# A Spinor indices on Yukawa coupling terms in electroweak sector

1. Jan 19, 2017

### spaghetti3451

In the electroweak sector, we define the left-handed Weyl fields $l$ and $\bar{e}$ in the representations $(2,-1/2)$ and $(1,+1)$ of $SU(2) \times U(1)$. Here, $l$ is an $SU(2)$ doublet: $l = \begin{pmatrix} \nu\\ e \end{pmatrix}.$

The Yukawa coupling in the electroweak sector is of the form $-y\epsilon^{ij}\phi_{i}l_{j}\bar{e} + \text{h.c.},$ where $\phi$ is the HIggs field in the representation $(2,-1/2)$.

After spontaneous symmetry breaking, in the unitary gauge, the Higgs field becomes $\phi = \frac{1}{\sqrt{2}}\begin{pmatrix} v+H\\ 0 \end{pmatrix}$ and

the Yukawa coupling becomes $-\frac{1}{\sqrt{2}}y(v+H)(e\bar{e}+\bar{e}^{\dagger}e^{\dagger}) = -\frac{1}{\sqrt{2}}y(v+H)\bar{\varepsilon}\varepsilon,$

where we have defined a Dirac field for the electron, $\varepsilon = \begin{pmatrix} e\\ e^{\dagger} \end{pmatrix}.$

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My questions are the following:

1. I notice that $\bar{e}$ is a left-handed Weyl field whereas $\bar{e}^{\dagger}$ is a right-handed Weyl field. Does this mean that taking the hermitian conjugate change the handedness of a Weyl field?

2. Observe the Yukawa coupling after spontaneous symmetry breaking: $e\bar{e}+\bar{e}^{\dagger}e^{\dagger}$. How do I make sense of the spinor indices here: is $e$ a row vector or a column vector? Is $\bar{e}$ a row vector or a column vector? What about their hermitian conjugates?

3. I notice that the Dirac field for the electron is $\varepsilon = \begin{pmatrix} e\\ e^{\dagger} \end{pmatrix}.$ $e$ and $\bar{e}^{\dagger}$ appear to be column vectors. Does this mean that $\bar{e}$ is a row vector? But then, $e\bar{e}$ becomes a matrix and is not a scalar, as is expected for a term in a Lagrangian!!!

2. Jan 23, 2017

bummpppp!!!!