A Lagrangian relationship between the PST and the Sen formulations of chiral forms

In this paper, we show the equivalence between the PST and Sen formulations for chiral forms at the Lagrangian level. This is by discussing how an action in the Sen formulation for chiral forms can be separated into two PST actions, one with the correct sign the other has the wrong sign, reflecting the feature of the Sen formulation that it contains two chiral form fields, one physical the other unphysical, decoupled from each other. The key idea is to add extra terms to the Sen action such that the modified action contains a PST scalar and is also equivalent with the original one. Then after eliminating a self-dual field $Q$ and making appropriate field redefinitions, we obtain a sum of two PST actions. We also consider alternative actions in which extra terms are added to the Sen action to either eliminate the physical or unphysical degrees of freedom and such that the resulting action is identified with the PST action either with correct or incorrect sign. Generalisation to non-linear theories coupling to external source is discussed. The realisation of the Sen M5-brane action as containing the PST M5-brane action is also explicitly shown.