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I want to understand when a given theory is renormalizable and how to find renormalizable theories for different dimensions (the latter will become clearer later on).

To do so, we work through an example: the following real scalar field theory in $d$ spacetime dimensions

\begin{equation*} \mathcal{L} = \frac 1 2 \partial_{\mu} \phi \partial^{\mu} \phi - \frac 1 2 m^2 \phi^2 - \frac{\lambda_3}{3!} \phi^3 - \frac{\lambda_4}{4!} \phi^4 - \frac{\lambda_5}{5!} \phi^5 - \frac{\lambda_6}{6!} \phi^6 \end{equation*}

Let us work with $3 \leq n \leq 6$. We first determine the coupling constants

\begin{align*} [\mathcal{L}] = d, \quad [\partial_{\mu}]=1, \quad [m] = 1,\quad \Rightarrow \quad& [\phi^n] = \frac 1 2 n(d-2), \quad [\lambda_n] = d- \frac 1 2 n(d-2) \\ &[\phi] = \frac 1 2 (d-2), \quad [\phi^2] = d-2 \end{align*}

I learned that a theory with $[\lambda_n] < 0$ is nonrenormalizable (from M. Srednicki's beautiful book, chapter 18). Hence, from $[\lambda_n] = d- \frac 1 2 n(d-2)$, we get the following condition

\begin{equation} [\lambda_n]<0 \iff n> \frac{2d}{d-2} \tag{*} \end{equation}

From $(*)$ we see that the renormalizability of the theory depends on the powers of $\phi$ and the dimension $d$. In particular, we see that our theory is renormalizable if we work with $d = 1$ and $d=3$. For $d > 3$ our theory becomes nonrenormalizable

Let me ask some questions

  • How to check whether our theory is renormalizable when $d=2$ or not? Our formula $n> \frac{2d}{d-2}$ breaks down for such a case.

  • Let's say we want to find renormalizable theories for higher dimensions. Based on $(*)$, for $d=4$ we see we would need to drop $\phi^5$ and $\phi^6$ terms. For $d=5,6$ we see we would need to drop $\phi^4$, $\phi^5$ and $\phi^6$. For $d=7,8,9,10,..,10^6,...$ (I did not check further than $d=10^6$ but it seems that the relation holds for further dimensions) we see we would need to drop $\phi^3$, $\phi^4$, $\phi^5$ and $\phi^6$. Does this mean that the simplest scalar field theory $\mathcal{L} = \frac 1 2 \partial_{\mu} \phi \partial^{\mu} \phi - \frac 1 2 m^2 \phi^2$ is renormalizable for all dimensions?

PS: Please note that I am aimed at understanding under which conditions is a theory renormalizable and not solving this one in particular.

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    $\begingroup$ $\frac{2d}{d-2}$ is monotonic decreasing so no need to check further than $d=10^6$ -- you've done more than enough. $\endgroup$ Apr 1, 2021 at 12:01

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Your statement (*) is valid only for $d > 2$ (you have to take care when dividing an inequality by a non-positive number). Based on the original formula for $[\lambda_n]$, renormalizability holds for all $n$ if $d \le 2$.

You are correct that the "simplest" (free) scalar field theory is renormalizable for all $d$.

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  • $\begingroup$ Oh so that means that we can only state that the sample theory is renormalizable for $d=3$. How could we check if our theory is renormalizable for $d=1,2$? $\endgroup$
    – JD_PM
    Apr 1, 2021 at 10:54
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    $\begingroup$ @JD_PM Just directly compute $[\lambda_n] = d- \frac 1 2 n(d-2)$ and check that it's non-negative. $\endgroup$
    – nanoman
    Apr 1, 2021 at 10:57

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