Extensions by ${\mathbf{K}}_2$ and factorization line bundles

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 ${\mathrm{Gr}}_G$. Our result affirms a conjecture of D. Gaitsgory and S. Lysenko and classifies factorization line bundles on ${\mathrm{Gr}}_G$.