  Weyl nodes in three-dimensional Weyl semimetals break the Liouville equation,
leading to the Liouville anomaly. Here we present a new approach to derive the
semiclassical action and equations of motion for Weyl fermions in the presence
of electromagnetic fields and torsions from the quantum field theory: combining
the Wigner transformation with band projection operation. It has been shown
that the Liouville anomaly, including a new pure torsion anomaly term, a mixing
term between the electromagnetic fields and torsions as well as the
conventional chiral anomaly, entirely differs from the counterpart of axial
gauge fields. We find various torsional responses and reproduce the chiral
vortical effect and the torsional chiral magnetic effect. A new torsion
modified anomalous Hall effect due to the mixing term in the Liouville anomaly
is predicted and its implementation is also discussed. Therefore, our work not
only provides new insights into the torsional responses for Weyl fermions but
also acts as a starting point to investigate their topological responses.
