An (already unfolded

PHYSICS · Multi-part · Fermi University Physics · Particle Physics & Quantum Fields · Particles, interactions and experiment · Electroweak theory and the Standard Model

Symmetry and Flavour, Ch Resonances: the relativistic Breit-Wigner, unitarity and the Z line shape

Problem

An (already unfolded, i.e. with initial-state radiation removed) scan of the hadronic cross section near the

Z

pole gives

\sigma_{\rm had}

in nb at each

\sqrt s

in GeV:

\sqrt s=88.69

8.30

;

89.94

20.75

;

91.19

41.50

;

92.44

20.75

;

93.69

8.30

. The lepton partial width is

\Gamma_\ell=83.90\,\mathrm{MeV}

and the measured ratio

R_\ell=\Gamma_{\rm had}/\Gamma_\ell=20.767

. ((a)) Read off

M_Z

(the peak position) and

\Gamma_Z

(the full width at half maximum, using the fact that the half-maximum points are a distance

\Gamma_Z/2

from the peak in

\sqrt s

) from the table. ((b)) Using

R_\ell

and

\Gamma_\ell

, find

\Gamma_{\rm had}

. ((c)) Find

\Gamma_{\rm inv}=\Gamma_Z-\Gamma_{\rm had}-3\Gamma_\ell

and the number of light neutrino species

N_\nu=\Gamma_{\rm inv}/165.9\,\mathrm{MeV}

(the width per species, from Chapter 2). ((d)) The table has already had initial-state radiation unfolded. Explain qualitatively why, if it had not been, the raw measured peak would sit at a cross section below

41.50\,\mathrm{nb}

and at a

\sqrt s

above

M_Z

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