# Lévy α-Stable Model for the Non-Exponential Low-|t| Proton–Proton Differential Cross-Section

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## Abstract

**:**

^{2}≤ −t ≤ 1.2 GeV

^{2}. Considering, instead of Gaussian, a more general Lévy $\alpha $-stable shape for the parton distributions of the constituent quark and diquark inside the proton and for the relative separation between them, a generalized description of data is obtained, where the ReBB model corresponds to the $\alpha =2$ special case. Extending the model to $\alpha <2$, we conjecture that the validity of the model can be extended to a wider kinematic range, in particular, to lower values of the four-momentum transfer $-t$. We present the formal Lévy $\alpha $-stable generalization of the Bialas–Bzdak model and show that a simplified version of this model can be successfully fitted, with $\alpha <2$, to the non-exponential, low $-t$ differential cross-section data of elastic proton–proton scattering at $\sqrt{s}=8$ TeV.

## 1. Introduction

^{2}≤ −t ≤ 1.2 GeV

^{2}in a statistically non-excludible manner, i.e., with a confidence level greater than or equal to 0.1%.

^{2}, −t > 1.2 GeV

^{2}) is anticipated.

^{2}$\le -t\le 1.2$ GeV

^{2}interval at LHC energies includes the region of the characteristic minimum–maximum structure of the $pp$ elastic differential cross-section. In the 0.01 GeV

^{2}$\lesssim -t\lesssim 0.15$ GeV

^{2}interval, another characteristic structure, a non-exponential behavior is observed. A significant non-exponential behavior was measured by TOTEM at CERN LHC at 8 and 13 TeV center of mass energies [8,9]. Similar behavior was observed also at the CERN ISR accelerator in the 1970s [10], where measurements were made in the 20 GeV $\lesssim \sqrt{s}\lesssim 60$ GeV energy region.

## 2. From Gaussian to Lévy $\mathit{\alpha}$-Stable $\mathit{p}=(\mathit{q},\mathit{d})$ BB Model

## 3. A Simple Lévy $\mathit{\alpha}$-Stable Model

## 4. The t = 0 Measurable Quantities and the BB Model Parameters

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Csörgő, T.; Hegyi, S.; Szanyi, I.
Lévy *α*-Stable Model for the Non-Exponential Low-|*t*| Proton–Proton Differential Cross-Section. *Universe* **2023**, *9*, 361.
https://doi.org/10.3390/universe9080361

**AMA Style**

Csörgő T, Hegyi S, Szanyi I.
Lévy *α*-Stable Model for the Non-Exponential Low-|*t*| Proton–Proton Differential Cross-Section. *Universe*. 2023; 9(8):361.
https://doi.org/10.3390/universe9080361

**Chicago/Turabian Style**

Csörgő, Tamás, Sándor Hegyi, and István Szanyi.
2023. "Lévy *α*-Stable Model for the Non-Exponential Low-|*t*| Proton–Proton Differential Cross-Section" *Universe* 9, no. 8: 361.
https://doi.org/10.3390/universe9080361