Exploring Excited Hadrons

Transcription

1 Exploring Excited Hadrons Colin Morningstar (Carnegie Mellon University) Lattice 2008: Williamsburg, VA July 15,

2 The frontier awaits experiments show many excited-state hadrons exist significant experimental efforts to map out QCD resonance spectrum JLab Hall B, Hall D, ELSA, etc. great need for ab initio calculations lattice QCD 2

3 The challenge of exploration! most excited hadrons are unstable (resonances) excited states more difficult to extract in Monte Carlo calculations correlation matrices needed operators with very good overlaps onto states of interest must extract all states lying below a state of interest as pion get lighter, more and more multi-hadron states best multi-hadron operators made from constituent hadron operators with well-defined relative momenta need for all-to-all quark propagators disconnected diagrams 3

4 Outline Resonances in a box Extracting excited stationary state energies recent results Outlook operator technology field smearing symmetry all-to-all quark propagators variance reduction with dilutions a new development 4

5 Resonances in a box 5

6 Resonances in a box: an example Consider simple 1D quantum mechanics example Hamiltonian H = 1 p + V( x) V( x) = ( x 3) e x /2 6

7 1D example spectrum Spectrum has two bound states, two resonances for E<4 transmission coefficient 7

8 Scattering phase shifts define even- and odd-parity phase shifts δ ± phase between transmitted and incident wave 8

9 Spectrum in box (periodic b.c.) spectrum is discrete in box (momentum quantized) narrow resonance is avoided level crossing, broad resonance? Dotted curves are V=0 spectrum 9

10 Unstable particles (resonances) our computations done in a periodic box momenta quantized discrete energy spectrum of stationary states single hadron, 2 hadron, how to extract resonance info from box info? approach 1: crude scan if goal is exploration only ferret out resonances spectrum in a few volumes placement, pattern of multi-particle states known resonances level distortion near energy with little volume dependence short-cut tricks of McNeile/Michael, Phys Lett B556, 177 (2003) 10

11 Unstable particles (resonances) approach 2: phase-shift method if goal is high precision work much harder! relate finite-box energy of multi-particle model to infinite-volume phase shifts evaluate energy spectrum in several volumes to compute phase shifts using formula from previous step deduce resonance parameters from phase shifts early references B. DeWitt, PR 103, 1565 (1956) (sphere) M. Luscher, NPB364, 237 (1991) (ρ-ππ in cube) approach 3: histogram method recent work for pion-nucleon system: V. Bernard et al, arxiv: [hep-lat] 11

12 1D example: phase-shift method periodic boundary condition of box leads to condition ( il E ) = ( iδ ± E ) exp 2 exp 2 ( ) calculate shifts δ ± (E) from finite-box energies 12

13 Mass and width of ρ Schierholz et al (this conference) Breit-Wigner fit: Γ = 200 MeV ρ

14 New histogram method use of probability distribution to study resonant structure reference: V. Bernard et al., arxiv: [hep-lat] count times eigenvalue occurs in particular momentum bin normalize to get probability distribution W(p) 14

15 Histogram method test tested on synthetic data to study Δ resonance Before subtracting free result After subtracting free result no prior theoretical bias possibility of seeing resonant structure even when avoid level crossing washed out by broad resonance 15

16 1D example: histogram method even parity channel 16

17 1D example: histogram method odd parity channel 17

18 Excited stationary states 18

19 Excited-state energies from Monte Carlo extracting excited-state energies requires matrix of correlators for a given N N correlator matrix Cαβ () t = 0 O () t O + α β (0)0 one defines the N principal correlators t, t ) as the eigenvalues of 1/ 2 1/ 2 C( t0 ) C( t) C( t0) where t 0 (the time defining the metric ) is small ( t t0 ) Eα tδeα can show that limt λ α ( t, t0) = e (1 + e ) eff λα ( t, t0) N principal effective masses defined by mα ( t) = ln λα ( t + 1, t0) now tend (plateau) to the N lowest-lying stationary-state energies analysis: fit each principal correlator to single exponential optimize on earlier time slice, matrix fit to optimized matrix both methods as consistency check λ α ( 0 19

20 Recent excited-state results Lattice QCD determination of patterns of excited baryon states, S. Basak, R. Edwards, G. Fleming, K. Juge, A. Lichtl, C. Morningstar, D. Richards, I. Sato, S. Wallace, Phys Rev D76, (2007) quenched first results for nucleons/deltas and quenched anisotropic Wilson, a s ~0.1 fm, a s /a t ~3, m π ~490 MeV Derivative sources in lattice spectroscopy of excited mesons, C. Gattringer, L. Glozman, C. Lang, D. Mohler, S. Prelovsek, arxiv: [hep-lat] quenched chiral-improved fermion, LW gauge, a s ~0.15 fm, range of m π 20

21 Nucleon spectrum: first results 200 quenched configs, anisotropic Wilson lattice, a s ~0.1 fm, a s /a t ~3, m π ~700 MeV (A. Lichtl thesis) 21

22 Delta spectrum: first results 200 quenched configs, anisotropic Wilson lattice, a s ~0.1 fm, a s /a t ~3, m π ~700 MeV (J. Bulava) 22

23 Inclusion of quark loops Left: N f =0 m π =700 MeV Right: N f =2 m π =360 MeV (Spectrum collaboration, preliminary) 23

24 Standard hadrons Preliminary results from QCD Spectrum collaboration (Edwards talk) Scale setting using mass ratios (Ω, K: Peardon talk) 24

25 0 -+ first-excited meson Gattringer et al. 25

26 0 -+ second-excited meson Gattringer et al. 26

27 1 -- ground state meson Gattringer et al. 27

28 1 -- first-excited state meson Gattringer et al. 28

29 1 -- second-excited state meson Gattringer et al. 29

30 1 ++ ground and first-excited state mesons Gattringer et al. 30

31 Search for light scalar tetraquarks with I=0, ½, 1 Sasa Prelovsek (Spectrum, Tuesday afternoon) Simulation: diquark anti-diquark interpolators 3 different smearings: 3x3 correlation matrix variational analysis: 3 states extracted Chiraly Improved f., mpi= MeV, L=12 & 16 Motivation: still not known whether some of the scalar resonances below 2 GeV correspond to qq or tetraquarks Results: ground-state eigenvalue: tower of scattering states excited states: m>2 GeV no indication for light tetraquarks found

32 Operator design issues statistical noise increases with temporal separation t use of very good operators is crucial or noise swamps signal recipe for making better operators crucial to construct operators using smeared fields link variable smearing quark field smearing spatially extended operators use large set of operators (variational coefficients) 32

33 Three stage approach ( PRD72:094506,2005 ) concentrate on baryons at rest (zero momentum) operators classified according to the irreps of G G, G, G, H, H (1) basic building blocks: smeared, covariant-displaced quark fields ~ ( p) ( D ~ ψ ( x)) p - link displaceme nt ( j = 0, ± 1, ± 2, ± 3) (2) construct elemental operators (translationally invariant) F F ( p) ( p) ( p) B ( x) = φ ε ( D ψ( x)) ( D ψ( x)) ( D ψ( x)) j 1g, 1u 2g 2u Aaα flavor structure from isospin color structure from gauge invariance g u O h ABC abc i Aaα j Bbβ k Ccγ (3) group-theoretical projections onto irreps of ΛλF dλ ( Λ) F Bi ( t) Dλλ ( R) U RBi ( t) U g D 33 O h + = R O D h R O h

34 Incorporating orbital and radial structure displacements of different lengths build up radial structure displacements in different directions build up orbital structure Δ-flux baryons Y-flux mesons operator design minimizes number of sources for quark propagators useful for mesons, tetraquarks, pentaquarks even! can even incorporate hybrid meson operators 34

35 Spin identification and other remarks spin identification possible by pattern matching total numbers of operators assuming two different displacement lengths total numbers of operators is huge uncharted territory ultimately must face two-hadron scattering states 35

36 Quark- and gauge-field smearing smeared quark and gluon fields fields dramatically reduced coupling with short wavelength modes link-variable smearing (stout links PRD69, (2004)) define C + ( x) = ρ U ( x) U ( x + ˆ) ν U ( x ˆ) μ μ μν ν μ ν + ± ( ν μ ) ρ jk = ρ, ρ4 k = ρk 4 = 0 spatially isotropic νˆ x exponentiate traceless Hermitian matrix + i + i + Ω μ = CU μ Qμ = ( Ωμ Ωμ) Tr μ 2 2N ( Ωμ Ωμ) ( n+ 1) ( n) ( n) iterate U = exp( iqμ ) U μ μ (1) ( n ) ~ U U U U μ μ quark-field smearing (covariant Laplacian uses smeared gauge field) nσ σ s 2 ψ () x = 1 + Δ ψ () x 4n σ μ μ μˆ 36

37 Importance of smearing Nucleon G 1g channel effective masses of 3 selected operators noise reduction from link variable smearing, especially for displaced operators quark-field smearing reduces couplings to high-lying states σ s n ρ = 4.0, nσ = 32 ρ = 2.5, n = 16 less noise in excited states using σ = 3.0 s ρ (t) 1.5 Single-Site 1 (t) M i M i (t) 0.5 Quark Smearing Only t / a t 1 (t) M i M i (t) 0.5 Link Smearing Only t / a t 1 (t) M i M i 0.5 Both Quark and Link Smearing t / a t Singly-Displaced (t) M i t / a t (t) M i t / a t (t) M i 1.5 Triply-Displaced-T t / a t t / a t t / a t t / a t 37

38 Operator selection rules of thumb for pruning operator sets noise is the enemy! prune first using intrinsic noise (diagonal correlators) prune next within operator types (single-site, singly-displaced, etc.) based on condition number of prune across all operators based on condition number best to keep a variety of different types of operators, as long as condition numbers maintained ˆ Cij () t Cij () t =, t = 1 Cii () t Cjj () t typically use 16 operators to get 8 lowest lying levels 38

39 Nucleon G 1g effective masses 200 quenched configs, anisotropic Wilson lattice, a s ~0.1 fm, a s /a t ~3, m π ~700 MeV nucleon G 1g channel green=fixed coefficients, red=principal 39

40 Nucleon H u effective masses 200 quenched configs, anisotropic Wilson lattice, a s ~0.1 fm, a s /a t ~3, m π ~700 MeV nucleon H u channel green=fixed coefficients, red=principal 40

41 Spatial summations baryon at rest is operator of form 1 B( p= 0, t) = ϕb ( x, t) V baryon correlator has a double spatial sum 1 0 B( p = 0, t) B( p = 0,0) 0 = 0 (, ) (,0) 0 2 ϕb x t ϕb y V computing all elements of propagators exactly not feasible x translational invariance can limit summation over source site to a single site for local operators 1 0 B( p = 0, t) B( p = 0,0) 0 = 0 ϕb( x, t) ϕb(0,0) 0 V xy, x April 21, 2008 C. Morningstar 41

42 All-to-all stochastic quark propagators good baryon-meson operator of total zero momentum has form 1 B( p,) t M( p,) t ϕ ( x,) t ϕ ( y,) t e ip x y = 2 V xy, B M i( ) cannot limit source to single site for multi-hadron operators disconnected diagrams (scalar mesons) will also need many-to-many quark propagators estimates of all quark propagator elements are needed! April 21, 2008 C. Morningstar 42

43 Matrix inversion quark propagator is just inverse of Dirac matrix M noise vectors η satisfying E(η i )=0 and E(η i η j *)=δ i j are useful for stochastic estimates of inverse of a matrix M Z 4 noise is used { 1, i, 1, i} define X(η)=M -1 η then EX ( η ) = E M η η = M E( η η ) = M k k k δ = M i j ik k j ik k j ik kj ij if can solve M X (r) = η (r) for each of N R noise vectors η (r) then we have a Monte Carlo estimate of all elements of M -1 : M 1 R N 1 ( r) ( r) ij Xi η j NR r= 1 variances in above estimates usually unacceptably large introduce variance reduction using source dilution 43

44 Source dilution for single matrix inverse dilution introduces a complete set of projections: P P = δ P, P = 1, P = P a observe that define so that ( a) ( b) ab ( a) ( a) ( a) ( a) M = M δ = M P = M P δ P ( a) 1 ( a) ( a) ij ik kj ik kj ik kk k j j j a a Monte Carlo estimate is now ( ηη ) ( ηη ) = M P E P = M E P P 1 ( a) ( a) 1 ( a) ( a) ik kk k j j j ik kk k j j j a a [ ] ( ) [ ] ( ) [ ] 1 [ ],, k = P kk k j jpjj Xk M = = kj j 1 ( r)[ a] ( r)[ a] Mij Xi η j NR r= 1 a η [ a] η [ a] has same expected value as ηη a i j i j, but reduced variance (statistical zeros exact) η η η η η a a ( ) M = E X η 1 [ ] [ ] ij i j a NR 1 44

45 Dilution schemes for spectroscopy Time dilution (particularly effective) ( B) Pa ; b ( x,; t y, t ) = δabδ δ ( x, y) δbtδ Bt, B= 0,1,, Nt 1 Spin dilution ( B) Pa ; b ( x,; t y, t ) = δabδb δb δ ( x, y) δ tt, B= 0,1,2,3 Color dilution ( B) Pa ; b ( x,; t y, t ) = δbaδbbδ δ ( x, y) δ tt, B= 0,1,2 Spatial dilutions? α β αβ α β α β α β αβ even-odd 45

46 Source-sink factorization baryon correlator has form stochastic estimates with dilution define C ll C = c c Q Q Q () l ( l ) A B C ll ijk ijk ii jj kk 1 () l ( l ) A A cijk cijk i i R r d d d N Γ = A B C ( Ar)[ d ] ( Ar)[ d ] ( ϕ η ) ( ( Br)[ d ] ( )[ ] )( ( )[ ] ( )[ ] ) B Br db Cr dc Cr dc ϕj ηj ϕk η c ( r)[ dadbdc] () l ( Ar )[ d A] ( Br)[ db] ( Cr)[ dc] l ijk i j k Ω = c ϕ ϕ ϕ η η η ( r)[ dadbdc] () l ( Ar)[ d A] ( Br )[ db] ( Cr)[ dc] l ijk i j k correlator becomes dot product of source vector with sink vector C ll 1 A B C A B C R Γ Ω N r d d d A B C ( r)[ d d d ] ( r)[ d d d ] l l store ABC permutations to handle Wick orderings April 21, 2008 C. Morningstar 46 k

47 Dilution tests (see J. Bulava talk) 100 quenched configs, anisotropic Wilson lattice, a s ~0.1 fm, a s /a t ~3, m π ~700 MeV three representative operators: SS,SD,TDT nucleon operators April 21, 2008 C. Morningstar 47

48 Dilution tests (continued) 100 quenched configs, anisotropic Wilson lattice April 21, 2008 C. Morningstar 48

49 Dilution tests (continued) 100 quenched configs, anisotropic Wilson lattice SS,SD,TDT nucleon operators same conclusions for other t values of C(t), fitted mass April 21, 2008 C. Morningstar 49

50 Future directions testing new method: clever choice of quark-field smearing makes exact computations with all-to-all quark propagators possible!! will work for disconnected diagrams will compare with stochastic with dilutions method 50

51 QCD Spectrum Collaboration A. Lichtl (Brookhaven Nat. Lab.) J. Bulava, C. Morningstar, J. Foley (Carnegie Mellon U.) R. Edwards, B. Joo, H.W. Lin, D. Richards (Jefferson Lab.) E. Engelson, S. Wallace (U. Maryland) K.J. Juge (U. of Pacific) N. Mathur (Tata Institute) M. Peardon, S. Ryan (Trinity Coll. Dublin) April 21, 2008 C. Morningstar 51

52 Configuration generation significant time on USQCD (DOE) and NSF computing resources anisotropic clover fermion action (with stout links) and anisotropic improved gauge action tunings of couplings, aspect ratio, lattice spacing in progress (Justin Foley, Robert Edwards talks) anisotropic Wilson configurations generated during clover tuning current goal: three lattice spacings: a = fm, 0.10 fm, 0.08 fm three volumes: V = (3.2 fm) 4, (4.0 fm) 4, (5.0 fm) flavors, m π ~ 350 MeV, 220 MeV, 180 MeV USQCD Chroma software suite April 21, 2008 C. Morningstar 52

53 QCD Spectrum Collaboration talks J. Bulava, Stochastic all-to-all propagators for baryon correlators E. Engelson, Lattice QCD determination of patterns of excited baryon masses J. Foley, Tuning improved anisotropic actions in lattice perturbation theory R. Edwards, Three flavor anisotropic clover fermions K.J. Juge, Multi-hadron operators with all-to-all quark propagators N. Mathur, Cascade baryon spectrum from lattice QCD M. Peardon, Determining bare quark masses for Nf = 2+1 dynamical simulations April 21, 2008 C. Morningstar 53

54 Summary discussed issues with unstable hadrons (resonances) discussed extraction of excited states in Monte Carlo calculations correlation matrices needed operators with very good overlaps onto states of interest must extract all states lying below a state of interest as pion get lighter, more and more multi-hadron states multi-hadron operators relative momenta need for all-to-all quark propagators disconnected diagrams exploration of excited hadrons is well underway 54