Past exam of the particle physics course of the University of Oxford / 2015 / 8 / d

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The CKM matrix appears in the charged current of b. For a tree semileptonic transition, a schematic extraction is

Γ = G_{F}^{2} ∣ V_{ij} ∣^{2} \times phase space \times hadronic factor \times (1 + corrections)

. Measure an absolute rate or differential spectrum, calculate or independently determine the hadronic factor, and solve for the modulus. The process is not a direct count of quark flavours in a detector: form factors, decay constants, nuclear structure and radiation can limit the result.

$∣ V_{u d} ∣$ : superallowed nuclear $0^{+} \to 0^{+}$ beta decay, with neutron and pion beta decay cross-checks; correct for nuclear and radiative effects.
$∣ V_{u s} ∣$ : $K \to π ℓ ν$ spectra with the $K π$ form factor; compare $K \to μν$ and $π \to μν$ using decay constants.
$∣ V_{u b} ∣$ and $∣ V_{c b} ∣$ : exclusive $B \to π ℓ ν$ and $B \to D^{(*)} ℓ ν$ , or inclusive charmless/charmed semileptonic spectra; separate the rare charmless component and control the hadronic calculation.
$∣ V_{c d} ∣$ and $∣ V_{cs} ∣$ : charm semileptonic or leptonic decays with form factors or decay constants, and charm production by neutrinos; charm-tagged W boson decays give a complementary constraint.
$∣ V_{t b} ∣$ : single-top quark production and the ratio of $t \to Wb$ to all $t \to W q$ decays, under stated assumptions about allowed interactions.
$∣ V_{t d} ∣$ and $∣ V_{t s} ∣$ : neutral B-meson mixing frequencies and rare loop-induced decays, with loop dynamics and hadronic inputs. Mixing-frequency ratios cancel some common uncertainties.

Rate moduli alone do not determine the complex phase. CP-asymmetric decay distributions and interference in neutral-meson systems, or tree-level

B \to DK

interference, constrain the phase and the CKM unitarity triangle. Unitarity supplies consistency checks and indirect determinations, but a fit that imposes it must be distinguished from a direct measurement of an element.

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Past exam of the particle physics course of the University of Oxford / 2015 / 8 / d

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