# 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

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**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