# High-Energy Lepton Scattering and Nuclear Structure Issues

## Abstract

**:**

## 1. Introduction

## 2. Kinematics for Semi-Inclusive Scattering

#### 2.1. Basic Variables for Semi-Inclusive Electron Scattering

**inclusive**scattering where no further information about the final state is assumed, or

**semi-inclusive**scattering where the final state is assumed to be divided into two pieces (see Figure 1), one a specific particle “x” that is assumed to be detected, having 4-momentum ${P}_{x}^{\mu}=\left({E}_{x},{\mathbf{p}}_{x}\right)$, where ${E}_{x}={({p}_{x}^{2}+{M}_{x}^{2})}^{1/2}$, together with the undetected (“missing”) parts of the final state having 4-momentum ${P}_{m}^{\mu}=\left({E}_{m}^{tot},{\mathbf{p}}_{m}\right)$ with missing energy ${E}_{m}^{tot}$, missing momentum ${\mathbf{p}}_{m}$, and invariant mass ${W}_{m}={({\left({E}_{m}^{tot}\right)}^{2}-{p}_{m}^{2})}^{1/2}$. Note: for the total missing energy we use ${E}_{m}^{tot}$, since we reserve the notation ${E}_{m}$ to denote a different, but related quantity (see below). Of course even more complicated reactions having more than two particles detected can be studied; however, for the present discussions we shall restrict our attention to inclusive and semi-inclusive reactions. Moreover, note that particle x has been left unspecified: often for nuclear physics this is a nucleon and for semi-inclusive scattering we mean reactions of the sort $(e,{e}^{\prime}p)$ or $(e,{e}^{\prime}n)$. However, other cases are also of interest such as $(e,{e}^{\prime}\alpha )$ (see [4]) or $(e,{e}^{\prime}\pi )$.

#### 2.2. Physical Region and Scaling Variables

**target rest frame**is the relevant one (see below), although in the future one anticipates experiments being performed at a collider—say involving colliding beams of electrons with ${}^{3}$He nuclei—in

**collider kinematics**where then $p\ne 0$. Moreover, it is well known that for some reactions where exclusive final states are reached the virtual-photon/target

**center-of-momentum frame**plays a special role; see, for instance, studies of pion electroproduction for kinematics where simply a pion and a nucleon define the final state. Or, finally, sometimes the

**Breit frame**is advantageous to use.

## 3. Nuclear Physics Issues

#### 3.1. General Shape of the “Nuclear Landscape”

#### 3.2. Constraints from Inclusive Scattering

#### 3.3. Further Comments

## 4. Summary

## Funding

## Acknowledgments

## Conflicts of Interest

## Note

1 | We use the conventions of [3] in this work. We also employ the conventions previously used by us and others in many previous studies. Namely, we denote 4-vectors by capital letters and 3-vectors by lower case letters, ${A}^{\mu}=({A}^{0},\mathbf{a})$, ${B}^{\mu}=({B}^{0},\mathbf{b})$, etc. The scalar product of two 4-vectors is then $A\xb7B={A}^{0}{B}^{0}-\mathbf{a}\xb7\mathbf{b}$ and, therefore, the scalar product of a 4-vector with itself is ${A}^{2}={\left({A}^{0}\right)}^{2}-{a}^{2}$ where $a\equiv \left|\mathbf{a}\right|$. One potential point of confusion can occur with these conventions, viz. for the momentum transfer 4-vector ${Q}^{\mu}=({Q}^{0},\mathbf{q})=(\omega ,\mathbf{q})$ we have ${Q}^{2}={\omega}^{2}-{q}^{2}$ which for electron scattering is space-like, and accordingly ${Q}^{2}<0$. One should be careful not to confuse our sign convention for this quantity with the so-called SLAC convention which has the opposite sign ${Q}_{SLAC}^{2}=-{Q}^{2}>0.$ |

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**Figure 1.**Feynman diagram for semi-inclusive electron scattering; figure from [1]. The 4-momenta here are discussed in the text. In particular, particle x is assumed to be detected in coincidence with the scattered electron and thus ${P}_{x}^{\mu}$ is assumed to be known. Since the total final-state momentum ${{P}^{\prime}}^{\mu}$ is known (see text) this implies that the missing 4-momentum is also known via the relationship ${P}_{m}^{\mu}={{P}^{\prime}}^{\mu}-{P}_{x}^{\mu}$.

**Figure 2.**Physically allowed region for the situation where $y<0$. The upper boundary labelled ${\mathcal{E}}_{m}^{0}\left({p}_{m}\right)$ has $cos{\theta}_{m}=0$. (The variables employed here are discussed in the text; figure from [1]).

**Figure 3.**Physically allowed region for the situation where $y>0$. The upper boundary labeled ${\mathcal{E}}_{m}^{0}\left({p}_{m}\right)$ has $cos{\theta}_{m}=0$ whereas that labeled ${\mathcal{E}}_{m}^{0}(-{p}_{m})$ has $cos{\theta}_{m}=\pi $. (The variables employed here are discussed in the text; figure from [1]).

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Donnelly, T.W.
High-Energy Lepton Scattering and Nuclear Structure Issues. *Universe* **2023**, *9*, 196.
https://doi.org/10.3390/universe9040196

**AMA Style**

Donnelly TW.
High-Energy Lepton Scattering and Nuclear Structure Issues. *Universe*. 2023; 9(4):196.
https://doi.org/10.3390/universe9040196

**Chicago/Turabian Style**

Donnelly, Thomas W.
2023. "High-Energy Lepton Scattering and Nuclear Structure Issues" *Universe* 9, no. 4: 196.
https://doi.org/10.3390/universe9040196