Jia-Jun Wu. Collaborate: Derek B. Leinweber, Ross D. Young, and James M. Zanotti. The University of Adelaide CSSM

Transcription

1 Jia-Jun Wu Collaborate: Derek B. Leinweber, Ross D. Young, and James M. Zanotti The University of Adelaide CSSM

2 Outline Motivation New method Pion-Pion with I= Summary and Outlook

3 Motivation : How important of High partial wave state Regge Trajectory High momentum cross section σ ~ p l1 Nucleon-Nuclei scattering P wave phase shifts are the key of A y -puzzle in nucleon-deuteron scattering.

4 Motivation : How to get High partial wave state in the finite volume?? Symmetry SO(3) L p = 0, 1 -,, 3 -, O h A 1, A, E, T 1, T

5 Motivation : Infinite Finite SO(3) O h 0 A T - 1 E T 3 - A - T 1- T - 4 A 1 E T 1 T 5 - E - T 1- T - 6 A 1 A E T 1 T 7 - A - E - T 1- T - 8 A 1 E T 1 T 9 - A 1- A - E - 3T 1- T - O h A 1 A E T 1 T A - 1 A - E - T - 1 T - SO(3) 0, 4, 8,. 6,., 4, 6, 8,. 4, 6, 8,., 4, 6, 8,. 9 -,. 3 -, 7 -, 9-,. 5 -, 7 -, 9-,. 1 -, 3 -, 5-, 7 -, 9-,. 3 -, 5-, 7 -, 9-,.

6 Motivation : Infinite Finite SO(3) O h 0 A T - 1 E T 3 - A - T 1- T - 4 A 1 E T 1 T 5 - E - T 1- T - 6 A 1 A E T 1 T 7 - A - E - T 1- T - 8 A 1 E T 1 T 9 - A 1- A - E - 3T 1- T - O h A 1 A E T 1 T A - 1 A - E - T - 1 T - SO(3) 0, 4, 8,. 6,., 4, 6, 8,. 4, 6, 8,., 4, 6, 8,. 9 -,. 3 -, 7 -, 9-,. 5 -, 7 -, 9-,. 1 -, 3 -, 5-, 7 -, 9-,. 3 -, 5-, 7 -, 9-,. For the High Partial Wave scattering amplitudes, we need the spectra of various irreducible representations.

7 New Method: Rotation of O h Question: How to express the rotation elements of O h group? There are 48 rotation elements which can connect 48 vectors in the coordinate space. R Rotation Axis Rotation Angle (x 1,x,x 3 ) R 1 Any 0 o (x 1,x,x 3 ) R (1,1,1) -10 o (x,x 3,x 1 ) R 4 (0,0,1) -180 o (-x 1,-x,x 3 ) R 5 = πr (-x 1,-x,-x 3 ) R 6 = πr - - (-x,-x 3,-x 1 ) R 48 = πr (x 1,x,-x 3 ) T. Luu and M. J. Savage, PRD (011)

8 New Method: Two body Operator x A δ/ Rotation Elements: R = {R 1, R,, R 48 } B A R 1 δ/ R 1 x B There are 48 different two particles operator, ϕ R.

9 New Method: Two body Operator x A δ/ Rotation Elements: R = {R 1, R,, R 48 } B A R 1 δ/ R 1 x B There are 48 different two particles operator, ϕ R.

10 New Method: Two body Operator x A δ/ Rotation Elements: R = {R 1, R,, R 48 } B A R 1 δ/ Simplifies continuum rotation symmetry R 1 x B There are 48 different two particles operator, ϕ R.

11 New Method: Basis State (1) Matrix Γ reg (R X ) is one of the matrices of representation of O h group, and it is matrix. Obviously, there are 48 matrices of Γ reg corresponding to 48 rotation elements of O h group. The representation matrices Γ reg use all the elements of the group as multipliers, and the dimension of it equal to the order of the group. This representation is called regular representation. Regular representation Γ reg is a reducible representation, it can be reduced to several irreducible representations through similarity transformation: S 1 Γ reg (R X ) S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T

12 New Method: Basis State () S 1 Γ reg S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T 48 = This block-diagonal matrices will lead to a new set of basis states: e.g. i: 1, Γ: E n: 1, i: 1,, 3 Γ: T n: 1,, 3 Φ i,γ,n δ

13 New Method: Basis State () S 1 Γ reg S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T 48 = This block-diagonal matrices will lead to a new set of basis states: e.g. i: 1, Γ: E n: 1, i: 1,, 3 Γ: T n: 1,, 3 Φ i,γ,n δ φ R δ Φ i,γ,n δ

14 New Method: Correlation Function(1) The correlation function is: 48 48

15 New Method: Correlation Function(1) The correlation function is: 48 48

16 New Method: Correlation Function(1) The correlation function is: E, π 8C 3, 8C 3 π 3C, 3C π 6C 4, 6C 4 π 6C, 6C π The operators in the same class has the same value of correlation function

17 New Method: Correlation Function(1) The correlation function is: E, π 8C 3, 8C 3 π 3C, 3C π 6C 4, 6C 4 π 6C, 6C π The operators in the same class has the same value of correlation function

18 New Method: Correlation Function() The correlation function is:

19 New Method: Correlation Function() The correlation function is:

20 New Method: Correlation Function() The correlation function is: Diagonal and Just rely on Γ Character Number of Γ h and l Γ are the orders of Oh group and irreducible representation Γ.

21 New Method: Correlation Function() The correlation function is: Diagonal and Just rely on Γ Character Number of Γ Shell Sink Two source locations as x δ and x δ h and l Γ are the orders of Oh group and irreducible representation Γ.

22 New Method: Boost Frame O h Subgroup D 4 Irreducible Representations of D4: A1, A, B1, B, E R Rotation Axis Rotation Angle (x 1,x,x 3 ) R 1 Any 0 o (x 1, x, x 3 ) R 14 (0,0,1) -90 o (x, -x 1, x 3 ) R 15 (0,0,1) 90 o (-x, x 1, x 3 ) R 4 (0,0,1) -180 o (-x 1, -x, x 3 ) R 4 = πr 18 (1,1,0) -180 o (-x, -x 1, x 3 ) R 43 = πr 19 (1,-1,0) -180 o (x, x 1, x 3 ) R 46 = πr (1,0,0) -180 o (-x 1, x, x 3 ) R 47 = πr 3 (0,1,0) -180 o (x 1, -x, x 3 )

23 Pion-Pion with I=: Correlation function π - L = 0,, 4,, and here we neglect L > 4 π - E, π 8C 3, 8C 3 π 3C, 3C π 6C 4, 6C 4 π 6C, 6C π L A , 4 E , 4 T T , 4 Thick lines:`uu Thin lines: `dd

24 Pion-Pion with I=: Spectra The result of with a = 0.07 fm, m π = 900 MeV 376 configurations and 16 source locations p A1 L π p E L π p T L π = ~0 = ~1 = ~

25 Pion-Pion with I=: Lüscher Equation Γ (i,j) (s, C i,j,sγ ) A 1 (1,1) (0,1) But only 3 equations, but 6 different phase shifts, δ 0 p A1, δ 4 p A1, δ p E, δ 4 p E, δ p T, δ 4 p T. M. Lüscher NPB (1991) (1,) (4, ) (,) (0,1), (4, 34 ), (6, (8, 560 ) 143 E (1,1) (0,1), (4, 18 ) 7 11 ), (1,) (4, ),(6, 30 3 ) 11 (,) (4, ),(6, 64), 11 (8, 39 ) 143 T 1 (1,1) (0,1), (4, 1 7 ) (1,) (4, ),(6, ) (,) (0,1), (4, ), (6,0 11 )

26 Pion-Pion with I=: Toy Model By fitting experimental data up to p = 600 MeV, for L=0,, 4 partial wave, the phase shifts can be expressed as W. Hoogland, ect NPB (1977) D. H. Cohen, ect PRD (1973) A. Zieminski, ect NPB (1974) N. Durusa, ect PLB (1973) J. J. Dudek, ect PRD (01)

27 Pion-Pion with I=: Toy Model By fitting experimental data up to p = 600 MeV, for L=0,, 4 partial wave, the phase shifts can be expressed as W. Hoogland, ect NPB (1977) D. H. Cohen, ect PRD (1973) A. Zieminski, ect NPB (1974) N. Durusa, ect PLB (1973) J. J. Dudek, ect PRD (01) δ 0 p A1, a, a 4

28 Pion-Pion with I=: Phase Shift

29 Results for PL/p=(0,0,1) case E {R 14, R 15 } {R 4, R 43 } {R 46, R 47 } {R 4 } L A , 4 E 0 0 0, 4 B , 4 (0,0,0) (0,0,1) (0,-1,0) (0,1,1) (0,-1,0) (0,1,1)

30 Summary In the coordinate space, we develop a new method to extract spectra of irreducible representation in the finite volume. It just cost two source inversions per measurement. And it can generate all spectra of irreducible representation for any boost momentum. We apply this method for ππ system with I= in the rest frame. It successfully generates signals of ground state of A1, E, and T representation. Then we can get phase shifts through Lüscher equation and toy model.

31 Outlook For the excited states in the each irreducible representation, we need to develop more two particle operators and use variational method. Operators basis distinguished by different δ.

32 Thanks for attention

33 New Method: Basis State () φ R x, δ, t Ω = φ R x, δ, t ~ φ R δ R X φ R δ = φ RX R δ = φ R δ R [Γ reg (R X )] R R Γ reg is regular representation. It can reduced to several irreducible representations : S 1 Γ reg S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T This block-diagonal matrices will lead to a new set of basis states: Φ i,γ,n δ i: 1, Γ: E n: 1, i: 1,, 3 Γ: T n: 1,, 3 R X Φ i,γ,n δ = Φ i,γ,n δ [S 1 Γ reg (R X ) i,γ,n S] i Γ n, i Γ n

34 New Method: Basis State (3) φ R x, δ, t Ω = φ R x, δ, t ~ φ R δ R X φ R δ = φ RX R δ = φ R δ R [Γ reg (R X )] R R Γ reg is regular representation. It can reduced to several irreducible representations : S 1 Γ reg S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T This reduced matrix will lead new basic state: R X Φ i,γ,n δ = Φ i,γ,n δ [S 1 Γ reg (R X ) i,γ,n S] i Γ n, i Γ n φ R δ Φ i,γ,n δ = φ R δ [ R S] R, i Γ n Φ i,γ,n δ

35 New Method: Basis State (4) Summary, now we have two sets of basis states, related by the transformation matrix S. Φ i,γ,n δ = φ R δ [ R S] R, i Γ n The representation matrix of these states are satisfied: R X φ R δ = φ R δ R [Γ reg (R X )] R R R X Φ i,γ,n δ = Φ i,γ,n δ [S 1 Γ reg (R X ) i,γ,n S 1 Γ reg S = A 1 A E 3T 1 3T A 1 A E 3T 1 3T We can make matrix S as an unitary matrix, S = S -1. Then the normalization is satisfied: S] i Γ n, i Γ n

36 New Method: Correlation Function(1) The correlation function is: