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Is the Zee model neutrino mass matrix ruled out?

A very economic model of generating small neutrino masses is the Zee model. This model has been studied extensively in the literature with most of the studies concentrated on the simplest version of the model where all diagonal entries in the mass matrix are zero. SNO, and KamLAND data disfavor this simple version, but only when one also combines information from atmospheric and K2K data, can one rule out this model with high confidence level. We show that the simplest version of Zee model is ruled out at $3σ$ level. The original Zee model, however, contains more than enough freedom to satisfy constraints from data. We propose a new form of mass matrix by naturalness consideration. This new form of mass matrix predicts that $m_{ν_3} = 0$, and $\tan^2θ_{solar}$ increases with $|V_{e3}|$. For the best fit value of $\tan^2θ_{solar}$, $|V_{e3}|$ is sizeable but below the upper bound.