The decay modes used in this analysis are the lepton + jets modes. As illustrated in Figure 2.8, this final state contains a lepton, a neutrino, two b-jets, and two additional jets from the hadronic W decay. In addition, it is not unlikely that there will be still more jets arising from QCD radiation from either one of the incoming partons or one of the final state quarks.
The neutrino can not be detected directly; however, its presence can
be inferred from a momentum imbalance in the plane transverse to the
beam. (Note that one cannot apply momentum balance arguments in the
direction along the beam, since the spectator quarks in the 
collision will always carry off a large amount of energy, most of
which escapes down the beam pipe.) The experimental signature
thus consists of one high-
isolated lepton, a
substantial imbalance in total transverse momentum (indicating a
neutrino), and several energetic jets.
There are two major backgrounds to this channel. The first is
single-W production with additional radiated jets.
The second is QCD multijet production, where one
jet is misidentified as an electron, and, in addition,
the missing 
is substantially mismeasured.
The former can be reduced by cutting on
the number and energies of the jets as well as on various topological
quantities; and the latter can be reduced by tightening the
electron identification cuts and by raising the missing threshold.
It is also worth noting that each top decay will contain two b-jets.
One might then ask if b-jets can be distinguished from other types
of jets (this is called tagging). This would be useful
for separating top decays from background since
the W + jets and QCD backgrounds
do not contain many heavy quarks.
It would also help in the problem of
properly assigning jets to parent partons within a top decay.
One way of doing this tagging
is to exploit the fact that the B-meson lifetime is large enough
(about 1.5 ps [3, p. 1207]) that the b-decay vertex can
sometimes be separated from the primary event vertex. (A B with
an energy of 
20 GeV
will have 
4, so it can cover a distance of about
2 mm in the lab frame in a 1.5 ps proper time.) The precision
required of this measurement usually demands a silicon vertex
detector, and is difficult under any conditions in the crowded
environment of a top decay.
Another way of identifying b-jets is to notice that a b decay will have a muon in the final state about 22% of the time. Thus, one can identify b-jets by looking for muons embedded in jets. (One can in principle do the same thing for electrons; however, it is much more difficult to identify a nonisolated electron.)
In the lepton + jets final states, if one knows the proper assignment of jets, one can completely reconstruct the decay, even though one component of the neutrino is not measured. In fact, the problem is overconstrained by two equations. Let's stop and enumerate the variables and constraints in this type of event.
Referring back to Figure 2.8, one can identify 13 particles:
p, 
; t, b, 
, 
, 
;
, 
, 
, q, 
; and the pseudoparticle X.
Each particle has four
kinematic variables, so the total number of variables is 
= 52.
Now, what do we know? First of all, we directly measure the energy and direction of most of the final-state particles. Also assume that the masses of these particles are known.
We also know the initial four-vectors of the p and 
:
We have total four-momentum conservation at each of the five internal vertices:
We know the masses of the W's and the 
:
Finally, we know that the two tops have the same mass:
Adding these up, we get
Thus, we have 52 variables and 54 constraints; i.e., the system is overconstrained by two.
If one solves out all trivial constraints, one is left with the 17 measured
variables (three variables each for the lepton and the four jets, and
two components of 
) and one unmeasured variable (the z-component
of the neutrino momentum). There are then three constraints left: the two
W-mass constraints, and the 
= 
constraint.
decay