Let $ \mathcal{M}$ be the moduli stacks of vector bundles of degree $ 0$ and rank $ n$ over smooth curve $ X$ , $ \mathcal{M}_{\mathcal{O}}$ be the moduli stack of vector bundles with fixed determinant $ \mathcal{O}$ and $ J$ be the stacks of line bundles of degree $ 0$ on X. There is a etale morphism $ f:\mathcal{M}_{\mathcal{O}}\times J \rightarrow \mathcal{M}$ . We denote by $ \theta_o$ , $ \theta$ the theta bundles (determiant of cohomology of universal bundles) of $ \mathcal{M}_{\mathcal{O}}$ and $ \mathcal{M}$ respectively. Using the etale morphism it follows that $ Pic~ \mathcal{M}\subset Pic~(\mathcal{M}_{\mathcal{O}}\times J)=\mathbb{Z}\cdot \theta_o \oplus Pic ~J$ .

Note: Since we are using stacks, the universal bundle is uniquely defined (not like what happens in moduli schemes).

My question: We should be able to write the line bundle $ \theta$ in terms of $ \theta_0$ and some line bundle on $ J$ . What is this expression? Is it known?