# Calculation of the weak coupling constant

There are two ways to calculate the coupling constant of the weak interaction $g$.
1) From the electromagnetic coupling constant and the weak mixing angle, using the relation $${\sf e} = g\sin(\theta_W)$$ where ${\sf e}=\sqrt{4\pi\alpha}$ is the elementary electric charge in natural units ($\alpha$ being the fine structure constant). Using the CODATA values of $\sin^2(\theta_W)=0.2223$ and $\alpha=7.297\cdot 10^{-3}$ this gives $$g=0.641.$$ 2) From the definition of the Fermi coupling constant $$G_F = \frac{\sqrt{2}}{8}\frac{g^2}{m_W^2}$$ where $m_W$ is the mass of the W boson. Using the CODATA value of $G_F=1.166\cdot 10^{-5}\;{\rm GeV}^{-2}$ and the PDG value of $m_W=80.385\;{\rm GeV}$ this gives $$g=0.653.$$ The difference is not big but nonetheless is significant. How can this be explained, since all the parameters used in the calculation are known with high precision?

• Your vision of a single $\theta_W$ is simply not realistic. Table 10.2 in the PDG review will clarify the issue for you. Pulling bland numbers out of CODATA rarely works. – Cosmas Zachos Jun 15 '18 at 15:26

Thank you @Cosmas Zachos for your reference to PDG review. Indeed the explanation after Table 10.2 is very useful. Actually, I used in the above calculation the so-called "on shell scheme" for the value of $\theta_W$; the caveat is that in the definition of $G_F$ the radiative corrections are not accounted for.

If one includes the radiative corrections $\frac{1}{\sqrt{1-\Delta r}}$ where $\Delta r=0.03648$ (as stated in the PDG review), one finds $1.019=\frac{0.653}{0.641}$, i.e. the ratio between the two calculated values of $g$. So, the difference is due to not taking in account the radiative corrections for $G_F$. Unluckily, this is not made also when stating the Higgs field vacuum expectation value $\upsilon=246.2\;\rm{GeV}$.

This same issue is addressed in the book "Quantum field theory and the Standard Model" by Matthew D. Schwartz (see (29.17) p. 588 for the calculation of $g$, (29.75) p. 604 for the calculation of $\upsilon$, (31.3) p. 642 for the radiative corrections of $G_F$).

Alas, when speaking about QFT results, one has always to ask which renormalization scheme is being considered.

Given you used multiple sources and parameters, the discrepancy likely stems from this. Recall that say $G_F$, would have been determined from scattering processes at some scale, say $\mu_1$.

Now, $\alpha$ could have been evaluated from entirely different experiments, at some scale $\mu_2$. The values calculated from each, $g(\mu_1)$ and $g(\mu_2)$ must differ due to the renormalisation group flow.

We have something like, $$\frac{\mathrm dg}{\mathrm d \log \mu} = -\frac{(22 - n_f-n_s)g^3}{48\pi^2} + \mathcal O (g^5)$$

where $n_f$ is the number of chiral fermions and $n_s$ the number of scalars. The point is though, regardless of the beta function's form, since $\beta \neq 0$, the couplings determined at different scales will necessarily have different values. It's not to do with precision but rather you've calculated two different but related quantites, despite both being "$g$".

Regardless, stay in the habit of propagating uncertainties. Sometimes a calculation should have two things agreeing, but within some margin of error

• I guessed the difference might be due to different energy scale of the parameters. The sources I used are the best publicly available, so I don't think any error stemming from uncertainty in the data (that are very low). Anyhow, I'm speaking about the universal constant of the weak interaction, that is fixed (like the fine structure constant for the electromagnetic interaction), not of the running coupling. I add that for the calculation of the Higgs field expectation value the 2nd value of $g$ is used $\upsilon = \frac{2m_W}{g} = 246.2\;{\rm GeV}$. – Stefano Zunino Nov 27 '17 at 13:39
• Of course the coupling constant I am referring to is the number that enters in the Lagrangian of the weak interaction after spontaneous symmetry breaking, before any perturbative approach, renormalization scheme, etc. is applied. That is obviously fixed and invariable (physics.stackexchange.com/q/11119), like the vacuum expectation value of the Higgs field that depends on it. – Stefano Zunino Nov 28 '17 at 4:32