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1 Isospin mixing and parity- violating electron scattering O. Moreno, P. Sarriguren, E. Moya de Guerra and J. M. Udías (IEM-CSIC Madrid and UCM Madrid) T. W. Donnelly (M.I.T.),.), I. Sick (Univ. Basel)

2 Summary Introduction Theoretical framework Results Conclusions

3 Introduction: parity violation in electron scattering (PWBA) Interesting for... Standard Model coupling constants Nucleon strangeness content Nuclear isospin Neutron distribution in nuclei

4 Theoretical formalism: PV asymmetry PWBA J π = 0 + Elastic scatt. N=Z T=0 g.s. Actual asymmetry: Asymmetry deviation:

5 Theoretical formalism: Form factors in s.h.o. basis Coulomb monopole form factors ratio: Coulomb monopole operator matrix element: Coulomb monopole matrix element between two s.h.o. states: Spherical part of the density matrix in the s.h.o. basis: And equivalently for WNC form factors but using G E, defined as:

6 Theoretical formalism: structure of nuclear target HF: Axially deformed Hartree-Fock mean field using a Skyrme nucleon-nucleon effective interaction (SLy4). BCS: pairing interactions treated within BCS approx. with fixed pairing gaps p,n =1 MeV. Occupations and number equation recomputed after each HF iteration. Expansion coefficients in s.h.o. basis of the HF+BCS single particle state i: Occupation probability of the HF+BCS single particle state i

7 Theoretical formalism: kinematics Relative error of the asymmetry: Figure-of-merit (FOM):

8 Theoretical formalism: summary of effects Summary of the effects on PV asymmetry under study Nuclear isospin mixing Nucleon strangeness Coulomb distortion Nuclear deformation Strong N-N interaction Nuclear mass

9 Results: elastic electron scattering cross sections Theory (line) vs. experiment (dots)

10 Results: Isospin mixing & coulomb distortion effects

11 Results: strangeness ρ s =+1.5 ρ s =0 ρ s = < ρ s <

12 Results 32 S

13 Results 28 Si

14 Results 24 Mg

15 Results 12 C

16 Results: optimal kinematic ranges for experiment Momentum transfer (fm -1 ) Incident energy at 10º (MeV) Scattering angle at 1 GeV (º)

17 Results: comparative (A dependence)

18 Results: influence of the N-N N interaction Skyrme force Pairing parameters Nuclear deformation

19 Results 208 Pb

20 Conclusions Study of PV elastic electron scattering off the N=Z, J π =0 + nuclei 12 C, 24 Mg, 28 Si, 32 S. Analysis of experimental feasibility: maximize figure-of-merit & asymmetry deviation. Nuclear ground states obtained from a deformed HF+BCS mean field New features included: COLLECTIVE EFFECTS Isospin mixing Deformation Pairing

21 Conclusions Effects on asymmetry deviation under study: isospin mixing, strangeness, Coulomb distortion We find LARGER isospin-mixing-induced PVasymmetry deviations with respect to previous shell-model calculations Why? We use 11 major shells and each single quasiparticle state is a mixture of radial quantum numbers n of the s.h.o. basis

22 Conclusions PV asymmetry is important in the experimental determination of: - Standard Model coupling constants - Nucleon strange content - Nuclear isospin structure - Neutron distribution in nuclei (PREX experiment)...

23 Isospin mixing and parity- violating electron scattering O. Moreno, P. Sarriguren, E. Moya de Guerra and J. M. Udías (IEM-CSIC Madrid and UCM Madrid) T. W. Donnelly (M.I.T.),.), I. Sick (Univ. Basel)

24 APPENDIX

25 Theoretical formalism Coulomb multipole operators

26 Theoretical formalism Spin-orbit term

27 Results Spin-orbit term

28 Strangeness contributions to PV electron scattering Isospin-mixing contribution to PV electron scattering G En 0 G (s) 0

29 Neutron distribution from PV asymmetry in e - scatt. 1

30 Standard Model coupling constants

31 Nuclear deformations

32

33 Results Multipole (l,j) analysis of isovector contributions

34 Results: strangeness contribution

35 Results 208 Pb

36 Nucleon form factors G Ep, G Mp, G En, G Mn : Höhler et al., Nucl. Phys. B 114 (1976) 505 G E (s), G M (s) :

37 Isospin mixing calculation Exact: Approx.: Expectation value of T perp. squared:

38 Results: isospin mixing % %

39 Results: densities

40 Results: form factors