#
Uncertainty Analysis of ^{208}Pb Neutron Skin Predictions with Chiral Interactions

## Abstract

**:**

^{208}Pb using chiral two- and three-body interactions at increasing orders of chiral effective field theory and varying resolution scales. Closely related quantities, such as the slope of the symmetry energy, are also discussed. The sensitivity of the skin to just pure neutron matter pressure when going from order 2 to order 4 of chiral effective theory is singled out in a set of calculations that employ an empirical equation of state for symmetric nuclear matter.

## 1. Introduction

^{0}boson couples primarily to neutrons [3]. From the first electroweak observation of the neutron skin in a neutron-rich heavy nucleus, a value of ${0.33}_{-0.18}^{+0.16}$ for the neutron skin of

^{208}Pb was determined [4], but the next PREX experiment aims to measure the skin within an uncertainty smaller by a factor of 3 (see [4] and references therein).

^{208}Pb at different orders of chiral EFT and changing resolution scale. Implementation of the full three-nucleon force (3NF) in nuclear matter at N

^{3}LO presents considerable challenges and has not yet been accomplished. Thus, at N

^{3}LO of the two-nucleon force (2NF), we include only the leading (N

^{2}LO) 3NF. Of course, both the 2NF and the 3NF should be complete at each order to explore definite order-by-order convergence, and we make no claim to have accomplished that. However, the result of the present approximation, which is widely used in the current literature, can provide valuable information and insight. Additional discussion of this procedure, particularly with regard to its application and validity in neutron matter, is included in the next section.

_{i}low-energy constants (LECs) within the range allowed by πN scattering data and apply those variations in the three-nucleon force (3NF). We believe that changes in the πN LECs should be applied consistently in the two- and the three-body force, refitting accordingly both the NN phase shifts and the A=3 system. Our values for the c

_{i}are the same in the 2NF and the 3NF.

_{0}. The second approximate equality is due to the vanishing of the first derivative of the energy per particle in SNM at ρ

_{0}, leaving a term proportional to the pressure in neutron matter. Nevertheless, L depends sensitively on the saturation density, which can be quite different from model to model, particularly when considering different chiral orders and regulators. In other words, theoretical predictions of L carry larger EFT uncertainties than the ones of just neutron matter pressure at some fixed density. To explore this point further, we will also compare predictions and uncertainties with those obtained using a phenomenological EoS for SNM consistent with the empirical saturation point.

## 2. Predictions of Symmetry Pressure and Neutron Skin

#### 2.1. Predictions with Microscopic EoS for NM and SNM

_{I}are the usual isoscalar and isovector densities, given by ρ

_{n}+ ρ

_{p}and (ρ

_{n}− ρ

_{p}), respectively, α is the neutron asymmetry parameter, α = ρ

_{I}/ρ, and e(ρ, α) is the energy per particle in isospin-asymmetric nuclear matter. The constant f

_{0}in Equation (2) is approximately 70 MeV fm

^{5}, whereas the magnitude of β is about 1/4 [11]. (Even with variations of β between −1 and +1, we found that the contribution from that term was negligibly small, so we disregarded its contribution.)

_{sym}, is defined as the strength of the quadratic term in an expansion of the energy per particle in asymmetric matter with respect to the asymmetry parameter α:

^{2}has been confirmed by many microscopic calculations (see for instance [12] and more recently [13,14]). It justifies the common approximation of neglecting powers beyond α

^{2}in the expansion above and thus defining the symmetry energy as the difference between the energy per particle in neutron matter and symmetric nuclear matter. Therefore,

^{40}Ca,

^{90}Zr, and

^{208}Pb with some of the Bonn meson-exchange potentials [15].

^{2}LO chiral three-body force. This effective interaction is obtained by summing one particle line over the occupied states in the Fermi sea. Neglecting small contributions from terms depending on the center-of-mass momentum, the resulting NN interaction can be expressed in analytical form with operator structures identical to those of free-space NN interactions. For symmetric nuclear matter all three-body forces contribute, while for pure neutron matter only terms proportional to the low-energy constants c

_{1}and c

_{3}are nonvanishing [16].

^{2}LO), respectively. The blue band is the result of a calculation employing next-to-next-to-next-to-leading order (N

^{3}LO) NN potentials together with 3NFs at N

^{2}LO. The pressure is proportional to the slope of the various curves which make up the corresponding bands shown in Figure 1b. We observe moderate cutoff dependence except at NLO and a slow convergence tendency with increasing order.

**Figure 1.**(

**a**) Pressure in pure neutron matter as a function of density, ρ. The yellow and red bands represent the uncertainties in the predictions due to cutoff variations as obtained in complete calculations at NLO and N

^{2}LO, respectively. The blue band is the result of a calculation employing N

^{3}LO NN potentials together with N

^{2}LO 3NFs. The dashed line shows the upper limit of the yellow band. (

**b**) Energy per particle in pure neutron matter. The meaning of the bands is the same as in (

**a**).

_{0}. The latter changes dramatically from order to order as well as with changing cutoff, which can be clearly seen from Figure 2. In Figure 3, we show the L parameter as a function of density, i.e.,

**Figure 2.**As in Figure 1b for symmetric nuclear matter.

^{208}Pb are summarized in Table 1, along with the corresponding values of the L parameter at the appropriate saturation density, different in each case and also reported in Table 1. Note that we do not show predictions at NLO because, at this low order, only the EoS with the largest cutoff (of 600 MeV) displays some (late) saturating behavior, cf. Figure 2. The upper and lower errors are the distances of the largest and smallest values (when changing the cutoff) from the average.

**Table 1.**Neutron skin thickness, S, in

^{208}Pb at the specified order of chiral EFT as explained in the text. The corresponding values of the L parameter and the saturation density are given in the last two columns.

Order | S(fm) | L(ρ_{0})(MeV) | ρ_{0}(fm^{−3}) |
---|---|---|---|

N^{2}LO | ${0.21}_{-0.02}^{+0.04}$ | ${77.4}_{-16.2}^{+31.2}$ | ${0.167}_{-0.022}^{+0.043}$ |

N^{3}LO | ${0.17}_{-0.01}^{+0.02}$ | ${39.9}_{-15.7}^{+17.2}$ | ${0.144}_{-0.032}^{+0.032}$ |

^{2}LO to be about 0.04 fm. A similar estimate at N

^{3}LO would require knowledge of the prediction at N

^{4}LO, which is not available. Assuming a (pessimistic) truncation error at N

^{3}LO of similar size as the one at N

^{2}LO, we then summarize our predictions for the skin as 0.17 ± 0.04 fm, where the error is likely to be smaller assuming a reasonable convergence rate. In fact, if one takes the cutoff variation as a realistic estimate of the error (as it is approximately the case at N

^{2}LO, cf. Table 1), then our N

^{3}LO prediction carries an error of 0.02 fm.

#### 2.2. Using a Phenomenological EoS for Symmetric Nuclear Matter

^{3}LO include the leading 3NF. For pure neutron matter, we expect the contribution from the 3NF at N

^{3}LO to be very small, as it was shown in [19] for the potential of [20] (about −0.5 MeV at normal density). Thus, it is likely that the set of calculations we report below shows a realistic convergence pattern of the skin from NLO to N

^{3}LO as determined by the corresponding pattern in neutron matter. The impact of the 3NF at N

^{3}LO is larger (attractive and about 3 MeV at normal density) if the chiral NN potential of [21] is used instead. We note, though, that a different power counting scheme is used by the authors of those interactions, and thus a comparison, particularly within the context of examining order-by-order pattern, would be inconsistent. We also observe that, in [19], the 3NF at N

^{2}LO and at N

^{3}LO are applied at the Hartree–Fock level. The low-energy constants c

_{i}are extracted from πN analyses at the respective orders, with uncertainties estimated by applying variations of those c

_{i}in the 3NF but not in the corresponding 2NF. The resulting impact of the 3NF contribution at N

^{3}LO is an enhancement of about 3 MeV with the potential of [20]. We end these comments by stressing again the importance of complete calculations at each order beyond the Hartree–Fock approximation in order to reach definite conclusions on the convergence pattern of the neutron skin.

_{0}= 0.16 fm

^{−3}with energy per particle equal to −16.0 MeV. The corresponding findings are displayed in Table 2. For this test, we also show the results at NLO, since the saturation point can be defined for all cases. Although the midvalues are reasonably consistent with those in Table 1, the uncertainties are much smaller, particularly for the L parameter, as to be expected based on the previous observations. The much smaller uncertainty at N

^{3}LO reflects the negligible cutoff dependence of neutron matter pressure at that order, see Figure 1.

**Table 2.**As Table 1, but employing a phenomenological model for the EoS of SNM. See text for details.

Order | S(fm) | L(ρ_{0})(MeV) |
---|---|---|

NLO | ${0.126}_{-0.003}^{+0.004}$ | ${20.4}_{-6.3}^{+8.8}$ |

N^{2}LO | ${0.20}_{-0.01}^{+0.01}$ | ${70.6}_{-8.0}^{+4.1}$ |

N^{3}LO | ${0.172}_{-0.005}^{+0.002}$ | ${44.9}_{-5.4}^{+3.8}$ |

^{2}LO as the difference between the prediction at this order and the one at the next order, which gives approximately 0.03. Assuming a similar uncertainty at N

^{3}LO, we estimate the skin thickness at N

^{3}LO, when adopting an empirical parametrization for the EoS of SNM, to be 0.17 ± 0.03. We note, again, that this reflects the uncertainty in pure neutron matter at the low densities probed by the skin. Such uncertainty is small, consistent with the low-density behavior seen in Figure 1a.

## 3. Conclusions

^{208}Pb with two- and three-body chiral interactions. The neutron and proton density functions are obtained in a simple approach based on the semi-empirical mass formula. We observed that, in fully microscopic calculations, model dependence from the details of SNM at the saturation point does impact predictions of the symmetry pressure and, to a lesser extent, the neutron skin.

^{4}LO are needed for a better quantification of the truncation error at N

^{3}LO, and thus a reliable comparison of the EFT error with the target uncertainty set by future PREX experiments. Concerning the latter, from [4] we learn that the target uncertainty of PREX II is a factor of 3 smaller than the one from the first PREX experiment, thus approximately ±0.05. If accomplished, this small uncertainty, along with the measured central value, will allow to discriminate between theoretical predictions. For instance, the present EFT predictions would not be consistent with a measurement such as 0.33 (the current central value) ±0.05.

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Sammarruca, F.
Uncertainty Analysis of ^{208}Pb Neutron Skin Predictions with Chiral Interactions. *Symmetry* **2015**, *7*, 1646-1654.
https://doi.org/10.3390/sym7031646

**AMA Style**

Sammarruca F.
Uncertainty Analysis of ^{208}Pb Neutron Skin Predictions with Chiral Interactions. *Symmetry*. 2015; 7(3):1646-1654.
https://doi.org/10.3390/sym7031646

**Chicago/Turabian Style**

Sammarruca, Francesca.
2015. "Uncertainty Analysis of ^{208}Pb Neutron Skin Predictions with Chiral Interactions" *Symmetry* 7, no. 3: 1646-1654.
https://doi.org/10.3390/sym7031646