  Let $X$ be a smooth, geometrically connected curve over a perfect field $k$.
Given a connected, reductive group $G$, we prove that central extensions of $G$
by the sheaf $\mathbf K_2$ on the big Zariski site of $X$, studied by J.-L.
Brylinski and P. Deligne, are equivalent to factorization line bundles on the
Beilinson-Drinfeld affine Grassmannian $\operatorname{Gr}_G$. Our result
affirms a conjecture of D. Gaitsgory and S. Lysenko and classifies
factorization line bundles on $\operatorname{Gr}_G$.
