First Order Relativistic Three-Body Scattering. Ch. Elster

Transcription

1 First Order Relativistic Three-Body Scattering Ch. Elster T. Lin, W. Polyzou W. Glöcle 4//7 Supported by: U.S. DOE, NERSC,OSC

2 Nucleons: Binding Energy of H NN Model E t [MeV] Nijm I -7.7 Nijm II AV CD-Bonn -8. Experiment Discrepancy in E t : NF Relativistic Effects

3 W.P. Abfalterer et al, PRL 81, 57 (1998)

4 Three Nucleon Observables: Three-Body Forces: Needed to get the binding energies of H and He General practice: Model for N force (TM and Urbana most common) Adjust parameters to fit H Describe bul properties (bound states & cross sections) Reasonably well χpt: up to N 4 LO - N & N forces consistent Relativistic Effects: Bound state: Effect ~.5 MeV & sign under debate Scattering

5 Relativistic Effects at higher energies: What is necessary? N and 4N systems: standard treatment based on pw projected momentum space successful (N scattering up to 5 MeV) but rather tedious N: j max =5, N: J max =5/ `channels Computational maximum today: N: j max =7, N: J max =1/ Solution: NO partial waves

6 Example: NN scattering 1 Elab= MeV: J=16 needed for convergence in cross section Smarter: NO partial waves

7 Roadmap for N problem without PW Scalar NN model Realistic NN Model NN scattering bound state N bound state N bound state NF N scattering: Full Faddeev Calculation Elastic scattering Below and above brea-up Brea-up Relativistic Calculation First Order in t NN scattering deuteron Potentials AV18 and Bonn-B Brea-up in first order: (p,n) charge exchange Max. Energy 5 MeV Relativistic inematics Full Faddeev Calculation NN interactions High energy limits

8 Three-Body Scattering - General Initial channel state r ϕ d Transition operators elastic scattering breaup

9 Three-Body Scattering - General Transition operator for elastic scattering Transition amplitude Brea-up operator 1 U = PG PtGU T U = tp tg PT Faddeev E. = ( 1 P) tgu = (1 P) T Here: Consider Spinless Interaction

10 Faddeev Euation for N Scattering T = tp tgpt NN t-matrix Free N Propagator Nasty Singularity Structure: Moving Singularities Multiple Scattering Series: T = tp tg PtP L 1 st Order in tp P = P 1 P P 1 P Permutation Operator

11 The Faddeev Euation in momentum space by using Jacobi Variables ( ) = = ) ( p p -Body Transition Amplitude (NR) d d d PT tg tp T ϕ ϕ ϕ = ( ) ( ) ( ) ( ) ε ϕ ε ϕ ϕ i E E T i E E t d E t T d m d m m s m s d d = " ˆ "," " " ", p, ˆ ", p, ˆ ˆ p s tˆ symmetrized -body t-matrix 1

12 Variables for D Calculation distinct vectors in the problem: p 5 independent variables: p = p, = x x p = pˆ ˆ, = ˆ ˆ = ( ) ( p) p x system : z system : z Variables invariant under rotation: freedom to choose coordinate system for numerical calculation

13 Elastic Scattering: (nonrelativistic) T φ = tp φ tgpt φ 1 st order rescattering All calculations use a Malfliet-Tjon type potential

14 Comparison with Realistic NN Potential in first order (NR) 495 MeV, θ=18 o

15 Relativistic Faddeev Calculations 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 euations same operator form but different ingredients Kinematics Lorentz transformations between frames Dynamics Baamjian-Thomas Scheme: Mass Operator M=M V Interaction embedded in -body space V M M

16 Relativistic Kinematics: Phase Space Factors ( ) 4 ˆ ) ( ) ( d d d n el U W E E d ϕ ϕ π σ Ω = ) ( 4 m p m W = NR: (m/) Invariant Mass 4 4 ) ( d cm p NR br U me d d d m ϕ φ π σ Ω Ω = 4 ) 4( 4 ) ( ) ( ) ( d u u p d n br U p m p d d d W E E ϕ φ π σ Ω Ω = u 1 p m W m W =

17 Kinematics: Poincaré-Jacobi momenta 1 Nonrelativistic (Galilei) p Relativistic (Lorentz)

18 Kinematics: Poincaré-Jacobi Coordinates ( ) ( ) ( ) ( ) = = p E E E E E E N c.m. frame: 1,, with 1 = K= Kp ) ( ) ( )] ( ) ( )[ ( Kp ) ( p) ( 1 1 E E E E p E K = = Poincarė-Jacobi Coordinates: All expressions related to permutations much more complicated Depend on vector variables => angle dependent

19 Permutation Operator: P=P1PP1P

20 Permutation Operator: P=P 1 P P 1 P Explicit expressions:

21 Kinematic Relativistic Effects: Lorentz transformation Lab c.m. frame (-body) Phase space factors in cross sections Poincarė-Jacobi momenta Permutations

22 Quantum Mechanics Galilei Invariant: K H = h ; h = h V M g NR Poincaré Invariant: H = K M ; M = M V1 V V 1 V ij = M ij M = ( m v ), ij ij m, ij Two-body interaction embedded in the -particle Hilbert space m, ij = m i p ij m j p ij M = m, ij m

23 Two-Body Input: T 1 -operator embedded in -body system T 1 (p', p;) = V (p', p;) d " (E( p')) V (p',";) T (",p;) 1 (E( ")) iε Potential: -body potential in c.m. frame Attempt via spectral expansion in Hard to compute! Kamada, Glöcle, Gola, Elster, PRC (). Comment: wors BUT not well enough New Suggestions Kamada-Glöcle nucl-th/71: Solve for V numerically via iteration --- not tested in N calculation

24 Instead: Obtain fully off-shell matrix elements T 1 (,,W) from half shell transition matrix elements by Solving a 1 st resolvent type euation T 1 (W) = T 1 (W ) T 1 (W) [g (W) - g (W )] T 1 (W ) For every single off-shell momentum point Proposed in Keister & Polyzou, PRC 7, 145 (6) Carried out for the first time now

25 Obtain embedded N t-matrix T 1 (,,W): Solution of the relativistic N LS euation with -body potential

26 Explicit Euation for T1

27 Approximations to the boosted potential relativistic interaction in the c.m. frame Remar to calculations: The relativistic potential v(p,p ) is phase-shift euivalent to the nonrelativistic potential Details on this issue later!!

28 Deuteron Binding Energy

29 Total Cross Section for Elastic Scattering

30 Elastic Scattering: Differential Cross Section

31 Inclusive Scattering Measured: Energy of one ejected particle as function of angle

32 Inclusive Breaup Scattering

33

34 Inclusive Breaup Elab=5 MeV

35 Inclusive Breaup Elab=495 MeV

36 Inclusive Breaup Scattering

37 Exclusive Breaup Scattering

38 Exclusive Breaup Cross Section - QFS Elab = 5 MeV: x = -1, x p = 1, φ p =

39 x = 1 x p = φ p = x = / x p = -.5 φ p = x p = -.5 x = -.9 φ p =

40 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 euivalent Four options: Start from relativistic LS euation (natural option) If non-relativistic LS euation is used: Refit of parameters (maybe time consuming in practice) Transformation of Kamada-Glöcle PRL 8, 547 (1998) Transformation of Coester-Piper-Serdue as given in Polyzou PRC 58, 91 (1998)

41 Kamada-Glöcle (KG) Unitary rescaling of momentum variables to change the nonrelativistic inetic energy into the relativisitic inetic energy: m = m m h ( ) d d ' vr = h( ') ( ') v NR ( ) h( ) Relativistic and nonrelativistic phase shifts are functions of invariant energy E Relativistic and nonrelativistic bound states have identical binding energy.

42 Coester-Piper-Serdue (CPS) (PRC11, 1 (1975)) Add interaction to suare 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 4 v M v u m m u m h mh u M M = = = m t e e m e t NR R ) ( ' ') ( ) ( 4 )) ( ( ' =

43 Total Cross Section Elastic Scattering

44 Inclusive MeV 4 deg 6 deg

45 Inclusive 5 MeV 1 deg 4 deg

46 Relativistic 1 st Order Faddeev Calculations Kinematics Phase space factors Lorentz Transformation from Lab to c.m. frame Above determines shifts of peas Lorentz Transformation of Jacobi Coordinates Always reduces effects of phase space factors Dynamics Exact calculation of the two-body interaction embedded in the three-particle Hilbert space Approximation V uite good up to ~ 5 MeV Observables show some sensitivity to the construction of the phase-shift euivalent relativistic interaction.

47 Roadmap - Summary For higher energies : NO partial waves Solve Faddeev euation for -particle scattering in vector variables Investigate convergence of multiple scattering series as function of energy for different observables and configuarations 1 st Order Relativistic Faddeev in tp Calculations at intermediate energy show that relativistic effects are uite visible. 1 st Order = Born term determines ernel of Faddeev E.