Poincaré Invariant Three-Body Scattering

Transcription

1 Poincaré Invariant Three-Body Scattering Ch. Elster T. Lin W. Polyzou, W. Glöckle 10/9/008 Supported by: U.S. DOE, OSC, NERSC

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

3 Bound State: 3 H - 3 He Scattering: Elastic Inelastic (Breakup) Energy Scale: kev MeV GeV

4 Challenges in 3N Physics Test of nuclear forces in the simplest nuclear environment (over a large energy range!) Two-body forces Genuine three-body forces Reaction mechanisms Example: deuteron breakup, (p,n) charge exchange. Higher Energy: Lorentz vs. Galilean Invariance Check of commonly used approximations (Glauber) Three-body decays of particles (e.g. η or η )

5 Relativistic Effects at Higher Energies Computational Challenge: 3N and 4N systems: Solution: standard treatment based on pw projected momentum space successful (3N scattering up to 50 MeV) but rather tedious N: j max =5, 3N: J max =5/ 00 `channels Computational maximum today: N: j max =7, 3N: J max =31/ NO partial wave decomposition of basis states

6 Roadmap for 3N problem without PW Scalar NN model Realistic NN Model NN scattering bound state 3N bound state 3N bound state 3NF 3N scattering: Full Faddeev Calculation Elastic scattering Below and above break-up Break-up Poincarė Invariant Faddeev Calculations NN scattering deuteron Potentials AV18 and Bonn-B Break-up in first order: (p,n) charge exchange Max. Energy 500 MeV Lorentz kinematics Exact Faddeev Calculation NN interactions High energy limits

7 Variables for 3D Scattering Calculation 3 distinct vectors in the problem: q 0 q p 5 independent variables: p = p, q = q x x p = pˆ qˆ, = qˆ qˆ 0 = (q q) (q p) q pq 0 0 x q 0 0 q system : z q q 0 system : z q 0 Numerical calculation: Three-body transition amplitude is a function of 5 variables

8 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 Bakamjian-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

9 Three-Body Scattering Transition operator for elastic scattering U = PG -1 0 PT Transition operator for breakup 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

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

11 Kinematics: Poincaré-Jacobi momenta Nonrelativistic (Galilei) 1 q 3 p Relativistic (Lorentz) 1 ( Kpq) k1k k3 = Kpq (k k ) 3 0

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

13 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 k m 0, ij q k 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 k m k q k

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

15 Two-Body Input: T1-operator embedded in 3-body system T 1 (p', p; q) = V (p', p; q) d 3 k " ( E ( V (p', k" p ' )) q ; q) T 1 (k", p; q) ( E ( k " )) q iε Do not solve for V! Obtain fully off-shell matrix elements T 1 (k,k,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)]

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

17 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-Glöckle PRL 80, 547 (1998) Transformation of Coester-Piper-Serduke as given in Polyzou PRC 58, 91 (1998)

18 Phase equivalent -body t-matrices: Coester-Pieper-Serduke (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 k { }, 4 with v M v u m m u m k h mh u M M = = = k m k t k k e k e m k k e t k NR R ) ( ' ') ( ) ( 4 )) ( ( ' =

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

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

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

22 Elastic Scattering: Differential Cross Section

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

24 Breakup 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

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

26 Exclusive Breakup Scattering E lab = 508 MeV (asymmetric configuration) QFS

27 Exclusive Breakup Scattering E lab = 508 MeV QFS

28 Exclusive Breakup Scattering Space-Star E lab = 508 MeV

29 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 peak positions in break-up observables Dynamics Exact calculation of the two-body interaction embedded in the three-particle Hilbert space The dynamic effects act in general opposite kinematic effects

30 Poincaré Invariant Faddeev Calculations Carried out up to GeV for elastic and breakup 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 kinematics determines QFS peak positions in inclusive and exclusive breakup Breakup: 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