# Universal Scaling of Distribution of Particles in Relativistic Nuclear Collisions

###### Abstract

With the experimental data from STAR, PHENIX and BRAHMS on the centrality and rapidity dependence of the spectrum in Au+Au and d+Au collisions, we show that there exists a scaling distribution which is independent of the colliding system, centrality and rapidity. The parameter for the average transverse momentum increases from peripheral to central d+Au collisions. This increase accounts for the enhancement of particle production in those collisions. A non-extensive entropy is used to derive the scaling function.

###### pacs:

25.75.DwThe distributions of produced particles play very important role in the study of new state of matter produced in relativistic nuclear collisions. From the relationship between the distributions at different centralities and rapidities one can see whether there is a suppression or enhancement, thus dynamical mechanism for those particle production can be revealed. In earlier works hy1 , a scaling distribution for produced pions in Au+Au collisions in mid-rapidity region at RHIC energies was found that is independent of the centrality and colliding energy. Similar scaling behaviors have been found in zscal ; cgc . The existence of such scaling behaviors of the spectrum for a species of particle may be an indication of universal underlying particle production mechanism in the collisions. One may ask two questions now. (1) Can the scaling be extended to non-central rapidity region? (2) Can a scaling behavior similar to that shown in hy1 be found for d+Au collisions at different centralities and rapidities? If the answers to above two questions are yes, one should investigate the difference between the scaling functions for different colliding systems. In this paper, we look for the scaling behavior of particle distributions in Au+Au collisions in noncentral rapidity region, in d+Au collisions at different centralities and rapidities at RHIC energies. The same universal scaling behavior is found for both Au+Au and d+Au collisions at different centralities and rapidities. A possible origin for the universal scaling distribution is given from the principle of maximum non-extensive entropy.

Generally, the invariant particle distribution for a given species of particles depends on the collision centrality , (pseudo)rapidity (or ) and the transverse momentum . When we focus on the particle production in a collision with given centrality in a small rapidity region around , we should distinguish two different factors. One is the number density in rapidity, which tells us how many particles are produced in the region. The other is the probability for a produced particle having transverse momentum . It is this probability that dictates the shape of the transverse momentum distributions. In this paper, we are interested only in such a probability which can be defined as

(1) |

where satisfies the normalization condition is the multiplicity of particles produced in a unit rapidity region. So

(2) |

In the definition of , the centrality and rapidity dependence is written explicitly. Experimental data show similar shape of the distributions for different centralities and rapidities. Because of the similarity in the shape, those distributions of pions produced in Au+Au collisions in the mid-rapidity region at difference centralities have been put together in hy1 to the same curve by shrinking by a factor and shifting the distribution by another factor . Then a scaling variable is defined as , and the normalized distribution is obtained as a function . Here the average transverse momentum is defined as

(3) |

The scaling function reads

(4) |

The obtained is universal in the sense that it is independent of the centrality at mid-rapidity in Au+Au collisions.

Now we extend the discussion to the particle production in non-central rapidity region in Au+Au collisions at 200 GeV. For this purpose, we work on the experimental data exp1 for negatively charged particles at large with exactly the same procedures as in hy1 and try to put all the data points to the curve for the scaling function . It turns out that the scaling function given by Eq. (4) can also describe the charged particle production at 2.6 and 3.2 for different centralities, as can be seen in Fig. 1. The shrinking factor for is given in TABLE 1.

Fig.1 | Fig. 2 | Fig. 3 () | ||||
---|---|---|---|---|---|---|

(%) | (%) | PHENIX | (%) | |||

0-10 | 0.4340 | 0.3448 | 0-20 | 0.563 | 0-20 | 0.3228 |

10-20 | 0.4359 | 0.3452 | 20-40 | 0.572 | 30-50 | 0.3151 |

20-40 | 0.4363 | 0.3458 | 40-60 | 0.601 | 60-80 | 0.3108 |

40-60 | 0.4366 | 0.3462 | 60-88 | 0.601 |

The second observation in this paper is that the same scaling distribution can describe the distributions of charged particles produced in d+Au collisions exp2 and data exp3 in central rapidity region at different centralities, as shown in Fig. 2. The fitted values of parameter for the PHENIX data are also tabulated in TABLE I.

To investigate the rapidity dependence of the distribution in d+Au collisions, we take the data for and at for different centrality cuts hongy and try to put the data points to the obtained scaling curve with exactly the same procedure as mentioned above. The agreement is quite good, as shown in Fig. 3. The fitted values of the parameter are also given in TABLE I. Then we can conclude that the same scaling function can describe pion and charged particle production in both Au+Au and d+Au collisions at different centralities and rapidities. In this sense, we can say that the obtained scaling distribution is universal.

To show the degree of agreement between the experimental data and the fitted results, we define as the ratio of the data to the fitted results. For illustration, , calculated from results in Fig. 3, as a function of is shown in Fig.4. Obviously, values of are very close to 1, indicating that the scaling of the produced pion spectrum is true with high accuracy.

Once we have found a scaling law for the distributions for Au+Au and d+Au collisions for different centralities and rapidities, we need to dig out the physical implications behind the scaling behavior. From the scaling function , one can see that the only parameter characterizing the normalized distribution is the average transverse momentum which depends on the centrality, rapidity and the colliding energy and system. Once is known, both the soft part with low and the hard part with high are determined from . Traditionally it was assumed that the bulk of particles is produced by soft processes which determine . If the soft and hard processes can be separated, as used in the two-component model xnwang , and each part has different dynamical origin, it is extremely hard to imagine that they can be united into a single scaling function. Thus, we conclude that the hard process must contribute considerably to the average and is a coherent part of particle production. In fact, we have known from the study of particle correlations in central Au+Au collisions at RHIC energies that the lost energy from hard partons must heat the medium and increase the mean transverse momentum.

On the other hand, we have known from experiments that the particle production is suppressed, called jet quenching quench , in central Au+Au collisions because of the possible formation of dense hot medium but enhanced in d+Au collisions due to the Cronin effect cronin , possibly because of the broadening from multiple soft collisions before a hard one multi , or from the recombination of soft partons with those from showers recom . One can show that the enhanced production in d+Au collisions can be accounted for from our fitted results that the mean transverse momentum increases from peripheral to central collisions. We can define a new variable to measure the suppression or enhancement from the scaling function as

(5) |

An advantage of this definition of over the traditional is that it involves only the probability for a produced particle having certain transverse momentum and is independent of the total multiplicity. So the overall enhancement in the backward region () in d+Au collisions cannot be shown in . In calculating the value of we do not need to use the number of binary hard collisions, which is model dependent. is shown in Fig. 5 as a function of for d+Au collisions at mid-rapidity and it increases slowly to about 1.33. Thus the production of high pions is enhancement by about 30 percent. This is in agreement with experimental observations enhan . In this way the enhancement or suppression behavior can be shown clearly in a model independent way.

Finally, we investigate nucleon-nucleon collisions at GeV to see whether the particle spectrum can be described by the same scaling function. The experimental data on the distribution in the central rapidity region in p+p collisions can be parameterized hage as

(6) |

where and can be determined from fitting the data. It is astonishing that this parameterization for pion distribution in p+p collisions, with the parameters suitably chosen, can also be put to the scaling curve for by shrinking horizontally and shifting vertically. As shown in Fig. 6 the spectrum in p+p collisions can be fitted to the scaling function. Then we come to our final conclusion that the same scaling function can be used, for different colliding systems at different centralities and rapidities, to give the probability for a produced particle having once the parameter for the average transverse momentum is known, thus the scaling function is universal.

Since is universal, the centrality and rapidity dependence in is totally encoded in . Straightforwardly, we can have

(7) | |||||

Therefore, the ratio is a constant independent of the colliding system, the centrality, and rapidity. This consequence can be easily checked experimentally. For and 4, the ratio equals to 1.65, 4.08 and 14.39, respectively.

We are now in a position to derive an expression for the particle distribution from some statistical consideration. One method is from the principle of maximum entropy. The scaling function is not an exponential form, therefore the well-known Shannon entropy cannot be used for our purpose. Instead, as suggested in biro , a non-extensive entropy may be used, considering the complex nonlinear interactions and the possible presence of fractal chaotic behavior in nuclear collisions, to derive an expression for the particle distribution. The entropy we will use here is the Tsallis entropy tsi which reads for our case

(8) |

with parameter as a measure of the non-extensiveness of the system. The maximum entropy under conditions demanded by Eqs. (2) and (3) gives, after the same shrinking and shifting as above,

(9) |

with and . So there is one parameter to determine the shape of the distribution. The functional form of derived from the maximum entropy method is very different from that given by Eq. (4) but is the same as that used in parameterizing the p+p data. Then one can compare the distributions from our fitting procedure and from the maximum entropy. As shown in Fig. 6, they agree with each other extremely well in a very wide range of when . There in log scale one can hardly see any difference between them. Small difference can be seen if we show a ratio between them in a linear scale. As can be seen from the insert in Fig. 6 the largest difference is less than ten percent, though the distribution covers six orders of magnitudes in the shown range of . This agreement indicates that the produced particle system, including both the soft and hard particles, may be a non-extensive fractal system and the fractal property is independent the colliding system. The parameter characterizing the fractal property of the system is then , not too far from 1.

In summary, a scaling distribution for pions is found for Au+Au, d+Au and p+p collisions at RHIC energy. The scaling distribution is universal, since it does not depend on the colliding system, the centrality and rapidity. What characterizes the dynamical difference in the particle production is the value of the average transverse momentum . A simple prediction which can be checked experimentally is that is independent of the colliding system, the colliding centrality and rapidity. Possible statistical origin for the scaling distribution of pions is discussed from a non-extensive entropy. Different particles may have different scaling properties.

This work was supported in part by the National Natural Science Foundation of China under grant No. 10475032.

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