# Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron Scattering

## Abstract

**:**

## 1. Introduction

- Is the input pomeron a simple Regge pole? What are the alternatives, if any?
- Do resonances terminate, abruptly replaced by a continuum or they gradually fade, their peaks becoming progressively wider and lower? This issue is connected with the Hagedorn spectrum and possible phase transition between hadronic and quark-gluon matter.
- Origin of the diffraction (dip-bump) pattern in elastic hadron scattering;
- Role of unitarization in producing the dip-bump structure;
- Are there two (or more) pomerons—a “soft” and a “QCD-inspired”, “hard” one?
- Can the Regge-pomeron pole be ${Q}^{2}-$dependent?

## 2. Hadron-Hadron Scattering

#### 2.1. Elastic Proton-Proton Scattering

#### 2.2. Regge Trajectories

#### 2.3. Unitarity

#### 2.3.1. “U-matrix” Unitarization

#### 2.3.2. Eikonal

## 3. Nuclear Structure: From Deeply Virtual Compton Scattering (DVCS) to Generalized Parton Distributions (GPDs)

#### 3.1. Deeply Virtual Compton Scattering

#### 3.2. Relating DVCS Observables to GPDs

## 4. Modelling DVCS

#### Simple Model of DVCS

## 5. Reggeometry

#### Two-Component Reggeometric Pomeron

#### Fitting the Two-Component Pomeron to VMP and DVCS HERA Data

## 6. Balancing between “Soft” and “Hard” Dynamics

- sub-leading Regge contributions must be included in any extension of the model to lower energies (below 30 GeV);
- the $\tilde{{Q}^{2}}$ dependence of the scattering amplitude, introduced empirically has to be compared with the results of unitarization and/or QCD evolution.
- as seen from Section 6, the “soft” component of the pomeron dominates in the region of small t and small $\tilde{{Q}^{2}}$. Hence, any parameter responsible for the “softness” and/or “hardness” of processes, should be a combination of t and ${Q}^{2}$. A simple solution was suggested in Ref. [44] with the introduction of the variable $z=t-{Q}^{2}$. The interplay of these two variables remains an important open problem that requires further investigation.

## 7. Spin of the Proton in Terms of Its Constituents

#### Ji’s and J-M’s Decompositions

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Diagram of elastic scattering with t-channel exchange containing a branch point at $t=4{m}_{\pi}^{2}$.

**Figure 2.**(

**a**) Typical shape of the high-energy differential (diffractive) cross section, and (

**b**) Its impact-parameter image. A “dictionary” relating the shape of the diffraction cone (in t), left icon to its impact-paramter image (rignt), and highlighting their two silent features: the “break”, origin of nucleon’s atmosphere and the dip-bump, resulting from absorptions (unitarity!) at small values of the impact parameter, see [13].

**Figure 3.**Deeply virtual Compton scattering (DVCS) and Bethe–Heitler (BH) amplitudes contributing to the photon leptoproduction cross section in leading order approximation of QED.

**Figure 4.**Interference in DVCS (Equation (34)) provides a holographic picture of the proton similar to the classical one.

**Figure 5.**Generalized parton distributions (GPD) (

**right**) unify in a non-trivial way the the impact parameter image of a nucleon (

**left**) with the x dependence of DIS structure functions (

**middle**).

**Figure 6.**Factorization of the DVCS amplitude to leading order in perturbative QCD and to leading twist-two accuracy.

**Figure 9.**Fit of Equation (60) to the data on the elastic cross section ${\sigma}_{el}\left(W\right)$ for ${\rho}^{0}$, for different values of ${Q}^{2}$.

**Figure 10.**Fit of Equation (60) to the data on the elastic cross section ${\sigma}_{el}\left(W\right)$ for $\varphi $, for different values of ${Q}^{2}$.

**Figure 11.**Fit of Equation (60) to the data on the elastic cross section ${\sigma}_{el}\left(W\right)$ for $J/\psi $, for different values of ${Q}^{2}$.

**Figure 12.**Fit of Equation (60) to the data on the elastic cross section ${\sigma}_{el}\left(W\right)$ for ${\rm Y}$, for different values of ${Q}^{2}$.

**Figure 13.**Fit of Equation (59) to the data on the differential elastic cross section $d{\sigma}_{el}/dt$ for ${\rho}^{0}$, for different values of ${Q}^{2}$ and W.

**Figure 14.**Experimental data on the slope B as function of $\tilde{{Q}^{2}}$ for ${\rho}^{0},\varphi ,J/\psi $, ${\rm Y}$ and $\mathsf{\Psi}$(2S), and our theoretical predictions from Equation (63).

**Figure 15.**Interplay between soft (green line), hard (blue line) and interference (yellow line) components of the cross section ${\sigma}_{i,el}$ (left plot) and ${R}_{i}(\tilde{{Q}^{2}},t)$ (right plot) as functions of $\tilde{{Q}^{2}}$, for $W=70$ GeV.

**Figure 16.**

**Left column**: soft (

**upper surface**), hard (

**middle surface**) and interference (

**bottom surface**) components of the ratio ${R}_{i}(\tilde{{Q}^{2}},W,t)$ are shown as functions of $\tilde{{Q}^{2}}$ and t, for $W=70$ GeV.

**Right column**: some representative curves of the surfaces projected on the ($t,\tilde{{Q}^{2}}$) plane.

${\mathit{A}}_{0}$$\left[\frac{\sqrt{\mathbf{nb}}}{\mathbf{GeV}}\right]$ | $\tilde{{\mathit{Q}}_{0}^{2}}$$\left[{\mathbf{GeV}}^{2}\right]$ | n | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}^{\prime}$$\left[\frac{1}{{\mathbf{GeV}}^{2}}\right]$ | a | b | ${\tilde{\mathit{\chi}}}^{2}$ | |
---|---|---|---|---|---|---|---|---|

${\rho}^{0}$ | 344 ± 376 | 0.29 ± 0.14 | 1.24 ± 0.07 | 1.16 ± 0.14 | 0.21 ± 0.53 | 0.60 ± 0.33 | 0.9 ± 4.3 | 2.74 |

$\varphi $ | 58 ± 112 | 0.89 ± 1.40 | 1.30 ± 0.28 | 1.14 ± 0.19 | 0.17 ± 0.78 | 0.0 ± 19.8 | 1.34 ± 5.09 | 1.22 |

$J/\psi $ | 30 ± 31 | 2.3 ± 2.2 | 1.45 ± 0.32 | 1.21 ± 0.09 | 0.077 ± 0.072 | 1.72 | 1.16 | 0.27 |

${\rm Y}$(1S) | 37 ± 100 | 0.93 ± 1.75 | 1.45 ± 0.53 | 1.29 ± 0.25 | 0.006 ± 0.6 | 1.90 | 1.03 | 0.4 |

$DVCS$ | 14.5 ± 41.3 | 0.28 ± 0.98 | 0.90 ± 0.18 | 1.23 ± 0.14 | 0.04 ± 0.71 | 1.6 | 1.9 ± 2.5 | 1.05 |

${\mathit{A}}_{0\mathit{s},\mathit{h}}$$\left[\frac{\sqrt{\mathbf{nb}}}{\mathbf{GeV}}\right]$ | $\tilde{{\mathit{Q}}_{\mathit{s},\mathit{h}}^{2}}$$\left[{\mathbf{GeV}}^{2}\right]$ | ${\mathit{n}}_{\mathit{s},\mathit{h}}$ | ${\mathit{\alpha}}_{0\phantom{\rule{0.166667em}{0ex}}\mathit{s},\mathit{h}}$ | ${\mathit{\alpha}}_{\mathit{s},\mathit{h}}^{\prime}$$\left[\frac{1}{{\mathbf{GeV}}^{2}}\right]$ | ${\mathit{b}}_{\mathit{s},\mathit{h}}$$\left[\frac{1}{{\mathbf{GeV}}^{2}}\right]$ | |
---|---|---|---|---|---|---|

soft | 2104 ± 1749 | 0.29 ± 0.20 | 1.63 ± 0.40 | 1.005 ± 0.090 | 0.32 ± 0.57 | 2.93 ± 5.06 |

hard | 44 ± 22 | 1.15 ± 0.52 | 1.34 ± 0.16 | 1.225 ± 0.055 | 0.0 ± 17 | 2.22 ± 3.09 |

**Table 3.**Values of ${\tilde{\chi}}^{2}$ of the fit and the numbers of degrees of freedom (number of data points) for different observables (i.e., ${\sigma}_{el}\left(W\right)$, ${\sigma}_{el}\left({Q}^{2}\right)$ or $d{\sigma}_{el}\left(t\right)/dt$), and values of ${\tilde{\chi}}_{i}^{2}$ for different reactions (VMP or DVCS).

Meson | ${\mathit{\sigma}}_{\mathbf{el}}\left(\mathit{W}\right)$ | ${\mathit{\sigma}}_{\mathbf{el}}\left({\mathit{Q}}^{2}\right)$ | $\frac{\mathit{d}{\mathit{\sigma}}_{\mathbf{el}}}{\mathbf{dt}}$ | ||||
---|---|---|---|---|---|---|---|

Production | ${\tilde{\mathit{\chi}}}^{2}$ | ${\mathit{N}}_{\mathit{d}.\mathit{o}.\mathit{f}.}$ | ${\tilde{\mathit{\chi}}}^{2}$ | ${\mathit{N}}_{\mathit{d}.\mathit{o}.\mathit{f}.}$ | ${\tilde{\mathit{\chi}}}^{2}$ | ${\mathit{N}}_{\mathit{d}.\mathit{o}.\mathit{f}.}$ | ${\tilde{\mathit{\chi}}}_{\mathit{i}}^{2}$ |

${\rm Y}$ | 0.47 | 4 | 0.00 | 1 | 0.00 | 1 | 0.469 |

$J\psi $ | 0.47 | 43 | 0.47 | 16 | 2.37 | 92 | 1.105 |

$\omega $ | 0.10 | 3 | 0.09 | 4 | 0.33 | 7 | 0.174 |

$\varphi $ | 1.19 | 46 | 1.42 | 22 | 1.10 | 85 | 1.238 |

$\rho $ | 1.49 | 112 | 0.97 | 64 | 3.85 | 94 | 2.104 |

$DVCS$ | 1.83 | 89 | 2.20 | 38 | 1.41 | 84 | 1.815 |

${\mathit{A}}_{0\mathit{s},\mathit{h}}$$\left[\frac{\sqrt{\mathbf{nb}}}{\mathbf{GeV}}\right]$ | $\tilde{{\mathit{Q}}_{\mathit{s},\mathit{h}}^{2}}$$\left[{\mathbf{GeV}}^{2}\right]$ | ${\mathit{n}}_{\mathit{s},\mathit{h}}$ | ${\mathit{\alpha}}_{0\phantom{\rule{0.166667em}{0ex}}\mathit{s},\mathit{h}}$ | ${\mathit{\alpha}}_{\mathit{s},\mathit{h}}^{\prime}$$\left[\frac{1}{{\mathbf{GeV}}^{2}}\right]$ | ${\mathit{b}}_{\mathit{s},\mathit{h}}$$\left[\frac{1}{{\mathbf{GeV}}^{2}}\right]$ | |
---|---|---|---|---|---|---|

soft | 807 ± 1107 | 0.46 ± 0.70 | 1.79 ± 0.79 | 1.08 | 0.25 | 3.41 ± 2.48 |

hard | 47.9 ± 46.9 | 1.30 ± 1.12 | 1.33 ± 0.26 | 1.20 | 0.01 | 2.15 ± 1.14 |

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Jenkovszky, L.
Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron Scattering. *Symmetry* **2020**, *12*, 1784.
https://doi.org/10.3390/sym12111784

**AMA Style**

Jenkovszky L.
Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron Scattering. *Symmetry*. 2020; 12(11):1784.
https://doi.org/10.3390/sym12111784

**Chicago/Turabian Style**

Jenkovszky, László.
2020. "Spin and Polarization in High-Energy Hadron-Hadron and Lepton-Hadron Scattering" *Symmetry* 12, no. 11: 1784.
https://doi.org/10.3390/sym12111784