The solution with constant $n_{\alpha}$ has a number of corrections which in the present approach can be studied in a systematic way. Ignoring them for them the moment we arrive at the universal scaling function which takes the following form:$$F_{\alpha}(\tau)=\frac{\gamma_{\alpha}b_{\alpha}}{2\kappa_{\alpha}^{2}}\left[
1+\frac{\tau^{1/(2+\lambda)}}{n_{\alpha}\,\kappa_{\alpha}}\right]
^{-n_{\alpha}}.\label{Falpha}%$$

The solution for $T_{\alpha}$ given by Eq.(\[Tsol\]) can be interpreted in terms of the saturation scale, $Q_{\text{s}}$ (\[Qsdef\]) which for $\tilde{m}_{\mathrm{T}}$–scaling takes the following form:$$Q_{\text{s}}(\tilde{m}_{\mathrm{T}})=Q_{0}\left(  \frac{\tilde{m}_{\mathrm{T}%
}}{W}\right)  ^{-\lambda/2}.$$ For quantities integrated over transverse momentum one introduces another saturation scale, $\bar{Q}_{\text{s}}$ which has a meaning of an average transverse momentum, or in this case average value of $\tilde{m}_{\mathrm{T}}%
$, and can be thought of as a solution of an equation [@Kharzeev:2004if]:$$\bar{Q}_{\text{s}}=Q_{\text{s}}(\bar{Q}_{\text{s}})$$ which gives$$\bar{Q}_{\text{s}}=Q_{0}\left(  \frac{W}{Q_{0}}\right)  ^{\lambda/(2+\lambda
)}. \label{aveQsat}$$ We see therefore that parameter $T_{\alpha}$, Tsallis temperature (\[Tsol\]), is proportional to the average saturation scale $\bar{Q}_{\text{s}}$ with proportionality constant $\kappa_{\alpha}$ which depends on particle species $\alpha$. Constants $\kappa_{\alpha}$ have been fitted to thermal distributions in Ref. [@McLerran:2013oju] and they are of the order of 0.1. Similar solution for the unidentified spectra has been discussed recently in Ref. [@Rybczynski:2012pn].

Constants $a_{\alpha},b_{\alpha}$ and $c_{\alpha}$ can be calculated analytically for $F_{\alpha}(\tau)$ given by Eq.(\[Falpha\]): $$\begin{aligned}
a_{\alpha}  & =\frac{\gamma_{\alpha}b_{\alpha}}{2 \kappa_{\alpha}^{2}}%
(n_{\alpha}\,\kappa_{\alpha})B(1,n_{\alpha}-1),\nonumber\\
b_{\alpha}  & =\frac{\gamma_{\alpha}b_{\alpha}}{ 2\kappa_{\alpha}^{2}}%
