Extending Precision Perturbative QCD with Track Functions.

Collider experiments often exploit information about the quantum numbers of final state hadrons to maximize their sensitivity, with applications ranging from the use of tracking information (electric charge) for precision jet substructure measurements, to flavor tagging for nucleon structure studies. For such measurements, perturbative calculations in terms of quarks and gluons are insufficient, and nonperturbative track functions describing the energy fraction of a quark or gluon converted into a subset of hadrons (e.g., charged hadrons) must be incorporated. Unlike fragmentation functions, track functions describe correlations between hadrons and therefore satisfy complicated nonlinear evolution equations whose structure has so far eluded calculation beyond the leading order. In this Letter, we develop an understanding of track functions and their interplay with energy flow observables beyond the leading order, allowing them to be used in state-of-the-art perturbative calculations for the first time. We identify a shift symmetry in the evolution of their moments that fixes their structure, and we explicitly compute the evolution of the first three moments at next-to-leading order, allowing for the description of up to three-point energy correlations. We then calculate the two-point energy correlator on charged particles at O(α_{s}^{2}), illustrating explicitly that infrared singularities in perturbation theory are absorbed by moments of the track functions and also highlighting how these moments seamlessly interplay with modern techniques for perturbative calculations. Our results extend the boundaries of traditional perturbative QCD, enabling precision perturbative predictions for energy flow observables sensitive to the quantum numbers of hadronic states.

Entropy of a Jet.

Scattering processes often inevitably include the production of infrared states, which are highly correlated with the hard scattering event, and decohere the hard states. This can be described using the entropy of the hard reduced density matrix, which is obtained from tracing over infrared states. We determine this entropy for an asymptotically free gauge theory by separating the Hilbert space into hard and infrared states, and calculate it in a leading-logarithmic approximation for jets. We find that the entropy increases when the resolution scales defining the hard radiation are lowered, that this entropy is related to the subjet multiplicity, and explore connections to using jet images for machine learning, and the forward-scattering density matrix of partons in a nucleon probed in deep-inelastic scattering.

Transverse-Momentum-Dependent Distributions with Jets.

We investigate the use of jets to measure transverse-momentum-dependent distributions (TMDs). The example we use to present our framework is the dijet momentum decorrelation at lepton colliders. Translating this momentum decorrelation into an angle θ≪1, we analyze the factorization of the cross section for the cases θ≫R, θ∼R, and θ≪R, where R is the jet radius. Critically, for the winner-take-all axis, the jet TMD has the same double-scale renormalization group evolution as TMD fragmentation functions for all radii R. TMD fragmentation functions in factorization theorems may then simply be replaced by the jet TMDs we calculate, and all ingredients to perform the resummation to next-to-next-to-leading logarithmic accuracy are available. Our approach also applies to semi-inclusive deep inelastic scattering, where a jet instead of a hadron is measured in the final state, and we find a clean method to probe the intrinsic transverse momentum of quarks and gluons in the proton that is less sensitive to final-state nonperturbative effects.

A case study of quark-gluon discrimination at NNLL ' in comparison to parton showers.

Predictions for our ability to distinguish quark and gluon jets vary by more than a factor of two between different parton showers. We study this problem using analytic resummed predictions for the thrust event shape up to NNLL ' using e + e - → Z → q q ¯ and e + e - → H → g g as proxies for quark and gluon jets. We account for hadronization effects through a nonperturbative shape function, and include an estimate of both perturbative and hadronization uncertainties. In contrast to previous studies, we find reasonable agreement between our results and predictions from both Pythia and Herwig parton showers. We find that this is due to a noticeable improvement in the description of gluon jets in the newest Herwig 7.1 compared to previous versions.

Dissecting soft radiation with factorization.

An essential part of high-energy hadronic collisions is the soft hadronic activity that underlies the primary hard interaction. It includes soft radiation from the primary hard partons, secondary multiple parton interactions (MPI), and factorization-violating effects. The invariant mass spectrum of the leading jet in Z+jet and H+jet events is directly sensitive to these effects, and we use a QCD factorization theorem to predict its dependence on the jet radius R, jet p_{T}, jet rapidity, and partonic process for both the perturbative and nonperturbative components of primary soft radiation. We prove that the nonperturbative contributions involve only odd powers of R, and the linear R term is universal for quark and gluon jets. The hadronization model in Pythia8 agrees well with these properties. The perturbative soft initial state radiation (ISR) has a contribution that depends on the jet area in the same way as the underlying event, but this degeneracy is broken by dependence on the jet p_{T}. The size of this soft ISR contribution is proportional to the color state of the initial partons, yielding the same positive contribution for gg→Hg and gq→Zq, but a negative interference contribution for qq[over ¯]→Zg. Hence, measuring these dependencies allows one to separate hadronization, soft ISR, and MPI contributions in the data.

Jet charge at the LHC.

Knowing the charge of the parton initiating a light-quark jet could be extremely useful both for testing aspects of the standard model and for characterizing potential beyond-the-standard-model signals. We show that despite the complications of hadronization and out-of-jet radiation such as pileup, a weighted sum of the charges of a jet's constituents can be used at the LHC to distinguish among jets with different charges. Potential applications include measuring electroweak quantum numbers of hadronically decaying resonances or supersymmetric particles, as well as standard model tests, such as jet charge in dijet events or in hadronically decaying W bosons in tt[over ¯] events. We develop a systematically improvable method to calculate moments of these charge distributions by combining multihadron fragmentation functions with perturbative jet functions and pertubative evolution equations. We show that the dependence on energy and jet size for the average and width of the jet charge can be calculated despite the large experimental uncertainty on fragmentation functions. These calculations can provide a validation tool for data independent of Monte Carlo fragmentation models.

Calculating track-based observables for the LHC.

By using observables that only depend on charged particles (tracks), one can efficiently suppress pileup contamination at the LHC. Such measurements are not infrared safe in perturbation theory, so any calculation of track-based observables must account for hadronization effects. We develop a formalism to perform these calculations in QCD, by matching partonic cross sections onto new nonperturbative objects called track functions which absorb infrared divergences. The track function Ti(x) describes the energy fraction x of a hard parton i which is converted into charged hadrons. We give a field-theoretic definition of the track function and derive its renormalization group evolution, which is in excellent agreement with the pythia parton shower. We then perform a next-to-leading order calculation of the total energy fraction of charged particles in e+ e-→ hadrons. To demonstrate the implications of our framework for the LHC, we match the pythia parton shower onto a set of track functions to describe the track mass distribution in Higgs plus one jet events. We also show how to reduce smearing due to hadronization fluctuations by measuring dimensionless track-based ratios.

Beam thrust cross section for Drell-Yan production at next-to-next-to-leading-logarithmic order.

At the LHC and Tevatron strong initial-state radiation (ISR) plays an important role. It can significantly affect the partonic luminosity available to the hard interaction or contaminate a signal with additional jets and soft radiation. An ideal process to study ISR is isolated Drell-Yan production, pp → Xℓ+ ℓ- without central jets, where the jet veto is provided by the hadronic event shape beam thrust τB. Most hadron collider event shapes are designed to study central jets. In contrast, requiring τ B << 1 provides an inclusive veto of central jets and measures the spectrum of ISR. For τ B << 1 we carry out a resummation of α s(n)ln(m)τ B corrections at next-to-next-to-leading-logarithmic order. This is the first resummation at this order for a hadron-hadron collider event shape. Measurements of τ B at the Tevatron and LHC can provide crucial tests of our understanding of ISR and of τ B's utility as a central jet veto.

N jettiness: an inclusive event shape to veto jets.

Jet vetoes are essential in many analyses at the LHC and Tevatron. Typical signals have a specific number of hard jets or leptons, while backgrounds have additional jets. Vetoing undesired jets efficiently discriminates signal and background. For a sample with ≥N jets, the veto to give N energetic jets defines an "exclusive" N-jet cross section. This strongly restricts the phase space and causes large double logarithms in perturbation theory that must be summed. Jet vetoes are typically implemented using jet algorithms, yielding complicated phase-space restrictions, and reliance on leading-log parton-shower Monte Carlo simulations. We introduce a global event shape "N jettiness" τN, which is defined for events with N signal jets. Requiring τN≪1 constrains radiation between the signal jets and provides a theoretically well-controlled jet veto. N jettiness yields a factorization formula with inclusive jet and beam functions.