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We present a detailed measurement of charged two-pion correlation functionsin 0%-30% centrality $\sqrt{s_{_{NN}}}=200$ GeV Au$+$Au collisions by thePHENIX experiment at the Relativistic Heavy Ion Collider. The data are welldescribed by Bose-Einstein correlation functions stemming from L\'evy-stablesource distributions. Using a fine transverse momentum binning, we extract thecorrelation strength parameter $\lambda$, the L\'evy index of stability$\alpha$ and the L\'evy length scale parameter $R$ as a function of averagetransverse mass of the pair $m_T$. We find that the positively and thenegatively charged pion pairs yield consistent results, and their correlationfunctions are represented, within uncertainties, by the same L\'evy-stablesource functions. The $\lambda(m_T)$ measurements indicate a decrease of thestrength of the correlations at low $m_T$. The L\'evy length scale parameter$R(m_T)$ decreases with increasing $m_T$, following a hydrodynamicallypredicted type of scaling behavior. The values of the L\'evy index of stability$\alpha$ are found to be significantly lower than the Gaussian case of$\alpha=2$, but also significantly larger than the conjectured value that maycharacterize the critical point of a second-order quark-hadron phasetransition.

L\'evy-stable two-pion Bose-Einstein correlations in $\sqrt{s_{_{NN}}}=200$ GeV Au$+$Au collisions