Excited States of the Nucleon in Lattice QCD

Transcription

1 Collaborators: Alan Ó Cais, Waseem Kamleh, Ben G. Lasscock, Derek B. Leinweber and Anthony G. Williams CSSM, University of Adelaide Adelaide

2 Outline Introduction 1 Introduction 2 3 4

3 2 point Correlation Function Two point correlation function: G ij (t, p) = x e i p. x Ω T {χ i (x) χ j (0)} Ω. (1) Inserting completeness B, p,s B, p,s = I Then B, p,s G ij (t, p) = B + λ B + λb +e E B +t γ.p B + +M B + 2E B + + λ B λ B e E B t γ.p B M B. (2) 2E B B

4 2 point Correlation Function λ B ±, λ B ± are the couplings of χ(0) and χ(0) with B ± defined by Ω χ(0) B +, p,s = λ B + B +, p,s χ(0) Ω = λ B + and for the negative parity states, Ω χ(0) B, p,s = λ B B, p,s χ(0) Ω = λ B M B + E B + u B +( p,s), M B + ū E B +( p,s), B + M B E B γ 5 u B ( p,s), M B ū E B ( p,s)γ 5. B

5 2 point Correlation Function At p = 0 G ± ij (t, 0) = Tr sp [Γ ± G ij (t, 0)] = B ± λ ± i λ ± j e M B ±t. (3) Parity projection operator, And Γ ± = 1 2 (1 ± γ 0). G ± ij (t, 0) t = λ ± i0 λ ± j0 e M 0 ±t. (4)

6 2 point Correlation Function Interpolators: χ 1 (x) = ǫ abc (u Ta (x)cγ 5 d b (x))u c (x), χ 2 (x) = ǫ abc (u Ta (x)cd b (x))γ 5 u c (x), χ 4 (x) = ǫ abc (u Ta (x)cγ 5 γ 4 d b (x))u c (x). χ 1 χ 1 χ 2 χ 2 χ 4 χ 4

7 Consider N interpolating fields, then φ α = φ α = N ui α χ i, i=1 N i=1 v α i χ i, such that, B β,p,s φ α Ω = δ αβ z α ū(α,p,s), Ω φ α B β,p,s = δ αβ z α u(α,p,s),

8 Then a two point correlation function matrix for p = 0, G ij (t)u α j = ( x Tr sp {Γ ± Ω χ i χ j Ω })u α j There is no sum over α = λ α i z α e mαt. (5) t dependence only in the exponential term

9 Then one can have a recurrence relation at time (t + t), G ij (t + t)u α j = e mα t G ij (t)u α j. Multiplying by [G ij (t)] 1 from left, [(G(t)) 1 G(t + t)] ij u α j = c α u α i, (6) where c α = e mα t is the eigenvalue. Similarly, it can also be solved for the left eigenvalue equation for v α eigenvector, v α i [G(t + t)(g(t)) 1 ] ij = c α v α j. (7)

10 The vectors u α j and v α i diagonalize the correlation matrix at time t and t + t making the projected correlation matrix, v α i G ij (t)u β j = δ αβ z α z β e mαt. (8) The projected correlator, is then analyzed to obtain masses of different states, v α i G ± ij (t)u α j G α ±, (9) We construct the effective mass ( Meff α (t) = ln G±(t, α ) 0) G± α (t + 1,. (10) 0)

11 Simulation Details lattice volume lattice spacing fm We use FLIC fermion action and quenched QCD Analysis is performed for 10 different pion masses: 797,729,641,541, 430,380,327,295,249,224 MeV. We use varieties of Gaussian smearing sweeps (number of sweeps 1,3,7,12,16,26,35,48,65) 2 2, 3 3, 4 4, 6 6 and 8 8 correlation matrices are analyzed To analyze data we use fitting robot

12 2x2, for point source, for χ 1 χ 2 Projected Mass Vs Mass From Eigenvalue

13 Eigenvectors - Point Source, for χ 1 χ 2 Left Eigenvectors Right Eigenvectors

14 For 3 3, of χ 1 χ 2 χ 4 Projected Mass Vs Mass From Eigenvalue

15 Eigenvectors Introduction

16 Smearing Introduction To create a comprehensive basis of interpolating fields we consider source smearing, ψ i (x,t) = x F(x,x )ψ i 1 (x,t), (11) where, F(x,x ) = (1 α)δ x,x + α 6 3 [U µ (x)δ x,x+ˆµ µ=1 +U µ(x ˆµ)δ x,x ˆµ], (12) Fixing α = 0.7, the procedure is repeated N sm times.

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18 2x2, for smeared source, for χ 1 χ 2 Projected Mass Vs Mass From Eigenvalue

19 Eigenvectors - Smeared source, for χ 1 χ 2 Left Eigenvectors Right Eigenvectors

20 Smeared Source Introduction

21 Smeared-Smeared Introduction M.S. Mahbub et al., Phys. Rev. D 80, (2009), [arxiv:hep-lat/ ].

22 Introduction Roper resonance (P 11 ) is the first positive parity excited state of the nucleon Observed in 1960 s from πn scattering The state is puzzling due to its lower mass (1440 MeV) from its nearest negative parity (S 11 ) excited state (1535 MeV). In constituent quark model, Roper state is 100 MeV heavier than the S 11 (1535 MeV) state. This state appeared too high in all previous attempts using variational method in lattice QCD.

23 4x4 bases of χ 1 χ 1 We use smeared smeared correlation functions Varieties of smearing sweeps Sweeps Basis No. Bases

24 4x4, For 4 th basis (3, 12, 26, 35) Projected Mass Vs Mass From Eigenvalue

25 4x4 bases of χ 1 χ 1 Sweeps Basis No. Bases

26 For all 4 4 bases Introduction

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28 3 rd basis (1,12,26,48) 4 th basis (3,12,26,35) 5 th basis (3,12,26,48) 6 th basis (12,16,26,35) t 1 t 2 χ am χ t dof 1 t 2 am χ t dof 1 t 2 am χ t dof 1 t 2 am dof (Roper) (Roper) (Roper) (Roper) (41) (39) (44) (40) (43) (41) (46) (39) (39) (45) (40) (42) (44) (51) (45) (46) (50) (51) (51) (52) (60) (55) (57) (60) (69) (67) (71) (75) (82) (81) (82) (85) (10) (10) (11) (11) (13) (12) (13) (13) 0.70

29 Roper from 4 4 Introduction Mahbub et al., Phys. Lett. B 679, 418 (2009), [arxiv:hep-lat/ ].

30 6x6 bases of χ 1 χ 1 Sweeps Basis No. Bases

31 For all 6x6 bases Introduction

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33 6x6 bases of χ 1 χ 2 Sweeps Basis No. Bases

34 For all 6x6 bases Introduction

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36 8x8 bases of χ 1 χ 2 Sweeps Basis No. Bases

37 For all 8x8 bases Introduction

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39 Story of excited states

40 Story of excited states

41 Story of excited states

42 Story of excited states

43 Story of excited states

44 Story of excited states

45 Story of excited states

46 2 states for 2x2,4x4,6x6,8x8

47 Roper state: Compilation of existing results

48 between Roper (1440 MeV) P 11 and N 1 2 (1535 MeV) S 11 states. Projected Mass 3 3 Vs 4 4

49 3 3 and 4 4 results

50 Roper (1440 MeV) and N 1 2 (1535 MeV) states

51 : Simulation details Lattice volume: a =0.125 fm 200 configurations, nf = 2, pion mass = 634 MeV. FLIC fermion action Collaborators:, Peter Moran,

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53 Introduction Various dimensions of the correlation matrices have been analyzed. Varieties of smearing sweeps have been used in constructing correlation matrices. We observed smearing dependency of the excited states given that the ground state is independent on smearing. A low-lying Roper state has been identified for the first time using variational method. For consistency and reliability check we considered several 4 4, 6 6, 8 8 matrices.

54 Introduction We have shown how excited states are split up with the dimension of the correlation matrices. We have shown the importance of using smeared-smeared correlation functions and larger correlation matrices for the reliable extraction of excited states mass. A level crossing between the Roper (1440 MeV) and N 1 2 (1535 MeV) states has been observed for the first time in variational approach. The Roper results in quenched and dynamical QCD are in very good agreement.

55 Thanks