![](https://csdnimg.cn/release/download_crawler_static/12266152/bg9.jpg)
Note that soft drop multiplicity is a special case (κ = 0) of this more general observable,
with the same criteria for IRC safety. We study the weighted multiplicity in detail in
appendix A, but find its quark/gluon discrimination power to be inferior to the discrete
κ = 0 case. In fact, LL reasoning leads one to expect the soft drop multiplicity n
SD
to have
the best discrimination power of any observable defined on the (z
n
, θ
n
) values; see the end
of section 3.3 for a short discussion.
Nevertheless, several other promising variants of soft drop multiplicity might prove
useful:
• The weighted soft drop multiplicity in eq. (2.8) only refers to the momentum fractions
z
n
in the sum over emissions. One could also consider an angle-weighted variant
X
n
z
κ
n
θ
α
n
, (2.9)
or indeed any function of z
n
and θ
n
. The potential advantage of including θ
n
infor-
mation is that even for θ
cut
= 0, such observables would be collinear safe for α > 0.
• Instead of counting emissions only from the trunk of the C/A tree, we could extend the
sum to include all branchings down to the angular resolution θ
cut
. This multiplicity
variant would require a modification of the ISD algorithm: in step 4
0
, the recursion
would be applied to both subjets i and j, not just the harder one. This is a step in
similarity towards full hadron multiplicity, reducing to it exactly when z
cut
= θ
cut
= 0.
This variant of soft drop multiplicity is more difficult to study analytically, however,
due to the nonlinear structure of the recursion. Moreover, it is not clear that this
variant would provide a performance advantage over n
SD
. While gluons emitted
from the hard core of a quark (gluon) jet give rise to factors of C
F
(C
A
), subsequent
emissions from those gluons give rise to factors of C
A
regardless of the jet flavor; this
might wash out quark/gluon discrimination power.
• The original soft drop algorithm uses a C/A tree to mimic the angular-ordered struc-
ture of the parton shower. One could also study variants based on reclustering with
the generalized-k
t
algorithm with exponent p [49, 50]. The C/A algorithm used above
corresponds to p = 0, while the k
t
algorithm uses p = 1. For this variant, it would
make sense to replace the angular cut θ
cut
with a cut d
cut
on the generalized distance
measure d
ij
.
This last k
t
variant is of particular interest, given the discussion below in section 3.3.
Nonperturbative physics typically dominates when k
t
' Λ
QCD
, so it makes sense to use a
clustering algorithm where the clustering scale is “parallel” to the nonperturbative scale.
This variant of n
SD
would then allow the nonperturbative phase space to be clearly sepa-
rated from the perturbative region and avoided. This would open up as much perturbative
phase space for measured emissions as possible. We note that it is possible to mimic some
of the LL structure of the k
t
variant by using ISD with β = −1, though there would be
differences going to NLL order.
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