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Electroweak Theory

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Electroweak Theory is a Quantum Field Theory which unifies Electromagnetism and the Weak Nuclear Force in a single Gauge Field with gauge group SU(2)_L \times U(1)_Y, where SU(2)_L acts only on Left-Handed Fermions, and U(1)_Y is "hypercharge". The Higgs Mechanism gives mass to the W and Z components of this gauge field, and breaks the symmetry to U(1)_{electromagnetism}, as described by Quantum Electrodynamics.

Lagrangian DensityEdit this section

It can be easily shown like as for other Quantum Field Theoryies, that the Dirac Equation for Electroweak Theory is: \left( i\hbar \not{\partial }-{{m}_{Weak}}{{c}_{0}}-\frac{{{g}_{W}}\alpha \not{W}+{{g}_{F}}Y\not{F}}{2} \right)\psi =0

The Lagrangian Density excluding field strength is given by:

{{\mathsf{\mathcal{L}}}_{EW}}={{c}_{0}}\bar{\psi }\left( i\hbar \not{\partial }-{{m}_{Weak}}{{c}_{0}}-\frac{{{g}_{W}}\alpha \not{W}+{{g}_{F}}Y\not{F}}{2} \right)\psi


Here, {{m}_{\text{Weak}}} is the mass of the W/Z boson, W is the Weak Field, F is the Photon Field, and Y is the Weak Hypercharge.


If one lets  \mathcal{Q}=\left[ \begin{matrix}
   {{g}_{W}}\alpha   \\
   {{g}_{F}}Y  \\
\end{matrix} \right]

and

\mathsf{\mathcal{V}}=\left[ \begin{matrix}
   {\not{W}}  \\
   {\not{F}}  \\
\end{matrix} \right]

then the Lagrangian Density (with a field strength I

{{\mathsf{\mathcal{L}}}_{EW}}=-\frac{{{I}^{\mu \nu \rho }}{{I}_{\mu \nu \rho }}}{4}+{{c}_{0}}\bar{\psi }\left( i\hbar \left( \not{\partial }+\frac{i}{\hbar }\frac{\cdot \mathsf{\mathcal{V}}}{2} \right)-{{m}_{\mathrm{Weak}}}{{c}_{0}} \right)\psi =-\frac{1}{4}{{I}^{\mu \nu \rho }}{{I}_{\mu \nu \rho }}+i\hbar {{c}_{0}}\bar{\psi }{{\not{\nabla }}_{EW}}\psi -\bar{\psi }{{m}_{EW}}c_{0}^{2}\psi

Dirac EquationEdit this section

From the result of the last section, we see that

i\hbar \not\nabla_{EW}\psi -m_\mathrm{Weak}c_0\psi = 0

Electroweak Symmetry BreakingEdit this section

Main Article: Higgs Mechanism

As per Experimental Results, Photons are massless, whereas Weak Force mediating W and Z Bosons are massive. If these W and Z Bosons are massive, then this means that Electroweak Symmetry is broken. This Electroweak Symmetry Breaking is the result of the Higgs Mechanism.

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