Coupling constants with fractional dimensions

[hilbert2]

Most QFT texts, such as Peskin&Schroeder and D. Tong's lecture notes, contain a mention that the renormalizability of an interacting theory requires the coupling constants to have correct dimensions, making scalar fields with $\phi^5 , \phi^6, \dots$ interactions uninteresting. Maybe there are similar limitations for vector and spinor field interactions, but someone more familiar with QFT must answer that.

Now, if instead of an integer power of the scalar field, I make a field equation that has something like $\phi^{7/2}$ or similar in it, does this lead to some kind of unphysical behavior too? In theories of physics, you don't really see constants of nature that have fractional powers of kilogram or second in their units (not sure why). Is this kind of a fractional interaction equivalent to a non-renormalizable one because the term with fractional order can be expanded to a power series with arbitrarily high integer powers of the field variable?

 

[dextercioby]

Can you build a Feynman vertex out of 3,5 scalar neutral particles, such as Higgs bosons? How would you do it, draw 3 full lines and the 4th reduced to half length?

 

[hilbert2]

A perturbation expansion containing products of three creation/annihilation operators with one square-root creation/annihilation operator would definitely be a bit difficult, and maybe it would cause problems similar to when you try to take the square root of the Klein-Gordon operator. It's just not immediately obvious to me that there's no way to do that.