Hints on Triviality

Since we have results for both $v_{0}$ and $m_{\mathfrak{N}}$ as functions of $j_{0}$, it is conceivable that these results can give us some hints as to the question of triviality. As we shall see, trying to do this does in fact provide some rather crude hints, but most of all it puts in evidence the limitations of the calculational technique.

Our Gaussian-Perturbative result for the relation between $j_{0}$ and $v_{0}$ in either phase of the model, as shown in Equation (4), may be written for the purposes of the continuum limit as

$$j_{0} = v_{0} \left\{ \lambda v_{0}^{2} + \left[ \a... ... \lambda (\mathfrak{N}+2) \sigma_{0}^{2} \right] \right\}.$$

On the other hand, our result for the renormalized mass parameter $\alpha_{\mathfrak{N}}$, also valid in either phase of the model, in the presence of the external source, as shown in Equation (6), may be written for the purposes of the continuum limit as

$$\alpha_{\mathfrak{N}} = 3\lambda v_{0}^{2} + \left[ \alpha + \lambda (\mathfrak{N}+2) \sigma_{0}^{2} \right].$$

It follows that we may combine these two results, and write a relation involving the renormalized quantities $v_{0}$ and $\alpha_{\mathfrak{N}}$,

$$j_{0} = v_{0} \left( \alpha_{\mathfrak{N}} - 2\lambda v_{0}^{2} \right).$$

This relation is valid for all $d\geq 3$, for all $\mathfrak{N}\geq 1$, and all explicit references to $\alpha$ are gone. In the continuum limit both sides of this equation approach zero. In order to analyze the limit, it is necessary to rewrite everything in terms of the corresponding finite and possibly non-zero dimensionfull quantities. Doing this with the use of the scalings given in Section 2, Equation (1), we get

$$J_{0} = V_{0} \left( m_{\mathfrak{N}}^{2} - 2\lambda\, a^{d-4} V_{0}^{2} \right).$$

This behaves differently in each dimension $d$. We will analyze separately the cases $d=3$, $d=4$ and $d\geq 5$.