2.4 Identifying Particle Decays with Jet Substructure
31
2.4 Identifying Particle Decays with Jet Substructure
Once the dynamics of an event in a high energy collision have been determined
using a suitable jet algorithm, analysing the substructure of these jets reveals valuable
information on the particles produced and their decays. Jet substructure is ubiquitous
in the identification of boosted heavy SM particles in their hadronic decays, but can
also be used in the search for exotic decays of BSM particles.
2.4.1 Jet Mass
The jet mass is the most important observable for identifying jets from heavy particle
decays. It is defined as the square root of the Lorentz-invariant product P μ P
μ ,
where the four-momentum P μ is obtained by summing the four-momenta of all jet
constituents. While partons can be massless, jets always have mass due to perturbative
radiation and hadronisation effects. For a massless parton the mass generated by the
collinear 1 → 2 splitting can be approximated by [59]
m
2
= ( p 1 + p 2 )
2
≈ p T,1 p T,2 R
2
12
(2.29)
where p i are the four-momenta of the massless particles 1 and 2 with transverse
momenta p T,i and R 12 is the geometrical distance in η and φ between them. A jet
acquires mass by a sequence of splittings, followed by non-perturbative hadronisation. The partonic cross section is proportional to 1/m and thus features singularities
in the collinear (R 12 → 0) and soft ( p T,i → 0) regime for fixed-order calculations.
The singularities can be overcome by resumming no-splitting probabilities, resulting in Sudakov form factors similar as in predictions for inclusive observables, like
event shapes in e
+ e
− collisions. However, a similar precision as obtained for inclusive observables is more difficult to achieve for jet substructure observables because
of non-global logarithms which complicate matters [205–207]. A resummation of
leading logarithms from subsequent collinear splittings results in a Sudakov peak in
the jet mass distribution, which position in m increases roughly linear with jet p T .
There are additional contributions from the hadronisation and the underlying event,
which can be estimated analytically similarly to the changes in the jet p T (Sect. 2.5.2).
The leading power corrections for m
2 scale with p T R for hadronisation corrections
and with p T R
4 for contributions from the underlying event. Hence, the hadronisation
corrections dominate by three powers of R for R < 1. Note that this approximation
is only valid to the right of the Sudakov peak and breaks down in the vicinity of
the peak [208]. An example is shown in Fig. 2.12, where results from a resummed
calculation of the m/ p T distribution with two different choices of the hadronisation
scale are compared to results from Pythia 8 and Herwig++. The results from
the analytical calculations agree well with predictions from parton showers, giving
confidence in the calculations and showing that the modelling of non-perturbative
31
2.4 Identifying Particle Decays with Jet Substructure
Once the dynamics of an event in a high energy collision have been determined
using a suitable jet algorithm, analysing the substructure of these jets reveals valuable
information on the particles produced and their decays. Jet substructure is ubiquitous
in the identification of boosted heavy SM particles in their hadronic decays, but can
also be used in the search for exotic decays of BSM particles.
2.4.1 Jet Mass
The jet mass is the most important observable for identifying jets from heavy particle
decays. It is defined as the square root of the Lorentz-invariant product P μ P
μ ,
where the four-momentum P μ is obtained by summing the four-momenta of all jet
constituents. While partons can be massless, jets always have mass due to perturbative
radiation and hadronisation effects. For a massless parton the mass generated by the
collinear 1 → 2 splitting can be approximated by [59]
m
2
= ( p 1 + p 2 )
2
≈ p T,1 p T,2 R
2
12
(2.29)
where p i are the four-momenta of the massless particles 1 and 2 with transverse
momenta p T,i and R 12 is the geometrical distance in η and φ between them. A jet
acquires mass by a sequence of splittings, followed by non-perturbative hadronisation. The partonic cross section is proportional to 1/m and thus features singularities
in the collinear (R 12 → 0) and soft ( p T,i → 0) regime for fixed-order calculations.
The singularities can be overcome by resumming no-splitting probabilities, resulting in Sudakov form factors similar as in predictions for inclusive observables, like
event shapes in e
+ e
− collisions. However, a similar precision as obtained for inclusive observables is more difficult to achieve for jet substructure observables because
of non-global logarithms which complicate matters [205–207]. A resummation of
leading logarithms from subsequent collinear splittings results in a Sudakov peak in
the jet mass distribution, which position in m increases roughly linear with jet p T .
There are additional contributions from the hadronisation and the underlying event,
which can be estimated analytically similarly to the changes in the jet p T (Sect. 2.5.2).
The leading power corrections for m
2 scale with p T R for hadronisation corrections
and with p T R
4 for contributions from the underlying event. Hence, the hadronisation
corrections dominate by three powers of R for R < 1. Note that this approximation
is only valid to the right of the Sudakov peak and breaks down in the vicinity of
the peak [208]. An example is shown in Fig. 2.12, where results from a resummed
calculation of the m/ p T distribution with two different choices of the hadronisation
scale are compared to results from Pythia 8 and Herwig++. The results from
the analytical calculations agree well with predictions from parton showers, giving
confidence in the calculations and showing that the modelling of non-perturbative