So suppose I have a random variable $X$ which is $\sigma$-sub-gaussian, i.e. $\log \mathbb Ee^{\lambda X} \leq \lambda^2 \sigma^2/2$. We also know such a r.v. has a norm associated with it called the sub-gaussian norm defined as:
$$ \|X\|_{\psi_2} = \inf\left\{t>0:\mathbb{E}\exp\left( \frac{X^2}{t^2}\right)\le2\right\}. $$
My question is, how do we convert between $\sigma \Longleftrightarrow \|X\|_{\psi_2}$ without the use of universal constants? I want to move back and forth between these terms without losing any information.