# Constraints on Nuclear Symmetry Energy Parameters

## Abstract

**:**

## 1. Introduction

## 2. The Nuclear Symmetry Energy

#### 2.1. Nuclear Mass Fitting

#### 2.2. Neutron Matter Theory

#### 2.3. The Unitary Gas Conjecture

## 3. Neutron Skin Thickness Constraints

#### 3.1. Neutron Skin Measurements and Correlations

**Table 4.**${}^{208}$Pb neutron skin measurements and theoretical predictions with $1\sigma $ uncertainties.

${}^{208}$Pb Experiment | Reference | ${\mathit{r}}_{\mathit{np}}^{208}$ (fm) |
---|---|---|

Coherent ${\pi}^{0}\gamma $ production | [77] | ${0.15}_{-0.04}^{+0.03}$ |

Pionic atoms | [73] | $0.15\pm 0.08$ |

Pion scattering | [73] | $0.11\pm 0.06$ |

$\overline{p}$ annihilation | [78,79] | $0.18\pm 0.06$ |

Elastic polarized p scattering | [70] | $0.16\pm 0.05$ |

Elastic polarized p scattering | [80] | ${0.211}_{-0.063}^{+0.054}$ |

Elastic p scattering | [81] | $0.197\pm 0.042$ |

Elastic p scattering | [72] | $0.119\pm 0.045$ |

Parity-violating ${e}^{-}$ scattering (PREX I+II) | [17] | $0.283\pm 0.071$ |

${}^{208}$Pb experimental weighted mean | $0.166\pm 0.017$ | |

Pygmy dipole resonances | [82] | $0.180\pm 0.035$ |

${r}_{np}^{\mathrm{Sn}}$ | [83] | $0.175\pm 0.020$ |

Anti-analog giant dipole resonance | [84] | $0.216\pm 0.048$ |

Symmetry energy ${}^{208}$Pb | [85] | $0.158\pm 0.014$ |

Dispersive optical model | [86] | ${0.18}_{-0.12}^{+0.25}$ |

Dispersive optical model | [67] | $0.25\pm 0.05$ |

Coupled cluster expansion | [66] | $0.17\pm 0.03$ |

${r}_{np}^{48}$ | [63,64], this paper | $0.128\pm 0.040$ |

${\alpha}_{D}^{208}$ | [62], this paper | $0.154\pm 0.019$ |

${\alpha}_{D}^{208}$ | [20,64], this paper | $0.188\pm 0.017$ |

${}^{208}$Pb theoretical weighted mean | $0.170\pm 0.008$ |

**Table 5.**${}^{48}$Ca neutron skin measurements and theoretical predictions with $1\sigma $ uncertainties. * Uncertainty scaled upwards as per Ref. [87].

${}^{48}$Ca Experiment | Reference | ${\mathit{r}}_{\mathit{np}}^{48}$ (fm) |
---|---|---|

Elastic polarized p scattering | [70] | $0.229\pm 0.050$ |

Elastic p scattering | [76] | $0.10\pm 0.03$ |

Elastic p scattering | [72] | $0.098\pm 0.043$ |

Elastic p scattering | [71] | ${0.168}_{-0.028}^{+0.025}$ |

Pionic atoms | [73] | $0.13\pm 0.06$ |

Pion scattering | [74] | $0.11\pm 0.04$ |

$\alpha $ scattering | [75] | $0.171\pm 0.050$ |

Parity-violating ${e}^{-}$ scattering (CREX) | [18] | $0.121\pm 0.035$ |

${}^{48}$Ca experimental weighted mean | $0.137\pm 0.015$ * | |

Coupled-cluster expansion | [65] | $0.135\pm 0.015$ |

Dispersive optical model | [68] | $0.249\pm 0.023$ |

${r}_{np}^{208}$ | [63], this paper | $0.173\pm 0.018$ |

${}^{48}$Ca theoretical weighted mean | $0.17\pm 0.03$ * |

#### 3.2. Parity-Violating Electron Scattering Measurements

^{48}$Ca.$

## 4. Other Nuclear Methods

#### 4.1. Correlations from Nuclear Dipole Polarizabilities

#### 4.2. Correlations from Heavy Ion Collisions

## 5. Astrophysical Considerations

#### 5.1. Neutron Star Radii

_{s}, one should be wary of making predictions about neutron stars from them. Nevertheless, as shown in Figure 11, there is a clear correlation between L and R

_{1.4}. The correlation between P

_{NSM}(n) and R

_{1.4}turns out to be greatest at densities in the range 1.5–2n

_{s}[48], low enough that interaction models are still reliable. Note, however, that these models should not be used to constrain neutron star maximum masses, which are most sensitive to P

_{NSM}for n ≳ 3n

_{s}[48].

#### 5.2. Tidal Deformabilities and Radii

^{2}. It therefore follows that

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NS | neutron star | BNS | binary neutron stars |

EOS | equation of state | PSR | pulsar |

SNM | symmetric nuclear matter | PNM | pure neutron matter |

$\chi $EFT | chiral effective field theory | RMF | relativistic mean field |

UGC | Unitary Gas Conjecture | UGPC | Unitary Gas Pressure Conjecture |

NICER | Neutron Star Interior Composition ExploreR |

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**Figure 1.**${S}_{V}$ and L data from individual Skyrme (black filled circles, Ref. [34]), relativistic mean field (RMF, black open circles, Ref. [36]) forces, both interaction types (Tagami 2022, red triangles, Ref. [37]), and all tabulated interactions (combined); corresponding 68.3% confidence ellipses are shown. The green hatched confidence ellipse is taken from the UNEDF collaboration [38] using $\sigma =1.2$ MeV (see text). The bounds provided by the Unitary Gas Conjecture (UGC, [35]) and the Unitary Gas Pressure Conjecture are shown as dotted curves.

**Figure 2.**The same as Figure 1 but for correlations between ${K}_{N}$ and L [

**left panel**] and ${Q}_{n}$ and L [

**right panel**]. The Unitary Gas Pressure Conjecture restricts allowable regions to the right of the dotted lines labelled UGPC.

**Figure 3.**Correlations among symmetric energy parameters of forces in compilations of Refs. [34,36]. The left and right panels show the $B-{n}_{s}$ and $L-{n}_{s}$ correlations, respectively. Individual interactions are shown by filled circles. 68.3% and 95% confidence ellipses for Skyrme (RMF) interactions are shown by black (red) solid and dashed ellipses, respectively; green ellipses show the confidence ellipses for the combined force models. 68.3% and 95.5% confidence regions determined from $\chi $EFT calculations of SNM plus PNM (see Section 2.2) are shown by the orange solid and dotted curves, respectively.

**Figure 4.**The same as Figure 3 except the left and right panels show the $L-{K}_{1/2}$ and $L-{Q}_{1/2}$ correlations, respectively.

**Figure 5.**Black correlation ellipses for ${S}_{V}-L$ (

**left panel**) and ${K}_{N}-L$ (

**right panel**) use model interaction data [34,36,37], with (solid) and without (dashed) application of Unitary Gas Constraints [35] boundaries (dotted). The blue confidence ellipse shows UNEDF [38] results assuming $\sigma =1.2$ MeV. The red (brown) confidence ellipses are from chiral EFT studies [39] using PNM results with empirical saturation properties (combined PNM+SNM results). The red-dashed quadrilateral are limits determined from elliptic flows in heavy-ion collisions [50].

**Figure 7.**Neutron skin thicknesses of ${}^{48}$Ca (red) and ${}^{208}$Pb (black) from interactions compiled in Ref. [37] (filled circles) and Refs. [18,60,63,64] (open circles). Means (1 standard deviations) of linear correlations are shown as solid (dashed) lines. The horizontal shaded bands indicate the 1 standard deviation ranges of the averaged experimental results. The dotted black (red) lines indicate the 1 standard deviation range of ${r}_{np}^{208}$ (${r}_{np}^{48}$) from PREX I+II [17] (CREX [18]).

**Figure 9.**Neutron skin thicknesses of ${}^{48}$Ca and ${}^{208}$Pb from interactions compiled in Ref. [37] (filled circles) and Refs. [18,60,63,64] (triangles). Colors indicate L values where known; black triangles indicate points where L values are unspecified. Standard deviations of a linear correlation Equation (21) are shown as dashed lines. The red (blue) confidence ellipses are from PREX I+II [17] and CREX [18] (mean of all experiments); solid (dashed) ellipses are 68% (90%) confidence.

**Figure 10.**Symmetry parameters ${S}_{V}-L$ (

**left panel**) and ${K}_{N}-L$ (

**right panel**) jointly satisfying parity-violating experiments to within (exceeding) 90% confidence are shown as red and black filled (black open) circles; filled black circles violate Unitary Gas constraints (dotted boundary). Red (blue) confidence ellipses are for models satisfying Unitary Gas constraints weighted by their two-dimensional Gaussian probability defined by the parity-violating (red) and average (blue) experimental ${r}_{np}^{48}$ and ${r}_{np}^{208}$ measurements and uncertainties. The black confidence ellipse shows PNM $\chi $EFT results.

**Figure 11.**The same as Figure 10 except showing ${R}_{1.4}$ versus L (

**left panel**) and ${\Lambda}_{1.4}$ versus L (

**right panel**) for subsets of forces from Refs. [18,37,60,63,64]. Black solid and dashed curves in the left panel show the ${R}_{1.4}-L$ correlation and standard deviations derived as in the text. The shaded green bands are 68% and 95% confidence intervals from a joint analysis of GW170817 and PSR J0030+0451 and PSR J0740+6620 by Ref. [120] (

**left panel**) and from GW170817 Bayesian analyses posteriors [21], corrected for $\Lambda $ priors chosen so as to reflect uniform R priors (

**right panel**).

Method/${\mathit{S}}_{\mathit{V}}-\mathit{L}$ | ${\mathit{S}}_{\mathit{V}0}$ (MeV) | ${\mathit{L}}_{0}$ (MeV) | ${\mathit{\sigma}}_{{\mathit{S}}_{\mathit{V}}}$ (MeV) | ${\mathit{\sigma}}_{\mathit{L}}$ (MeV) | r |
---|---|---|---|---|---|

UNEDF [38] | 30.5 | 45.1 | 1.9 | 24.0 | 0.970 |

Skyrme [34] | 30.9 | 41.5 | 2.25 | 27.2 | 0.812 |

RMF [36] | 33.1 | 85.8 | 2.12 | 17.4 | 0.625 |

Tagami [37] | 32.0 | 57.7 | 2.37 | 25.2 | 0.702 |

Combined [34,36,37] | 32.1 | 62.2 | 2.45 | 30.6 | 0.783 |

Combined + UGC/UGPC | 32.5 | 57.7 | 2.09 | 20.7 | 0.920 |

$\chi $EFT (SNM+PNM) [39] | 31.7 | 59.8 | 1.1 | 4.2 | 0.715 |

$\chi $EFT (SNM+PNM) | 31.7 | 60.4 | 2.4 | 8.1 | 0.913 |

$\chi $EFT (PNM) | 32.0 | 51.9 | 1.1 | 7.9 | 0.978 |

neutron skin (CREX+PREX) | 32.2 | 52.9 | 1.7 | 13.2 | 0.820 |

neutron skin (other) | 31.0 | 42.1 | 1.2 | 8.2 | 0.729 |

Method/${\mathit{K}}_{\mathit{N}}-\mathit{L}$ | ${\mathit{K}}_{\mathit{N}\mathbf{0}}$ (MeV) | ${\mathit{L}}_{\mathbf{0}}$ (MeV) | ${\mathsf{\sigma}}_{{\mathit{K}}_{\mathit{N}}}$ (MeV) | ${\mathsf{\sigma}}_{\mathit{L}}$ (MeV) | $\mathit{r}$ |

Skyrme [34] | 73.3 | 41.6 | 98.9 | 27.2 | 0.952 |

RMF [36] | 234.0 | 85.8 | 63.6 | 17.4 | 0.666 |

Tagami 2022 [37] | 161.9 | 57.9 | 99.5 | 25.6 | 0.757 |

Combined [34,36,37] | 147.7 | 60.4 | 113.7 | 30.6 | 0.899 |

Combined [34,36,37] + UGC/UGPC | 137.3 | 57.7 | 74.8 | 20.7 | 0.745 |

$\chi $EFT (SNM+PNM) | 172.1 | 60.4 | 27.4 | 8.1 | 0.558 |

$\chi $EFT (PNM) | 152.3 | 51.9 | 35.1 | 7.9 | 0.993 |

neutron skin (CREX+PREX) | 141.6 | 52.9 | 73.2 | 13.2 | 0.530 |

neutron skin (other) | 104.8 | 42.1 | 75.4 | 8.2 | 0.590 |

Method/${\mathit{Q}}_{\mathit{N}}-\mathit{L}$ | ${\mathit{Q}}_{\mathit{N}\mathbf{0}}$ (MeV) | ${\mathit{L}}_{\mathbf{0}}$ (MeV) | ${\mathsf{\sigma}}_{{\mathit{Q}}_{\mathit{N}}}$ (MeV) | ${\mathsf{\sigma}}_{\mathit{L}}$ (MeV) | $\mathit{r}$ |

Skyrme [34] | 75.3 | 41.6 | 178.5 | 27.2 | −0.843 |

RMF [36] | −211.9 | 85.8 | 421.4 | 17.4 | −0.017 |

Combined [34,36] | −53.4 | 61.2 | 341.2 | 32.1 | −0.498 |

Combined [34,36] + UGC/UGPC | 7.86 | 58.2 | 297.1 | 21.6 | −0.378 |

$\chi $EFT (SNM+PNM) | −123.3 | 60.4 | 381.3 | 8.1 | −0.686 |

$\chi $EFT (PNM) | 112.8 | 51.9 | 90.7 | 7.9 | 0.398 |

**Table 2.**Coefficients for the relation ${r}_{np}^{208}=\tilde{a}/\mathrm{fm}+\tilde{b}{\tilde{L}}_{1}/\mathrm{MeV}$ and inferred values of ${\tilde{L}}_{1}$ from the error-weighted mean experimental value ${r}_{np}^{208}=0.166\pm 0.017$ fm. Also given are two estimated values of ${\tilde{L}}_{1}$ from neutron skin measurements of Sn isotopes. ${}^{\u2020}$ 0.01 fm uncertainty introduced for consistency.

Reference | $\tilde{\mathit{a}}$ | $\tilde{\mathit{b}}$ | ${\tilde{\mathit{L}}}_{1}$ (MeV) |
---|---|---|---|

[55] | 0.0 | $0.00378\pm 0.00014$ | $43.9\pm 4.8$ |

[57] | $0.00994\pm 0.01000$ | 0.0036 | $43.4\pm 5.5$ |

[58] | $0.0101\pm 0.01{\phantom{\rule{3.33333pt}{0ex}}}^{\u2020}$ | 0.00377 | $41.4\pm 5.2$ |

[1] | $0.0148\pm 0.0100$ | 0.00414 | 3$6.5\pm 4.8$ |

[20] | $0.0590\pm 0.0028$ | 0.00313 | $34.2\pm 10.5$ |

[59] | Sn isotopes | $42.9\pm 4.1$ | |

[58] | Sn isotopes | $43.7\pm 5.3$ | |

Error-weighted mean | $41.7\pm 2.0$ |

**Table 3.**Slopes, intercepts, and standard deviations of linear fits ${r}_{np}/\mathrm{fm}=a\pm \Delta a+bL/\mathrm{MeV}$.

Reference | a | b |
---|---|---|

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

[37] | $0.0963\pm 0.0041$ | 0.001566 |

[18] | $0.1028\pm 0.0115$ | 0.001617 |

[60] | $0.0964\pm 0.0039$ | 0.001563 |

[20] | $0.0865\pm 0.0124$ | 0.001837 |

[61] | $0.0967\pm 0.001447$ | 0.00145 |

[62] | $0.0986\pm 0.0137$ | 0.001537 |

Mean | $0.0996\pm 0.0096$ | 0.001518 |

${}^{48}$Ca | ||

[37] | $0.1250\pm 0.0028$ | 0.000873 |

[18] | $0.1261\pm 0.0056$ | 0.000990 |

[60] | $0.1290\pm 0.0037$ | 0.000791 |

Mean | $0.1255\pm 0.0052$ | 0.000882 |

**Table 6.**Dipole polarizabilities with $1\sigma $ uncertainties. ${}^{\u2020}$${\alpha}_{D}$ values corrected as per Ref. [90].

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Lattimer, J.M. Constraints on Nuclear Symmetry Energy Parameters. *Particles* **2023**, *6*, 30-56.
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