- #1
Geonaut
- TL;DR Summary
- I am having some trouble understanding the notation used to describe fermion mass terms in Peskin and Schroeder's book on QFT.
I'm currently looking at how fermion masses are produced via the Higgs mechanism in "An Introduction to Quantum Field Theory" by Peskin and Schroeder. It all makes a lot of sense and I've been fine with it so far, but I ended up getting stuck on something that's driving me nuts. I feel silly asking this since it's just a question about notation, but after hours of searching for a clue my brain is turning to mush. You see, to generate fermion masses in the Standard Model they use the equation
$$ \mathcal{L}_m =-\lambda^{ij}_d \overline Q^i_L \cdot \phi d^j_R - \lambda^{ij}_u \epsilon^{ab}\overline Q^i_{La} \phi^{\dagger}_b u^j_R + h.c.$$
With the following equations given in the book it's clear what's going on here for the most part:
$$ \overline Q^i_L =
\begin{pmatrix}
u^i \\
d^i
\end{pmatrix}_L =
\begin{pmatrix}
\begin{pmatrix}
u \\
d
\end{pmatrix}_L
,&
\begin{pmatrix}
c \\
s
\end{pmatrix}_L
,&
\begin{pmatrix}
t \\
b
\end{pmatrix}_L
\\
\end{pmatrix}
$$
$$u^i_R =
\begin{pmatrix}
u_R ,& c_R ,& t_R
\end{pmatrix}
$$
$$d^i_R =
\begin{pmatrix}
d_R ,& s_R ,& b_R
\end{pmatrix}
$$
The only thing I don't seem to understand is the notation in $$\epsilon^{ab}\overline Q^i_{La} \phi^{\dagger}_b$$. To be specific, what does the "a" in the subscript for $$\overline Q^i_{La}$$ represent? What does the "b" in $$\phi^{\dagger}_b$$ represent? Where does $$\epsilon^{ab}$$ even come from? I would guess that $$\epsilon^{ab} = -\epsilon^{ba}=1$$, but I don't see where this stuff is explained. I would guess that "a" and "b" represent a set of numbers, namely, "a = 1, 2, 3" and "b = 1, 2, 3"... but something isn't adding up. Can someone show me what's going on here? Searching for information as trivial as notation in dense books on QFT is beginning to feel like torture.
$$ \mathcal{L}_m =-\lambda^{ij}_d \overline Q^i_L \cdot \phi d^j_R - \lambda^{ij}_u \epsilon^{ab}\overline Q^i_{La} \phi^{\dagger}_b u^j_R + h.c.$$
With the following equations given in the book it's clear what's going on here for the most part:
$$ \overline Q^i_L =
\begin{pmatrix}
u^i \\
d^i
\end{pmatrix}_L =
\begin{pmatrix}
\begin{pmatrix}
u \\
d
\end{pmatrix}_L
,&
\begin{pmatrix}
c \\
s
\end{pmatrix}_L
,&
\begin{pmatrix}
t \\
b
\end{pmatrix}_L
\\
\end{pmatrix}
$$
$$u^i_R =
\begin{pmatrix}
u_R ,& c_R ,& t_R
\end{pmatrix}
$$
$$d^i_R =
\begin{pmatrix}
d_R ,& s_R ,& b_R
\end{pmatrix}
$$
The only thing I don't seem to understand is the notation in $$\epsilon^{ab}\overline Q^i_{La} \phi^{\dagger}_b$$. To be specific, what does the "a" in the subscript for $$\overline Q^i_{La}$$ represent? What does the "b" in $$\phi^{\dagger}_b$$ represent? Where does $$\epsilon^{ab}$$ even come from? I would guess that $$\epsilon^{ab} = -\epsilon^{ba}=1$$, but I don't see where this stuff is explained. I would guess that "a" and "b" represent a set of numbers, namely, "a = 1, 2, 3" and "b = 1, 2, 3"... but something isn't adding up. Can someone show me what's going on here? Searching for information as trivial as notation in dense books on QFT is beginning to feel like torture.