Effective lagrangian for axial anomaly and its applications in Dirac and Weyl semimetals

A gauge invariant effective lagrangian for the fermion axial anomaly is constructed. The dynamical degree of freedom for fermion field is preserved. Using the anomaly lagrangian, the scattering cross section of pair production γγ → e^-e⁺ in Dirac or Weyl semimetal is computed. The result is compared with the corresponding result from Dirac lagrangian. It is found that anomaly lagrangain and Dirac lagrangian exhibit the same E ⋅ B pattern, therefore the E ⋅ B signature may not serve a good indicator of the existence of axial anomaly. Because anomaly generates excessive right-handed electrons and positrons, pair production can give rise to spin current by applying gate voltage and charge current with depositing spin filters. These experiments are able to discern genuine anomaly phenomena.