Structure and renormalizability of massive Yang-Mills field theories

We observe that the theory of the Yang-Mills field with bare mass is included in the framework of spontaneously broken gauge theories, except that the Higgs field is realized nonlinearly. With the nonlinear symmetry associated with the Higgs field taken into account, the $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ massive Yang-Mills theory and the associated ${\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}}_L\times {\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}}_R$ nonlinear $\sigma$ model are shown to be renormalizable and asymptotically free theories in two dimensions.