Massive SLE${}_4$, massive CLE${}_4$ and the massive planar GFF

We construct a coupling between a massive GFF and a random curve in which the curve can be interpreted as the level line of the field and has the law of massive SLE${}_4$. This coupling is obtained by reweighting the law of the standard coupling GFF-SLE${}_4$ and our result can be seen as a conditional version of the path-integral formulation of the massive GFF. We then show that by reweighting the law of the coupling GFF-CLE${}_4$ in a similar way, one obtains a coupling between a massive GFF and a random countable collection of simple loops, that we call massive CLE${}_4$. Using this coupling, we relate massive CLE${}_4$ to the massive Brownian loop soup with intensity $1\mathrm{/}2$, thus proving a conjecture of Camia. As the law of the massive GFF, the laws of massive SLE${}_4$ and massive CLE${}_4$ are conformally covariant.