Can a Lagrangian in QFT be Renormalizable?

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In quantum field theory (QFT), the renormalizability of a Lagrangian is influenced by the mass dimensions of its terms. A term like g/φ^2 is considered renormalizable based on power counting, with the coupling g having a mass dimension of 2(D-1). For four-dimensional space-time (D=4), this results in a mass dimension of 6, which is above the threshold for renormalizability. Terms with mass dimensions greater than 5 are typically non-renormalizable, leading to challenges in perturbative renormalization. This aligns with the behavior observed in theories like gravity and the Fermi model, which operate as effective theories with energy cutoffs.
Giuseppe Lacagnina
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Possibly very silly question in QFT. Consider the Lagrangian for a scalar field theory.
A term like

g/φ^2

should be renormalizable on power counting arguments. The mass dimension of g should be

2 (D-1)

where D is the number of space-time dimensions.Does this make sense?
 
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Isn't there a rule where if the mass dimension is greater than 5, the term is non-renormalizable?
2(D-1)=6, if D=4.
 
As far as I know, a lagrangian term is not perturbatively renormalizable if it involves a coupling with negative mass dimension.
Like it happens for gravity or the Fermi model of weak interactions, which works as an effective theory with an energy cutoff.
 

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