# Coupling Constant

Perhaps I'm confusing two different things but I've read online (http://en.wikipedia.org/wiki/Gravitational_coupling_constant) that the gravitational coupling constant is dimensionless and proportional to Newton's constant G.

However, I have also read that the gravitational coupling constant is proportional to the square root of G, and since in a 4d theory, G has mass dimension -2 (can see from an Einstein Hilbert action), the coupling will have dimension -1 and this is the reason GR can't be renormalised.

My questions are:

1, Which of these are correct?

2, How do we derive the relationship between the coupling and G?

Thanks.

It depends on the definition. You can see that Wikipedia defines the coupling constant as $\alpha_G = m_e / m_p$, whereas normally you would call $1/m_p$ as the coupling constant, as it's what's in front of the interaction term in the Lagrangian.
It depends on the definition. You can see that Wikipedia defines the coupling constant as $\alpha_G = m_e / m_p$, whereas normally you would call $1/m_p$ as the coupling constant, as it's what's in front of the interaction term in the Lagrangian.