an>

Actually ,this work could attain added significance, had we been able to compute the average transverse momentum values based on Equation (1) and compare them with the values obtained by the somewhat empirical formula given by Equation (8). But due to gross uncertainty in the values of q (which would in all cases be entirely arbitrary) we keep ourselves limited to expression (8) alone for the calculational purposes of the average transverse momentum () values.

The work presented here is essentially based on some fundamental physical ideas: 1) the idea of large- scaling (or -scaling as was defined earlier), 2) the validity of the factorization hypothesis, 3) the constraint of exclusivity of reactions on inclusivity as was studied and pointed out by Brodsky et al. [2] who set the limit on the value of the exponent n, given by n ≤ 20.

3. Results

The results are presented here in graphical plots and the accompanying tables for the values of the used parameters. In Figures 1(a) and (b) the differential cross-sections for negative and positive pion, kaon and proton production cases in Pb + Pb collisions at SPS energies (at 20A GeV) are reproduced by the used theoretical framework. The figures in all the cases have been appropriately labeled and the parameters are shown in the table (Table 1). The plots in Figures 2(a) and (b) are for positive and negative pion, positive and negative kaon in Pb + Pb collisions at 30A GeV. The plots presented in Figure 2(c) for proton-antiproton production in Pb + Pb interaction at SPS energies, specifically at 17.3 GeV. The parameter values used to obtain the nature of fits for Figure 2 are shown in Table 2. The graphs in Figures 3 and 4 present the fits to the invariant cross-sections for the secondaries, and proton-antiproton produced in the Au + Au collisions at 19.6 GeV and the corresponding parameters are depicted in Table 3. Against the data points, the plots shown in Figure 5 by solid lines depict the fits based on the same model to the invariant cross-sections for various major varieties of secondaries produced in Pb + Pb collisions at 40A GeV, though data on production of were not available. The parameter values used are given in Table 4. In Figure 6, we demonstrate the fits to the invariant cross-sections for various major varieties of the secondaries produced in central (0% - 10%) Au + Au collisions at 19.6 GeV and the corresponding parameters are depicted in Table 5. The fits to the charge ratio-values of the

(a)(b)

Figure 1. Transverse mass spectra of π+, K+, P (left) and π, K (right) produced in central Pb + Pb collision at 20A GeV. The lines are fits of equation power law model. The statistical errors are smaller than the symbol size, for which no errors are shown in the figure. Parameter values are taken from Table 1. The experimental data are taken from References [3] and [4].

Table 1. Numerical values of the fit parameters of power law equation for Pb + Pb collisions (Reference Figure 1).

(a)(b)(c)

Figure 2. Transverse mass spectra of π+, K+ (left) and π, K (middle) and P, (right) produced in central Pb + Pb collision at 30A GeV and 17.3 GeV. The lines are fits of equation of power law model. The statistical errors are smaller than the symbol size, for which no errors are shown in the figure. Parameter values are taken from Table 2. The experimental data are taken from References [3] and [5].

Table 2. Numerical values of the fit parameters of power law equation for Pb + Pb collisions (Reference Figure 2).

(a)(b)(c)(d)

Figure 3. The transverse mass spectra of π+ (upper left), π (upper right) and K+ (lower left), K (lower right) from STAR experiment at 19.6 GeV in Au + Au collisions and the results of SPS experiments NA44, NA49, WA98 at 17.3 GeV in Pb + Pb collisions. The line is fit of power law model with all the STAR and SPS experiment. Parameter values are taken from Table 3. The experimental data are taken from Reference [6] and all errors are only of statistical nature.

(a)(b)

Figure 4. The transverse mass spectra of P (left) and (right) from STAR experiment at 19.6 GeV in Au + Au collisions and the results of SPS experiments NA44, NA49, WA98 at 17.3 GeV in Pb + Pb collisions. The line is fit of power law model with all the STAR and SPS experiment. Parameter values are taken from Table 3. The experimental data are taken from Reference [6]. The statistical errors are smaller than the symbol size, for which no errors are shown in the figure.

Table 3. Numerical values of the fit parameters for pion, kaon, proton and antiproton using power law model for Au + Au collisions at 19.6 GeV (Reference Figures 3 and 4).

Figure 5. Transverse mass spectra of K+, K, P, and π produced in central Pb + Pb collision at 40A GeV. The lines are fits of equation of power law model. The statistical errors are smaller than the symbol size, for which no errors are shown in the figure. Parameter values are taken from Table 4. The experimental data are taken from [4].

Figure 6. Transverse mass spectra of identified hadrons measured at midrapidity. The results at = 19.6 GeV for the production of π+, π, P, , K+ and K for 0% - 10% centrality in Au + Au collisions. The solid curves provide the power law model based results. Parameter values are taken from Table 5. The experimental data are taken from Reference [7].

Table 4. Numerical values of the fit parameters for negative pion, positive and negative kaon, proton and antiproton using power law model for Pb + Pb collisions at 40A GeV (Reference Figure 5).

Table 5. Numerical values of the fit parameters for pion, kaon, proton and antiproton using power law model for Au + Au collisions at 19.6 GeV for 0% - 10% centrality (Reference Figure 6).

pionic, kaonic and baryonic particles are drawn in Figure 7. These charge-ratio values offer a cross-check of the results arrived at for the invariant cross-sections. Tables 6 to 10 indicate the calculation of average transverse momentum in Pb + Pb collisions at 20A GeV, 30A GeV, 17.3 GeV, 19.6 GeV, 40A GeV respectively and Table 11 in Au + Au collisions at 19.6 GeV. At last, the plots shown in Figures 8(a) and (b) (Tables 12 and 13) show the nature of the average transverse momentum values, which form the core of this work and its central theme.

(a)(b)(c)

Figure 7., and ratios vs. for 0% - 10% centrality in Au + Au collisions at = 19.6 GeV. The solid curves provide the power law model based results. Data are taken from Reference [7]. All errors are only of statistical nature.

Table 6. Calculation of average transverse momentum for Pb + Pb collisions at = 20A GeV (Data are taken from Table 1).

Table 7. Calculation of average transverse momentum for Pb + Pb collisions at = 30A GeV (Data are taken from Table 2).

Table 8. Calculation of average transverse momentum for Pb + Pb collisions at = 17.3 GeV (Data are taken from Table 2).

Table 9. Calculation of average transverse momentum for Au + Au collisions at = 19.6 GeV (Data are taken from Table 3).

Table 10. Calculation of average transverse momentum for Pb + Pb collisions at = 40A GeV (Data are taken from Table 4).

Table 11. Calculation of average transverse momentum for Au + Au collisions at = 19.6 GeV for 0% - 10% centrality (Data are taken from Table 5).

(a)(b)

Figure 8. Average transverse momenta vs. c.m. energy plots. Parameter values are taken from Tables 12 and 13.

Table 12. Data for drawing graph vs. average transverse momentum () (Data are taken from Tables 6, 7 and 9) for (Reference Figure 8(a)).

Table 13. Data for drawing graph vs. average transverse momentum () (Data are taken from Tables 6, 7 and 9) for (Reference Figure 8(b)).

4. Discussion and Conclusions

By all indications, the results manifested in the measured data on the specific observables chosen here are broadly consistent with the power law model put into work here. This is modestly true of even the nature of charge-ratios which provide virtually a cross-check of the model utilised here. Of course, some comments on our modelbased plots, especially the plots on the charge-ratios are in order here. The lack of the predictivity of the used model is caused only by the circumstances, i.e. the lack of measured data at the successive and needed intervals; the problem can be remedied by supplying the necessary and reliable data from the arranged laboratory experiments at high-to-very high energies. However, the problem of constraining the parameters still remains. The other observations are: as is expected for two symmetric collisions of neighbouring values of mass numbers at very close energies do not reveal any significant differences with respect to the observables chosen by the experimental groups. This work demonstrates somewhat convincingly that the power law models can easily take care of data; so the notion of compartmentalisation between the possible applicability of the power law models and of the exponential models is only superficial and also questionable. Thus, the power law models which establish them as more general ones obtain a clear edge over the exponential models. Reliable data on various other related observables are necessary for definitive final conclusions. By all indications, the experimentally observed nature of -scaling is found to remain valid in the studied low- range of this paper.

Now follow a few comments on the nature of the average transverse momenta in nucleus-nucleus collisions: Firstly, the average values for the secondaries, especially for the pion secondaries, show very slow rising nature with center of mass energy. Secondly, the ranges of the averages -values do not differ any prominently between simple PP reactions and heavy nucleus-nucleus collisions at the similar ranges of energy. Thirdly, differences between the Au + Au reactions at 19.6 GeV and Pb + Pb interaction at 17.3 GeV, 20A GeV, 30A GeV, 40A GeV are quite insignificant. This is, in fact, the main content of the “universality” property of high energy interactions. Fourthly, the nature of average transverse momenta of the secondaries does not show any major difference between “soft” (small-pT) and “hard” (large-) reactions. Lastly and finally, that the values of are insensitive to the collisions-specifies like Au + Au or Pb + Pb at the neighboring energies is only natural. Such minor differences between “A”-values do hardly introduce any significant changes in the magnitudes of the average transverse momenta.

REFERENCES

  1. B. B. Back, “Consequences of Energy Conservation in Relativistic Heavy-Ion Collisions,” Physical Review C, Vol. 72, No. 6, 2005, Article ID: 064906. doi:101103/PhysRevC.72.064906
  2. S. J. Brodsky, H. J. Pirner and J. Raufeisen, “Scaling Properties of High pT Inclusive Hadron Production,” Physics Letters B, Vol. 637, No. 1-2, 2006, pp. 58-63. doi:10.1016/j.physletb.2006.03.079
  3. C. Alt, et al., “Pion and Kaon Production in Central Pb + Pb Collisions at A-20 and A-30 GeV: Evidence for the Onset of Deconfinement,” Physical Review C, Vol. 77, No. 2, 2008, Article ID: 024903. doi:10.1103/PhysRevC.77.024903
  4. C. Alt, et al., “Bose-Einstein Correlations of π-π-Pairs in Central Pb + Pb Collisions at A-20, A-30, A-40, A-80 and A-158 GeV,” Physical Review C, Vol. 77, No. 6, 2008, Article ID: 064908. doi:10.1103/PhysRevC.77.064908
  5. C. Alt, et al., “High Transverse Momentum Hadron Spectra at = 17.3 GeV in Pb + Pb and p + p Collisions,” Physical Review C, Vol. 77, No. 3, 2008, Article ID: 034906. doi:10.1103/PhysRevC.77.034906
  6. D. Cebra, “Charged Hadron Results from Au + Au at 19.6 GeV,” Invited Talk Presented at Quark Matter, 2008.
  7. R. Picha, “Charged Hadron Distribution in 19.6 GeV Au + Au Collision,” Ph.D. Thesis, University of California, Davis, 2005.

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