# Chiral Soliton Models and Nucleon Structure Functions

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## Abstract

**:**

## 1. Introduction

## 2. Framework of Deep Inelastic Scattering

## 3. The Chiral Quark Model

## 4. Self-Consistent Soliton

## 5. Hadron Tensor for the Nucleon as Soliton

## 6. Numerical Results

#### 6.1. Unpolarized Structure Functions

#### 6.2. Polarized Structure Functions

#### 6.3. Boosting to the Infinite Momentum Frame

#### 6.4. DGLAP Evolution

## 7. Related Approaches

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Feynman diagram describing the kinematical set-up, where k and ${k}^{\prime}$ are the momenta of the initial and final electrons, respectively, while p is the momentum of the incoming proton, typically taken in the rest frame. The set of final hadrons, X is not detected and summed over, cf. Equation (3).

**Figure 2.**Two photon coupling to fermion loop. Thick lines are the full fermion propagators ${{\mathbf{D}}^{\left(\pi \right)}}^{-1}$ (or ${{\mathbf{D}}^{\left(\pi \right)}}_{5}^{-1}$) without any perturbation expansion. The thin line in the loop represents a free (massless) fermion propagator, $\partial \phantom{\rule{-6.00006pt}{0ex}}{/}^{-1}$. Dashed lines denote Cutkosky cuts as discussed after Equation (58).

**Figure 3.**Model prediction (with $m=400\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$) for the isoscalar unpolarized structure function in the nucleon rest frame. Dotted and dotted-dashed lines refer to the positive and negative frequency contributions, respectively.

**Figure 4.**Same as Figure 3 for the isovector unpolarized structure function. Observe the logarithmic scale for the Bjorken variable x.

**Figure 5.**Model prediction of the unpolarized structure function ${f}_{2}^{p}\left(x\right)-{f}_{2}^{n}\left(x\right)$ for the constituent quark mass of $m=400\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$.

**Figure 6.**Model prediction ($m=400\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$) for the isovector longitudinal polarized structure functions. For the valence and vacuum contributions we separately display the positive (dotted) and negative (dotted-dashed) frequency contributions.

**Figure 7.**Same as Figure 6 for the isoscalar longitudinal polarized structure functions.

**Figure 8.**Model prediction of the isoscalar structure function, ${g}_{2}$, for the constituent quark mass of $m=400\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$.

**Figure 9.**Model prediction of the isovector polarized structure functions, ${g}_{2}$, frame for the constituent quark mass of $m=400\phantom{\rule{3.33333pt}{0ex}}\mathrm{MeV}$.

**Figure 10.**Model prediction for the longitudinal polarized proton structure functions. Left panel: ${g}_{1}^{p}\left(x\right)$; right panel: ${g}_{1}^{{}^{3}\mathrm{He}}\left(x\right)$. These functions are “DGLAP” evolved from ${\mu}^{2}=0.4\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{2}$ to ${Q}^{2}=3\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{2}$ after being projected to the IMF. Data are from Refs. [83,84] for the proton and from Ref. [85] for helium. In the latter case E refers to the electron energy.

**Figure 11.**Model prediction for the polarized proton structure functions ${g}_{2}^{p}\left(x\right)$. This function is “DGLAP” evolved from ${\mu}^{2}=0.4\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{2}$ to ${Q}^{2}=5\phantom{\rule{0.166667em}{0ex}}{\mathrm{GeV}}^{2}$ after being projected to the IMF. Data are from Ref [90].

**Figure 12.**Model prediction ($m=400\phantom{\rule{0.166667em}{0ex}}\mathrm{MeV}$) for the unpolarized structure function that enters the Gottfried sum rule, Equation (86). This function is “DGLAP” evolved from ${\mu}^{2}=0.4\phantom{\rule{3.33333pt}{0ex}}{\mathrm{GeV}}^{2}$ to ${Q}^{2}=4\phantom{\rule{3.33333pt}{0ex}}{\mathrm{GeV}}^{2}$ after transformation to the IMF. Data are from Ref. [80].

**Table 1.**Projection operators which extract the leading large ${Q}^{2}$ components from the hadron tensor. The projectors given in the spin independent cases presume the contraction of ${W}_{\rho \sigma}$ with ${S}^{\mu \nu \rho \sigma}={g}^{\mu \rho}{g}^{\nu \sigma}+{g}^{\rho \nu}{g}^{\mu \sigma}-{g}^{\mu \nu}{g}^{\rho \sigma}$. The last row denotes the required spin orientation of the nucleon.

${\mathit{f}}_{1}$ | ${\mathit{f}}_{2}$ | ${\mathit{g}}_{1}$ | ${\mathit{g}}_{\mathit{T}}={\mathit{g}}_{1}+{\mathit{g}}_{2}$ |
---|---|---|---|

$-\frac{1}{2}{g}^{\mu \nu}$ | $-x{g}^{\mu \nu}$ | $\frac{\mathrm{i}}{2{M}_{N}}{\u03f5}^{\mu \nu \rho \sigma}\frac{{q}_{\rho}{p}_{\sigma}}{q\xb7s}$ | $\frac{-\mathrm{i}}{2{M}_{N}}{\u03f5}^{\mu \nu \rho \sigma}{s}_{\rho}{p}_{\sigma}$ |

$\genfrac{}{}{-1.0pt}{}{\mathrm{spin}}{\mathrm{independent}}$ | $\genfrac{}{}{-1.0pt}{}{\mathrm{spin}}{\mathrm{independent}}$ | $\overrightarrow{s}\Vert \overrightarrow{q}$ | $\overrightarrow{s}\perp \overrightarrow{q}$ |

$\mathit{m}\left[\mathbf{MeV}\right]$ | ${\mathit{m}}^{0}\left[\mathbf{MeV}\right]$ | $\mathbf{\Lambda}\left[\mathbf{GeV}\right]$ | ${\mathit{E}}_{\mathbf{tot}}\left[\mathbf{MeV}\right]$ | ${\mathit{\alpha}}^{2}[1/\mathbf{GeV}]$ | $\mathbf{\Delta}\mathit{M}\left[\mathbf{MeV}\right]$ | ${\mathit{g}}_{\mathit{a}}$ |
---|---|---|---|---|---|---|

350 | 7.9 | 0.77 | 1267 | 8.65 | 173 | 0.85 |

400 | 8.4 | 0.74 | 1269 | 5.89 | 255 | 0.80 |

450 | 8.5 | 0.73 | 1257 | 4.82 | 311 | 0.77 |

**Table 3.**The Gottfried sum rule for various values of m. The subscripts “v” and “s” denote the valence and vacuum contributions, respectively. The fourth column contains their sums.

$\mathit{m}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{MeV}\right]$ | ${\left[{\mathcal{S}}_{\mathit{G}}\right]}_{\mathbf{v}}$ | ${\left[{\mathcal{S}}_{\mathit{G}}\right]}_{\mathbf{s}}$ | ${\mathcal{S}}_{\mathit{G}}$ | Emp. Value |
---|---|---|---|---|

400 | $0.214$ | $0.000156$ | $0.214$ | |

450 | $0.225$ | $0.000248$ | $0.225$ | $0.235\pm 0.026$ [80] |

500 | $0.236$ | $0.000356$ | $0.237$ |

**Table 4.**Axial isovector and isoscalar charges for various values of the constituent quark mass m from integrating the longitudinal structure functions. Subscripts are as in Table 3. Data in parenthesis give the numerical results as obtained from the coordinate space representation, cf. Equation (47) and Table 2.

$\mathit{m}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{MeV}\right]$ | $[{\mathit{g}}_{\mathit{A}}{]}_{\mathbf{v}}$ | $[{\mathit{g}}_{\mathit{A}}{]}_{\mathbf{s}}$ | ${\mathit{g}}_{\mathit{A}}$ | emp. Value | $[{\mathit{g}}_{\mathit{A}}^{0}{]}_{\mathbf{v}}$ | $[{\mathit{g}}_{\mathit{A}}^{0}{]}_{\mathbf{s}}$ | ${\mathit{g}}_{\mathit{A}}^{0}$ | emp. Value |
---|---|---|---|---|---|---|---|---|

400 | $0.734$ | $0.065$ | $0.799$ ($0.800$) | $1.2601$ | $0.344$ | $0.0016$ | $0.345$ ($0.350$) | |

450 | $0.715$ | $0.051$ | $0.766$ ($0.765$) | $\pm 0.0025$ | $0.327$ | $0.0021$ | $0.329$ ($0.332$) | $0.33\pm 0.06$ |

500 | $0.704$ | $0.029$ | $0.733$ ($0.733$) | [62] | $0.316$ | $0.0028$ | $0.318$ ($0.323$) | [81] |

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Weigel, H.; Takyi, I.
Chiral Soliton Models and Nucleon Structure Functions. *Symmetry* **2021**, *13*, 108.
https://doi.org/10.3390/sym13010108

**AMA Style**

Weigel H, Takyi I.
Chiral Soliton Models and Nucleon Structure Functions. *Symmetry*. 2021; 13(1):108.
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**Chicago/Turabian Style**

Weigel, Herbert, and Ishmael Takyi.
2021. "Chiral Soliton Models and Nucleon Structure Functions" *Symmetry* 13, no. 1: 108.
https://doi.org/10.3390/sym13010108