Relativity and Three-Body Systems

Transcription

1 Relativity and Three-Body Systems Ch. Elster T. Lin W. Polyzou, W. Glöcle 6/17/008 Supported by: U.S. DOE, OSC, NERSC

2 A Few-Body Theorist s view of the Nuclear Chart

3 Relativistic Three-Body Problem Context: Poincarė Invariant Quantum Mechanics Poincarė invariance is exact symmetry, realized by a unitary representation of the Poincarė group on a fewparticle Hilbert space Instant form Faddeev equations same operator form but different ingredients Kinematics Lorentz transformations between frames Dynamics Baamjian-Thomas Scheme: Mass Operator M=M 0 V replaces Hamiltonian H=H 0 v Connect Galilean two-body v with Poincarė two-body v Construct V := M q M 0 q

4 Three-Body Scattering Transition operator for elastic scattering U = PG -1 0 PT Transition operator for breaup scattering U 0 = (1 P)T U U 0 Faddeev equation (Multiple Scattering Series) T 0 = tp tg PtP L 1 st Order in tp t = v vg 0 t =: NN t-matrix P = P 1 P 3 P 13 P 3 Permutation Operator

5 Kinematic Relativistic Ingredients: Lorentz transformation Lab c.m. frame (3-body) Phase space factors in cross sections Poincarė-Jacobi momenta Permutations for identical particles

6 Kinematics: Poincaré-Jacobi momenta Nonrelativistic (Galilei) 1 q 3 p Relativistic (Lorentz) 1 ( Kpq) 1 3 = Kpq ( ) 3 0

7 Relativistic inematics: IA (1 st order) T = tp Lorentz transformation Lab c.m. frame) (3-body) Phase space factors in cross sections Poincarė-Jacobi momenta Permutations

8 Quantum Mechanics Galilei Invariant: K NR NR NR H = h ; h = h0 v1 v13 v3 M g Poincaré Invariant: H = K M ; M = M 0 V1 V3 V 31 V ij = M ij M 0 = ( m v ) 0, ij ij q m 0, ij q Two-body interaction embedded in the 3-particle Hilbert space m 0, ij = m i p ij m j p ij M 0 = m 0, ij q m q

9 V ij embedded in the 3-particle Hilbert space ij ij ( m v ) q m q V = M M = 0 0, ij ij 0, ij need matrix elements :

10 Two-Body Input: T1-operator embedded in 3-body system T 1 (p', p; q) = V (p', p; q) d 3 " ( E ( V (p', " p ' )) q ; q) T 1 (", p; q) ( E ( " )) q iε Do not solve for V! Obtain fully off-shell matrix elements T 1 (,,q) from half shell transition matrix elements by Solving a 1 st resolvent type equation: T 1 (q) = T 1 (q ) T 1 (q) [g 0 (q) - g 0 (q )] T 1 (q ) For every single off-shell momentum point Proposed in Keister & Polyzou, PRC 73, (006) Carried out for the first time here [PRC 76, (007)]

11 Obtain embedded N t-matrix T 1 (,,z ) halfshell in -body c.m. frame first : Solution of the relativistic N LS equation with -body potential

12 Consideration for two-body t-matrix Relativistic and non-relativistic t-matrix should give identical observables for determining relativistic effects Or two-body t-matrices should be phase-shift equivalent Four options: Start from relativistic LS equation natural option employed for NN interactions fit to 1 GeV If non-relativistic LS equation is used: Refit of parameters (maybe time consuming in practice) Transformation of Kamada-Glöcle PRL 80, 547 (1998) Transformation of Coester-Piper-Serdue as given in Polyzou PRC 58, 91 (1998)

13 Phase equivalent -body t-matrices: Coester-Pieper-Serdue (CPS) (PRC11, 1 (1975)) Add interaction to square of non-interacting mass operator NO need to evaluate v directly, since M, M, h have the same eigenstates Relation between half-shell t-matrices Relativistic and nonrelativistic cross sections are identical functions of the invariant momentum { }, 4 with v M v u m m u m h mh u M M = = = m t e e m e t NR R ) ( ' ') ( ) ( 4 )) ( ( ' =

14 Total Cross Section for Elastic Scattering: 1 st Order T= t P

15 Faddeev Equation as Multiple Scattering Series T = tp tg0pt T 0 = tp tg PtP L 1 st Order or IA

16 Convergence of the Faddeev Multiple Scattering Series E lab [GeV]

17 Elastic Scattering: Differential Cross Section

18 Differential Cross Section: Convergence of the Faddeev Multiple Scattering Series

19 Breaup Scattering Exclusive: Measure energy & angles of two ejected particles V.Punjabi et al. PRC 38, 78 (1998) TRIUMF 508 MeV Outgoing protons are measured in the scattering plane

20 Exclusive Breaup Scattering E lab = 508 MeV (symmetric configuration) (V.Punjabi et al. PRC 38, 78 (1998) QFS

21 Exclusive Breaup Scattering E lab = 508 MeV (asymmetric configuration) QFS

22 Exclusive Breaup Scattering E lab = 508 MeV QFS

23 Exclusive Breaup Scattering Space-Star E lab = 508 MeV

24 Poincaré Invariant Faddeev Calculations Kinematics Phase space factors Lorentz Transformation from Lab to c.m. frame Lorentz Transformation of Jacobi Coordinates Always reduces effects of phase-space factors Kinematics determines pea positions in brea-up observables Dynamics Exact calculation of the two-body interaction embedded in the three-particle Hilbert space The dynamic effects act in general opposite inematic effects

25 Poincaré Invariant Faddeev Calculations Carried out up to GeV for elastic and breaup scattering Solved Faddeev equation in vector variables = NO partial waves Relativistic effects are important at 500 MeV and higher Relativistic total elastic cross section increases up to 10% compared to the non-relativistic Relativistic inematics determines QFS pea positions in inclusive and exclusive breaup Breaup: Relativistic effects very large dependent on configuration Above 800 MeV projectile energy: multiple scattering series converges after ~ iterations Future Systematic studies of selected cross sections & high energy limits Long term: include Spin