Is the fine-structure constant a parameter of the standard model?

1,004

Your guess is correct. After electroweak symmetry breaking, the coupling constant for the residual $U(1)_\textrm{EM}$ gauge group can be written as a function of the couplings of the broken $SU(2)_L \times U(1)_\textrm{Y}$ gauge groups: $$ \alpha = \frac{1}{4\pi}\frac{g^2 g\prime^2}{g^2+g\prime^2} = \frac{e^2}{4\pi} $$

These couplings, however, are running parameters, defined at a particular energy scale. In your table, the energy scale is $\mu=M_Z$. If you plugged in the numbers from your table, you would calculate $\alpha$ at $\mu=M_Z$, which is $\alpha(\mu=M_Z) \approx 1/128$.

The fine-structure constant is usually considered to be the IR fixed-point of $\alpha$, which is $\alpha(\mu = m_e) = 1/137$, i.e., $\alpha$ at low energy. To calculate this from your table, you would have to run $e$ to a lower energy scale, with the $\beta$-function: $$ \frac{\partial e(\mu)}{\partial \log \mu} \equiv \beta(e) = \frac{e(\mu)^3}{12\pi^2} $$ With this, and your knowledge of $\alpha(\mu=M_Z)$, you could recover $\alpha(\mu = m_e)= 1/137$.

1,004

Peter de Rivaz

Working at Argon Design.

Updated on August 01, 2022

Comments

Peter de Rivaz 5 months

According to the wikipedia entry on the fine-structure constant:

In fact, α is one of the about 20 empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model.

but, the wikipedia list of parameters does not mention α:

me  Electron mass       511 keV
mμ  Muon mass       105.7 MeV
mτ  Tau mass        1.78 GeV
mu  Up quark mass   μMS = 2 GeV     1.9 MeV
md  Down quark mass     μMS = 2 GeV     4.4 MeV
ms  Strange quark mass  μMS = 2 GeV     87 MeV
mc  Charm quark mass    μMS = mc    1.32 GeV
mb  Bottom quark mass   μMS = mb    4.24 GeV
mt  Top quark mass  On-shell scheme     172.7 GeV
θ12     CKM 12-mixing angle         13.1°
θ23     CKM 23-mixing angle         2.4°
θ13     CKM 13-mixing angle         0.2°
δ   CKM CP-violating Phase      0.995
g1 or g'    U(1) gauge coupling     μMS = mZ    0.357
g2 or g     SU(2) gauge coupling    μMS = mZ    0.652
g3 or gs    SU(3) gauge coupling    μMS = mZ    1.221
θQCD    QCD vacuum angle        ~0
v   Higgs vacuum expectation value      246 GeV
mH  Higgs mass      ~ 125 GeV (tentative)

Is α one of the basic parameters of the Standard Model?

If not, then is there a simple formula for α in terms of these other parameters?

(My guess is that α can be derived from g1/g2/g3. However, I have been unable to find an explicit formula so far.)

Peter de Rivaz over 9 years

Thanks, this is exactly what I've been searching for! Do you happen to have a link to a web page where I should have been able to find these formulae myself?
innisfree over 9 years

A nice fist course on the Standard Model, arxiv.org/abs/hep-ph/0001283, which will contain the first equation (p65), but it won't cover the $\beta$-functions. For that you will need a course on QFT, see p395 of web.physics.ucsb.edu/~mark/qft.html, but that reference is quite advanced...