Partonshower effects on Higgs boson production via vectorboson fusion in association with three jets
Abstract:
We present an implementation of Higgs boson production via vectorboson fusion in association with three jets at hadron colliders in the POWHEG BOX, a framework for the matching of NLOQCD calculations with partonshower programs. Our work provides the means to precisely describe the properties of extra jet activity in vectorboson fusion reactions that are used for the suppression of QCD backgrounds by central jet veto techniques. For a representative setup at the CERN LHC we verify that uncertainties related to partonshower effects are mild for distributions related to the third jet, in contrast to what has been observed in calculations based on vectorboson fusion induced Higgs production in association with two jets.
1 Introduction
With the discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) [1, 2] particle physics has entered a new era. Both LHC collaborations, ATLAS and CMS, have confirmed the existence of a boson with a mass of about 126 GeV and properties consistent with those of the scalar CPeven particle predicted by the Standard Model [3, 4]. In order to fully establish the nature of the Higgs boson, a precise determination of its couplings to fermions and gauge bosons is essential [5, 6, 7].
A rather clean environment for such coupling measurements is provided by the vectorboson fusion (VBF) production mode [8, 9, 10, 11, 12, 13], where the Higgs boson is produced via quarkscattering mediated by weak gauge boson exchange in the channel, . Because of the low virtuality of the exchanged weak bosons, the tagging jets emerging from the scattered quarks are typically located in the forward and backward regions of the detector, while the centralrapidity region exhibits little jet activity due to the colorsinglet nature of the channel exchange. These features can be exploited to efficiently suppress QCD backgrounds with a priori large cross sections at the LHC. In the context of centraljet veto (CJV) techniques, events are discarded if they exhibit one or more jets in between the two tagging jets. To quantitatively employ such selection strategies, a precise knowledge of the VBF cross section with an additional jet, i.e. the reaction , is crucial.
Nexttoleading order (NLO) QCD corrections to VBFinduced production have first been computed in [14], yielding results with only small residual scale uncertainties of order 10% or less. In particular, in that approach the survival probability for the Higgs signal has been estimated to exhibit a perturbative accuracy of about 1%. The calculation of [14] is implemented in the VBFNLO package [15, 16, 17] in the form of a flexible partonlevel MonteCarlo program. More recently, an NLOQCD calculation for electroweak production has been presented [18], where several approximations of Ref. [14] have been dropped.
In this work, we merge the partonlevel calculation of [14] with a partonshower MonteCarlo in the framework of the POWHEG formalism [19, 20], a method for the matching of an NLOQCD calculation with a transversemomentum ordered partonshower program. For our implementation we are making use of version 2 of the POWHEG BOX [21, 22], a repository that provides the processindependent ingredients of the POWHEG method. The code we develop yields precise, yet realistic predictions for VBFinduced production at the LHC in a public framework that can easily be used by the reader for further phenomenological studies.
2 Technical details of the implementation
The implementation of production via VBF in the context of the POWHEG BOX requires, as major building blocks, the matrix elements for all relevant partonic scattering processes at Born level and at nexttoleading order. These have first been calculated in [14] and are publicly available in the VBFNLO package [15]. We extracted the matrix elements from VBFNLO and adapted them to the format required by the POWHEG BOX.
At leading order (LO), processes of the type and all crossingrelated channels are taken into account, if they include the exchange of a weak boson in the channel . Some representative Feynman diagrams are depicted in Fig. 1.
The gaugeinvariant class of diagrams involving weakboson exchange in the channel is considered as part of the Higgsstrahlung process, and disregarded in the context of our work on VBFinduced Higgs production. The interference of channel with channel diagrams in flavor channels with quarks of the same type is neglected. Once VBFspecific selection cuts are imposed, these approximations are well justified [23]. Throughout, we assume a diagonal CKM matrix. We refer to the electroweak production process at order within these approximations as “VBF production”.
The virtual corrections to this reaction comprise the interference of the Born amplitudes with oneloop diagrams where a virtual gluon is attached to a single fermion line [c.f. Fig. 2 (a)–(g)], and diagrams where a virtual gluon is exchanged between the two different fermion lines, see Fig. 2 (h),(i).
As discussed in some detail in [14], the latter contributions are strongly suppressed by color factors and due to the VBF dynamics. They can be neglected, if the respective color structures of the realemission contributions are disregarded as well, as these would serve to cancel the infrared singularities of the pentagon and hexagon contributions that we drop.
The realemission contributions involve subprocesses with four external (anti)quarks and two gluons such as , as well as pure quark scattering processes of the type , and all crossingrelated channels with channel weak boson exchange. Because of the approximations we have employed in the virtual contributions, where we dropped colorsuppressed contributions giving rise to pentagon and hexagon integrals, the respective color structures have to be disregarded in the realemission contributions as well. In practice this means to neglect interference terms between diagrams where a given gluon is emitted once from the upper and once from the lower quark line in the Feynman graphs. For example, interference terms like , with as depicted in Fig. 3, are dropped,
while their squares or the squares of the topologies sketched in Fig. 4 are fully considered.
Representative diagrams for a pure quark subprocess are depicted in Fig. 5. For this class of subprocesses we require that the pair stems from a gluon. Contributions involving the hadronic decay of a weak boson, , such as graph 5 (c), are disregarded within our VBF setup.
While in [14] soft and collinear singularities have been taken care of by a dipole subtraction procedure, the POWHEG BOX makes use of the socalled FKS subtraction scheme [24]. From the color and spincorrelated amplitudes provided by the user, the POWHEG BOX internally constructs the counterterms that are needed to cancel soft and collinear singularities in the realemission contributions. Because we are disregarding certain colorsuppressed contributions in the virtual and realemission amplitudes, we have to make sure that only the counterterms relevant for our setup are constructed. This is achieved by passing only those color and spincorrelated Born amplitudes to the POWHEG BOX that correspond to the color structures we consider within our approximations.
We have carefully tested that the counter terms constructed in this way approach the realemission amplitudes in the soft and collinear limits. Additionally, we have compared the treelevel and real emission amplitudes for selected phase space points with code generated by MadGraph [25] that has been adapted to match the approximations of our calculation. We found agreement at the level of more than ten digits. The virtual amplitudes have been compared to VBFNLO, again showing full agreement for single phase space points. We note that some care has been necessary in this latter check, as finite parts of the subtraction terms are included in the virtual amplitudes in the default setup of VBFNLO. To verify the entire setup of our code, we have compared cross sections and distributions for various sets of selection cuts at LO and NLOQCD accuracy as obtained with the POWHEG BOX with respective results of VBFNLO. We found full agreement for all considered scenarios.
We note that special care is needed when performing the phasespace integration of VBF production in the framework of the POWHEG BOX. In contrast to the VBFinduced production cross section that is entirely finite at leading order, the inclusive VBF cross section diverges already at leading order when a pair of partons becomes collinear or a soft gluon is encountered in the final state. While such divergent contributions disappear after phenomenologically sensible selection cuts are imposed, their presence considerably reduces the efficiency of the numerical phase space integration. This effect can be avoided by appropriate phasespace cuts at generation level, or by a socalled Bornsuppression factor that dampens the integrand whenever singular configurations in phasespace are approached. In order to ensure that our results are independent of technical cuts in the phasespace integration, we recommend the use of a Bornsuppression factor. In our POWHEG BOX implementation we provide two alternative versions of Bornsuppression factors:

In our first, multiplicative, approach, the factor is of the form
(1) where the and respectively denote the transverse momenta and invariant masses of the three finalstate partons of the underlying Born configuration. The and are cutoff parameters that are typically set to values of a few GeV.

Following the procedure suggested for the related case of trijet production in the framework of the POWHEG BOX [26], we use an exponential suppression factor of the form
(2) with
(3) for the suppression of infrared divergent configurations in the underlying Born kinematics, accompanied by a factor
(4) where
(5) The factor serves to suppress configurations where all partons are having small transverse momenta, and at the same time increase the fraction of events generated with large transverse momenta. Combining with , we construct
(6)
For the generation of the phenomenological results shown below we are using a Born suppression factor of the form given in Eq. (6) with GeV and GeV, supplemented by a small generation cut on the transverse momenta of the three outgoing partons of the underlying Born configuration, GeV.
To make sure our results do not depend on these technical parameters, in addition to our default setup we ran our code using the Born suppression factor of Eq. (1) with GeV and, again, GeV. The results in the two setups are in full agreement with each other and, at fixed order, also with respective results obtained with VBFNLO that is using an entirely different phasespace generator.
3 Phenomenological results
Our implementation of VBF production at the LHC is made publicly available in version 2 of the POWHEG BOX, and can be obtained as explained at the project webpage, http://powhegbox.mib.infn.it/.
Here, we are providing phenomenological results for a representative setup at the LHC with a centerofmass energy of TeV. We are using the CT10 fixedfourflavor set [27] for the parton distribution functions of the proton as implemented in the LHAPDF library [28] and the accompanying value of the strong coupling, . Jets are reconstructed via the anti algorithm with a resolution parameter of , with the help of the FASTJET package [29, 30, 31]. As electroweak input parameters we are using the masses of the weak gauge bosons, GeV and GeV, and the Fermi constant, GeV. Other electroweak parameters are computed thereof via treelevel relations. The widths of the massive gauge bosons are set to GeV and GeV, respectively. For the Higgs boson, we are using GeV and MeV. The renormalization and factorization scales are identified as . In order to assess uncertainties that remain after matching the NLO calculation with a parton shower program, we consider three different tools: PYTHIA 6.4.25 with the Perugia 0 tune [32], HERWIG++ 2.7.0 [33, 34] with its default angularordered shower, and with a transversemomentum ordered dipole shower [35] which we tag as PYT, HER, and DS++, respectively. We note that wideangle, soft radiation that is in principle needed when matching an NLO calculation with a partonshower program using the POWHEG method, is missing in the default angularordered HERWIG++ shower. The impact of this missing piece on observables can only be estimated by a comparison with predictions obtained with transverse momentum ordered showers, such as the DS++ version of HERWIG++. We do not consider hadronization, QED radiation, multiple parton interactions, and underlying event effects in this work.
In order to define a event, we demand at least three wellobservable jets with
(7) 
In addition, we impose VBFspecific selection cuts. The two hardest jets, referred to as “tagging jets”, are required to fulfill
(8) 
and be wellseparated from each other,
(9) 
The kinematics of the Higgs boson is not restricted.
With these cuts, we obtain a cross section of fb at fixed order, where the error is the statistical error of the Monte Carlo calculation. After matching the NLO result with a parton shower, some of the events fail to pass the cuts, resulting in slightly smaller cross sections of fb, fb, and fb, respectively. Apart from this change in normalization the impact of the parton shower on observables related to the tagging jets is very mild, as illustrated in Fig. 6
for the transverse momentum distribution of the hardest tagging jet and the invariant mass of the tagging jet pair.
In contrast to NLO calculations for VBF production, where the third jet can be described only with LO accuracy, our calculation is NLO accurate in distributions related to the third jet. In Fig. 7, NLO+PS results for the transverse momentum and the rapidity distribution of the third jet are shown for different parton shower programs together with the fixedorder NLO result.
For all considered parton showers, the difference between the NLO and the NLO+PS results is small. However, PYTHIA tends to produce slightly more jets in the centralrapidity region of the detector, while HERWIG++ preferentially radiates in the collinear region between the two tagging jets and the beam axis. We will see below that this effect is more pronounced in the case of subleading jets.
Larger differences between the fixedorder and the various matched predictions occur in distributions related to the fourth jet. In the partonlevel NLO calculation a fourth jet can only stem from the realemission contributions, and can thus be described only at treelevel accuracy. Larger theoretical uncertainties are therefore expected for observables related to the fourth jet. Fig. 8
illustrates the effect of the POWHEGSudakov factor on the transverse momentum of the fourth jet and clarifies how extra radiation in the VBF setup is distributed by the different parton shower programs via the variable. This quantity is defined as
(10) 
in order to parameterize the rapidity of the fourth jet relative to the two hard tagging jets. The respective distribution shows, more pronouncedly than in the case of the third jet, that PYTHIA and HERWIG++ tend to produce radiation in different regions of phase space. The differences between the various NLO+PS curves can thus be considered as inherent uncertainty of the matched prediction.
4 Conclusions
In this work, we have presented an implementation of VBF production in version 2 of the POWHEG BOX repository. We have performed the matching of an existing NLOQCD calculation with partonshower programs using the POWHEG formalism and presented phenomenological results for a representative setup at the LHC. The code we developed is publicly available and can be adapted to the user’s need in a straightforward manner.
We have shown that theoretical uncertainties associated with the description of the third jet by genuinely different partonshower programs are mild at NLO+PS level, contrary to what is observed in studies based on matrix elements for VBF production that are only LO accurate in the third jet. Our implementation thus provides an important improvement in the theoretical assessment of centraljet veto observables that are crucial for VBF analyses at the LHC.
Acknowledgments
We are grateful to Carlo Oleari for help with implementing the code in the POWHEG BOX repository. The work of B. J. is supported by the Institutional Strategy of the University of Tübingen (DFG, ZUK 63). F. S. is supported by the “Karlsruher Schule für Elementarteilchen und Astroteilchenphysik: Wissenschaft und Technologie (KSETA)” and D. Z. by the BMBF under “Verbundprojekt 05H2012 – Theorie”.
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