# How to determine the coupling parameters in SM or beyond SM?

Dear everyone,
For example, the simplest action for a nonminimally coupled scalar field is
$$S=\int d^{4}x\sqrt{-g}\left[ \frac{1}{2}g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi -V(\phi )+\frac{\xi }{2}R\phi ^{2}\right].$$
$\xi=0$ leads to the scalar field minimally coupled to the gravity; $\xi=\frac{1}{6},V=0$ leads to a theory which is invariant under conformal transformations
$$\widetilde{g}_{\mu\nu}={\Omega}^{2}(x)g_{\mu\nu}.$$
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 ${\xi}RA_{\mu}A^{\mu}.$
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!

It depends on the theory you are considering. In general (at least as far as I know) if you have a fundamental theory (say QED) the couplings of the theory are free parameters and therefore they must be experimentally measured. Consider, for example, QED and suppose that we still don't know what the value of the electric charge, i.e. of the coupling, is. What you could do is, starting from your theoretical Lagrangian, compute the cross section for a certain process, e.g. compton scattering. This cross section will be a function of the electric charge. Then, measuring the value of the cross section with a suitable experiment you can fit the value of e.

However, if the theory is not fundamental, you can sometimes deduce the value of the coupling directly from theoretical arguments. Right now I have in my mind the 4-Fermi effective theory. The coupling for that Lagrangian is given by the Fermi coupling, ##G_F##. However, since this is just a low energy expansion of the complete electro-weak theory (when your energy scale is much lower that the mass of the W boson) one can show that the coupling is actually given by:

$$G_F=\frac{\sqrt{2}}{8}\frac{g_W^2}{M_W^2},$$

where ##g_W## is the weak coupling constant and ##M_W## is the mass of the W boson. So, in principle, if these quantities are know you can theoretically compute the value of ##G_F##. Of course, now the problem is simply shifted to the determination of the electro-weak coupling constant (which is the coupling of a fundamental theory, i.e. a free parameter).

I hope I didn't write anything wrong and I hope this is useful.
Cheers

• yancey
It depends on the theory you are considering. In general (at least as far as I know) if you have a fundamental theory (say QED) the couplings of the theory are free parameters and therefore they must be experimentally measured. Consider, for example, QED and suppose that we still don't know what the value of the electric charge, i.e. of the coupling, is. What you could do is, starting from your theoretical Lagrangian, compute the cross section for a certain process, e.g. compton scattering. This cross section will be a function of the electric charge. Then, measuring the value of the cross section with a suitable experiment you can fit the value of e.

However, if the theory is not fundamental, you can sometimes deduce the value of the coupling directly from theoretical arguments. Right now I have in my mind the 4-Fermi effective theory. The coupling for that Lagrangian is given by the Fermi coupling, ##G_F##. However, since this is just a low energy expansion of the complete electro-weak theory (when your energy scale is much lower that the mass of the W boson) one can show that the coupling is actually given by:

$$G_F=\frac{\sqrt{2}}{8}\frac{g_W^2}{M_W^2},$$

where ##g_W## is the weak coupling constant and ##M_W## is the mass of the W boson. So, in principle, if these quantities are know you can theoretically compute the value of ##G_F##. Of course, now the problem is simply shifted to the determination of the electro-weak coupling constant (which is the coupling of a fundamental theory, i.e. a free parameter).

I hope I didn't write anything wrong and I hope this is useful.
Cheers
Thank you, Einj. You explained my confusion clearly!