  We consider an $f(Q,T)$ type gravity model in which the scalar non-metricity
$Q_{\alpha \mu \nu}$ of the space-time is expressed in its standard Weyl form,
and it is fully determined by a vector field $w_{\mu}$. The field equations of
the theory are obtained under the assumption of the vanishing of the total
scalar curvature, a condition which is added into the gravitational action via
a Lagrange multiplier. The gravitational field equations are obtained from a
variational principle, and they explicitly depend on the scalar nonmetricity
and on the Lagrange multiplier. The covariant divergence of the matter
energy-momentum tensor is also determined, and it follows that the
nonmetricity-matter coupling leads to the nonconservation of the energy and
momentum. The energy and momentum balance equations are explicitly calculated,
and the expressions of the energy source term and of the extra force are found.
We investigate the cosmological implications of the theory, and we obtain the
cosmological evolution equations for a flat, homogeneous and isotropic
geometry, which generalize the Friedmann equations of standard general
relativity. We consider several cosmological models by imposing some simple
functional forms of the function $f(Q,T)$, and we compare the predictions of
the theory with the standard $\Lambda$CDM model.
