If $\alpha = \lim _ { x \rightarrow 0 ^ { + } } \left( \frac { \mathrm { e } ^ { \sqrt { \tan x } } - \mathrm { e } ^ { \sqrt { x } } } { \sqrt { \tan x } - \sqrt { x } } \right)$ and $\beta = \lim _ { x \rightarrow 0 } ( 1 + \sin x ) ^ { \frac { 1 } { 2 } \cot x }$ are the roots of the quadratic equation $a x ^ { 2 } + b x - \sqrt { \mathrm { e } } = 0$, then $12 \log _ { \mathrm { e } } ( \mathrm { a } + \mathrm { b } )$ is equal to $\_\_\_\_$