Wave functions of the Nucleon

Transcription

1 Wave functions of the Nucleon Samuel D. Thomas (1) Collaborators: Waseem Kamleh (1), Derek B. Leinweber (1), Dale S. Roberts (1,2) (1) CSSM, University of Adelaide, (2) Australian National University LHPV, Cairns 2015 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

2 Introduction What do we mean by Wave function? As a state in quantum field theory, the nucleon doesn t have a simple wave function in the naive quantum mechanic sense. Lattice QCD interpolating fields naturally correspond to a fixed number of quarks This leads to a description using the Nambu-Bethe-Saltpeter wavefunction of three quarks Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

3 Introduction Consider the standard nucleon interpolating fields, but with a displaced d quark χ 1 (x, z; t) = ɛ abc [u T a (x; t)cγ 5 d b (x + z; t)]u c (x; t) χ 2 (x, z; t) = ɛ abc [u T a (x; t)cd b (x + z; t)]γ 5 u c (x; t) Could also consider displacement of the u quarks as long as we consider an interchange u(x + z) u(x) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

4 Introduction Define the wave function given by the two-point correlator of this displaced operator with a standard source operator: W (p, z, t) = x V e ipx χ(x, z; t) χ(x 0, 0; t 0 ) Not gauge invariant - fix to Landau gauge Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

5 Lattice Parameters (2+1) flavour PACS-CS collaboration ( Aoki et al [arxiv: ] ) via ILDG a= fm, al x =2.9 fm (68MeV) 1 κ ud m π N MeV MeV MeV Gaussian smeared fermion sources, α=0.7 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

6 Previous Work ( Roberts et al [hep-lat: ], [hep-lat: ] ) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

7 Previous Work Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

8 Spherical Harmonics z 0 not restricted to zero angular momentum Recall the spherical harmonics Y l,m : Y 1,0 = 1 2 Y 0,0 = π 3 π cosθ = 1 3 z 2 π r Y 1,±1 = π sinθe±iφ = 1 3 x ± iy 2 π r Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

9 Spherical Harmonics Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

10 Y 0,0 : (i s = 1, j s = 1; i s = 2, j s = 2) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

11 Y 1,0 : (i s = 1, j s = 3; i s = 2, j s = 4) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

12 Y 1, 1 : (i s = 1, j s = 4) (real part) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

13 Y 1, 1 : (i s = 1, j s = 4) (imaginary part) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

14 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

15 Variational Analysis From the standard 2-point correlation function, projected to definite parity: G ij ( p, t) = x e i p x Tr sp { 1 2 (γ 0 + I) χ i (x) χ j (0) (1) The right eigenvector is defined by the generalized eigenvalue problem: G ij (t 0 + δt)v j = e mδt 0 G ij (t)v j Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

16 Variational Analysis t 0 =17, dt=3 ; basis N sm =[16, 35, 100, 200] Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

17 Normalization These eigenvectors are calculated with a normalized correlation matrix G ij (t) G ij (t) Gii (0)G jj (0) This makes the components of G similar in magnitude Otherwise the generalized eigenvalue is badly-behaved Then normalize the eigenvectors u, v by their vector norm Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

18 Normalization In this case, we can t create a full correlation matrix for the wave function ( sink smearing would destroy the spatial information we care about) However, the sum v j χ j has been determined to be the best linear combination of the operators to create a single state from the vacuum. The standard normalization for v is then v j v j / G jj (0). W proj (z, t) = j W j (z, t)v j Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

19 Normalization Consider the probability density for positive parity P j (z, t) = Tr Sp Γ + W j (z, t) 2. An alternative normalization could be to scale W j (z, t) such that for each individual time value P j (z, t) = 1 How would this change the eigenstate projection? z Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

20 Normalization Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

21 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

22 Projected results Ground state (lightest pion mass): Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

23 Projected results First posity parity excited state: Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

24 Projected results Second posity parity excited state: Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

25 Projected results Ground state (heavier pion mass): Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

26 Projected results First posity parity excited state: Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

27 Projected results Second posity parity excited state: Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

28 χ 2 (z) χ 1 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

29 Higher spin Examining off-diagonal Dirac elements of the wave function has let us obtain angular momenta of m l = ±1. To access higher values we need a interpolating field for the spin-32 nucleon. There is one obvious choice - the interpolator χ µ : χ µ (x, z) = ɛ abc [u T a (x)cγ 5 γ µ d b (x + z)]γ 5 u c (x + z) The local version of this operator (z=0) has both spin- 1 2 and spin- 3 2 contributions. Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

30 Higher spin The spin- 3 2 components must obey the Rarita-Schwinger equations (p µ u µ = 0, γ µ u µ = 0 ) as well as the Dirac equation Also, P munu must be idempotent ( P 2 = P ) This gives a simple spin- 3 2 projection operator (at p = 0): P µν = δ ij I 1 3 γ iγ j (µ, ν 0) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

31 Spin projection The spin- 1 2 projection operators are : P µν = δ µ0 δ ν0 Equivalent to using χ = ɛ abc [u T a Cγ 0 γ 5 d b ]γ 5 u c at both the source and sink and P µν = 1 3 γ iγ j (µ, ν 0) Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

32 Spin projection The explicit form of the projection operator depends on the basis for the gamma matrices At zero momentum, the projection operator is block-diagonal in the dirac indices (it does not mix upper and lower components). Considering only the upper components and lorentz indices 1,2,3; this gives a 4-dimension eigenspace with eigenvalue 1 (spin-3/2), and a 2-dimension eigenspace with eigenvalue 0 (spin-1/2). Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

33 Spin projection This 4-dimensional space is spanned by the Clebsch-Gordan addition of a spin-1 vector and dirac spinor: ψ(+3/2) = 1 2 { ψ 1 ( ) + iψ 2 ( )} ψ(+1/2) = 1 6 { ψ 1 ( ) + iψ 2 ( ) + 2ψ 3 ( )} ψ( 1/2) = 1 6 {ψ 1 ( ) + iψ 2 ( ) + 2ψ 3 ( )} ψ( 3/2) = 1 2 {ψ 1 ( ) + iψ 2 ( )} Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

34 Spin projection Similarly, the two-dimensional space of the spin- 1 2 component has basis: ψ(+1/2) = 1 3 {ψ 1 ( ) iψ 2 ( ) + ψ 3 ( )} ψ( 1/2) = 1 3 { ψ 1 ( ) iψ 2 ( ) + ψ 3 ( )} Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

35 ψ 1 ( ) (real part) Corresponds to s = 3 2, m s = +3 2, 1 2 and to s = 1 2, m s = 1 2 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

36 ψ 1 ( ) (imaginary part) Corresponds to s = 3 2, m s = +1 2, 3 2 and to s = 1 2, m s = +1 2 Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37

37 Pψ 1 ( ) = P 1ν ψ ν Samuel D. Thomas (CSSM) Wave functions of the Nucleon LHPV, Cairns / 37