Elastic and quasi-elastic electron scattering off nuclei with neutron excess

Transcript

1 Elastic and quasi-elastic electron scattering off nuclei with neutron excess Matteo Vorabbi Dipartimento di Fisica - Università di Pavia INFN - Sezione di Pavia tranto May 31 Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 1 / 27

2 Introduction Relativistic model for ground-state observables Elastic electron scattering Inclusive quasi-elastic electron scattering Parity violating asymmetry Numerical results Collaborators: C. Giusti, A. Meucci, P. Finelli, F. D. Pacati Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 2 / 27

3 Relativistic model for ground-state observables Lagrangian L = ψ(iγ µ µ M)ψ µ σ µ σ U(σ) g σ σ ψψ 1 4 Ωµν Ω µν m2 ω ωµ ω µ g ω ω µ ψγ µ ψ 1 4 R µν Rµν m2 ρ ρ µ ρ µ g ρ ρ µ ψγ µ τψ 1 4 Fµν µ F µν ea µ ψγ (1 τ 3) ψ 2 Nonlinear sigma potential U(σ) = 1 2 m2 σ g 2σ g 3σ 4 Field tensors Ω µν = µ ω ν ν ω µ R µν = µ ρ ν ν ρ µ g ρ ( ρ µ ρ ν ) F µν = µ A ν ν A µ Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 3 / 27

4 Relativistic model for ground-state observables Gap parameter: for open shell nuclei is employed the constant gap approximation with empirical (5) (N) = 1 [ ] M(N+ 2) 4M(N+ 1)+6M(N) 4M(N 1)+M(N 2) 8 Five different isotopes d 5 2 1p 12 1p 32 1s 12 Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 4 / 27

5 Elastic electron scattering - Plane-Wave Born Approximation ne photon exchange Neglect the effect of the nuclear Coulomb field on incoming and outgoing electrons Differential cross section ( ) dσ dω = σ M F p (q) 2 EL Charge form factor F p (q) = dr j (qr)ρ p (r) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 5 / 27

6 Inclusive Quasi-Elastic electron scattering Inclusive differential cross section ( ) dσ dε dω = σ M [v L R L + v T R T ] Coefficients v L = QE ( Q 2 ) 2 q 2 v T = tan 2 θ 2 Q2 2 q 2 Q 2 = q 2 ω 2 Longitudinal and transverse response functions R L (q,ω) = W (q,ω) Hadron tensor W µµ (q,ω) = i f R T (q,ω) = W 11 (q,ω)+w 22 (q,ω) Ψf Ĵµ (q) Ψ 2 δ(e +ω E f ) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 6 / 27

7 Relativistic Green s function model Equivalent expression for the hadron tensor W µµ (q,ω) = 1 π Im Ψ J µ (q)g(e f )J µ (q) Ψ Final expression for the hadron tensor [ W µµ (q,ω) = n Re T µµ n (E f ε n, E f ε n ) 1 π P 1 de M E f ε n E T n µµ (E, E) = λ n ϕ n j µ (q) Im Tµµ n (E, E f ε n ) 1 V (E) χ ( ) E (E) χ ( ) E (E) 1 V (E)j µ (q) ϕ n ] Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 7 / 27

8 Parity-violating asymmetry - Elastic scattering Dirac equation [α p+u ± (r)]ψ ± = EΨ ± Total potential U(r) ± = V(r)±γ 5 A(r) Axial potential A(r) = G F 2 2 ρ W(r) Weak charge density [ ] ρ W (r) = dr G E ( r r ) ρ n (r )+(1 4sin 2 Θ W )ρ p (r ) Electric form factor G E (r) Λ3 8π e Λr Parity-violating asymmetry in Born approximation Λ = 4.27 fm 1 A pv = dσ + dω dσ dω dσ + dω + dσ dω = G F Q 2 [ 4 4 sin 2 Θ W 1+ F ] n(q) 2 πα F p (q) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 8 / 27

9 Numerical Results xygen neutron and proton densities ρ n (fm -3 ) ρ p (fm -3 ) r (fm) r (fm) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 9 / 27

10 Numerical Results lcium neutron and proton densities ρ n (fm -3 ) ρ p (fm -3 ) r (fm) r (fm) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 1 / 27

11 Numerical Results - Experimental elastic differential cross sections Electron energy: ε = MeV for, ε = MeV for 4 σ EL = σ M F p (q) 2 dσ/dω [mb/sr] data data θ [deg] Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

12 Numerical Results xygen elastic differential cross sections at ε = MeV dσ/dω [mb/sr] θ [deg] Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

13 Numerical Results lcium elastic differential cross sections at ε = MeV dσ/dω [mb/sr] θ [deg] 36 Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

14 Numerical Results Experimental QE differential cross sections σ [nb/(mev sr)] (a) data RGF ω [MeV] (e,e ) ( ) dσ dε dω (e, e ) QE ε = 18 MeV θ = 32 o = σ M [v L R L + v T R T ] σ [nb/(mev sr)] (e,e ) (b) 4 data RGF 4 (e, e ) ε = 841 MeV θ = 45.5 o ω [MeV] Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 / 27

15 Numerical Results xygen inclusive QE differential RPWIA cross sections σ [nb/(mev sr)] (a) RPWIA ω [MeV] 28 (e, e ) ε = 18 MeV θ = 32 o σ [nb/(mev sr)] (b) proton proton 18 proton 22 proton 28 proton neutron neutron 18 neutron 22 neutron 28 neutron RPWIA ω [MeV] Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

16 Numerical Results xygen inclusive QE differential RGF cross sections at ε = 18 MeV, θ = 32 o σ [nb/(mev sr)] ω [MeV] RGF Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 / 27

17 Numerical Results lcium inclusive QE differential RPWIA cross sections at ε = 56 MeV, θ = 6 o σ [nb/(mev sr)] ω [MeV] RPWIA Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

18 Numerical Results lcium inclusive QE differential RGF cross sections at ε = 56 MeV, θ = 6 o σ [nb/(mev sr)] ω [MeV] RGF Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

19 Numerical Results xygen parity violating asymmetry A pv MeV 1 2 θ (deg) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

20 Numerical Results lcium parity violating asymmetry A pv MeV 1 2 θ (deg) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May 31 2 / 27

21 Numerical Results First minimum positions of A pv θ min (deg) xygen lcium 85 MeV = (N- Z)/Z Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

22 Numerical Results Neutron density and A pv for 22 and 24 : possible candidates for bubble nuclei, nuclear systems with a strong depleted central density. ρ n (fm -3 ) (no gap) A pv 85 MeV r (fm) θ (deg) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

23 Numerical Results - P-REX Collaboration Physical Review C 85, 3251(R) (212) Physical Review Lett. 18, (212) Asymmetry averaged over the acceptance A pv = dθ sinθapv(θ) dσ dω ǫ(θ) dθ sinθ dσ dω ǫ(θ) Exp. Result: A Pb pv =.656 ±.6(stat) ±.(syst) ppm Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

24 Numerical Results - Weak charge density and A pv ( 28 Pb) -ρ Weak (fm -3 ).1.5 ρ p ε = 1.63 GeV 28 Pb A pv <A pv > =.712 ppm 5 1 r (fm) 5 1 θ (deg) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

25 Numerical Results - Parity violating asymmetry ( 48 ) Proposal to Jefferson Lab C-REX: Parity-violating measurement of the weak charge distribution of 48 at 2.2 GeV ε = 2.2 GeV A pv θ (deg) Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

26 Conclusions Elastic and quasi-elastic electron scattering give information on the global properties of nuclei and on the single particle aspects of the nucleus Parity-violating asymmetry parameter investigates the neutron skin Good agreement with PREX measurement on 28 Pb Prediction for the future experiment CREX on 48 Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27

27 Relativistic model for ground-state observables - Couplings Couplings of σ and ω mesons to the nucleon Coupling of ρ to the nucleon g i (ρ) = g i (ρ sat )f i (x) for Bare values for ω and ρ masses i = σ,ω 1+b i (x + d i ) 2 f i (x) = a i 1+c i (x + d i ) 2 x = ρ ρ sat =.152 fm 3 ρ sat g ρ (ρ) = g ρ (ρ sat ) exp[ a ρ (x 1)] m ω = 783 MeV m ρ = 763 MeV Matteo Vorabbi (Università di Pavia) Elastic and quasi-elastic scattering tranto May / 27