Form of axion in DFSZ models with $U(1{)}_{PQ}$ symmetry

Form of axion state in the DFSZ model is determined by exacted diagonalizing of the mass mixing matrix of $CP$-odd sector. After applying $PQ$ mechanism to determine $PQ$ charges of particles of the model, these $PQ$ charges are used to define the general form of axion. Both of these two methods give consistent and reasonable physical state of axion. The constraint of $PQ$ charge in $DFSZ-I$ model is also pointed out. Then, anomalous couplings of axion-photon-photon is re-examined. Triple-couplings of axion with two fermions are studied. The decay of and axion into a pair of photon is also considered via re-normalizable interactions arising from kinetic terms of scalar fields.