# Precise QCD predictions for the production

of dijet final states in deep inelastic scattering

###### Abstract

The production of two-jet final states in deep inelastic scattering is an important QCD precision observable. We compute it for the first time to next-to-next-to-leading order (NNLO) in perturbative QCD. Our calculation is fully differential in the lepton and jet variables and allows one to impose cuts on the jets both in the laboratory and the Breit frame. We observe that the NNLO corrections are moderate in size, except at kinematical edges, and that their inclusion leads to a substantial reduction of the scale variation uncertainty on the predictions. Our results will enable the inclusion of deep inelastic dijet data in precision phenomenology studies.

###### pacs:

13.87.-a, 13.60.Hb, 12.38Bx^{†}

^{†}preprint: IPPP/16/38, ZU-TH 17/16, NSF-KITP-16-064

Our understanding of the inner structure of the proton has been shaped through a long series of deep-inelastic lepton-nucleon experiments, which have established the partonic structure of the proton and provided precision measurements of parton distribution functions disbook (). Specific combinations of the quark distributions can be probed in inclusive deep inelastic scattering (DIS), where the gluon distribution only enters indirectly as a correction and through scaling violations. A direct probe of the gluon distribution, which is less well constrained than the quark distributions, requires the selection of specific hadronic final states in deep inelastic scattering Newman:2013ada (), such as heavy quarks or jets.

Dijet final states in DIS are formed lo () through the two basic scattering processes and , which vary in relative importance depending on the kinematical region. Especially at low invariant masses of the dijet system, the gluon-induced process is largely dominant. The interplay of lepton and dijet kinematics in this region allows the gluon distribution to be probed over a substantial range. The same process also provides a direct measurement of the strong coupling constant .

The DESY HERA electron-proton collider provided a large data set of hadronic final states in DIS at GeV. Dijet final states have been measured to high precision over a large kinematical range by the H1 h11 (); h12 (); h1jet () and ZEUS zeus1 (); zeus2 () experiments, that have also used these measurements in the determination of the strong coupling constant. The reconstruction of jets is performed in the Breit frame, defined by the direction of the virtual photon and incoming proton, while the jet rapidity coverage is limited by the detector’s geometry in the laboratory frame. Consequently, the definition of the fiducial phase space used in a jet measurement typically combines information from both frames.

The interpretation of HERA data on dijet production in DIS relies at present on theoretical predictions at next-to-leading order (NLO) in perturbative QCD graudenz (); mirkes (); nagy (). The uncertainty associated with the NLO predictions (as estimated through the variation of renormalization and factorization scales) is the main limitation to precision studies based on these data. In particular, they can not be included in a consistent manner in state-of-the-art determinations of parton distributions abm (); mmht (); nnpdf (); ct (), which typically require their input data to be described at next-to-next-to-leading order (NNLO) QCD accuracy.

In this letter, we present the first calculation of the next-to-next-to-leading order (NNLO) QCD prediction to dijet production in DIS. The QCD corrections at this order involve three types of scattering amplitudes: the two-loop amplitudes for two-parton final states Z3p2l (), the one-loop amplitudes for three-parton final states Z4p1l () and the tree-level amplitudes for four-parton final states Z5p0l (). The contribution from each partonic final state multiplicity contains infrared divergences from soft and collinear real radiation and from virtual particle loops; these infrared singularities cancel only once the different multiplicities are summed together for any infrared-safe final state definition sw (). To implement the different contributions into a numerical program, a procedure for the extraction of all infrared singular configurations from each partonic multiplicity is needed. Several methods have been developed for this task at NNLO: sector decompostion secdec (), -subtraction qtsub (), antenna subtraction ourant (), sector-improved residue subtraction stripper (), -jettiness subtraction njettiness () and colorful subtraction trocsanyi ().

Our calculation is based on the antenna subtraction method ourant (), which constructs the subtraction terms for the real radiation processes out of antenna functions that encapsulate all color-ordered unresolved parton emission in between a pair of hard radiator partons, multiplied with reduced matrix elements of lower partonic multiplicity. By factorizing the final state phase space accordingly, it is possible to analytically integrate the antenna functions to make their infrared pole structure explicit, such that the integrated subtraction terms can be combined with the virtual corrections to yield a finite result. In the case of jet production in deep inelastic scattering, we need to use antenna functions with both hard radiators in the final state ourant () and with one radiator in the initial and one in the final state gionata (). The combination of real radiation contributions and unintegrated antenna subtraction terms is numerically finite in all infrared limits, such that all parton-level contributions to two-jet final states at NNLO can be implemented into a numerical program (parton-level event generator). This program can then incorporate the jet algorithm used in the experimental measurement as well as any type of event selection cuts. A substantial part of the infrastructure of our program is common to other NNLO calculations of jet production observables within the antenna subtraction method eerad3 (); nnlo2j (); nnlott (); nnlohj (); nnlozj (), which are all part of a newly developed code named NNLOJET. To validate our implementation of the tree-level and one-loop matrix elements, we compared the NLO predictions for dijet and trijet production against SHERPA sherpa () (in DIS kinematics hoeche ()), which uses OpenLoops openloops () to automatically generate the one-loop contributions at NLO. The antenna subtraction is then verified by testing the convergence of subtraction terms and matrix elements in all unresolved limits (as documented for example in joao ()) and by the infrared pole cancellation between the integrated subtraction terms and the two-loop matrix elements.

As a first application of our calculation, we consider the recent measurement by the H1 collaboration h1jet () of dijet production in DIS at high virtuality . The measurement was performed on data taken at the DESY HERA electron proton collider at a centre-of-mass energy of GeV. Deep inelastic scattering events are selected by requiring the range of lepton scattering variables: exchanged boson virtuality 150 GeV15000 GeV and energy transfer in the proton rest system . The hadronic final state is boosted to the Breit frame of reference, where the jet clustering is performed using the inclusive hadronic algorithm hadkt () with recombination. To ensure that the jets are contained in the calorimeter coverage, a cut on their pseudorapidity is applied in the HERA laboratory frame: . Jets are accepted in the inclusive dijet sample if their transverse momentum in the Breit frame is 5 GeV GeV and are ordered in this variable. The event is retained if the invariant mass of the two leading jets is GeV. The H1 collaboration provides double differential distributions in and either the average transverse momentum of the two leading jets or the variable where is the Bjorken variable reconstructed from the lepton kinematics. At leading order, can be identified with the proton momentum fraction carried by the parton that initiated the hard scattering process.

The theoretical predictions use the NNPDF3.0 parton distribution functions nnpdf () with and are evaluated with default renormalization and factorization scales , . The uncertainty on the theoretical prediction from missing higher orders is estimated by varying these scales by a factor between and . The electromagnetic coupling is also evaluated at a dynamical scale as according to QED evolution, with . The theoretical predictions are corrected bin-by-bin for hadronization and electroweak effects using the tables provided in h1jet ().

Figure 1 displays the distribution in six bins. For better visibility, the same plots are normalized to the NLO prediction in Figure 2, excluding the LO contribution which is typically considerably below the NLO curve and is associated with a large error. We observe that for all but the first bins in , the NNLO predictions are inside the NLO uncertainty band and that their inclusion leads to a substantial reduction of the theory uncertainty to typically 5% or less (especially at high ), which is now below the statistical and systematical uncertainty on the experimental data. We observe that the theoretical NNLO predictions tend to be above the experimental data. This feature points to the potential impact that the inclusion of these data could have in a global determination of parton distributions and of the strong coupling constant at NNLO accuracy. The tension between data and NNLO predictions is largest at lower values of , where the data is most accurate and the gluon-induced subprocess dominates the dijet production cross section.

The first bins in display a larger correction, often at the upper boundary of the NLO band, and only a mild reduction in scale uncertainty. They already have very large NLO corrections, typically with a NLO/LO ratio of about 2. This feature can be understood from a sophisticated interplay of the GeV cut with the other jet cuts. The cut forbids a substantial part of the phase space relevant to the first bin in the distribution to be filled by the leading order process. This results in a perturbative instability sudakov () starting below GeV, which leads to a destabilization of the perturbative series for the first bin.

To further illustrate this issue, we display the distribution in the lowest bin in in Figure 3. The same perturbative instability is present, now spread more uniformly over the first two bins. It is more pronounced than in the distribution due to the fact that an even larger fraction of the phase space is forbidden at leading order, since jets down to GeV are accepted in this distribution, while maintaining the GeV cut. The resulting instability can already be seen in going from LO to NLO, with substantial corrections outside the nominal scale variation band. In the bins with larger , events with low close to the cut are of lower importance, resulting in a better perturbative convergence and a more reliable prediction.

In this letter, we presented the first calculation of dijet production in deep inelastic scattering to NNLO in QCD. Our results are fully differential in the kinematical variables of the final state lepton and the jets. We applied our calculation to the kinematical situation that is relevant to a recent dijet measurement by the H1 collaboration h1jet (). Except for jet production at low transverse momentum (where the experimental event selection cuts destabilize the perturbative convergence), we observe the NNLO corrections to be moderate in size, and overlapping with the scale uncertainty band of the previously available NLO calculation. Especially at lower , the NNLO predictions tend to be above the data, which could provide important new information on the gluon distribution at NNLO. The residual uncertainty on the NNLO results is of the order of 5% or less, and below the errors on the experimental data. Our results enable the inclusion of deep inelastic jet data into precision phenomenology studies of the structure of the proton and of the strong coupling constant.

We would like to thank Nigel Glover, Alexander Huss and Thomas Morgan for many interesting discussions throughout the whole course of this project, Stefan Höche and Marek Schönherr for help with the NLO comparisons against SHERPA and Daniel Britzger for useful clarifications on the H1 jet data. This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200020-162487, in part by the UK Science and Technology Facilities Council as well as by the Research Executive Agency (REA) of the European Union under the Grant Agreement PITN-GA-2012-316704 (“HiggsTools”), the ERC Advanced Grant MC@NNLO (340983) and by the National Science Foundation under grant NSF PHY11-25915.

## References

- (1) R. Devenish and A. Cooper-Sarkar, Deep inelastic scattering, Oxford University Press (Oxford, 2004).
- (2) P. Newman, M. Wing, Rev. Mod. Phys. 86 (2014) 1037.
- (3) K.H. Streng, T.F. Walsh and P.M. Zerwas, Z. Phys. C 2 (1979) 237; R. D. Peccei and R. Rückl, Nucl. Phys. B 162 (1980) 125; C. Rumpf, G. Kramer and J. Willrodt, Z. Phys. C 7 (1981) 337.
- (4) A. Aktas et al. [H1 Collaboration], Phys. Lett. B 653 (2007) 134.
- (5) F. D. Aaron et al. [H1 Collaboration], Eur. Phys. J. C 67 (2010) 1.
- (6) V. Andreev et al. [H1 Collaboration], Eur. Phys. J. C 75 (2015) 65.
- (7) S. Chekanov et al. [ZEUS Collaboration], Phys. Lett. B 547 (2002) 164.
- (8) H. Abramowicz et al. [ZEUS Collaboration], Eur. Phys. J. C 70 (2010) 965.
- (9) E. Mirkes and D. Zeppenfeld, Phys. Lett. B 380 (1996) 205.
- (10) D. Graudenz, Phys. Rev. D 49 (1994) 3291; hep-ph/9710244.
- (11) Z. Nagy and Z. Trocsanyi, Phys. Rev. Lett. 87 (2001) 082001.
- (12) S. Alekhin, J. Blümlein and S. Moch, Phys. Rev. D 89 (2014) 054028.
- (13) L.A. Harland-Lang, A.D. Martin, P. Motylinski and R.S. Thorne, Eur. Phys. J. C 75 (2015) 204.
- (14) R. D. Ball et al., JHEP 1504 (2015) 040.
- (15) S. Dulat et al., Phys. Rev. D 93 (2016) 033006.
- (16) L.W. Garland, T. Gehrmann, E.W.N. Glover, A. Koukoutsakis and E. Remiddi, Nucl. Phys. B 627 (2002) 107; Nucl. Phys. B 642 (2002) 227; T. Gehrmann and E. Remiddi, Nucl. Phys. B 640 (2002) 379; T. Gehrmann and E.W.N. Glover, Phys. Lett. B 676 (2009) 146.
- (17) E. W. N. Glover and D. J. Miller, Phys. Lett. B 396 (1997) 257; Z. Bern, L. J. Dixon, D. A. Kosower and S. Weinzierl, Nucl. Phys. B 489 (1997) 3; J. M. Campbell, E. W. N. Glover and D. J. Miller, Phys. Lett. B 409 (1997) 503; Z. Bern, L. J. Dixon and D. A. Kosower, Nucl. Phys. B 513 (1998) 3.
- (18) K. Hagiwara and D. Zeppenfeld, Nucl. Phys. B 313 (1989) 560; F. A. Berends, W. T. Giele and H. Kuijf, Nucl. Phys. B 321 (1989) 39; N. K. Falck, D. Graudenz and G. Kramer, Nucl. Phys. B 328 (1989) 317.
- (19) G. F. Sterman and S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436.
- (20) T. Binoth and G. Heinrich, Nucl. Phys. B 693 (2004) 134; C. Anastasiou, K. Melnikov and F. Petriello, Phys. Rev. D 69 (2004) 076010.
- (21) S. Catani and M. Grazzini, Phys. Rev. Lett. 98 (2007) 222002.
- (22) A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, JHEP 0509 (2005) 056; Phys. Lett. B 612 (2005) 49; Phys. Lett. B 612 (2005) 36; J. Currie, E. W. N. Glover and S. Wells, JHEP 1304 (2013) 066.
- (23) M. Czakon, Phys. Lett. B 693 (2010) 259; R. Boughezal, K. Melnikov and F. Petriello, Phys. Rev. D 85 (2012) 034025.
- (24) R. Boughezal, C. Focke, X. Liu and F. Petriello, Phys. Rev. Lett. 115 (2015) 062002; R. Boughezal, X. Liu and F. Petriello, Phys. Rev. D 91 (2015) 094035; J. Gaunt, M. Stahlhofen, F. J. Tackmann and J. R. Walsh, JHEP 1509 (2015) 058.
- (25) G. Somogyi and Z. Trocsanyi, JHEP 0808 (2008) 042; V. Del Duca, C. Duhr, A. Kardos, G. Somogyi and Z. Trocsanyi, arXiv:1603.08927.
- (26) A. Daleo, T. Gehrmann and D. Maitre, JHEP 0704 (2007) 016.
- (27) A. Daleo, A. Gehrmann-De Ridder, T. Gehrmann and G. Luisoni, JHEP 1001 (2010) 118.
- (28) A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, JHEP 0711 (2007) 058; Comput. Phys. Commun. 185 (2014) 3331.
- (29) A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and J. Pires, Phys. Rev. Lett. 110 (2013) 162003; J. Currie, A. Gehrmann-De Ridder, E.W.N. Glover and J. Pires, JHEP 1401 (2014) 110.
- (30) G. Abelof, A. Gehrmann-De Ridder and I. Majer, JHEP 1512 (2015) 074.
- (31) X. Chen, T. Gehrmann, E.W.N. Glover and M. Jaquier, Phys. Lett. B 740 (2015) 147.
- (32) A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, arXiv:1507.02850; arXiv:1605.04295.
- (33) T. Gleisberg, S. Höche, F. Krauss, M. Schönherr, S. Schumann, F. Siegert and J. Winter, JHEP 0902 (2009) 007.
- (34) T. Carli, T. Gehrmann and S. Höche, Eur. Phys. J. C 67 (2010) 73.
- (35) F. Cascioli, P. Maierhöfer and S. Pozzorini, Phys. Rev. Lett. 108 (2012) 111601.
- (36) E.W.N. Glover and J. Pires, JHEP 1006 (2010) 096; A. Gehrmann-De Ridder, E.W.N. Glover and J. Pires, JHEP 1202 (2012) 141.
- (37) S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Nucl. Phys. B 406 (1993) 187; G.P. Salam, Eur. Phys. J. C 67 (2010) 637.
- (38) S. Catani and B.R. Webber, JHEP 9710 (1997) 005.