Question 1 :
A top quark decays into a bottom quark, whose momentum
is measured, and W, which decays to a muon (mu)
and a muon neutrino (nu).
The momentum of the mu is measured and the
momentum of the neutrino transverse to the beam direction
is inferred from the missing transverse momentum in the event.
If the transverse and longitudinal components of the
muon momentum are called p_mu(perp) and p_mu(para), find the
two possible values of the longitudinal momentum of the
neutrino. When is there no solution for the longitudinal
momentum of the neutrino ?
Indirect Top Quark Searches
Before the top quarks were discovered and thus
the top quark mass was unknown, the W width measurement
was one way of
predicting the top quark mass or setting lower limits
on the mass.
In 1994, we had the following information.
R = 10.90 +- 0.32 (stat.) +- 0.29 (syst.) from CDF
where R is the ratio of p p-bar -> W -> e nu cross section
to p p-bar -> Z -> ee cross section.
the theoretical calculation of p p-bar -> W cross section
to p p-bar -> Z cross section = 3.33 +- 0.03
Gamma(Z) = 2.492 +- 0.007 GeV from LEP
Gamma(Z -> ee) = 83.33 +- 0.30 MeV from LEP
Question 2 :
Extract the W branching ratio into e nu from the information
above.
Assuming M_bottom = 0, one can derive the width of W.
Gamma_0 = Gamma(W -> lnu), l = e,mu,tau
Gamma(W -> ud) = 3 Gamma_0
Gamma(W -> cs) = 3 Gamma_0
Gamma(W -> tb) =
3 Gamma_0 x (1 - 3r/2 + r^3/2)
where r = (M_top / M_W)^2,
if M_top < M_W
0 if M_top > M_W
Question 3 :
Plot the inverse of Br(W -> enu), that is, Gamma(W) /
Gamma(W -> enu), as a function of M_top between 0 and 100 GeV,
and overlay the answer of your Question 2 to this plot
Question 4 :
predict M_top or
set a lower limit on M_top
at 95% CL.