f(W c.m. )] (fm) Im[f Dbar D *,D bar D * ] Im[f ρη c,ρη c ] Im[f πj/ψ,πj/ψ ] x W c.m. (GeV) Authors conclude Z c (3900) not a usual resonance but a threshold cusp Analysis of S-matrix pole structure is consistent with this picture Structure comes from strong πj/ψ DD coupling Analysis at close-to-physical pion mass planned Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

59 Outline 1 A lesson from the ρ resonance 2 Spectroscopy of Heavy-Baryons 3 Light mesons and coupled-channel scattering Light scalar mesons: σ, κ, a 0 4 Some evidence for hybrid mesons and baryons 5 Heavy-light p-wave mesons: D s0 (2317) and B s states 6 Positive parity B s states 7 The X(3872) 8 Charged charmonium-like states 9 Tetraquarks with heavy mesons Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

60 Recent simulations of charm or beauty tetraquarks Searches for charmed tetraquarks Doubly charmed and charmed-strange tetraquarks with the HALQCD method Ikeda et al. PLB (2014) Search for doubly charmed tetraquarks on CLS lattices (preliminary) Guerrieri et al. arxiv: HHLL systems with static heavy quarks Tetraquark bound states in heavy-light heavy-light systems Brown and Orginos PRD (2012) Lattice QCD results for a bottom-bottom tetraquark Bicudo and Wagner PRD (2013) Search for ud b b ss b b and cc b b tetraquarks Bicudo et al., PRD (2015) BB interactions with static bottom quarks Bicudo, Cichy, Peters, Wagner, PRD (2016) Doubly bottom strong-interaction tetraquarks Francis, Hudspith, Lewis, Maltman, arxiv: Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

61 Search for charmed tetraquarks by HALQCD Ikeda et al. PLB (2014) Search for bound states or resonances in DD, KD, DD and KD interactions with flavor structure ccū d and csū d These contain no quark line diagrams with quark annihilation Uses 2+1 flavor gauge configurations with a = 0.907(13) and m π = 410, 570, 700 HALQCD method Ishii et al. PLB 712, 437 (2012) Calculate a potential as a function of distance r Solve Schrödinger equation with given V(r) and determine scattering phase shifts Uses variant of the Fermilab method (relativistic heavy quark action) Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

62 Tetraquarks with the HALQCD method: Results Ikeda et al. PLB (2014) Repulsive interaction in all I = 1 channels considered Attractive interaction in all I = 0 channels considered δ[deg] (c) D-D* phase shift 10 M π =700MeV 5 M π =570MeV 0 M π =410MeV W c.m. - M D - M D* [MeV] a [fm] Scattering lengths a D-D* a K bar -D* a K bar -D M π [GeV ] No bound states or resonances at simulated m π Attraction becomes more prominent at light pion masses Authors have some indication that BB with IJ P = 01 + is bound Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

63 BB interactions with static bottom quarks Bicudo, Cichy, Peters, Wagner, PRD (2016) Potentials of two static antiquarks in the presence of two light quarks Search for bound states (rather than resonances) Lattices with a = fm and m π 650, 480, 340 Fit function used for the lattice QCD potentials V(r) = α ( r ) p ) ( r exp + V 0 d 0.1 qq = (ud-du)/ 2 qq= uu, (ud+du)/ 2, dd α MeV -20 MeV Resulting binding energy: d in fm r min =3a vector isotriplet extrapolation vector isotriplet B40 vector isotriplet B85 vector isotriplet B150 r min =2a scalar isosinglet extrapolation scalar isosinglet B40 scalar isosinglet B85 scalar isosinglet B MeV -100 MeV E B = MeV Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

64 Doubly bottom tetraquarks with NRQCD b-quarks Francis, Hudspith, Lewis, Maltman, arxiv: Study at a single lattice spacing and three pion masses Authors obtain bound 4-quark states for both ud b b and ls b b Potential issues Binding energies extracted from ratios can be misleading For a bound state, excited state naively expected above threshold Finite volume effects alter the binding energy Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55

65 Backup slides Daniel Mohler (HIM) Bound states on the lattice Admont, February, / 55