Mass Renormlization in the Nelson Model

The asymptotic behavior of the effective mass $m_{\rm eff}(\Lambda)$ of the so-called Nelson model in quantum field theory is considered, where $\Lambda$ is an ultraviolet cutoff parameter of the model. Let $m$ be the bare mass of the model. It is shown that for sufficiently small coupling constant $|\alpha|$ of the model, $m_{{\rm eff}}(\Lambda)/m$ can be expanded as $m_{{\rm eff}}(\Lambda)/m= 1+\sum_{n=1}^\infty a_n(\Lambda) \alpha^{2n}$. A physical folklore is that $a_n(\Lambda)\sim [\log \Lambda]^{(n-1)}$ as $\Lambda\to \infty$. It is rigorously shown that $$0<\lim_{\Lambda\to\infty}a_1(\Lambda)<C,\quad C_1\leq \lim_{\Lambda\to\infty}a_2(\Lambda)/\log\Lambda\leq C_2$$ with some constants $C$, $C_1$ and $C_2$.