# Renormalizability conditions for a real scalar field in d dimensions

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• JD_PM
In summary, the conversation discusses the real scalar field theory in ##d## spacetime dimensions, as presented in M. Srednicki QFT's draft book, chapter 18. The Lagrangian for this theory includes polynomial interactions of degree less than or equal to 6, and the dimensions of various components are determined. The concept of superficial degree of divergence is introduced, which is defined as the powers of momentum in the numerator minus the powers of momentum in the denominator. The formula for calculating the superficial degree of divergence is given, and it is explained that a theory is non-renormalizable if any coefficient of any term in the Lagrangian has negative mass dimension. The maximal value of ##d## for which this theory is
JD_PM
TL;DR Summary
I want to understand how the ##n## powers of ##\phi## are limited (in a renormalized theory) depending on the ##d## number of dimensions we consider as well as how to "render" our theory renormalizable for higher dimensions
I am studying the real scalar field theory in ##d## spacetime dimensions as beautifully presented by M. Srednicki QFT's draft book, chapter 18 (actually, for the sake of simplicity, let us include polynomial interactions of degree less than or equal to 6 only)

\begin{equation*}
\mathcal{L} = −\frac 1 2 Z_{\phi}\partial_{\mu} \phi \partial^{\mu} \phi - \frac 1 2 Z_m m^2 \phi^2 - \sum_{n \geq 3}^6 Z_n \frac{\lambda_n}{n!} \phi^n
\end{equation*}

We see that

\begin{equation*}
\end{equation*}

Where we noted that ##[\phi^n] = \frac 1 2 n (d-2)##.

In order to have the whole picture, we need to work with the so called superficial degree of divergence, which is defined as the powers of momentum in the numerator minus the powers of momentum in the denominator. If ##D \geq 0## the diagram diverges, otherwise it does not. As an example, take the Saturn diagram for ##\phi^4## and obtain the Feynman amplitude. You'll see that it diverges as ##D=8-6>0##.

I understand (after studying example 32.4 of the enlightening book by Lancaster & Blundell: QFT for the gifted amateur; btw I wish I have encountered this book years ago, it is beautiful!) that ##D## is given by the following formulas

$$D=dL-I$$

$$D=d-E$$

Where ##L##, ##I## and ##E## stand for the number of loops, internal lines and external lines respectively.

OK, but Srednicki does not present ##D## in any of the above forms. He argues the superficial degree of divergence as follows (he uses as notation ##g_n## instead of ##\lambda_n##).

Based on (18.6) he argues that given any ##[\lambda_n]<0##, our diagram diverges as ##D>0## (where he assumed that the dimension of the tree diagram ##[g_E] > 0## always). So we conclude that any theory with negative coupling constant is non-renormalizable.

Once we have made such conclusion and recalling ##[\lambda_n] = d- \frac 1 2 n(d-2)##, we are ready to check how the ##n## powers of ##\phi## are limited depending on the ##d## number of dimensions we consider.

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

Questions

1) Why does he assume that the dimension of the tree diagram is ##[g_E] > 0## always? He states that "a theory is nonrenormalizable if any coefficient of any term in the lagrangian has negative mass dimension" but he does so after assuming ##[g_E] > 0##, so I am confused.

2) What is the maximal value of ##d## for which this theory is renormalizable? Our model has terms up to ##n=6##. Based on ##(*)##, I would say that ##d=3##. So if we wish to have ##\phi^6## power terms in a renormalizable theory, we need to work in ##3## dimensions.

3) Is it possible to kind of "render" our theory renormalizable for higher dimensions (say for ##d=4,5,6,7,8,9,10##)? I guess that the answer is yes, by means of introducing polynomial potential terms (with integer powers of ##n##). However I am not sure how this "render machinery" works (are these polynomial potential terms "counterterms"?). Might you please shed some light on it?

Thank you

Alright, I acknowledge I went a bit off on a tangent here . Please check this more on-point version and let me reformulate the 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?

Have a look at Sect. 5.5 in the following manuscript

https://itp.uni-frankfurt.de/~hees/publ/lect.pdf

where the power counting is done for ##\phi^4## theory. You can easily generalize this to theories with higher powers and vertices with derivative couplings.

JD_PM

## 1. What are renormalizability conditions for a real scalar field in d dimensions?

Renormalizability conditions for a real scalar field in d dimensions refer to the mathematical criteria that must be satisfied in order for a theory involving this type of field to be considered physically meaningful. These conditions ensure that the theory is free from divergences and can be used to make accurate predictions.

## 2. Why is it important to have renormalizability conditions for a real scalar field?

Having renormalizability conditions for a real scalar field is important because it allows us to make accurate predictions about physical phenomena and ensure that our theories are consistent and well-defined. Without these conditions, the theory would be plagued with infinities and would not be able to accurately describe real-world phenomena.

## 3. What are some examples of renormalizability conditions for a real scalar field in d dimensions?

Some examples of renormalizability conditions for a real scalar field in d dimensions include the requirement that the theory is invariant under certain transformations, such as Lorentz invariance, and that it has a finite number of divergent terms in its perturbative expansion. Additionally, the theory must have a well-defined ultraviolet limit, meaning that it can be extrapolated to arbitrarily high energies without encountering infinities.

## 4. How do renormalizability conditions affect the predictions of a theory?

Renormalizability conditions have a significant impact on the predictions of a theory. By ensuring that the theory is free from divergences and well-defined, these conditions allow for accurate calculations and predictions about physical phenomena. Without these conditions, the theory would not be able to make meaningful predictions and would not be considered a valid theory.

## 5. Are there any limitations to renormalizability conditions for a real scalar field in d dimensions?

While renormalizability conditions are crucial for a theory to be considered physically meaningful, there are some limitations. For instance, these conditions only apply to perturbative theories, meaning that they may not be applicable in non-perturbative regimes. Additionally, renormalizability conditions do not guarantee the ultimate validity of a theory, as they only ensure that it is free from divergences and well-defined at a certain level of approximation.

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