Lattice Studies of Baryon Resonances

Transcription

1 Lattice Studies of Baryon Resonances HADRON SPECTRUM COLLABORATION J. Bulava, J. Dudek, R. Edwards, E. Engelson, Justin Foley, Bálint Joó, J. Juge, A. Lichtl,H.- W Lin, N. Mathur, C. Morningstar, D. Richards, and S. Wallace Goals: Solve QCD Theoretically determine the mass spectrum baryons, meson, hybrids, Means: Lattice QCD Lots of interpolating field operators Anisotropic lattices: a t = 1 3 a s Variational method Diagonalize matrices of correlations functions Extract energies from eigenvalues of matrices. 1

2 I= 1 2 BARYON SPECTRUM USQCD Resources & Lattices Spin on the lattice A lattice view of the physical spectrum Variational method Pattern of low-lying states at two pion masses. Evidence for spin 5 2 Summary 2

3 USQCD Collaboration of collaborations N f = 2 simulations for m π = 416 MeV used QCDOC Brookhaven National Laboratory BlueGene Teragrid Resource San Diego Supercomputer Center N f = 2 simulations for m π = 572 MeV used Jaguar Cray XT3 National Center for Computational Science, Oak Ridge National Laboratory. and the Chroma software system ( Edwards et al.) 3

4 Lattices for N f = 2 SPECTRUM Szymanzik improved action u & d quarks with equal masses m π = 416 MeV anisotropic lattice (362 configs) a s /a t = 3 anisotropy a s =.18(7) fm, m π = 572 MeV anisotropic lattice (43 configs) a s /a t = 3 anisotropy a s =.113(7) fm, Smeared links, smeared quark fields 4

5 Quantum numbers of states Spins J 2 and J z are not diagonal. Irreducible representations (irreps) of double octahedral group are diagonal. Recover states of good J 2 and J z patterns of octahedral irreps. x T B (Λλ) k (x,t)b (Λ,λ ) k () = C (Λ,λ) kk (t)δ ΛΛ δ λλ as 5

6 Double octahedral group, O D and subduction of J. IR IR Parity Dimension 1 2 G 1g H g G 2g G 1u H u G 2u Spin 1 2 : Isolated G 1 state, Spin 3 2 : Isolated H state. Spin 5 2 : Degenerate G 2 and H states as a Spin 7 2 : Degenerate G 1, H and G 2 states as a 6 J

7 Physical positive-parity spectrum subduced G1g G2g Hg 2 15 N(171) 1/2+ N(144) 1/2+ N(199) 7/2+ N(168) 5/2+ N(19) 3/2+ N(172) 3/2+ 1 N(939) 1/2+ G1g G2g Hg 7

8 Physical negative-parity spectrum subduced G1u G2u Hu 2 15 N(165) 1/2- N(1535) 1/2- N(1675) 5/2- N(17) 3/2- N(152) 3/2-1 G1u G2u Hu 8

9 Need for nonlocal operators. Three spin 1 2 local operators can produce S = 1 2 or S = 3 2. Example: N 121 = (u 1 d 2 d 1 u 2 )u 1 is S = 1 2 Nucleon operator No higher spins can be produced. No G 2 irreps can be produced. Nonlocal operators: provide L = 1, L = 2,... L = 1 S = 3 2 J = 5 2 G 2 9

10 Types of nonlocal operators single-site singly-displaced doubly-displaced-i doubly-displaced-l triply-displaced-t 1

11 Quark and link smearing (t) 1 Single-Site (t) 1 Singly-Displaced (t) 1 Triply-Displaced-T M i M i M i Quark Smearing Only t / a t t / a t t / a t (t) 1 (t) 1 (t) 1 M i M i M i Link Smearing Only t / a t t / a t t / a t (t) 1 (t) 1 (t) 1 M i M i M i Both Quark and Link Smearing 1 2 t / a t 1 2 t / a t 1 2 t / a t Effective masses M(t) for unsmeared ( circles) and smeared (triangles) operators: single site (left), singly displaced (center), triply displaced T (right). Top row: quark-field smearing only. Middle row: link-variable smearing only. Bottom row: both quark and link smearing. 11

12 Pruning to sets of 16 operators 179 G 1g operators Delete ones with high intrinsic noise in self correlators Delete ones that are too parallel Delete ones with high condition numbers: λ max /λ min Keep 16 most promising operators 12

13 Lattice Charge-Conjugation Symmetry [ C (Λλ) kk (t) = P k δ P, n n θ(t) B (Λλ) k n n B (Λλ) e E nt θ( t) B (Λλ) k n n B(Λλ) k e E nt ], k (1) Charge conjugation relations n = C n e iφ produce a relation between correlation functions [ (t) =δ P, θ(t)p (Λ) k B (Λλ) k n n B (Λλ) k C (Λλ) kk n η t θ(t t)p (Λ c) k B (Λ cλ c ) k n n B (Λ cλ c ) k e E n(t t) ]. e E nt (2) The forward propagating signal of a correlation function is equal to the backward propagating signal of the parity-reversed, complex-conjugated correlation function within the factor η t, i.e., C (Λλ) kk (t) = η t C (Λ cλ c ) kk (T t). (3) 13

14 Improved statistics For + parity operators use B (Λ). For - parity operators use B (Λ c). Each provides a correlation function in < t < T/2 for BOTH parities. They are uncorrelated samples: 1 time slices apart. Roughly double the number of gauge configurations. 14

15 Variational method Diagonalize matrices of correlation functions C kk (t) to extract spectrum of energies. 1.) Calculate matrices of correlation functions x T B (Λλ) k (x,t)b (Λ,λ ) k () = C (Λ,λ) kk (t)δ ΛΛ δ λλ 2.) Solve generalized eigenvalue eq. k C(Λ) kk (t)v (n) k = α (n) (t,t ) k C(Λ) kk (t )v (n) k, 3.) Obtain principal eigenvalues ( ) α (n) (t,t ) e E n(t t ) 1 + O(e δe t ), 15

16 Principal correlators Eigenvectors diagonalize the correlation matrix: v (n)t k (t,t ) C (Λ) kk (t)v (n ) k (t,t ) = α (n) (t,t )δ nn. Eigenvalue is principal correlator: C (Λ) nn (t) = α (n) (t,t ). based on optimized operators: O (Λλ) n = k ṽ (n) k (t,t )B (Λλ) k. 16

17 Fit principal correlators two exponential fits for all but G 1u channel C fit (t) = Ae E (t t ) + (1 A)e E(t t ) three-exponential fits in G 1u channel C fit (t) = Ae E (t t ) + (1 A B)e E(t t ) +Be E G 1g (t t ) The backward state in the G 1u channel is the nucleon state and it can contribute significantly because it is less massive and the lattice length is short (owing to anisotropy). 17

18 m π = 4 MeV Effective masses M eff a t M eff a t t/a t t/a t Left: G 1g Right: G 1u.3.3 M eff a t M eff a t t/a t t/a t Left: H g ; Right: H u M eff a t M eff a t t/a t t/a t Left: G 2g ; Right: G 2u 18

19 m π = 572 MeV Effective masses M eff a t t/a t M eff a t 1..9 t/a t.9 Left: G 1g Right: G 1u M eff a t M eff a t t/a t t/a t Left: H g ; Right: H u M eff a t M eff a t.9.9 t/a t t/a t Left: G 2g ; Right: G 2u 19

20 Pattern of lowest energies in each channel m π = 572 MeV E (MeV) M G 1g H g G 2g G 1u H u G 2u 2

21 Pattern of lowest energies in each channel m π = 4 MeV $E$ (MeV) M π G1g Hg G2 g G1u Hu G 2 u 21

22 Comparison of m π = 4 & 572 MeV $E$ (MeV) E (MeV) M π 5 M G1g Hg G2 g G1u Hu G 2 u G 1g H g G 2g G 1u H u G 2u m π = 4 MeV (left panel) and m π = 572 MeV (right panel) 22

23 Extrapolations: E = a + bm 2 π 3 25 E (MeV) m π 2 (GeV 2 ) Lowest G 1g, first excited G 1u, lowest H u, lowest G 2u. Lowest G 1u state using E = a + m π + bm 2 π. Degenerate G 1u (2) and H u (1) energies matches 3 2 (152) 1 2 (1535) 23

24 Signature of 5 2 state m π State E 4 H u (2) 1943(24) 4 G 2u (1) 1955(31) 572 H u (2) 2236(36) 572 G 2u (1) 2234(42) Second H u energy is degenerate with first G 2u energy at both pion masses Pattern E(H u (1)) < E(H u (2)) = E(G 2u (1)) matches 3 2 (152) < 5 2 (1675). 24

25 Progress N f = 2 QCD at m π = 4 and 572 MeV, a s.1 fm. Lowest I = 1 2 energies Many operators pruned to sets of energies at each m π Pattern of lowest energies for negative parity is similar to the pattern of lowest physical resonance states. Evidence for 5 2 state. Partner G 2u and H u states Energies above thresholds for multihadron states; Excited G 1g high. 25

26 Progress on related fronts Form factor of G 1g first excited state at m π =72 MeV. arxiv: First Lattice Study of the N-P 11 (144) Transition Form Factors Huey- Wen Lin, Saul D. Cohen, Robert G. Edwards, David G. Richards E2/M1 ratio for (1232) arxiv: ; arxiv:71621: Deltabaryon electromagnetic form factors in lattice QCD C. Alexandrou, et al. Charmonium arxiv: Charmonium excited state spectrum in lattice QCD J.J. Dudek, R.G. Edwards, N. Mathur, D.G. Richards 26

27 N f = QCD Prospectus Multihadron operators Include pion & baryon operators. All-to-all propagators needed, e.g., for annihilation of q New methods: pattern-to-pattern propagators for many patterns Additional lattice volumes Lower pion masses: close to physical limit. 27

28 16 Ξ* 12 a 1 b 1 Σ Ξ Λ Σ* ϕ MeV 8 ρ K* a N η 4 N f = masses versus experiment. 28

29 12 8 N Percent error deviation from exp 4-4 η ρ K* ϕ a a 1 b 1 Σ Ξ Λ Σ* Ξ* N f = masses versus experiment. 29

30 .5 ξ f = 2.979(28).4.3 E 2 (p) p 2 ξ f = 3.45(35) E 2 (p) p 2 Dispersion relation: π and ρ 3