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Interaction of the Higgs doublet with fermions and generation of fermion masses

The mass terms for fermions in the electroweak theory are generated via their $SU(2)\times U(1)$ invariant Yukawa interaction with Higgs doublet. Namely, the Yukawa Lagrangian

$$- \frac{m_1 \sqrt{2}}{\phi_0} \left( \bar{\Psi}^L_i \epsilo... ...^R_2 + \bar{\Psi}^R_2 \stackrel{*}{\Phi^{i}} \Psi^L_i \right)$$

produces the mass terms for doublet components

$$-m_1 \bar{\Psi}_1 \Psi_1 -m_2 \bar{\Psi}_2 \Psi_2\;,$$

which are accompanied by vertices of interactions of fermions with Goldstone fields:

$$- \frac{ m_1\, g_2}{2 M_W} \left ( \frac{i}{\sqrt{2}}(W^-_f \bar{\P... ...5) \Psi_2) + H \bar{\Psi}_1 \Psi_1 + i Z_f \bar{\Psi}_1 \gamma^5 \Psi_1 \right)$$

$$- \frac{ m_2\, g_2}{2 M_W} \left ( \frac{i}{\sqrt{2}}(W^+_f \bar{\P... ... \Psi_1) + H \bar{\Psi}_2 \Psi_2 -i Z_f \bar{\Psi}_2 \gamma^5 \Psi_2 \right)\;.$$ (20)

If there are several doublets with the same hypercharge, then a general form of Yukawa Lagrangian contains a product of terms from different doublets. Such terms form two mass matrices: one for upper and another for lower fermions. Each of these matrices can be diagonalized by means of the unitary transformation of doublets, but this cannot be done for both of them at the same time. In this case the basis of doublets is chosen in such a way to present one of these matrices, for example, for upper fermions, in the diagonal form. Then the physical particles correspond to linear combinations of lower doublet fields realized by some unitary matrix which is called a mixing matrix.

Generally the Lagrangian is written down in terms of fermion fields which directly correspond to particles. Interactions of such fields with $A$, $Z$, $H$, and $Z_f$ are the same as defined by (19) and (20), whereas interactions with $W^\pm$ and $W^\pm_f$ contain elements of the mixing matrix.