24
2 Phenomenology of Jet Substructure
Table 2.3 Numerical values of ρ and α for top quark decays for the approximation in (2.19). Shown
are the values for the minimum value that max can take, and the 70th, 80th and 90th percentiles
of max . All values are obtained for a quark threshold of p T,q > 20 GeV
a requirement of p T,q > 30 GeV leads to a drop of the efficiency to 70%, and with
p T,q > 40 GeV the efficiency is only 60%.
Similar to the case of vector boson and H decays, a p T threshold on the quarks
leads to a change in the distribution of the angular distance between the quarks.
Since there are three quarks involved, the relevant quantity is the maximum angular
distance
max = max
q), ,R(b, q
), ,R(q, q
)
,
(2.18)
which represents a proxy for fully merged final states. Obviously, approximation (2.14) will not yield an accurate prediction for max . Surprisingly, it still gives
a relatively good estimate for the MPV of max , as can be seen in Fig. 2.10. However, the 1/ p T scaling does not give an accurate description of the minimum value
max can take, and also cannot be used to predict a given percentile of the max
distribution. A better phenomenological approximation is obtained by modifying the
scaling with an exponent α, similar to (2.16),
max =
ρ
p
α
T
.
(2.19)
Numerical values for ρ and α are given in Table 2.3, for the three percentiles shown
in Fig. 2.10. The values given approximate the shape of the three percentiles to within
1% in the range 200 < p T < 1500 GeV.
Fig. 2.10 Maximum angular
distance max of the three
quarks from the hadronic top
quark decay, as a function of
the top quark p T . The
transverse momenta of the
three quarks b, q and q
are
required to be
p T,q > 20 GeV. The fraction
of events contained within a
given interval in max are
shown by shaded areas, the
MPV is depicted by a dashed
line. For comparison, also
shown are the expressions
2m t / p T (solid line) and
(800 GeV)/ p T (dotted line)
[GeV]
T
p
200
400
600
800
1000 1200 1400
')
q
(bq
max
R
Δ
0
1
2
3
'
q
bq
→
bW
→
t
> 20 GeV:
T,q
Percentiles for p
70%
80%
90%
MPV
1
−
T
p
t
2m
1
−
T
(800 GeV) p
2 Phenomenology of Jet Substructure
Table 2.3 Numerical values of ρ and α for top quark decays for the approximation in (2.19). Shown
are the values for the minimum value that max can take, and the 70th, 80th and 90th percentiles
of max . All values are obtained for a quark threshold of p T,q > 20 GeV
a requirement of p T,q > 30 GeV leads to a drop of the efficiency to 70%, and with
p T,q > 40 GeV the efficiency is only 60%.
Similar to the case of vector boson and H decays, a p T threshold on the quarks
leads to a change in the distribution of the angular distance between the quarks.
Since there are three quarks involved, the relevant quantity is the maximum angular
distance
max = max
q), ,R(b, q
), ,R(q, q
)
,
(2.18)
which represents a proxy for fully merged final states. Obviously, approximation (2.14) will not yield an accurate prediction for max . Surprisingly, it still gives
a relatively good estimate for the MPV of max , as can be seen in Fig. 2.10. However, the 1/ p T scaling does not give an accurate description of the minimum value
max can take, and also cannot be used to predict a given percentile of the max
distribution. A better phenomenological approximation is obtained by modifying the
scaling with an exponent α, similar to (2.16),
max =
ρ
p
α
T
.
(2.19)
Numerical values for ρ and α are given in Table 2.3, for the three percentiles shown
in Fig. 2.10. The values given approximate the shape of the three percentiles to within
1% in the range 200 < p T < 1500 GeV.
Fig. 2.10 Maximum angular
distance max of the three
quarks from the hadronic top
quark decay, as a function of
the top quark p T . The
transverse momenta of the
three quarks b, q and q
are
required to be
p T,q > 20 GeV. The fraction
of events contained within a
given interval in max are
shown by shaded areas, the
MPV is depicted by a dashed
line. For comparison, also
shown are the expressions
2m t / p T (solid line) and
(800 GeV)/ p T (dotted line)
[GeV]
T
p
200
400
600
800
1000 1200 1400
')
q
(bq
max
R
Δ
0
1
2
3
'
q
bq
→
bW
→
t
> 20 GeV:
T,q
Percentiles for p
70%
80%
90%
MPV
1
−
T
p
t
2m
1
−
T
(800 GeV) p