Scale-invariant supergravity theory in component formulation

We formulate $N=1$ supergravity theory in four-dimensions with local scale invariance in semi-on-shell component formulation in four dimensions. The algebra we adopt has the generators $(P_m,M_{mn},Q_{\alpha },S)$ consisting of the generators in super-Poincaré algebra: $(P_m,M_{mn},Q_{\alpha })$ with $4+6+4$ components, and that of local scale-transformation $S$ with one additional component. Our field content consists of the on-shell multiplet of supergravity $(e_{\mu }^m,\psi _{\mu }^{\alpha })$, the tensor multiplet $(B_{\mu \nu },{\chi }_{\alpha },\phi )$ and the 'Weylon multiplet' $(S_{\mu },{\rho }_{\alpha },C_{\mu \nu \rho })$, where $S_{\mu }$ is the Weylon that gauges local scale symmetry, ${\psi }_{\mu },\chi$ and ρ are Majorana spinors, while $C_{\mu \nu \rho }$ is a tensor auxiliary field. Similar to the non-supersymmetric case that has no minimal coupling between the $S_{\mu }\text{-}$field and spin 1/2 field, the Weylon does not minimally couple to spin 3/2 gravitino field in our locally supersymmetric generalization. By going to the Einstein frame from the original Jordan frame, we show that our system is equivalent to a system with the gauged dilaton-shift symmetry. This implies that our system bypasses persistent dilaton-related problems, avoiding a long-distance force for a massless dilaton, a minimum value of its potential, or a run-away vacuum.