# -Jettiness: An Inclusive Event Shape to Veto Jets

###### Abstract

Jet vetoes are essential in many Higgs and new-physics analyses at the LHC and Tevatron. The signals are typically characterized by a specific number of hard jets, leptons, or photons, while the backgrounds often have additional jets. In such cases vetoing undesired additional jets is an effective way to discriminate signals and background. Given an inclusive event sample with or more jets, the veto to have only energetic jets defines an “exclusive” -jet cross section. This strongly restricts the phase space of the underlying inclusive -jet cross section and causes large double logarithms in perturbation theory that must be summed to obtain theory predictions. Jet vetoes are typically implemented using jet algorithms. This yields complicated phase-space restrictions and one often relies on parton-shower Monte Carlos, which are limited to leading-logarithmic accuracy. We introduce a global event shape “-jettiness”, , which is defined for events with signal jets and vanishes in the limit of exactly infinitely narrow jets. Requiring constrains radiation between the signal jets and vetoes additional undesired jets. This provides an inclusive method to veto jets and to define an exclusive -jet cross section that can be well-controlled theoretically. -jettiness yields a factorization formula with inclusive jet and beam functions.

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^{†}preprint: MIT–CTP 4139 April 14, 2010

## Introduction.

At the LHC or Tevatron, hard interactions involving Higgs or new-physics particles are identified by looking for signals with a characteristic number of energetic jets, leptons, or photons exp . The backgrounds come from Standard Model processes producing the same signature of hard objects possibly with additional jets. An example are top quarks decaying into plus -jet, which is a major background for Aaltonen:2010yv . When reconstructing masses and decay chains of new-physics particles additional jets can cause large combinatorial backgrounds. Standard Model processes can also fake a signal when a jet is misidentified as lepton or photon, a typical example being .

Thus, a veto on additional undesired jets is an effective and sometimes necessary method to clean up the events and discriminate signal and the various backgrounds. More generally, one would like to measure an “exclusive” -jet cross section, , to produce signal jets where the remaining contains no hard (central) jets. Here, and denotes the hard leptons or photons required as part of the signal.

We introduce an inclusive event shape “-jettiness”, denoted and defined below in Eq. (1). For an event with at least energetic jets, provides an inclusive measure of how -jet-like the event looks. In the limit the event contains exactly infinitely narrow jets. For the event has hard radiation between the signal jets. Requiring constrains the radiation outside the signal and beam jets, providing an inclusive way to veto additional central jets. It yields an inclusive definition of an exclusive -jet cross section with a smooth transition between the case of no jet veto, , and the extremely exclusive case .

Vetoing additional jets imposes a phase-space restriction on the underlying inclusive -jet cross section to produce or more jets with the same . Irrespective of its precise definition, the jet veto introduces a jet resolution scale that characterizes this restriction, i.e. the distinction between and jets. Hence, the exclusive -jet cross section contains phase-space logarithms , where and is the scale of the hard interaction. For , . Generically there is always a hierarchy , which becomes larger the stronger the restrictions are. These large logarithms must be summed to obtain reliable predictions.

Jet vetoes are typically implemented by using a jet algorithm to find all jets in the event and veto events with too many energetic jets. Jet algorithms are good tools to identify the signal jets. However, they are not necessarily well-suited to veto unwanted jets, because the corresponding phase-space restrictions are complicated and depend in detail on the algorithm. This makes it difficult to incorporate the jet veto into explicit theoretical calculations and inhibits a systematic summation of the resulting large logarithms. In this case, usually the only way to predict the corresponding exclusive -jet cross section is to rely on parton shower Monte Carlos to sum the leading logarithms (LL). For particular jet algorithms, the resolution defines the transition from to jets. Next-to-leading logarithms for this and other hadron-collider event shapes were summed in Ref. Banfi:2004nk .

Vetoing jets by cutting on an inclusive variable like has several advantages. First, we can go beyond LL order, because the logarithms from the phase-space restriction, , are simple enough to allow their systematic summation to higher orders. Moreover, the theory predictions with factorization can be directly compared with experiment without having to utilize Monte Carlos for parton showering or hadronization. Experimentally, reduces the dependence on jet algorithms and might help improve the background rejection.

## Definition.

-jettiness is defined as

(1) |

As we discuss below, this definition of yields a factorization formula with inclusive jet and beam functions and allows the summation of logarithms to next-to-next-to-leading logarithmic (NNLL) order. The sum over in Eq. (1) runs over the momenta of all measured (pseudo-)particles in the final state excluding the signal leptons or photons in . (Any other leptons or photons, e.g. from hadronic decays, are included in the sum.) For simplicity we take all to be massless. The , , and , …, are a fixed set of massless reference momenta for the two beams and the signal jets,

(2) |

The and correspond to the energies and directions of the signal jets (for both massive and massless jets). Their choice is discussed below. The beam reference momenta and are the large momentum components of the colliding partons along the beam axis (taken to be the axis). They are defined by

(3) |

and analogously for with . Here, is the total momentum of the non-hadronic signal . In Eq. (1), is the hard interaction scale, and the distance of a particle with momentum from the jets or beams is measured by . If contains missing energy, so and are not known, one can use a modified distance measure as we discuss below Eq. (Generalizations.).

The minimum for each in Eq. (1) associates the particle with the closest beam or jet, appropriately dividing the hadronic initial-state radiation (ISR) and final-state radiation (FSR). Soft particles and energetic particles near any jet or beam only give small contributions to the sum. For scattering of massless partons, . Energetic particles far away from all jets and beams give large contributions. Hence, for the final state has jets, two forward beam jets, and only soft radiation between them. In this limit are the momentum fractions of the annihilated partons, and is the boost of the partonic center-of-mass frame.

## for jets.

In collisions there is no hadronic ISR, so we drop the entries in Eq. (1). Now is the total invariant mass of the leptons and . In the two-jet limit, the jet directions are close to the thrust axis , defined by the thrust . Hence we can choose

(4) |

as reference momenta, and Eq. (1) becomes

(5) |

where is the angle between and . The minimum divides all particles into the two hemispheres perpendicular to as shown in Fig. 1(a). For , the total invariant mass in each hemisphere is much smaller than , so the final state contains two narrow jets. In this limit, , and a factorization theorem exists for , which can be used to sum logarithms of ee . For a given jet algorithm with resolution parameter , the value marks the transition between and jets. Thus requiring also vetoes events with jets.

## for Drell-Yan.

Next, consider the isolated Drell-Yan process, with no hard central jets, shown in Fig. 1(b). We now have ISR from the incoming partons, but no FSR from jets. From Eq. (3) we have

(6) |

where and are the dilepton invariant mass and transverse momentum, and equals the dilepton rapidity. Now, and Eq. (1) becomes

(7) |

where and are the transverse momentum and rapidity of . The and dependence in Eq. (1) explicitly accounts for the boost of the partonic center-of-mass frame. For , the minimum in Eq. (7) divides all particles into two hemispheres perpendicular to the beam axis (analogous to above). For , the hemispheres are boosted with their boundary now at , and the beam jet in the direction of the boost is narrower than the other, as depicted in Fig. 1(b). Contributions to from large rapidities are exponentially suppressed by , so particles beyond the detector’s rapidity reach give negligible contributions.

Beam thrust Stewart:2009yx is given by . It is obtained by choosing in case and are measured rather than the longitudinal components in Eq. (6). For the hadronic final state can only contain soft radiation plus energetic radiation in the forward directions, so and . A factorization theorem for at small was derived and used to sum logarithms of to NNLL in Refs. Stewart:2009yx .

## General case.

For we have both ISR and FSR. We select candidate signal events by measuring and running a jet algorithm to find the signal jets and their momenta . The conditions on the jets and that define the signal are encoded in the cross section by a measurement function . Generically, will enforce that there are at least energetic jets that are sufficiently separated from each other and the beams. We now use the measured jet energies and directions to define the massless reference momenta in Eq. (Definition.),

(8) |

while and are given by Eqs. (Definition.) and (3).

Taking the minimum in Eq. (1) combines the previous cases in Eqs. (5) and (7). It divides all particles into jet and beam regions that are unique for a given set of reference momenta and whose union covers all of phase space, as illustrated in Fig. 1(c). The boundary between any two neighboring regions is part of a cone and is such that the sum of the total invariant masses in the regions is minimized (or in case of a beam region the virtuality of the incoming colliding parton).

For events with small all jet algorithms should agree how energetic radiation is split up between the jets and beams, and only differ in their treatment of softer particles. Thus, they all give the same and up to power corrections, while the split up of the soft radiation is determined by itself. Hence, the dependence of on the jet algorithm is formally power suppressed, , as seen in Eq. (Factorization formula.) below.

To measure , we still rely on having a suitable jet algorithm to find the signal jets but not more so than if we were not measuring . Imagine the jet size in the algorithm is chosen too small such that the algorithm divides what should be a single signal jet into several narrow jets ^{1}^{1}1This can be tested by comparing the total energy in each region defined by with the energy from the jet algorithm. If these are very different, but at the same time is small, then there are additional energetic particles near the signal jets that the algorithm should have included..
In this case, the jet algorithm yields a poorly reconstructed signal irrespective of measuring .

Since the jet veto is now provided by , this situation can be avoided because we do not have to rely on the jet algorithm to identify additional jets and so can use an algorithm that can be forced to always yield at most jets. This is in fact the most natural thing to do when one is looking for jets. Therefore, using as jet veto could also help improve the signal reconstruction.

## Generalizations.

We can generalize to

(9) |

where can be any infrared-safe distance measure. In Eq. (1), with

(10) |

Here, and are the rapidity and azimuthal distances between and . If these are small, the factor in brackets reduces to the familiar .

Different measures that are boost-invariant along the beam axis can be obtained by modifying the dependence on rapidity, , and in Eq. (Generalizations.). A geometric measure, which is independent of , is

(11) |

It evenly divides the area rather than invariant mass between neighboring regions, such that more energetic jets also get more invariant mass.

If contains missing energy, then in Eq. (3) and thus and are not known. For , one can use any hard scale, like the of the hardest jet or leave it out, since it only serves as an overall normalization. In the beam measures we can simply set , which defines them in the hadronic center-of-mass frame.

-jettiness does not split events into , , , etc. jets like a traditional jet algorithm. But we can consider using to define an “exclusive -jet algorithm” as follows: First, we use a geometric measure and find the directions and boost that minimize , analogous to finding for . This might actually allow one to get an estimate of even in the case of missing energy by exploiting the asymmetry in the beam jets. Second, we determine the jet energies by summing over the particles in each jet region. (To reduce the sensitivity to the underlying event and pile-up, one can weigh the sum over energies by the distance from .)

## Factorization formula.

We now use again as defined in Eq. (1). For , QCD ISR and FSR can be described in soft-collinear effective theory SCET at leading power by independent sectors for collinear particles close to each with and a separate sector for soft particles. By power counting, -collinear particles are closest to , so for the -collinear sector

(12) |

where (up to power corrections) is the total invariant mass in the -collinear sector. Similarly, the sum over the beam collinear sectors yields the total (transverse) virtuality of the colliding partons, and . Therefore,

(13) |

The sum in the last term is now restricted to the soft sector. Combining
Eq. (13) with the analyses in Refs. Bauer:2008jx ; Stewart:2009yx
yields the factorization formula for -jettiness ^{2}^{2}2Here,
enforces distinct collinear sectors with
and . We assume only depends on the large components
of the jet momenta, , and that only
couples to the QCD subprocess via a hard interaction. We also assume that Glauber gluons
do not spoil this factorization.

(14) |

Here, contains the underlying hard interaction , where , , and denote parton types, and the sum over is over all relevant partonic channels. It is a matrix in color space given by the IR-finite parts (in pure dim. reg.) of the squared partonic matrix elements in each channel. The -body phase space for the massless momenta is denoted , and that for by .

The inclusive jet and beam functions, and , describe the final and initial state radiation emitted by the outgoing and incoming partons from the hard interaction. The latter also determine the momentum fractions of the colliding partons and are given by parton distribution functions as Fleming:2006cd ; Stewart:2009yx

(15) |

The are perturbative coefficients describing collinear ISR, and at tree level . The last term in Eq. (13) is the contribution to from soft particles in the underlying event. It is described by the soft function , which depends on the jet’s angles and energy fractions . Like , it is a color matrix, and the trace in Eq. (Factorization formula.) is over .

In Eq. (Factorization formula.), all functions are evaluated at the same renormalization scale . Large logarithms of in are summed by first computing , , , at the scales , , , where the functions contain no large logarithms, and then evolving them to the scale . This evolution is known analytically evolution and the required anomalous dimensions are already known to NNLL NNLL ; Stewart:2009yx , because we have inclusive jet and beam functions. NNLL also requires the corrections for each function, which are known for and . The hard function is determined by the one-loop QCD matrix elements. For , can be computed perturbatively and will be given in a future publication.

We thank C. Lee, K. Tackmann, and J. Thaler for comments and discussions. This work was supported by the Office of Nuclear Physics of the U.S. Department of Energy, under the grant DE-FG02-94ER40818.

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