Analyses of $d{N}_{ch}/d\eta$ and $d{N}_{ch}/dy$ distributions of BRAHMS Collaboration by means of the Ornstein-Uhlenbeck process

Abstract

Interesting data on $d{N}_{\mathrm{ch}}/d\eta$ in Au-Au collisions ($\eta =-\ln \tan (\theta /2)$) with the centrality cuts have been reported by BRAHMS Collaboration. Using the total multiplicity $N_{\mathrm{ch}}=\int (d{N}_{\mathrm{ch}}/d\eta )d\eta$, we find that there are scaling phenomena among $(N_{\mathrm{ch}}{)}^{-1}d{N}_{\mathrm{ch}}/d\eta =dn/d\eta$ with different centrality cuts at $\sqrt{s_{NN}}=$ 130 GeV and 200 GeV, respectively. To explain these scaling behaviors of $dn/d\eta$, we consider the stochastic approach named the Ornstein-Uhlenbeck process with two sources. The following Fokker-Planck equation is adopted for the present analyses, $$\frac{\partial P(x,t)}{\partial t}=\gamma [\frac{\partial }{\partial x}x+\frac{1}{2}\frac{{\sigma }^2}{\gamma }\frac{{\partial }^2}{\partial x^2}]P(x,t)$$ where $x$ means the rapidity (y) or pseudo-rapidity ($\eta$). $t$, $\gamma$ and ${\sigma }^2$ are the evolution parameter, the frictional coefficient and the variance, respectively. Introducing a variable of $z_r=\eta /{\eta }_{\mathrm{rms}}$ (${\eta }_{\mathrm{rms}}=\sqrt{<{\eta }^2>}$) we explain the $dn/d{z}_r$ distributions in the present approach. Moreover, to explain the rapidity (y) distributions from $\eta$ distributions at 200 GeV, we have derived the formula as $$\frac{dn}{dy}=J^{-1}\frac{dn}{d\eta },$$ where $J^{-1}=\sqrt{M(1+{\sinh }^{2}y)}/\sqrt{1+M{\sinh }^{2}y}$ with $M=1+(m/p_{\mathrm{t}}{)}^2$. Their data of pion and all hadrons are fairly well explained by the O-U process. To compare our approach with another one, a phenomenological formula by Eskola et al. is also used in calculations of $dn/d\eta$.

12 pages, 8 figures, Latex2e