High order chiral Lagrangians with vector mesons in different approaches

The chiral Lagrangians with vector mesons are constructed in different approaches, including the next-to-leading order Lagrangian in the vector-field approach, the next-to-next-to-leading order Lagrangians in the tensor-field and the hidden local symmetry approaches. Some redundant terms are found at the next-to-leading order in the tensor-field and the hidden local symmetry approaches. The corresponding relationships between the next-to-next-to-leading order pseudoscalar mesonic low-energy constants and the ones in the hidden local symmetry approach are obtained at the tree level. The equivalence between the tensor-field approach and the hidden local symmetry approach is also discussed.