 Solution

ID: past-exam-of-the-particle-physics-course-of-the-university-of-oxford/2016/5/3/b/solution

u \overset{ˉ}{d}

and

u \overset{s}{ˉ}

V_{u d}

and

V_{u s}

. Exclusive tau lepton decays into a charged pseudoscalar meson give

\frac{B ( τ \to K ν )}{B ( τ \to π ν )} = \frac{f _{K}^{2} ∣ V _{u s} ∣ ^{2}}{f _{π}^{2} ∣ V _{u d} ∣ ^{2}} \frac{( 1 - m _{K}^{2} / m _{τ}^{2} ) ^{2}}{( 1 - m _{π}^{2} / m _{τ}^{2} ) ^{2}},

⟨ 0∣ \overset{q}{ˉ} γ^{μ} γ^{5} u ∣ P (p)⟩ = i f_{P} p^{μ}

0.70/10.83 = 0.06464

, and the kinematic factor is approximately

0.8622

. Hence the directly determined combination is

\frac{f _{K} ∣ V _{u s} ∣}{f _{π} ∣ V _{u d} ∣} ≃ 0.2738 .

With an independent, representative lattice input

f_{K} / f_{π} ≃ 1.193

, this becomes

∣ V_{u s} / V_{u d} ∣ ≃ 0.2295

, close to the source's rounded expectation

0.224/0.975 = 0.2297

. Conversely that expectation and the hadronic input predict a branching fraction ratio near

0.0648

. Without the decay-constant input the table does not isolate a pure Cabibbo-Kobayashi-Maskawa matrix ratio. Setting

f_{K} = f_{π}

would give approximately

0.0455

, demonstrating the importance of flavor symmetry breaking.^[ref]

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