Renormalizable parametrization for the real and complex Grassmannian σ models: SO(N)SO(p)×SO(N−p) and SU(N)S(U(p)×U(N−p))

We present a nonstandard parametrization for the Grassmannian $\ensuremath{\sigma}$ models. It is first explained, for the real model with $p=1$ (the "old" $\ensuremath{\sigma}$ model), how the renormalizability proof can be worked out in this unusual parametrization of the fields. This analysis is shown to generalize successfully for the real model in the more difficult case $p\ensuremath{\ge}2$, a model for which Cremmer-Scherek-Macfarlane (CSM) parametrization fails to be renormalizable. The most remarkable aspect of the nonstandard parametrization used here is that it is also renormalizable for the complex model, but in this case the use of CSM parametrization is much simpler and should be preferred.