A sample of mesons ( ) is reconstructed in a silicon vertex detector

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

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

Problem

A sample of

B_s^0

mesons (

m_{B_s^0} = 5.366\,\mathrm{GeV}

) is reconstructed in a silicon vertex detector. The tracks have mean lab-frame momentum

\langle p\rangle = 45\,\mathrm{GeV}

, and an exponential fit to the distribution of production-to-decay vertex separations gives a mean decay length

\langle L\rangle = 3.77\,\mathrm{mm}

. The measured semileptonic branching fraction is

\mathcal{B}_{\rm sl} = 10.7

percent. Take the

B_s^0

to decay by the same spectator mechanism as the

B^0

b \to c\,\ell\,\bar\nu

with the light quarks unaffected, so that

\Gamma_{\rm sl} = \frac{G_F^2 m_b^5}{192\pi^3}\,f(x)\,|V_{cb}|^2, \qquad x = \frac{m_c^2}{m_b^2},

with

m_b = 4.8\,\mathrm{GeV}

m_c = 1.5\,\mathrm{GeV}

, and

f(x) = 0.492

for this mass ratio. Use

G_F = 1.166\times10^{-5}\,\mathrm{GeV^{-2}}

\hbar = 6.582\times10^{-25}\,\mathrm{GeV\,s}

and

\hbar c = 0.1973\,\mathrm{GeV\,fm}

. ((a)) Using the relativistic relation between lab decay length and proper time, find the mean proper lifetime

\tau_{B_s^0}

implied by these vertex data, in picoseconds. ((b)) Using this lifetime and the measured semileptonic branching fraction, extract

|V_{cb}|

, and compare it with the value

A\lambda^2 \approx 0.042

that Chapter 10 used.

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