We investigate the effect of strong electronic correlation on the massless Dirac fermion system, $alpha$-(BEDT-TTF)$_2$I$_3$, under pressure. In this organic salt, one can control the electronic correlation by changing pressure and access the quantum critical point between the massless Dirac fermion phase and the charge ordering phase. We theoretically study the electronic structure of this system by applying the slave-rotor theory and find that the Fermi velocity decreases without creating a mass gap upon approaching the quantum critical point from the massless Dirac fermion phase. We show that the pressure-dependence of the Fermi velocity is in good quantitative agreement with the results of the experiment where the Fermi velocity is determined by the analysis of the Shubnikov-de Haas oscillations in the doped samples. Our result implies that the massless Dirac fermion system exhibits a quantum phase transition without creating a mass gap even in the presence of strong electronic correlations.
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