Parity-Violating Asymmetry for 208 Pb
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1 Parity-Violating Asymmetry for 208 Pb Matteo Vorabbi Dipartimento di Fisica - Università di Pavia INFN - Sezione di Pavia Rome January 15 Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 1 / 32
2 Outline Introduction Parity-violating asymmetry Nuclear structure models Symmetry energy at saturation density Results for 208 Pb Predictions for 48 Ca Collaborators C. Giusti (Pavia), A. Meucci (Pavia) P. Finelli (Bologna) Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 2 / 32
3 Introduction Purpose Our goal is present and discuss numerical predictions for the neutron density distribution of 208 Pb and compare the results with the experimental data obtained from different reactions Theoretical background Relativistic and non-relativistic mean-field models for the nuclear structure Parity-violating asymmetry parameter for elastic electron scattering is computed at the kinematics of the PREX experiment and is obtained averaging the theoretical results over the experimental acceptance function Theoretical predictions of the neutron skin thickness are obtained from mean field models Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 3 / 32
4 Introduction - Experiments Osaka measurement - Phys. Rev. C, 82, (2010) Cross sections and analyzing power for polarized proton elastic scattering Measurement of neutron density distribution and neutron skin thickness PREX measurement - Phys. Rev. Lett. 108, (2012) Parity-violating asymmetry A pv from 208 Pb Information about the difference between the radii of the neutron and proton distributions R n R p First electroweak observation of the neutron skin Mainz measurement - Phys. Rev. Lett. 112, (2014) Coherent pion photoproduction from 208 Pb Information on the size and shape of the neutron skin Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 4 / 32
5 Parity-violating asymmetry Dirac equation Total potential: Axial potential: [α p + U ± (r)]ψ ± = EΨ ± U(r) ± = V (r) ± γ 5 A(r) A(r) = G F 2 ρ 2 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 Λ = 4.27 fm 1 Parity-violating asymmetry in Born approximation A pv = G F Q 2 [ 4 4 sin 2 Θ W 1 + F ] n(q) 2 πα F p (q) Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 5 / 32
6 Nuclear Structure Models Relativistic models Nonlinear RMF model (NL) Density-dependent RMF model (DD) Nonlinear point-coupling RMF model (PC) Density-dependent point-coupling RMF model (DDPC) Non-relativistic models Skyrme interaction (SIII, SKM*, SLY4, SLY5) Gogny interaction (D1S) Approximations Mean field description Spherical symmetry No-sea approximation (relativistic models) Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 6 / 32
7 Relativistic Models - Meson exchange Nuclei are described as a system of Dirac nucleons interacting by the exchange of mesons and the electromagnetic field Nonlinear RMF model Lagrangian: L = L N + L m + L int L m includes terms like: ω µ ω µ, (ω µ ω µ ) 2,... L int contains the interaction terms between nucleons and mesons Density-dependent RMF model Lagrangian: L = L N + L m + L int L int contains the interaction terms between nucleons and mesons with density-dependent coupling constants g i (ρ) = g i (ρ sat )f i (x) for i = σ, ω, ρ, x = ρ/ρ sat ρ sat = fm 3 denotes the nucleon density at saturation in symmetric nuclear matter Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 7 / 32
8 Relativistic Models - Point coupling The point-coupling models are defined by a Lagrangian density that only consists of nucleon fields Nonlinear point-coupling RMF model Lagrangian: L = L free + L 4f + L hot + L der L 4f includes 4-fermion terms: ( ψγ µ ψ)( ψγ µ ψ) L hot includes higher order terms: [( ψγ µ ψ)( ψγ µ ψ)] 2 L der includes derivative terms: ( ν ψγµ ψ)( ν ψγ µ ψ) Density-dependent point-coupling RMF model Lagrangian: L = L free + L 4f + L der L 4f includes 4-fermion terms: ( ψγ µ ψ)( ψγ µ ψ) L der represents derivatives in the nucleon scalar density Coupling constants are density-dependent Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 8 / 32
9 Non-relativistic Models Skyrme interaction Interaction: V = i<j V (ij) + i<j<k V (ijk) Short-range expansion for the two-body interaction: V (1, 2) = V 0 δ(r 1 r 2 ) + V 1 [δ(r 1 r 2 )ˆk 2 + ˆk 2 ] δ(r 1 r 2 ) + V 2 ˆkδ(r 1 r 2 )ˆk + iw 0 (σ (1) + σ (2) )ˆk δ(r 1 r 2 )ˆk Zero-range force for three-body interaction: V (1, 2, 3) = V 3 δ(r 1 r 2 ) δ(r 2 r 3 ) Gogny interaction Zero-range forces of Skyrme interaction might not be able to simulate the long range parts of the realistic effective interaction V 0, V 1 and V 2 of Skyrme force are replaced by a sum of two Gaussians with spin-isospin exchange mixtures Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January 15 9 / 32
10 Nuclear symmetry energy EOS of isospin asymmetric nuclear matter E(ρ, α) = E(ρ, α = 0) + E sym (ρ) α 2 + O(α 4 ) Nuclear symmetry energy and isospin asymmetry: E sym (ρ) = 1 2 E(ρ, α) 2 α 2 α=0 α = ρ n ρ p ρ n + ρ p Nuclear symmetry energy at saturation density E sym (ρ) E sym (ρ 0 ) + L ( ) ρ ρ0 + K ( sym ρ ρ0 3 ρ 18 ρ Parameters of expansion: L = 3ρ 0 E sym (ρ) ρ 2 K sym = 9ρ 2 E sym (ρ) 0 ρ=ρ0 ρ 2 ) 2 ρ=ρ0 Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
11 Elastic Scattering off 208 Pb Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
12 dσ/d Ω [cm 2 /sr] 1e-26 1e-27 1e-28 1e-29 1e-30 1e-31 1e Pb 250 MeV Phys. Rev. Lett.,19, 242, (1967) NL3 DDME1 PCF1 SLY θ [deg] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
13 dσ/dω [mb/sr] Pb Phys. Rev. C, 82, (2010) RIA + edai potential RIA + democratic potential DDME1 295 MeV θ [deg] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
14 1 analyzing power RIA + edai potential RIA + democratic potential DDME1 Phys. Rev. C, 82, (2010) θ [deg] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
15 Results for 208 Pb Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
16 0.6 SLY5 (mod) diffuseness [fm] D1S SKM* SLY5 SLY4 DDME1 DDPC1 MAINZ DDME2 MEAN PKDD PC NL3-II NL3 SIII (mod) SIII NL-SH L neutron half-height radius [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
17 - ρ Weak [fm -3 ] Pb NL3 DDME1 PCF1 SLY4 NL-SH SIII MEAN r [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
18 0.76 A pv [ppm] D1S SLY4 SKM* DDPC1 DDME2 SLY5 SLY5 (mod) SIII SIII (mod) L2 DDME1 MEAN PKDD NL-SH NL3-II NL R rms neutron [fm] PC PREX Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
19 A pv [ppm] SIII D1S SIII (mod) SLY4 SLY5 SLY5 (mod) SKM* MAINZ ( γ,π 0 ) DDPC1 DDME2 DDME1 MEAN PKDD NL-SH OSAKA-RCNP ( p, p ) neutron skin [fm] L2 PC NL3-II NL3 PREX Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
20 Measured values of neutron skin thicknesses of 208 Pb PREX (e,e) arxiv: [nucl-th] (α,α ) GDR 120 MEV (1991) (α,α ) GDR 200 MEV (2004) antiproton absorption (2007) pygmy resonance (2007) (p,p) 295 MeV (2010) pygmy resonance (2012) ( p, p ), EDS (2012) (2012) (γ,π 0 ) Mainz (2013) R skin [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
21 neutron skin [fm] NL-ME DD-ME NL-PC DD-PC Non-rel. Skyrme Non-rel. Gogny PREX OSAKA-RCNP ( p, p ) MAINZ ( γ,π 0 ) L [MeV] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
22 neutron skin [fm] NL-ME DD-ME NL-PC DD-PC Non-rel. Skyrme Non-rel. Gogny PREX OSAKA-RCNP ( p, p ) MAINZ ( γ,π 0 ) K sym [MeV] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
23 Predictions for 48 Ca Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
24 R rms proton [fm] Ca NL-ME DD-ME NL-PC DD-PC Non-rel. Skyrme Non-rel. Gogny R rms neutron [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
25 Ca - Rskin [fm] NL-ME DD-ME NL-PC DD-PC Non-rel. Skyrme Non-rel. Gogny R skin = R skin Pb - Rskin [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
26 A pv NL3 DDME1 PC-F1 SLY4 E = 2.2 GeV 48 Ca θ [deg] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
27 2.7 A pv [ppm] D1S 48 Ca SIII SLY4 E = 2.2 GeV θ = 4 o SLY5 SKM* DDPC1 DDME2 DDME1 L2 NL-SH PKDD NL3 NL3-II PC R skin [fm] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
28 Conclusions Determination of neutron distribution in nuclei Great experimental and theoretical efforts have been devoted to achieve this goal Parity-violating electron scattering is an accurate and model-independent tool for probing neutron properties More information on nuclear structure More stringent constraints of the range of variation of nuclear matter expansion parameters Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
29 Backup Slides Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
30 Symmetrized Fermi Density Two-parameter distribution sinh(r/a) ρ(r) = ρ 0 cosh(r/a) + cosh(r/a) ρ 0 = 3 [ ( πa ) 2 ] 1 4πR R Properties d 3 r ρ(r) = 1 Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
31 Asymmetry parameter and acceptance function Asymmetry parameter Acceptance function acceptance function A pv = dθ sin θapv (θ) dσ dω ɛ(θ) dθ sin θ dσ dω ɛ(θ) Phys. Rev. Lett., 108, (2012) Supplemental material PREX θ [deg] Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
32 Best fit lines Slope Intercept Chi-Square R rms A pv R skin A pv L R skin K sym R skin Matteo Vorabbi (Università di Pavia) Parity-Violating Asymmetry for 208 Pb Rome January / 32
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