# Spinor indices on Yukawa coupling terms in electroweak sector

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• spaghetti3451
In summary, in the electroweak sector, we have left-handed Weyl fields ##l## and ##\bar{e}## in representations ##(2,-1/2)## and ##(1,+1)## of ##SU(2) \times U(1)##. The Yukawa coupling in this sector is given by ##-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, the Higgs field takes on the form ##\phi = \frac{1}{\
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!

bummpppp!

## What is the significance of spinor indices in the Yukawa coupling terms in the electroweak sector?

The spinor indices in the Yukawa coupling terms indicate the coupling strength between the Higgs boson and the fermion fields in the electroweak sector. They also determine the chirality (left- or right-handedness) of the fermion fields involved.

## How do spinor indices affect the interactions between particles in the electroweak sector?

The spinor indices play a crucial role in determining the strength and type of interactions between particles in the electroweak sector. They allow for the exchange of virtual particles, such as the W and Z bosons, which mediate the weak interactions.

## What is the mathematical representation of spinor indices in the Yukawa coupling terms?

The spinor indices are represented by Greek letters, such as α, β, γ, etc., and are often accompanied by a bar or dagger symbol to indicate the conjugate or adjoint representation of the spinor. They can also be represented using Dirac spinor notation.

## How do spinor indices relate to the concept of parity in the electroweak sector?

Spinor indices are essential in determining the chirality of fermion fields, which is closely related to the concept of parity in the electroweak sector. Parity is a fundamental symmetry that describes the behavior of particles under spatial reflections, and it is violated in weak interactions due to the presence of spinor indices.

## Are there any experimental implications of spinor indices on Yukawa coupling terms in the electroweak sector?

Yes, the spinor indices can have experimental implications as they affect the decay rates and branching ratios of particles in the electroweak sector. By studying these quantities, researchers can test the predictions of the Standard Model and search for new physics beyond it.

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