# Bayesian inferences of Skyrme Energy Density Functionals

Pietro Klausner, Gianluca Colò, Xavier Roca-Maza, Enrico Vigezzi

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Pietro Klausner, Gianluca Colò, Xavier Roca-Maza, Enrico Vigezzi. Bayesian inferences of Skyrme Energy Density Functionals. 14th International Spring Seminar on Nuclear Physics: "Cutting-edge developments in nuclear structure physics", May 2025, Ischia, Italy. pp.01006, 10.1051/epj-conf/202534201006. hal-05381767

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# Bayesian inferences of Skyrme Energy Density Functionals

Pietro Klausner<sup>1,2,3,*</sup>, Gianluca Colò<sup>1,2,**</sup>, Xavier Roca-Maza<sup>1,2,4,5,***</sup>, and Enrico Vigezzi<sup>2,****</sup>

<sup>1</sup>Dipartimento di Fisica “Aldo Pontremoli”, Università degli Studi di Milano, 20133 Milano, Italy

<sup>2</sup>INFN, Sezione di Milano, 20133 Milano, Italy

<sup>3</sup>Laboratoire de Physique Corpusculaire L.P.C., CNRS, ENSICAEN, UMR6534, Université de Caen Normandie, CEDEX, 14050 Caen, France

<sup>4</sup>Departament de Física Quàntica i Astrofísica, Martí i Franqués, 1, 08028 Barcelona, Spain

<sup>5</sup>Institut de Ciències del Cosmos, Universitat de Barcelona, Martí i Franqués, 1, 08028 Barcelona, Spain

**Abstract.** Nuclear Density Functional Theory (DFT) is the microscopic nuclear model with the broadest range of applicability; at the same time, several ways to parameterise an Energy Density Functional (EDF) exist, and improvements in the performances of the different EDFs seem to have slowed down in recent years. In this contribution, results of different Bayesian inferences of the Skyrme parameters will be discussed; in particular, we will focus on the differences related to the use of different target observables.

## 1 Introduction

Nuclear EDFs have a long success story because they are based on a formal rigorous framework, namely DFT that dates back to the work by Hohenberg and Kohn, and at the same time they have the flexibility associated with the different *Ansätze* for the form of the EDF that one can use as a start, as well as with the different parameters and fitting procedures. Typical nuclear EDFs are Skyrme, Gogny or covariant ones. Review papers and books exist on this topic [1–4]. In short, nuclear EDFs have of the order of ≈ 10 free parameters, that are fit typically on some tenths of observables and can be used to study and predict many more like e.g. hundreds of masses and radii, in addition to excited states, reactions involving more nuclei, decay properties etc. The way to fix the parameters is the phenomenological part of nuclear DFT while, as said, the formal ground is rigorous; in fact, in the Coulomb case, DFT is said to be an *ab initio* approach.

Advances in devising EDFs with better accuracy and predictive power are currently reported in the literature. At the same time, there exist indications that current EDFs may be reaching their limits. Such a claim, in the case of the Skyrme EDFs, has been made when the UNEDF set has been built [5]. Within the same Skyrme framework, it has been shown that extending the form of the EDF, namely including fourth-order derivative terms on top of the usual second-order ones, does not lead to very significant improvements [6]. A symptom of the obstacles in further improving functionals can be found in the difficulty of reproducing

<footnote>*e-mail: pietro.klausner@unimi.it</footnote>

<footnote>**e-mail: gianluca.colo@mi.infn.it</footnote>

<footnote>***e-mail: xavier.roca.maza@fqa.ub.es</footnote>

<footnote>****e-mail: enrico.vigezzi@mi.infn.it</footnote>
new, different observables. EDFs tend to loose their predictive power in regions far from those in which they have been fit, like e.g. for exotic nuclei close to the drip line (Fig. 2 in [7] may be quoted as an example).

Different strategies can be thought to be helpful for making a leap and designing a new generation of better EDFs that overcome the problems we have mentioned. In Coulomb DFT, since several decades, there are functionals whose exchange and correlation part is derived from more fundamental approaches like Quantum Monte Carlo for the electron gas and this goes hand in hand with what we said above, namely that Coulomb DFT is said to be *ab initio*. Merging nuclear DFT with *ab initio* should be definitely pursued [8–13] but this is not the subject of this contribution. Here, we focus on the attempt to ameliorate the determination of the EDFs parameters using advanced techniques like Bayesian inference.

Bayesian inference techniques allow estimating the entire probability distributions of the parameter values, while the more traditional $\chi^2$-test only provides the best estimates for these parameters. The Bayesian framework allows a neat analysis of correlations between parameters as well as between parameters and observables. Obviously, the features and performances of an EDF strongly depend on the observables that are used in the fit and on the weight that the observables are given. We would like to highlight this point in this short contribution. We will review a recent Bayesian inference that we have performed and published in Ref. [14]. Then, we will specifically analyse some parameters that result from the inference and highly debated at present (they are those controlling the density dependence of the symmetry energy): we will show that by changing the input for the inference, namely the pool of target data, the posterior distributions for these parameters are shifted.

We will briefly recall the theoretical framework, namely the formalism of the Skyrme EDFs and the Bayesian setup that we specifically employ, in Secs. 2 and 3, respectively. Then, we will discuss the results of [14] and of the new inference that we have performed afterwards, in Sec. 4.

## 2 The Skyrme quasi-local EDF

The Skyrme EDF is described in detail in the references that we have already cited. There are two main classes of EDFs. On the one hand, there are those which are derived from the traditional Skyrme force and are characterised by the standard parameters $t_i$, $x_i$, $\sigma$ and $W_0$ like in Ref. [15]. We use such standard form for the sake of simplicity. Other EDFs are written directly without reference to an underlying Hamiltonian like those of the UNEDF family that we mentioned in the Introduction. In either case, the total energy is a *quasi-local* functional. In fact, the total energy is an integral of the energy density that can be split in four parts for the sake of clarity:

$$E = \int d^3\mathbf{r} \left( \mathcal{E}_{\text{kin}} + \mathcal{E}_{\text{Skyrme}} + \mathcal{E}_{\text{Coul}} + \mathcal{E}_{\text{pair}} \right). \quad (1)$$

While the kinetic energy is that of independent particles, the most important term, that is the Skyrme energy density, is

$$\mathcal{E}_{\text{Skyrme}}(\vec{r}) = f[\tau_t(\vec{r}), \rho_t(\vec{r}), \mathbf{J}_t(\vec{r})], \quad (2)$$

namely the energy density at point $\vec{r}$ is a function of the density $\rho$, the kinetic energy density $\tau$, and the spin-orbit density $\mathbf{J}$ at the very same point. Since derivatives of $\rho$ also appear in $\mathcal{E}_{\text{Skyrme}}$ the functional is called quasi-local. The index $t$ labels isoscalar ($t=0$) and isovector ($t=1$) densities (no charge-mixing is assumed). The dependence of $\mathcal{E}_{\text{Skyrme}}$ on these densities is given in terms of parameters $C$ that are defined in Ref. [16]. Echoing what is said above, these can be taken as the basic EDF parameters or, in the case of an EDF extracted from an
effective Hamiltonian, are functions of the parameters $t_i, x_i, \sigma$ and $W_0$. In the Coulomb term, the exchange part is made local by the use of the Slater approximation. The pairing energy density is that obtained from a zero-range volume pairing interaction and, consequently, a further parameter is the strength of such interaction $v_0$.

As discussed in the next Section, the observables that we aim to reproduce are either ground-state properties or properties associated with the nuclear response or with Giant Resonances. The ground-state properties are extracted from Skyrme Hartree-Fock (HF) in the case of closed-shell nuclei or HF+BCS in the case of open-shell nuclei. Nuclear excited states are evaluated using the Random Phase Approximation or RPA (or its extension, the Quasiparticle RPA or QRPA) in the case of closed-shell (or open-shell) systems. We use the code described in [17] (see also Ref. [18] where more details are provided).

Instead of performing the Bayesian inference starting from either the C parameters or from $(t_i, x_i, \sigma, W_0)$, we resort to combinations that have a clearer physical meaning. Seven of them are well-known parameters of the nuclear Equation of State (EoS). There are: the nuclear saturation density of symmetric nuclear matter $n_{sat}$ that is implicitly defined by the fact that the pressure $P$ at saturation density vanishes; the energy at saturation density, $E_{sat}$; the nuclear incompressibility $K_{sat}$; the symmetry energy at saturation $E_{sym}$; the slope parameter $L_{sym}$; the isoscalar and the isovector effective masses. The relationships with the EDF parameters are given explicitly in [14]. We need to add the remaining four parameters that are relevant for finite nuclei and do not appear in the EoS. Three of them are: the isoscalar surface parameter $G_0 = -C_0^{\Delta\rho}/2$; its isovector counterpart, $G_1 = -C_1^{\Delta\rho}/2$; the spin-orbit strength $W_0$. Finally, the last parameter is the strength of the pairing interaction $v_0$ that has been introduced in the first part of this Section.

## 3 The Bayesian setup and the target observables

As in many other applications of the Bayes' theorem in physics, we are interested in the following question. We have a model $M$ characterised by a set of parameters that we call $\theta$. This set has been spelled out at the end of the previous Section. Given a set of data $X$ (that are our “target observables” and will be discussed shortly), we are interested in the conditional probability $p(\theta|X)$, that is the probability distribution of the parameters given the data. Using the Bayes' theorem we can write

$$p(\theta|X) = \frac{p(X|\theta)p(\theta)}{p(X)} \quad (3)$$

In Bayesian jargon, each term in this equation has a precise name.

* $p(\theta|X)$ is called the posterior probability distribution (also known simply as *posterior*);

* $p(X|\theta)$ is called the *likelihood* (sometimes indicated with the symbol $\mathcal{L}$);

* $p(\theta)$ is the prior probability distribution (also simply referred to as *prior*);

* $p(X)$, called *evidence*, is the normalization factor, as it is $\theta$-independent.

We will not be concerned with $p(X)$, as we will calculate only the numerator of Eq. (3). We will make the rather agnostic choice of priors that are flat within a given interval. The interval is, at first, guessed based on the values explored by the Skyrme EDFs in use. During our inference process, however, we will check that the posterior distributions do not explore significantly the region outside of the interval (and, in case, we readjust it). The intervals are shown in Table 1.

The data $X$ that have been used are reported in Table 2. The ground-state properties of closed-shell nuclei and the isoscalar/isovector properties of excited states are, to a large
<table><caption>**Table 1.** Lower and upper limits of the interval of the uniform prior distribution of each model parameter.</caption><thead><tr><th></th><th>Units</th><th>Lower limit</th><th>Upper limit</th></tr></thead><tbody><tr><td>n<sub>sat</sub></td><td>[fm<sup>-3</sup>]</td><td>0.150</td><td>0.175</td></tr><tr><td>E<sub>sat</sub></td><td>[MeV]</td><td>-16.50</td><td>-15.50</td></tr><tr><td>K<sub>sat</sub></td><td>[MeV]</td><td>180.00</td><td>260.00</td></tr><tr><td>E<sub>sym</sub></td><td>[MeV]</td><td>24.00</td><td>40.00</td></tr><tr><td>L<sub>sym</sub></td><td>[MeV]</td><td>-20.00</td><td>120.00</td></tr><tr><td>G<sub>0</sub></td><td>[MeV fm<sup>5</sup>]</td><td>90.00</td><td>170.00</td></tr><tr><td>G<sub>1</sub></td><td>[MeV fm<sup>5</sup>]</td><td>-90.00</td><td>70.00</td></tr><tr><td>W<sub>0</sub></td><td>[MeV fm<sup>5</sup>]</td><td>60.00</td><td>190.00</td></tr><tr><td>m<sup>*</sup><sub>0</sub>/m</td><td>[-]</td><td>0.70</td><td>1.10</td></tr><tr><td>m<sup>*</sup><sub>1</sub>/m</td><td>[-]</td><td>0.60</td><td>0.90</td></tr><tr><td>v<sub>0</sub></td><td>[MeV fm<sup>3</sup>]</td><td>150</td><td>350</td></tr></tbody></table>

extent, the same as in Ref. [14]. It should be noted, though, that we have *excluded* magic nuclei with $N = Z$. We refer to that paper for a thorough discussion, and for the list of references from which these quantities have been extracted. Here we only remind a few facts. B.E., $R_{ch}$ and $\Delta E_{SO}$ label binding energies, charge radii and spin-orbit splittings of specific levels (the neutron 2p splitting in $^{48}$Ca and the proton 2f splitting in $^{208}$Pb). The energies of the second part of the Table refer to the isoscalar monopole and quadrupole resonances. In the third part, $\alpha_D$, $m(1)$ and $A_{PV}$ denote, respectively, the dipole polarizability from (p,p') scattering, the energy-weighted sum rule (EWSR) for the isovector dipole strength, and the parity-violating asymmetry. For these latter three quantities, experimental errors are adopted. For the other quantities the errors are taken as theoretical errors and reflect the capability of typical EDFs to reproduce these observables and/or their weight in the inference process. A key point to be kept in mind is that the error that has been assumed should be comparable to the width of the posterior distributions. Again, the reader is invited to consult Ref. [14] where all aspects are discussed in detail.

Compared to Ref. [14], the values (and errors) in bold in the Table have been updated. We have updated the experimental values of the excitation energy of the monopole resonance in $^{90}$Zr because of the new analysis of the raw data that has been published in Ref. [19]. In the case of the parity-violating asymmetry, $A_{PV}$, we have taken into account the fact that QED corrections need to be added to the theory as argued in [20]; as these corrections are model-independent, we dealt with them by simply correcting the experimental values. The most important change with respect to [14] is, however, the inclusion of open-shell nuclei. As these call for the inclusion of the pairing interaction, the standard neutron pairing gap $\Delta_n$ of $^{120}$Sn has been included among the target data.

The posterior distribution of the parameters is sampled through the Metropolis-Hastings algorithm, which is a Markov chain Monte Carlo method that is frequently employed in contexts like the present one. Given a likelihood function $\mathcal{L}$, this implementation of the random walk explores the parameter space, favouring regions having higher likelihood; in our case, the likelihood is a product of different Gaussian distributions, whose means and widths are the adopted values and errors of the observables. The algorithm requires a large number of model evaluations ($\approx 10^6 - 10^7$) to achieve a satisfactory sampling of the posterior distributions of the parameters. To overcome this problem, we resorted to the MADAI package, an emulator software based on Gaussian Processes, built for Bayesian inferences with slow
<table><caption>**Table 2.** Observables used for the inference. See the main text for details.</caption><thead><tr><th colspan="4">Ground-state properties of magic nuclei with N ≠ Z</th></tr><tr><th></th><th>B.E. [MeV]</th><th>R<sub>ch</sub> [fm]</th><th>ΔE<sub>SO</sub> [MeV]</th></tr></thead><tbody><tr><td><sup>208</sup>Pb</td><td>1636.4 ± 2.0</td><td>5.50 ± 0.05</td><td>2.02 ± 0.50</td></tr><tr><td><sup>48</sup>Ca</td><td>416.0 ± 2.0</td><td>3.48 ± 0.05</td><td>1.72 ± 0.50</td></tr><tr><td><sup>68</sup>Ni</td><td>590.4 ± 2.0</td><td>-</td><td>-</td></tr><tr><td><sup>132</sup>Sn</td><td>1102.8 ± 2.0</td><td>4.71 ± 0.05</td><td>-</td></tr><tr><td><sup>90</sup>Zr</td><td>783.9 ± 2.0</td><td>4.27 ± 0.05</td><td>-</td></tr></tbody></table>

<table><thead><tr><th colspan="3">Isoscalar resonances</th></tr><tr><th></th><th>E<sup>IS</sup><sub>GMR</sub> [MeV]</th><th>E<sup>IS</sup><sub>GQR</sub> [MeV]</th></tr></thead><tbody><tr><td><sup>208</sup>Pb</td><td>13.5 ± 0.5</td><td>10.9 ± 0.5</td></tr><tr><td><sup>90</sup>Zr</td><td><b>18.7 ± 0.5</b></td><td>-</td></tr></tbody></table>

<table><thead><tr><th colspan="4">Isovector properties</th></tr><tr><th></th><th>α<sub>D</sub> [fm<sup>3</sup>]</th><th>m(1) [MeV fm<sup>2</sup>]</th><th>A<sub>PV</sub> [ppb]</th></tr></thead><tbody><tr><td><sup>208</sup>Pb</td><td>19.60 ± 0.60</td><td>961 ± 22</td><td><b>528 ± 18</b></td></tr><tr><td><sup>48</sup>Ca</td><td>2.07 ± 0.22</td><td>-</td><td><b>2550 ± 113</b></td></tr></tbody></table>

<table><thead><tr><th colspan="4">Ground-state properties of open-shell nuclei</th></tr><tr><th></th><th>B.E. [MeV]</th><th>R<sub>ch</sub> [fm]</th><th>Δ<sub>n</sub> [MeV]</th></tr></thead><tbody><tr><td><sup>50</sup>Ca</td><td>427.5 ± 2.0</td><td>3.52 ± 0.05</td><td>-</td></tr><tr><td><sup>46</sup>Ca</td><td>398.8 ± 2.0</td><td>-</td><td>-</td></tr><tr><td><sup>44</sup>Ca</td><td>381.0 ± 2.0</td><td>-</td><td>-</td></tr><tr><td><sup>42</sup>Ca</td><td>361.9 ± 2.0</td><td>-</td><td>-</td></tr><tr><td><sup>120</sup>Sn</td><td>1020.5 ± 2.0</td><td>4.65 ± 0.05</td><td>1.3 ± 0.2</td></tr><tr><td><sup>112</sup>Sn</td><td>953.5 ± 2.0</td><td>-</td><td>-</td></tr><tr><td><sup>124</sup>Sn</td><td>1050.0 ± 2.0</td><td>-</td><td>-</td></tr></tbody></table>

models. This tool was developed by the MADAI collaboration (Models and Data Analysis Initiative) [21].

## 4 Results

The published results of Ref. [14] were in line with previous EDF fits and with a few similar analysis, with the significant exception of the symmetry energy parameters $E_{sym}$ and especially $L_{sym}$. Since these are isovector properties, the fact that we had included in the observables the ground-state properties of only few nuclei, with a significant fraction of nuclei with $N = Z$, may have played a role. Nuclei with $N = Z$ have been found to display specific correlations that are not captured by the widely used EDFs: in fact, in mass models, terms like the Wigner term that is prominent at $N \approx Z$ and can mimic proton-neutron pairing or clustering, are often included. Therefore, taking care of the bias that $N = Z$ nuclei may induce, we have excluded them; and, considering that recent EDF fits have included many more nuclei than in the past (like in [5] and references therein), we have added open-shell nuclei to the pool of observables, as described in the previous Section. In total, we have added ten more observables with respect to [14]; the goal of this Section is to highlight the changes that this produces in the results of the inference.

Fig. 1 has the same format as Fig. 2 in [14] and, in principle, allows comparing the results directly (except for the case of $v_0$ that has been included only in the new inferences
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Marginalized posterior distributions of all parameters. (a) is the result published in [14]. (b) includes open-shell nuclei and the updated monopole energy but not the updated values of $A_{PV}$. (c) is the result that includes all observables listed in Table 2. Finally, (d) singles out the effect of the $A_{PV}$ of $^{48}$Ca by excluding the $A_{PV}$ of $^{208}$Pb.</figcaption></figure>

reported here). However, the scale prevents from understanding some fine details. We have reproduced in row (a) the final result of [14] and displayed the new results in rows (b-d). We aim at understanding separately the effects on isovector properties brought, respectively, by the larger amount of nuclei with $N > Z$ and the updated values of $A_{PV}$. This is the reason for distinguishing the cases (b) and (c). The updated monopole energy is an isoscalar observable and has been included in all new inferences (b-d). The rationale behind the case (d) is discussed below.

One of the main conclusions that can be drawn is that the values of $E_{sym}$ and $L_{sym}$ are shifted upward in the new inferences, as seen from rows (b) and (c). The effect of a larger pool of nuclei, and of the exclusion of those with $N = Z$, is that of bringing the symmetry energy parameters in better agreement with the commonly accepted values (although they might be still lower than some estimates in the literature). In quantitative terms, when considering all the data from Table 2, we find that $E_{sym}$ peaks at $\approx 32$ MeV, while $L_{sym}$ peaks at $\approx 40$ MeV. Of course, the subtle balance between the inclusion of more nuclei, and the competing effects of the dipole polarizability and the parity-violating asymmetry, cannot be fully appreciated from Fig. 1 and will be further discussed below. A few more comments related to the results of Fig. 1 are in order. The new monopole energy in $^{90}$Zr, that is higher than the one from the previous analysis, leads clearly to higher values of $K_{sat}$. This has a side effect. The incompressibility is known to scale accordingly to the power $\sigma$ of the $t_3$ term in the Skyrme force ($K_{sat} \sim \sigma$, see e.g. p. 183 of [22]). A larger value of $\sigma$ leads also to a smaller $n_{sat}$. Finally, the distributions of $G_0$, $G_1$, $W_0$ and of both effective masses have not changed much from the results of [14].

Some details are more clear in Fig. 2 as compared with Fig. 1. We can see better that $K_{sat}$ is shifted upward, while as a consequence $n_{sat}$ is slightly shifted downward, when moving from the inference published in Ref. [14] to the new ones. $E_{sat}$ is also slightly shifted downward. We can also confirm that the isovector effective mass has not changed at all and that the changes in the posterior distributions of the isoscalar effective mass, and of $G_0$, $G_1$ and $W_0$,
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Marginalized posterior distributions as in Fig. 1 but on a different scale. The four curves correspond to the four cases already shown, but here the legend should be self-explanatory. (a) is Ref. [14], (b) is Ref. [19], and (c) is Ref. [20].</figcaption></figure>

are not large. On the other hand, the significant shift of $E_{sym}$ and $L_{sym}$ towards larger values can be well appreciated in this figure.

Fig. 2 allows also to comment more on the parity-violating asymmetry $A_{PV}$. As shown in Fig. 2 of [23] all models have difficulty to reconcile the reproduction of the $A_{PV}$ values in $^{48}$Ca and $^{208}$Pb. If they manage to reproduce the value in $^{208}$Pb, the value in $^{48}$Ca is too large; if they manage to reproduce the value in $^{48}$Ca, the value in $^{208}$Pb is too small. In other words, the $A_{PV}$ of $^{208}$Pb points to larger values of $L_{sym}$ as compared to $^{48}$Ca: these large values are in contrast with what is demanded by other observables like the dipole polarizability or the masses and, accordingly, the Skyrme EDFs that were most likely, in probabilistic terms, in Ref. [14], were able to reproduce all observables except the $A_{PV}$ of $^{208}$Pb. We can now discuss what happens after the corrections of [20]. The new corrected value in $^{208}$Pb, $A_{PV} = 528$ p.p.b., exacerbates the tension with other observables (in particular, with $\alpha_D$). By itself, it would demand a very large value of $L_{sym}$ of the order of 160 MeV. This is why we have performed the last inference, marked by (d) in Fig. 2, in which we have removed the $A_{PV}$ of $^{208}$Pb. One can see that the $A_{PV}$ of $^{208}$Pb is the main responsible for drifting the symmetry energy parameters, while the $A_{PV}$ of $^{48}$Ca, by itself, has no more impact that the open-shell nuclei.

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