Analyses of $dN_{ch}/dη$ and $dN_{ch}/dy$ distributions of BRAHMS Collaboration by means of the Ornstein-Uhlenbeck process

Abstract

Interesting data on $dN_{\rm ch}/dη$ in Au-Au collisions ($η=-\ln \tan (θ/2)$) with the centrality cuts have been reported by BRAHMS Collaboration. Using the total multiplicity $N_{\rm ch} = \int (dN_{\rm ch}/dη)dη$, we find that there are scaling phenomena among $(N_{\rm ch})^{-1}dN_{\rm ch}/dη= dn/dη$ with different centrality cuts at $\sqrt{s_{NN}} =$ 130 GeV and 200 GeV, respectively. To explain these scaling behaviors of $dn/dη$, we consider the stochastic approach named the Ornstein-Uhlenbeck process with two sources. The following Fokker-Planck equation is adopted for the present analyses, $$ \frac{\partial P(x,t)}{\partial t} = γ[\frac{\partial}{\partial x}x + \frac 12\frac{σ^2}γ\frac{\partial^2}{\partial x^2}] P(x, t) $$ where $x$ means the rapidity (y) or pseudo-rapidity ($η$). $t$, $γ$ and $σ^2$ are the evolution parameter, the frictional coefficient and the variance, respectively. Introducing a variable of $z_r = η/η_{\rm rms}$ ($η_{\rm rms}=\sqrt{< η^2 >}$) we explain the $dn/d z_r$ distributions in the present approach. Moreover, to explain the rapidity (y) distributions from $η$ distributions at 200 GeV, we have derived the formula as $$ \frac{dn}{dy}=J^{-1}\frac{dn}{d η}, $$ where $J^{-1}=\sqrt{M(1+\sinh^2 y)}/\sqrt{1+M\sinh^2 y}$ with $M = 1 + (m/p_{\rm t})^2$. 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 $dn/dη$.