We explore the physical properties of a unified microscopic theory for the coexistence of superconductivity and charge density waves in two-dimensional transition metal dichalcogenides. In the case of particle-hole symmetry the elementary particles are Dirac fermions at the nodes of the charge density wave gap. When particle-hole symmetry is broken electron (hole) pockets are formed around the Fermi surface. The superconducting ground state emerges from the pairing of nodal quasi-particles mediated by acoustic phonons via a piezoelectric coupling. We calculate several properties in the s-wave superconducting phase, including specific heat, ultra-sound absorption, nuclear magnetic relaxation, thermal, and optical conductivities. In the case with particle-hole symmetry, the specific heat jump at the transition deviates strongly from ordinary superconductors. The nuclear magnetic response shows an anomalous anisotropy due to the broken time-reversal symmetry of the superconducting gap, induced by the triple charge density wave state. The loss of lattice inversion symmetry in the charge density wave phase leads to anomalous coherence factors in the optical conductivity and to the appearance of an absorption edge at the optical gap energy. Furthermore, optical and thermal conductivities display anomalous peaks in the infrared when particle-hole symmetry is broken.
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