# Isospin Symmetry Breaking Effects on the Mass-Radius Relation of a Neutron Star

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Finite Size Effects

#### 2.2. Electromagnetic Spin-Orbit

#### 2.3. Coulomb Exchange

#### 2.4. Vacuum Polarization Correction

#### 2.5. Charge Symmetry Breaking and Charge Independence Breaking Potentials

^{3}and ${s}_{0}=-26.3\left(7\right)$ MeV fm

^{3}determined from nuclear data and pseudo-data as previously explained. Summing and re-writing the previous terms,

## 3. Results

^{−3}, where it becomes smaller, but still its trend is softer than SAMi for most of the density range shown. This behaviour is due to the fine-tuning of the SAMi-ISB parameters since ISB terms alone would produce attraction in neutron matter (cf. Equation (16)) and, thus, the opposite trend for $\rho \le $ 0.5 fm

^{−3}. Hence, the small ISB contributions cannot be disentangled from the effect resulting from the re-fit of the parameters between SAMi and SAMi-ISB (cf. previous Section). In the inset, the same neutron matter EoSs are shown and compared with the results of two representative ab initio approaches for the low density part. The details of those calculations can be found in Refs. [39,40]. It is seen from the figure that SAMi-ISB agrees better than SAMi with the ab initio calculations shown here.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**panel) ISB contributions to the symmetric matter equation of state as predicted by the BHF calculations based on AV18 potential (black circles) of Ref. [25]. The solid line corresponds to SAMi-ISB. (

**Right**panel) IAS energy for Sn isotopes as predicted by SAMi and SAMi-ISB, compared to experimental data. Figure taken from the supplemental material of Ref. [21].

**Figure 2.**(

**Left**panel) EoS of pure neutron and symmetric nuclear matter as predicted by SAMi (dashed) and SAMi-ISB (dash-dotted). In the inset only the neutron matter EoSs are shown and compared with the results from the ab initio approaches of Ref. [39] (brown shaded area) and Ref. [40] (grey shaded area). (

**Right**panel) Mass-Radius relation of a neutron star as predicted by SAMi and SAMi-ISB. The brown shaded area encloses SAMi-ISB predictions with the predictions of the same model if ${u}_{0}={s}_{0}=0$. The maximum observed mass of the millisecond pulsar J1614-2230 ($1.97\pm 0.04{M}_{\mathrm{sun}}$) is also shown (grey horizontal band) [41] as well as the simultaneous estimation of the mass and radius of the millisecond pulsar PSR J0030+0451 [43] (dot) and [44] (square). Errors correspond to the 1$\sigma $ estimation.

Observable | Nucleus | SAMi | SAMi-ISB | Experiment | Property | SAMi | SAMi-ISB | |
---|---|---|---|---|---|---|---|---|

B (MeV) | ${}^{48}$Ca | 415.61 | 417.67 | 415.99 | ${\rho}_{0}$ (fm^{−3}) | 0.159(1) | 0.1613(6) | |

${}^{90}$Zr | 781.26 | 783.60 | 783.89 | ${e}_{0}$ (MeV) | −15.93(9) | −16.03(2) | ||

${}^{132}$Sn | 1103.09 | 1102.75 | 1102.85 | ${m}_{\mathrm{IS}}^{*}/m$ | 0.6752(3) | 0.730(19) | ||

${}^{208}$Pb | 1636.61 | 1635.78 | 1636.43 | ${m}_{\mathrm{IV}}^{*}/m$ | 0.664(13) | 0.667(120) | ||

J (MeV) | 28(1) | 30.8(4) | ||||||

L (MeV) | 44(7) | 50(4) | ||||||

${K}_{0}$ (MeV) | 245(1) | 235(4) | ||||||

$<{r}_{\mathrm{ch}}^{2}{>}^{1/2}$(fm) | ${}^{48}$Ca | 3.51 | 3.49 | 3.47 | ||||

${}^{90}$Zr | 4.27 | 4.26 | 4.27 | |||||

${}^{132}$Sn | 4.73 | 4.73 | – | |||||

${}^{208}$Pb | 5.50 | 5.50 | 5.50 |

**Table 2.**Some basic properties of a neutron star as predicted by SAMi, SAMi-ISB and SAMi-ISB with ${u}_{0}={s}_{0}=0$ functionals. Specifically, the maximum mass (${M}_{\mathrm{max}}$), radius (${R}_{\mathrm{max}}$) and central density (${\rho}_{1.4}^{c}$), radius (${R}_{1.4}$), tidal deformability (${\mathrm{\Lambda}}_{1.4}$) and compactness (${\xi}_{1.4}$) for a $1.4{M}_{\mathrm{sun}}$ neutron star.

${\mathit{M}}_{\mathbf{max}}/{\mathit{M}}_{\mathbf{sun}}$ | ${\mathit{R}}_{\mathbf{max}}$ [km] | ${\mathit{\rho}}_{1.4}^{\mathit{c}}$ [fm^{−3}] | ${\mathit{R}}_{1.4}$ | ${\mathbf{\Lambda}}_{1.4}$ [km] | ${\mathit{\xi}}_{1.4}$ | |
---|---|---|---|---|---|---|

SAMi | 2.03 | 9.8 | 0.54 | 11.2 | 301 | 0.18 |

SAMi-ISB | 1.88 | 9.8 | 0.59 | 11.2 | 261 | 0.19 |

SAMi-ISB | 1.86 | 9.9 | 0.61 | 11.0 | 242 | 0.19 |

(${u}_{0}={s}_{0}=0$) |

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**MDPI and ACS Style**

Selva, G.; Roca-Maza, X.; Colò, G.
Isospin Symmetry Breaking Effects on the Mass-Radius Relation of a Neutron Star. *Symmetry* **2021**, *13*, 144.
https://doi.org/10.3390/sym13010144

**AMA Style**

Selva G, Roca-Maza X, Colò G.
Isospin Symmetry Breaking Effects on the Mass-Radius Relation of a Neutron Star. *Symmetry*. 2021; 13(1):144.
https://doi.org/10.3390/sym13010144

**Chicago/Turabian Style**

Selva, Giovanni, Xavier Roca-Maza, and Gianluca Colò.
2021. "Isospin Symmetry Breaking Effects on the Mass-Radius Relation of a Neutron Star" *Symmetry* 13, no. 1: 144.
https://doi.org/10.3390/sym13010144