Electroweak Theory

Quantum Field Theory
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Related
Related	De Donder-Weyl Theory

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 Density

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 Equation

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 Breaking

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.