Nucleon Spectroscopy with Multi-Particle Operators

Transcription

1 Nucleon Spectroscopy with Multi-Particle Operators Adrian L. Kiratidis Waseem Kamleh, Derek B. Leinweber CSSM, The University of Adelaide 21st of July - LHP V Cairns, Australia 1/41

2 Table of content 1 Correlation matrix techniques 2 Previous results A toy model 5-quark operators 3 Methodology Test results 4 5 2/41

3 Correlation matrix techniques Table of content 1 Correlation matrix techniques 2 Previous results A toy model 5-quark operators 3 Methodology Test results 4 5 3/41

4 Correlation matrix techniques Correlation Matrix Techniques Begin by constructing an N N basis of cross correlation functions G ± ij ( p, t) = [ e i p x Tr sp Γ ± Ω χi ( x, t) χ j ( 0, t src ) ] Ω x = α λ α i λ α j e mαt α enumerates the energy eigenstates of mass m α and parity ± that we have projected with Γ ± = (γ 0 ± 1)/2, λ α i and λ α j are the couplings of our creation and annihiliation operators χ j and χ i at the source and sink respectively. 4/41

5 Correlation matrix techniques Correlation Matrix Techniques (cont.) We then search for a linear combinations of operators φ α = i χ i v α i and φα j = j χ j u α j such that φ and φ couple to a single energy eigenstate. One can then see from our cross correlation matrix equation that G ij (t 0 + t)uj α = e mα t G ij (t 0 )uj α Hence the required values for uj α and vi α can be obtained from solving the eigenvalue equations [ G 1 (t 0 ) G(t 0 + t) ] ij uα j = c α ui α vi α [ G(t0 + t) G 1 (t 0 ) ] ij = cα vj α, 5/41

6 Correlation matrix techniques Correlation Matrix Techniques (cont.) As our correlation matrix is diagonalised at t 0 and t 0 + t by the eigenvectors uj α and vi α we can obtain the eigenstate projected correlator G α ± = v α i G ± ij uα j which is then use to extract a mass. 6/41

7 Correlation matrix techniques Typical mass fit M(GeV) Euclidean Time 7/41

8 Previous results A toy model 5-quark operators Table of content 1 Correlation matrix techniques 2 Previous results A toy model 5-quark operators 3 Methodology Test results 4 5 8/41

9 Previous results A toy model 5-quark operators 3-Quark Operator χ 1 = ɛ abc (u at Cγ 5 d b )u c and χ 2 = ɛ abc (u at Cd b )γ 5 u c M. S. Mahbub et al. [CSSM Lattice Collaboration], Phys. Rev. D 87, (2013). 9/41

10 Previous results A toy model 5-quark operators 3-Quark Operator (cont.) M. S. Mahbub, W. Kamleh, D. B. Leinweber and A. G. Williams, Annals Phys. 342, 270 (2014) 10/41

11 Previous results A toy model 5-quark operators 3-Quark Operator (cont.) t 0 t t max M 1 M 2 λ 1 λ 2 χ 2 /dof (25) 2.45(41) 1.83(1.95) 6.22(1.23) (39) 2.36(50) 1.60(2.83) 6.19(2.02) (43) 2.37(60) 1.75(3.38) 6.02(2.48) (30) 2.38(40) 1.48(2.02) 6.43(1.28) (49) 2.26(41) 1.00(2.53) 6.60(1.77) (56) 2.25(49) 1.05(3.04) 6.52(2.20) (85) 1.95(11) 0.12(0.77) 16.25(0.97) (99) 1.97(20) 0.25(2.54) 16.31(1.58) (68) 1.93(06) 0.04(0.20) 16.05(0.92) (85) 1.93(08) 0.06(0.40) 16.09(1.03) 0.10 no prediction 11/41

12 Previous results A toy model 5-quark operators Toy Model Consider a simple 2-component toy model with QCD eigen-states given by a = cos θ 1 + sin θ 2 b = sin θ 1 + cos θ 2 where 1 and 2 denote a single-hadron and meson-baryon type component respectively, while θ is some arbitrary mixing. Now suppose we have a three quark operator χ 3 that has substantial overlap with 1 but not 2 Ω χ3 1 C and Ω χ3 2 C. So χ 3 acting on the vacuum creates 1 = cos θ a sin θ b. 12/41

13 Previous results A toy model 5-quark operators Toy Model (cont.) No operator sensitive to 2 no way to disentangle energy-eigenstates. Concern of not being able to see states with high 2 component and contamination of extracted state. In our work we therefore utilize 5-quark operators which are expected to have higher overlap with meson-baryon type states. 13/41

14 Previous results A toy model 5-quark operators Toy Model (cont.) It is now known (from meson studies for example) that scattering states can be extracted by explicitly projecting the momentum of interest of each state. Rather than performing this projection, the question we endeavour to address is what role do five-quark operators (without explicitly projected momentum) have on the mass spectrum? 14/41

15 Previous results A toy model 5-quark operators 5-quark operators Using the Clebsch-Gordan coefficients we can therefore write down five quark operators 2 χ 5 (x) = nπ + 1 p 3 π = 1 { 2 3 ɛabc 2 ( u Ta (x) Γ 1 d b (x) ) [ ] Γ 2 d c (x) d e (x) γ 5 u e (x) ( u Ta (x) Γ 1 d b (x) ) [ ] Γ 2 u c (x) d e (x) γ 5 d e (x) + ( u Ta (x) Γ 1 d b (x) ) [ Γ 2 u c (x) ū(x) e γ 5 u (x)] } e, where χ 5 and χ 5 correspond to (Γ 1, Γ 2 ) = (Cγ 5, I) and (Γ 1, Γ 2 ) = (C, γ 5 ) respectively. 15/41

16 Previous results A toy model 5-quark operators 5-quark operators (cont.) Now need to calculate the more computationally intense loop propagators S(x, x). 16/41

17 Methodology Test results Table of content 1 Correlation matrix techniques 2 Previous results A toy model 5-quark operators 3 Methodology Test results /41

18 Methodology Test results Methodology Proceed by generating an ensemble of random independent Z 4 noise vectors η 1... η N performing full dilution in spin, colour, and time as a means of variance reduction ηα( x, a t) = ( x, t). where η ab,t αβ b,β,t η ab,t αβ ( x, t) = δ αβδ ab δ tt η a α( x, t). (No summation). The stochastic estimate of S(y, x) for a single noise vector is then given by Sγα( y, ca x) = χ cb,t γβ ( y, t)η ab,t αβ ( x, t). b,β,t 18/41

19 Methodology Test results Test We now test the robustness of method by calculating correlators with stochastically estimated propagators and comparing them with correlators that use standard S(x, 0) propagators. Replace only one of the propagators present with a stochastic one. Smearing of stochastically estimated propagators can be done post inversion. 19/41

20 Methodology Test results Pion Correlator log Gπ n s =, stochastic n s =, standard n s = 100, stochastic n s = 100, standard n s =, stochastic n s =, standard Euclidean Time 20/41

21 Methodology Test results Nucleon Correlator log GN n s =, stochastic n s =, standard n s = 100, stochastic n s = 100, standard n s =, stochastic n s =, standard Euclidean Time 21/41

22 Table of content 1 Correlation matrix techniques 2 Previous results A toy model 5-quark operators 3 Methodology Test results /41

23 Configuration Details PACS-CS flavour dynamical-fermion configurations made available through the ILDG Non-perturbatively O(a)-improved Wilson fermion action, and the Iwasaki gauge action. Lattice size is with a spacing of fm providing a volume of (2.90 fm) 3. β = 1.90, the light quark mass is set by the hopping parameter κ ud = which gives a pion mass of m π = 293 MeV, while the strange quark mass is set by κ s = Make use of 720 configurations. 23/41

24 Table of Operators Basis Number Operators Used 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 24/41

25 Correlation Matrix n s = 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) Basis Number 25/41

26 Correlation Matrix (cont.) State 1 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters 26/41

27 Correlation Matrix (cont.) State 2 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters 27/41

28 Correlation Matrix (cont.) State 3 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters 28/41

29 Correlation Matrix (cont.) State 1 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 29/41

30 Correlation Matrix (cont.) State 2 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 30/41

31 Correlation Matrix (cont.) State 3 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 31/41

32 Correlation Matrix (cont.) State 2 Eigenmass State 3 Eigenmass Eigenmass comparison 2.8 M(GeV) Basis Number/Variational Parameters 32/41

33 Correlation Matrix 6 5 S-wave N + π n s = + 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) Basis Number 33/41

34 Correlation Matrix (cont.) State 0 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters 34/41

35 Correlation Matrix (cont.) State 1 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters /41

36 Correlation Matrix (cont.) State 2 Eigenvector Component u χ1 u χ1 u χ2 u χ2 u χ5 u χ5 u χ 5 u χ Basis Number/Variational Parameters 36/41

37 Correlation Matrix (cont.) State 0 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 37/41

38 Correlation Matrix (cont.) State 1 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 38/41

39 Correlation Matrix (cont.) State 2 Mass from projected correlator fit Eigenmass Extracted mass used M(GeV) Basis Number/Variational Parameters 39/41

40 Summary Developed method to smear stochastically estimated loop propagators. Introduced five-quark operators and performed correlation matrix analyses with them. χ 5 seems to be important in accessing energies in the region of scattering states. Fitting a single state ansatz to eigenstate projected correlators enables reliable extraction of energies across qualitatively different variational bases. In particular, using the techniques described herein, one doesn t need access to low-lying states in order to reliably extract energies closely related to the resonances of Nature. 40/41

41 Future Work Further work will include explicitly specifying the momentum of particles present in scattering states and analysis in other channels. Thanks for Listening! 41/41