= Solution
The <action> is dimensionless and $[d^4x]=-4$, so $[\mathcal L]=4$. A canonical <scalar> <kinetic term> then requires $2+2[\phi]=4$, and the electromagnetic <kinetic term> requires $2[F_{\mu\nu}]=4$. Thus
$$
\boxed{[\Phi]=[a]=[b]=1,\qquad[F_{\mu\nu}]=2.}
$$
The brackets here denote <mass dimension>. The <Lagrangian> has no explicit coordinate dependence, and all indices are contracted as <Lorentz scalars>. Its continuous spacetime <symmetry> is the <Poincaré group>. <Noether's theorem> gives conserved <four-momentum> from translations and conserved <angular momentum>, including the boost generators, from <Lorentz transformations>. The nonzero <mass> $M$ and dimensionful <couplings> prevent a dilation <symmetry>.
\b[Sign qualification:] the printed electromagnetic <kinetic term> has the opposite healthy relative sign to the <scalar> <kinetic terms>. With the stated mostly-minus <Minkowski metric>, a physical <Maxwell Lagrangian> is $-F_{\mu\nu}F^{\mu\nu}/4$. The printed plus sign instead gives <photon> <Hamiltonian density> $-(\mathbf E^2+\mathbf B^2)/2$; a nonzero purely electric configuration already has negative <energy>, unbounded below as its amplitude grows. The physical <decay widths> below use the healthy Maxwell sign with the given interaction normalization. The dimensional and spacetime-symmetry conclusions themselves do not depend on this sign correction.
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