Understanding $\mathscr{L}_Y^{\text{down} }=-\sqrt{2}\sum _{a,q}\overline{\psi }_{iL}Y^{\text{down} }_{iq}q'_RH+\text{h.c.}$

Currently reading Bilenky's "Introduction to the Physics of Massive and Mixed Neutrinos." (2nd ed.)

On page 56 (Section 3.4 on the Standard model) elaborates

"Let us assume that in the total Lagrangian of the SM there is the following Lagrangian of the Yuwaka interaction of the quark and Higgs fields" $$\mathscr{L}_Y^{\text{down} }=-\sqrt{2}\sum _{a,q}\overline{\psi }_{iL}Y^{\text{down} }_{iq}q'_RH+\text{h.c.} \tag{Eq. 3.126}$$

Where

• $$Y^{\text{down} }_{iq}$$ is a complex $$3\times 3$$ matrix.

• $$q(x)$$ is the quark field.

• $$\psi _{iL}$$ and $$H$$ are $$SU(2)$$ doublets.

• $$q'_R$$ are singlets.

I can't seem to understand the equation at all. And more importantly, I can't seem to see how is the product defined at all. If $$Y$$ is a $$3\times 3$$ matrix, how do we do the product with doublets?

Also, why is the sum over $$a$$ and $$q$$? I would expect $$q$$ to be there but what's $$a$$ all about? Maybe a typo?

I'm also unsure as to what $$\psi$$ means here. My best guess is that it's the doublets presented at the beginning of the section (3.4):

• $a$ seems to be a typo indeed, I think it should be replaced by $i$. Thus, $Y^\text{down}_{iq}$ is not a matrix, but the components of a $3\times 3$ matrix. Commented Oct 24, 2023 at 5:50
• That makes a lot of sense. I really wish this great book had an errata online... Commented Oct 24, 2023 at 5:52

in the script I'm usually using, this expression would be (without h.c.):

where $$\phi=(v+h)/\sqrt{2}$$ (doublet initially), and the $$Y^{d/u}_{ij}$$ are complex 3x3 matrices. Now to the $$Q^{I}_{Li}$$:

$$I$$ stands for the flavor basis (interaction basis).

$$L$$ for the left-handed doublet.

$$i$$ really is the family/ generation, yes.

So the matrix really sums over the different flavors. $$a$$ should be a typo.

Here's the definition of $$Q^{I}_{Li}$$ used in this script:

I realize now that this particular script is not available online, so I can't give you a link. It should be based on the book Quarks and Leptons by F. Halzen & A. Martin however. Hope that helped a little bit.

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