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

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    $\begingroup$ Coupling constants are dimensionless $\endgroup$ – dukwon May 9 at 12:55
  • $\begingroup$ @dukwon Should probably be an answer, with maybe a short explanation as to where the units come in when we do calculations. $\endgroup$ – probably_someone May 9 at 13:20
  • $\begingroup$ @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 $d=4$. $\endgroup$ – AccidentalFourierTransform May 9 at 14:42
  • $\begingroup$ Related. $\endgroup$ – Cosmas Zachos May 9 at 16:34
  • $\begingroup$ Answer obscure? $\endgroup$ – Cosmas Zachos May 15 at 13:20
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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 $\sqrt{\epsilon_0 ħc}$.

At the core of your curiosity question, however is the 500lb gorilla question of why ? The weak interactions are microscopic, and you'll never monitor something about them with your ammeter, galvanometer, whatever. The microscopic world of HEP has precious few conceptual linkages with engineering. You will probably use the SM to compute lifetimes, in seconds, or cross sections in barns, etc... in which the dimensions of the couplings would be in the way. All you'd need do is throw in the suitable powers of ħ and c in redimensionalizing.

For any intuition, stay natural and compare with your yardstick e=0.30282212....

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