I am trying to understand the Atiyah–Singer index formula for pseudo-differential operators. As far as I understood, the Fredholm index of the operator $A$ on a manifold can be computed just from the knowledge of its associated principal symbol $\sigma_p(A)$, at least for elliptic operators. However, I am interested in non-local operator of the form
$$ A(f)(x)= \int_{\mathbb R} \tilde A(x-x') f(x')\,\mathrm d x' $$ with $f$ in the Schwartz space.

Suppose $B$ acts on $\mathcal C^{\infty}(\mathbb R,\mathbb R)$ as: $$ B(f)(x)= \int_{\mathbb R} e^{-(x-x')^2/2} f(x')\,\mathrm d x' $$ and has symbol $$ \sigma(B)(x,\xi) = e^{-\xi^2/2},\quad(x,\xi)\in\mathbb R^2 $$ (notice, no dependence on $x$).

$\sigma(B)$ is a Hörmander symbol in the class $\mathcal S^{m}_{1,0}$, $\forall\, m\in \mathbb R$.

Question: Does $\sigma_p(B)$ exist? Is it a generic feature of non-local operator not to have a principal symbol? Can the Atiyah–Singer index formula be applied to such operators?

N.B. For the notion of Hörmander symbol one can look at the Wikipedia page on pseudo-differential operators.