Should moderator variables be always a multiplication of two main predictors? While studying moderation, I have always seen that a moderator is a multiplication of two main predictors. In the below example, $x_1*x_2$ is a moderator.
$Y=b_0 + b_1x_1 + b_2x_2 + b_3(x_1*x_2) + e$
I wonder if the moderator variables can have a form other than multiplication of two main predictors. If they can, showing some examples would be great.
 A: From Robins & Greenland (1992) and Pearl (2001), we also have the idea of natural direct effects ($NDE$) and natural indirect effects ($NIE$), which add (not multiply) to the total effect ($TE$). 
Using the variables you defined above — where Y is our outcome, $X_1$ is our (I'll assume binary) treatment, and $M$ is our mediator (for notational simplicity instead of $X_2$) — we have potential outcomes $Y(x_1,\ M(x_1))$. These are the outcomes we'd observe if both the treatment is $x_1$ ($0$ or $1$) and the mediator is what we'd observe when the treatment is $x_1$. So then we define
$$ TE = NIE + NDE,$$
with 
$$ TE = E\{Y(1,\ M(1))\} - E\{Y(0,\ M(0))\} $$
$$ NIE = E\{Y(1,\ M(1))\} - E\{Y(1,\ M(0))\} $$
$$ NDE = E\{Y(1,\ M(0))\} - E\{Y(0,\ M(0))\}. $$
How do we interpret these quantities? The total effect is the difference between the expected potential outcome when all subjects are assigned to treatment and the expected potential outcome when all subjects are assigned to control. The natural indirect effect is the portion of the total effect attributable to hypothetical changes in the value of the mediator (note that the potential outcomes in the second NIE expectation will be unobserved for all subjects). And the natural direct effect is the portion of the total effect attributable only to changes in treatment, holding the mediator constant at its value under the control.
On the other hand, the linear model that you mention (with multiplicative moderation) comes from Baron & Kenny (1986). Vansteelandt has some nice slides that explain these issues in more detail.
