# O(N) sigma model renormalization

Does anyone know, is a model with lagrangian $\mathcal{L} = \frac{(\partial_{\mu}\phi_a)^2}{2}-\frac{m^2 \phi_a^2}{2}-\frac{\lambda}{8N}(\phi_a \, \phi_a)^2$ renormalizable? I'm using BPHZ scheme and everything is OK in one loop. But it seems to me (may be I'm mistaken) that the scheme breaks down even in two loops. I will be grateful for links on books or lectures where such a model is considered.

• Renormalizability depends on space-time dimension. If $d\leq 4$, you're fine. – Adam Jun 3 '14 at 16:15
• Yes, I'm considering this model in d=4 space-time. Couldn't you please tell me where can I read more about this model? – user43283 Jun 3 '14 at 16:17
• Have a look at Zinn-Justin's book, you'll find everything you want about the O(N) model... – Adam Jun 3 '14 at 17:38
• Thanks a lot for advicing this book, but as far as I understood, Zinn-Justin uses non-perturbative approach to this problem. Do you know any article or lecture where this model is considered in perturbative way? I just need to check my calculations. Anyway, thanks for Zinn-Justin :) – user43283 Jun 3 '14 at 19:45
• In "Quantum Field Theory and Critical Phenomena", he does everything ;-) Chapter 11 might be what you're looking for. – Adam Jun 3 '14 at 19:50