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CKM Mixing Unitarity Triangle

What you are seeing: the CKM unitarity condition VudVub+VcdVcb+VtdVtb=0V_{ud}V^*_{ub} + V_{cd}V^*_{cb} + V_{td}V^*_{tb} = 0 drawn as three complex side-vectors added tip-to-tail, closing the triangle with vertices (0,0)(0,0), (1,0)(1,0), (ρˉ,ηˉ)(\bar\rho, \bar\eta) and angles α,β,γ\alpha, \beta, \gamma. The enclosed area is the Jarlskog invariant, so the CP-asymmetry panel shows the B0B^0 vs Bˉ0\bar B^0 golden-mode rates (sin2β\propto \sin 2\beta): a flat triangle gives equal rates (no CP violation), a fat one a large asymmetry. Drag ρˉ,ηˉ\bar\rho, \bar\eta to move the apex.

Figure 1. CKM matrix and unitarity triangle.
ρ̄0.157
η̄0.355

WHAT TO TRY

  • Drag the apex (rho-bar, eta-bar): the three CKM side-vectors stay tip-to-tail and the triangle keeps closing, because unitarity forces the sum to zero. The angles alpha, beta, gamma always add to 180 degrees.
  • Flatten the triangle toward the real axis (eta-bar to 0): the area collapses and so does the Jarlskog invariant, the single measure of CP violation. A degenerate triangle means no CP violation.
  • Compare the apex against the CP-asymmetry bars: a non-zero height eta-bar is what makes the B0 and B0-bar decay rates differ, the matter-antimatter asymmetry the experiments measure.