Neutrino Magnetic Moment Behaviour in a Renormalizable Model

We have shown in hep-ph/9608266 that a neutrino magnetic moment form factor $\mu_{\nu} (q^2)$ could be considerably amplified at low momentum transfer,$q^2 \leq m_N^2$, at the cost of introducing an extra light neutral fermion $N$ with mass $m_N$ and nonzero magnetic moment. It was assumed that the magnetic moment of $N$ would originate in a renormalizable way at a heavy scale $M$. While the enhancement of the neutrino magnetic moment was unambiguous, we stressed that in this effective Lagrangian approach an uncertainty persisted about the behaviour of $\mu_{\nu} (q^2)$ in the interval $m_N^2 \ll q^2 \ll M^2$. This is not unexpected in presence of a nonrenormalizable effective theory (a particle with bare magnetic moment). We show in a simple renormalizable model for the magnetic moment of particle $N$ how a 2 loop calculation solves the ambiguity. In the domain $m_N^2 \ll q^2 \ll M^2$ we confirm the result obtained in hep-ph/9608266 using a sharp cut-off. It is amusing that the correct results are given, as expected, through dimensional regularization in the full theory, but not in the effective lagrangian approach.