Question about Standard Model Lagrangian

Question

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?

Answer 1

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.
4. Probably not. Neutrino oscillations, whilst unequivocally observed, are typically excluded from the Standard Model Lagrangian on account that we don't know the exact nature of the neutrino interactions (or indeed the nature of the right-handed neutrinos themselves). However, this Lagrangian is schematic enough that they could be considered part of the final term, which would give the neutrinos a (Dirac) mass.
5. Every PhD student with an afternoon to kill has probably tried to LaTeX the Standard Model Lagrangian before. Whilst the formula is correct (although all such formulae are somewhat schematic, so it is difficult to be unequivocally wrong), it is written in such a way as to suggest to me that it doesn't come from a book or any kind of authoritative source.