Calculation of the weak coupling constant

Question

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?

Answer 1

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.

Answer 2

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

Answer 3

As far as I know, the coupling constant of the weak force is much smaller than all the above. It should be at 10^-7. According to my calculations: α_W = 1.016 × 10^(-7). Less precise calculations yield: 2.739 × 10^(-7).

Reference: http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/couple.html