Can a Lagrangian in QFT be Renormalizable?

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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