Information content of a new observable Witold Nazarewicz UWS, May 25, 2010

1. Information content of a new observable Witold Nazarewicz UWS, May 25, 2010 Nuclei communicate with us through a great variety of observables. Some are easy to measure, some take a considerable effort and experimental ingenuity. In this study, we show how to assess the uniqueness and usefulness of an observable, i.e., its information content with respect to current theoretical models. We also quantify the meaning of a correlation between different observables and discuss how to estimate theoretical statistical uncertainties. The methodology used in this work should be of interest to any theoretical framework that contains parameters adjusted to measured data.

2. Early attempts to employ statistical methods of linear-regression and error analysis have been revived recently and been applied to determine the correlations between model parameters, parameter uncertainties, and the errors of calculated observables. This is essential for providing predictive capability and extrapolability, and estimate the theoretical uncertainties. • G.F. Bertsch et al., Phys. Rev. C 71, 054311 (2005). • M. Kortelainen et al., Phys. Rev. C 77, 064307 (2008). • J. Toivanen et al., Phys. Rev. C 78, 034306 (2008). • P. Klüpfel et al., Phys. Rev. C 79, 034310 (2009). • P.-G. Reinhard and W. Nazarewicz, Phys. Rev. C, in press (2010). • M . Kortelainen et al., to be published (2010). Examples of some DFT-based work M . Kortelainen et al., Phys. Rev. C 77, 064307 (2008)

3. Based on: P.G. Reinhard and WN, Phys. Rev. C (R) 2010; arXiv:1002.4140) M. Kortelainen et al., 2010 To what extent is a new observable independent of existing ones and what new information does it bring in? Without any preconceived knowledge, all different observables are independent of each other and can usefully inform theory. On the other extreme, new data would be redundant if our theoretical model were perfect. Reality lies in between. Consider a model described by coupling constants Any predicted expectation value of an observable is a function of these parameters. Since the number of parameters is much smaller than the number of observables, there must exist correlations between computed quantities. Moreover, since the model space has been optimized to a limited set of observables, there may also exist correlations between model parameters. How to confine the model space to a physically reasonable domain? Statistical methods of linear-regression and error analysis Objective function fit-observables (may include pseudo-data)

4. Consider a model described by coupling constants The optimum parameter set The reasonable domain is defined as that multitude of parameters around minimum that fall inside the covariance ellipsoid : Uncertainty in variable A: Correlation between variables A and B:

5. Product-moment correlation coefficient between two observables/variables A and B: =1: full alignment/correlation =0: not aligned/statistically independent Some of the previous theoretical papers have explored the dependence between observables by explicit variation of selected properties (e.g. symmetry energy) within the given model. The present covariance analysis is the least biased and most exhausting way to find out the correlations between all conceivable observables. There remain, however, what is called systematic errors which are here hidden constraints and limitations of the given model. Such systematic errors can only be determined by comparing different models or sufficiently flexible variants of a model. A comparison of different models is thus an instructive complement. But the use of different models is not appropriate to quantitatively assess the correlation between observables. For that reason, our study is based on the covariance analysis within the framework of one model.

6. Nuclear Density Functional Theory and Extensions Input NN+NNN interactions Density Matrix Expansion Density dependent interactions Energy Density Functional Optimization • Fit-observables • experiment • pseudo data DFT variational principle HF, HFB (self-consistency) Symmetry breaking Symmetry restoration Multi-reference DFT (GCM) Time dependent DFT (TDHFB) Observables • two fermi liquids • self-bound • superfluid (ph and pp channels) • self-consistent mean-fields • broken-symmetry generalized product states • Direct comparison with experiment • Pseudo-data for reactions and astrophysics

7. The model used: DFT (EDF + fitting protocol) The fit-observables embrace nuclear bulk properties (binding energies, surface thicknesses, charge radii, spin-orbit splittings, and pairing gaps) for selected semi-magic nuclei which are proven to allow a reasonable DFT description. SV-min Skyrme functional P. Klüpfel et al, Phys. Rev. C79, 034310 (2009) RMF-d-t RMF functional Includes isoscalar scalar, vector, isovector vector, tensor couplings of vector fields, isovector scalar field with mass 980 MeV, and the Coulomb field; the density dependence is modeled only by non-linear couplings of the scalar field. Since the resulting NMP of this model (K=197MeV, asym=38MeV,m*/m=0.59) strongly deviate from the accepted values, we use this model only to discuss the robustness of our certain predictions and to illustrate the model dependence of the statistical analysis.

8. Quantities of interest… bulk equilibrium symmetry energy symmetry energy at surface density slope of binding energy of neutron matter dipole polarizability low-energy dipole strength neutron skin

9. Neutron-rich Matter in the Heavens and on Earth 132Sn Equation of state Laboratory observables In-medium interactions Many-body theory Neutron star crust Astronomical observables Microphysics (transport,…)

10. Various correlations reported… Typel and Brown, Phys. Rev. C 64, 027302 (2001) Furnstahl, Nucl. Phys. A 706, 85 (2002) Klimkiewicz et al., Phys. Rev. C 76, 051603(R) (2007) Yoshida and Sagawa, Phys. Rev. C 69, 024318 (2004)

11. An example… • A.Veyssiere et al., Nucl. Phys. A 159, 561 (1970) • E. Lipparini and S. Stringari, Phys. Rep. 175, 103 (1989) A 10% experimental uncertainty due to statistical and photon-beam calibration errors makes it impossible to use the current best value of aD as an independent check on neutron skin.

12. Good isovector indicators Poor isovector indicators

13. To estimate the impact of precise experimental determination of neutron skin, we generated a new functional SV-min-Rn by adding the value of neutron radius in 208Pb, rn=5.61 fm, with an adopted error 0.02 fm, to the set of fit observables. With this new functional, calculated uncertainties on isovector indicators shrink by about a factor of two.

15. Assessing impact of new measurements We also carried out calculations with a new EDF obtained by a new fit where the neutron-rich nuclei have been given more weight (a factor 2 to 3 for the three outermost neutron-rich isotopes in most chains). The purpose of this exercise is to simulate the expected increased amount of data on neutron-rich nuclei. While the correlations seem to change very little, the extrapolation uncertainties in neutron observables shrink by a factor of 1.5–2.0. For instance, with this new functional, the predicted neutron skin in 208Pb is 0.191(0.024) fm, as compared to the SV-min value of 0.170(0.037) fm. This exercise demonstrates that detailed conclusions of the statistical analysis depend on a chosen model and a selected set of fit observables.

16. Summary (1) We propose to use a statistical least-squares analysis to identify the impact of new observables, quantify correlations between predicted observables, and assess uncertainties of theoretical predictions. To illustrate the concept, we studied the neutron radius of 208Pb. By means of covariance analysis we identified a set of good isovector indicators. An indicator that is particularly attractive, as it can be measured in finite nuclei, is dipole polarizability. Unfortunately, the current best experimental value of αD in 208Pb is not known precisely enough to offer an independent check on the neutron skin or to provide a quality constraint on EDF. We also demonstrate that low-energy E1 strength and giant resonance energies are poor indicators of isovector properties, at least those related to neutron skin. While we have good reason to believe that our general conclusion about good and poor isovector indicators is robust, predictions for individual observables are obviously model dependent. This is an important point: even the best statistical analysis is not going to eliminate systematic errors due to incorrect theoretical assumptions. The methodology used in this work should be of interest to any theoretical framework that contains parameters fine-tuned to experiment. Examples include fits of nucleon-nucleon forces to scattering and few-body data, adjustments of shell-model matrix elements, and fits of coupling constants of symmetry-dictated Hamiltonians.

17. UNEDFpre functional optimization fit-observables

18. New model-based derivative-free optimizer

19. Binding energies Deformation properties

20. Sensitivity analysis

21. Sensitivity analysis with respect to individual data points

22. Summary (2) The purpose of this study was to optimize the standard Skyrme functional based on a set of experimental data (masses, charge radii, and odd-even mass differences) pertaining to 72 spherical and deformed nuclei amenable to a mean-field description. The new model-based optimization algorithm was compared with other standard derivative-free optimization methods such as Nelder-Mead and was found to be significantly better in terms of speed, accuracy, and precision. The optimization was carried out at the fully self-consistent, deformed Hartree-Fock-Bogoliubov level. Here, we took advantage of the efficient DFT solver hfbtho optimized in the first phase of the project. The resulting parameter set UNEDFpre gives good agreement with experimental masses, radii, and deformations. This functional is expected to work well for heavy nuclei and should be considered as a reference against which more advanced EDFs will be benchmarked. We have applied full-fledged regression diagnostics, focusing on statistical correlations between model parameters and the sensitivity of parameters to variations in fit observables. To this end, we computed and analyzed the correlation and sensitivity matrices at the optimal parameter set. This kind of nonlinear regression analysis is expected to be helpful when designing next generation EDFs. Moreover, the statistical tools presented in this study can be used to pinpoint specific nuclear observables that are expected to strongly affect the developments of the nuclear universal density functional.