On Self-Similarity of Top Production at Tevatron

This paper presents the results of analysis of the DØ 1.0 fb 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 observed in pp and   z  pp interactions over a wide collision energy range s = 19 1960 GeV has been verified. This property of   z    was found for different hadrons—from π-mesons up to Υ particles. The flavor independence of is used as indication on self-similarity of the top-quark production. A tendency to saturation of at low z for top-quark production has been confirmed. Momentum fraction x1 of the incoming (anti)protons as a function of the scaled transverse momentum  z 


Introduction
The measurements of the top-quark transverse momentum distribution have been performed at the Fermilab Tevatron collider at s = 1800 and 1960 GeV by the CDF [1] and DØ [2] Collaborations, respectively.The integrated luminosities of CDF and DØ data samples are 106 pb -1 and 1 fb -1 .The top-quark is the heaviest known elementary particle and was discovered at the Tevatron pp collider in 1995 by the CDF and DØ Collaborations [3,4] at a mass of around 170 GeV.It is expected [5][6][7][8][9][10][11][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 pp 0 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 [17], S some physical parameters.The shape of the scaling function was found to be independent of the collision energy, multiplicity density of particles, detection angle and hadron type.The power behavior of function   z z    was established in high-z (high-p T ) range.At low z (lowp T ), saturation of the scaling function was found down to a value of z  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 K -meson [18], and jet [19,20] production are presented in [21][22][23].The method allows us to perform systematic analysis of data on inclusive cross sections of hadrons, direct photons, and jets under different kinematic conditions.Scaling function pp 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 and scaling variable z are expressed via experimentally measurable quantities: inclusive and total inelastic cross sections, multiplicity density, momenta and masses of colliding and produced particles and using  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 F  which increases together with the hadron mass.
In this paper we have analyzed the data [2] on transve rse momentum spectra of the top-quark production in pp collisions at energy s = 1960 GeV in the middle rapi y th dity range obtained b e DØ Collaboration at the Tevatron.The measurements select the events with an isolated lepton having transverse momentum p 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 m ng transverse en of 20 GeV was applied.Furthermore at least four jets were required with p T > 20 GeV/c and issi ergy   2.5, an additional cut of p T > 40 GeV/c was applied he 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 se for t ctio ns are compared with Tevatron data [24][25][26] on J/ψ, D 0 , B, and Υ particle spectra at s = 1960 and 1800 GeV in the z-presentation.We ha verified the flavor inde-ve in thi at systematic measurements of the inclusi

z-Scaling
the basic ideas of the z-scaling approach pendence of including the inclusive top-quark measurements s 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 th ve 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 [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 x M x M m y   in the final state.oduced seco nsform 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: The pr ndary objects tra 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 1 Quantity Ω is proportional to the relative number of all co nstituent configurations in the inclusive reaction, which contain the configuration defined by fractions 1 2 , , ,

Scaling Variable z and Scaling Function Ψ(z) z z
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 m , and  is the maximal value of ( 2 as follows [15]: Here s is the square of the center-of-mass energy and J is  the corresponding Jacobian.The multiplicity density d d N  in (4) depends on the center-of-mass energy, ty, 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 as a probability densi e incl

Flavor Independence of Ψ(z) and
Th on means ty of the production of th usive particle with the corresponding value of variable z.

Self-Similarity of Top Production
e flavor independence of hadron producti that spectra of particles with a different flavor content can be described by universal scaling function [15,16].Our previous analysis is 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 i ing the heavy quarkonia, J/ψ [24], and Υ [26] F F (7) to compare the shape of scaling functi  which can be approximated by a constant.
In th oss section of the top-quark production measured by the DØ Collaboration at the Tevatron as a function of the transverse momentum p 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 .It preserves the normalization Equation (6) and ergy, angular and multiplicity independences of the z-presentation of particle spectra.As it is seen 5.
from Figure 1(b), the z-presentation of the top-quark transverse momentum distribution follows the shape of the z-scaling in pp ( pp ) collisions for other particles sufficiently well.Note t t the top-spectrum is in the limited kinematic region and the error bars of the data are large enough.We would like to stress that existing analyses were performed with p T -distributions of the inclusive cross sections 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 We ass hat the data on the top-qua ume t rk differential inclusive cross section over a wider range of p T and collision energy s at the Tevatron and LHC could be of interest to verify the flavor independence of z-scaling and self-similarity of top-quark production.

1  and 2  2 . 1 .
. The constituents of the incoming objects adrons 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 b y ) of the momentum of the object produced in the co tituent collision in the observed direction.Its fragmentation is characterized by parameter a  .The fragmentation in the recoil direction is describe by parameter b  and momentum fraction b y .Multiple interactions of the constituents are considd to be similar.This property reflects the self-similarity of the hadron interactions at the constituent level.Momentum Fractions x 1 , x 2 , y a and y b a binaryThe elementary sub-process is considered to be collision of the constituents with masses x 1 M 1 and x 2 M 2 resulting in the scattered and recoil objects with masses 2 , , a x x y , and b y which determine the underlying elemen -process The structure of colliding objects and fragmentation of tary sub .the systems formed in scattered and recoil directions are characterized by parameters 1 2 ,   , and , a b   , respectively.The parameters are relat ith the co sponding momentum fractions by function and b y .The Ω is interpreted as a relative volume which occupies these configurations in the space of the momentum fractions.Parameters 1 2 , taken as fractal dimensions in th e of the momentum fractions which correspond to the colliding objects and fragmentation processes, respectively.For the given values of1 2   , e parts of the spac   , and , and b y are determined in such a way to tion Ω, simultaneously fulfilling condition (1).In the maximize the func case of pp ( pp ) interactions we have  .It is assumed that on of the o ving in the scattered and recoil directions can be described by the same pawhich depends on the type of the inclusive e values of parameters particle.Th  and F  are determined according to the self-similar y requiments of experimental data in z-presentation.They were found to have constant values in pp and it re pp collisions at high energies.
) with(1).For he above inclusive reaction the quantity z is proportional to the transverse kinetic energy nditi T s of the constituent subprocess consumed for the uction of the inclusive prod particle and its counterpart with masses m 1 and m 2 , respectively.The quantity 0 d d ch N  is the corresponding multiplicity density of the central region of the reaction at the pseudo-rapidity 0 charged particles produced in   .Parameter c characterizes properties of the promedium.It is interpreted as "specific heat".The constant m N is taken to be a nucleon mass.
, measured at the Tevatron energies based nclud s = 1960 and 1800 GeV.The property of   z  observed at very small values of z was  10 −3 .In gion z the re  0.1 we observe a saturation of scaling function   z is paper we analyze the data [2] on the differential cr

Figure 1 (FFigure 1 .
Figure 1.The flavor independence of z-s g.The spectra calin of (a) J/ψ, D 0 , B, Υ mesons; and (b) top-quarks produced in pp collisions at the Tevatron in z-presentation.The specof π -mesons were measured in pp collisions at ISR.The data are taken from [2,24-27].The solid line in (b) is the same as in (a).

2 .
the width of the bins.The condition (6) is satisfied with the normalization This corresponds to two entries per event the total normalization to the with tt production cross sec- tion tt  = 8.31 pb[2].Th alues of the fr e v actal dimension  = 0.5 and "s e pecific heat" c = 0.25 are the same as us d in the previous analyses of the inclusive spectra [15,16,21].We have set 0 top   in the case of the top-quark since no energy los umed in the elementary s is ass tt production process.This choice corresponds to 1 a b y y   in the whole p T -range.The value of F  in rmation (7) is found to be top the transfo   0.004 No additional parame- ters were used.

Figure 2 .
Figure 2. The momentum fraction x 1 as a function of the scaled transverse momentum p T /m of hadrons produced in pp collisions at s = 1800 and 1960 GeV in the middle idity region.rap ith extra large energy dissipation in the final state ac-w companied by production of this particle.For a fixed value of T p m the fraction x 1 decreases while increasing collision energy s .The kinematic limit of the reaction 1 2 P P p    corresponds to 1 2 1 x x   at any co d for any type of ive particle.X n energy an the inclus fraction x 1 reveal similar dependences aled transverse momentum on the sc T p m as for the heavy mesons.