A meson ( ) decays when its quark turns into an or a quark

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

Symmetry and Flavour, Ch Heavy flavour: charm and beauty decays and the magnitudes of the CKM elements

Problem

A D+D^+ meson (cdˉc\bar d) decays when its cc quark turns into an ss or a dd quark, emitting a W+W^+ that becomes udˉu\bar d or usˉu\bar s. Take λ=0.225\lambda = 0.225. ((a)) Classify D+→Kˉ0π+D^+ \to \bar K^0\pi^+, D+→π+π0D^+ \to \pi^+\pi^0 and D+→K+π0D^+ \to K^+\pi^0 as Cabibbo-favoured, singly suppressed or doubly suppressed, and give the power of λ\lambda in each rate relative to the favoured one. ((b)) The measured magnitudes are ∣Vcd∣=0.221±0.004|V_{cd}| = 0.221 \pm 0.004, ∣Vcs∣=0.975±0.006|V_{cs}| = 0.975 \pm 0.006 and ∣Vcb∣=0.040±0.002|V_{cb}| = 0.040 \pm 0.002. Treating the three as independent, find the second-row sum S=∣Vcd∣2+∣Vcs∣2+∣Vcb∣2S = |V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 and its uncertainty σS\sigma_S, and say which element dominates σS\sigma_S. ((c)) A hypothetical fourth-generation down-type quark b′b' would add an element Vcb′V_{cb'} to the row, so that unitarity would read S+∣Vcb′∣2=1S + |V_{cb'}|^2 = 1. Taking ∣Vcb′∣2≤1−S+2σS|V_{cb'}|^2 \le 1 - S + 2\sigma_S as the largest value the data allow, find the upper limit on ∣Vcb′∣|V_{cb'}| and place it between powers of λ\lambda. ((d)) Suppose future measurements left SS at exactly 11 and made ∣Vcs∣|V_{cs}| exact, with the other two uncertainties unchanged. Could the row then exclude ∣Vcb′∣=λ2|V_{cb'}| = \lambda^2 by the same criterion?

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