Bottom quark

​Vud​Vcd​Vtd​​Vus​Vcs​Vts​​Vub​Vcb​Vtb​​

​, with four independent parameters: three mixing angles and one CP-violating phase. A convenient approximation is the Wolfenstein parametrization, which expands the matrix in powers of the small Cabibbo angle parameter $\lambda \approx |V_{us}| \approx 0.22$λ≈∣Vus​∣≈0.22, along with $A \approx 0.81$A≈0.81, $\rho$ρ, and $\eta$η. This form highlights the hierarchical structure, where transitions involving the bottom quark (third generation) are suppressed, such as $|V_{ub}| \sim 3.7 \times 10^{-3}$∣Vub​∣∼3.7×10−3 and $|V_{cb}| \sim 0.041$∣Vcb​∣∼0.041.^[23][24] One key unitarity relation from the CKM matrix is $V_{ud} V_{ub}^* + V_{cd} V_{cb}^* + V_{td} V_{tb}^* = 0$Vud​Vub∗​+Vcd​Vcb∗​+Vtd​Vtb∗​=0, which, when normalized by $|V_{cd} V_{cb}^*|$∣Vcd​Vcb∗​∣, defines the unitarity triangle in the complex plane. The triangle's interior angles are $\alpha \equiv \arg\left(-\frac{V_{td} V_{tb}^*}{V_{ud} V_{ub}^*}\right)$α≡arg(−Vud​Vub∗​Vtd​Vtb∗​​), $\beta \equiv \arg\left(-\frac{V_{cd} V_{cb}^*}{V_{td} V_{tb}^*}\right)$β≡arg(−Vtd​Vtb∗​Vcd​Vcb∗​​), and $\gamma \equiv \arg\left(-\frac{V_{ud} V_{ub}^*}{V_{cd} V_{cb}^*}\right)$γ≡arg(−Vcd​Vcb∗​Vud​Vub∗​​), with current measurements yielding $\alpha \approx 84^\circ$α≈84∘, $\beta \approx 22^\circ$β≈22∘, and $\gamma \approx 66^\circ$γ≈66∘. The base of the triangle corresponds to $|V_{cd} V_{cb}^*| \approx 1$∣Vcd​Vcb∗​∣≈1, while the closure is tested through precise determinations of $|V_{ub}|$∣Vub​∣ and $|V_{cb}|$∣Vcb​∣, providing stringent constraints on the apex coordinates $(\rho, [\eta](/page/Eta))$(ρ,[η](/page/Eta)).^[24] The bottom quark plays a central role in constraining the CKM matrix, as its decays dominate measurements of the third-generation mixing elements. Semileptonic decays such as $B \to \pi \ell \nu$B→πℓν probe $|V_{ub}|$∣Vub​∣, while $B \to D \ell \nu$B→Dℓν determine $|V_{cb}|$∣Vcb​∣, with inclusive and exclusive analyses yielding consistent values that anchor global fits to the unitarity triangle. Additionally, rare flavor-changing neutral-current processes like $b \to s \gamma$b→sγ, observed in $B \to X_s \gamma$B→Xs​γ with a branching ratio of $(3.49 \pm 0.19) \times 10^{-4}$(3.49±0.19)×10−4, are loop-suppressed in the Standard Model and depend on CKM factors such as $|V_{ts} V_{tb}^*|$∣Vts​Vtb∗​∣; deviations from predictions could signal new physics beyond the CKM framework.^[24][25]

CP Violation and Heavy Quark Symmetry

Experimental Studies

Production and Observation

Recent Measurements and Applications