2.4 Identifying Particle Decays with Jet Substructure
33
procedure is usually used,
8 where seed values for the subjet axes are obtained by
running the k T algorithm in the exclusive mode. The value of τ N is then minimised
in using these seeds as initial values.
The value of N -subjettiness ranges between 0 and 1 and assesses the degree to
which the jet constituents are localised near N axes. While a single value of τ N has
little discrimination power between one and N prong decays, as the value of τ N
can always be reduced by adding an additional axis, ratios of N -subjettiness values,
τ N /τ M , often written as τ NM , show excellent discrimination power. One important
feature of N -subjettiness ratios is that these are analytically calculable [211], such
that cross section measurements can be safely defined using selections based on τ NM .
Additionally, by comparisons with predictions from event generators, the accuracy
of parton shower models can be tested and improved systematically.
The ratio τ 21 is commonly used to identify two prong decays such as hadronic W
and Z decays. Note that this observable is only Sudakov-safe [212, 213], but can be
made IRC safe by either requiring a minimum jet mass or a minimum value of τ 1 . It
has also been pointed out that this observable has a complex singular structure [214],
which leads to shoulders in the τ 21 distribution. This implies that higher order calculations are necessary to achieve reliable theoretical predictions. The presence of
singularities also leads to large non-perturbative corrections, making this observable
susceptible to different tunes of MC event generators. While no analytical studies
for three prong decays τ 32 exist, it is expected that the increase to three axes will
result in an even more complicated singular structure for the resolved τ 32 → 1. A
theoretically less complex variable with similar discriminative power as τ NM should
therefore be favoured in experimental analyses.
Energy Correlation Functions
A computational drawback of N -Subjettiness is the explicit identification of subjet
axes, which also depends on the specifics of the minimisation algorithm used. Generalised energy correlation functions [215] are substructure variables independent of
subjet axes. These are a generalisation of the event shape parameter C [216, 217],
and the IRC-safe N -point energy correlation function (ECF) is defined as
ECF(N , β) =
i 1 N
a=1
p Ti a
N −1
b=1
N
c=b+1
R i b i c
β
(2.31)
for hadron colliders, where i a is the index of particle a inside the jet and N J is the total
number of particles in the jet. At e
+ e
− colliders, where energies and angles are the
natural observables, p T is replaced with E and R with θ in (2.31). A dimensionless
variable that can be used to identify if a jet has N hard prongs is the ratio of two ECFs,
ECF(N + 1, β)/ECF(N , β), since ECF(N + 1, β) will be significantly smaller than
8 This is in contrast to a multi-pass minimisation, where 100 or more minimisations are done, with
different initial axes in each attempt. While this procedure is more likely to find the global minimum,
the one-pass minimisation gives reasonable results even though it only finds a local minimum.
33
procedure is usually used,
8 where seed values for the subjet axes are obtained by
running the k T algorithm in the exclusive mode. The value of τ N is then minimised
in using these seeds as initial values.
The value of N -subjettiness ranges between 0 and 1 and assesses the degree to
which the jet constituents are localised near N axes. While a single value of τ N has
little discrimination power between one and N prong decays, as the value of τ N
can always be reduced by adding an additional axis, ratios of N -subjettiness values,
τ N /τ M , often written as τ NM , show excellent discrimination power. One important
feature of N -subjettiness ratios is that these are analytically calculable [211], such
that cross section measurements can be safely defined using selections based on τ NM .
Additionally, by comparisons with predictions from event generators, the accuracy
of parton shower models can be tested and improved systematically.
The ratio τ 21 is commonly used to identify two prong decays such as hadronic W
and Z decays. Note that this observable is only Sudakov-safe [212, 213], but can be
made IRC safe by either requiring a minimum jet mass or a minimum value of τ 1 . It
has also been pointed out that this observable has a complex singular structure [214],
which leads to shoulders in the τ 21 distribution. This implies that higher order calculations are necessary to achieve reliable theoretical predictions. The presence of
singularities also leads to large non-perturbative corrections, making this observable
susceptible to different tunes of MC event generators. While no analytical studies
for three prong decays τ 32 exist, it is expected that the increase to three axes will
result in an even more complicated singular structure for the resolved τ 32 → 1. A
theoretically less complex variable with similar discriminative power as τ NM should
therefore be favoured in experimental analyses.
Energy Correlation Functions
A computational drawback of N -Subjettiness is the explicit identification of subjet
axes, which also depends on the specifics of the minimisation algorithm used. Generalised energy correlation functions [215] are substructure variables independent of
subjet axes. These are a generalisation of the event shape parameter C [216, 217],
and the IRC-safe N -point energy correlation function (ECF) is defined as
ECF(N , β) =
i 1 N
a=1
p Ti a
N −1
b=1
N
c=b+1
R i b i c
β
(2.31)
for hadron colliders, where i a is the index of particle a inside the jet and N J is the total
number of particles in the jet. At e
+ e
− colliders, where energies and angles are the
natural observables, p T is replaced with E and R with θ in (2.31). A dimensionless
variable that can be used to identify if a jet has N hard prongs is the ratio of two ECFs,
ECF(N + 1, β)/ECF(N , β), since ECF(N + 1, β) will be significantly smaller than
8 This is in contrast to a multi-pass minimisation, where 100 or more minimisations are done, with
different initial axes in each attempt. While this procedure is more likely to find the global minimum,
the one-pass minimisation gives reasonable results even though it only finds a local minimum.