2.2 General Considerations
21
the W boson, resulting in a small value of quark p T and a large angular distance
R between the two quarks. Consequently, when requiring the decay quarks to
have a minimum p T , a cut-off at large values of the R distribution is introduced,
and vice-versa. Since the decay angle distributions dσ/d|cos θ
∗
| are identical for W
and Z bosons, the results from Fig. 2.7 (left) are also valid for Z bosons, where the
small correction due to the mass difference m W − m Z leads to a negligible change.
In Fig. 2.7 (right) the detection efficiency is shown for H → bb decays. Since the
distribution dσ/d|cos θ
∗
| is flat for spin-0 particles, there are more collinear decays
than in the W L case, but less than for W T bosons. This is reflected in the detection
efficiency, which lies between the efficiencies obtained for W L and W T decays.
The angular distance R between the two decay quarks is expected to decrease
proportional to 1/ p T of the parent particle. While the relation (2.14) gives the minimum value of R, the tails of the R distribution are more difficult to obtain,
especially if detection thresholds on the quarks are included. The scaling of the R
distribution as a function of p T can be readily estimated, though. Since any given
decay angle θ
∗ will transform proportional to 1/γβ, see (2.11), any given point of the
R distribution will scale as 1/ p T . Once the R distribution is known for a given
value value of p T , this scaling can be used to calculate an expected value of R as
long as no additional kinematic requirements on the decay quarks are imposed. The
relation
R 90
0
dR
dσ
dR
∞
0
dR
dσ
dR
= 0.9
(2.15)
defines the 90th percentile R 90 , which denotes the value below which 90% of the
values of R can be found. For example, in the case of W L decays, R 90 = 0.46
for p T = 500 GeV and hence 90% of all possible values of R lie in the interval
[(160 GeV)/ p T , (230 GeV)/ p T ]. Since these values depend on the nature and polarisation of the decaying boson, the replacement of 2M with ρ 90 in (2.14),
R 90 =
ρ 90
p
α
T
,
(2.16)
can help to estimate the expected range of R for a given boson decay. The parameter
α has been introduced to allow for modifications from the 1/ p T behaviour once p T
thresholds on the quarks are introduced. An overview of the ρ 90 values for W , Z and
H bosons and different polarisation states is given in Table 2.2.
The effect a p T detection threshold has on the R distribution is shown in Fig. 2.8,
where both quarks from the W T (left) and H (right) boson decay are required to have
a minimum transverse momentum, p T,q > 20 GeV. For comparison, the expected
1/ p T scaling for the R 90 percentile is shown, as obtained for the ρ 90 value determined at p T = 1500 GeV. While this function describes the behaviour of the R
distribution well if no kinematic requirements on the decay quarks are imposed,
significant deviations from the 1/ p T scaling are seen once a detection threshold is
introduced. Especially at small values of p T , the naive 1/ p T scaling overestimates
the angular distance significantly. When allowing for deviations of the form 1/ p
α
T ,
21
the W boson, resulting in a small value of quark p T and a large angular distance
R between the two quarks. Consequently, when requiring the decay quarks to
have a minimum p T , a cut-off at large values of the R distribution is introduced,
and vice-versa. Since the decay angle distributions dσ/d|cos θ
∗
| are identical for W
and Z bosons, the results from Fig. 2.7 (left) are also valid for Z bosons, where the
small correction due to the mass difference m W − m Z leads to a negligible change.
In Fig. 2.7 (right) the detection efficiency is shown for H → bb decays. Since the
distribution dσ/d|cos θ
∗
| is flat for spin-0 particles, there are more collinear decays
than in the W L case, but less than for W T bosons. This is reflected in the detection
efficiency, which lies between the efficiencies obtained for W L and W T decays.
The angular distance R between the two decay quarks is expected to decrease
proportional to 1/ p T of the parent particle. While the relation (2.14) gives the minimum value of R, the tails of the R distribution are more difficult to obtain,
especially if detection thresholds on the quarks are included. The scaling of the R
distribution as a function of p T can be readily estimated, though. Since any given
decay angle θ
∗ will transform proportional to 1/γβ, see (2.11), any given point of the
R distribution will scale as 1/ p T . Once the R distribution is known for a given
value value of p T , this scaling can be used to calculate an expected value of R as
long as no additional kinematic requirements on the decay quarks are imposed. The
relation
R 90
0
dR
dσ
dR
∞
0
dR
dσ
dR
= 0.9
(2.15)
defines the 90th percentile R 90 , which denotes the value below which 90% of the
values of R can be found. For example, in the case of W L decays, R 90 = 0.46
for p T = 500 GeV and hence 90% of all possible values of R lie in the interval
[(160 GeV)/ p T , (230 GeV)/ p T ]. Since these values depend on the nature and polarisation of the decaying boson, the replacement of 2M with ρ 90 in (2.14),
R 90 =
ρ 90
p
α
T
,
(2.16)
can help to estimate the expected range of R for a given boson decay. The parameter
α has been introduced to allow for modifications from the 1/ p T behaviour once p T
thresholds on the quarks are introduced. An overview of the ρ 90 values for W , Z and
H bosons and different polarisation states is given in Table 2.2.
The effect a p T detection threshold has on the R distribution is shown in Fig. 2.8,
where both quarks from the W T (left) and H (right) boson decay are required to have
a minimum transverse momentum, p T,q > 20 GeV. For comparison, the expected
1/ p T scaling for the R 90 percentile is shown, as obtained for the ρ 90 value determined at p T = 1500 GeV. While this function describes the behaviour of the R
distribution well if no kinematic requirements on the decay quarks are imposed,
significant deviations from the 1/ p T scaling are seen once a detection threshold is
introduced. Especially at small values of p T , the naive 1/ p T scaling overestimates
the angular distance significantly. When allowing for deviations of the form 1/ p
α
T ,