Properties of ground and excited state hadrons from lattice QCD

Transcription

1 Properties of ground and excited state hadrons from lattice QCD Daniel Mohler TRIUMF, Theory Group Vancouver, B.C., Canada Newport News, March Co-Workers: Christof Gattringer, Christian Lang, Georg Engel, Markus Limmer, Leonid Glozman, Sasa Prelovsek, Richard Woloshyn Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

2 Outline Excited state spectroscopy Excited states and the lattice The variational method Suitable sources and sinks 2 Spectroscopy with Chirally Improved quarks Light-quark mesons Spotlight: Scalar mesons Light tetraquark states? 3 Baryon axial charges Results for Nucleon and Hyperon axial charges Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

3 Motivation: ground state spectrum Recent (impressive) results: Postdiction of the ground state spectrum What about excited states? BMW-collaboration 2008 Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

4 Euclidean space correlators Euclidean correlator of two Hilbert-space operators Ô and Ô2. Ô2 (t)ô(0) = tr (e T Ĥe tĥô 2 e tĥô ) T Z T T e 0 Ô2 n ten n Ô 0 n Can also be expressed as a Euclidean path integral Ô2 (t)ô(0) = D[ψ, T Z ψ, U]e S E O 2 [ψ, ψ, U]O [ψ, ψ, U], T Z T = D[ψ, ψ, U]e S E. No field operators appear on the right. Simple integral over the classical Euclidean action Can be evaluated with an (importance sampling) Markov chain Monte Carlo Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

5 Euclidean space correlators Euclidean correlator of two Hilbert-space operators Ô and Ô2. Ô2 (t)ô(0) = tr (e T Ĥe tĥô 2 e tĥô ) T Z T T e 0 Ô2 n ten n Ô 0 n Can also be expressed as a Euclidean path integral Ô2 (t)ô(0) = D[ψ, T Z ψ, U]e S E O 2 [ψ, ψ, U]O [ψ, ψ, U], T Z T = D[ψ, ψ, U]e S E. No field operators appear on the right. Simple integral over the classical Euclidean action Can be evaluated with an (importance sampling) Markov chain Monte Carlo Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

6 Euclidean space correlators Euclidean correlator of two Hilbert-space operators Ô and Ô2. Ô2 (t)ô(0) = tr (e T Ĥe tĥô 2 e tĥô ) T Z T T e 0 Ô2 n ten n Ô 0 n Can also be expressed as a Euclidean path integral Ô2 (t)ô(0) = D[ψ, T Z ψ, U]e S E O 2 [ψ, ψ, U]O [ψ, ψ, U], T Z T = D[ψ, ψ, U]e S E. No field operators appear on the right. Simple integral over the classical Euclidean action Can be evaluated with an (importance sampling) Markov chain Monte Carlo Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

7 The problem with excited states From the analysis of Euclidean correlators we found: Ô2 (t)ô(0) e ten < 0 Ô2 n >< n Ô 0 > T n The whole tower of states contributes Ground state is dominant at large t Exited states appear as sub-leading exponentials Noisy background from limited statistics... E 3 E 2 E E 0 Fit to several exponentials leads to poor results/ is often unstable Advanced methods needed for excited states! Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

8 Variational method for hadron masses Variational method (Michael; Lüscher and Wolff; Blossier et al.) Matrix of correlators projected to fixed momentum (will assume 0) C(t) ij = e ten 0 O i n n O j 0 n Solve the generalized eigenvalue problem: C(t) ψ (k) = λ (k) (t)c(t 0 ) ψ (k) ( ( )) λ (k) (t) e te k + O e t E k At large time separation: only a single mass in each eigenvalue. Eigenvectors can serve as a fingerprint. Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

9 Example operators Need: Interpolating field operator that creates states with correct quantum numbers. Example I: Pseudoscalar Mesons with IJ PC = 0 + O () π O (2) π = ūγ 5 d = ū D γ i γ t γ 5 d Example II: Nucleon O N = ǫ abc Γ u a ( u T b Γ 2 d c d T b Γ 2 u c ) In practice: Many (slightly different) constructions possible! Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

10 Example operators Need: Interpolating field operator that creates states with correct quantum numbers. Example I: Pseudoscalar Mesons with IJ PC = 0 + O () π O (2) π = ūγ 5 d = ū D γ i γ t γ 5 d Example II: Nucleon O N = ǫ abc Γ u a ( u T b Γ 2 d c d T b Γ 2 u c ) In practice: Many (slightly different) constructions possible! Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

11 Angular momentum (mesons) Reminder: No unique spin assignment on the lattice. Five irreducible representations: Irrep of O J Spinors in irrep A 0,4,...,γ t,γ 5,γ t γ 5 A 2 3,6,... E 2,4,5,... T,3,4,5,... γ i,γ t γ i, γ 5 γ i,γ t γ 5 γ i T 2 2,3,4,5,... Classification of interpolator basis by representations Identify spin by degeneracies/ through continuum extrapolation Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

12 A variational basis for meson spectroscopy Jacobi smeared quark sources, e.g., u s (S u) x N S = M S 0 with M = κ n H n H( n, m ) = 3 j= n=0 ( ( U j n, 0 ) ( ) δ n + ĵ, m + U j ( n ĵ, 0 ) δ ( n ĵ, m) ). Combination of different widths allows nodes in the interpolating operators Derivative quark sources W di : D i ( x, y) = U i ( x, 0)δ( x + î, y) U i ( x î, 0) δ( x î, y), W di = D i S w. Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

13 Dynamical Chirally Improved fermions 6 D mn = Γ α cp α α= p Pm,n α l p U l δ n,m+p Insert above ansatz into Ginsparg-Wilson-equation Truncate the length of the contributions Set of algebraic equations Wilson s s 2 s 3 s γ v + v 2 + v 3... µ + + a γ µ γ ν t γ µ γ ν γ ρ + γ 5 p... (Gattringer, Hip, Lang, 200) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

14 CI fermions - simulation details n f = 2 mass degenerate flavors of CI quarks Lüscher- Weisz gauge action Hybrid Monte Carlo simulation Mass preconditioning with 2 pseudofermions Chronological inverter Mixed precision inverter (Dürr et al. PRD ) Multiple ensembles for set β LW m 0 # config s a[fm] m π [MeV] m AWI [MeV] A /00 0.5(2) 525(7) 42.8(4) B / () 470(4) 34.(2) C / () 322(5) 5.3(4) We are currently analyzing further ensembles and extending statistics Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

15 Example I: 0 + and channels For the lack of a systematic approach: linear fits to guide the eye More data needed in all channels! (in progress) mass [GeV] A from 3,8, B from 3,8, C from 3,8, π(300) mass [GeV] ρ(450) ρ(770) A B C M π [GeV ] M π [GeV ] Figure: First excited states in the pion and vector meson channels Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

16 Example I: 0 + and channels For the lack of a systematic approach: linear fits to guide the eye More data needed in all channels! (in progress) 2.5 ρ meson mass [GeV] A from 3,8, B from 3,8, C from 3,8, π(300) mass [GeV] A from,4 B from,4 C from,4 ρ(770) quenched M π [GeV ] M π [GeV ] Figure: First excited states in the pion and vector meson channels Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

17 Example II: 2 ++ channel mass [GeV] a 2 (320) A from,2,3,4 B from,2,3,4 C from 2,3 mass [GeV] a 2 (320) A from,2,5 B from,2,5 C from,2, M π [GeV ] M π [GeV ] Figure: Ground state of the a 2 from the T 2 (lhs.) and E (rhs.) irreducible representations Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

18 Light hadron masses mass [GeV] mass [GeV] CI lattice results 0.5 T 2 irrep E irrep 0 π a 0 ρ a b 0 π 2 a 2 ρ 2 Errors are purely statistical and systematical effects are not negligible Excited states only in the and 0 + channels. Quenching effects are visible in the a 0 and ρ channels Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

19 Isovector scalar mesons: Some lattice history Most quenched results indicated the ground state to be consistent with the a 0 (450) Group m a0 [GeV] Bardeen et al..34(9) Mathur et al..42(3) Burch et al..45 M S [GeV] systematic error a 0 (450) a 0 (980) 8,0, M π 2 [GeV 2 ] Gattringer et al., PRD 78 (2008) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

20 Recent results I a 0 meson a 0 meson mass [GeV] 0.5 a 0 (980) run A from 8 run B from 8 run C from 8 mass [GeV] 0.5 a 0 (980) run A from 8 run B from 8 run C from 8 quenched from 5,7, M π [GeV ] M π [GeV ] Caveat: Broad range of values for different interpolators/ensembles Role of scattering states? Supporting evidence: m b m a MeV Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

21 A different view: The scalar meson puzzle Low lying scalars could be tetraquark states mass I=0, Observed scalars (below GeV) a0(980) f0(980) mass I=0, Tetraquark nonet ussu dssd ussd I=0, mass qq nonet (vector meson case) ss φ I=/2 I=0?? κ(800) σ(600) I=/2 I=0 udud udds I=/2 I=0 uu dd us ud K * ρ,ω /2 0 /2 I 3 /2 0 /2 I 3 /2 0 /2 I 3 quark models would place qq with L = above GeV m κ < m a0 hard to reconcile with ūs and ūd a 0 (980) couples well with K K Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

22 Light scalars as tetraquark states? Question: Do the light scalars have a sizable q qqq component? In the following, we do not distinguish between tetraquark states and mesonic molecules A succesful lattice identification will need to simultaneously identify the lowest scattering states E P P 2 E P (k) + E P2 ( k), E P (k) = mp 2 + k 2 k = 2π L n Results are of a qualitative nature and we do not strive to measure the width of light ressonances Will focus on I = 0 and I = 2 channels (I = channel contains two towers of low-lying scattering states) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

23 Scalar tetraquarks: Interpolating fields Isospin 0 (flavor content 2 duūd ūu dd + ūuūu + dd dd) O = PP ; O 2 = i V i V i ; O 3 = i A i A i O 4 = [ q Cγ 5 q 2 ][q 3 Cγ 5 q 4 ]; O 5 = [ q γ 5 q 2 ][q 3 γ 5 q 4 ] Isospin 2 (flavor content du du) O = PP ; O 2 = i V i V i ; O 3 = i A i A i When performing the contractions we neglect single and double annihilation diagrams. No q qqq qq vac glue mixing (a) (b) (c) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

24 Scalar tetraquarks: Results at a glance E [GeV] I=0 dynamical simulation I=2 n= n=2 n= m π [GeV] m π [GeV] m π [GeV] S.Prelovsek et al., in preparation I=0 quenched simulation, L=6 I= m π [GeV] Possible interpretation as σ and κ or artifact from omissions? We checked several things but have no final answer Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

25 Scalar tetraquarks: Results at a glance E [GeV] n= n=2 n=3 I=/ m π [GeV] dynamical simulation I=3/ m π [GeV] I=/ m π [GeV] S.Prelovsek et al., in preparation quenched simulation, L=6 I=3/ m π [GeV] Possible interpretation as σ and κ or artifact from omissions? We checked several things but have no final answer Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

26 Scalar tetraquarks: Results at a glance E [GeV] n= n=2 n=3 I=/ m π [GeV] dynamical simulation I=3/ m π [GeV] I=/ m π [GeV] S.Prelovsek et al., in preparation quenched simulation, L=6 I=3/ m π [GeV] Possible interpretation as σ and κ or artifact from omissions? We checked several things but have no final answer Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

27 Scalar tetraquarks: Discussion of results Idea: Volume dependence to distinguish between one and two particle states Zi n (6) ( 2 6 )3/2 Zi n (2) when n is two-particle state P P 2 Zi n (6) Zi n (2) in case when n is a one-particle state (resonance) Condition to apply this problematic (See Niu et al PRD ): La should be much larger than the range of interaction between P and P 2 resonance width Γ E with energy spacing E The time dependence of Eigenavlues on a periodic lattice may help us to identify contributions from scattering states. In the presence of a scattering state we should have the form λ n (t) = w n [e Ent +e En(T t) ]+ w n [e m P t e m P 2 (T t) +e m P 2 t e m P (T t) ] This fit form leads to stable effective mass plateaus Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

28 Scalar tetraquarks: Discussion of results Idea: Volume dependence to distinguish between one and two particle states Zi n (6) ( 2 6 )3/2 Zi n (2) when n is two-particle state P P 2 Zi n (6) Zi n (2) in case when n is a one-particle state (resonance) Condition to apply this problematic (See Niu et al PRD ): La should be much larger than the range of interaction between P and P 2 resonance width Γ E with energy spacing E The time dependence of Eigenavlues on a periodic lattice may help us to identify contributions from scattering states. In the presence of a scattering state we should have the form λ n (t) = w n [e Ent +e En(T t) ]+ w n [e m P t e m P 2 (T t) +e m P 2 t e m P (T t) ] This fit form leads to stable effective mass plateaus Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

29 Scalar tetraquarks: Discussion of results Idea: Volume dependence to distinguish between one and two particle states Zi n (6) ( 2 6 )3/2 Zi n (2) when n is two-particle state P P 2 Zi n (6) Zi n (2) in case when n is a one-particle state (resonance) Condition to apply this problematic (See Niu et al PRD ): La should be much larger than the range of interaction between P and P 2 resonance width Γ E with energy spacing E The time dependence of Eigenavlues on a periodic lattice may help us to identify contributions from scattering states. In the presence of a scattering state we should have the form λ n (t) = w n [e Ent +e En(T t) ]+ w n [e m P t e m P 2 (T t) +e m P 2 t e m P (T t) ] This fit form leads to stable effective mass plateaus Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

30 Scalar tetraquarks: Some further consistency checks Extracted energies E n and couplings Z n i = 0 O n i n in the I = 2 channel for several interpolator choices. Z n i = 0 O i n = k C ik(t) uk n (t) (t) C lm (t) um(t) n eent/2 lm un l Z i n n=, t=[7,0] choice n=2, t=[7,0] choice i= i=2 i=3 i= i= n=3, t=[6,9] choice E a I=/2, dyn. simulation, m π =469 MeV, fit t=[7,0] choice Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

31 Outline Excited state spectroscopy Excited states and the lattice The variational method Suitable sources and sinks 2 Spectroscopy with Chirally Improved quarks Light-quark mesons Spotlight: Scalar mesons Light tetraquark states? 3 Baryon axial charges Results for Nucleon and Hyperon axial charges Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

32 Baryon axial charges Axial charge: value of the axial form factor at zero momentum transfer G a,bb (q 2 = 0) < B A µ (q) B > ( = ū B (p ) γ µ γ 5 G a,bb (q 2 G p (q 2 ) ) ) + γ 5 q µ u B (p)e iq x 2M B No disconnected contributions in isovector combinations Nucleon G a is related to polarized quark distributions in the proton (assuming CVC): G a = u d Interesting issues χpt description lattice data Effects from excited states? Axial charges of negative parity nucleon excitations speculations about chiral restauration Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

33 Variational method II - three point functions We extract the axial charge from ratios of three-point functions G A = Z A Z V l i j ψ(k) i T A (t, t ) ij ψ (k) j m ψ(k) l T V (t, t ) lm ψ (k) m T A and T V are the three-point functions with axial and vector current insertions. Burch et al.prd79:04504,2009 The eigenvectors ψ are obtained from the variational analysis of the two-point functions Taking one set of eigenvectors, one assumes translation invariance and reflection symmetry Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

34 Nucleon effective masses Example plot for effective masses (run B).4 m=m-0.06 Effective masses - am Nucleon groundstate t Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

35 Results for Σ and Ξ masses 2 Σ masses 2 Ξ masses.5.5 mass [GeV] 0.5 GS run A GS run B GS run C Experimental value mass [GeV] 0.5 GS run A GS run B GS run C Experimental value M π [GeV ] M π [GeV ] Reminder: Strange quark mass set with the Omega Baryon Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

36 Preliminary results: Nucleon axial charge.2 G A flavor CI 2.4 fm 2+ flavor domain wall 2.7fm 2+ flavor domain wall.8fm Experiment M π [GeV ] Domain Wall results from Yamazaki et al., PRL 00, 7602 (2008) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

37 Preliminary results: Σ and Ξ axial chrages 0 G A G A flavor CI 2 flavor mixed action flavor CI 2 flavor mixed action M π [GeV ] M π [GeV ] Mixed action results from Lin & Orginos, PR D (2009) Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

38 Summary We used the variational method to extract excited states and to suppress contaminations from excited states Good signals for most ground states; weak signals for excited states towards small pion masses Basis with non standard interpolators significantly improves the results in the scalar and pseudovector channels Higher spin states can be obtained with derivative sources We studied light scalar mesons using tetraquark interpolators The variational method can be used for baryon three-point functions Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30

39 Summary We used the variational method to extract excited states and to suppress contaminations from excited states Good signals for most ground states; weak signals for excited states towards small pion masses Basis with non standard interpolators significantly improves the results in the scalar and pseudovector channels Higher spin states can be obtained with derivative sources We studied light scalar mesons using tetraquark interpolators The variational method can be used for baryon three-point functions Thank you! Daniel Mohler (TRIUMF) Hadron properties from LQCD Newport News, March / 30