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Chapter 16 - Test of the Standard Model - Coggle Diagram
Chapter 16 - Test of the Standard Model
Z resonance
\(M_Z \propto \frac{g_Z}{q^2 - m_Z^2}\); seems diverge, but need to account for finite lifetime of Z!
replace \(m_Z \rightarrow m_Z - i\Gamma_Z/2\)
then propagator is \(\frac{1}{q^2 - m_Z^2 + im_Z\Gamma_Z}\)
For s channel, \(\sigma\propto |M|^2 \propto |\frac{1}{s - m_Z^2+im_Z\Gamma_Z}|^2 = \frac{1}{(s - m_Z^2)^2 + m_Z^2\Gamma_Z^2}\)
For a
resonance production
, cross section as a function of \(E_{CM} = \sqrt{s}\) is the "lineshape" of resonance, described by the
Breit-Wigner (BW) function
.
Lineshape is the
same
for
all
decay channels!
FWHM
: \(\sqrt{s} = m_Z \pm \frac{\Gamma_Z}{2}\)
In collider, \(\Gamma(e^+e^- \rightarrow Z \rightarrow f\bar{f}) = \frac{12 \pi s}{m_Z^2} \frac{\Gamma_{ee}\Gamma_{ff}}{(s - m_Z^2)^2 + m_Z^2\Gamma_Z^2}\)
Measure lineshape -> find \(m_Z, \Gamma_Z\) -> find \(Gamma\) for individual decay channels.
LEP
: \(e^-e^+\) collider at CERN (1989 -> 2000). 4 detectors: ALEPH, DELPHI, OPAL, L3. Studied Z resonance and WW production.
Corrected for Initial State Radiation (
ISR
, a higher-order QED process) which
decreased
the effective CM energy.
2 MeV uncertainty in energy!!
Result: \(m_Z = 91\) GeV, \(\Gamma_Z = 2.5\) GeV, \(\tau_Z = 10^{-25}s\)
Testing 4-th generation of light neutrino: \(\Gamma_Z = 3 \Gamma_{ll} + \Gamma_{hadron} + N_\nu \Gamma^{SM}_{\nu\nu}\)
\(N_\nu = \frac{\Gamma_Z - 3\Gamma_{ll} - \Gamma_{hadron}}{\Gamma^{SM}_{\nu\nu}}\)
Result
consistent
with 3 generations.
Measure \(\theta_W\)
For \(Z \rightarrow l^-l^+\), \(\frac{c_V}{c_A} = \frac{I_W^{(3)} - 2Q_l \sin^2(\theta_W)}{I_W^{(3)}}\) = \(1 - 4 \sin^2(\theta_W)\)
To measure the ratio, use
forward-backward asymmetry
\(A_{FB}^l = \frac{\sigma_F - \sigma_B}{\sigma_F + \sigma_B} = \frac{N_F- N_B}{N_F - N_B}\)
"Forward" means \(\cos\theta \in [0, 1]\)
\(\frac{d\sigma}{d\Omega} (e^+e^- \rightarrow Z \rightarrow \mu^+\mu^-) = a(1 + \cos^2\theta) + 2b\cos\theta\)
\(a = \Big((c_L^e)^2 + (c_R^e)^2\Big)\Big((c_L^\mu)^2 + (c_R^\mu)^2\Big)\)
\(b = \Big((c_L^e)^2 - (c_R^e)^2\Big)\Big((c_L^\mu)^2 - (c_R^\mu)^2\Big)\)
If b=0
(Z couples to LH and RH chiral states equally), then angular distribution is
symmetric
!
Integrate out the angular distribution to obtain \(\sigma_{F/B}\), then \(A_{FB}^l = \frac{3b}{4a}\). Alternatively:
\(A_{FB}^\mu =\frac{3}{4}A_e A_\mu\)
\(A_f = \frac{(c_L^f)^2 - (c_R^f)^2}{(c_L^f)^2 + (c_R^f)^2} = \frac{2c_V^fc_A^f}{(c_V^f)^2 + (c_A^f)^2} = \frac{2 c_V/c_A}{1 + (c_V/c_A)^2}\)
\(A_{FB}^l\) is measured for multiple lepton final states, e.g. \(ee, \mu\mu, \tau\tau\)
This can be translated to measuring
individual asymmetry parameter
\(A_f\), thus \(c_V/c_A\), thus \(\theta_W\)
Result: \(\theta_W = 0.23\)
\(A_e, A_\mu, A_\tau\) turns out to be similar, indicating
lepton universality
.
W boson
Production at LEP: \(e^+e^- \rightarrow Z^* \rightarrow WW \). Decay to hadron/lepton.
W mass and decay rate are obtained through
direct reconstruction
of final products.
Results: \(m_W = 80\) GeV, \(\Gamma_W = 2.1\) GeV.
The physical W boson mass has quantum loop corrections due to top quark; implied top quark mass ~ 175 GeV.
Top quark
Property
First discovered in \(p\bar{p})\) collision at
Tevatron
(CDF+Dzero exp, 1995); usually produced in \(t\bar{t}\)
pairs
; short lifetime -> could not form bound state.
Almost entirely
decays to \(t \rightarrow W^+ b\) due to large \(|V_{tb}|\).
Tevatron: \(p\bar{p} \rightarrow t\bar{t} \rightarrow bW^+ bW^+\); one W decays hadronically, one decays leptonically.
Propagators: 2 for t-quark, 2 for W (close to on-shell): \(|M|^2 \propto \frac{1}{(q_1^2 - m_t^2)^2 + m_t^2\Gamma_t^2} \times \frac{1}{(q_2^2 - m_t^2)^2 + m_t^2\Gamma_t^2}\)
Decay
\(-iM = -i\frac{g_W}{\sqrt{2}}\epsilon_\mu^*(p_W)\bar{u}(p_b)\gamma^\mu\frac{1}{2}(1-\gamma^5)u(p_t)\)
for \(t \rightarrow W^+b\),
b quark is LH
(neglecting b mass), and W could only be either RH or Longitudinal.
\(\Gamma_{t\rightarrow W^+b} = \frac{G_Fm_t^3}{8\sqrt{2}\pi}(1 - \frac{m_W^2}{m_t^2})^2(1 + \frac{2m_W^2}{m_t^2})\)
Results: \(m_t = 173\) GeV, \(\Gamma_t = 1.5\) GeV, \(\tau_t = 10^{-25} s\)
Short lifetime ->
decay before hadronizing
!
