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This paper presents the results of analysis of the D? 1.0 fb
^{-1} data on top-quark differential cross section measurements at the Fermilab Tevatron collider at √s= 1960 GeV in the framework of
z-scaling approach. The flavor independence of scaling function Ψ(z)observed in
pp and
pp interactions over a wide collision energy range √s= 19-1960 GeV has been verified. This property of Ψ(z) was found for different hadrons – from
π-mesons up to
Υ particles. The flavor independence of Ψ(z) is used as indication on self-similarity of the
top-quark production. A tendency to saturation of Ψ(z) at low
z for
top-quark production has been confirmed. Momentum fraction
x_{1} of the incoming (anti)protons as a function of the scaled transverse momentum
p_{T}/m and masses of heavy mesons is studied. We anticipate that the data on low- and high-
p_{T} inclusive spectra of the
top-quark production at the Tevatron and LHC energies could be of interest to verify self-similarity over a wide range of masses and different flavor content of produced particles.

The measurements of the top-quark transverse momentum distribution have been performed at the Fermilab Tevatron collider at = 1800 and 1960 GeV by the CDF [^{–1} and 1 fb^{–1}. The top-quark is the heaviest known elementary particle and was discovered at the Tevatron collider in 1995 by the CDF and DØ Collaborations [3,4] at a mass of around 170 GeV. It is expected [5-12] that the top physics is extremely important for scientific search for new phenomena.

In the given paper we have analyzed the DØ data using the method known as z-scaling [13,14]. Main features of the approach in pp and interactions at FNAL, CERN, and BNL (RHIC) energies were presented and discussed in [15,16]. Some results of analysis of the LHC data on the charged hadron [_{T}) range. At low z (lowp_{T}), saturation of the scaling function was found down to a value of 10^{−3} [16,22]. It has been concluded that z-scaling reflects self-similarity of the hadron structure, constituent interactions and hadronization process. The analyzed experimental data cover a wide range of the collision energies, transverse momenta and angles of the produced particles. The energy, angular and multiplicity independence of scaling function gives strong constraints on the values of parameters δ, c, and ε entering in definition of z. The parameter c which controls the behavior of at low z has analogy with the “specific heat” of the produced medium associated with the inclusive particle production. The scaling in pp and collisions is consistent with the constant value of c = 0.25. A possible change of this parameter could be assumed to be an indication of the phase transition of the matter produced in high energy collisions. The structure of the interacting objects produced at high momenta is characterized by parameter interpreted as a nucleon fractal dimension. The scaling is consistent with the constant value of δ = 0.5 for all types of the analyzed inclusive hadrons. The fragmentation process is parameterized in terms of dimension which increases together with the hadron mass.

In this paper we have analyzed the data [_{T} of at least 20 GeV/c and a pseudo-rapidity of 1.1 (e + jets) or 2.0 (μ + jets). A cut on the missing transverse energy of 20 GeV was applied. Furthermore at least four jets were required with p_{T} > 20 GeV/c and 2.5, an additional cut of p_{T} > 40 GeV/c was applied for the leading jet. Finally at least one jet needs to be identified as a b-jet. Additional constraints are used to reconstruct the event kinematics: the masses of two W bosons are constrained to 80.4 GeV. The masses of the two reconstructed top quarks are assumed to be equal.

The results of analysis of the top inclusive cross sections are compared with Tevatron data [24-26] on J/ψ, D^{0}B, and Υ particle spectra at = 1960 and 1800 GeV in the z-presentation. We have verified the flavor independence of including the inclusive top-quark measurements in this region. A microscopic scenario of hadron production in the z-scaling approach is used to estimate the energy loss and recoil mass at a constituent level in the dependence on transverse momentum of an inclusive particle. This gives the specific dependence of the momentum fraction x_{1} characteristic for different types of produced hadrons. The p_{T}-behavior of the fraction x_{1 }for the top-quark production is compared with other particles.

We expect that systematic measurements of the inclusive differential spectra of the top-quark as a function of the transverse momentum at LHC energies could give new information on self-similarity of the heavy flavor production in the super high energy domain.

Here we follow the basic ideas of the z-scaling approach [15,16]. It is assumed that the collision of extended objects (hadrons, nuclei) at sufficiently high energies could be considered to be an ensemble of individual interactions of their constituents (partons, quarks, gluons). Structures of the colliding objects are characterized by parameters and. The constituents of the incoming objects (hadrons or nuclei) with masses M_{1}, M_{2} carry away fractions x_{1}, x_{2} of their momenta P_{1}, P_{2}. The inclusive particle has a fraction (denoted by) of the momentum of the object produced in the constituent collision in the observed direction. Its fragmentation is characterized by parameter. The fragmentation in the recoil direction is described by parameter and momentum fraction. Multiple interactions of the constituents are considered to be similar. This property reflects the self-similarity of the hadron interactions at the constituent level.

The elementary sub-process is considered to be a binary collision of the constituents with masses x_{1}M_{1} and x_{2}M_{2} resulting in the scattered and recoil objects with masses and in the final state. The produced secondary objects transform into real particles after the constituent collisions. The registered particle with mass m_{1} and 4-momentum p is produced with its hadron counterpart with mass m_{2} carrying the momentum fractions of the produced recoil. The momentum conservation law of the constituent sub-process is written in the following form:

Here M_{X} is the recoil mass and

.

The production of the associated particle with mass m_{2} ensures conservation of the additive quantum numbers. Equation (1) is an expression of the locality of the hadron interaction at a constituent level. It represents a kinematic constraint on momentum fractions, and which determine the underlying elementary sub-process.

The structure of colliding objects and fragmentation of the systems formed in scattered and recoil directions are characterized by parameters, and, respectively. The parameters are related with the corresponding momentum fractions by function

Quantity Ω is proportional to the relative number of all constituent configurations in the inclusive reaction, which contain the configuration defined by fractions and. The Ω is interpreted as a relative volume which occupies these configurations in the space of the momentum fractions. Parameters, and are taken as fractal dimensions in the parts of the space of the momentum fractions which correspond to the colliding objects and fragmentation processes, respectively. For the given values of, and the fractions and _{ }are determined in such a way to maximize the function Ω, simultaneously fulfilling condition (1).

In the case of pp () interactions we have and set. It is assumed that the fragmentation of the objects moving in the scattered and recoil directions can be described by the same parameter which depends on the type of the inclusive particle. The values of parameters and are determined according to the self-similarity requirements of experimental data in z-presentation. They were found to have constant values in pp and collisions at high energies.

The self-similarity of hadron interactions reflects the property that hadron constituents and their interactions are similar. The self-similarity variable z is defined as follows:

where, and is the maximal value of (2) with condition (1). For the above inclusive reaction the quantity z is proportional to the transverse kinetic energy of the constituent subprocess consumed for the production of the inclusive particle and its counterpart with masses m_{1} and m_{2}, respectively. The quantity is the corresponding multiplicity density of charged particles produced in the central region of the reaction at the pseudo-rapidity. Parameter c characterizes properties of the produced medium. It is interpreted as “specific heat”. The constant m_{N} is taken to be a nucleon mass.

Scaling function is expressed in terms of the experimentally measured inclusive cross section, the multiplicity density, and the total inelastic cross section as follows [

Here s is the square of the center-of-mass energy and J is the corresponding Jacobian. The multiplicity density in (4) depends on the center-of-mass energy, centrality, and on the production angles at which the inclusive spectra were measured. The above expression can be rewritten in the central interaction region into the following form:

The scaling function is normalized as follows:

It allows us to interpret as a probability density of the production of the inclusive particle with the corresponding value of variable z.

The flavor independence of hadron production means that spectra of particles with a different flavor content can be described by universal scaling function in z-presentation [15,16]. Our previous analysis is based on the observation that simultaneous energy, angular and multiplicity independence of the z-scaling for negative pions, kaons, and anti-protons produced in proton-proton collisions gives the same shape of the scaling function. The flavor independence of was also confirmed for other inclusive particles including the heavy quarkonia, J/ψ [^{−3}. In the region 0.1 we observe a saturation of scaling function which can be approximated by a constant.

In this paper we analyze the data [_{T} at a middle rapidity. We have shown that the flavor independence of the z-presentation of hadron spectra is valid for the top-quark production as well. We exploit the scaling transformation

to compare the shape of scaling function for different hadron species. Parameter is the scale independent quantity. The transformation does not destroy the shape of. It preserves the normalization Equation (6) and the energy, angular and multiplicity independences of the z-presentation of particle spectra.

^{0}, B, and Υ) [24-26] obtained in collisions at the Tevatron energies = 1960 and 1800 GeV in the central rapidity region. The experimental data are shown by symbols. The data include measurements up to small transverse momenta (p_{T} ≈ 125 MeV/c for charmonia, p_{T} ≈ 290 MeV/c for bottomia, and p_{T} ≈ 500 MeV/c for B-mesons). The data on the -meson spectra at = 53 GeV [_{T}, and). The scale factors are constants. It allows us to describe z-presentation of the spectra for different hadron species with a single function. The collapse of data points onto a single curve corresponds to the estimated errors of at the level of 20%.

_{T}-bin. The

horizontal errors refer to the width of the bins. The condition (6) is satisfied with the normalization and n = 2. This corresponds to two entries per event with the total normalization to the production cross section = 8.31 pb [

The values of the fractal dimension = 0.5 and “specific heat” c = 0.25 are the same as used in the previous analyses of the inclusive spectra [15,16,21]. We have set in the case of the top-quark since no energy loss is assumed in the elementary production process. This choice corresponds to in the whole p_{T}-range. The value of in the transformation (7) is found to be 0.0045. No additional parameters were used.

As it is seen from _{T}-distributions of the inclusive cross sections which reveal strong dependence on the energy, angle, multiplicity and type of the produced particles. Based on the above comparison we have concluded that the Tevatron data on inclusive spectra of the top-quark production measured by the DØ Collaboration support the flavor independence of scaling function over the range of z = 0.02 - 2. This result gives us an indication on self-similarity of top-quark production in collisions at = 1960 GeV.

The determination method of the momentum fractions allows us to analyze kinematics of the constituent interactions in the framework of the developed approach. Unlike other particles, the null value of ε_{top} means that the energy loss of the top-quark production is zero or negligible. This result is expressed by condition and by the value of recoil mass M_{X} which is practically equal to the mass of the top-quark. The kinematics of the underlying sub-process is fully determined by momentum fractions x_{1} and x_{2} in this case. The fractions characterize the amount of the energy (momentum) of the incoming protons (antiprotons) carried by the interacting constituents which cause the inclusive particle production. In the general case of other particles, x_{1} and x_{2} are functions of y_{a} and y_{b} [_{1} and x_{2} are equal to each other. A comparison of fraction x_{1} for the top-quark production with other heavy particles is shown in _{1} growth with. The value of x_{1} is larger for the production of heavy particles as compared with the light ones. The exception for J/ψ meson was observed (see discussed in [

with extra large energy dissipation in the final state accompanied by production of this particle. For a fixed value of the fraction x_{1} decreases while increasing collision energy. The kinematic limit of the reaction corresponds to at any collision energy and for any type of the inclusive particle.

We have presented the results of analysis of the data on inclusive spectra of the top-quark production in collisions at the energy = 1960 GeV measured by the DØ Collaboration at the Tevatron. The transverse momentum spectra in z-presentation are compared with the data obtained for the heavy mesons J/ψ, D^{0}, B, and Υ at the Tevatron energies = 1960 and 1800 GeV in the central rapidity range. Based on the results presented here we conclude that the data on the transverse momentum distribution of the top-quark production in collision are in good agreement with flavor independence of the z-scaling. The result also supports the energy independence of the scaling function in the middle rapidity region. A tendency to saturation at low z for the top-quark production is confirmed as well. The momentum fraction x_{1} of the incoming protons for the top-quark was compared with the corresponding values for the heavy mesons measured at the Tevatron. Though production of the top-quark is characterized by no energy loss and constant recoil mass, the fraction x_{1} reveal similar dependences on the scaled transverse momentum as for the heavy mesons.

We assume that the data on the top-quark differential inclusive cross section over a wider range of p_{T} and collision energy at the Tevatron and LHC could be of interest to verify the flavor independence of z-scaling and self-similarity of top-quark production.

These investigations have been supported by the IRP AVOZ10480505, by the Ministry of Education of the Czech Republic grants LA08002, LA08015.