  We study the effect of a small fermion mass in the formulation of the
on-shell effective field theory (OSEFT). This is our starting point to derive
small mass corrections to the chiral kinetic theory. In the massless case, only
four Wigner functions are needed to describe positive and negative energy
fermions of left and right chirality, corresponding to the vectorial components
of a fermionic two-point Green's function. As soon as mass correction are
introduced, tensorial components are also needed, while the scalar components
strictly vanish in the OSEFT. The tensorial components are conveniently
parametrized in the so-called spin coherence function, which describe quantum
coherent mixtures of left-right and right-left chiral fermions, of either
positive or negative energy. We show that, up to second order in the energy
expansion, vectorial and tensorial components are decoupled, and obey the same
dispersion law and transport equation, depending on their respective chirality.
We study the mass modifications of the reparametrization invariance of the
OSEFT, and check that vector and tensorial components are related by the
associated symmetry transformations. We study how the macroscopic properties of
the system are described in terms of the whole set of Wigner functions, and
check that our framework allows to account for the mass modifications to the
chiral anomaly equation.
