The coupling constants (in the gauge theory) fix an inner product on the lie algebra of the gauge group and we use it to define strength of the fields. we are using ad-invariant inner products which are determined by some numbers. In other words, the set of all ad-invariant inner products form a space of more than one dimension and to fix a specific inner product we need to choose some numbers which are the coupling constants. This is the story which happens over each point of the space-time. Mathematically one can produce a theory in which these numbers (and so inner product) changes from a point to another (similar to Riemann metric on a general Riemannian manifold). In other words we can have coupling fields rather that constants.
What is the physical reasoning which disallow us having such a theory (with coupling fields)? And what would be the physical implications if the nature follows such a model?
Answer
You may always promote "couplings constants" (charge, mass, etc...) to fields. Now, as a physicist, you need to make some contact with reality. So you have to tell why and which field you are using (for instance the Higgs field (up to a constant), which has a $SU(2)$ charge, is used to replace a constant mass coupling in the interaction $m (\bar e_R e_L + \bar e_L e_R)$, this is because the left electron has a $SU(2)$ charge while the right electron has not), and how an experiment can test your hypothesis.
Finally, even coupling constants, in a Quantum field theOry, are not really constant, and depenD on the energy scale $e = e(\Lambda)$
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