- #1
spaghetti3451
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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!
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!