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the global structure of multiple hadroproduction in annihilation , has been well understood with remarkable results concerning the total cross section and angular distribution calculations . perturbative quantum electrodynamics and perturbative quantum chromodynamics provide a good description of the initial process involving short distances @xmath2 quark - antiquark ( @xmath3 ) and @xmath2 quark - antiquark - gluon ( @xmath4 ) interactions , main responsible for the global characteristics of this kind of process . the quarks produced off - shell initialize a cascade of several quarks and gluons with complex interactions themselves . this stage is in general described in probabilistic terms , based on leading - log quantum chromodynamics approximation . the outgoing colored partons are transformed into color singlet hadrons through the soft hadronization process , forming a jet of particles traveling approximately in the initial parton direction . however , hadronization require the quantum chromodynamics in the soft regime , where the coupling constant become large and there is no easy way to understand it from first principles .    for more than a decade several different models making use of monte carlo techniques were developed to describe such processes @xcite . these methods avoid the microscopic complexity by being able to define probability distributions for each stage of the jet evolution . in these approaches , all known aspects involving quantum chromodynamics were included , the unknown aspects like hadronization being modeled phenomenologically . each model has a set of free parameters which enables it to reproduce many distributions over a wide range of energy . in particular , the transverse momenta of the hadrons with respect to the jet direction , main subject of this paper , has been accurately parameterized . nevertheless , there is no fundamental understanding of the matter @xcite . one alternative approach to understand the transverse momentum distribution of the charged hadrons produced in annihilation is to look at the whole process in a macroscopic way , trying to extract information through a thermodynamical treatment , not taking into account the microscopy interactions between partons , governed by the quantum chromodynamics . and @xmath5 because we are looking at the global characteristics of the events ] fermi @xcite was the pioneer in this kind of approach , followed by hagedorn @xcite who provided the most consistent and well succeeded model . nowadays , the thermodynamical formalism is widely used in heavy ions collisions @xcite and has also been applied in @xmath6 , proton - proton ( @xmath7 ) and proton - antiproton ( @xmath8 ) interactions to determine the multiplicity of hadrons produced , with noteworthy results @xcite . according to hagedorn , a high energy collision produces an _ unlimited _ and _ undetermined _ number of different excited hadron _ fireballs _ that reach a thermodynamical equilibrium . in this relativistic statistical _ ansatz _ , not only the number of particles but also the number of available types of particles itself grows with energy . the immediate and important consequence of this production mechanism is that _ the temperature is independent of the primary energy _ @xcite . a similar statement was also made , a few years later , by field and feynman @xcite . one should expect that this temperature governs the transverse momentum ( @xmath0 ) distribution of the outgoing particles with respect to the jet axis .    in a modern ansatz @xcite , one can assume that the thermodynamical equilibrium is reached in each jet produced in @xmath9 interaction . we consider in this paper that the thermalization begins just after the hadronization process takes place and finishes when the hadrons are no longer interacting . since the range of this interaction is of order of a few fermi , the resonances produced , with a typical lifetime of order of @xmath10 s , should decay while the particles are still interacting . thus , the particles produced from these decays should also participate in the thermalization process . as most charged particles observed in the detector are produced prompt or through the decay of hadronic resonances , we could expect that the charged particles observed in the detectors have their distributions dictated by the previous ` thermodynamical ' equilibrium . it is then important to determine the thermodynamical properties of the event , without any kinematical effect caused by the fast relative motion between the hadrons . using the boltzmann - gibbs statistics in this large grand - canonical ensemble for a relativistic gas , the transverse momentum distribution with temperature @xmath11 , can be written as ( see hagedorn @xcite ) :    @xmath12    where @xmath13 means the longitudinal momentum , @xmath14 and @xmath15 is the mass ( @xmath16 ) @xcite . after the integration indicated in eq . ( [ eq : pl ] ) and for @xmath17 we obtain an exponential asymptotic distribution for the transverse momentum @xcite :    @xmath18    as the temperature should not depend on the center - of - mass energy ( @xmath19 ) , we can expect that eq . ( [ eq : ptb ] ) presents the same shape for all @xmath20 energies spectrum varying only the parameter @xmath21 , which depends on the average multiplicity of the events and consequently on the @xmath19 . the measured transverse momentum distributions of charged hadrons with respect to the jet axis , defined as the sphericity axis @xcite , for center - of - mass energies in the range of 14 gev to 161 gev are displayed in figure 1 . one notices the clear experimental deviation , for high values of @xmath0 , from the exponential behavior predicted by eq . ( [ eq : ptb ] ) and indicated by a dotted line . notice that we could also get decent fits for low values of data point of @xmath0 ( @xmath22 gev ) using the eq . ( [ eq : ptb ] ) , but in this case the best values for the parameter @xmath23 would clearly increase with energy , a feature which disagrees with hagedorn s physically reasonable assumption . the boltzmann - gibbs statistics , used to derive eq . ( [ eq : ptb ] ) , is suitable for an enormous number of systems , but it is known that it could lead to unphysical results for some systems that present long - range interactions ( lr ) . in fact , lr interaction is the usual interpretation for the intermittency phenomena @xcite , large particle density fluctuations @xcite , present in hadroproduction . long - range ( lr ) interactions can occur in many situations , particularly in processes where the interacting volume is of order of the dimension of the hadrons themselves , resulting in a high hadronic density . in this way the gradual deviation of the observed distribution from the exponential function , shown in figure 1 , could be interpreted as a gradual increase of the multiplicity and the hadronic density in the thermalization mechanism . therefore it is useful , to look for another statistics that , unlike the boltzmann - gibbs , could treat these systems . recently tsallis @xcite proposed a generalized statistics whose formalism @xcite has been able to describe many systems having as common features the presence of lr interactions and/or non - markovian processes @xcite . this formalism is based on the entropic form @xmath24 where @xmath25 is the density matrix ( state ) of the system and @xmath26 is a real positive parameter . when @xmath27 the above entropy tends to the well - known boltzmann - gibbs - shannon ( bgs ) entropy @xmath28 . in this sense the @xmath29 entropy extends the bgs entropy to the nonextensive cases corresponding to @xmath30 . in fact , in this case the additivity property of the entropy fails because it appears a new term proportional to @xmath31 . for example , if we look the way the composed entropy of two sub - systems @xmath32 and @xmath33 , @xmath34 , is related to the entropy of each of the two sub - systems , @xmath35 and @xmath36 , when the sub - systems @xmath32 and @xmath33 are statistically independents , one gets the expression @xmath37 . it is easy here to see that the case @xmath38 reobtains the standard result ( additivity ) from boltzmann - gibbs entropy @xcite . the extremization of the @xmath29 entropy with adequate constraints lead us to a probability distribution that exhibits a power - law decay ( instead of the exponential decay presented by the bgs entropy ) . examples of application of this statistics are available in several papers @xcite , whose studied systems are , in general , nonextensive . thus , assuming a thermal equilibrium of a relativistic gas ( the _ fireball _ ) that obeys the tsallis statistics we obtain , observing the derivation made by @xcite , the following equation for the transverse momentum distribution ( which replaces eq . ( [ eq : pl ] ) ) :    @xmath39^{\frac{q}{1-q } }   \hspace{0.1 cm } , \label{eq : pt}\ ] ]    where @xmath26 is related to the degree of non - extensivity of the system @xcite . @xmath11 , @xmath13 , @xmath21 and @xmath40 have the same meaning as in the previous equations . the boltzmann - gibbs limit - eq . ( [ eq : pl ] ) - is achieved as @xmath41 . the exact expression of the transverse momentum distribution , eq . ( [ eq : pt ] ) , is written in the appendix . with this expression we fit the experimental data leaving @xmath26 , @xmath23 and @xmath21 as free parameters and the best fits are presented in the figure 1 . we can see that the asymptotic behaviour of the new distribution , for larger values of @xmath0 , represents very - well the deviation from the exponential for _ all _ center - of - mass energies .    . [ cols="^,^,^,^",options="header " , ]     in figure 2 we present @xmath26 and @xmath11 values as functions of the center - of - mass energy . their numerical values are shown in table 1 . we can see that the @xmath26 parameter increases smoothly with the energy , saturating around @xmath42 . on the other hand , the temperature has a pretty constant behaviour , @xmath43 gev , for the higher energy range . we note that this constant value for the temperature is not an ad hoc hypothesis ; rather , it was obtained by the fit , indicating that all the energy given to the system is being used to create new _ fireballs _ , as predicted by hagedorn @xcite . we note that our @xmath11 value is lower than the value obtained in reference @xcite , and this difference is probably due to the fact that , unlike us , the author considered the thermal equilibrium just after the hadronization , i.e. , before the decay of the hadronic resonances . in our process , the interaction volume continue to expand , lowering the temperature , until the particles no longer interact among themselves . also , the statistics used in both papers are different . the smooth increase of @xmath26 with the center of mass energy can be understood as the expected increase of the influence of the lr interactions among the hadronic particles produced in the event .    the deviation from an exponential behaviour observed in the transverse momentum distribution in the interaction , is also observed for hadrons produced in other interactions like @xmath44 or heavy ions . it indicates a universal characteristic of the dynamics of hadron production , probably due to the relevance of the lr regime , related to the high hadronic density . the thermodynamical treatment using the tsallis statistics discussed here presented a remarkable agreement with the experimental data in more than 4 orders of magnitude in the differential cross section - emphasized with the predicted constant behavior of the temperature @xmath11 . therefore , the results of this approach suggest an alternative statistical treatment for the transverse momentum distribution within the accepted theory of strong interaction .    acknowledgments . we thank dr . oliver passon for detailed delphi experimental data and carlos aguiar for useful discussion . one of us , e.m.f.c . , thanks pronex / mct for the partial financial support . e - mail : bediaga@cbpf.br , eme@cbpf.br , jussara@lafex.cbpf.br    99    marchesini , g. and webber , b. r. , nucl . b * 310 * , 461 - 526 ( 1988 ) . sjostrand , t. , int . phys . a * 3 * , 751 - 823 ( 1988 ) . chun , s. and buchanan , c. , phys . rep . * 292 * , 239 - 317 ( 1998 ) . fermi , e. , phys . * 81 * , 683 - 687 ( 1951 ) . hagedorn , r. , suppl . nuovo cimento * 3 * , 147 - 186 ( 1965 ) . schnedermann , e. , sollfrank , j. and heinz , u. , phys . c * 48 * , 2462 - 2475 ( 1993 ) . becattini , f. , z. phys . c * 69 * , 485 - 492 ( 1996 ) . becattini , f. , giovannini , a. and lupia , s. , z. phys . c * 72 * , 491 - 496 ( 1996 ) . becattini , f. and heinz , u. , z. phys . c * 76 * , 269 - 286 ( 1996 ) . field , r.d . and feynman , r.p . , n. phys . b * 136 * , 1 - 76 ( 1978 ) . althoff , m. et al . 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( [ eq : pt ] ) was obtained by the software mathematica 3.0 and the transverse momentum distribution can be written as :
we show that the inclusion of long distance effect , expected in strong interactions , through a nonextensive thermodynamical approach is able to explain the experimental distribution of the transverse momentum of the hadrons with respect to the jet axis ( @xmath0 ) @xmath1 reaction . 
 the observed deviation from the exponential behavior , predicted by the boltzmann - gibbs thermodynamical treatment , is automatically recovered by the nonextensive tsallis statistics used here . 
 we fitted the observed @xmath0 spectrum in the range of 14 gev to 161 gev and obtained , besides a good fit , the theoretical important fact that the temperature becomes independent of the primary energy . 
 xxxxxxxxxxxxxxxxxx=    high energy ; hadroproduction ; fireball ; statistical models . 
 + 24.10.pa , 13.65.+i , 13.60.hb ,