  We study the anomalous Hall effect (AHE) in tilted Weyl metals with weak
Gaussian disorder under the Kubo-Streda formalism in this work. To separate the
three different contributions, namely the intrinsic, side jump and skew
scattering contribution, it is usually considered necessary to go to the
eigenstate (chiral) basis of the Kubo-Streda formula. However, it is more
straight-forward to compute the total Hall current in the spin basis. For the
reason, we develop a systematic and transparent scheme to separate the three
different contributions in the spin basis for relativistic systems by building
a one-to-one correspondence between the Feynman diagrams of the different
mechanisms in the chiral basis and the products of the symmetric and
anti-symmetric part of the polarization operator in the spin basis. We obtain
the three contributions of the AHE in tilted Weyl metals by this scheme and
found that the side jump contribution exceeds both the intrinsic and skew
scattering contribution for the low-energy effective Hamiltonian. We compared
the anomalous Hall current obtained from our scheme with the results from the
semi-classical Boltzmann equation approach under the relaxation time
approximation and found that the results from the two approaches agree with
each other in the leading order of the tilting velocity.
