Convolution of Gaussian with function/distribution to "half-Gaussian"

I am looking for a function or Schwarz distribution $gamma$, so that for an $x = (x_1,…,x_m,…,x_n) in mathbb R^n$, which consists out of concatenation of $x_{mathrm{a}=(x_1,…,x_m)$ and $x_{mathrm{b}=(x_{m+1},…,x_n)$

begin{align}
x in mathbb R^n
g(x) = frac1{sqrt(2pi)^ncdot|Sigma|} cdot exp(-0.5dot x^T Sigma^{-1} x )
zeta(x) = (g triangleq gamma)(x) = int_{-infty}^{infty} cdots int_{-infty}^{infty} g(t) cdot gamma(t-x) d t
zeta(x) propto exp(-0.5dot (x_{mathrm{a}})^T Sigma^{-1} x_a )
1 = int_{-infty}^{infty} cdots int_{-infty}^{infty} zeta(x) dx
begin{end}