In this paper, one discusses the effects of the Tsallis entropy on the radial pressure distribution in the proton. Using a damped confinement potential the pressure distribution is obtained from the Tsallis entropy approach, being the entropic-index $ω$ connected with the proton temperature concerning some transition temperature. Then, the approach allows the study of the proton thermal evolution up to the Quark-Gluon Plasma regime. The von Laue stability condition, arising from the pressure distribution results in positive and negative energy regions. An analogy between the results for the radial pressure distribution and the proton-proton and the antiproton-proton total cross section is performed. The negative energy region is identified with the odderon exchange while the positive represents the pomeron exchange dominance above some transition energy $\sqrt{s_c}$. The hollowness effect is also discussed in terms of the results obtained and the proposed analogy.
A detailed analysis is presented of the precise values of the Tsallis parameters obtained in $p-p$ collisions for identified particles, pions, kaons and protons at the LHC at three beam energies $\sqrt{s} = 0.9, 2.76$ and $7$ TeV. Interpolated data at $\sqrt{s} = $ 5.02 TeV have also been included. It is shown that the Tsallis formula provides reasonably good fits to the $p_T$ distributions in $p-p$ collisions at the LHC using three parameters $dN/dy$, $T$ and $q$. However, the parameters $T$ and $q$ depend on the particle species and are different for pions, kaons and protons. As a consequenc…
The charged particle transverse momentum ($p_T$) spectra measured by the ATLAS and CMS collaborations in proton - proton collisions at sqrt(s) = 0.9 and 7 TeV have been studied using Tsallis thermodynamics. A thermodynamically consistent form of the Tsallis distribution is used for fitting the transverse momentum spectra at mid-rapidity. It is found that the fits based on the proposed distribution provide an excellent description over 14 orders of magnitude with $p_T$ values up to 200 GeV/c.
This is a concise introduction to the topic of nonextensive Tsallis statistics meant especially for those interested in its relation to high energy proton-proton, proton-nucleus, and nucleus-nucleus collisions. The three types of Tsallis statistics are reviewed. Only one of them is consistent with the fundamental hypothesis of equilibrium statistical mechanics. The single particle distributions associated with it, namely Boltzmann, Fermi-Dirac, and Bose-Einstein, are derived. These are not equilibrium solutions to the conventional Boltzmann transport equation which must be modified in a rather…
A systematic analysis of transverse momentum and rapidity distributions measured in high-energy proton - proton (pp) collisions for energies ranging from 53 GeV to 7 TeV using Tsallis thermodynamics is presented. The excellent description of all transverse momentum spectra obtained in earlier analyses is confirmed and extended. All energies can be described by a single Tsallis temperature of 68 +/- 5 MeV at all beam energies and particle types investigated (43 in total). The value of the entropic index, q, shows a wider spread but is always close to q approx 1.146. These values are then used t…
Numerous papers have appeared recently showing fits to transverse momentum ($p_T$) spectra measured at the Large Hadron Collider (LHC) in proton - proton collisions.This talk focuses on the fits extending to very large values of the transverse momentum with $p_T$ values up to 200 GeV/c as measured by the ATLAS and CMS collaborations at $\sqrt{s}$ = 0.9 and 7 TeV. A thermodynamically consistent form of the Tsallis distribution is used for fitting the transverse momentum spectra at mid-rapidity. The fits based on the proposed distribution provide an excellent description over 14 orders of magnit…
In recent years the Tsallis statistics is gaining popularity in describing charged particle produc- tion and their properties, in particular pT spectra and the multiplicities in high energy particle collisions. Motivated by its success, an analysis of the LHC data of proton-proton collisions at ener- gies ranging from 0.9 TeV to 7 TeV in different rapidity windows for charged particle multiplicities has been done. A comparative analysis is performed in terms of the Tsallis distribution, the Gamma distribution and the shifted-Gamma distribution. An interesting observation on the inapplicability…
The possibility of formation of Bose-Einstein Condensation (BEC) is studied in $pp$ collisions at $\sqrt s$ = 7 TeV at the Large Hadron Collider. A thermodynamically consistent non-extensive formulation of the identified hadron transverse momentum distributions is used to estimate the critical temperature required to form BEC of charged pions, which are the most abundant species in a multi-particle production process in hadronic and nuclear collisions. The obtained results have been contrasted with the systems produced in Pb-Pb collisions to have a better understanding. We observe an explicit …
We present an overview of a proposal in relativistic proton-proton ($pp$) collisions emphasizing the thermal or kinetic freeze-out stage in the framework of the Tsallis distribution. In this paper we take into account the chemical potential present in the Tsallis distribution by following a two step procedure. In the first step we used the redundancy present in the variables such as the system temperature, $T$, volume, $V$, Tsallis exponent, $q$, chemical potential, $μ$, and performed all fits by effectively setting to zero the chemical potential. In the second step the value $q$ is kept fixed…
The latest, high-accuracy identified hadron spectra measurements in high-energy nuclear collisions led us to the investigation of the strongly interacting particles and collective effects in small systems. Since microscopical processes result in a statistical Tsallis-Pareto distribution, the fit parameters $q$ and $T$ are well suited for identifying system size scalings and initial conditions. Moreover, parameter values provide information on the deviation from the extensive, Boltzmann-Gibbs statistics in finite-volumes. We apply here the fit procedure developed in our earlier study for proton…
In high-energy collisions the number of the created particles is far less than the thermodynamic limit, especially in small colliding systems (e.g. proton-proton). Therefore final-state effects and fluctuations in the one-particle energy distribution are appreciable. As a consequence the characterization of identified hadron spectra with the Boltzmann\,--\,Gibbs thermodynamical approach is insufficient. Instead particle spectra measured in high-energy collisions can be described very well with Tsallis\,--\,Pareto distributions, derived from non-extensive thermodynamics. Using the Tsallis q-ent…
Valence Quark Distributions in Pions: Insights from Tsallis Entropy
South China Normal University（华南师范大学）；University of Chinese Academy of Sciences（中国科学院大学）；Chinese Academy of Sciences（中国科学院）；Lanzhou University（兰州大学）；Guizhou University of Finance and Economics（贵州财经大学）
In the framework of a multi-source thermal model at the partonic-level, we have analyzed transverse momentum spectra of hadrons measured by the ALICE Collaboration in proton-proton ($pp$ or $p$-$p$) collisions at the center-of-mass energy of $\sqrt{s}=7$ and 13 TeV, proton-lead ($p$-Pb) collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and lead-lead (Pb-Pb) collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV. For meson(baryon), the contributions of two(three) constituent quarks are considered, in which each quark contributes to hadron transverse momentum to obey the revised phenomenological Tsallis transvers…
We investigate the transverse momentum ($p_{\rm T}$) spectra of identified hadrons in minimum-bias proton-proton (pp) collisions at a centre-of-mass energy ($\sqrt{s}$) of 0.9, 2.76, 5.02, 7 and 13 TeV in the framework of Tsallis-blast wave (TBW) model. It is found that the model describes well the particle spectra up to 10 GeV/c. The radial flow ($\langle β\rangle$) increases with the collision energy. The degrees of non-equilibrium ($q$) and the Tsallis temperature parameter ($T$) show a similar behaviour, but with a much weaker trend. With this dependence of the freeze-out parameters on the…
The experimental data from the RHIC and LHC experiments of invariant pT spectra in A+A and p + p collisions are analysed with Tsallis distributions in different approaches. The information about the freeze-out surface in terms of freeze-out volume, temperature, chemical potential and radial flow velocity for different particle species are obtained. Further, these parameters are studied as a function of the mass of the secondary particles. A mass-dependent differential freeze-out is observed which does not seem to distinguish between particles and their antiparticles. Further a mass-hierarchy i…
The Tsallis distribution has been tested to fit the all particle spectra at mid-rapidity from central events produced in d+Au, Cu+Cu, Au+Au collisions at RHIC and p+Pb, Pb+Pb collisions at LHC. Even though there are strong medium effects in Cu+Cu and Au+Au collisions, the results show that the Tsallis distribution can be used to fit most of particle spectra in the collisions studied except in Au+Au collisions where some deviations are seen for proton and $Λ$ at low $p_T$. In addition, as the Tsallis distribution can only fit part of the particle spectra produced in Pb+Pb collisions where $p_T$…
We explore a hadron resonance gas model using Tsallis non-extensive distribution to study the energy dependence of the product of the moments, $Sσ$ and $κσ^2$ of net-proton multiplicity distributions of published STAR data in \auau collisions at relativistic heavy-ion collider (RHIC) energies. While excellent agreements are found between model predictions and measurements of $Sσ$ and $κσ^2$ of most peripheral collisions and $Sσ$ of most central collisions, the $κσ^2$ for most central collisions deviates significantly from the predictions particularly at \sqsn = 19.6 GeV and $27$ GeV. This coul…
We analyse the transverse momentum ($p_{\rm T}$)-spectra as a function of charged-particle multiplicity at midrapidity ($|y| < 0.5$) for various identified particles such as $π^{\pm}$, $K^{\pm}$, $K_S^0$, $p+\overline{p}$, $ϕ$, $K^{*0} + \overline {K^{*0}}$, and $Λ$ + $\barΛ$ in proton-proton collisions at $\sqrt{s}$ = 7 TeV using Boltzmann-Gibbs Blast Wave (BGBW) model and thermodynamically consistent Tsallis distribution function. We obtain the multiplicity dependent kinetic freeze-out temperature ($T_{\rm kin}$) and radial flow ($β$) of various particles after fitting the $p_{\rm T}$-distri…
The transverse momentum distributions measured in $p-p$ collisions at the LHC determine the kinetic freeze-out stage of the collision. The parameters deduced from these distributions differ from those determined at chemical freeze-out. The present investigation focuses on the chemical potentials at kinetic freeze-out, these are not necessarily zero as they are at chemical freeze-out, the only constraint is that they should be equal for particles and antiparticles at LHC energies. The thermodynamic variables are determined in the framework of the Tsallis distribution. The chemical potentials in…
In this paper, one uses the Tsallis entropy in the impact parameter space to study $pp$ and $\bar{p}p$ inelastic overlap function and the energy density filling up mechanism responsible by the so-called black disk limit as the energy increases. The Tsallis entropy is non-additive and non-extensive and these features are of fundamental importance since the internal constituents of $pp$ and $\bar{p}p$ are strongly correlated and also the existence of the multifractal character of the total cross-section. The entropy approach presented here takes into account a phase transition occurring inside t…
In this work we study the pseudorapidity spectra o charged particles produced in proton+proton and proton+antiproton interactions in a wide energy range using the non-extensive Tsallis approach. We evaluate the inelasticity coefficients of the discussed reactions which remain approximately independent of the collision energy.
Final-state effects on the transverse-momentum spectra of charged hadrons in $p+Pb$ and $Pb+Pb$ collisions at $\sqrt {s_{\rm{NN}}}~=~5.02$~TeV using the modified Tsallis distribution
Noida International University（诺伊达国际大学）；Graphic Era Hill University（格拉菲克时代山岳大学）；Academia Sinica（中央研究院）；Ewing Christain College（尤因基督学院）；Gujarat Arts and Science College（古吉拉特艺术与理科学院）；Central University of South Bihar（比哈尔南部中央大学）
The Bose-Einstein and Fermi-Dirac statistics of the identified hadrons were verified on the basis of the transverse momentum distributions of bosons and fermions created in the $pp$ collisions at high energies using the Tsallis-factorized statistics and the two-component distributions of the Boltzmann-Gibbs statistics. The main differences between the parameters of the Tsallis-factorized statistics and the Boltzmann-Gibbs statistics were identified. The results of the Boltzmann-Gibbs statistics are reasonable and suggest that the soft and hard hadrons may be produced from the two different mac…
Comparing effective temperatures in standard and Tsallis distributions from transverse momentum spectra in small collision systems
Shanxi University（山西大学）；Xinzhou Normal University（忻州师范学院）；Physical-Technical Institute of Uzbekistan Academy of Sciences（乌兹别克斯坦科学院物理技术研究所）；National University of Science and Technology MISIS (NUST MISIS), Almalyk Branch（国立科技大学MISIS阿拉木克分校）
The thermodynamic properties of matter created in high-energy heavy-ion collisions have been studied in the framework of the non-extensive Tsallis statistics. The transverse momentum ($p_{\rm T}$)~spectra of identified charged particles (pions, kaons, protons) and all charged particles from the available experimental data of Au-Au collisions at the Relativistic Heavy Ion Collider (RHIC) energies and Pb-Pb collisions at the Large Hadron Collider (LHC) energies are fitted by the Tsallis distribution. The fit parameters, $q$ and $T$ measure the degree of deviation from an equilibrium state and th…
Transverse momentum spectra of direct (prompt) photons from two-body processes in high-energy proton-proton (p+p) collisions are analyzed in this paper. We collected the experimental invariant cross-sections at mid-rapidity in p+p collisions measured by the UA6, CCOR, R806, R110, PHENIX, NA24, CMS, ALICE, and ATLAS Collaborations over a center-of-mass energy range from 24.3 GeV to 13 TeV. In fitting the data, we used different kinds of functions which include the revised Tsallis--Pareto-type function (the TP-like function) at the particle level, the convolution of two TP-like functions at the …
Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions over a center-of-mass energy, $\sqrt{s}$, range from a few GeV to above 10 TeV are analyzed by the blast-wave fit with Boltzmann (Tsallis) distribution. The blast-wave fit results are well fitting to the experimental data measured by several collaborations. In a particular superposition with Hagedorn function, both the excitation functions of kinetic freeze-out temperature ($T_0$) of emission source and transverse flow velocity ($β_T$) of p…
Neglecting spin effects, one introduces here a subtle approximation for the scattering angle, which allows the obtaining of a logarithmic leading Regge pole, consistent with the Froissart-Martin bound. A simple parameterization is also introduced for the proton-proton total cross section. Fitting procedures are implemented only for the highest energy experimental data available. The intercept for a linear approach is obtained, indicating the presence of a soft pomeron. The Tsallis entropy in the impact parameter space is calculated using the Regge pole formalism. This entropy depends on a free…
In the framework of a multisource thermal model, we describe experimental results of the transverse momentum spectra of final-state light flavour particles produced in gold-gold (Au-Au), copper-copper (Cu-Cu), lead-lead (Pb-Pb), proton-lead ($p$-Pb), and proton-proton ($p$-$p$) collisions at various energies, measured by the PHENIX, STAR, ALICE, and CMS Collaborations, by using the Tsallis-standard (Tsallis form of Fermi-Dirac or Bose-Einstein), Tsallis, and two- or three-component standard distributions which can be in fact regarded as different types of "thermometers" or "thermometric scales…
The transverse momentum spectra of charged pions, kaons, and protons produced at mid-rapidity in central nucleus-nucleus (AA) collisions at high energies are analyzed by considering particles to be created from two participant partons which are assumed to be contributors from the collision system. Each participant (contributor) parton is assumed to contribute to the transverse momentum by a Tsallis-like function. The contributions of the two participant partons are regarded as the two components of transverse momentum of the identified particle. The experimental data measured in high-energy AA…
We investigate how the non-extensive approach works in high-energy physics. Transverse momentum ($p_T$) spectra of several hadrons are fitted by various non-extensive momentum distributions and by the Boltzmann--Gibbs statistics.~It is shown that some non-extensive distributions can be transferred one into another.~We find explicit hadron mass and center-of-mass energy {scaling both in the temperature and in the non-extensive parameter, $q$,} in proton--proton and heavy-ion collisions. We find that the temperature depends linearly, but the Tsallis $q$ follows a logarithmic dependence on the co…
Hadron spectra measured in high-energy collisions present distributions which can be derived from the non-extensive statistical and thermodynamical phenomena. Based on earlier theoretical developments, it seems, the methods are very applicable for jets hadronization processes in electron-positron, proton-proton, and even in heavy-ion collisions. Here, we present what can was learnt from the recent theoretical and phenomenological developments: transverse momentum spectra and azimuthal anisotropy ($v_2$) of charge averaged pions, kaons and protons stemming from central Pb+Pb collisions at $\sqr…
We analyze the transverse momentum ($p_{T}$) spectra of lepton pairs ($\ell\bar \ell$) generated in the Drell-Yan process, as detected in proton-nucleus (pion-nucleus) and proton-(anti)proton collisions by ten collaborations over a center-of-mass energy ($\sqrt{s_{NN}}$ or $\sqrt{s}$ if in a simplified form) range from $\sim20$ GeV to above 10 TeV. Three types of probability density functions (the convolution of two Lévy-Tsallis functions, the two-component Erlang distribution, and the convolution of two Hagedorn functions) are utilized to fit and analyze the $p_{T}$ spectra. The fit results a…
In this work, one proposes the use of a damped confinement potential to mimic the proton internal energy. The internal pressure is calculated taking into account the occurrence of a phase transition in the $pp$ and $\bar{p}p$ total cross section. The model predicts the inversion of the positive and negative pressure regions depending on the squared energy $s_0$, where the total cross section achieves its minimum value. Considering energies below the phase transition the expected results are in accordance with the recent proton internal pressure measurement. For energies above $s_0$, the model …
Transverse momentum spectra of different types of identified charged particles in central Gold-Gold (Au-Au) collisions, and inelastic (INEL) or non-single-diffrative (NSD) proton-proton (pp) collisions at the Relativistic Heavy Ion Collider (RHIC), as well as in central and peripheral Lead-Lead (Pb-Pb) collisions, and INEL or NSD pp collisions at the Large Hadron Collider (LHC) are analyzed by the blast wave model with Tsallis statistics. The model results are approximately in agreement with the experimental data measured by STAR, PHENIX and ALICE Collaborations in special transverse momentum …
A classical model of a stable particle of finite size is studied. The model parameters can be chosen such that the described particle has the mass and radius of a proton. Using the energy-momentum tensor (EMT), we show how the presence of long-range forces alters some notions taken for granted in short-range systems. We focus our attention on the D-term form factor. The important conclusion is that a more careful definition of the D-term may be required when long-range forces are present.
We used blast wave model with Tsallis statistics to analyze the experimental data measured by ALICE Collaboration in proton-proton collisions at Large Hadron Collider and extracted the related parameters (kinetic freeze-out temperature, transverse flow velocity and kinetic freeze-out volume of emission source) from transverse momentum spectra of the particles. We found that the kinetic freeze-out temperature and kinetic freeze-out volume are mass dependent. The former increase while the latter decrease with the particle mass which is the evidence of a mass as well as volume differential kineti…
Kinetic Freeze-Out Properties from Transverse Momentum Spectra of kaon, Pion, and (anti-)proton production in U+U collisions at $\sqrt{s_{NN}}$ = 193 GeV
We review transverse momentum distributions of various identified charged particles stemming from high energy collisions fitted by various non-extensive distributions as well as by the usual Boltzmann-Gibbs statistics. We investigate the best-fit formula with the obtained $χ^2/ndf$ values. We find that the physical mass and $\sqrt{s}$ scaling become more explicit with heavier produced hadrons in both proton-proton and heavy-ion collisions. The spectral shape parameters, in particular the temperature $T$ and the non-extensive Tsallis parameter $q$, do exhibit an almost linear dependence with th…
The differential cross-section in squared momentum transfer of $ρ$, $ρ^0$, $ω$, $ϕ$, $f_{0}(980)$, $f_{1}(1285)$, $f_{0}(1370)$, $f_{1}(1420)$, $f_{0}(1500)$, and $J/ψ$ produced in high energy virtual photon-proton ($γ$$^{*} p$), photon-proton ($γp$), and proton-proton ($pp$) collisions measured by the H1, ZEUS, and WA102 Collaborations are analyzed by the Monte Carlo calculations. In the calculations, the Erlang distribution, Tsallis distribution, and Hagedorn function are separately used to describe the transverse momentum spectra of the emitted particles. Our results show that the initial- …
We study the internal structure of the proton and propose a shell structure model of the proton. Based on the mass distribution and mass radius of proton we analyze the experiments at different energy scales and conclude that there should be shell structure inside proton, and the quark and gluon shells are energy-scale-dependent. We successfully explain the dip-like structure in the proton-proton elastic scattering cross section distribution from TOTEM Collaboration using the mass decomposition of proton and the proton shell structure at different energy scales.
In the quantum chromodynamics, the mass source of the proton is decomposed into four parts by the energy momentum tensor : quark energy term, gluon energy term, quark mass term and trace anomaly term. And the trace anomaly term is the most crucial contribution for studying the internal structure of the proton. In this work, under the definition of the vector meson dominant model, the trace anomaly contribution of the proton is extracted from the experimental datas of light vector mesons $ρ$, $ω$ and $ϕ$ photoproduction at the near-threshold, which are ($1.15\pm0.08$)$\%$, ($2.70\pm0.04$)$\%$ a…
Originally it was thought that the proton is not divisible into smaller particles. However, from collision experiments it follows that it consists of a 'soup' of countless 'colored' elementary particles such as quarks and gluons that are kept together by the strong nuclear force. The particles constituting the proton are also referred to as partons and their distributions are described by parton distribution functions (PDFs). As these functions cannot be calculated, they must be extracted from experiment. More knowledge on PDFs teaches us more about the inner workings of protons and enables us…
We suggest 3P0 quantum state of virtual (ss') pairs in the nucleon can be polarised by the internal fields permeating the volume of the nucleon (proton or neutron). Due to the quadratic Zeeman interaction, 3P0 wavefunction of virtual (qq') pairs acquires the admixture of 1P10 quantum state in the magnetic field, which generates the antiparallel polarization of s and s' quarks (in the nucleon). Considering the internal magnetic fields of neutron and proton (originating from their measured magnetic dipole moments), we suggest the induced s-quark polarization in the neutron to be of the oposite d…
Transverse momentum spectra of negative and positive pions produced at mid-(pseudo)rapidity in inelastic or non-single-diffractive proton-proton collisions and in central nucleus-nucleus collisions over an energy range from a few GeV to above 10 TeV are analyzed by a (two-component) blast-wave model with Boltzmann-Gibbs statistics and with Tsallis statistics respectively. The model results are in similarly well agreement with the experimental data measured by a few productive collaborations who work at the Heavy Ion Synchrotron (SIS), Super Proton Synchrotron (SPS), Relativistic Heavy Ion Coll…
