Chiral Effective Lagrangian With Vector Mesons

Chiral perturbation theory uses the chiral symmetry produced by the chiral limit of light quark mass to zero to study low energy hadronic physics.Chiral perturbation theory first needs to construct chiral Lagrangian.In this paper,three commonly used representations,vector-field,tensor-field and hidden local symmetry representation,are considered.When the vector meson is regarded as a resonant state to study the properties of the pseudoscalar meson,the low-order Lagrangian of the vector meson is enough.At present,the low-order Lagrangian of the vector meson under three representations has been obtained.However,higher-order Lagrangian is needed to more accurately study the self-properties of the vector meson such as vector meson mass and radiative decay.Different representations of vector mesons are suitable for dealing with different problems,and the study of equivalence is helpful to further understand the relationship between these different problems.Therefore,in this paper,the construction of the higher order Lagrangian of vector meson and the strong equivalence between tensor-field and hidden local symmetry representation are discussed.In the vector-field and tensor-field representation,the equation of motion of the vector field is combined with the pseudo-scalar meson constraint condi-tion,and the chiral Lagrangian is constructed to the next-to-leading order and the next-to-next-to-leading order respectively.At the next-to-leading order of the tensor-field representation,it is found that there are four and three re-dundant terms in the results of normal and anormalous terms in the literature respectively.In the hidden local symmetry representation,in the next-to-next-to-leading order,it is found that there is one redundant term in the literature,and the chiral Lagrangian is constructed to the secondary leading order by in-troducing new building blocks and constraint condition.In order to facilitate the study of the properties of pseudoscalar mesons and the weak equivalence of vector mesons,the vector field is integrated and the corresponding relationship-s 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.Finally,by adding auxiliary fields,the strong equivalence of vector mesons between tensor-field and hidden local symmetry is discussed,and it is found that in order to discuss the complete strong equivalence of the two representations,all number of vector meson vertices should be included.