# What is the dimension of the weak gauge field couplin constant NOT in natural units?

What is the dimension of g and g', NOT in natural units, but in terms of mass, length, time, and permittivity?

• Coupling constants are dimensionless – dukwon May 9 at 12:55
• @dukwon Should probably be an answer, with maybe a short explanation as to where the units come in when we do calculations. – probably_someone May 9 at 13:20
• @dukwon Coupling constants are not (generically) dimensionless. The gauge coupling constant in $d\neq 4$ is dimensionful, and so is e.g. $\lambda \phi^6$ in $d=4$. – AccidentalFourierTransform May 9 at 14:42
• – Cosmas Zachos May 9 at 16:34
• Answer obscure? – Cosmas Zachos May 15 at 13:20

In natural units (first: there is an overwhelming reason and method in their madness!) these are both related to the electric charge e through the Weak mixing angle. See the triangle figure: $$g=e/\sin\theta_W, \qquad g'= e/\cos\theta_w, \\ \sin^2 \theta _\text{W}=1-(m_\text{W}/m_\text{Z})^2=0.2223(21).$$
All you need then is to convert the natural units e to the engineering units e you are seeking through the dimensionless fine structure constant, $$α= {\mathbf{e}} ^2 /4π \epsilon_0 ħc = 0.0072973525664(17) ,$$ or something... Insert your new/accurate CODATA values at your leisure.
In natural units, $$\alpha = e^2/ 4 \pi$$ since 1=ε0 = c = ħ = 1, so you redimensionalize the above g and g' by multiplying them with $$\sqrt{\epsilon_0 ħc}$$.