# Question about Standard Model Lagrangian

Hi, I really like this form of the Standard Model Lagrangian, but I have a couple of questions to it.

• In the last line: What is $G_1$ and $G_2$? I assume they are the Yukawa couplings, but does the index refer to leptons vs quarks, or "up-type" vs "down-type" or something else?

• In the same line: Why does the Higgs field have a c as index for $G_2$, but not for $G_1$ and what does it stand for?

• The $W_{\mu}$ is written as a bold vector. What are its components?

• Does this formulation also include neutrino-oscillations?

• Does anybody know if there is some reference for this formula? I.e. does it come from a book? And is it actually correct?

• – JakobH Nov 22 '17 at 10:15

1. $G_1$ and $G_2$ are Yukawa couplings as you say. More reasonably these should be grouped into three terms – one which couples the left-handed lepton doublet to the right-handed leptons, one which couples the left-handed quark doublet to the right-handed down-type quarks, and one which couples the left-handed quark doublet to the right-handed up-type quarks. The first two of these involve the Higgs $\phi$, whilst the latter involves its charge conjugate $\phi_c$.
2. The charge conjugate of the Higgs is defined by $(\phi_c)^\alpha = \epsilon^{\alpha \beta} (\phi^\dagger)_\beta$, where $\alpha,\beta$ are $\mathrm{SU}(2)$ (weak) indices. This is needed to couple the left-handed quark doublet to the right-handed up-type quarks in order to give a hypercharge neutral term in the Lagrangian. Note that using $\phi^\dagger$ doesn't work for this purpose, since it doesn't have the correct $\mathrm{SU}(2)$ transformation properties to form an $\mathrm{SU}(2)$ singlet with $\bar{L}$.
3. The $W_{\mu \nu}$ are labelled by a gauge index which runs from 1 to 3 (this is the dimension of the group $\mathrm{SU}(2)$). This is the statement that there are three weak gauge bosons. The gluon field strength is also labelled by such an index (which runs from 1 to 8 in this case). The author of this image has decided to include these indices explicitly for the gluons and leave them as a 'vector dot product' for the weak bosons. There is no reason to do this.