Abstract
In collider physics, jet algorithms are a ubiquitous tool for clustering particles into discrete jet objects. Event shapes offer an alternative way to characterize jets, and one can define a jet multiplicity event shape, which can take on fractional values, using the framework of "jets without jets". In this paper, we perform the first analytic studies of fractional jet multiplicity $\tilde{N}_{\rm jet}$ in the context of $e^+e^-$ collisions. We use fixed-order QCD to understand the $\tilde{N}_{\rm jet}$ cross section at order $\alpha_s^2$, and we introduce a candidate factorization theorem to capture certain higher-order effects. The resulting distributions have a hybrid jet algorithm/event shape behavior which agrees with parton shower Monte Carlo generators. The $\tilde{N}_{\rm jet}$ observable does not satisfy ordinary soft-collinear factorization, and the $\tilde{N}_{\rm jet}$ cross section exhibits a number of unique features, including the absence of collinear logarithms and the presence of soft logarithms that are purely non-global. Additionally, we find novel divergences connected to the energy sharing between emissions, which are reminiscent of rapidity divergences encountered in other applications. Given these interesting properties of fractional jet multiplicity, we advocate for future measurements and calculations of $\tilde{N}_{\rm jet}$ at hadron colliders like the LHC.
Highlights
For almost forty years, we have known that high energy particle collisions can produce jets [1, 2]
Though the observation of three-jet structure in e+e− collisions strongly hinted at the existence of gluons [8, 9], an unambiguous discovery at PETRA [10,11,12,13] required the use of event shapes like thrust [14]
Note that this does not fully capture the correct higher-order terms (for example, we do not get any term at O(αs3) from this exponentiation), but we will see in section 7 that it is enough to reproduce some of the higher-order effects observed in parton shower Monte Carlo generators
Summary
For almost forty years, we have known that high energy particle collisions can produce jets [1, 2]. The soft logarithms in Njet are purely non-global [26, 27], in the sense that they arise from soft emissions in a restricted region of phase space (see section 2.3).3 These features would seem to preclude any standard factorization theorem, especially given the non-additive nature of Njet. That theoretical calculations of Njet for hadronic collisions must contend with additional effects such as the underlying event and pileup contamination It will be non-trivial to include nonperturbative effects, power-suppressed terms, and higher-order perturbative effects such as the resummation of non-global logarithms [26, 27, 51,52,53,54].
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