Consider the pion leptonic decays and

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Electroweak theory and the Standard Model · Flavour, neutrinos and CP violation

Symmetry and Flavour, Ch Quark mixing: the Cabibbo angle, the GIM mechanism and the CKM matrix

Problem

Consider the pion leptonic decays

\pi^+ \to e^+\nu_e

and

\pi^+ \to \mu^+\nu_\mu

, with

m_\pi = 139.57\,\mathrm{MeV}

m_\mu = 105.66\,\mathrm{MeV}

m_e = 0.511\,\mathrm{MeV}

, and measured

\mathcal{B}(\pi^+\to e^+\nu_e) = 1.230\times10^{-4}

\mathcal{B}(\pi^+\to \mu^+\nu_\mu) = 99.988\%

. ((a)) Using the two-body decay-rate formula with the

m_\ell^2(1-m_\ell^2/m_\pi^2)^2

helicity-suppression factor (whose origin is the fixed neutrino helicity), and taking

|V_{ud}|

and

f_\pi

common to both channels, predict

\Gamma(\pi^+\to e^+\nu)/\Gamma(\pi^+\to \mu^+\nu)

and compare it with the measured value. ((b)) As an idealisation, suppose a strangeness-changing semileptonic rate is

1/18

of a strangeness-conserving one at equal phase space, with both running through the same left-handed vertex rule so that only the quark coupling differs. Extract

\theta_C

from this ratio and compare with the value

13.0^\circ

obtained from kaon decays.

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