Mass Renormlization in the Nelson Model

TL;DR

This paper rigorously analyzes the asymptotic behavior of the effective mass in the Nelson model, showing specific bounds and limits for the coefficients in its perturbative expansion as the ultraviolet cutoff grows large.

Contribution

It provides a rigorous mathematical proof of the asymptotic behavior of the effective mass coefficients in the Nelson model, confirming and refining physical folklore.

Findings

The first coefficient's limit is positive and finite as or large rac;

The second coefficient's growth is logarithmic in or large rac;

Established bounds for the asymptotic limits of the coefficients.

Abstract

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 $C_2$.