Possible Implication of a Single Nonextensive $p_T$ Distribution for Hadron Production in High-Energy $pp$ Collisions

Multiparticle production processes in $pp$ collisions at the central rapidity region are usually considered to be divided into independent"soft"and"hard"components. The first is described by exponential (thermal-like) transverse momentum spectra in the low-$p_T$ region with a scale parameter $T$ associated with the temperature of the hadronizing system. The second is governed by a power-like distributions of transverse momenta with power index $n$ at high-$p_T$ associated with the hard scattering between partons. We show that the hard-scattering integral can be approximated as a nonextensive distribution of a quasi-power-law containing a scale parameter $T$ and a power index $n=1/(q -1)$, where $q$ is the nonextensivity parameter. We demonstrate that the whole region of transverse momenta presently measurable at LHC experiments at central rapidity (in which the observed cross sections varies by $14$ orders of magnitude down to the low $p_T$ region) can be adequately described by a single nonextensive distribution. These results suggest the dominance of the hard-scattering hadron-production process and the approximate validity of a"no-hair"statistical-mechanical description of the $p_T$ spectra for the whole $p_T$ region at central rapidity for $pp$ collisions at high-energies.


Introduction
Particle production in pp collisions comprises of many different mechanisms in different parts of the phase space. We shall be interested in particle production in the central rapidity region where it is customary to divide the multiparticle production into independent soft and hard processes populating different parts of the transverse momentum space separated by a momentum scale p 0 . As a rule of thumb, the spectra of the soft processes in the low-p T region are (almost) exponential, F(p T )∼exp(−p T /T ), and are usually associated with the thermodynamical description of the hadronizing system, the fragmentation of a flux tube with a transverse dimension, or the production of particles by the Schwinger mechanism [1][2][3][4][5]. The p T spectra of the hard process in the high-p T region are regarded as essentially power-like, F(p T )∼p −n T , and are usually associated with the hard scattering process [6][7][8][9][10]. However, it was found already long time ago that both description could be replaced by simple interpolating formula [11], ⋆ Presented by G.Wilk a e-mail: wongc@ornl.gov b e-mail: wilk@fuw.edu.pl c e-mail: cirto@cbpf.br d e-mail: tsallis@cbpf.br that becomes power-like for high p T and exponential-like for low p T . Notice that for high p T , where we are usually neglecting the constant term, the scale parameter p 0 becomes irrelevant, whereas for low p T it becomes, together with power index n, an effective temperature T = p 0 /n. The same formula re-emerged later to become known as the QCD-based Hagedorn formula [12]. It was used for the first time in the analysis of UA1 experimental data [13] and it became one of the standard phenomenological formulas for p T data analysis.
In the mean time it was realized that Eq. (1) is just another realization of the nonextensive distribution [14] with parameters q and T , and a normalization constant A, that has been widely used in many other branches of physics. For our purposes, both formulas are equivalent with the identification of n = 1/(q−1) and p 0 = nT , and we shall use them interchangeably. Because Eq. (2) describes nonextensive systems in statistical mechanics, the parameter q is usually called the nonextensivity parameter. As one can see, Eq. (2) becomes the usual Boltzmann-Gibbs exponential distribution for q → 1, with T becoming the temperature. Both Eqs. (1) and (2) have been widely used in the phenomenological analysis of multiparticle productions (cf., for example [15][16][17][18][19][20][21][22][23][24][25][26][27][28]) 1 .
We shall demonstrate here that, similar to the original ideas presented in [11,12], the whole region of transverse momenta presently measurable at LHC experiments (which spans now enormous range of ∼14 orders of magnitude in the measured cross-sections down to the low-p T region) [17][18][19] can be adequately described by a single quasi-power law distribution, either Eq. (1) or Eq. (2). We shall offer a possible explanation of this phenomenon by showing that the hard-scattering integral can be cast approximately into a non-extensive distribution form and that the description of a single nonextensive p T distribution for the p T spectra over the whole p T region suggests the dominance of the hard-scattering process at central rapidity for high-energy pp collisions.

Nonextensive distribution for p T spectra in pp collisions
The possibility of two components in the transverse spectra implies that its complete description will need two independent functions with different sets of parameters, each dominating over different regions of the transverse momentum space. The presence of two different components will be indicated by gross deviations when the spectrum over the whole transverse space is analyzed with only a single component. An example for the presence of two (or more) components of production processes can be clearly seen in Fig. 1 of [31], in the p T spectra in central (0-6%) PbPb collisions at √ s NN =2.76 TeV from the ALICE Collaboration, where two independent functions are needed to describe the whole spectra as described in [32,33]. For our purposes in studying produced hadrons in pp collisions, where the high p T hard-scattering component is expected to have a power-law form with a power index n, either (1) or (2) can be written as where n is the power index, T is the 'temperature' parameter, and m and m T = m 2 + p 2 T are the rest mass and transverse mass of the produced hadrons which are taken to be the dominant particles, the pions. It came as a surprise to us that for pp collisions at √ s NN = 7 TeV, the p T spectra within a very broad range, from 0.5 GeV up to 181 GeV, in which cross section varies by 14 orders of magnitude, can still be described well by a single nonextensive formula with power index n = 6.6 [34]. The good fits to the p T spectra over such a large range of p T with only three parameters, (A, n, T ), raise intriguing questions : • Why are there only three degrees of freedom in the spectra over such a large p T domain? Does it imply that there  is only a single component, the hard scattering process,  contributing dominantly over the whole p T domain? If  so, are there supporting experimental evidences from  other correlation measurements? • Mathematically, the power index n is related to the parameter q = 1 + 1/n in non-extensive statistical mechanics [14]. What is the physical meaning of n? If n is related to the power index of the parton-parton scattering law, then why is the observed value so large, n∼7, rather than n ∼ 4 as predicted naively by pQCD?
• Are the power indices for jet production different from those for hadron production? If so, why ?
• Do multiple parton collisions play any role in modifying the power index n?
• In addition to the power law 1/p n T , does the differential cross section contain other additional p T -dependent factors? If they are present, how do they change the power index?
These questions were discussed and, at least partially, answered in [35]. Before proceeding to our main point of phenomenological considerations we shall first recapitulate briefly the main results of this attempt to reconcile, as far as possible, the nonextensive distribution with the QCD where, as shown in [35], the only relevant ingredients from QCD are hard scatterings between constituents resulting in the production of jets which further undergo fragmentation, showering, and hadronization to become the observed hadrons.

Approximate Hard-Scattering Integral
The answers to the questions posed above will be facilitated with an approximate analytical form of the hardscattering integral. We start with the relativistic hardscattering model as proposed in [6] 2 and examined in [5,10,35]. We consider the collision of projectiles A and B in the center-of-mass frame at an energy √ s in the reaction A + B → c + X, with c coming out at midrapidity, η∼ 0. Upon neglecting the intrinsic transverse momentum and rest masses, the differential cross section in the lowest-order parton-parton elastic collisions is given by The parton-parton invariant cross section is related to dσ(ab→cX ′ )/dt by In the infinite momentum frame the momenta can be written as We denote light-cone variable x c of the produced parton c as x c = (c 0 + c z )/ √ s. The constraint ofŝ +t +û = 0 gives We consider only the special case of c coming out at . After integrating over x a , we obtain To integrate over x b , we use the saddle point method, write , and expand f (x b ) about its minimum at x b0 . We obtain then that For simplicity, we assume G a/A and G b/B to have the same form. At θ c ∼ 90 0 in the CM system, the minimum value of f (x b ) is located at and we get the hard-scattering integral For the case of If the basic process ab → cX ′ is gg → gg or ab → cX ′ is qq ′ → qq ′ , the cross sections at θ c ∼ 90 o [36] are In both cases, the differential cross section behave as

Parton Multiple Scattering
As the collision energy increases, the value of x c gets smaller and the number of partons and their density increase rapidly. Thus the total hard-scattering cross section increases as well [8]. The presence of a large number of partons in the colliding system results in multiple hardscatterings of projectile parton on partons from target nucleon. We find that for the process of a → c in the collision of a parton a with a target of A partons in sequence without a centrality selection, the c T -distribution is given by [35] where the terms on the right-hand side correspond to collisions of the incident parton with one, two and three target partons, respectively. Here, the quantity A is the number of partons in the nucleon as a composite system and is the integral of the parton density over the parton momentum fraction. This result shows that without centrality selection in minimum-biased events, the differential cross section will be dominated by the contribution from a single parton-parton scattering that behaves as α 2 s /c 4 T (cf. previous analysis on the multiple had-scattering process in [37][38][39]). Multiple scatterings with N > 1 scatterers contribute to terms of order α 2N

The Power Index in Jet Production
From the above results one gets the approximate analytical formula for hard-scattering invariant cross section σ inv , for The power index n has here the value 4 + 1/2. Its value can be extracted by plotting (ln σ inv ) as a function of (ln c T ) (then the slope in the linear section gives the value of n, and the variation of (ln σ inv ) at large (ln c T ) gives the value of g a and g b ). One can also consider for this purpose a fixed x c and look at two different energies (as suggested in [40]), We follow an alternative method and analyze the p T spectra using a running coupling constant, where we have chosen Λ QCD to be 0.25 GeV to give α s (M 2 Z ) = 0.1184 [41]. We identify Q as c T and have chosen C=10 both to give α s (Q∼Λ QCD ) ∼ 0.6 in hadron spectroscopy studies [42] and to regularize the coupling constant for small values of Q(c T ). We search for n by writing the invariant cross section Eq. (16) for jet production as  [47] pp at 7 TeV 0.5 |η| < 0. 5 5.39 In the literature [43,44] the index g a for the structure function of a gluon varies from 6 to 10. Following [43] we shall take g a = 6. As shown in Fig. 1 and Table I [46], the power index is n=4.78 for R = 0.2, and is n=4.98 for R = 0.4 (Table I). The power index is n=5.39, for CMS jet differential cross section in pp collisions at √ s = 7 TeV at the LHC within |η| < 0.5 and R = 0.5 [47]. This latter n value exceeds slightly the expected value of n = 4.5.
Except for the CMS data at 7 TeV that may need further re-examination, the power indices extracted for hadron jet production and listed in Table I are in approximate agreement with the value of n=4.5 in Eq. (16) and with previous analysis of Arleo et al. [40], indicating the approximate validity of the hard-scattering model for jet production in hadron-hadron collisions, with the predominant α 2 s /c 4 T parton-parton differential cross section as predicted by pQCD.

Production to Hadron Production
The results in the last section indicates that the simple hard-scattering model, i.e., Eq. (18), adequately describes the power index of n ∼ 4.5 for jet production in high-energy pp collisions. However, the power index for hadron production is considerable greater, in the range of n∼ 6 − 10 [34,40]. What is the origin of the increase in the power index n?
A jet c evolves by fragmentation, showering, and hadronization to turn the jet into a large numbers of hadrons in a cone along the jet axis. The showering of the partons will go through many generations of branching. If we label the (average) momentum of the i-th generation parton by p (i) T , the showering can be represented as T . Each branching will kinematically degrade the momentum of the showering parton by a momentum fraction, ζ=p (i+1) T . At the end of the terminating λ-th generation of the showering, and hadronization, the p T of a produced hadron is related to the c T of the parent parton jet by It is easy to prove that if the generation number λ and the fragmentation fraction z are independent of the jet c T , then the power law and the power index for the p T distribution are unchanged [35]. We note however that in addition to the kinematic decrease of p T as described by (20), the showering generation number λ is governed by an additional criterion on the virtuality, which measures the degree of the off-themass-shell property of the parton. From the different parton showering schemes in the PYTHIA [48], the HERWIG [49], and the ARIADNE [50], we can extract a general picture that the initial parton with a large initial virtuality Q decreases its virtuality by showering until a limit of Q 0 is reached. The downgrading of the virtuality will proceed as There is a one-to-one mapping of the initial virtuality Q with the transverse momentum c T of the evolving parton as Q(c T ) (or conversely c T (Q)). Because of such a mapping, the decrease in virtuality Q corresponds to a decrease of the corresponding mappedc T as c T =c (0) . The cut-off virtuality Q 0 maps into a transverse momentum c T 0 =c T (Q 0 ). In each successive generation of the showering, the virtuality decreases by a virtuality fraction which corresponds, at least approximately, in terms of the corresponding mapped parton transverse momentumc (i) T , to a decrease by a corresponding transverse momentum fraction,ζ=c (i+1) T . The showering will end in λ generations such that We can infer a relation between c T and the number of generations, λ, Thus, the showering generation number λ depends on the magnitude of c T . On the other hand, kinematically, the showering processes degrades the transverse momentum of the parton c T to that of the p T of the produced hadron as given by Eq. (20), depending on the number of generations λ. The magnitude of the transverse momentum p T of the produced hadron is related to the transverse momentum c T of the parent parton jet by We can solve the above equation for p T as a function of c T and obtain where µ = ln ζ/lnζ, and µ is a parameter that can be searched to fit the data. As a result of the virtuality ordering and virtuality cut-off, the hadron fragment transverse momentum p T is related to the parton momentum c T nonlinearly by an exponent 1 − µ.
After the fragmentation and showering of the parent parton c T to the produced hadron p T , the hard-scattering cross section for the scattering in terms of hadron momentum p T becomes Upon substituting the non-linear relation (24) between the parent parton moment c T and the produced hadron p T in Eq. (24), we get Therefore under the fragmentation from c to p, the hardscattering cross section for AB → pX becomes where Thus, the power index n for jet production can be significantly changed to n ′ for hadron production because the greater the value of the parent jet c T , the greater the number of generations λ to reach the produced hadron, and the greater is the kinematic energy degradation. By a proper tuning of µ, the power index can be brought to agree with the observed power index in hadron production. For example, for µ=0.4 one gets n ′ =6.2 and for µ = 0.6 one gets n ′ =8.2. Because the parton branching probability, parton kinematic degradation, and parton virtuality degradation depend on the coupling constant and the coupling constant depends on the parton energy, we expect the quantity µ to depend on the pp collision energy. Consequently, n ′ may change significantly with the collision energy.

Regularization of the Hard-Scattering Integral
The power-law (28) has been obtained for high p T . In order to apply it to the whole range of E T , we need to regularize it by the replacement, The quantity m T 0 measures the average transverse mass of the detected hadron in the hard-scattering process. The differential cross section d 3 σ(AB → pX)/dyd p T in (28) is then regularized as In the above equation for the production of a hadron with a transverse momentum p T , the variable c T (p T ) refers to the transverse momentum of the parent jet c T before fragmentation. We can relate p T with c T by using the empirical fragmentation function of Ref. [51] and we get [35] This can be regarded as a linearized approximation of Eq. CMS [17], ATLAS [18], and ALICE Collaborations [19] are shown in Fig. 2. We find that the experimental data gives n ′ =5.69 and m T 0 =0.804 GeV for √ s=7 TeV and n ′ =5.86 and m T 0 =0.634 GeV for √ s=0. 9 TeV. This indicates that there is indeed a systematic change of the power index n from jet production to a larger value n ′ in hadron production. The fits to the low p T region for the ALICE data can be improved, with a larger power index n ′ as we shall see below in Section 9.

Further Approximation of the Hard-Scattering Integral
We would like to simplify further the p T dependencies of the structure function in Eq. (31) and the running coupling constant as additional power indices in such a way that will facilitate subsequent phenomenological comparison. For parton c coming at mid-rapidity, the quantities x a0 , x b0 , and x c in Eqs. (10) and (31) are The structure function factor and the denominator factor in Eq. (31) can be approximated for high energies with √ s ≫ c T as We can relate c T with p T by Eq. (32) and further approximate the right-hand side of the above equation in a form that is advantageous for subsequent purposes. For high energy with large √ s, we make the approximation We therefore estimate that n g ∼ 0.04 and 0.007 for √ s = 0.9 and 7 TeV respectively.
The running coupling constant α s is a monotonically decreasing function of Q(c T ). It can be written approximately as where n α can be chosen to minimize errors by matching α s at two points of p T . If we match α s (p T ) at p T =Λ QCD =0.25 GeV and at p T =100 GeV, then n α = 0.36. If we match α s at p T =Λ QCD and at p T =20 GeV, then n α = 0.46. As a consequence of the above simplifying approximations, we can write the hard-scattering integral Eq. (31) in the approximate form where n = n ′ + n g + n α , and n ′ is the power index after taking into account the fragmentation process, n g the power index from the structure function, and n α from the coupling constant. We note that the predominant change of the power index from jet production to hadron production arises from the fragmentation process because n g and n α are relatively small. In reaching the above equation, we have approximated the hard-scattering integral F(p T ) that may not be exactly in the form of 1/[1 + m T /m T 0 ] n into such a form. It is easy then to see that upon matching F(p T ) with A/[1 + m T /m T 0 ] n according to some matching criteria, the hard-scattering integral F(p T ) will be in excess of 1/[1 + m T /m T 0 ] n in some region, and will be in deficit in some other region. As a consequence, the ratio of the hardscattering integral F(p T ) to the fitting 1/[1+m T /m T 0 ] n will oscillate as a function of p T . This matching between the physical hard-scattering outcome that contains all physical effects with the approximation of Eq. (37) may be one of the origin of the oscillations of the experimental fit with the non-extensive distribution (as can be seen below in Fig.  3).

Nonextensive Distribution as a Lowest-Order Approximation of the Hard-scattering Integral
In the hard-scattering integral Eq. (37), if we identify and consider produced particles to be relativistic so that m T ∼E T ∼p T and E T ∼E at mid-rapidity, then we will get the nonextensive distribution of Eq. (2) as the lowest-order approximation for the QCD-based hard-scattering integral.
The convergence of Eq. (37) and Eq. (2) can be considered from the viewpoint of the reduction of a microscopic description to a statistical-mechanical description. From the microscopic perspective, the hadron production in a pp collision is a very complicated process, as evidenced by the complexity of the evolution dynamics in the evaluation of the p T spectra in explicit Monte Carlo programs, for example, in [48][49][50]. If one starts from the initial condition of two colliding nucleons, there are many intermediate and complicated processes entering into the dynamics, each of which contain a large set of microscopic and stochastic degrees of freedom. Along the way, there are stochastic elements in the picking of the degree of inelasticity, in picking the colliding parton momenta from the parent nucleons, the scattering of the partons, the showering evolution of scattered partons, the hadronization of the fragmented partons. Some of these stochastic elements cannot be definitive and many different models, sometimes with untestable assumptions, have been put forth. In spite of all these complicated stochastic dynamics, the final result of Eq. (37) of the single-particle distribution can be approximated to depend only on three degrees of freedom, after all is done, put together, and integrated. The simplification can be considered as a "no hair" reduction from the microscopic description to nonextensive statistical mechanics in which all the complexities in the microscopic description "disappear" and are subsumed behind the stochastic processes and integrations. In line with statistical mechanics and in analogy with the Boltzmann-Gibbs distribution, we can cast the hard-scattering integral in the non-extensive form in the lowest-order approximation as [52] where E= m 2 + p 2 and E=E T =m T at y=0. Here, the parameter q is related physically to the power index n, the parameter T related to m T 0 and the average transverse momentum, and the parameter A related to the multiplicity (per unity rapidity) after integration over p T . Given a physically determined invariant cross section in the loglog plot of the cross section as a function of the transverse hadron energy as in Fig. 3, the slope at large p T gives the power index n (and q), the average of E T gives T (and m T 0 ), and the integral over p T gives A. Fig. 3 gives the comparisons of the results from Eq. (40) with the experimental p T spectra at central rapidity obtained by different Collaborations [17][18][19]. In these calculations, the effective temperature parameter is set equal to T =0.13 GeV, and the parameters of A, q and the corresponding n are given in Table 2. The dashed line (an ordinary exponential of E T for q → 1) illustrates the large discrepancy if the distribution is described by Boltzmann-Gibbs distribution. The results in Fig. 3 shows that Eq. (40) adequately describes the hadron p T spectra at central rapidity in high-energy pp collisions. We verify that q increases slightly with the beam energy, but, for the present Herein the temperature is set to be the same for all curves and equal T = 0.13 GeV, and the normalization constant in units of GeV −2 /c 3 . The corresponding Boltzmann-Gibbs (purely exponential) fit is illustrated as the dashed curve. For a better visualization both the data and the analytical curves have been divided by a constant factor as indicated. The ratios data/fit are shown at the bottom, where a roughly log-periodic behavior is observed on top of the q-exponential one. Data are taken from [17][18][19].
energies, remains always q ≃ 1.1, corresponding to a power index n in the range of 6-8 that decreases as a function of √ s. What interestingly emerges from the analysis of the data in high-energy pp collisions is that the good agreement of the present phenomenological fit extends to the whole p T region (or at least for p T greater than 0.2 GeV/c, where reliable experimental data are available) [34]. This is being achieved with a single nonextensive distribution. On the other hand, theoretical analysis demonstrates that the hard-scattering integral can be written as a nonextensive distribution with only three degrees of freedom, in the lowest-order approximation. It is reasonable to infer that the dominant mechanism of hadron production over the whole range of p T at central rapidity and high energies is the hard-scattering process.
The dominance of hard-scattering also for the production of low-p T hadron in the central rapidity region is supported by two-particle correlation data where the twoparticle correlations in minimum p T -biased data reveals that a produced hadron is correlated with a "ridge" of particles along a wide range of ∆η on the azimuthally away side centering around ∆φ ∼ π [16,53,54]. The ∆φ ∼ π (back-to-back) correlation indicates that the correlated pair is related by a collision, and the ∆η correlation in the shape of a ridge indicates that the two particles are partons from the two nucleons and they carry fractions of the longitudinal momenta of their parents, leading to the ridge of ∆η at ∆φ ∼ π.

Conclusions and Discussions
Particle production in high-energy pp collisions at central rapidity is a complex process that can be viewed from two different and complementary perspectives. On the one hand, there is the successful microscopic description involving perturbative QCD and nonperturbative hadronization at the parton level where one describes the detailed mechanisms of parton-parton hard scattering, parton structure function, parton fragmentation, parton showering, the running coupling constant and other QCD processes. On the other hand from the viewpoint of statistical mechanics, the single-particle distribution can be cast into a form that exhibit all the essential features of the process with only three degrees of freedom. The final result of the process can be summarized, in the lowest-order approximation, by a power index n which can be represented by a nonextensivity parameter q=(n + 1)/n, the average transverse momentum m T 0 which can be represented by an effective temperature T =m T 0 /n, and an multiplicity constant A that is related to the multiplicity per unit rapidity when integrated over p T . Such a reduction from microscopic description to a statistical mechanical description can be shown both from theoretical considerations by obtaining a simplified and approximate hard-scattering integral, and also by comparing with experimental data. In the process, we uncover the dominance of the hard-scattering hadronproduction and the approximate validity of a "no-hair" statistical-mechanical description for the whole transverse momentum region in pp collision at high-energies. We emphasize also that, in all cases, the temperature turns out to be one and the same, namely T = 0.13 GeV.
What we may extract from the behavior of the experimental data is that scenario proposed in [11,12] appears to be essentially correct excepting for the fact that we are not facing thermal equilibrium but a different type of stationary state, typical of violation of ergodicity (for a discussion of the kinetic and effective temperatures see [55, 56])).
As a concluding remark, we note that the data/fit plot in the bottom part of Fig. 3 exhibit an intriguing rough log-periodicity oscillations, which suggest corrections to the lowest-order approximation of Eq. (37) and some hierarchical fine-structure in the quark-gluon system where hadrons are generated. This behavior is possibly an indication of some kind of fractality in the system. Indeed, the concept of self-similarity, one of the landmarks of fractal structures, has been used by Hagedorn in his definition of fireball, as was previously pointed out in [21] and found in analysis of jets produced in pp collisions at LHC [57]. This small oscillations have already been preliminary discussed in Section 8 and in [58,59], where the authors were able to mathematically accommodate these observed oscillations essentially allowing the index q in the very same Eq. (40) to be a complex number 4 (see also Refs. [60,61]; more details on this phenomenon, including also discussion of its presence in recent AA data, can be found in [33]).