# The Symmetry Energy: Current Status of Ab Initio Predictions vs. Empirical Constraints

## Abstract

**:**

## 1. Introduction

## 2. Symmetry Energy: General Aspects

## 3. Theoretical Tools

#### 3.1. Energy per Nucleon in Infinite Matter

#### 3.2. Chiral Effective Field Theory

#### 3.2.1. Quantifying Errors in Chiral EFT

#### 3.3. The Two-Nucleon Force

#### 3.4. The Three-Nucleon Force

## 4. Ab Initio Predictions in Infinite Matter

#### 4.1. Symmetric Nuclear Matter

#### 4.2. Neutron Matter and the Symmetry Energy

#### 4.3. Impact on the Neutron Skin

## 5. Impact of the Isovector Part of the Free-Space $\mathit{NN}$ Force

## 6. Conclusions and Future Perspective: Where Do We Go from Here?

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagrammatic representation of the Brueckner integral equation. Intermediate nucleon lines with the double slash represent in-medium particle states.

**Figure 2.**Diagrams of the leading 3NF: (

**a**) the long-range 2PE, depending on the ${c}_{1,3,4}$; (

**b**) the medium-range 1PE, involving ${c}_{D}$; and (

**c**) the short-range contact, proportional to ${c}_{E}$.

**Figure 3.**Selected diagrams of the subleading 3NF, each representative of a specific topology: (

**a**) 2PE; (

**b**) 2P1PE; (

**c**) ring; (

**d**) 1P-contact; (

**e**) 2P-contact.

**Figure 5.**(Color online) Energy per neutron in NM vs. density, from leading (black dashes) to fourth order (solid red).

**Figure 6.**(Color online) The symmmetry energy vs. density, from leading (black dashes) to fourth order (solid red).

**Figure 7.**(Color online) The proton fraction in $\beta $-stable matter vs. density, from leading (black dashes) to fourth order (solid red).

**Figure 8.**(Color online) Discrepancy between the CREX and PREX measurements and theoretical predictions. See text and ref. [80] for more details.

**Figure 9.**(Color online) Phase shifts for selected isospin-1 partial waves as a function of the laboratory energy. Solid black: original potential; Red dashes: modified potential.

**Figure 10.**(Color online) Energy per neutron as a function of NM density. Solid black: original potential; Red dashes: modified potential.

**Table 1.**The LECs used in our calculations. n is the exponent of the regulator, Equation (25). The LECs ${c}_{1,3,4}$ are given in units of GeV${}^{-1}$, and ${C}_{S}$ and ${C}_{T}$ are in units of GeV${}^{-2}$.

Order | n | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ${\mathit{C}}_{\mathit{S}}$ | ${\mathit{C}}_{\mathit{T}}$ |
---|---|---|---|---|---|---|

N${}^{2}$LO | 2 | −0.74 | −3.61 | 2.44 | ||

N${}^{3}$LO | 2 | −1.07 | −5.32 | 3.56 | −118.13 | −0.25 |

**Table 2.**Predictions for the saturation properties of SNM from the final PPD of ref. [58]. Shown are the medians, 68% credible regions (CR), and 90% CR. The last column contains our predictions [57]. The energy per nucleon, E/A, and the incompressibility, K, are in units of MeV. The saturation density, ${\rho}_{0}$, is in units of fm${}^{-3}$.

Observable | Median | 68% CR | 90% CR | Our Predictions |
---|---|---|---|---|

$E/A$ | −15.2 | [−16.3, −13.9] | [17.1, −13.4] | −14.98 ± 0.85 |

${\rho}_{0}$ | 0.163 | [0.147, 0.176] | [0.140, 0.186] | 0.161 ± 0.015 |

${K}_{0}$ | 264 | [219, 300] | [202, 336] | 216 ± 33 |

**Table 3.**The energy per neutron, the symmetry energy, the L parameter, and the pressure at N${}^{3}$LO at selected densities, $\rho $, in units of ${\rho}_{0}$ = 0.155 fm${}^{-3}$. L is defined in Equation (7) and calculated at the specified density. The values in parentheses are from ref. [66]. The constraint for L in the third row ($\rho =0.67{\rho}_{0}$) applies to $\rho $ = 0.1 fm${}^{-3}$. The constraint at $\rho =0.31{\rho}_{0}$ is from ref. [70].

$\mathit{\rho}$$\left({\mathit{\rho}}_{0}\right)$ | $\frac{\mathit{E}}{\mathit{N}}\left(\mathit{\rho}\right)$ (MeV) | ${\mathit{e}}_{\mathit{sym}}\left(\mathit{\rho}\right)$ (MeV) | $\mathit{L}\left(\mathit{\rho}\right)$ (MeV) | ${\mathit{P}}_{\mathit{NM}}\left(\mathit{\rho}\right)$ (MeV/fm${}^{3}$) |
---|---|---|---|---|

1 | 15.56 ± 1.10 | 31.57 ± 1.53 (33.3 ± 1.3) | 49.58 ± 8.47 (59.6 ± 22.1) | 2.562 ± 0.438 (3.2 ± 1.2) |

0.72 (0.72 ± 0.01) | 11.52 ± 0.43 | 26.46 ± 0.82 (25.4 ± 1.1) | 44.91 ± 3.40 | 1.05 ± 0.13 |

0.67 (0.66 ± 0.04) | 10.81 ± 0.41 | 25.25 ± 0.72 (25.5 ± 1.1) | 44.65 ± 3.23 (53.1 ± 6.1) | 0.859 ± 0.120 |

0.63 (0.63 ± 0.03) | 10.39 ± 0.41 | 24.47 ± 0.66 (24.7 ± 0.8) | 43.81 ± 3.11 | 0.748 ± 0.116 |

0.31 (0.31 ± 0.03) | 6.715 ± 0.086 | 15.43 ± 0.12 (15.9 ± 1.0) | 32.35 ± 0.55 | 0.174 ± 0.008 |

0.21 (0.22 ± 0.07) | 5.472 ± 0.039 | 11.73 ± 0.05 (10.1 ± 1.0) | 27.57 ± 0.11 | 0.106 ± 0.002 |

**Table 4.**The neutron skin of ${}^{208}$Pb, S, calculated as described in ref. [76] using the given symmetry energy, J, and its slope at ${\rho}_{0}$, L.

J (MeV) | L (MeV) | S (fm) | Source for J, L |
---|---|---|---|

31.3 ± 0.8 | 52.6 ± 4.0 | [0.13, 0.17] | [34] |

(31.1, 32.5) | [44.8, 56.2] | [0.12, 0.17] | [44] |

(28, 35) | [20, 72] | [0.078, 0.20] | [77] |

(27, 43) | [7.17, 135] | [0.055, 0.28] | [78] |

38.29 ± 4.66 | 109.56 ± 36.41 | [0.17, 0.31] | [33] |

**Table 5.**The energy per neutron, its slope, and the pressure at a density of 0.155 fm${}^{-3}$ with the originl and the modified potentials. Only the 2NF is included.

Calculated Quantity | N${}^{3}$LO(450) | Modified Potential |
---|---|---|

${e}_{n}\left({\rho}_{0}\right)$ (MeV) | 11.11 | 13.88 |

${\left(\frac{\partial {e}_{n}\left(\rho \right)}{\partial \rho}\right)}_{{\rho}_{0}}$ (MeV/fm${}^{-3}$) | 39.79 | 70.03 |

$P\left({\rho}_{0}\right)$ (MeV/fm${}^{3}$) | 0.956 | 1.68 |

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**MDPI and ACS Style**

Sammarruca, F.
The Symmetry Energy: Current Status of Ab Initio Predictions vs. Empirical Constraints. *Symmetry* **2023**, *15*, 450.
https://doi.org/10.3390/sym15020450

**AMA Style**

Sammarruca F.
The Symmetry Energy: Current Status of Ab Initio Predictions vs. Empirical Constraints. *Symmetry*. 2023; 15(2):450.
https://doi.org/10.3390/sym15020450

**Chicago/Turabian Style**

Sammarruca, Francesca.
2023. "The Symmetry Energy: Current Status of Ab Initio Predictions vs. Empirical Constraints" *Symmetry* 15, no. 2: 450.
https://doi.org/10.3390/sym15020450