Hadron Transverse Momentum Distributions Research Articles

Hadron transverse momentum distributions in the Tsallis statistics with escort probabilities

Abstract The exact and approximate hadron transverse momentum distributions for the Fermi–Dirac, Bose–Einstein and Maxwell-Boltzmann statistics of particles in the framework of the Tsallis statistics with escort probabilities (the Tsallis-3 statistics) have been derived. The classical and quantum transverse momentum distributions in the zeroth term approximation and the quantum transverse momentum distributions in the factorization approximation introduced in the zeroth term approximation were found. The transverse momentum distributions in the zeroth term approximation and in the factorization approximation of the zeroth term approximation are the same in the Tsallis-3, Tsallis-2 and q-dual statistics. The well-known classical phenomenological Tsallis distribution exactly coincides with the classical transverse momentum distribution of the Tsallis-3 statistics in the zeroth term approximation for which the entropy of the system is zero in the whole range of state variables. However, the quantum phenomenological Tsallis distribution does not coincide with either the exact or approximate transverse momentum distributions of the Tsallis-3 statistics. The exact Tsallis-3 classical distribution and the classical phenomenological Tsallis distribution were applied to describe the experimental spectra of the charged pions produced in the proton-proton collisions at high energies. The values of the parameters (T, q) for both these model distributions differ in the whole energy range. Thus, the classical phenomenological Tsallis distribution is an unsatisfactory approximation for the exact classical transverse momentum distribution of the Tsallis-3 statistics.

Bose-Einstein condensation of pions in proton–proton collisions at the Large Hadron Collider using non-extensive Tsallis statistics

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 dependency of BEC critical temperature and number of particles in the pion condensates on the non-extensive parameter q, which is a measure of degree of non-equilibrium – as q decreases, the critical temperature increases and approaches to the critical temperature obtained from Bose-Einstein statistics without non-extensivity. Studies are performed on the final state multiplicity dependence of number of particles in the pion condensates in a wide range of multiplicity covering hadronic and heavy-ion collisions, using the inputs from experimental transverse momentum spectra.

QCD resummation on single hadron transverse momentum distribution with the thrust axis

We derive the transverse momentum dependent (TMD) factorization and resummation formula of the unpolarized transverse momentum distribution (jT) for the single hadron production with the thrust axis in an electron-positron collision. Two different kinematic regions are considered, including small transverse momentum limit jT « Q, and joint transverse momentum and threshold limit jT « Q(1 − zh) « Q, where Q and zh are the hard scattering energy and the observed hadron momentum fraction. Using effective theory methods, we resum logarithms ln(Q/jT) and ln(1 − zh) to all orders. In the end, we present the differential cross sections and Gaussian widths calculated for the inclusive charged pion production and find that our results are consistent with the measurements reported by the Belle collaboration.

Equivalence of the phenomenological Tsallis distribution to the transverse momentum distribution of q-dual statistics

In the present work, we have found that the phenomenological Tsallis distribution (which nowadays is largely used to describe the transverse momentum distributions of hadrons measured in pp collisions at high energies) is consistent with the basis of the statistical mechanics if it belongs to the q-dual nonextensive statistics instead of the Tsallis one. We have defined the q-dual statistics based on the q-dual entropy which was obtained from the Tsallis entropy under the multiplicative transformation of the entropic parameter qrightarrow 1/q. We have found that the phenomenological Tsallis distribution is equivalent to the transverse momentum distribution of the q-dual statistics in the zeroth term approximation. Since the q-dual statistics is properly defined, it provides a correct link between the phenomenological Tsallis distribution and the second law of thermodynamics.

Hadron transverse momentum distributions of the Tsallis normalized and unnormalized statistics

The exact analytical formulas for the transverse momentum distributions of the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles with nonzero mass in the framework of the Tsallis normalized and Tsallis unnormalized (also known as Tsallis-1 and Tsallis-2) statistics have been consistently derived. The final exact results were expressed in terms of the series expansions in the integral representation. The zeroth term approximation to both quantum and classical statistics of particles has been introduced. We have revealed that the phenomenological classical Tsallis distribution (widely used in high energy physics) is equal to the distribution of the Tsallis unnormalized statistics in the zeroth term approximation, but the phenomenological quantum Tsallis distributions (introduced by definition on the basis of the generalized entropy of the ideal gas) do not correspond to the distributions of the Tsallis statistics. We have found that in the ranges of the entropic parameter relevant to the processes of high-energy physics ($q<1$ for Tsallis-1 and $q>1$ for Tsallis-2) the Tsallis statistics is divergent. Therefore, to obtain physical results, we have regularized the Tsallis statistics by introducing an upper cut-off in the series expansion. The exact numerical results for the Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann statistics of particles in the Tsallis normalized and unnormalized statistics have been obtained. We observed that the exact results of the Tsallis statistics strongly enhanced the production of high-$p_{T}$ hadrons in comparison with the usual phenomenological Tsallis distribution function at the same values of $q$. The $q$-duality of the Tsallis normalized and unnormalized statistics for the massive particles was studied.

Jet fragmentation functions for Z-tagged jets

Recently the LHCb collaboration has measured both longitudinal and transverse momentum distribution of hadrons produced inside Z-tagged jets in proton-proton collisions at the Large Hadron Collider. These distributions are commonly referred to as jet fragmentation functions and are characterized by the longitudinal momentum fraction zh of the jet carried by the hadron and the transverse momentum j⊥ with respect to the jet direction. We derive a QCD formalism within Soft-Collinear Effective Theory to describe these distributions and find that the zh-dependence provides information on standard collinear fragmentation functions, while j⊥-dependence probes transverse momentum dependent (TMD) fragmentation functions. We perform theoretical calculations and compare our results with the LHCb data. We find good agreement for the intermediate zh region. For j⊥-dependence, we suggest binning in both zh and j⊥, which would lead to a more direct probing of TMD fragmentation functions.

Supersymmetric and Conformal Features of Hadron Physics

The QCD Lagrangian is based on quark and gluonic fields—not squarks nor gluinos. However, one can show that its hadronic eigensolutions conform to a representation of superconformal algebra, reflecting the underlying conformal symmetry of chiral QCD. The eigensolutions of superconformal algebra provide a unified Regge spectroscopy of meson, baryon, and tetraquarks of the same parity and twist as equal-mass members of the same 4-plet representation with a universal Regge slope. The predictions from light-front holography and superconformal algebra can also be extended to mesons, baryons, and tetraquarks with strange, charm and bottom quarks. The pion q q ¯ eigenstate has zero mass for m q = 0 . A key tool is the remarkable observation of de Alfaro, Fubini, and Furlan (dAFF) which shows how a mass scale can appear in the Hamiltonian and the equations of motion while retaining the conformal symmetry of the action. When one applies the dAFF procedure to chiral QCD, a mass scale κ appears which determines universal Regge slopes, hadron masses in the absence of the Higgs coupling. One also predicts the form of the nonperturbative QCD running coupling: α s ( Q 2 ) ∝ e − Q 2 / 4 κ 2 , in agreement with the effective charge determined from measurements of the Bjorken sum rule. One also obtains viable predictions for spacelike and timelike hadronic form factors, structure functions, distribution amplitudes, and transverse momentum distributions. The combination of conformal symmetry, light-front dynamics, its holographic mapping to AdS 5 space, and the dAFF procedure thus provide new insights, not only into the physics underlying color confinement, but also the nonperturbative QCD coupling and the QCD mass scale.

The transverse momentum distribution of hadrons within jets

We study the transverse momentum distribution of hadrons within jets, where the transverse momentum is defined with respect to the standard jet axis. We consider the case where the jet substructure measurement is performed for an inclusive jet sample pp → jet + X. We demonstrate that this observable provides new opportunities to study transverse momentum dependent fragmentation functions (TMDFFs) which are currently poorly constrained from data, especially for gluons. The factorization of the cross section is obtained within Soft Collinear Effective Theory (SCET), and we show that the relevant TMDFFs are the same as for the more traditional processes semi-inclusive deep inelastic scattering (SIDIS) and electron-positron annihilation. Different than in SIDIS, the observable for the in-jet fragmentation does not depend on TMD parton distribution functions which allows for a cleaner and more direct probe of TMDFFs. We present numerical results and compare to available data from the LHC.

Applications of the Tsallis Statistics in High Energy Collisions

We analytically investigate the thermodynamic variables of a hot and dense system in the framework of the Tsallis non-extensive classical statistics in the massless and in the massive cases. In addition to that, we study the effect of Tsallis Power Law distribution on the multi-particle production in the high-energy collisions. The effect of the Tsallis q parameter on the experimentally measured nuclear suppression factor has been investigated, and an attempt to describe the transverse momentum distribution of hadrons produced in high-energy collisions has been made.

Directed flow in asymmetric nucleus-nucleus collisions and the inverse Landau-Pomeranchuk-Migdal effect

It is proposed to identify a strong electric field - created during relativistic collisions of asymmetric nuclei - via the observation of pseudorapidity and transverse momentum distributions of hadrons with the same mass but opposite charge. The results of detailed calculations within the Parton-Hadron String Dynamics (PHSD) approach for the charge-dependent directed flow $v_1$ are presented for semi-central Cu+Au collision at $\sqrt{s_{NN}}=200$ GeV incorporating the inverse Landau-Pomeranchuk-Migdal (iLPM) effect, which accounts for a delay in the electromagnetic interaction with the charged degree of freedom. Including the iLPM effect we achieve a reasonable agreement of the PHSD results for the charge splitting in $v_1(p_T)$ in line with the recent measurements of the STAR Collaboration for Cu+Au collisions at $\sqrt{s_{NN}}=200$ GeV while an instant appearance and coupling of electric charges at the hard collision vertex overestimates the splitting by about a factor of 10. We predict that the iLPM effect should practically disappear at energies of $\sqrt{s_{NN}} \approx$9 GeV, which should lead to a significantly larger charge splitting of $v_1$ at the future FAIR/NICA facilities.

Transverse momentum distributions of hadrons in the Tsallis-1 and Tsallis-2 statistics

We considered the ultrarelativistic transverse momentum distributions of the Tsallis-1 and Tsallis-2 statistics using two regularization schemes. It was revealed that the cut-off parameter strongly influences the behavior of the transverse momentum distribution in both statistics. We have also found that the ultrarelativistic transverse momentum distribution of the Tsallis-1 statistics is transformed to the momentum distribution of the Tsallis-2 statistics by identifying q → 1/ q c .

Comparison of Tsallis statistics with the Tsallis-factorized statistics in the ultrarelativistic pp collisions

The Tsallis statistics was applied to describe the experimental data on the transverse momentum distributions of hadrons. We considered the energy dependence of the parameters of the Tsallis-factorized statistics, which is now widely used for the description of the experimental transverse momentum distributions of hadrons, and the Tsallis statistics for the charged pions produced in $pp$ collisions at high energies. We found that the results of the Tsallis-factorized statistics deviate from the results of the Tsallis statistics only at low NA61/SHINE energies when the value of the entropic parameter is close to unity. At higher energies, when the value of the entropic parameter deviates essentially from unity, the Tsallis-factorized statistics satisfactorily recovers the results of the Tsallis statistics.

Evidence for creation of strong electromagnetic fields in relativistic heavy-ion collisions

It is proposed to identify a strong electric field created during relativistic collisions of asymmetric nuclei via observation of pseudorapidity and transverse momentum distributions of hadrons with the same mass but opposite charges. The detailed calculation results for the directed flow within the Parton-Hadron String Dynamics model are given for Cu-Au interactions at the NICA collision energies of $\sqrt{s_{NN}}=9$ and $5$ GeV. The separation effect is observable at 9 GeV as clearly as at 200 GeV

How many of the scaling trends inppcollisions will be violated atsNN=14 TeV? Predictions from Monte Carlo quark-gluon string model

Multiplicity, rapidity and transverse momentum distributions of hadrons produced both in inelastic and nondiffractive $pp$ collisions at energies from $\sqrt{s} = 200$\,GeV to 14\,TeV are studied within the Monte Carlo quark-gluon string model (QGSM). Good agreement with the available experimental data up to $\sqrt{s} = 13$\,TeV is obtained, and predictions are made for the collisions at top LHC energy $\sqrt{s} = 14$\,TeV. The model indicates that Feynman scaling and extended longitudinal scaling remain valid in the fragmentation regions, whereas strong violation of Feynman scaling is observed at midrapidity. The Koba-Nielsen-Olesen (KNO) scaling in multiplicity distributions is violated at LHC also. The origin of both maintenance and violation of the scaling trends is traced to short range correlations of particles in the strings and interplay between the multistring processes at ultrarelativistic energies.

Scaling behaviours of the pT spectra for identified hadrons in pp collisions

We extend the scaling behaviour observed in the inclusive charged hadron transverse momentum (pT) distributions to the pT spectra of pions, kaons and protons produced in proton–proton (pp) collisions with center of mass energies () at 0.9, 2.76 and 7 TeV. This scaling behaviour arises when a linear transformation, is applied on the pion, kaon or proton pT spectra. The scaling parameter K depends on and is determined by a new method, the quality factor method, which does not rely on the shape of the scaling function. We argue that the pions, kaons and protons originate from different distributions of clusters which are formed by strings overlapping, and the scaling behaviours of these identified particles pT spectra could be understood with the colour string percolation model in a quantitative way simultaneously.

Pre-equilibrium evolution effects on heavy-ion collision observables

To investigate the importance of pre-equilibrium dynamics on relativistic heavy-ion collision observables, we match a highly nonequilibrium early evolution stage, modeled by free-streaming partons generated from the Monte Carlo Kharzeev-Levin-Nardi (MC-KLN) and Monte Carlo Glauber (MC-Glb) models, to a locally approximately thermalized later evolution stage described by viscous hydrodynamics and study the dependence of final hadronic transverse momentum distributions, in particular their underlying radial and anisotropic flows, on the switching time between these stages. Performing a three-parameter fit of the measured values for the average transverse momenta $\ensuremath{\langle}{p}_{\ensuremath{\perp}}\ensuremath{\rangle}$ for pions, kaons, and protons, as well as the elliptic and triangular flows of charged hadrons ${v}_{2,3}^{\mathrm{ch}}$, with the switching time ${\ensuremath{\tau}}_{s}$, the specific shear viscosity $\ensuremath{\eta}/s$ during the hydrodynamic stage, and the kinetic decoupling temperature ${T}_{\mathrm{dec}}$ as free parameters, we find that the preferred ``thermalization'' times ${\ensuremath{\tau}}_{s}$ depend strongly on the model of the initial conditions. MC-KLN initial conditions require an earlier transition to hydrodynamic behavior (at ${\ensuremath{\tau}}_{s}\ensuremath{\approx}0.13 \mathrm{fm}/c$), followed by hydrodynamic evolution with a larger specific shear viscosity $\ensuremath{\eta}/s\ensuremath{\approx}0.2$, than MC-Glb initial conditions, which prefer switching at a later time (${\ensuremath{\tau}}_{s}\ensuremath{\approx}0.6 \mathrm{fm}/c$) followed by a less viscous hydrodynamic evolution with $\ensuremath{\eta}/s\ensuremath{\approx}0.16$. These new results including pre-equilibrium evolution are compared to fits without a pre-equilibrium stage where all dynamic evolution before the onset of hydrodynamic behavior is ignored. In each case, the quality of the dynamical descriptions for the optimized parameter sets, as well as the observables which show the strongest constraining power for the thermalization time, are discussed.

Erratum to: Hadron transverse momentum distributions in muon deep inelastic scattering at 160 GeV/ $$c$$ c

Author(s): Adolph, C; Alekseev, MG; Alexakhin, VY; Alexandrov, Y; Alexeev, GD; Amoroso, A; Andrieux, V; Austregesilo, A; Badelek, B; Balestra, F; Barth, J; Baum, G; Bedfer, Y; Berlin, A; Bernhard, J; Bertini, R; Bicker, K; Bieling, J; Birsa, R; Bisplinghoff, J; Boer, M; Bordalo, P; Bradamante, F; Braun, C; Bravar, A; Bressan, A; Buchele, M; Burtin, E; Capozza, L; Chiosso, M; Chung, SU; Cicuttin, A; Crespo, ML; Dalla Torre, S; Dasgupta, SS; Dasgupta, S; Denisov, OY; Donskov, SV; Doshita, N; Duic, V; Dunnweber, W; Dziewiecki, M; Efremov, A; Elia, C; Eversheim, PD; Eyrich, W; Faessler, M; Ferrero, A; Filin, A; Finger, M; Finger Jr, M; Fischer, H; Franco, C; du Fresne von Hohenesche, N; Friedrich, JM; Frolov, V; Garfagnini, R; Gautheron, F; Gavrichtchouk, OP; Gerassimov, S; Geyer, R; Giorgi, M; Gnesi, I; Gobbo, B; Goertz, S; Grabmuller, S; Grasso, A; Grube, B; Gushterski, R; Guskov, A; Guthorl, T; Haas, F; von Harrach, D; Heinsius, FH; Herrmann, F; Hes, C; Hinterberger, F; Hoppner, C; Horikawa, N; d’Hose, N; Huber, S; Ishimoto, S; Ivanshin, Y; Iwata, T; Jahn, R

Systematic analysis of pT -distributions in p + p collisions

A systematic analysis of transverse momentum distribution of hadrons produced in ultra-relativistic p + p collisions is presented. We investigate the effective temperature and the entropic parameter from the non-extensive thermodynamic theory of strong interaction. We conclude that the existence of a limiting effective temperature and of a limiting entropic parameter is in accordance with experimental data.

Measurement of neutral mesons inp+pcollisions ats=200 GeVand scaling properties of hadron production

The PHENIX experiment at the Relativistic Heavy Ion Collider has measured the invariant differential cross section for production of K^0_S , \omega, \eta prime, and \phi mesons in p + p collisions at = 200 GeV. Measurements \omega and \phi production in different decay channels give consistent results. New results for the \phi are in agreement with previously published data and extend the measured pT coverage. The spectral shapes of all hadron transverse momentum distributions measured by PHENIX are well described by a Tsallis distribution functional form with only two parameters, n and T, determining the high-pT and characterizing the low-pT regions of the spectra, respectively. The values of these parameters are very similar for all analyzed meson spectra, but with a lower parameter T extracted for protons. The integrated invariant cross sections calculated from the fitted distributions are found to be consistent with existing measurements and with statistical model predictions.

Transverse momentum broadening in semi-inclusive DIS on nuclei

Using a three stage model of hadron formation we calculate the change of the transverse momentum distribution of hadrons produced in semi-inclusive deep inelastic scattering (SIDIS) on nuclei. In the first stage after its interaction with the virtual photon, the struck quark propagates quasi free in the nuclear environment undergoing multiple collisions with nucleons. During this stage it can acquire transverse momentum. In the second stage a prehadron is formed which has a very small elastic cross section with the nucleons. In the third stage the prehadron turns into a hadron. For HERMES energies, prehadron elastic scatterings contribute little to p ⊥ -broadening. The acquired extra Δ p ⊥ 2 of hadrons can therefore be deduced entirely from the first stage of quasi free quark propagation with quark–nucleon collisions. We use this model to describe π-production on Ne, Kr, Xe and compare with the most recent HERMES data.