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For example, the simplest action for a nonminimally coupled scalar field is

[tex]

S=\int d^{4}x\sqrt{-g}\left[ \frac{1}{2}g^{\mu \nu }\partial _{\mu }\phi

\partial _{\nu }\phi -V(\phi )+\frac{\xi }{2}R\phi ^{2}\right].

[/tex]

[itex]\xi=0[/itex] leads to the scalar field minimally coupled to the gravity; [itex]\xi=\frac{1}{6},V=0[/itex] leads to a theory which is invariant under conformal transformations

[tex]

\widetilde{g}_{\mu\nu}={\Omega}^{2}(x)g_{\mu\nu}.

[/tex]

So the coupling parameter ξ is free, are there any constraints on it? How to get these constraints if they exist?

The same thing happens to a genaral vector field case [itex]{\xi}RA_{\mu}A^{\mu}.[/itex]

In QED, the coupling constant characterize the interaction between the pin-1/2 field and the electromagnetic field is the electric charge of the bi-spinor field, how dose people determine this constant? by theoretical prediction or by experimental measurements?

How does the constraints on the form of the Lagrangian leads to constraints on coupling parameters?

Any documents can resolve my confusion will be welcome.

Thanks!