Someone argues: " proceeds through a tree diagram carrying the CKM combination

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Flavour, neutrinos and CP violation · Flavour, neutrinos and CP violation

Symmetry and Flavour, Ch B mesons: mixing, time-dependent CP asymmetries and the unitarity triangle

Problem

Someone argues: "B0→K+π−B^0 \to K^+\pi^- proceeds through a tree diagram carrying the CKM combination VubVus∗V_{ub}V_{us}^*, which has a large weak phase γ\gamma. Writing the amplitude as A=∣A∣eiγA = |A|e^{i\gamma} and the conjugate-channel amplitude as Aˉ=∣A∣e−iγ\bar A = |A|e^{-i\gamma}, it follows that ∣A∣2−∣Aˉ∣2|A|^2 - |\bar A|^2 must be non-zero, of order sin⁡γ\sin\gamma, so this single diagram already gives direct CP violation." ((a)) Find the error in this argument. ((b)) State, using the general two-amplitude expression ∣A∣2−∣Aˉ∣2=−4∣A1∣∣A2∣sin⁡(δ1−δ2)sin⁡(ϕ1−ϕ2)|A|^2 - |\bar A|^2 = -4|A_1||A_2|\sin(\delta_1-\delta_2)\sin(\phi_1-\phi_2), the minimum ingredients a decay amplitude must have before any direct CP violation is possible.

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