Electron Sca ering off 4 He

Transcription

1 Electron Sca ering o He Sonia Bacca CAADA'S ATIOAL LABORATORY FOR PARTICLE AD UCLEAR PHYSICS Owned and operated as a joint venture by a consortium o Canadian universities via a contribution through the ational Research Council Canada. LABORATOIRE ATIOAL CAADIE POUR LE RECHERCHE E PHYSIQUE UCLEAIRE ET E PHYSIQUE DES PARTICULES Propriété d'un consortium d'universités canadiennes, géré en co-entreprise à partir d'une contribution administrée par le Conseil national de recherches Canada

2 Electron scattering reaction Virtual Photon k µ P µ (ω, q) k µ µ µ µ q = k k µ q = ( ω, q) µ P can vary independently d σ dωdω = σ M Inclusive cross section A(e,e )X [ Q q R L(ω, q) + ( Q q + tan θ with Q = qµ = q ω and scattering angle ) ] R T (ω, q) θ and σ M Mott cross section

3 Electron scattering reaction ) R L (ω, q) = Ψ ρ(q) Ψ δ (E E ω + q M ) R T (ω, q) = Ψ J T (q) Ψ δ (E E ω + q M In a non relativistic limit: the charge is a one-body operator ρ 1 (q) the current is a one-body J 1 (q) plus a two-body operator J (q), required by gauge invariance J (x) = i [V,ρ 1 (x)] m + Electron scattering o He was investigated in the past by Carlson and Schiavilla with the Laplace transorm using the GFMC approach: E(q, τ) = ω th dωe τ(ω q m ) R L/T (q, ω) G E p (q, ω) π π m (1) In longitudinal response the FSI is undamental () MEC lead to an enhancement o % o the transverse response 3

4 Electron scattering reaction ) R L (ω, q) = Ψ ρ(q) Ψ δ (E E ω + q M ) R T (ω, q) = Ψ J T (q) Ψ δ (E E ω + q M In a non relativistic limit: the charge is a one-body operator ρ 1 (q) (1) the current is a one-body J 1 (q) plus a two-body operator J (q), required by gauge invariance J (x) = i [V,ρ 1 (x)] m + Electron scattering o He was investigated in the past by Carlson and () Schiavilla with the Laplace transorm using the GFMC approach: E(q, τ) = ω th dωe τ(ω q m ) R L/T (q, ω) G E p (q, ω) π π m PRC 9 R88 (199) (1) In longitudinal response the FSI is undamental () MEC lead to an enhancement o % o the transverse response 3

5 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Eros, Leidemann, Orlandini, PLB 338 (199) 13 Γ

6 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand R(ω) = ψ ψ (ω σ) + Γ ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea

7 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed and calculate ψ ψ or dierent Γ σ

8 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed Γ and calculate ψ ψ or dierent umerical inversion o the transorm L(σ) σ R(ω)

9 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed Γ and calculate ψ ψ or dierent umerical inversion o the transorm Best it method or L(σ) = n c n χ α n(σ) c n R(ω) = n L(σ) c n χ α n(ω) σ R(ω)

10 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed Γ and calculate ψ ψ or dierent umerical inversion o the transorm Best it method or L(σ) = n c n χ α n(σ) c n R(ω) = n L(σ) c n χ α n(ω) σ R(ω) Redo or dierent Γ

11 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed Γ and calculate ψ ψ or dierent umerical inversion o the transorm Best it method or L(σ) = n c n χ α n(σ) c n R(ω) = n L(σ) c n χ α n(ω) σ R(ω) Redo or dierent Γ Obtain a response unction independent on Γ

12 The Lorentz Integral Transorm -Summary- Introduced yesterday by S.Quaglioni R(ω) = L(σ, Γ) = ψ Ô ψ δ(e E ω) dω R(ω) = ψ ψ (ω σ) + Γ (H E σ + iγ) ψ = Ô ψ σ Bound-state method to expand ψ, ψ Eros, Leidemann, Orlandini, PLB 338 (199) 13 in terms o a complete set o basis state Γ Hyper-spherical Harmonics ρ, Ω =( θ, φ, ϑ 1, ϕ 1, ϑ, ϕ, ϑ 3, ϕ 3 ) ρ = i r i EIHH talk by.barnea For ixed Γ and calculate ψ ψ or dierent umerical inversion o the transorm Best it method or L(σ) = n c n χ α n(σ) Redo or dierent Γ Obtain a response unction independent on Γ L(σ) c n R(ω) = n c n χ α n(ω) σ R(ω) One can compare directly with experiment!!

13 Electron scattering reaction o He R L (ω, q) to investigate the eect o 3F First step Study (no MEC) Potentials used: orce AV18 3 orce UIX 5

14 He(e,e )X (E E ω + q ) R L (ω, q) = Ψ ρ(q) Ψ δ M ρ(q) = k e iq r k 1+τ 3 k = J C S J (q)+c V J (q) Distribution o the total strength among multipoles: irst sum rule Isoscalar 1.5% Isovector 89.5% 3 Isoscalar.1% Isovector 57.9% 3 Isoscalar 5.1% Isovector 5.9% Strength [%] q=1 MeV/c Strength [%] 1 q=3 MeV/c Strength [%] 1 q=5 MeV/c J J J 6

15 He(e,e )X (E E ω + q ) R L (ω, q) = Ψ ρ(q) Ψ δ M ρ(q) = k e iq r k 1+τ 3 k = J C S J (q)+c V J (q) Distribution o the total strength among multipoles: irst sum rule 3 Isoscalar.1% Isovector 57.9% 3 Isoscalar 1.9% Isovector 58.1% Strength [%] q=3 MeV/c Strength [%] AV18 q=3 MeV/c J J 7

16 He(e,e )X (E E ω + q ) R L (ω, q) = Ψ ρ(q) Ψ δ M ρ(q) = k e iq r k 1+τ 3 k = J C S J (q)+c V J (q) Distribution o the total strength among multipoles: irst sum rule 3 Isoscalar.1% Isovector 57.9% 3 Isoscalar 1.9% Isovector 58.1% Strength [%] q=3 MeV/c Strength [%] AV18 q=3 MeV/c J CS.535 CV.8155 < J CS.6 CV.836 7

17 Convergence o the calculations Isovector Response R L [1-3 MeV -1 ] 8 6 K max K max - Sum Lits and Invert q=5 MeV/c q=5 MeV/c Maximal error in shown energy range is.5% or q=5 MeV/c and 3% or q=5 MeV/c 8

18 Convergence o the calculations Isoscalar Response 9

19 Convergence o the calculations Isoscalar Response Monopole resonance E Width kev 9

20 Convergence o the calculations Isoscalar Response E [MeV] Γ[MeV] Diagonalizing the Hamiltonian VMC 1. - EIHH EIHH AV Monopole resonance E Width kev 9

21 Convergence o the calculations Isoscalar Response E [MeV] Γ[MeV] Diagonalizing the Hamiltonian VMC 1. - EIHH EIHH AV Monopole resonance Width kev E Typical inversion basis Inverting LIT imposing a resonance 8 χ n (ω) =ω l+1 e αω/n χ(ω) = 1 1 π + χ n (ω) Γ (ω E ) + Γ Total Longitudinal Response unstable Inversions imposing resonance kev R L [1-3 MeV -1 ] 6 q= MeV/c

22 Convergence o the calculations Isoscalar Response E [MeV] Γ[MeV] Diagonalizing the Hamiltonian VMC 1. - EIHH EIHH AV Monopole resonance Width kev E Typical inversion basis Inverting LIT imposing a resonance 8 χ n (ω) =ω l+1 e αω/n χ(ω) = 1 1 π + χ n (ω) Γ (ω E ) + Γ Total Longitudinal Response unstable Inversions imposing resonance kev 6 Still large error in height o the peak irst 5 MeV rom not enough reliable R L [1-3 MeV -1 ] q= MeV/c

23 Convergence o the calculations Isoscalar Response E [MeV] Γ[MeV] Diagonalizing the Hamiltonian VMC 1. - EIHH EIHH AV Monopole resonance Width kev E Indetermination on monopole does not aect the prediction o the quasi elastic peak! Typical inversion basis Inverting LIT imposing a resonance R L [1-3 MeV -1 ] 8 6 χ n (ω) =ω l+1 e αω/n χ(ω) = 1 1 π + χ n (ω) Γ (ω E ) + Γ Total Longitudinal Response Inversions imposing resonance unstable q= MeV/c kev Final error is only 1.5% or energies > 5 MeV 9

24 He(e,e )X Comparison with experiment Calculation o R L (ω, q) with the LIT/EIHH method Medium-q kinematics 1 R L [1-3 MeV -1 ] AV18 Saclay Bates q=3 MeV/c R L [1-3 MeV -1 ] AV18 Saclay q= 35 MeV/c R L [1-3 MeV -1 ] 8 6 AV18 Saclay Bates q= MeV/c The comparison with experiment improves with addition o 3F 3F mainly in quasi-elastic peak region 1

25 He(e,e )X Comparison with the Laplace Calculation o R L (ω, q) with the LIT/EIHH method q=3 MeV/q kinematics R L [1-3 MeV -1 ] AV18 Saclay Bates q=3 MeV/c Eucledian Response He q=3 MeV/c AV18 GFMC AV8+UVIII Saclay Bates Longitudinal ! [MeV -1 ] E(q, τ) = w th dωe τ(ω q m ) R L(q, ω) G E p (q, ω) The two calculations are consistent, but with the Laplace transorm one looses resolution 11

26 He(e,e )X Calculation o R L (ω, q) with the LIT/EIHH method Low-q kinematics 1 8 AV18 15 AV18 R L [1-3 MeV -1 ] 6 q=1 MeV/c PRELIMIARY R L [1-3 MeV -1 ] 1 5 q= MeV/c PRELIMIARY This observable is very sensitive to 3F Quest or new precise measurements! Proposal in MAMI@Mainz 1

27 He(e,e )X Comparison with experiment R L [1-3 MeV -1 ] 8 6 Calculation o R L (ω, q) with the LIT/EIHH method High-q kinematics AV18 Saclay q= 5 MeV/c R L [1-3 MeV -1 ] AV18 Saclay Bates q= 5 MeV/c The comparison with experiment improves with addition o 3F on the quasi-elastic peak The orce model does not reproduce experiment at large momenta and energy 13

28 He(e,e )X Comparison with experiment Relativistic orm actors Calculation o R L (ω, q) with the LIT/EIHH method High-q kinematics R L [1-3 MeV -1 ] 6 Rel FF on Rel FF Saclay q= 5 MeV/c R L [1-3 MeV -1 ] Rel FF on Rel FF Saclay Bates q= 5 MeV/c Relativity does not change much the situation The reason o discrepancy must be something else 1

29 Conclusion and Outlook The LIT is a very powerul method to an exact study o the electron scattering reaction o ew-body systems, which enables to accomplish many goals: Shed more light on role o 3F Intersection theory-experiment Investigate the role o relativity Future: Investigate the transverse response Perorm calculations with chiral EFT potential and consistent two-body currents Extend these calculations to heavier systems: halo 6 He ELISE@FAIR 15