Structure of the Roper in Lattice QCD

Transcription

1 Structure of the Roper in Lattice QCD Waseem Kamleh Collaborators Dale Roberts, Derek Leinweber, Adrian Kiratidis Selim Mahbub, Peter Moran and Tony Williams CSSM, University of Adelaide APFB 2014

2

3

4 Roper Resonance Quark model: N = 2 radial excitation of the nucleon. Much lower in mass than simple quark model predictions.

5 Roper Resonance

6 Roper Resonance Quark model: N = 2 radial excitation of the nucleon. Much lower in mass than simple quark model predictions. Experiment: Lighter than N = 1 radial excitation of the nucleon, the negative parity S 11 (1535). Exotic in nature.

7 Roper Resonance It has proven difficult to isolate this state on the lattice. Consider the nucleon interpolators, χ 1 (x) = ɛ abc (u Ta (x) Cγ 5 d b (x)) u c (x), χ 2 (x) = ɛ abc (u Ta (x) C d b (x)) γ 5 u c (x), χ 4 (x) = ɛ abc (u Ta (x) Cγ 5 γ 4 d b (x)) u c (x). Historically thought Roper couples to χ 2.

8 Roper Resonance It has proven difficult to isolate this state on the lattice. Consider the nucleon interpolators, χ 1 (x) = ɛ abc (u Ta (x) Cγ 5 d b (x)) u c (x), χ 2 (x) = ɛ abc (u Ta (x) C d b (x)) γ 5 u c (x), χ 4 (x) = ɛ abc (u Ta (x) Cγ 5 γ 4 d b (x)) u c (x). Historically thought Roper couples to χ 2. We will see that this is wrong!

9 Roper Resonance It has proven difficult to isolate this state on the lattice. Consider the nucleon interpolators, χ 1 (x) = ɛ abc (u Ta (x) Cγ 5 d b (x)) u c (x), χ 2 (x) = ɛ abc (u Ta (x) C d b (x)) γ 5 u c (x), χ 4 (x) = ɛ abc (u Ta (x) Cγ 5 γ 4 d b (x)) u c (x). Historically thought Roper couples to χ 2. We will see that this is wrong! Key to isolating this elusive state is an appropriate variational basis. Phys.Lett. B707 (2012) , Roper Resonance in 2+1 Flavor QCD

10 Correlation Functions Start with the two point correlation function: G(t, p) = x e i p. x Ω T {χ(x) χ(0)} Ω.

11 Correlation Functions Start with the two point correlation function: G(t, p) = x e i p. x Ω T {χ(x) χ(0)} Ω. Insert completeness, project parity ± and p = 0 to obtain a tower of (excited) states G ± (t, 0) = α λ α λ α e Mαt.

12 Correlation Functions Start with the two point correlation function: G(t, p) = x e i p. x Ω T {χ(x) χ(0)} Ω. Insert completeness, project parity ± and p = 0 to obtain a tower of (excited) states G ± (t, 0) = α λ α λ α e Mαt. Asymptotically, Euclidean time evolution isolates the ground state G ± (t, 0) t = λ 0 λ 0 e M 0t.

13 Variational Method How can we access the excited states?

14 Variational Method How can we access the excited states? Construct an n n correlation matrix, G ij (t, p) = x e i p. x Ω T {χ i (x) χ j (0)} Ω.

15 Variational Method How can we access the excited states? Construct an n n correlation matrix, G ij (t, p) = x e i p. x Ω T {χ i (x) χ j (0)} Ω. Solve a generalised eigenproblem to find the linear combination of interpolating fields, φ α = N ui α χ i, φ α = i=1 N i=1 vi α χ i such that the correlation matrix is diagonalised, v α i G ij (t)u β j = δ αβ z α z β e mαt.

16 Eigenstate-Projected Correlators The left and right vectors are used to define the eigenstate-projected correlators v α i G ± ij (t)u α j G α ±(t).

17 Eigenstate-Projected Correlators The left and right vectors are used to define the eigenstate-projected correlators v α i G ± ij (t)u α j G α ±(t). Effective masses of different states are then analysed from the eigenstate-projected correlators in the usual way.

18 Eigenstate-Projected Correlators The left and right vectors are used to define the eigenstate-projected correlators v α i G ± ij (t)u α j G α ±(t). Effective masses of different states are then analysed from the eigenstate-projected correlators in the usual way. Not able to access the Roper using χ 1, χ 2 ( or χ 4 ) alone.

19 Eigenstate-Projected Correlators The left and right vectors are used to define the eigenstate-projected correlators v α i G ± ij (t)u α j G α ±(t). Effective masses of different states are then analysed from the eigenstate-projected correlators in the usual way. Not able to access the Roper using χ 1, χ 2 ( or χ 4 ) alone. Solution: Use different levels of gauge-invariant quark smearing to expand the operator basis.

20 Eigenstate-Projected Correlators The left and right vectors are used to define the eigenstate-projected correlators v α i G ± ij (t)u α j G α ±(t). Effective masses of different states are then analysed from the eigenstate-projected correlators in the usual way. Not able to access the Roper using χ 1, χ 2 ( or χ 4 ) alone. Solution: Use different levels of gauge-invariant quark smearing to expand the operator basis. Analogy: Can expand any radial function using a basis of Gaussians of different widths f ( r ) = i c i e ε i r 2.

21 Simulation Details PACS-CS Configs (via ILDG) S. Aoki, et al., Phys. Rev. D79 (2009) flavour dynamical-fermion QCD Lattice volume: a = fm, (2.9 fm) 3 m π = { 156, 293, 413, 572, 702 } MeV Combined 8 8 correlation matrix analysis using χ 1, χ 2 and χ 1, χ 4 with 4 different levels (n = 16, 35, 100, 200) of smearing. Corresponds to RMS radii of 2.37, 3.50, 5.92 and 8.55 lattice units.

22 N + spectrum

23 Eigenvector analysis State 1 (Ground) Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

24 Eigenvector analysis State 2 (First Excited) Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

25 Eigenvector analysis First Excited State Dominant contribution is from χ 1, n = 200. Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

26 Eigenvector analysis First Excited State Opposite contribution from a mix of χ 1, n = 100 and χ 1, n = 35. Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

27 Eigenvector analysis First Excited State Negligible contribution from χ 1, n = 16 and all χ 2 operators. Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

28

29

30

31 Eigenvector analysis State 4 Eigenvector Component χ 1 χ n = 16 n = 35 n = 100 n = 200

32 Eigenvector analysis First positive-parity excited state couples strongly to χ 1. Large smearing values are critical. χ 2 coupling to the Roper is negligible. Transition from scattering state to resonance as quark mass drops. The 3-quark coupling to meson-baryon scattering states is suppressed by the lattice volume 1/ V. At light quark mass the Roper mass is pushed up due to finite volume effects? How can we learn more? Study multiple lattice volumes.

33 Eigenvector analysis First positive-parity excited state couples strongly to χ 1. Large smearing values are critical. χ 2 coupling to the Roper is negligible. Transition from scattering state to resonance as quark mass drops. The 3-quark coupling to meson-baryon scattering states is suppressed by the lattice volume 1/ V. At light quark mass the Roper mass is pushed up due to finite volume effects? How can we learn more? Study multiple lattice volumes. Expensive.

34 Eigenvector analysis First positive-parity excited state couples strongly to χ 1. Large smearing values are critical. χ 2 coupling to the Roper is negligible. Transition from scattering state to resonance as quark mass drops. The 3-quark coupling to meson-baryon scattering states is suppressed by the lattice volume 1/ V. At light quark mass the Roper mass is pushed up due to finite volume effects? How can we learn more? Study multiple lattice volumes. Expensive. Look at the excited state structure via the wave function.

35

36 Hydrogen S states

37 Wave function of the Roper We explore the structure of the nucleon excitations by examining the Bethe-Salpeter amplitude. The baryon wave function is built by giving each quark field in the annihilation operator a spatial dependence, χ 1 ( x, y, z, w) = ɛ abc ( u T a ( x + y) Cγ 5 d b ( x + z) ) u c ( x + w). The creation operator remains local. The resulting construction is gauge-dependent. We choose to fix to Landau gauge.

38 Wave function of the Roper Non-local sink operator cannot be smeared. Construct states using right eigenvector u α only. Eigenvectors from 4 4 CM analysis using χ 1 only. The position of the u quarks is fixed and we measure the d quark probability distribution.

39 Ground state probability distribution

40 Ground state probability distribution

41 Ground state probability distribution

42 Ground state probability distribution

43 Ground state probability distribution

44 First excited state probability distribution

45 First excited state probability distribution

46 First excited state probability distribution

47 First excited state probability distribution

48 First excited state probability distribution

49 Second excited state probability distribution

50 Second excited state probability distribution

51 Second excited state probability distribution

52 Second excited state probability distribution

53 Second excited state probability distribution

54

55

56

57

58

59 Wave Function

60 Wave Function

61 Wave Function

62 Wave Function

63 Quark Model comparison Compare to a non-relativistic constituent quark model. One-gluon-exchange motivated Coulomb + ramp potential. Spin dependence in R. K. Bhaduri, L. E. Cohler and Y. Nogami, Phys. Rev. Lett. 44 (1980) The radial Schrodinger equation is solved with boundary conditions relevant to the lattice data. The derivative of the wave function is set to vanish at a distance L x /2. Two parameter fit to the nucleon radial wave function yields: String tension σ = 400 MeV Constituent quark mass m q = 360 MeV These parameters are held fixed for the excited states.

64 Ground state comparison

65 Ground state comparison

66 Ground state comparison

67 Ground state comparison

68 Ground state comparison

69 First excited state comparison

70 First excited state comparison

71 First excited state comparison

72 First excited state comparison

73 First excited state comparison

74 Second excited state comparison

75 Second excited state comparison

76 Second excited state comparison

77 Second excited state comparison

78 Second excited state comparison

79 Quark Model comparison Ground state QM agrees well (as expected).

80 Quark Model comparison Ground state QM agrees well (as expected). First excited state shows a node structure.

81 Quark Model comparison Ground state QM agrees well (as expected). First excited state shows a node structure. Consistent with N = 2 radial excitation. QM predicts node position fairly well. QM disagrees near the boundary.

82 Quark Model comparison Ground state QM agrees well (as expected). First excited state shows a node structure. Consistent with N = 2 radial excitation. QM predicts node position fairly well. QM disagrees near the boundary. Reveals why an overlap of two broad Gaussians with opposite sign is needed to form the Roper.

83 Quark Model comparison Ground state QM agrees well (as expected). First excited state shows a node structure. Consistent with N = 2 radial excitation. QM predicts node position fairly well. QM disagrees near the boundary. Reveals why an overlap of two broad Gaussians with opposite sign is needed to form the Roper. Second excited state shows a double node structure. Consistent with N = 3 radial excitation. Similar story for QM comparison.

84 Quark Model comparison Ground state QM agrees well (as expected). First excited state shows a node structure. Consistent with N = 2 radial excitation. QM predicts node position fairly well. QM disagrees near the boundary. Reveals why an overlap of two broad Gaussians with opposite sign is needed to form the Roper. Second excited state shows a double node structure. Consistent with N = 3 radial excitation. Similar story for QM comparison. Finite volume effects?

85 Ground state probability distribution

86 Ground state probability distribution

87 Ground state probability distribution

88 Ground state probability distribution

89 Ground state probability distribution

90 First excited state probability distribution

91 First excited state probability distribution

92 First excited state probability distribution

93 First excited state probability distribution

94 First excited state probability distribution

95 Second excited state probability distribution

96 Second excited state probability distribution

97 Second excited state probability distribution

98 Second excited state probability distribution

99 Second excited state probability distribution

100 Quark Model comparison Wave function should be spherically symmetric. Outer shell of Roper wave function clearly reveals distortion due to finite volume. Effective field theory arguments suggest the small volume will drive up the energy.

101 5-quark operators What if the Roper has a large 5-quark component? Dynamical gauge fields can create q q from glue. Maybe we can do a better job by introducing 5-quark operators? Take χ 1 and χ 2 operators and couple a π to get N quantum numbers: χ 1 + π χ 5 χ 2 + π χ 5 Preliminary results at m π = 293 MeV with two smearings n = 35, 200.

102 N+ spectrum with 5 quark operators n s = S-wave N + π + π P-wave N + π 1 => χ 1 + χ 2 2 => χ 1 + χ 2 + χ 5 3 => χ 1 + χ 2 + χ 5 4 => χ 1 + χ 2 + χ 5 + χ 5 5 => χ 1 + χ 5 + χ 5 6 => χ 2 + χ 5 + χ 5 7 => χ 5 + χ 5 M(GeV) Correlation Matrix Number

103 N+ spectrum with 5 quark operators

104 n s = State 1 E-vector component u χ1 35 u χ1 200 u χ2 35 u χ2 200 u χ5 35 u χ5 200 u χ 5 35 u χ Correlation Matrix Number

105 n s = State 2 E-vector component u χ1 35 u χ1 200 u χ2 35 u χ2 200 u χ5 35 u χ5 200 u χ 5 35 u χ Correlation Matrix Number

106 Summary A basis of multiple Gaussian smearings is well-suited to isolating radial excitations of the nucleon. The variational method allows us to access a state that is consistent with the N = 2 Roper excitation with standard three-quark interpolators. Probing the Roper wave function reveals a nodal structure. χ 2 has negligible coupling to the Roper. Multiple χ 1 operators at large smearings are critical to form the correct structure.

107 Summary Qualitative agreement with QM predictions for the Roper radial wave function. Finite volume effects clearly evident in the Roper probability distribution. Larger lattice volumes needed! Preliminary results do not indicate a strong coupling to 5-quark operators.