Constrains for non-standard statistical models of particle creations by identified hadron multiplicity results at LHC energies

0 Cosmic Ray Laboratory, National Centre for Nuclear Research , Lodz, Poland 1 Department of Physics, University of dz , Pomorsta 149/153, 90-236 Lodz, Poland We analyzed the identified hadron multiplicity predictions of the modified thermodynamical model of the multiparticle production processes with non-extensive statistic. The replacement of the standard Boltzmann exponential factor by the eventually much more slowly falling Tsallis one is suggested by the analysis of the transverse momentum distributions measured at high energies. The increase of high transverse momenta should accord with the abundance of heavy secondary particles, in particular multistrange baryons. The introduction to the thermodynamical model of suppression factors similar to the ones in quark jet fragmentation models is discussed. The identified hadron ratios have been measured with all LHC detectors and results were compared with high-energy event generators available in the market [1-5]. The comparison, in general, is not very satisfactory. In the present paper we would like to use data from the ALICE experiment performed with pp interaction of s 7 TeV available energy [3-8] to test the particle creation description based on a thermodynamical approach. The standard statistical picture is known to work well in the soft, low p, sector of the particle creation process, where the exponential fall of the transverse momentum distribution is observed. The hard inelastic scattering leads to the quark jet fragmentation with the power-law transverse momentum (transverse mass) distributions. Detailed studies of the measured charged particle transverse momentum (transverse mass) distributions suggested already some time ago that the very good agreement of the invariant differential cross section in the whole transverse momentum range can be obtained with an empirical formula inspired by QCD - p + p0 (see, e.g., [10] for further discussion and references). It has been shown [11] that not only the fit of the simple form of Eq. (1) works well but the whole theoretical model of particle creation which stands behind it could be successfully applied to the highest available energy data on charged particle transverse momentum [12]. The model parameters found in [12] define the occupation of phase space for given charged particle transverse momentum. If the picture is self-consistent, the same set of parameters should give correct yields of different kinds of created particles. It is well known that the multiplicities of new created heavy particles are described to some extent by the Boltzmann statistical model (e.g., [13,14]). The Tsallis modification undoubtedly increases the high ps, and, obviously, the high transverse mass particle abundances. This should lead to the overabundance of heavy particles. We would like to look for the possibility to suppress this effect in a consistent way and to see if satisfactory results could be obtained. 2 Thermodynamical model The thermodynamical picture of the particle creation process in hadronic collisions was the first and quite successful attempt to describe it. The elaborated and complete theory was presented in a series of papers by Hagedorn (see [1517] and references therein). The idea of the fireball together with the proposition that all fireballs are equal gives considerable predictions concerning the produced particle spectra. One of the predictions was that the temperature of the hadronic soup (precisely defined) could not exceed a universal constant T0 of order of 160 MeV. This value comes not as a result of the procedure of parameter adjusting using multiparticle production (e.g., transverse momenta) data, but from an examination of the elementary particle mass spectrum. The Hagedorn theory had been abundant for some time, when more sophisticated, jet- or QCD-based ideas appeared [18]. One of the reasons was the failure of the high transverse momenta description. The temperature of the fireball is defined as the parameter in the classical Boltzmann exponential term of the probability weights for phase space average occupation numbers. This gives the (asymptotic) form of the distribution of transverse momenta of the particles created from decaying fireballs. It was found that at high and very high interaction energies the predicted exponential fall does not agree with the observed high p behavior. Successes of QCD-based description of the hard processes gave deep insight into the nature of physics involved, and belief that this is just the right theory of the strong interactions, making the thermodynamical approach a very approximate, simple, and naive tool of limited applicability and thus of limited significance. But on the other hand, the simplicity of the theory and notorious constant lack of an effective QCD theory of soft hadronization processes give hope that the fireball idea can be enriched, modified and can become important again. The Hagedorn idea was used again to describe the identified particle multiplicities in hadronization, both in e+e annihilation and hadronic collisions. The grand canonical formalism of Hagedorn was replaced in the series of papers by Becattini et al. [19] by the canonical one, very relevant for studies of small systems like primary created fireballs for which the requirement of the exact conservation of some quantum numbers seems important. In general, the thermodynamics of the system is determined by the partition function which can be written as Z(Q0) = where P is the classical Boltzmann factor and j and k enumerate the particle types and momentum cells, Q0 is the initial fireball quantum number vector and Q is the respective vector of the particular state, and jk is the occupation number. Introducing the Fourier transform of (and reducing the vector Q to 3-dimensional: charge, baryon number, and strangeness) Eq. (2) becomes j=1 where q j is the quantum number vector of the particle j and w j is the weight factor associated with the particle of the type j . The first guess is that it should be equal to (2 J j + 1) and counts spin states. However, this does not seem to be so simple (see, e.g., [2022]) and other solutions introducing factors responsible for some wave-function normalization, which should disfavor heavier states, were found to be preferable by measurements. We will discuss this point later on. With Eq. (3) we are ready for detailed numerical calculations. 2.1 Average multiplicities With the known partition function Z the average characteristics of the system can be obtained in the usual way. For the average multiplicity we have d3 p [eE/T ei q j 1]1, where the upper sign is for fermions and the lower is for bosons. Because the eE/T factor is expected to be small (for all particles except pions), n j d3 p eE/T . The conventional BoltzmannGibbs description shown above could be, in principle, modified to allow for the description of the systems of not-completely-free particles: the correlat (...truncated)