Probing Nucleon Resonances on the Lattice

Transcription

1 Probing Nucleon Resonances on the Lattice Benjamin Owen Supervisors: Derek Leinweber, Waseem Kamleh April 7th, 2014 Benjamin Owen (Adelaide Uni) April 7th, / 27

2 Outline 1 Accessing states on the Lattice 2 Accessing individual states on the Lattice 3 Hadron structure and Matrix elements 4 Improved Ground State Isolation 5 Probing excited states: Nucleon Resonance Benjamin Owen (Adelaide Uni) April 7th, / 27

3 Accessing states on the Lattice Quantities of interest are correlators G( p, t) = x e i p x Ω χ(x) χ(0) Ω Inserting complete set of states, I = α α G( 0, t) α e Mα t Ω χ α α χ Ω The terms Ω χ α describe the coupling strength Z α of operator χ to state α Benjamin Owen (Adelaide Uni) April 7th, / 27

4 Result is a sum over exponentials of increasing mass G(t) Consider the effective mass e Mα t Z α 2 α=0 M eff log ( G(t) ) G(t + 1) At large times (t ), the ground state will dominate M eff M 0 Benjamin Owen (Adelaide Uni) April 7th, / 27

5 Effective Mass for the pion M GeV Benjamin Owen (Adelaide Uni) April 7th, / 27

6 Correlation Matrix methods Ideally we want interpolators φ α such that β φ α Ω δ αβ Seek a linear combination of operators χ j to produce φ α N φ α = χ j u α j j=1 Begin with a matrix of cross-correlators G ij (t) = x Ω χ i χ j Ω multiply on the right by u α j G ij (t)u α j = x Ω χ i χ j u α j Ω Benjamin Owen (Adelaide Uni) April 7th, / 27

7 Knowledge of the time dependence provides the recurrence relation G ij (t + δt)u α j = e Mα δt G ij (t)u α j Multiplying from the left by G 1 provides an eigenvalue equation for eigenvectors [ G 1 (t) G(t + δt) ] ij u α j = e Mα δt u α j Having solved for u α j, we can project the matrix of correlators to produce correlators for the state α G α (t) = v α i G ij (t)u α j Benjamin Owen (Adelaide Uni) April 7th, / 27

8 Pion effective mass with Correlation Matrix approach M GeV Benjamin Owen (Adelaide Uni) April 7th, / 27

9 Probing hadron structure on the Lattice To explore the structure a state, we must probe it with some external current χ j (0) χ i (x) Once again, the correlator will have contributions from a tower of states Benjamin Owen (Adelaide Uni) April 7th, / 27

10 Probing hadron structure on the Lattice We can use our optimised operators, determined via the two-point correlation function, to project out the three-point correlation function for an individual state u α j ( p) χ j(0) v α i ( p )χ i (x) One needs to take care of the source and sink momenta as the optimised operators are momentum dependent Benjamin Owen (Adelaide Uni) April 7th, / 27

11 Nucleon Axial Charge Nucleon axial charge has been quantity of significant interest on the lattice Despite relative simplicity, lattice determinations have been consistently low Finite volume effects are known to play a role, but are not the complete answer Excited state contamination has suggested as possible issue Can use correlation matrix methods to eliminate excited state contaminations B. J. Owen et al., Phys. Lett. B 723, (2013) Benjamin Owen (Adelaide Uni) April 7th, / 27

12 Axial charge of the nucleon standard approach Using single operator for source and sink 1.5 g A t S Benjamin Owen (Adelaide Uni) April 7th, / 27

13 Axial charge of the nucleon standard approach Using a different operator 1.4 g A t S Benjamin Owen (Adelaide Uni) April 7th, / 27

14 Axial charge of the nucleon Correlation matrix approach 1.4 g A t S Benjamin Owen (Adelaide Uni) April 7th, / 27

15 Axial charge of the nucleon Comparison between methods 1.4 g A t S Benjamin Owen (Adelaide Uni) April 7th, / 27

16 Projected Correlator for the first nucleon excitation Correlation Matrix approach is a method for studing excited states log G Benjamin Owen (Adelaide Uni) April 7th, / 27

17 The Sachs Electric form factor - Quark Sector comparison (u sector in the proton) GE Benjamin Owen (Adelaide Uni) April 7th, / 27

18 The Sachs Electric form factor - Quark Sector comparison (d sector in the proton) GE Benjamin Owen (Adelaide Uni) April 7th, / 27

19 The Sachs Magnetic form factor - Quark Sector comparison (u sector in the proton) GM ΜN Benjamin Owen (Adelaide Uni) April 7th, / 27

20 The Sachs Magnetic form factor - Quark Sector comparison (d sector in the proton) GM ΜN Benjamin Owen (Adelaide Uni) April 7th, / 27

21 Charge radii for the nucleon, + and first radial excitation of the nucleon (proton and neutron) Use dipole Ansatz to calculate charge-square radii 0.6 r 2 fm Π Benjamin Owen (Adelaide Uni) April 7th, / 27

22 Nucleon spectrum M. S. Mahbub et al., Phys. Lett. B 707, (2012) D. S. Roberts et al., arxiv: [hep-lat] Benjamin Owen (Adelaide Uni) April 7th, / 27

23 Wave Function of 1 st Nucleon Resonance - heaviest mass D. S. Roberts et al., arxiv: [hep-lat] Benjamin Owen (Adelaide Uni) April 7th, / 27

24 Wave Function of 1 st Nucleon Resonance - 2 nd heaviest mass D. S. Roberts et al., arxiv: [hep-lat] Benjamin Owen (Adelaide Uni) April 7th, / 27

25 Comparison of radii One can use wave function to calculate r Gauge dependent Consider ratio of r between states as comparison between methods FF: WF: r 1 r 0 = 1.26, 1.32 r 1 r 0 = 1.16, 1.08 Consistency, noting that WF method suffers significantly from finite volume effects which will tend to suppress the value Benjamin Owen (Adelaide Uni) April 7th, / 27

26 Magnetic moments for the nucleon, + and first radial excitation of the nucleon (proton and neutron) Use dipole Ansatz to calculate magnetic moment Μ ΜN Π Benjamin Owen (Adelaide Uni) April 7th, / 27

27 Conclusion Demonstrated how use correlation matrix methods to construct ideal operators Seen how correlation techniques can improve ground state overlap Clear plateau observed for Form Factors of the 1 st nucleon excitation Consistency with wave function results Magnetic moment is consistent with simple quark model expectation for s-wave excitation Further investigation required to determine dominant contribution to resonance at heavier masses In process of extending analysis down to light masses Benjamin Owen (Adelaide Uni) April 7th, / 27