1. **Problem Statement:** We are analyzing the mass terms of the quark doublet $(t,b)$ under strong and electroweak (EW) symmetry breaking, focusing on how the top quark mass $m_t$ and bottom quark mass $m_b$ relate to mixing angles and Yukawa interactions. 2. **Key Formula:** The mixing angle $\theta_t$ between the top and bottom quarks is approximately given by: $$\theta_t \approx \frac{m_t}{m_b}$$ This ratio indicates that if $m_t$ is much larger than $m_b$, the mixing angle becomes large, affecting the mass eigenstates. 3. **Yukawa Lagrangian:** The Yukawa interaction Lagrangian responsible for quark masses after symmetry breaking is: $$\mathscr{L}_Y = \bar{Q}_L Y_U \phi U_R + \bar{Q}_L Y_D \tilde{\phi} D_R + h.c.$$ where: - $Q_L$ is the left-handed quark doublet, - $Y_U$ and $Y_D$ are Yukawa coupling matrices for up-type and down-type quarks, - $\phi$ is the Higgs doublet, - $\tilde{\phi}$ is the charge-conjugated Higgs doublet, - $U_R$ and $D_R$ are right-handed up and down quark singlets. 4. **Interpretation:** - The large top quark mass $m_t$ arises from a large Yukawa coupling $Y_U$ after the Higgs field acquires a vacuum expectation value. - Both scalar and fermion quark doublets receive large masses from strong and EW symmetry breaking, potentially leading to very heavy quark states near or below the TeV scale. - This mass generation mechanism breaks the original symmetry and affects the participation of quarks in strong interactions. 5. **Summary:** The large mass difference between $t$ and $b$ quarks leads to a significant mixing angle $\theta_t$, and the Yukawa Lagrangian explains how these masses arise from symmetry breaking. This results in heavy quark doublets with masses influenced by both strong and electroweak forces. **Final expressions:** $$\theta_t \approx \frac{m_t}{m_b}$$ $$\mathscr{L}_Y = \bar{Q}_L Y_U \phi U_R + \bar{Q}_L Y_D \tilde{\phi} D_R + h.c.$$