Let $\Phi: \mathbb{R} \to [0,1]$ define the standard normal cdf function. I am trying to find some functions $f$ and $g$ -- if they exist -- such that $$\Phi(x)f(y)=g(x+y)$$ for all $(x,y)\in \mathbb{R}^2$. I don't care about an explicit form for $g$, but would like to find some $f$ so that the above holds. 

Any hint regarding the existence of such $f$ and $g$ or the form of $f$ would be much appreciated!