We now come to the Higgs sector with the Lagrangian[5]
(27.1)
 &bg=ffffff&s=2.5)
^{\dagger} \left (D^{\mu} \Phi \right ) &bg=ffffff&s=2.5)
whose main actor is the scalar doublet 
(27.2)
(10052024)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
quantum field theory calculations
(01032024)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
What follows then before we close this section is to write down a basic }_{\alpha} &bg=ffffff&s=0)
(29012024)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
What we have seen so far are the interactions of these lefthanded SU(2) fermion doublets with the Higgs boson and the way in which these fermions acquire their masses through their interactions with the vev of the Higgs field.
In  \times U(1) &bg=ffffff&s=0)
 &bg=ffffff&s=0)
(15102023)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
 \times U(1) &bg=ffffff&s=2.5)
Construction of The Electro-weak theory
We have just presented an incomplete construction of the electro-weak theory and now we come to a more inclusive construction to accommodate the other generations of the lepton family and the three generations of the quark family. For such chief purpose of this section, we have [5] as our reference guide in constructing the  &bg=ffffff&s=0)
 &bg=ffffff&s=0)
(26052023)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
A more general setup is where the Higgs boson carries a full four-momentum, }^{0}, \vec{P}_{(1)} \right ) &bg=ffffff&s=0)
} &bg=ffffff&s=0)
}^{0}, \vec{k}_{(i)} \right ) i = 1, 2 &bg=ffffff&s=0)
(19.9.1)
}^{0} k_{(2)}^{0} = 2 \vec{k}_{(1)} \bullet \vec{k}_{(2)} %2B m_{\eta}^{2} &bg=ffffff&s=2.5)
and if in the final state all leptons carry equal energies, }^{0} = k_{(2)}^{0} = \frac{1}{2} P_{(1)}^{0} &bg=ffffff&s=0)
(19.9.2)
} \bullet \left (\vec{k}_{(1)} %2D \vec{k}_{(2)} \right ) = m_{\eta}^{2} \left ( 1 %2D \frac{4m_{\psi}^{2}}{m_{\eta}^{2}} \right ) \ne 0 &bg=ffffff&s=2.5)
implying in turn that the fermion and anti-fermion (leptons) carry equal but opposite spatial momenta, } = %2D \vec{k}_{(2)} &bg=ffffff&s=0)
(19.9.3)
} \right \vert^{2} = m_{\eta}^{2} \left ( 1 %2D \frac{4m_{\psi}^{2}}{m_{\eta}^{2}} \right ) &bg=ffffff&s=2.5)
We take (19.1) to write
(19.9.4)
}(k_{(2)}) v^{(1)}(k_{(1)}) = \left ( \vec{k}_{(1)} %2D \vec{k}_{(2)} \right ) \bullet \hat{e}_{1} %2D \break \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent i \left ( \vec{k}_{(1)} %2D \vec{k}_{(2)} \right ) \bullet \hat{e}_{2} &bg=ffffff&s=2.5)
With this we may then write (19.8.4) as
(19.9.5)
}(k_{(2)}) v^{(1)}(k_{(1)}) \right \vert^{2} = 4 \left \vert \vec{k}_{(1)} \right \vert_{k^{3}_{(1)} = 0 }^{2} = m_{\eta}^{2} \left ( 1 %2D \frac{4m_{\psi}^{2}}{m_{\eta}^{2}} \right ) &bg=ffffff&s=2.5)
where in the result, the third spatial components of four-momenta are suppressed,
}^{3} = 0 &bg=ffffff&s=0)
Take note that the fermion and anti-fermion are all in the up spin states but directed in equal but opposite spatial momenta.
A cautionary remark must be duly noted here regarding the assumed consequence following (19.9.2) that the final state spatial momenta are directed opposite to each other, } = %2D \vec{k}_{(2)} &bg=ffffff&s=0)
}^{0} = m_{\eta}, \vec{P}_{(1)} = \vec{0} ) &bg=ffffff&s=0)
}^{0} = k_{(2)}^{0} = \frac{1}{2} m_{\eta} &bg=ffffff&s=0)
The other condition imposed to satisfy (19.9.2) is where } \ne %2D \vec{k}_{(2)} &bg=ffffff&s=0)
}^{1} = \vec{k}_{(2)}^{1} &bg=ffffff&s=0)
}^{2} = %2D \vec{k}_{(1)}^{2} &bg=ffffff&s=0)
}^{1} \vert^{2} = \vert \vec{k}_{(2)}^{1} \vert^{2} = \frac{1}{4} ( (P_{(1)}^{0})^{2} %2D m_{\eta}^{2}) &bg=ffffff&s=0)
}^{2} = %2D \vec{k}_{(1)}^{2} \to \vec{P}_{(1)}^{2} = \vec{0} &bg=ffffff&s=0)
(06052023)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
The final states, comprising of an out-going electron and an out-going positron are the physical states and they are represented by solid arrows coming out and into a vertex respectively, while the initial Higgs boson state is in broken line coming into the same vertex, where it is signified a delta function to conserve the four-momenta. The physical states are the detected particles but the summation of all four-momenta must take into account the four-momentum of the decayed initial state.
(12012023)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
From there we read off our interaction Hamiltonian of interest
(20.10.4)
}\left ( \bar{\psi}_2 , \psi_2 \right ) = y \int d^3 \vec{x}\bar{\psi}_2 \psi_2 \eta = \frac{m_\psi}{\beta}\int d^3 \vec{x}\bar{\psi}_2 \psi_2 \eta &bg=ffffff&s=2.5)
that we need as a field operator in (8), where we have dropped off the subscripts in the fermions for convenience.
Spotted right on is that such interaction is directly proportional to the fermion mass and the fermions involved here are lightweight so that the Yukawa coupling constant 
(15122022 updates)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
 = \frac{1}{2} \left ( \partial_{\mu} \eta \right)^2 %2D \frac{1}{2}m_{\eta}^{2} \eta^2 %2B 2Q^{\prime 2} \beta \left ( W_{\mu}^{(%2B)}W_{(%2D)}^{\mu} %2B \frac{1}{2cos^2 \alpha}Z_{\mu}Z^{\mu} \right ) \eta \break \indent \indent \indent \indent %2B Q^{\prime 2} \left ( W_{\mu}^{(%2B)}W_{(%2D)}^{\mu} \eta %2B \frac{1}{2cos^2 \alpha}Z_{\mu}Z^{\mu} \eta \right ) \eta %2D \frac{\lambda^2 \eta^4}{4}
\break \indent \indent \indent \indent %2D \lambda^2 \beta \eta^3 %2D y \eta \left ( \bar{\psi}^{R}_{2} \psi^{L}_{2} %2B \bar{\psi}^{L}_{2} \psi^{R}_{2} \right) &bg=ffffff&s=2.5)
From the Higgs sector of this rudimentary  \times U(1) &bg=ffffff&s=0)
(22112022 updates)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
(20.9.9.3)
} W_{(%2D)}^{\mu} %2B \frac{Q^{\prime 2} \beta^2}{2cos^2 \alpha} Z_{\mu}Z^{\mu} = \break \break \indent \indent \indent \indent \indent \indent \indent \frac{1}{2} Q^{\prime 2} \beta^2 W_{\mu}^{(1)} W_{(1)}^{\mu} %2B \frac{1}{2} Q^{\prime 2} \beta^2 W_{\mu}^{(2)} W_{(2)}^{\mu} %2B \break \break \indent \indent \indent \indent \indent \indent \indent \frac{Q^{\prime 2} \beta^2}{2cos^2 \alpha} Z_{\mu}Z^{\mu} &bg=ffffff&s=2.5)
(20.9.9.4)
}^2 = Q^{\prime 2} \beta^2 &bg=ffffff&s=2.5)
}^2}{cos^2 \alpha} &bg=ffffff&s=2.5)
The mass terms (20.9.9.3) don’t include the mass for the third vector gauge component }^\mu &bg=ffffff&s=0)
(29092022 updates)First Order Calculations of the Probability Amplitude for Higgs Boson to Electron-Positron Decay
We may now see that expanding the potential in terms of the field components would yield a correct mass term for a time-like particle, which is specifically the Higgs boson and we write the following expansion that is given by
(20.9.7.13)
 = \lambda^2 \beta^2 \eta^2 %2B \frac{\lambda^2 \eta^4}{4} %2B \frac{\lambda^2}{4} \left (\varepsilon^2 %2B \left \vert \varphi_1 \right \vert^2 \right )^2 %2B \lambda^2 \beta \eta \left (\eta^2 %2B \varepsilon^2 %2B \left \vert \varphi_1 \right \vert^2 \right ) %2B \frac{\lambda^2 \eta^2}{2} \left (\varepsilon^2 %2B \left \vert \varphi_1 \right \vert^2 \right ) &bg=ffffff&s=0)
This mass term is 
(20.9.7.14)
We might as well interpret this outcome that gives out the mass of the Higgs boson as due to the Higgs boson’s interaction with the vacuum expectation value or vev of the Higgs field, where in this interaction, the Higgs boson goes along with itself as it would interact with said vev just as the left-handed and right-handed electrons go together to interact with the vev to acquire a mass for a single electron entity. The difference between two cases is that for the Higgs boson it happens in the Higgs potential as it is expanded in terms of the component fields to give the correct mass term for a time-like particle, whereas for the fermions it is in the Yukawa interaction terms.
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