Inclusive Electron Scattering off 4 He

Transcription

1 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Inclusive Electron Scattering off He S. Bacca in coloration with: H. Arenhövel Johannes-Gutenberg Universität, Mainz N. Barnea The Racah Institute of Physics, The Hebrew University, Jerusalem W. Leidemann and G. Orlandini Dipartimento di Fisica, Università di Trento and INFN Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p.1/15

2 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Outline Inclusive Electron Scattering off He S. Bacca Introduction to Electron Scattering Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p./15

3 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Outline Inclusive Electron Scattering off He S. Bacca Introduction to Electron Scattering Longitudinal Response Function Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p./15

4 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Outline Inclusive Electron Scattering off He S. Bacca Introduction to Electron Scattering Longitudinal Response Function Transverse Response Function: one- and two-body current Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p./15

5 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Outline Inclusive Electron Scattering off He S. Bacca Introduction to Electron Scattering Longitudinal Response Function Transverse Response Function: one- and two-body current Siegert Operator: test / when is the Siegert approximation reliable? Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p./15

6 Gesellschaft für Schwerionenforschung, Darmstadt Theory Division Outline Inclusive Electron Scattering off He S. Bacca Introduction to Electron Scattering Longitudinal Response Function Transverse Response Function: one- and two-body current Siegert Operator: test / when is the Siegert approximation reliable? Conclusions Few Body 1, August 1-6, 6, Santos-S.Paulo, Brazil. p./15

7 Introduction to Electron Scattering k µ P f µ Virtual Photon k µ µ µ µ q = k k µ q = ( ω, q) µ P (ω,q) vary independently Inclusive Cross Section A(e,e )X d σ dωdω = σ M [ Q q R L(ω,q) + ( Q q + tan θ ) ] R T (ω,q) with Q = q µ = q µ q µ and θ scattering angle and σ M Mott cross section. p.3/15

8 Introduction to Electron Scattering Longitudinal and transverse response functions can be disentangled via Rosenbluth separation ) R L (ω,q) = Ψ f ρ(q) Ψ δ (E f E ω + q M f ) R T (ω,q) = Ψ f J T (q) Ψ δ (E f E ω + q M In a non relativistic framework: f The charge ρ(q) is a one-body operator ρ 1 (q) The current J T (q) includes a one-body operator J 1 (q), spin and convection current, and a two-body operator J (q), the so-called Meson Exchange Current required by gauge invariance J (x) = i[v, ρ 1 (x)]. p.3/15

9 Previous Work The quasi-elastic region ω q /m is seen as an interesting playground to study nuclear correlations.. p./15

10 Previous Work The quasi-elastic region ω q /m is seen as an interesting playground to study nuclear correlations. Technical difficulty of providing realistic initial and final scattering state wave functions. p./15

11 Previous Work The quasi-elastic region ω q /m is seen as an interesting playground to study nuclear correlations. Technical difficulty of providing realistic initial and final scattering state wave functions e Most of the calculations are performed in PWIA. They lead to: (1) Overestimation of the longitudinal response function () Slight underestimation of the transverse response function (1) is supposed to be due to the missing FSI () probably due to the missing Meson Exchange Currents e He γ * p 3 H. p./15

12 Previous Work The quasi-elastic region ω q /m is seen as an interesting playground to study nuclear correlations. Technical difficulty of providing realistic initial and final scattering state wave functions e Most of the calculations are performed in PWIA. They lead to: (1) Overestimation of the longitudinal response function () Slight underestimation of the transverse response function (1) is supposed to be due to the missing FSI () probably due to the missing Meson Exchange Currents For He the only realistic calculation with consistent two-body operators and FSI is performed with the Laplace Transform J. Carlson and R. Schiavilla, P. R. C 6, 36 (199) e He γ * p 3 H. p./15

13 Previous Work The quasi-elastic region ω q /m is seen as an interesting playground to study nuclear correlations. Technical difficulty of providing realistic initial and final scattering state wave functions Most of the calculations are performed in PWIA. They lead to: (1) Overestimation of the longitudinal response function () Slight underestimation of the transverse response function (1) is supposed to be due to the missing FSI () probably due to the missing Meson Exchange Currents For He the only realistic calculation with consistent two-body operators and FSI is performed with the Laplace Transform J. Carlson and R. Schiavilla, P. R. C 6, 36 (199) AV + UXIII Pion exchange current e He γ * e p 3 H. p./15

14 Previous Work Calculation for He with the LIT method and CHH R L [MeV 1 ] TN Potential FSI q=3 MeV/c.6 PWIA PWIA Bates Saclay. TN Potential FSI q=5 MeV/c Bates Saclay ω [MeV] ω [MeV] V. E. Efros, W. Leidemann and G. Orlandini, P.R.L. 7, 3 (1997).. p.5/15

15 Aim of the Present Work (1) Use a method powerful enough to deal with continuum to account for FSI Lorentz Integral Transfrom method L(σ,q) = Z 1 D E dωr(ω, q) (ω σ R ) + σi = eψ Ψ e ; L(σ,q) Inversion R(ω, q) E (H E + σ) eψ = O(q) Ψ LIT equation. p.6/15

16 Aim of the Present Work (1) Use a method powerful enough to deal with continuum to account for FSI Lorentz Integral Transfrom method L(σ,q) = Z 1 D E dωr(ω, q) (ω σ R ) + σi = eψ Ψ e ; L(σ,q) Inversion R(ω, q) E (H E + σ) eψ = O(q) Ψ LIT equation () Spectral resolution method: Effective Interaction Hyperspherical Harmonics 3 1 η 1 η r r 3 η ρ, Ω = (Θ,Φ, θ, φ, θ, φ, θ, φ ) EI: Lee-Suzuki approach to accelerate convergence. p.6/15

17 Aim of the Present Work (1) Use a method powerful enough to deal with continuum to account for FSI Lorentz Integral Transfrom method L(σ,q) = Z 1 D E dωr(ω, q) (ω σ R ) + σi = eψ Ψ e ; L(σ,q) Inversion R(ω, q) E (H E + σ) eψ = O(q) Ψ LIT equation () Spectral resolution method: Effective Interaction Hyperspherical Harmonics 3 1 η 1 η r r 3 η ρ, Ω = (Θ,Φ, θ, φ, θ, φ, θ, φ ) EI: Lee-Suzuki approach to accelerate convergence (3) Use an interaction where you can construct consistently MEC: simple MTI-III potential with spin-isospin dependence.. p.6/15

18 Aim of the Present Work (1) Use a method powerful enough to deal with continuum to account for FSI Lorentz Integral Transfrom method L(σ,q) = Z 1 D E dωr(ω, q) (ω σ R ) + σi = eψ Ψ e ; L(σ,q) Inversion R(ω, q) E (H E + σ) eψ = O(q) Ψ LIT equation () Spectral resolution method: Effective Interaction Hyperspherical Harmonics 3 1 η 1 η r r 3 η ρ, Ω = (Θ,Φ, θ, φ, θ, φ, θ, φ ) EI: Lee-Suzuki approach to accelerate convergence (3) Use an interaction where you can construct consistently MEC: simple MTI-III potential with spin-isospin dependence. To perform an exact and consistent calculation of R L/T (ω,q). p.6/15

19 Longitudinal Response Function Comparison with experiment.1.1 MTI III q=3 MeV/c.6.5 MTI III Bates Saclay 1 ) [MeV ] (ω R L Bates Saclay..3. q=5 MeV/c [MeV] ω [MeV] ω p.7/15

20 Longitudinal Response Function Comparison with experiment.1.1 MTI III q=3 MeV/c.6.5 MTI III Bates Saclay 1 ) [MeV ] (ω R L Bates Saclay..3. q=5 MeV/c [MeV] ω [MeV] ω With MTI-III we find a pronounced structure close to threshold mainly due to isoscalar monopole transition.. p.7/15

21 Transverse Response Function One-body current Spin Current J s 1 i m Convection Current J c 1 1 m X (σ k q)e iqr k k X {p k, e iq r k } k.1 ] 1 ) [MeV (ω q=3 MeV/c Spin Spin+Conv Bates Saclay ] 1 ) [MeV ( ω.1..6 q=5 MeV/c Spin Spin+Conv Bates Saclay R T. T R ω [MeV] ω [MeV] 5. p./15

22 Transverse Response Function One-body current Spin Current J s 1 i m Convection Current J c 1 1 m X (σ k q)e iqr k k X {p k, e iq r k } k.1 ] 1 ) [MeV (ω q=3 MeV/c Spin Spin+Conv Bates Saclay ] 1 ) [MeV ( ω.1..6 q=5 MeV/c Spin Spin+Conv Bates Saclay R T. T R ω [MeV] ω [MeV] 5 Is the missing strength due to the two-body current?. p./15

23 Siegert Operator Multipole decomposition of the current J µ (q) = J Ĵ [ ( ) ] δ µ C Jµ + µ TJµ el + µt mag Jµ δ µ 1 if q z axis Coulomb C Jµ dˆq ρ(q)y J µ (ˆq) Magnetic Electric T mag Jµ dˆq J(q) Y µ JJ1 (ˆq) T el Jµ dˆq T el Jµ J + 1 [ J+1 J q J(q) q }{{} [H,ρ(q)] q = ω q ρ(q) ω q C Jµ J }{{} Siegert Operator + Y J µ (ˆq)+ Ĵ J J(q) Y µ JJ+11 (ˆq) ] dˆq Ĵ J(q) Y µ JJ+11 (ˆq) J }{{} Correction Operator In the Siegert operator part of the MEC is implicitly included!. p.9/15

24 Siegert Operator: Estimation of MEC effect on isovector dipole MTI-III: Isospin dependence MEC R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator ω q C 1. p.1/15

25 Siegert Operator: Estimation of MEC effect on isovector dipole MTI-III: Isospin dependence MEC R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator ω q C 1 Siegert Part of Convection Current q J dˆq c 1 (q) q Y 1 (ˆq). p.1/15

26 Siegert Operator: Estimation of MEC effect on isovector dipole MTI-III: Isospin dependence MEC R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator ω q C 1 Siegert Part of Convection Current q J dˆq c 1 (q) q Y 1 (ˆq) The missing strength is due to the MEC!. p.1/15

27 Explicit Inclusion of MEC The MTI-III has the form V = V 1 (r) + V (r)τ 1 τ + V 3 (r)σ 1 σ + V (r)(τ 1 τ )(σ 1 σ ) where V (r) J m (r) with J m scalar meson propagator and r = r 1 r. p.11/15

28 Explicit Inclusion of MEC The MTI-III has the form V = V 1 (r) + V (r)τ 1 τ + V 3 (r)σ 1 σ + V (r)(τ 1 τ )(σ 1 σ ) where V (r) J m (r) with J m scalar meson propagator and r = r 1 r The consistent MEC has only meson in flight terms J (x,r 1,r ) (τ 1 τ ) 3 J m (r 1 x) x J m (x r ) m N N. p.11/15

29 Explicit Inclusion of MEC The MTI-III has the form V = V 1 (r) + V (r)τ 1 τ + V 3 (r)σ 1 σ + V (r)(τ 1 τ )(σ 1 σ ) where V (r) J m (r) with J m scalar meson propagator and r = r 1 r The consistent MEC has only meson in flight terms J (x,r 1,r ) (τ 1 τ ) 3 J m (r 1 x) x J m (x r ) m In momentum space J (q,r 1,r ) J (q,r,r) = 1 π 3 eir q( r I m (q,r) ) R = r 1 + r, r = r 1 r. N N. p.11/15

30 Explicit Inclusion of MEC The MTI-III has the form V = V 1 (r) + V (r)τ 1 τ + V 3 (r)σ 1 σ + V (r)(τ 1 τ )(σ 1 σ ) where V (r) J m (r) with J m scalar meson propagator and r = r 1 r The consistent MEC has only meson in flight terms J (x,r 1,r ) (τ 1 τ ) 3 J m (r 1 x) x J m (x r ) m In momentum space J (q,r 1,r ) J (q,r,r) = 1 π 3 eir q( r I m (q,r) ) R = r 1 + r, r = r 1 r. Approximation J (q,r,r) J A (q,r) = 1 π 3 ( r I m (q,r) ) N N. p.11/15

31 Siegert Operator Test the validity of the approximation R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator Siegert part of Convection Current q J dˆq c 1 (q) q Y 1 (ˆq). p.1/15

32 Siegert Operator Test the validity of the approximation R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv MEC 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator Siegert part of Convection Current Siegert part of MEC R dˆq q J (q) Y q 1 (ˆq) q J dˆq c 1 (q) q Y 1 (ˆq). p.1/15

33 Siegert Operator Test the validity of the approximation R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv MEC MEC+Conv 1 q=1 MeV/c 6 q= MeV/c ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c Siegert Operator Siegert part of Convection Current Siegert part of MEC R dˆq q J (q) Y q 1 (ˆq) q J dˆq c 1 (q) q Y 1 (ˆq) Siegert part of (Conv + MEC) R q (J c dˆq 1 (q)+j (q)) Y q 1 (ˆq). p.1/15

34 Siegert Operator Test the validity of the approximation R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv MEC MEC+Conv 1 q=1 MeV/c 6 q= MeV/c ~1% ω [MeV] ω [MeV] ω ω [MeV] [MeV] q=3 MeV/c ~% Siegert Operator Siegert part of Convection Current Siegert part of MEC R dˆq q J (q) Y q 1 (ˆq) q J dˆq c 1 (q) q Y 1 (ˆq) Siegert part of (Conv + MEC) R q (J c dˆq 1 (q)+j (q)) Y q 1 (ˆq) Effect of the approximation in peak height + small shift! Negligible effect for q 1 MeV/c. p.1/15

35 Siegert Operator: When is the Siegert theorem reliable for the total isovector dipole? R E1 1 (ω ) [MeV *1 ] q=3 MeV/c q=1 MeV/c q= MeV/c 16 1 Siegert q=3 MeV/c ω [MeV] ω [MeV] ω [MeV] ω [MeV] Siegert Operator. p.13/15

36 Siegert Operator: When is the Siegert theorem reliable for the total isovector dipole? R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv q=1 MeV/c ω [MeV] ω [MeV] ω [MeV] ω [MeV] 6 q= MeV/c q=3 MeV/c Siegert Operator Total Convection Current. p.13/15

37 Siegert Operator: When is the Siegert theorem reliable for the total isovector dipole? R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv MEC+Conv q=1 MeV/c ω [MeV] ω [MeV] ω [MeV] ω [MeV] 6 q= MeV/c q=3 MeV/c Siegert Operator Total Convection Current Total Convection Current + Total MEC. p.13/15

38 Siegert Operator: When is the Siegert theorem reliable for the total isovector dipole? R E1 1 (ω ) [MeV *1 ] 16 1 q=3 MeV/c Siegert Conv MEC+Conv q=1 MeV/c ω [MeV] ω [MeV] ω [MeV] ω [MeV] 6 q= MeV/c ~7% q=3 MeV/c ~5% Siegert Operator Total Convection Current Total Convection Current + Total MEC The Siegert operator is a good approximation of the total isovector dipole for q 1 MeV/c. At higher q the deviation from exact treatment increases!. p.13/15

39 Transverse Response Function 1-body + -body current R T (ω ) [MeV 1 ].3..1 q=1 MeV/c 1 body current Spin Spin+Conv..6.. q= MeV/c q=3 MeV/c ω [MeV] ω [MeV] ω [MeV] Spin current strongly dominates for q MeV/c. p.1/15

40 Transverse Response Function 1-body + -body current R T (ω ) [MeV 1 ].3..1 q=1 MeV/c 1 body current Spin Spin+Conv Spin+Siegert..6.. q= MeV/c q=3 MeV/c ω [MeV] ω [MeV] ω [MeV] Spin current strongly dominates for q MeV/c Strong MEC effect for q = 1 MeV/c. p.1/15

41 Transverse Response Function 1-body + -body current R T (ω ) [MeV 1 ].3..1 q=1 MeV/c 1 body current Spin Spin+Conv Spin+Siegert 1+ body current..6.. q= MeV/c q=3 MeV/c ω [MeV] ω [MeV] ω [MeV] Spin current strongly dominates for q MeV/c Strong MEC effect for q = 1 MeV/c. p.1/15

42 Transverse Response Function 1-body + -body current R T (ω ) [MeV 1 ] q=1 MeV/c 1 body current. q= MeV/c q=3 MeV/c Spin. Spin+Conv.6 Spin+Siegert.6 1+ body MeV ~%. MeV ~% ω [MeV] ω [MeV] ω [MeV] Spin current strongly dominates for q MeV/c Strong MEC effect for q = 1 MeV/c Siegert overestimation of the tail!. p.1/15

43 Transverse Response Function 1-body + -body current R T (ω ) [MeV 1 ] q=1 MeV/c 1 body current. q= MeV/c q=3 MeV/c Spin. Spin+Conv.6 Spin+Siegert.6 1+ body MeV ~%. MeV Comparison with experimental data ~% MTI III 6 1 q=3 6 MeV/c ω [MeV] ω [MeV] ω [MeV].1 1 body current 1+ body current Bates. Saclay Spin current strongly dominates for q MeV/c R T (ω) [MeV 1 ].6 Strong MEC effect for q = 1 MeV/c. Siegert overestimation of the tail! ω [MeV]. p.1/15

44 Conclusion and Outlook We have presented the study of electron scattering off He within the LIT and EIHH methods.. p.15/15

45 Conclusion and Outlook We have presented the study of electron scattering off He within the LIT and EIHH methods. In this framework we performed the first calculation of the transverse response function with consistent treatment of MEC in a simple potential model: (*) Strong MEC effects are found for low momentum transfer, where unfortunately no experimental data are availe (**) Siegert approximation for isovector dipole is reliable only for q 1 MeV/c. p.15/15

46 Conclusion and Outlook We have presented the study of electron scattering off He within the LIT and EIHH methods. In this framework we performed the first calculation of the transverse response function with consistent treatment of MEC in a simple potential model: (*) Strong MEC effects are found for low momentum transfer, where unfortunately no experimental data are availe (**) Siegert approximation for isovector dipole is reliable only for q 1 MeV/c Future: extend this investigation to more realistic potentials with consistent two-body currents in order to investigate tensor correlations and pionic degrees of freedom.. p.15/15