# Similarities in Multiparticle Production Processes in pp Collisions as Imprints of Nonextensive Statistics

^{*}

## Abstract

**:**

- (i)
- A Tsallis distribution successfully describes inclusive ${p}_{T}$ distributions of particles as well as jets produced in $pp$ interactions. The nonextensivity parameter in the case of jets, $q=1.14$, is comparable to $q=1.15$ describing inclusive distributions of transverse momenta of particles at the same energy 7 TeV.
- (ii)
- The values of q obtained in this case are roughly the same as those obtained from an analyzes of multiplicity distributions. In particular correspond to q values evaluated from $K/\pi $ ratio as shown in Figure 3.
- (iii)
- One observes a similarity of the corresponding nonextensivity parameters of multiplicity distributions $P\left(N\right)$ and transverse momentum distributions $f\left({p}_{T}\right)$ in multiparticle production processes. Being confirmed, this strengthens the nonextensive Tsallis approach.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Michael, C.; Vanryckeghem, L. Consequences of Momentum Conservation for Particle Production at Large Transverse Momentum. J. Phys. G
**1977**, 3, L151. [Google Scholar] [CrossRef] - Michael, C. Large Transverse Momentum and Large Mass Production in Hadronic Interactions. Prog. Part Nucl. Phys.
**1979**, 2, 1. [Google Scholar] [CrossRef] - Arnison, G.; Astbury, A.; Aubert, B.; Bacci, C.; Bernabei, R.; Bezaguet, A.; Bock, R.; Bowcock, T.J.V.; Calvetti, M.; Carroll, T.; et al. Transverse momentum spectra for charged particles at the CERN proton-antiproton collider. Phys. Lett. B
**1982**, 118, 167. [Google Scholar] [CrossRef] [Green Version] - Hagedorn, R. Multiplicities, p
_{T}Distributions and the Expected Hadron → Quark-Gluon Phase Transition. Riv. Nuovo Cimento.**1983**, 6, 1. [Google Scholar] [CrossRef] - Tsallis, C. Possible Generalization of Boltzmann-Gibbs Statistics. Stat. Phys.
**1988**, 52, 479. [Google Scholar] [CrossRef] - Tsallis, C. Nonadditive entropy: The Concept and its use. Eur. Phys. J. A
**2009**, 40, 257. [Google Scholar] [CrossRef] [Green Version] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Wilk, G.; Włodarczyk, Z. Power laws in elementary and heavy-ion collisions: A Story of fluctuations and nonextensivity? Eur. Phys. J. A
**2009**, 40, 299. [Google Scholar] [CrossRef] [Green Version] - Wilk, G.; Włodarczyk, Z. The Imprints of superstatistics in multiparticle production processes. Cent. Eur. J. Phys.
**2012**, 10, 568. [Google Scholar] [CrossRef] [Green Version] - Wilk, G.; Włodarczyk, Z. Consequences of temperature fluctuations in observables measured in high energy collisions. Eur. Phys. J. A
**2012**, 48, 161. [Google Scholar] [CrossRef] [Green Version] - Adams, J.; Aggarwal, M.M.; Ahammed, Z.; Amonett, J.; Anderson, B.D.; Arkhipkin, D.; Averichev, G.S.; Badyal, S.K.; Bai, Y.; Balewski, J.; et al. Strange particle production in p + p collisions at $\sqrt{s}$ = 200 GeV. Phys. Rev. C
**2007**, 75, 064901. [Google Scholar] - Adare, A.; Afanasiev, S.; Aidala, C.; Ajitanand, N.N.; Akiba, Y.; Al-Bataineh, H.; Alexander, J.; Aoki, K.; Aphecetche, L.; Armendariz, R.; et al. Measurement of neutral mesons in p + p collisions at $\sqrt{s}$ = 200 GeV and scaling properties of hadron production. Phys. Rev. D
**2011**, 83, 052004. [Google Scholar] [CrossRef] [Green Version] - Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Ero, J.; Friedl, M.; Fruhwirth, R.; Ghete, V.M.; et al. Observation of long-range, near-side angular correlations in proton-proton collisions at the LHC. JHEP
**2010**, 2, 41. [Google Scholar] [CrossRef] [Green Version] - Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Ero, J.; Fabjan, C.; Friedl, M.; Fruhwirth, R.; et al. Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at $\sqrt{s}$ = 7 Tev. Phys. Rev. Lett.
**2010**, 105, 022002. [Google Scholar] [CrossRef] [Green Version] - Aamodt, K.; Abel, N.; Abeysekara, U.; Abrahantes Quintana, A.; Abramyan, A.; Adamova, D.; Aggarwal, M.M.; Aglieri Rinella, G.; Agocs, A.G.; Aguilar Salazar, S.; et al. Transverse momentum spectra of charged particles in proton-proton collisions at $\sqrt{s}$ = 900 GeV with ALICE at the LHC. Phys. Lett. B
**2010**, 693, 53. [Google Scholar] [CrossRef] [Green Version] - Aamodt, K.; Abrahantes Quintana, A.; Adamova, D.; Adare, A.M.; Aggarwal, M.M.; Aglieri Rinella, G.; Agocs, A.G.; Aguilar Salazar, S.; Ahammed, Z.; Ahmad, N.; et al. Strange particle production in proton-proton collisions at $\sqrt{s}$ = 0.9 TeV with ALICE at the LHC. Eur. Phys. J. C
**2011**, 71, 1594–1655. [Google Scholar] [CrossRef] [Green Version] - Aad, G.; Abat, E.; Abdallah, J.; Abdelalim, A.A.; Abdesselam, A.; Abdinov, O.; Abi, B.A.; Abolins, M.; Abramowicz, H.; Acerbi, E.; et al. Charged-particle multiplicities in pp interactions measured with the ATLAS detector at the LHC. New J. Phys.
**2011**, 13, 053033. [Google Scholar] [CrossRef] [Green Version] - Shao, M.; Yi, L.; Tang, Z.; Chen, H.; Li, G.; Xu, Z. Examine the species and beam-energy dependence of particle spectra using Tsallis Statistics. J. Phys. G
**2010**, 37, 085104. [Google Scholar] [CrossRef] - Cleymans, J.; Worku, D. The Tsallis Distribution in Proton-Proton Collisions at sqrt(s)= 0.9 TeV at the LHC. J. Phys. G
**2012**, 39, 025006. [Google Scholar] [CrossRef] - Wong, C.Y.; Wilk, G. Tsallis Fits to p
_{t}Spectra for pp Collisions at LHC. Acta Phys. Pol. B**2012**, 43, 2047. [Google Scholar] [CrossRef] - Wong, C.Y.; Wilk, G. Tsallis fits to p
_{T}spectra and multiple hard scattering in pp collisions at the LHC. Phys. Rev. D**2013**, 87, 114007. [Google Scholar] [CrossRef] [Green Version] - Wong, C.-Y.; Wilk, G.; Cirto, L.J.L.; Tsallis, C. From QCD-based hard-scattering to nonextensive statistical mechanical descriptions of transverse momentum spectra in high-energy p p and p antip collisions. Phys. Rev. D
**2015**, 91, 114027. [Google Scholar] [CrossRef] [Green Version] - Cirto, L.J.L.; Tsallis, C.; Wong, C.-Y.; Wilk, G. Possible Implication of a Single Nonextensive p
_{T}Distribution for Hadron Production in High-Energy pp Collisions. Eur. Phys. J. Web Conf.**2015**, 90, 04002. [Google Scholar] - Wilk, G.; Włodarczyk, Z. Self-similarity in jet events following from pp collisions at LHC. Phys. Lett. B
**2013**, 727, 163. [Google Scholar] [CrossRef] [Green Version] - Rybczyński, M.; Włodarczyk, Z. Imprints of energy limitation in transverse momentum distributions of jets. Eur. Phys. J. A
**2015**, 51, 80. [Google Scholar] [CrossRef] [Green Version] - Chatrchyan, S.; Khachatryan, V.; Sirunyan, A.M.; Tumasyan, A.; Adam, W.; Bergauer, T.; Dragicevic, M.; Ero, J.; Fabjan, C.; Friedl, M.; et al. Measurement of the Inclusive Jet Cross Section in pp Collisions at $\sqrt{s}$ = 7 TeV. Phys. Rev. Lett.
**2011**, 107, 132001. [Google Scholar] [CrossRef] [Green Version] - Aaltonen, T.; Adelman1, J.; Akimoto, T.; Albrow, M.G.; Alvarez Gonzalez, B.; Amerio, S.; Amidei, D.; Anastassov, A.; Annovi, A.; Antos, J.; et al. Measurement of the Inclusive Jet Cross Section at the Fermilab Tevatron p anti-p Collider Using a Cone-Based Jet Algorithm. Phys. Rev. D
**2008**, 78, 052006. [Google Scholar] [CrossRef] [Green Version] - Abbott, B.; Abdesselam, A.; Abolins, M.; Abramov, V.; Acharya, B.S.; Adams, D.L.; Adams, M.; Alves, G.A.; Amos, N.; Anderson, E.W.; et al. Quasi-model-independent search for new physics at large transverse momentum. Phys. Rev. D
**2001**, 64, 032003. [Google Scholar] [CrossRef] [Green Version] - Bagnaia, P.; Banner, M.; Battiston, R.; Bloch, P.; Borer, K.; Borghini, M.; Burger, J.; Chollet, J.-C.; Clark, A.G.; Conta, C.; et al. Measurement of Very Large Transverse Momentum Jet Production at the CERN anti-p p Collider. Phys. Lett. B
**1984**, 138, 430. [Google Scholar] [CrossRef] [Green Version] - Bhattacharyya, S.; Haiduc, M.; Neagu, A.T.; Firu, E. A Comparative Study of K
^{±}/π^{±}Ratio in Proton-Proton Collisions at Different Energies: Experimental Results versus Model Simulation. Adv. High Energy Phys.**2018**, 2018, 6307205. [Google Scholar] [CrossRef] [Green Version] - Adam, J.; Adamova, D.; Aggarwal, M.M.; Aglieri Rinella, G.; Agnello, M.; Agrawal, N.; Ahammed, Z.; Ahmed, I.; Ahn, S.U.; Aimo, I.; et al. Measurement of pion, kaon and proton production in proton–proton collisions at $\sqrt{s}$ = 7 TeV. Eur. Phys. J. C
**2015**, 75, 226. [Google Scholar] [CrossRef] [Green Version] - Cleymans, J.; Lykasov, G.I.; Parvan, A.S.; Sorin, A.S.; Teryaev, O.V.; Worku, D. Systematic properties of the Tsallis Distribution: Energy Dependence of Parameters in High-Energy p-p Collisions. Phys. Lett. B
**2013**, 723, 351–354. [Google Scholar] [CrossRef] [Green Version] - Wibig, T. Non-extensivity Parameter of Thermodynamical Model of Hadronic Interactions at LHC energies. J. Phys. G
**2010**, 37, 115009. [Google Scholar] [CrossRef] [Green Version] - Szwed, R.; Wrochna, G.; Wroblewski, A.K. Mystery of the Negative Binomial Distribution. Acta Phys. Polon. B
**1988**, 19, 763. [Google Scholar] - Alexopoulos, T.; Anderson, E.W.; Biswas, N.N.; Bujak, A.; Carmony, D.D.; Erwin, A.R.; Gutay, L.J.; Hirsch, A.S.; Hojvat, C.; Kenney, V.P.; et al. The role of double parton collisions in soft hadron interactions. Phys. Lett. B
**1998**, 435, 453–457. [Google Scholar] [CrossRef] - Kittel, W.; Wolf, E.A.D. Soft Multihadron Dynamics; World Scientific: Singapore, 2005. [Google Scholar]
- Fiete Grosse-Oetringhaus, J.F.; Reygers, K. Charged-Particle Multiplicity in Proton-Proton Collisions. J. Phys. G
**2010**, 37, 083001. [Google Scholar] [CrossRef] [Green Version] - Wilk, G.; Włodarczyk, Z. Fluctuations, correlations and the nonextensivity. Physics A
**2007**, 376, 279–288. [Google Scholar] [CrossRef] [Green Version] - Wilk, G.; Włodarczyk, Z. Generalized thermodynamic uncertainty relations. Physics A
**2011**, 390, 3566–3572. [Google Scholar] [CrossRef] [Green Version]

**Figure 2.**Transverse spectra for jets [17,26] (full symbol) in comparison with spectra for hadrons [13,14] (open symbols) steaming from proton-proton collisions at $\sqrt{s}$ = 7 TeV (with arbitrary normalization at ${p}_{T}$ = 20 GeV). Full and dashed line show power dependencies with slope parameters n to be different by $\Delta n$ = 1.5.

**Figure 3.**Energy dependence of $q-1$ deduced from data on ${(K/\pi )}_{exp}$ ratio (charged particles) compared with values obtained from transverse momenta distributions, $f\left({p}_{T}\right)$ [33] and from the non-diffractive samples of the multiplicity distributions $P\left(N\right)$ [34,35] of the negative binomial form.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rybczynski, M.; Wlodarczyk, Z.
Similarities in Multiparticle Production Processes in *pp* Collisions as Imprints of Nonextensive Statistics. *Symmetry* **2020**, *12*, 1339.
https://doi.org/10.3390/sym12081339

**AMA Style**

Rybczynski M, Wlodarczyk Z.
Similarities in Multiparticle Production Processes in *pp* Collisions as Imprints of Nonextensive Statistics. *Symmetry*. 2020; 12(8):1339.
https://doi.org/10.3390/sym12081339

**Chicago/Turabian Style**

Rybczynski, Maciej, and Zbigniew Wlodarczyk.
2020. "Similarities in Multiparticle Production Processes in *pp* Collisions as Imprints of Nonextensive Statistics" *Symmetry* 12, no. 8: 1339.
https://doi.org/10.3390/sym12081339