The change in a call portfolio ($f$), derived from Itô's Lemma, is: \begin{align*} \left( \frac{\partial f}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2}\right)\mathrm{d}t &=r\left( f-rS\frac{\partial f}{\partial S}\right)\mathrm{d} t, \\ \implies\frac{\partial f}{\partial t}+rS\frac{\partial f}{\partial S}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 f}{\partial S^2} -rf&=0 \end{align*}
where $\frac{\partial f}{\partial t}$ denotes theta, $\frac{\partial f}{\partial S}$ denotes delta and $\frac{\partial^2 f}{\partial S^2}$ denotes gamma.
So gamma's PnL would be $\frac{1}{2}\Gamma \sigma^2 \mathrm{d}S^2$, where $\mathrm{d}S^2$ is the underlying price's change.
But why is gamma's PnL in reallity $\frac{1}{2}\Gamma \mathrm{d}S^2$, and not the previous formula? Why shouldn't volatility be included gamma's PnL?