Uncertainty evaluation of peak energy of giant dipole resonance propagated from uncertainty of parameters of effective interaction

. We evaluate uncertainty of giant dipole resonance (GDR) peak energy propagated from the uncertainty of effective interaction parameter. The Monte Carlo calculation of microscopic random phase approximation using randomized parameter sets is performed. Under the condition that correlations between the parameters are considered, the calculated GDR peak energy has an uncertainty of ∼ 1 MeV irrespective of nuclear mass and is strongly correlated with the parameters, in the present calculations. Our result serves as a guide for a new parametrization of effective interaction.


Introduction
Random phase approximation (RPA) calculation with effective interaction is one of the standard theoretical tools to calculate photoabsorption cross section in MeV region from the degree of freedom of nucleons, and has been applied to systematic calculations of nuclei covering wide range of the nuclear chart [1,2]. However, it is known that the RPA calculation with Skyrme effective interaction underestimates peak energies of giant dipole resonances (GDRs) in light nuclei by a few MeV. The parameters of effective interactions are determined to reproduce limited sets of experimental data of ground states and nuclear matter properties. This limitation raises the question of its predictive power especially for excited states. Therefore, uncertainty estimation of the calculated values and feedback to the parameters are strongly desired. In the last decade, many theoretical studies have estimated uncertainty of calculated values and correlations of calculated values and the parameters [3][4][5][6][7][8][9][10]. However, these results are not used to improve the parameters of the effective interactions. This is because the relations between the parameters and the calculated values are not known clearly.
In this study, we performed Monte Carlo calculation of RPA using randomized parameters in Skyrme interaction. We evaluated the uncertainty of the GDR peak energy, propagated from the uncertainties of Skyrme parameters. By explicit taking the correlations of the parameters in randomization, we found that there are strong correlations between the Skyrme parameters and the calculated GDR peak energy.

Method
* Corresponding author: inakura.t.aa@m.titech.ac.jp We employ Skyrme SLy5-min parameter set as the effective interaction. The SLy5-min parameters and their uncertainties Δ [8] are listed in Table 1. Note that parameters 2 , 0 , and are fixed when the SLy5-min parameters were determined and thus they do not have uncertainties. Table 2 shows a part of correlation matrix of the SLy5-min parameters. Other matrix elements of correlation are listed in Ref. [11].  For the Monte Carlo calculation using randomized SLy5-min parameter sets, we generate = 1000 random samples, ( ) ( = 1,2, … , ), which satisfy the correlation . Before generating the random samples, we introduce a one-dimensional normal distribution ( , ), where and are mean value and standard deviation of the distribution, respectively. And in preparation, the correlation matrix , which is positive semidefinite matrix, is factorized as = by the singular value decomposition.
First, we generate sets of random independent and is uncorrelated with each others. Secondly, we act the factorized matrix on each  Table 1. These are the correlated random parameter sets of SLy5-min.
Using these randomized parameter sets ( ) , we perform the RPA calculations times. The RPA solver is Skyrme-rpa [12]. The Skyrme-rpa solves the RPA equation for spherical nuclei in the coordinate representation. The calculation space is a sphere with radius 25 fm and mesh span 0.1 fm. The photoabsorption cross section is computed from the resulting dipole strength functions and smeared with a width Γ = 2 MeV. From this photoabsorption cross section, we extract the position of the GDR peak top of jth random parameter set, ( ) . This calculated ( ) is different form one that the original SL5-min parameter set produces because different parameter set ( ) is used. These are repeated = 1000 times. The mean value and the standard deviation of the calculated GDR peak energies are obtained by statistical process from these 1000 results. This is the uncertainty evaluation of propagated from the uncertainties of the parameters.

Results and discussion
The photoabsorption cross sections in 40 Ca calculated with the random parameter sets are plotted in Figs. 1. In Fig. 1(b), the red line shows the photoabsorption cross section calculated with the original SLy5-min parameter set, and black lines are those with the randomized parameter sets. Figure 1(a) shows histogram of the randomized GDR peak energies. Arrow denotes mean value of the peak energies and its standard deviation, 17.9 ± 1.5 MeV. Remarkably, the randomized GDR peak energies have two regions separated in their energy. In upper energy region, the randomized peak energies scatter around = 18.3 MeV that the original SLy5-min results. In the lower energy region at ≤ 17 MeV, the bunch is clearly separated from the upper one, while the number of the low-energy randomized peaks is ∼ 200. These peaks originate from a shoulder at 17 MeV of the original cross section denoted by led line in Fig. 1(b). Figure 1(c) shows averaged cross section and its standard deviations. If the parameter uncertainties are small, | −Δ | | | ≪ 1 , the averaged cross section is expected to be close to the original cross section. However, the averaged cross section is different from the original one. The GDR peak height is lowered, its width is broadened To see which parameter affects the GDR peak energy, we calculate the Pearson correlation coefficient of the GDR peak energies and the parameters . Figures 2 show relations of the randomized parameters and the corresponding randomized GDR peak energies. Note that the parameters 2 , 0 , and are fixed in the SLy5-min parameter set, and therefore the relations of 0 and are not plotted and always 2 = −1 . The randomized GDR peak energies have strong correlations with 0 , 1 , 2 , and 3 , and their Pearson correlation coefficients calculated from all randomized GDR peak energies are 0.8 or -0.8. Since the randomized GDR peak energies has two separated bunches [See Fig. 1(a)], the plotted relations are also separated into upper-and lower-energy regions. If we pick up the upper-energy region only, the correlations become stronger.
Similar values of these Pearson correlation coefficients are related to the strong correlations between 0 , 1 , 2 , and 3 , shown in Table 2. The moduli of the correlations between 0 , 1 , 2 , and 3 are larger than 0.95. This means that uncertainties of 1 , 2 , and 3 are almost redundant and can be approximated by the uncertainty of 0 . Similar is seen in the relations of the randomized GDR peak energies and the parameters 0 , 1 , and 3 . The moduli of the correlations between 0 , 1 , and 3 are larger than 0.93. The strong correlations between the randomized GDR peak energies and the parameter enable us to improve the calculated GDR peak energy by tuning the parameters.
Same Monte Carlo calculations are performed for spherical nuclei, 16 Table 3. The standard deviations of the GDR peak energies are 1.0-1.5 MeV, irrespective of nuclear mass. The Pearson correlation coefficients between the randomized GDR peak energies and the randomized parameters 0,1,2,3 in 16 O, 90 Zr, and 208 Pb are strong, similar to those in 40 Ca. These moduli are approximately 0.83, 0.86, 0.65, respectively. Also, the correlation coefficients of 0,1,3 are almost same as those in 40 Ca. The uncertainty propagation from the parameters to the GDR peak energies are not sensitive to nuclear mass.   We performed the Monte Carlo calculation to evaluate the uncertainty of the GDR peak energy, propagated from the uncertainties of effective interaction parameters. The RPA calculation with randomized parameters is applied to spherical nuclei, 16 O, 40 Ca, 90 Zr, and 208 Pb. In the case that SLy5-min parameters is employed with the correlations between the parameters, the standard deviations of the GDR peak energies are ∼ 1 MeV, irrespective of nuclear mass. The GDR peak energy has strong correlations with the parameters. These correlations enable us to handle the calculated GDR peak energy by tuning the parameters.
One of the promising ways to tune the parameters is by shifting some parameters in directions that calculated becomes higher, such as 0 → 0 + Δ 0 , with satisfying the correlation matrix ( Table 2). Here is a free parameter. We introduce mass dependence to push up larger (smaller) for lighter (heavier) nuclei. We have the plan to propose a new parameter set that reproduces properly.