Towards the hadron spectrum using spatially-extended operators in lattice QCD

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1 Towards the hadron spectrum using spatially-extended operators in lattice QCD Colin Morningstar Carnegie Mellon University 3rd Topical Workshop on Lattice Hadron Physics Jefferson Lab July 3, 26 July 3, 26 Hadron spectrum from lattice QCD

2 Lattice Hadron Physics Collaboration Lattice Hadron Physics Collaboration (LHPC) formed in 2 LHPC has several broad goals compute QCD spectrum (baryons, mesons, ) hadron structure (form factors, structure functions, ) hadron-hadron interactions current members of spectroscopy effort: Robert Edwards, George Fleming, Jimmy Juge,, Adam Lichtl,, CM, Nilmani Mathur,, David Richards, Ikuro Sato, Steve Wallace July 3, 26 Hadron spectrum from lattice QCD 2

3 LHPC spectroscopy efforts extracting spectrum of resonances is big challenge!! need sets of extended operators (correlator( matrices) multi-hadron operators needed too deduce resonances from finite-box energies anisotropic lattices inclusion of light-quark loops at realistically light quark mass ( a < a ) t s long-term project efforts divide into two categories operator technology Monte Carlo updating technology (light quark loops) this talk is an interim status report focus on baryons July 3, 26 Hadron spectrum from lattice QCD 3

4 Outline how to extract excited-state energies from Monte Carlo computations need for multi-hadron operators operator construction spatially-extended operators symmetry channels field smearing operator pruning milestone reached: extraction of nine or more levels in a symmetry channel!! outlook and conclusion July 3, 26 Hadron spectrum from lattice QCD 4

5 Excited states, resonances in lattice Monte Carlo July 3, 26 Hadron spectrum from lattice QCD 5

6 Principal correlators extracting excited-state energies described in C. Michael, NPB 259,, 58 (985) Luscher and Wolff, NPB 339,, 222 (99) exploits the variational method N N for correlator matrix C ( t) = O ( t) O () define N principal correlators t as eigenvalues of where (the time defining the metric )) is small ( t t ) Eα t Eα can show limt λ α (, ) = ( + ) λα (, tt) N principal effective masses Ω α () t = ln λ α ( t+, t) now tend (plateau) to N lowest-lying lying stationary-state state energies C( t λ α ( t, t) / 2 / 2 ) C( t) C( t) t t e αβ α β + e July 3, 26 Hadron spectrum from lattice QCD 6

7 Unstable particles (resonances) our computations done in a periodic box momenta quantized discrete energy spectrum of stationary states single hadron,, 2 hadron, scattering phase shifts resonance masses, widths (in principle) deduced from finite-box spectrum B. DeWitt, PR 3,, 565 (956) (sphere) M. Luscher,, NPB ,, 237 (99) (cube) first goal: get finite-box spectrum must extract multi-hadron state energies check volume dependences know masses of decay products placement and pattern of multi-particle states roughly known July 3, 26 Hadron spectrum from lattice QCD 7

8 Operator construction July 3, 26 Hadron spectrum from lattice QCD 8

9 Operator design issues must facilitate spin identification shun the usual method of operator construction which relies on cumbersome continuum space-time constructions focus on constructing operators which transform irreducibly under the symmetries of the lattice one eye on maximizing overlaps with states of interest other eye on minimizing number of quark-propagator sources use building blocks useful for baryons, mesons, multi-hadron operators must project onto definite spin polarizations or will observe many degeneracies July 3, 26 Hadron spectrum from lattice QCD 9

10 Three stage approach concentrate on baryons at rest (zero momentum) operators classified according to the irreps of () basic building blocks: smeared, covariant-displaced quark fields (2) construct elemental operators (translationally( invariant) F F ( p) ( p) ( p) B ( x) = φ ε ( D ψ( x)) ( D ψ( x)) ( D ψ( x)) ~ ( D ( p j ) ~ ψ ( x)) G, H, g, G u, G2g, G2u Aaα flavor structure from isospin,, color structure from gauge invariance (3) group-theoretical projections onto irreps of ΛλF dλ ( Λ) F + Bi ( t) = Dλλ ( R) U RBi ( t) U R g D D O R O h h Grassmann package in Maple to do these calculations details in PRD72, 9456 (25) July 3, 26 Hadron spectrum from lattice QCD g H p - link displacement ( j u O h =, ±, ± 2, ± 3) ABC abc i Aaα j Bbβ k Ccγ O h

11 Three-quark elemental operators three-quark operator Φ = ABC ( p) A ( p) B ( p) C αβγ, ijk ( t) εabc( Di ψ( x, t)) aα ( Dj ψ( x, t)) bβ ( Dk ψ( x, t)) cγ x covariant displacements ( p) D ( x, x ) = U ( x) U ( x+ ˆj) U ( x+ ( p ) ˆj) δ ( j =±, ± 2, ± 3) D j j j j x, x+ pj ˆ ( p) ( x, x ) = δ x, x Baryon ++ + Σ N + Ξ Λ Ω Operator Φ Φ uu u Φ αβγ,ijk us u Φ αβγ,ijk uud duu αβγ, ijk Φαβγ, ijk su s Φ αβγ,ijk uds dus αβγ, ijk Φαβγ, ijk ss s Φ αβγ,ijk July 3, 26 Hadron spectrum from lattice QCD

12 Incorporating orbital and radial structure displacements of different lengths build up radial structure displacements in different directions build up orbital structure single site mock up quarkdiquark singly-displaced -flux doubly-displaced-l triply-displaced-t Y-flux doubly-displaced-i triply-displaced-o minimizes number of sources for quark propagators useful for mesons, tetraquarks, pentaquarks even! can even incorporate hybrid mesons operator (in progress) July 3, 26 Hadron spectrum from lattice QCD 2

13 Enumerating the three-quark operators lots of operators (too many!) ++, Ω + Σ, Ξ N + Λ Single-site Singly-displaced Doubly-displaced displaced-i Doubly-displaced displaced-l Triply-displaced displaced-t Triply-displaced displaced-o July 3, 26 Hadron spectrum from lattice QCD 3

14 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 July 3, 26 Hadron spectrum from lattice QCD 4

15 Single-site operators choose Dirac-Pauli convention for γ-matrices July 3, 26 Hadron spectrum from lattice QCD 5

16 ++ single-site site operators July 3, 26 Hadron spectrum from lattice QCD 6

17 Single-site N+ operators July 3, 26 Hadron spectrum from lattice QCD 7

18 Quark-field and link-variable smearing issues July 3, 26 Hadron spectrum from lattice QCD 8

19 Run parameters run parameters for all results presented here anisotropic lattice fermion actions Wilson gauge, Wilson fermion lattice spacings as.fm, as/ at 3. quark masses such that m π 7MeV quenched correlator matrices averaged over irrep rows use of opposite-parity parity time-reversed propagators to double statistics number of configurations used 5 for operator smearing tests 2 for operator prunings July 3, 26 Hadron spectrum from lattice QCD 9

20 Quark- and gauge-field smearing smeared quark and gluon fields fields dramatically reduced coupling with short wavelength modes link-variable smearing (stout links PRD69 69,, 545 (24)) + define Cµ ( x) = ρµνuν ( x) U µ ( x + ˆ) ν Uν ( x + ˆ) µ µˆ ± ( ν µ ) νˆ spatially isotropic ρ jk = ρ, ρ4 k = ρk 4 = exponentiate traceless Hermitian matrix + i + CU Q i + Ω = = Ω Ω Tr Ω Ω iterate ( ) N ( ) ( n+ ) ( n) ( n) U = exp( iqµ ) U 2 2 µ µ µ µ µ µ µ µ U U U () µ µ µ µ µ ( n ) ~ U µ quark-field smearing (covariant Laplacian uses smeared 2 nσ gauge field) σ 2 s ψ () x = + ψ () x 4n σ x parameters to tune: σs, nσ, ρ, n ρ July 3, 26 Hadron spectrum from lattice QCD 2

21 Quark-field smearing tuning focus on three particular operators for smearing tests O SS a single-site site operator in the irrep a singly-displaced operator O SD with a particular choice of Dirac indices and 3-link 3 displacement length riply-displaced displaced-t T operator O TDT with a particular choice of the Dirac indices (3-link displacement lengths) Cii() t define effective mass as usual Mi () t = ln C ii( t a ) + t use M ( t = 4 a ) to compare different quark-field smearings i t smeared links ρ n = 2.5, n = 6 since displaced operators noisy ρ ρ G g n σ =,2,4,8, 6,32,64 July 3, 26 Hadron spectrum from lattice QCD 2

22 Link-variable smearing tuning first, used the effective mass Et= ( ) associated with the static tic quark-antiquark antiquark potential at spatial separation R= 5a s.5fm found that link-smearing did not appreciably alter values of baryon effective masses, but had dramatic effect on variance compared relative jackknife error η i( t = 4 at) of Mi( t = 4 at) for different link-smearing parameters ( σ = 4., = 32) lesson learned: preferred parameters from static potential produce smallest errors in baryon effective masses n ρ =,2,4,8,6,32 s n σ July 3, 26 Hadron spectrum from lattice QCD 22

23 Importance of smearing Nucleon Gg channel effective masses of the 3 selected operators noise reduction from link variable smearing, especially for displaced operators quark-field smearing reduces couplings to high-lying states σ = 4., n = 32 n s ρ σ ρ = 2.5, n = 6 ρ effect on excited states shows σ s = 3. better (t) M i (t) M i (t) M i.5.5 Single-Site Quark Smearing Only (t) M i 2.5 t /.5 Link Smearing Only (t) M i 2.5 t /.5 (t) M i Both Quark and Link Smearing 2 t /.5.5 Singly-Displaced (t) M i 2.5 t /.5 (t) M i 2.5 t /.5 (t) M i 2 t /.5.5 Triply-Displaced-T 2.5 t / t /.5 2 t / July 3, 26 Hadron spectrum from lattice QCD 23

24 Smearing and excited states previous tests involved lowest state only important to tune smearing for excited states as well lowest 4 principal effective masses for x matrix of DDI operators shown G g same link smearing ρ n = 2.5, n = 6 ρ quark-field smearings σ = 2. ( blue), 3. ( red), 4. (black) n σ = 32 ρ nρ = 5., nρ = 32 with σ s = 4. did not reduce errors preferred: s ρ n σ = 32, σ s = 3. July 3, 26 Hadron spectrum from lattice QCD 24

25 Smearing summary From our quenched study of the nucleon channel on small 3 lattices 2 48 for a.fm and a / a 3. and m π 7MeV, the preferred smearing parameters are s G g s t ρ n = 2.5, n = 6 ρ ρ n σ = 32, σ s = 3. factors still to consider: evidence for same smearing for other irreps expect same smearing for other isospin channels dependence on lattice spacing dependence on quark mass July 3, 26 Hadron spectrum from lattice QCD 25

26 Operator pruning issues July 3, 26 Hadron spectrum from lattice QCD 26

27 Operator plethora Number of operators given below ( displacement length) N + G g total of 79 operators in channel Gg G2g H g Single-site 3 Singly-displaced Doubly-displaced displaced-i Doubly-displaced displaced-l Triply-displaced displaced-t since 79x79 matrix too large to be practical, operator pruning is clearly necessary G g will focus on channel first July 3, 26 Hadron spectrum from lattice QCD 27

28 Operator plethora (Gg Nucleon) Single Site Operator Single Site Operator Three-Link Singly Displaced Operator 3 Three-Link Singly Displaced Operator t/ t/ t/ t/ Single Site Operator 2 Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator 5 Three-Link Singly Displaced Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator 2 Three-Link Singly Displaced Operator 7 Three-Link Singly Displaced Operator t/ t/ t/ t/ July 3, 26 Hadron spectrum from lattice QCD 28

29 Gg nucleon operators Three-Link Singly Displaced Operator 9 Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator 5 Three-Link Singly Displaced Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator 2 Three-Link Singly Displaced Operator 7 Three-Link Singly Displaced Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator 3 Three-Link Singly Displaced Operator 4 Three-Link Singly Displaced Operator 9 Three-Link Singly Displaced Operator t/ t/ t/ t/ July 3, 26 Hadron spectrum from lattice QCD 29

30 G2g nucleon operators Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator Three-Link Triply Displaced T Operator 34 Three-Link Triply Displaced T Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator 2 Three-Link Singly Displaced Operator 3 Three-Link Triply Displaced T Operator 36 Three-Link Triply Displaced T Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator 4 Three-Link Singly Displaced Operator 5 Three-Link Triply Displaced T Operator 38 Three-Link Triply Displaced T Operator t/ t/ t/ t/ July 3, 26 Hadron spectrum from lattice QCD 3

31 Hu nucleon operators Single Site Operator Three-Link Singly Displaced Operator Three-Link Triply Displaced T Operator 35 Three-Link Triply Displaced T Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator Three-Link Singly Displaced Operator 2 Three-Link Triply Displaced T Operator 37 Three-Link Triply Displaced T Operator t/ t/ t/ t/ Three-Link Singly Displaced Operator t/ Three-Link Singly Displaced Operator t/ Three-Link Triply Displaced T Operator t/ Three-Link Triply Displaced T Operator t/ July 3, 26 Hadron spectrum from lattice QCD 3

32 Pruning: step one look at effective masses of diagonal elements of correlation matrix and discard noisy operators examples shown below all 79 effective masses ( ρn = 2.5, n = 6, n = 32, σ = 4. used) ρ ρ σ s July 3, 26 Hadron spectrum from lattice QCD 32

33 Pruning: step one (continued) retain 64 operators out of the 79 SS:,, 2 SD: 3, 5, 6, 8,, 2, 4, 7, 9, 2, 22, 24, 25 DDI: 27, 29, 3, 3, 33, 36, 38, 4, 43, 44, 45, 48 DDL: 52, 54, 56, 57, 58, 59, 6, 6, 62, 72, 74, 76, 78, 85, 88, 94, 97, 98, 5, TDT: 6, 8, 9, 24, 25, 26, 32, 34, 36, 38 49, 58, 62, 63, 69, 74 July 3, 26 Hadron spectrum from lattice QCD 33

34 Noise, noise, noise, noise noise is the enemy!! intrinsic noise in each operator presence of small singular values indicates operator set not independent enough noise can creep in examine renormalized matrix at early time t= sort operators in order of increasing noise keep 6 least noisiest operators of each kind (SD,DDI, ) which yield acceptable matrix condition number from above set, choose 25 operators which still yield acceptable matrix condition number try to keep different operator types if condition number allows Cˆ () = C () / C () C () ij ij ii jj July 3, 26 Hadron spectrum from lattice QCD 34

35 Milestone: principal effective masses Gg nucleons (6x6 matrix) using 2 configurations world record: 9 levels extracted (even more possible!) effect of time-backward propagation of anti-baryons July 3, 26 Hadron spectrum from lattice QCD 35

36 Gu Nucleon channel can see effect of time-backward propagation of anti-baryons July 3, 26 Hadron spectrum from lattice QCD 36

37 G2g nucleon channel July 3, 26 Hadron spectrum from lattice QCD 37

38 G2u nucleon channel July 3, 26 Hadron spectrum from lattice QCD 38

39 Hg nucleon channel July 3, 26 Hadron spectrum from lattice QCD 39

40 Hu nucleon channel July 3, 26 Hadron spectrum from lattice QCD 4

41 Future work three-quark operator pruning to be completed all irreps,, all isospin channels different displacement lengths include mesons (in progress) include multi-hadron operators stochastic all-to to-all propagators with dilution, low eigenvectors will also facilitate disconnected diagrams goal: operator technology ready to go when unquenched simulations near realistically-light light quark masses become possible LHPC recently was awarded /6 of QCDOC to generate configurations on large lattices with dynamical quarks such that pion mass around 3 MeV July 3, 26 Hadron spectrum from lattice QCD 4

42 Summary progress report on ongoing efforts of LHPC to extract hadron spectrum with large sets of extended operators need for correlation matrices of good operators spin identification must be addressed multi-hadron operators will become important exploration of 3-quark 3 baryon operators nearly done study of meson operators beginning major milestone reached: can extract 9 or more states!! ultimately, MC updating with realistically light quark masses needed very challenging calculations will keep hammering at it! July 3, 26 Hadron spectrum from lattice QCD 42