Poincaré Invariant Three-Body Scattering at Intermediate Energies

Transcription

1 Poincaré Invariant Three-Boy Scattering at Intermeiate nergies Ch. lster T. Lin W. Polyzou, W. Glöcle 5/8/9 Supporte by: U.S. DO, OSC, NRSC

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3 A Few-Boy Theorist s view of the Nuclear Chart

4 Boun State: H - He Scattering: lastic Inelastic (Breaup) nergy Scale: ev MeV GeV

5 Challenges in N Physics Test of nuclear forces in the simplest nuclear environment (over a large energy range!) Two-boy forces Genuine three-boy forces Reaction mechanisms xamples: euteron breaup, (p,n) charge exchange, exclusive breaup (specific configurations) Higher nergy: Lorentz vs. Galilean Invariance Chec commonly use approximations (e.g. Glauber approach)

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

7 Relativistic ffects at Higher nergies Computational Challenge: N an 4N systems: Solution: stanar treatment base on pw projecte momentum space successful (N scattering up to 5 MeV) but rather teious N: j max 5, N: J max 5/ `channels Computational maximum toay: N: j max 7, N: J max 1/ NO partial wave ecomposition of basis states

8 Roamap for N problem without PW Scalar NN moel Realistic NN Moel NN scattering boun state N boun state N boun state NF N scattering: Full Faeev Calculation lastic scattering Below an above brea-up Brea-up Poincarė Invariant Faeev Calculations NN scattering euteron Potentials AV18 an Bonn-B Brea-up in first orer: (p,n) charge exchange Max. nergy 5 MeV Lorentz inematics xact Faeev Calculation NN interactions High energy limits

9 Three-Boy Scattering - General Transition operator for elastic scattering U PG -1 PT Transition operator for breaup scattering U (1 P)T T tp tg PT Faeev euation (Multiple Scattering Series) U U T tp tg PtP L 1 st Orer in tp t v vg t : NN t-matrix P P 1 P P 1 P Permutation Operator

10 The Faeev uation in momentum space by using Jacobi Variables ( ) ) ( 1 1 p 1 p -Boy Transition Amplitue (NR) PT tg tp T ϕ ϕ ϕ ( ) ( ) ( ) ( ) ε ϕ ε ϕ ϕ i T i t t T m m m s m s " ˆ "," " " ", p, ˆ ", p, ˆ ˆ p s tˆ symmetrize -boy t-matrix 1

11 Variables for D Calculation istinct vectors in the problem: p 5 inepenent variables: p p, x x p pˆ ˆ, ˆ ˆ ( ) ( p) p x system : z system : z Variables invariant uner rotation: freeom to choose coorinate system for numerical calculation

12 Relativistic Faeev Calculations Context: Poincarė Invariant Quantum Mechanics Poincarė invariance is exact symmetry, realize by a unitary representation of the Poincarė group on a fewparticle Hilbert space Instant form Faeev euations same operator form but ifferent ingreients Kinematics Lorentz transformations between frames Dynamics Baamjian-Thomas Scheme: Mass Operator MM V replaces Hamiltonian HH v Connect Galilean two-boy v with Poincarė two-boy v Construct V : M M

13 Lorentz Kinematics: Phase Space Factors ( ) 4 ˆ ) ( ) ( n el U W ϕ ϕ π σ Ω ) ( 4 m p m W NR: (m/) Invariant Mass 4 4 ) ( cm p NR br U m m ϕ φ π σ Ω Ω 4 ) 4( 4 ) ( ) ( ) ( u u p n br U p m p W ϕ φ π σ Ω Ω u 1 p m W m W

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

15 Kinematics: Poincaré-Jacobi Coorinates ( ) ( ) ( ) ( ) p N c.m. frame: 1,, with 1 K Kp ) ( ) ( )] ( ) ( )[ ( Kp ) ( p) ( 1 1 p K Poincarė-Jacobi Coorinates: All expressions relate to permutations much more complicate Depen on vector variables > angle epenent

16 Permutation Operator: PP1PP1P

17 Relativistic inematics: IA (1 st orer) T tp U PG 1 PT Lorentz transformation Lab c.m. frame) (-boy) Phase space factors in cross sections Poincarė-Jacobi momenta Permutations

18 Quantum Mechanics Galilei Invariant: K NR NR NR H h ; h h v1 v1 v M g Poincaré Invariant: H K M ; M M V1 V V 1 V ij M ij M ( m v ), ij ij m, ij Two-boy interaction embee in the -particle Hilbert space m, ij m i p ij m j p ij M m, ij m

19 V ij embee in the -particle Hilbert space ij ij ( m v ) m V M M, ij ij, ij nee matrix elements :

20 Two-Boy Input: T1-operator embee in -boy system T 1 (p', p; ) V (p', p; ) " ( ( V (p', " p ' )) ; ) T 1 (", p; ) ( ( " )) iε Do not solve for V! Obtain fully off-shell matrix elements T 1 (,,) from half shell transition matrix elements by Solving a 1 st resolvent type euation: T 1 () T 1 ( ) T 1 () [g () - g ( )] T 1 ( ) For every single off-shell momentum point Propose in Keister & Polyzou, PRC 7, 145 (6) Carrie out for the first time here [PRC 76, 1141 (7)]

21 Obtain embee N t-matrix T 1 (,,z ) halfshell in -boy c.m. frame first : Solution of the relativistic N LS euation with -boy potential

22 Consieration for two-boy t-matrix Relativistic an non-relativistic t-matrix shoul give ientical observables for etermining relativistic effects Or two-boy t-matrices shoul be phase-shift euivalent Four options: Start from relativistic LS euation natural option employe for NN interactions fit to 1 GeV If non-relativistic LS euation is use: Refit of parameters (maybe time consuming in practice) Transformation of Kamaa-Glöcle PRL 8, 547 (1998) Transformation of Coester-Piper-Serue as given in Polyzou PRC 58, 91 (1998)

23 Phase euivalent -boy t-matrices: Coester-Pieper-Serue (CPS) (PRC11, 1 (1975)) A interaction to suare of non-interacting mass operator NO nee to evaluate v irectly, since M, M, h have the same eigenstates Relation between half-shell t-matrices Relativistic an nonrelativistic cross sections are ientical 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 )) ( ( '

24 Total Cross Section for lastic Scattering: 1 st Orer T t P

25 Unitarity Relation ( ) ( ) φ φ ϑ π φ φ φ φ ϑ π φ φ φ φ φ φ 1 * * ' p ' ' ' U i U U i U U U p ( ) ( ) br el tot n U W σ σ σ ϕ ϕ π,1, Im 16 All calculations use a Malfliet-Tjon type potential

26 Total Cross Section an Unitarity Relation σ tot σ el σ br

27 Faeev uation as multiple scattering series T tp tgpt T tp tg PtP L 1 st Orer or IA

28 Convergence of the Faeev Multiple Scattering Series lab [GeV]

29 Convergence of the Faeev Multiple Scattering Series

30 lastic Scattering: Differential Cross Section

31 Breaup Scattering xclusive: Measure energy & angles of two ejecte particles V.Punjabi et al. PRC 8, 78 (1998) TRIUMF 58 MeV Outgoing protons are measure in the scattering plane

32 xclusive Breaup Scattering lab 58 MeV (symmetric configuration) (V.Punjabi et al. PRC 8, 78 (1998) QFS

33 xclusive Breaup Scattering lab 58 MeV (asymmetric configuration) QFS

34 xclusive Breaup Scattering lab 58 MeV QFS

35 xclusive Breaup Scattering Space-Star lab 58 MeV

36 xclusive Breaup Scattering : Coplanar Star QFS lab 58 MeV [MeV] [MeV]

37 Relevance of Stuy with Moel Interaction

38 Calcula tion: Henry Witala MeV

39 Results for Triton Bining nergy (FB ) 5-Channel Calculation CPS KG

40 Triton Bining nergy with CD-Bonn (arxiv:81.148) NR R Δ 5-ch (s-wave) ch (jm) ch (jm) ch (jm4) ch npnn ch (npnnwigner)

41 Computational uipment IBM Cluster P AMD Opteron ( TFlop) Jacuar: 56 P Opteron Cluster 56 P Itanium Cluster

42 Poincaré Invariant Faeev Calculations Kinematics Phase space factors Lorentz Transformation from Lab to c.m. frame Lorentz Transformation of Jacobi Coorinates Always reuces effects of phase-space factors Kinematics etermines pea positions in brea-up observables Dynamics xact calculation of the two-boy interaction embee in the three-particle Hilbert space The ynamic effects act in general opposite inematic effects

43 Poincaré Invariant Faeev Calculations Carrie out up to GeV for elastic an breaup scattering Solve Faeev euation in vector variables NO partial waves Relativistic effects are important at 5 MeV an higher Relativistic total elastic cross section increases up to 1% compare to the non-relativistic Relativistic inematics etermines QFS pea positions in inclusive an exclusive breaup Breaup: Relativistic effects very large epenent on configuration Above 8 MeV projectile energy: multiple scattering series converges after ~ iterations In breaup QFS conitions 1 st orer calculations sufficient

44 Poincaré Invariant Faeev Calculations Triton calculations: Difference in bining energy between relativistic an nonrelativistic calculation is.1 MeV Provie the CPS realization of a relativistic interaction is use. CPS is in a Hamiltonian context the correct way Future Systematic stuies of selecte cross sections & high energy limits Triton: Question about consistent inclusion of NF Long term: inclue Spin

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