  In his proof of the fundamental lemma, Ng\^o established the product formula
for the Hitchin fibration over the anisotropic locus. One expects this formula
over the larger generically regular semisimple locus, and we confirm this by
deducing the relevant vanishing statement for torsors over loop groups $R((t))$
from a general formula for $\mathrm{Pic}(R((t)))$. In the build up to the
product formula, we present general algebraization, approximation, and
invariance under Henselian pairs results for torsors, give short new proofs for
the Elkik approximation theorem and the Chevalley isomorphism $\mathfrak{g}//G
\cong \mathfrak{t}/W$, and improve results on the geometry of the Chevalley
morphism $\mathfrak{g} \rightarrow \mathfrak{g}//G$.
