Physics:PP/4/Jets

Jets

Concept and Definitions

Hadronic jets are the collimated spray of particles produced by a high momentum quark or gluon. Theorists can define jets by applying cluster algorithms (to be discussed later) on a list of partons (quarks and gluons) or stable particles after parton shower and hadronization. In experiments, jet can be constructed from calorimeter cells, tracks or more complex objects, such as particle flows reconstructed using the combined information from calorimeter clusters and tracks.

On a very basic level, jets can easily be found after a visual analysis of events in experiments. However, to compare observables based on jet characteristics with a theory, one needs an objective and a unambiguous jet definition to be used by experimentalists and theorists on an equal footing.

Jet definitions should satisfy the following requirements:

(1) Predictions for jets should be infrared and collider safe: i.e. a measured jet cross section should not change if the original parton radiates a soft parton or if it splits into two collinear partons;

(2) The decision on which jet algorithm to use has to be based on understanding of the size of high-order QCD corrections. At fixed-order QCD, an observable, A, can be expressed by a perturbation series in powers of the strong coupling constant, $\displaystyle{ A=A_{1}\; \alpha _{s}(\mu _{R}) + A_{2}\;\alpha _{s}^{2}(\mu _{R}) + B(\mu _{R}) }$, where $\displaystyle{ B(\mu _{R}) }$ denotes missing high-order QCD terms, $\displaystyle{ \mu _{R} }$ is the renormalisation scale used to deal with the ultraviolet divergences (A is independent of $\displaystyle{ \mu _{R} }$). To estimate the contribution from unknown $\displaystyle{ B(\mu _{R}) }$, $\displaystyle{ \mu _{R} }$ can be varied within some range. If the renormalisation scale is set to the jet transverse energy, $\displaystyle{ \mu _{R}=E_{\bot } }$, a typical variation presently adopted to estimate the renormalisation scale uncertainty is $\displaystyle{ 0.5E_{\bot }\lt \mu _{R}\lt 2E_{\bot } }$ (This range is considered as the convention.). An optimal algorithm has to have a small uncertainty associated with such variations. This gives an indication that missing high-order QCD contributions do not change significantly the fixed-order theoretical predictions;

(3) Close correspondence with the original parton direction, since the association of jets with hard partons is the basic assumption when the theoretical predictions are compared to the data. This property is essential when simple kinematical considerations are used to reconstruct heavy particles from the invariant mass of two or more jets;

(4) An optimal jet definition should have small hadronization corrections, as well as small hadronization uncertainties.

The hadronization correction factor, C, is evaluated as the ratio $\displaystyle{ \sigma _{hadrons}^{MC}/\sigma _{partons}^{MC} }$, where $\displaystyle{ \sigma ^{MC} }$ is the jet cross section obtained using Monte Carlo (MC) models generated for hadrons or partons. For an optimal jet algorithm, $\displaystyle{ C\sim 1 }$. Note that such correction factor used to multiply the fixed-order QCD cross sections is not fully justified for every observable: the parton level of MC models is fundamentally non-perturbative because of the QCD cut-off used to deal with divergent integrals, and the number of partons in MC models significantly exceeds the multiplicity of partons for fixed-order calculations.

The hadronization correction techniques was extensively used at HERA ep experiments where jet transverse momenta were relatively small (10- 70~GeV). Therefore, it was essential to apply the hadronization correction to theoretical models for partons. This correction was adopted when: a) a fixed-order QCD calculation and the corresponding parton-level MC prediction well agree ($\displaystyle{ \lt 5\% }$ difference); b) the hadronization correction is not large ($\displaystyle{ \lt 20\% }$); c) the hadronization uncertainties are small ($\displaystyle{ \lt 5\% }$). The latter can be found by comparing hadronization corrections evaluated using the Lund string fragmentation model with the cluster fragmentation models, which are both implemented in MC simulations. Numerous results from HERA indicated that measured jet cross sections better agree with the next-to-leading order (NLO) calculations corrected using the MC hadronization correction;

(5) Suppression of soft processes related to the beam remnants or minimum bias events from pileup events, i.e. multiple collisions of incoming particles per bunch crossing;

(6) Small experimental uncertainties, such as those related to jet resolution and energy scale;

(7) On a computational level, jet algorithms should be numerically fast and simple to use in experimental analyses and in theoretical calculations. Note that the same jet algorithm has to be uniquely defined for experimental and theoretical calculation inputs, without any additional modification.

To be done

Jet Reconstruction Algorithms

On an abstract level, jet finding algorithms solve the clustering problem, i.e. they attempt organization of final-state particles (or objects that represent particles) into groups (clusters) while keeping dissimilarity between groups as high as possible. Such algorithms belong to unsupervised learning, which is a common technique for statistical data analysis used in many fields.

As mentioned before, jet algorithms reduce information on the hadronic final state resulting from high-energy collisions: Instead of analysing a large number of hadrons produced in an event, one could focus on a relatively small number of jets that approximate characteristics (i.e. momenta, rapidity etc) of original quarks and gluons resulting from collisions. This helps concentrate on main features of the underlying physics, the theory of quantum chromodynamics (QCD) used to describe dynamics of quarks and gluons, as well as allows the reconstruction of heavy particles (Z, W bosons, top quarks, Higgs boson) by grouping their decay products.

A number of jet algorithms has been proposed in the past (see an overview [1] [2] [3]. It can be pointed out that there is no algorithm which is optimal for all possible jet-related studies. Usually, different jet algorithms have different emphasis. Some jet finders are preferable for precise comparisons with QCD theory, since the jet cross sections reconstructed with such algorithms have small fixed-order perturbative corrections, as well as small hadronization corrections. However, such jet algorithms may not be the most optimal for other tasks.

References

1. Moretti, StefanoLonnblad, LeifSjostrand, Torbjorn "New and old jet clustering algorithms for electron positron events" JHEP (1998) 08, p.001 arXiv: https://arxiv.org/abs/hep-ph/9804296
2. Chekanov, S. "Jet algorithms: a minireview" , (2002) , p. arXiv: https://arxiv.org/abs/hep-ph/0211298
3. Atkin, Ryan "Review of jet reconstruction algorithms" Journal of Physics: Conference Series (2015) 645, p.012008 https://doi.org/10.1088/1742-6596/645/1/012008