Higher-order utiyama invariant interaction

In this paper we prove higher-order version of Utiyama invariant interaction of a particle and a gauge field (a principal connection on a principal bundle). To describe the Utiyama interaction in order r ≥ 2 we have to use an auxiliary gravitational field (a linear symmetric connection on the base manifold). We prove that any natural (invariant) operator of order s in the gravitational field, of order r in the gauge field and of order k in the particle field, s, r ≥ k − 1, s ≥ r − 2, with values in a (1, 0)-order G-gauge-natural bundle factorizes through the curvature tensors of both connections, the particle field and their covariant differentials up to sufficiently high orders. The covariant differentials are considered with respect to both connections.