  We study scalar-tensor theories respecting the projective invariance in the
metric-affine formalism. The metric-affine formalism is a formulation of
gravitational theories such that the metric and the connection are independent
variables in the first place. In this formalism, the Einstein-Hilbert action
has an additional invariance, called the projective invariance, under a shift
of the connection. Respecting this invariance for the construction of the
scalar-tensor theories, we find that the Galileon terms in curved spacetime are
uniquely specified at least up to quartic order which does not coincide with
either the covariant Galileon or the covariantized Galileon. We also find an
action in the metric-affine formalism which is equivalent to class ${}^2$N-I/Ia
of the quadratic degenerated higher order scalar-tensor (DHOST) theory. The
structure of DHOST would become clear in the metric-affine formalism since the
equivalent action is just linear in the generalized Galileon terms and
non-minimal couplings to the Ricci scalar and the Einstein tensor with
independent coefficients. The fine-tuned structure of DHOST is obtained by
integrating out the connection. In these theories, non-minimal couplings
between fermionic fields and the scalar field may be predicted. We discuss
possible extensions which could involve theories beyond DHOST.

General Relativity and Quantum Cosmology;High Energy Physics - Theory