# Improved Superscaling in Quasielastic Electron Scattering with Relativistic Effective Mass

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of Superscaling Formalism

#### 2.1. Electromagnetic Response Functions

#### 2.2. Geometrical Interpretation

#### 2.3. Scaling

#### 2.4. SuSAM*

## 3. Averaged Single-Nucleon Response Functions

#### 3.1. RFG Extrapolation

#### 3.2. Longitudinal Single-Nucleon Response

#### 3.3. Transverse Single-Nucleon Response

#### 3.4. Alternative to the Extrapolated Single-Nucleon Responses

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

RFG | Relativistic Fermi gas |

RMF | Relativistic mean field |

SuSA | Superscaling analysis |

SuSAM* | Superscaling analysis with relativistic effective mass |

## Appendix A. Single-Nucleon Responses

#### Appendix A.1. Longitudinal Single-Nucleon Response

#### Appendix A.2. Transverse Single-Nucleon Response

## References

- West, G.B. Electron Scattering from Atoms, Nuclei and Nucleons. Phys. Rept.
**1975**, 18, 263–323. [Google Scholar] [CrossRef] - Saruis, A.M. Self-consistent HF-RPA description of electron and photon nuclear reactions with Skyrme forces. Phys. Rept.
**1993**, 235, 57–188. [Google Scholar] [CrossRef] - Benhar, O.; Day, D.; Sick, I. Inclusive quasielastic electron-nucleus scattering. Rev. Mod. Phys.
**2008**, 80, 189. [Google Scholar] [CrossRef] [Green Version] - Boffi, S.; Giusti, C.; Pacati, F.D.; Radici, M. Electromagnetic Response of Atomic Nuclei. In Oxford Studies in Nuclear Physics; Clarendon Press: Oxford, UK, 1996; Volume 20. [Google Scholar]
- Walecka, J.D. Theoretical Nuclear and Subnuclear Physics; Editorial World Scientific Pub. Co. Inc.: Singapore, 2004. [Google Scholar]
- Alvarez-Ruso, L.; Athar, M.S.; Barbaro, M.B.; Cherdack, D.; Christy, M.E.; Coloma, P.; Donnelly, T.W.; Dytman, S.; de Gouvêa, A.; Hill, R.J.; et al. NuSTEC White Paper: Status and Challenges of Neutrino-Nucleus Scattering. Prog. Part. Nucl. Phys.
**2018**, 100, 1. [Google Scholar] [CrossRef] [Green Version] - Mosel, U. Neutrino Interactions with Nucleons and Nuclei: Importance for Long-Baseline Experiments. Ann. Rev. Nuc. Part. Sci.
**2016**, 66, 171–195. [Google Scholar] [CrossRef] [Green Version] - Megias, G.D.; Barbaro, M.B.; Caballero, J.A.; Amaro, J.E.; Donnelly, T.W.; Simo, I.R.; Orden, J.W.V. Neutrino–oxygen CC0π scattering in the SuSAv2-MEC model. J. Phys. G
**2018**, 45, 013001. [Google Scholar] [CrossRef] [Green Version] - Alvarez-Ruso, L.; Hayato, Y.; Nieves, J. Progress and open questions in the physics of neutrino cross sections at intermediate energies. New J. Phys.
**2014**, 16, 075015. [Google Scholar] [CrossRef] [Green Version] - Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; González-Jiménez, R.; Megias, G.D.; Simo, I.R. Electron- versus neutrino-nucleus scattering. J. Phys. G
**2020**, 47, 124001. [Google Scholar] [CrossRef] - Ankowski, A.M.; Ashkenazi, A.; Bacca, S.; Barrow, J.L.; Betancourt, M.; Bodek, A.; Christy, M.E.; Dytman, L.D.S.; Friedland, A.; Hen, O.; et al. Electron Scattering and Neutrino Physics. arXiv
**2022**, arXiv:2203.06853. [Google Scholar] - Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; Donnelly, T.W. Quasielastic Charged Current Neutrino-nucleus Scattering. Phys. Rev. Lett.
**2007**, 98, 242501. [Google Scholar] [CrossRef] [Green Version] - Benhar, O.; Day, D.; Sick, I. An archive for quasi-elastic electron-nucleus scattering data. arXiv
**2006**, arXiv:nucl-ex/0603032. [Google Scholar] - Benhar, O.; Day, D.; Sick, I. Available online: http://faculty.virginia.edu/qes-archive/ (accessed on 13 May 2022).
- Donnelly, T.W.; Sick, I. Superscaling in inclusive electron—nucleus scattering. Phys. Rev. Lett.
**1999**, 82, 3212–3215. [Google Scholar] [CrossRef] [Green Version] - Donnelly, T.W.; Sick, I. Superscaling of inclusive electron scattering from nuclei. Phys. Rev. C
**1999**, 60, 065502. [Google Scholar] [CrossRef] [Green Version] - Rosenfelder, R. Quasielastic electron scattering from nuclei. Ann. Phys.
**1980**, 128, 188. [Google Scholar] [CrossRef] - Serot, B.D.; Walecka, J.D. Relativistic Nuclear Many-Body Theory. Adv. Nucl. Phys.
**1986**, 16, 1. [Google Scholar] - Drechselt, D.; Giannini, M.M. Electron scattering off nuclei. Rep. Prog. Phys.
**1989**, 52, 1083. [Google Scholar] [CrossRef] - Wehrberger, K. Electromagnetic response functions in quantum hadrodynamics. Phys. Rep.
**1993**, 225, 273. [Google Scholar] [CrossRef] - Alberico, W.M.; Molinari, A.; Donnelly, T.W.; Kronenberg, E.L.; Orden, J.W.V. Scaling in electron scattering from a relativistic Fermi gas. Phys. Rev. C
**1988**, 38, 1801–1810. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Cenni, R.; Donnelly, T.W.; Molinari, A. Inclusive versus exclusive EM processes in relativistic nuclear systems. Phys. Rev. C
**1997**, 56, 276–291. [Google Scholar] [CrossRef] [Green Version] - Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; Donnelly, T.W.; Gonzalez-Jimenez, R.; Megias, G.D.; Simo, I.R. Neutrino-nucleus scattering in the SuSA model. Eur. Phys. J.
**2021**, 230, 4321–4338. [Google Scholar] [CrossRef] - Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; Donnelly, T.W.; Molinari, A.; Sick, I. Using electron scattering superscaling to predict charge-changing neutrino cross sections in nuclei. Phys. Rev. C
**2005**, 71, 015501. [Google Scholar] [CrossRef] [Green Version] - Megias, G.D.; Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; Donnelly, T.W. Inclusive electron scattering within the SuSAv2 meson-exchange current approach. Phys. Rev. D
**2016**, 94, 013012. [Google Scholar] [CrossRef] [Green Version] - Martinez-Consentino, V.L.; Simo, I.R.; Amaro, J.E.; Arriola, E.R. Fermi-momentum dependence of relativistic effective mass below saturation from super-scaling of quasielastic electron scattering. Phys. Rev. C
**2017**, 96, 064612. [Google Scholar] [CrossRef] [Green Version] - Amaro, J.E.; Martinez-Consentino, V.L.; Arriola, E.R.; Simo, I.R. Global Superscaling Analysis of Quasielastic Electron Scattering with Relativistic Effective Mass. Phys. Rev. C
**2018**, 98, 024627. [Google Scholar] [CrossRef] [Green Version] - Simo, I.R.; Martinez-Consentino, V.L.; Amaro, J.E.; Arriola, E.R. Quasielastic charged-current neutrino scattering in the scaling model with relativistic effective mass. Phys. Rev. D
**2018**, 97, 116006. [Google Scholar] [CrossRef] [Green Version] - Maieron, C.; Amaro, J.E.; Barbaro, M.B.; Caballero, J.A.; Donnelly, T.W.; Williamson, C.F. Superscaling of non-quasielastic electron-nucleus scattering. Phys. Rev. C
**2009**, 80, 035504. [Google Scholar] [CrossRef] [Green Version] - Martinez-Consentino, V.L.; Simo, I.R.; Amaro, J.E. Meson-exchange currents and superscaling analysis with relativistic effective mass of quasielastic electron scattering from C12. Phys. Rev. C
**2021**, 104, 025501. [Google Scholar] [CrossRef] - Martinez-Consentino, V.L.; Amaro, J.E.; Simo, I.R. Semiempirical formula for electroweak response functions in the two-nucleon emission channel in neutrino-nucleus scattering. Phys. Rev. D
**2021**, 104, 113006. [Google Scholar] [CrossRef] - Martinez-Consentino, V.L.; Amaro, J.E.; Casale, P.R.; Simo, I.R. Semiempirical formula for two-nucleon emission induced by short-range correlations in electron and neutrino scattering. arXiv
**2022**, arXiv:2210.09982. [Google Scholar] - Caballero, J.A.; Amaro, J.E.; Barbaro, M.B.; Donnelly, T.W.; Maieron, C.; Udias, J.M. Superscaling in charged current neutrino quasielastic scattering in the relativistic impulse approximation. Phys. Rev. Lett.
**2005**, 95, 252502. [Google Scholar] [CrossRef] [Green Version] - Ivanov, M.V.; Antonov, A.N.; Megias, G.D.; Caballero, J.A.; Barbaro, M.B.; Amaro, J.E.; Simo, I.R.; Donnelly, T.W.; Udías, J.M. Realistic spectral function model for charged-current quasielastic-like neutrino and antineutrino scattering cross sections on
^{12}C. Phys. Rev. C**2019**, 99, 014610. [Google Scholar] [CrossRef] [Green Version] - Forest, T.D. Off-Shell electron Nucleon Cross-Sections. The Impulse Approximation. Nucl. Phys. A
**1983**, 392, 232–248. [Google Scholar] [CrossRef] - Amaro, J.E.; Arriola, E.R.; Simo, I.R. Scaling violation and relativistic effective mass from quasi-elastic electron scattering: Implications for neutrino reactions. Phys. Rev. C
**2015**, 92, 054607, Erratum in Phys. Rev. C**2019**, 100, 019904. [Google Scholar] [CrossRef] [Green Version] - Galster, S.; Klein, H.; Moritz, J.; Schmidt, K.H.; Wegener, D.; Bleckwenn, J. Elastic electron-deuteron scattering and the electric neutron form factor at four-momentum transfers 5 fm
^{−2}< q^{2}< 14 fm^{−2}. Nucl. Phys. B**1971**, 32, 221. [Google Scholar]

**Figure 1.**Integration path in momentum space of the initial nucleon corresponding to the integral (13) for different values of the energy transfer $\omega $ (indicated in MeV in the key for each panel) and for three values of the momentum transfer.

**Figure 2.**Super scaling analysis with relativistic effective mass (SuSAM*) of ${}^{12}$C data. Top panel: experimental scaling data ${f}_{exp}^{\ast}$ plotted against ${\psi}^{\ast}$. Middle panel: data surviving after cleanup of non-quasielastic sparse points. The red curve is Gaussian fit made in this work, ${f}_{QE}^{\ast}\left({\psi}^{\ast}\right)$. In the bottom panel, we compare the two scaling functions obtained with two different definitions of the averaged single-nucleon responses: using the extrapolated Fermi gas responses and performing the average with a Fermi distribution defined in Section 3.

**Figure 3.**Averaged and extrapolated longitudinal and transverse response functions for protons plus neutrons, as a function of $\omega $ and of the scaling variable ${\psi}^{\ast}$, for three values of the momentum transfer.

**Figure 4.**Averaged and extrapolated longitudinal and transverse response functions for protons and neutrons, as a function of the scaling variable and for three values of the momentum transfer.

**Figure 5.**Averaged and extrapolated transverse response functions for protons and neutrons, for ${G}_{M}^{\ast}=0$, as a function of the scaling variable and for three values of the momentum transfer. Averaged and extrapolated longitudinal response functions for protons and neutrons, for ${G}_{E}^{\ast}=0$, as a function of the scaling variable and for three values of the momentum transfer.

**Figure 6.**Longitudinal and transverse response functions separated for protons and neutrons in the SuSAM* model using the averaged and extrapolated single-nucleon responses for $q=500$ (two left panels) and 1000 MeV/c (two right panels).

**Figure 7.**Longitudinal and transverse response functions in the SuSAM* model using the averaged and extrapolated single-nucleon responses.

**Figure 8.**Quasielastic $(e,{e}^{\prime})$ cross section of ${}^{12}C$ as a function of $\omega $ for several values of the electron energy, $\u03f5$, and scattering angles $\theta $, computed with the present SuSAM* model (black lines) compared to the RFG with effective mass (blue lines). Experimental data (purple lines) are from refs. [13,14].

**Figure 9.**Quasielastic $(e,{e}^{\prime})$ cross section of ${}^{12}C$ as a function of $\omega $ for several values of the electron energy, $\u03f5$, and scattering angles $\theta $, computed with the present SuSAM* model (black lines) compared to the RFG with effective mass (blue lines). Experimental data (purple lines) are from refs. [13,14].

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**MDPI and ACS Style**

Casale, P.R.; Amaro, J.E.; Martinez-Consentino, V.L.; Simo, I.R.
Improved Superscaling in Quasielastic Electron Scattering with Relativistic Effective Mass. *Universe* **2023**, *9*, 158.
https://doi.org/10.3390/universe9040158

**AMA Style**

Casale PR, Amaro JE, Martinez-Consentino VL, Simo IR.
Improved Superscaling in Quasielastic Electron Scattering with Relativistic Effective Mass. *Universe*. 2023; 9(4):158.
https://doi.org/10.3390/universe9040158

**Chicago/Turabian Style**

Casale, Paloma Rodriguez, Jose E. Amaro, Victor L. Martinez-Consentino, and Ignacio Ruiz Simo.
2023. "Improved Superscaling in Quasielastic Electron Scattering with Relativistic Effective Mass" *Universe* 9, no. 4: 158.
https://doi.org/10.3390/universe9040158