# Cross Sections for Coherent Elastic and Inelastic Neutrino-Nucleus Scattering

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## Abstract

**:**

## 1. Introduction

## 2. Formalism

#### 2.1. CE$\nu $NS Cross Section

#### 2.2. Inelastic Cross Sections

## 3. Results and Discussion

#### Constraining ${}^{40}$Ar

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) Diagrammatic representation of the CE$\nu $NS process where a single ${Z}^{0}$ boson is exchanged between neutrino and target nucleus. (

**Right**) Diagrammatic representation of the inelastic neutrino-nucleus scattering where a single ${W}^{+}$ (CC) or ${Z}^{0}$ (NC) boson is exchanged between neutrino and target nucleus.

**Figure 2.**Panels (

**a**,

**b**) represent proton and neutron densities of different nuclei obtained using the HF–SkE2 approach. Panels (

**c**) through (

**d**) represent charge and weak form factors for the different nuclei.

**Figure 3.**The “weak-skin” form factor depicts the difference between the charge and weak form factors.

**Figure 4.**(

**Left**): the charge form factor of ${}^{208}$Pb compared with elastic electron scattering data of Ref. [29]. (

**Right**): the weak form factor of ${}^{208}$Pb along with the single point measured by the PREX collaboration at the momentum transfer of q = 0.475 fm${}^{-1}$ [34,35]. Both form factors are compared with relativistic mean–field predictions of Yang et al. [41].

**Figure 5.**Total CE$\nu $NS cross sections for a set of nuclear targets obtained within the HF–SkE2 approach.

**Figure 6.**CE$\nu $NS cross section strength compared to CCQE and NCQE scattering cross sections for several nuclei, above particle emission threshold.

**Figure 7.**(

**Left**) The ${}^{40}$Ar charge form factor predictions compared to elastic electron scattering data taken from Ref. [70], a comparison is also performed with the coupled–cluster theory predictions of Payne et al. [40] as well as with Klein–Nystrand [38] (standard and adapted) and Helm [39] form factors. (

**Right**) The ${}^{40}$Ar weak form factor predictions compared with calculations of Payne et al. [40], Yang et al. [41], Hoferichter et al. [43] and with the predictions of Klein–Nystrand [38] (standard and adapted) and Helm [39] form factors.

**Figure 11.**(

**Left**) The CE$\nu $NS cross section on ${}^{40}$Ar as a function of neutrino energy, recent flux–folded measurement by the COHERENT collaboration [5] is shown along with the flux-folded HF–SkE2 prediction. (

**Right**) Flux–averaged CE$\nu $NS cross sections as a function of neutron number for the ${}^{12}$C, ${}^{16}$O, ${}^{40}$Ar, ${}^{56}$Fe and ${}^{208}$Pb nuclei. We also show ${}^{40}$Ar data measured by COHERENT [5].

**Figure 12.**Charged-current (

**left**) and neutral-current (

**right**) inelastic cross section: total as a function of neutrino energy shown along with contributions from different multipoles (top panel), differential as a function of excitation energy (middle panel) and as a function of lepton scattering angle (bottom panel) for fixed neutrino energies, ${E}_{\nu}=$ 30 and 50 MeV.

**Table 1.**Single –particle energies in ${}^{40}$Ar, as provided by a HF calculation using the SkE2 interaction.

$\mathit{p}/\mathit{n}$ | i | ${\mathit{n}}_{\mathit{i}},{\mathit{l}}_{\mathit{i}},{\mathit{j}}_{\mathit{i}}$ | ${\mathit{\epsilon}}_{\mathit{i}}$ (MeV) | ${\mathit{v}}_{\mathit{i}}^{2}$ | # N |
---|---|---|---|---|---|

p | 1 | $1{s}_{1/2}$ | −43.7029 | 1.00 | 2 |

p | 2 | $1{p}_{3/2}$ | −31.4496 | 1.00 | 4 |

p | 3 | $1{p}_{1/2}$ | −27.3921 | 1.00 | 2 |

p | 4 | $1{d}_{5/2}$ | −17.7027 | 1.00 | 6 |

p | 5 | $2{s}_{1/2}$ | −12.0822 | 1.00 | 2 |

p | 6 | $1{d}_{3/2}$ | −10.9243 | 0.50 | 2 |

n | 1 | $1{s}_{1/2}$ | −48.3047 | 1.00 | 2 |

n | 2 | $1{p}_{3/2}$ | −35.2020 | 1.00 | 4 |

n | 3 | $1{p}_{1/2}$ | −31.0247 | 1.00 | 2 |

n | 4 | $1{d}_{5/2}$ | −21.1035 | 1.00 | 6 |

n | 5 | $2{s}_{1/2}$ | −16.1116 | 1.00 | 2 |

n | 6 | $1{d}_{3/2}$ | −14.0266 | 1.00 | 4 |

n | 7 | $1{f}_{7/2}$ | −7.2108 | 0.25 | 2 |

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**MDPI and ACS Style**

Van Dessel, N.; Pandey, V.; Ray, H.; Jachowicz, N.
Cross Sections for Coherent Elastic and Inelastic Neutrino-Nucleus Scattering. *Universe* **2023**, *9*, 207.
https://doi.org/10.3390/universe9050207

**AMA Style**

Van Dessel N, Pandey V, Ray H, Jachowicz N.
Cross Sections for Coherent Elastic and Inelastic Neutrino-Nucleus Scattering. *Universe*. 2023; 9(5):207.
https://doi.org/10.3390/universe9050207

**Chicago/Turabian Style**

Van Dessel, Nils, Vishvas Pandey, Heather Ray, and Natalie Jachowicz.
2023. "Cross Sections for Coherent Elastic and Inelastic Neutrino-Nucleus Scattering" *Universe* 9, no. 5: 207.
https://doi.org/10.3390/universe9050207