Quasiscaling in the analysis of the yield ratio ${\pi }^{-}\mathrm{/}{\pi }^{+}$: Mathematical structure and estimation of source size

Recently we have found that integral of the squared Coulomb wave function describing systemcomposed of charged pion and central charged fragment $Z_{eff}$ protons, $\mathrm{\mid }{\psi }_r(r){\mathrm{\mid }}^2$, times pion source function $\rho (r)$ (of the size $\beta$), $\text{intdr}\mathrm{\mid }{\psi }_r(r){\mathrm{\mid }}^2\rho (r)$, shows a quasiscaling behavior. This is approximately invariant under the following transformation: $(\beta ,Z_{eff})\to (\lambda \beta ,\lambda Z_{eff})$; $\lambda >0$. We called such behavior $\beta -Z_{eff}$ quasiscaling. We examine this quasiscaling behavior in detail. In particular we provide a semi-analytical examination of this behavior and confirm it for the exponential pionic source functions in addition to the Gaussian ones and for the production of K mesons as well. When combined with the results of the HBT, a result of the yield ratio allows us to estimate the size of the central charged fragment (CCF) to be $125\le Z_{eff}\le 150$ for Pb+Pb collisions at energy 158 GeV/nucleon. From our estimation, the baryon number density $0.024\le n_B\le 0.036$ [1/fm^3] is obtained.