  Recently it was shown that an impurity exchanging orbital angular momentum
with a surrounding bath can be described in terms of the angulon quasiparticle
[Phys. Rev. Lett. 118, 095301 (2017)]. The angulon consists of a quantum rotor
dressed by a many-particle field of boson excitations, and can be formed out
of, for example, a molecule or a nonspherical atom in superfluid helium, or out
of an electron coupled to lattice phonons or a Bose condensate. Here we develop
an approach to the angulon based on the path-integral formalism, which sets the
ground for a systematic, perturbative treatment of the angulon problem. The
resulting perturbation series can be interpreted in terms of Feynman diagrams,
from which, in turn, one can derive a set of diagrammatic rules. These rules
extend the machinery of the graphical theory of angular momentum - well known
from theoretical atomic spectroscopy - to the case where an environment with an
infinite number of degrees of freedom is present. In particular, we show that
each diagram can be interpreted as a 'skeleton', which enforces angular
momentum conservation, dressed by an additional many-body contribution. This
connection between the angulon theory and the graphical theory of angular
momentum is particularly important as it allows to systematically and
substantially simplify the analytical representation of each diagram. In order
to exemplify the technique, we calculate the 1- and 2-loop contributions to the
angulon self-energy, the spectral function, and the quasiparticle weight. The
diagrammatic theory we develop paves the way to investigate next-to-leading
order quantities in a more compact way compared to the variational approaches.
