I have the following model: $$y=\beta_{0}+\beta_{1}x_1+\beta_{2}x_2+\beta_{3}x_1x_2+u$$
where
$y$=dependent variable
$x_1$=standardized independent variable
$x_2$=dummy (binary) independent variable
$x_1x_2$=interaction term of $x_1$ and $x_2$.
The interpretation of standardized variable is a bit tricky. If we give (ceteris paribus) an additional one unit of standard deviation to $x_1$, $y$ will differ by $\beta_1$. What is the correct interpretation of the interaction term in this case? Does $\beta_3$ mean an additional unit or an additional unit of standard deviation (c.p.) affecting $y$?