Nucleon excited states on the lattice

Transcription

1 Nucleon excited states on the lattice C.B. Lang, Valentina Verduci Graz,

2 Overview Motivation: Investigate the nucleon spectrum studying the π N scattering. The approach: How do we extract excited states on the lattice? Results: What do we expect? What do we learn?

3 Excited states on the lattice Engel et al. arxiv: Lattice QCD successfully estimates the ground states of the baryon spectrum but excited states still represent an outstanding challenge.

4 Nucleon excited states N(1440)P 11 is the lightest excitation of the Nucleon. Constituent quark models based on SU(6) symmetry predict the spectrum of the nucleon to be arranged into successive bands of positive and negative parity.

5 Nucleon excited states on the lattice Lattice simulations have problems in reproducing the mass reversal order observed in nature.

6 Nucleon spectrum N(939) N(1440) Nπ 55-75% N(1535) Nπ 35-55% N(1650) Nπ 50-90% N π is the main decay channel of the Nucleon excited states.

7 N π scattering Including explicitly the pion and the nucleon on the lattice might help! Positive parity sector Positive parity sector N + Nπ N Nπ P wave S wave

8 N π scattering: the issues Many diagrams: large amount of cpu time needed. Disconnected diagrams: not affordable with traditional techniques. Many energy levels: how to extract them reliably?

9 Mass Spectroscopy on the lattice: Ingredients Action S = S Gauge + S fermion Gauge configurations with Boltzmann distribution e S Observable for the estimation of the masses of the QCD spectrum: The hadron correlator function.

10 Hadron Correlation Function The hadron correlation function in the Euclidean is defined as C ij (t) = χ i (t)χ j (0) = n 0 χ i n e Ent n χ j 0 = = a 1 e E 1t + a 2 e E 2t + a 3 e E 3t noise m 2 i = E 2 i p 2

11 Compute the Correlation Function The interpolators: χ B (x) = ɛ abc Γ A q a (x) q b (x) Γ B q c (x) χ M (x) = q a (x) Γ q a (x) The correlator involves terms like C(x, y) = D q a (x) q b (x) q c (x) q e (y) q }{{} f (y) q g (y). M 1 (x,y) It is required to solve N 3 s N t N c equations like M(x, y)φ(y) = η(x) φ(x) = M 1 (x, y)η(y) Determining and storing all the elements of M 1 is not possible!

12 Quark Propagator Point-to-all method: compute the propagator for one localized source at given time slice to all the lattice. It works for correlators of single hadron operators concerning connected diagrams. Disconnected diagrams are involved in the study of multi-hadron states. To evaluate backtracking loops we need to consider many sources on each time slice: all-to-all propagator. N 3 S N c inversions are needed: too expensive!

13 Distillation Method (Peardon et al, arxiv: ) Smeared sources + Cut measurement costs Smearing the quarks with a very low rank operator written in terms of eigenvectors of the 3D Laplacian q(x) S(x, x )q(x ) = N V i=1 v i (x)v i (x )q(x )

14 Distillation After the distillation τ ij = v i (x)m 1 v j (y) N v (N T N d ) inversions instead of M 1 (x, y) N 3 S N c(n T N d ) inversions

15 Extract the excited states: Variational method Consists in disentangling the states using several interpolators. Use several interpolators χ i to construct a basis with minimum overlap. Compute the cross correlations C ij (t) = χ i (t)χ j (0) Solve the generalized eigenvalue problem C(t)u (n) = λ (n) C(t 0 )u (n) Obtain energy levels from the eigenvalues lim t λ (n) (t, t 0 ) = e En(t t 0)

16 From 3 to 5 quarks: the challenge

17 Step 1: N N A good interpolator for the nucleon has the form χ i = P ± {ɛ abc Γ 1 u a [u T b Γ 2 d c d T b Γ 2 u c ]} χ 1 : (1, Cγ 5 ) χ 2 : (γ 5, C) χ 3 : (i1, Cγ 4 γ 5 ) P ± = (1 ± γ 0 )/2

18 Step 2: Nπ Nπ A good interpolator for the pion-nucleon system is Nπ = ɛ abc γ 5 Γ 1 d a u T b Γ 2 d c d γ 5 u An isospin projection is needed in order to overlap with the nucleon states 1/2 ± : Nπ = Pπ Nπ +

19 Simulation setting Fermion action: Wilson Clover action with 2 degenerate flavours. Configurations: 280 Lattice size: (a = 0.12 fm) Pion masses: 266 MeV

20 N N 2 terms

21 The positive parity sector

22 N N : N configs, 6 interpolators: 32, 64 source and sink eigenvectors

23 N N : N + All to all: 1.093(12) Point to all: 1.092(34)

24 The negative parity sector

25 N N : N 280 configs, 6 interpolators: 32, 64 source and sink eigenvectors All to all Point to all

26 N N : N

27 N decay channels N(1535) Nπ 35-55% N(1535) Nη 32-52% N(1650) Nπ 50-90% N(1650) Nη 5-15% N(1650) ΛK 3-11%

28 Nπ in S-wave Non interacting N π

29 Nπ in S-wave

30 Nπ in S-wave

31 The Nπ scattering Nπ N 4 terms

32 The Nπ scattering Nπ Nπ 7 terms

33 The Nπ scattering Nπ Nπ 12 terms

34 Nπ in S-wave 280 configs, 9 interpolators: 32, 64 source and sink eigenvectors

35 Nπ in S-wave 1st state 2nd state 3rd state

36 Nπ in S-wave 280 configs, 9 interpolators: 32, 64 source and sink eigenvectors N N Nπ Nπ

37 Nπ in S-wave

38 Phase shift analysis The Luescher method connects the discrete spectrum in finite volume with the scattering phase shift in infinite volume tan δ(q) = π3/2 q Z 00 (1, q 2 ) for P = 0

39 Phase shift analysis The phase shift profile can be fitted against the Breit-Wigner form in the vicinity of a resonance to evaluate the mass and width of the resonance. m R = GeV ρ(s) = s Γ(s) cot δ(s) = m 2 R s

40 Summary The nucleon spectrum is a not yet understood issue. The study of the pion-nucleon scattering provides new information. The distillation method represent a fundamental tool to deal with disconnected diagrams in a reasonable computer time. Thanks to the flexibility of the approach, it be used for the study of other baryons of the QCD spectrum.