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Quasiscaling in the analysis of the yield ratio $π^-/π^+$: 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, $|ψ_r(r)|^2$, times pion source function $ρ(r)$ (of the size $β$), $\intdr |ψ_r(r)|^2 ρ(r)$, shows a quasiscaling behavior. This is approximately invariant under the following transformation: $(β,Z_{eff}) \to (λβ,λZ_{eff})$; $λ>0$. We called such behavior $β-Z_{eff}$ 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 $0.024 \le n_{B}\le0.036$ [1/fm^3] is obtained.