Quartet-metric gravity, scalar-graviton dark substance and vacuum energy screening: extending GR vs. its WTDiff alternative

In the frameworks of the effective field theory of metric supplemented by some distinct dynamical coordinates parametrized, in turn, by a scalar quartet -- the so-called quartet-metric gravity -- the extension of tensor gravity through a massive scalar graviton in addition to the massless tensor one is consistently exposed. The field equations for the two realizations of such an extension originating from the classically equivalent prototype theories - General Relativity (GR) and its Weyl transverse (WTDiff) alternative - are derived and argued to be, generally, non-equivalent, with the pure-gravity case manifesting this explicitly in detail. A splitting of the cosmological constant onto the gravitating and non-gravitating parts, with a partial screening of the vacuum energy through an emergent scalar-graviton dark substance, is considered. A prior importance of treating the WTDiff gravity as a prototype one on par with GR, when looking for a putative next-to-GR extended theory of gravity with a scalar-graviton dark substance, is stressed.