  In order to extract maximal information about cosmology from the large-scale
structure of the Universe, one needs to use every bit of signal that can be
observed. Beyond the spatial distributions of astronomical objects, the spatial
correlations of tensor fields, such as galaxy spins and shapes, are ones of
promising sources that can be accessed in the era of large surveys in the near
future. The perturbation theory is a powerful tool to analytically describe the
behaviors and evolutions of correlation statistics on large scales for a given
cosmology. In this paper, we formulate a nonlinear perturbation theory of
tensor fields in general, based on the formulation of integrated perturbation
theory for the scalar-valued bias, generalizing it to include the tensor-valued
bias. To take advantage of rotational symmetry, the formalism is constructed on
the basis of the irreducible decomposition of tensors, identifying physical
variables which are invariant under the rotation of the coordinates system.
