Dijet production in diffractive deepinelastic scattering in nexttonexttoleading order QCD
Abstract
Hard processes in diffractive deepinelastic scattering can be described by a factorisation into partonlevel subprocesses and diffractive parton distributions. In this framework, cross sections for inclusive dijet production in diffractive deepinelastic electron–proton scattering (DIS) are computed to nexttonexttoleading order (NNLO) QCD accuracy and compared to a comprehensive selection of data. Predictions for the total cross sections, 40 singledifferential and four doubledifferential distributions for six measurements at HERA by the H1 and ZEUS collaborations are calculated. In the studied kinematical range, the NNLO corrections are found to be sizeable and positive. The NNLO predictions typically exceed the data, while the kinematical shape of the data is described better at NNLO than at nexttoleading order (NLO). A significant reduction of the scale uncertainty is achieved in comparison to NLO predictions. Our results use the currently available NLO diffractive parton distributions, and the discrepancy in normalisation highlights the need for a consistent determination of these distributions at NNLO accuracy.
1 Introduction
Diffractive processes in deepinelastic scattering, \(ep \rightarrow eXY\), where the final state systems X and Y are separated in rapidity, have been studied extensively at the electron–proton collider HERA [1]. The forward system Y consists of the leading proton, which stays intact after the collisions, but may also contain its low mass dissociation. Between the systems X and Y a depleted region without any hadronic activity is observed, the socalled large rapidity gap (LRG). This is a consequence of the vacuum quantum numbers of the diffractive exchange which is often referred to as a pomeron (\(I\!\!P\)). Experimentally, the diffractive events can be selected either by requiring a rapidity region in the direction of the proton beam without any hadronic activity (LRG method) or by direct detection of the leading proton using dedicated spectrometers. In the second case, the system Y is free of any diffractive dissociation.
Predictions for diffractive processes in DIS can be obtained in the framework of perturbative QCD (pQCD). According to the factorisation theorem for diffractive DIS (DDIS) [2], if the process is sufficiently hard, the calculation can be subdivided into two components: the hard partonic cross sections, \(\mathrm {d}\hat{\sigma }_n\), are calculable within pQCD in powers of \(\alpha _{\text {s}} (\mu _{\text {R}}) \), which need to be convoluted with soft diffractive parton distribution functions (DPDFs, \(f^D_a\)) that specify the contributing parton a inside the incoming hadron. DPDFs are universal for all diffractive deepinelastic processes [2], with the hardness of the process being ensured by the virtuality \(Q^2\) of the exchanged photon.
Up to now, predictions for diffractive processes, and in particular for diffractive dijet production, were performed only in nexttoleading order QCD (NLO). These predictions were able to describe the measured cross sections satisfactorily, both in shape and normalisation (for a review see e.g. Ref. [1]). However, due to their large theoretical uncertainties they did not achieve the precision of the data and thus did not allow for more stringent conclusions, i.e. about the underlying fundamental concepts of the diffractive exchange. Furthermore, the NLO predictions for dijet production were about two times higher than the leadingorder (LO) predictions. This raised the natural question concerning the size of contributions from even higher orders for such processes at the comparably low scales of the HERA data.
Here, we present the nexttonexttoleading (NNLO) perturbative QCD calculations for dijet production in diffractive DIS. These calculations are performed for the first time and constitute the first NNLO predictions for a diffractive process. We compare our predictions with several single, doubledifferential and total cross sections from six distinct measurements published by the H1 or ZEUS collaboration. A quantitative comparison of NLO and NNLO predictions with the data is presented. We further study the scale dependence of the NNLO predictions. Different DPDF parameterisations are studied and we provide additional studies about the sensitivity of the dijet data for future DPDF determinations.
2 NNLO predictions for dijet production in DDIS
In this case, a dijet system is characterised by at least two outgoing jets within a given pseudorapidity range (\(\eta _\mathrm{jet}^{*}\) or \(\eta _\mathrm{lab}^\mathrm{jet}\)) with sufficiently high transverse momenta \(p_{\mathrm {T}}^{*,\mathrm jet}\) in the \(\gamma ^*p\) rest frame.^{1} At HERA, particles are commonly clustered into jets using the \(k_t\) cluster algorithm [4]. The jet with the highest (second highest) \(p_{\mathrm {T}}^{*,\mathrm jet}\) is denoted as ‘leading jet’ (‘subleading jet’) and their average transverse momentum and invariant mass is calculated as \(\langle p_{\mathrm {T}}\rangle =(p_{\mathrm {T}}^{*,\mathrm jet1} +p_{\mathrm {T}}^{*,\mathrm jet2})/2\) and denoted by \(M_\mathrm {12}\), respectively.
The DPDFs have many properties similar to the nondiffractive PDFs, in particular they obey the DGLAP evolution equation [2, 5, 6, 7], however, DPDFs are constrained by the presence of the leading proton in the final state. In parameterised DPDFs the tdependence of the cross section is integrated out and in the considered measurements is restricted either by \(t<1\,\mathrm GeV ^2 \) or \(t<0.6\,\mathrm GeV ^2 \).
In this paper, the partonlevel jetproduction cross sections in DDIS are calculated up to NNLO. These calculations are identical to the NNLO calculations in the nondiffractive case [8, 9]. The NNLO correction involves three types of scattering amplitudes: the twoloop amplitudes for twoparton final states [10, 11, 12, 13], the oneloop amplitudes for threeparton final states [14, 15, 16, 17] and the treelevel amplitudes for fourparton final states [18, 19, 20]. These contributions contain implicit infrared divergences from soft and/or collinear realemission corrections as well as explicit divergences of both infrared and ultraviolet origin from the virtual loop corrections. When calculating predictions for an infraredsafe final state definition, these singularities cancel when the different parton multiplicities are combined [21]. The calculation employs the antenna subtraction method [22, 23, 24, 25]: For realradiation processes, the subtraction terms are constructed out of antenna functions, which encapsulate all colorordered unresolved parton emission inbetween pairs of hard radiator partons. To constitute a subtraction term, the antenna functions are then multiplied with reduced matrix elements of lower partonic multiplicity. By making the infrared pole structure explicit, the integrated subtraction terms can be combined with the virtual corrections in order to obtain a finite result. Relevant treelevel and oneloop matrix elements were verified against Sherpa [26, 27, 28] and nlojet++ [29, 30, 31]. Our computation is performed within the partonlevel event generator NNLOJET [32], which implements the antenna subtraction formalism and further provides a validation framework to ensure the correctness of the results. These tests comprise the analytic cancellation of all infrared poles and a numerical check of the behaviour of the subtraction terms to mimic the realemission matrix elements in all unresolved limits [33, 34]. All calculations are performed using the \(\overline{\mathrm{MS}}\) renormalisation scheme and for five massless quark flavors. The strong coupling constant is set to \(\alpha _{\text {s}} (M_{\text {Z}}) =0.118\) [35].
The calculation of NNLO partonic cross sections [8, 9] has recently been applied successfully to describe inclusive jet and dijet cross section data in nondiffractive DIS [8, 9, 36, 37]. Here, however, the hard coefficients are now convoluted with DPDFs for the first time and we present the first calculation of a diffractive jet production process to NNLO in \(\alpha _{\text {s}} (\mu _{\text {R}})\). For this reason our predictions are limited by the available DPDFs which have only been determined up to NLO so far.
For the convolutions with the DPDFs, the phase space integration of the matrix element squared has to be adopted for the integrations over the additional diffractive variables t and \(x_{I\!\!P} \). This has been reported, for instance, for implementations in the programs DISENT [38, 39], JetViP [40, 41, 42] and nlojet++ [29, 30, 43]. While these previous calculations commonly used the computationally very expensive Monte Carlo or the slicing method [43], here an improved convolution formalism is used. Our calculation thereby employs the fastNLO formalism [3, 44, 45] which has the advantage that the matrix elements have to be calculated only once and can then be used repeatedly for integrations of the DPDFs. The formalism will be briefly explained in the following.
The fastNLO based approach has advantages of a higher numerical accuracy of the \(x_{I\!\!P}\) integration, and, more importantly still, a significantly higher numerical accuracy is achieved in the calculation of the hard matrix elements for a given amount of computing time. This is of great importance for the calculation of the doublereal and realvirtual NNLO amplitudes, which are calculated here using several 100,000 hours of CPU time using stateoftheart CPUs. The numerical accuracy of the fastNLO interpolation technique is typically smaller than the numerical precision of the tabulated DPDFs, and thus can be neglected.
In order to avoid regions of the phase space where the predictions exhibit an enhanced infrared sensitivity [41, 46], the phase space definitions of all analyses have asymmetric cuts on the transverse momenta of the two leading jets. It was tested that the difference of \(\sim \! 1\, \text {GeV}\) between the cuts on the leading and subleading jet is sufficient to remove this region.

H1FitB [52] is the most widely used DPDF. It was determined from an NLO DGLAP QCD fit to reduced inclusive DDIS cross sections. The diffractive data was selected using the LRG method and, therefore, the DPDF includes proton dissociation into a lowmass hadronic state (\(M_\mathrm{Y} <1.6\,\mathrm GeV \)). The phase space of the selected data was restricted to \(\beta < 0.8\) and \(Q^2 >8.5\,\mathrm GeV ^2 \). The gluon DPDF at the starting scale of the evolution, \(\mu _0^2 = 1.75\,\mathrm GeV ^2 \), was assumed to be a constant, i.e. independent of the value of \(z_{I\!\!P}\).

H1FitA [52] is a variant of the H1FitB DPDF, which uses a more flexible parameterisation of the gluon distribution at the starting scale of the evolution. In comparison to the H1FitB DPDF, a significantly larger gluon DPDF is found although both, the H1FitA and the H1FitB DPDF, describe the shape of the data equally well, as inclusive DDIS cross sections are only weakly sensitive to the gluon DPDF. A detailed analysis of dijet data suggests [43] that the gluon component in the H1FitA DPDF is overestimated.

H1FitJets [43] is the first DPDF fitted based on the combination of inclusive and dijet data, using the same inclusive data sample as for H1FitB and H1FitA. The inclusion of dijet data, which is more sensitive to the gluon content, led to a slightly smaller gluon distribution compared to the H1FitB DPDF.

ZEUSSJ [53] is determined by a combined fit of inclusive and dijet data by the ZEUS collaboration. Compared to H1 fits, the proton dissociation has been subtracted using Monte Carlo (MC) estimates such that this DPDF is defined for elastic scattering (\(M_\mathrm{Y} =m_P\)).

The MRW DPDF [54] is based on the same data as the H1FitB DPDF. In contrast, however, Regge factorisation is only assumed at the starting scale and the evolution is performed using inhomogeneous evolution equations accounting for pomerontoparton splittings.
Similarly as in the definitions for DPDF fits, also the various measurements impose different definitions of \(M_\mathrm{Y}\). The LRG measurements by H1 are defined for \(M_\mathrm{Y} <1.6\,\mathrm GeV \), whereas ZEUS extrapolated its LRG measurement to \(M_\mathrm{Y} =m_P\). Two of the H1 measurements are based on proton spectrometers (FPS, VFPS), and thus these data do not contain any proton dissociation (\(M_\mathrm{Y} =m_P\)).
In order to compare the data with fixedorder predictions, correction factors accounting for hadronisation effects are applied. These are estimated using MC simulations and corresponding correction factors are provided together with the respective data as discussed in the next section.
3 Data sets and observables
Summary of the dijet data sets. The first column represents the data set label and the second shows the integrated luminosity and the number of events of the given data set. The other columns summarise the definition of the phase space of the given data. In cases, where the DIS phase space is defined in terms of W, the corresponding range in \(y=W^2/s\) is shown. All measurements have in common a requirement of \(n_\mathrm{jets} \ge 2\) in the given dijet range, which is applied after identifying the two leading jets
Data set  \(\varvec{\mathcal {L}}\)  DIS range  Dijet range  Diffractive range 

H1 FPS (HERA 2) [59]  156.6 \(\mathrm{pb}^{1}\)  \(4<Q^2 <110\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5\,\mathrm GeV \)  \(x_{I\!\!P} <0.1\) 
(581ev)  \(0.05< y < 0.7\)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<1\,\mathrm GeV ^2 \)  
\(1<\eta _\mathrm{lab}^\mathrm{jet} <2.5\)  \(M_\mathrm{Y} = m_P \)  
H1 VFPS (HERA 2) [60]  50 \(\mathrm{pb}^{1}\)  \(4<Q^2 <80\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5.5\,\mathrm GeV \)  \(0.010<x_{I\!\!P} <0.024\) 
(550ev)  \(0.2< y < 0.7\)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<0.6\,\mathrm GeV ^2 \)  
\(1<\eta _\mathrm{lab}^\mathrm{jet} <2.5\)  \(M_\mathrm{Y} = m_P \)  
H1 LRG (HERA 2) [3]  290 \(\mathrm{pb}^{1}\)  \(4<Q^2 <100\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5.5\,\mathrm GeV \)  \(x_{I\!\!P} <0.03\) 
(\(\sim \)15000ev)  \(0.1< y < 0.7\)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<1\,\mathrm GeV ^2 \)  
\(1<\eta _\mathrm{lab}^\mathrm{jet} <2\)  \(M_\mathrm{Y} <1.6 \, \mathrm GeV \)  
H1 LRG (HERA 1) [43]  51.5 \(\mathrm{pb}^{1}\)  \(4<Q^2 <80\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5.5\,\mathrm GeV \)  \(x_{I\!\!P} <0.03\) 
(2723ev)  \(0.1< y < 0.7\)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<1\,\mathrm GeV ^2 \)  
\(3<\eta ^{*\mathrm {jet}}<0\)  \(M_\mathrm{Y} <1.6 \, \mathrm GeV \)  
H1 LRG (\(300\,\mathrm GeV \)) [61]  18 \(\mathrm{pb}^{1}\)  \(4<Q^2 <80\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5\,\mathrm GeV \)  \(x_{I\!\!P} <0.03\) 
(322ev)  \(165< W < 242\,\mathrm GeV \)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<1\,\mathrm GeV ^2 \)  
\((0.30< y < 0.65)\)  \(1<\eta _\mathrm{lab}^\mathrm{jet} <2\)  \(M_\mathrm{Y} <1.6 \, \mathrm GeV \)  
\(3<\eta ^{*\mathrm {jet}}<0\)  
ZEUS LRG (HERA 1) [62]  61 \(\mathrm{pb}^{1}\)  \(5<Q^2 <100\,\mathrm GeV ^2 \)  \(p_{\mathrm {T}}^{*,\mathrm jet1} >5\,\mathrm GeV \)  \(x_{I\!\!P} <0.03\) 
(5539ev)  \(100< W < 250\,\mathrm GeV \)  \(p_{\mathrm {T}}^{*,\mathrm jet2} >4.0\,\mathrm GeV \)  \(t<1\,\mathrm GeV ^2 \)  
\((0.10< y < 0.62)\)  \(3.5<\eta ^{*\mathrm {jet}}<0\)  \(M_\mathrm{Y} = m_P\) 

The DIS kinematic variables: \(Q^2\), y and W;

The jet transverse momentum observables: \(p_{\mathrm {T}}^{*,\mathrm jet1}\), \(p_{\mathrm {T}}^{*,\mathrm jet2}\), \(\langle p_{\mathrm {T}}\rangle \) and \(p_{\mathrm {T}}^{*,\mathrm jet}\). Here \(p_{\mathrm {T}}^{*,\mathrm jet}\) refers to the \(p_\mathrm {T}\) of the leading and subleading jet;

The jet pseudorapidity observables: \(\langle \eta _\mathrm{lab}^\mathrm{jet}\rangle \), \(\eta _\mathrm{jet}^{*}\), \(\Delta \eta _\mathrm{lab}^\mathrm{jet}\), and \(\Delta \eta ^{*}\). Here \(\langle \eta _\mathrm{lab}^\mathrm{jet}\rangle \) denotes the average pseudorapidity \(\eta _\mathrm{jet}^{*}\) of the two leading jets and \(\Delta \eta _\mathrm{lab}^\mathrm{jet}\) and \(\Delta \eta ^{*}\) denote their separation in pseudorapidity;

Observables of the diffractive final state: \(x_{I\!\!P}\), \(z^\mathrm{obs}_{I\!\!P}\) and \(M_\mathrm{X}\);

Doubledifferential measurements as functions of \(z^\mathrm{obs}_{I\!\!P}\) or \(p_{\mathrm {T}}^{*,\mathrm jet1}\) for \(Q^2\) intervals, and as a function of \(z^\mathrm{obs}_{I\!\!P}\) for \(p_{\mathrm {T}}^{*,\mathrm jet1}\) intervals.
4 Results
4.1 Total dijet production cross section
Comparison of the measured and predicted total dijet cross sections for the six measurements. Listed are the data cross section, \(\sigma ^\mathrm{Data}\), the NLO and the NNLO predictions, \(\sigma ^\mathrm{NLO}\) and \(\sigma ^\mathrm{NNLO}\), respectively. For \(\sigma ^\mathrm{Data}\) the uncertainties denote the statistical and the systematic uncertainty. In case of H1 LRG (\(300\,\mathrm GeV \)), the total cross section is calculated by us from the singledifferential distributions. The uncertainty of the NLO or NNLO predictions denote the scale uncertainty obtained from a simultaneous variation of \(\mu _{\text {R}}\) and \(\mu _{\text {F}}\) by factors of 0.5 and 2. The last two columns show the DPDF uncertainty obtained from H1FitB for the NLO or NNLO predictions. In terms of a relative uncertainty, the DPDF uncertainty is almost identical for NLO and NNLO predictions
Data set  \(\sigma ^\mathrm{Data}\)  \(\sigma ^\mathrm{NLO}\)  \(\sigma ^\mathrm{NNLO}\)  \(\Delta _\mathrm{DPDF}^\mathrm{NLO}\)  \(\Delta _\mathrm{DPDF}^\mathrm{NNLO}\) 

[pb]  [pb]  [pb]  [pb]  [pb]  
H1 FPS (HERA 2)  \(254\pm 20\pm 27\)  \(296^{+92}_{57}\)  \(366^{+27}_{41}\)  \(^{+29}_{46}\)  \(^{+36}_{57}\) 
H1 VFPS (HERA 2)  \(30.5\pm 1.6\pm 2.8\)  \(29.3^{+11.2}_{6.7}\)  \(38.3^{+5.1}_{5.8}\)  \(^{+3.2}_{4.2}\)  \(^{+4.4}_{5.6}\) 
H1 LRG (HERA 2)  \(73\pm 2\pm 7\)  \(75.7^{+29.4}_{17.7}\)  \(98.6^{+13.2}_{15.4}\)  \(^{+8.5}_{10.9}\)  \(^{+11.7}_{14.7}\) 
H1 LRG (HERA 1)  \(51\pm 1^{+7}_{5}\)  \(63.4^{+25.2}_{15.1}\)  \(85.3^{+14.3}_{14.3}\)  \(^{+7.1}_{9.2}\)  \(^{+10.1}_{12.7}\) 
H1 LRG (300 Gev)  \(28.7\pm 1.8 \pm 3.0\)  \(32.5^{+13.7}_{7.9}\)  \(46.4^{+9.9}_{8.5}\)  \(^{+3.5}_{4.6}\)  \(^{+5.3}_{6.7}\) 
ZEUS LRG (HERA 1)  \(89.7\pm 1.2^{+6.0}_{6.4}\)  \(95.5^{+31.5}_{20.0}\)  \(114.9^{+7.1}_{13.8}\)  \(^{+10.5}_{13.4}\)  \({}^{+13.5}_{16.7}\) 
The NNLO predictions compared to the NLO predictions are higher by about 20–40 %. Since the kinematic ranges of different measurements are rather similar (Table 1), also the NNLO corrections are of similar size for the individual measurements. As found previously [3, 43, 59, 60, 61], the NLO predictions provide a good description for all of the data. In contrast, the NNLO predictions typically overshoot the data. This tension between NNLO and data may be attributed to inappropriate DPDFs, where we use the H1FitB DPDF set, which has been determined using NLO predictions. In particular, the gluon component in this DPDF appears to be too high for the usage with NNLO QCD coefficients, as this DPDF has been determined from inclusive DDIS cross section data using the respective NLO predictions only.
When compared to our common predictions, all measurements appear to be consistent with each other, although they use different techniques for the identification of the diffractive final states.
4.2 NNLO scale uncertainty and scale choice
The scale uncertainties, which are obtained by a simultaneous variation of \(\mu _{\text {R}}\) and \(\mu _{\text {F}}\) by factors of 0.5 and 2, are found to be reduced significantly for NNLO predictions in comparison to NLO predictions (see also Table 2 and Fig. 2). The typical size of the scale uncertainty of the total dijet cross sections at NNLO is about 15 %, whereas it is about 35 % in NLO. In case of the H1 LRG (HERA 2) total cross section for instance, the upward (downward) scale uncertainty is reduced from 39 % (23 %) at NLO to 13 % (16 %) at NNLO. This makes these uncertainties competitive with the data uncertainty (\(\sim \! 10\%\)). For all total cross section measurements, however, the differences between data and NNLO predictions are larger than respective theoretical scale uncertainties.
NNLO predictions for H1 LRG (HERA 2) using different choices for \(\mu _{\text {R}} ^2\) and \(\mu _{\text {F}} ^2\). The uncertainties denote the scale uncertainty from simultaneously varying \(\mu _{\text {R}}\) and \(\mu _{\text {F}}\) by factors of 0.5 or 2
Data set  \(\sigma ^\mathrm{Data}\)  \(Q^2 +\langle p_{\mathrm {T}}\rangle ^2\)  \(Q^2 \)  \(\langle p_{\mathrm {T}}\rangle ^2\)  \(\tfrac{Q^2}{4}+\langle p_{\mathrm {T}}\rangle ^2\)  \(\sqrt{Q^4+\langle p_{\mathrm {T}}\rangle ^4}\) 

[pb]  [pb]  [pb]  [pb]  [pb]  [pb]  
H1 LRG (HERA 2)  \(73\pm 7_\mathrm{exp}\)  \(98.6^{+13.2}_{15.4}\)  \(111.7^{43.4}_{11.5}\)  \(102.1^{+8.4}_{15.2}\)  \(101.1^{+10.6}_{15.4}\)  \(101.0^{+11.2}_{15.5}\) 
4.3 DPDF choice and uncertainties
NNLO predictions for H1 LRG (HERA 2) using different DPDFs. Mind, all DPDFs have been determined only in NLO accuracy. The uncertainties denote the DPDF uncertainty as provided by the respective DPDF sets
Data set  \(\sigma ^\mathrm{Data}\)  \(\sigma ^{\text{ H1FitA }}\)  \(\sigma ^{\text{ H1FitB }}\)  \(\sigma ^{\text{ H1FitJets }}\)  \(\sigma ^{\text{ MRW }}\)  \(\sigma ^{\text{ ZEUSSJ }}\) 

[pb]  [pb]  [pb]  [pb]  [pb]  [pb]  
H1 LRG (HERA 2)  \(73\pm 7_\mathrm{exp}\)  \(129.3^{+16.8}_{20.4}\)  \(98.6^{+11.7}_{14.7}\)  83.1  101.8  78.0 
Overview of the measured single and doubledifferential distributions
Histogram  H1 FPS  H1 VFPS  H1 LRG  H1 LRG  H1 LRG  ZEUS LRG 

(HERA 2)  (HERA 2)  (HERA 2)  (HERA 1)  (\(300\,\mathrm GeV \))  (HERA 1)  
\(Q^2 \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  
\(y ~[W]^*\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(*\)  \(*\) 
\(p_{\mathrm {T}}^{*,\mathrm jet1} ~[p_{\mathrm {T}}^{*,\mathrm jet} ]^*\)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(*\) 
\(\langle p_{\mathrm {T}}\rangle \)  \(\checkmark \)  
\(p_{\mathrm {T}}^{*,\mathrm jet2} \)  \(\checkmark \)  
\(\langle \eta _\mathrm{lab}^\mathrm{jet}\rangle ~[\eta _\mathrm{jet}^{*} ]^*\)  \(\checkmark \)  \(\checkmark \)  \(*\)  
\(\Delta \eta _\mathrm{lab}^\mathrm{jet} ~[\Delta \eta ^{*} ]^*\)  \(*\)  \(\checkmark \)  \(*\)  \(*\)  \(*\)  
\(M_\mathrm{X} \)  \(\checkmark \)  \(\checkmark \)  
\(x_{I\!\!P} \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \) 
\(z^\mathrm{obs}_{I\!\!P} \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  \(\checkmark \)  
\((Q^2;p_{\mathrm {T}}^{*,\mathrm jet1})\)  \(\checkmark \)  
\((Q^2;z^\mathrm{obs}_{I\!\!P})\)  \(\checkmark \)  \(\checkmark \)  
\((p_{\mathrm {T}}^{*,\mathrm jet1};z^\mathrm{obs}_{I\!\!P})\)  \(\checkmark \) 
4.4 Differential distributions
In total we computed 40 singledifferential distributions and four doubledifferential distributions for available measurements, which are summarised in Table 5.
The NNLO predictions as a function of \(Q^2\), \(\Delta \eta ^{*} \) (or \(\Delta \eta \)), \(p_{\mathrm {T}}^{*,\mathrm jet1}\) (or \(p_{\mathrm {T}}^{*,\mathrm jet}\)), \(\langle p_{\mathrm {T}}\rangle \), \(p_{\mathrm {T}}^{*,\mathrm jet2} \), \(M_\mathrm{X} \), \(\langle \eta _\mathrm{lab}^\mathrm{jet}\rangle \) (or \(\eta _\mathrm{jet}^{*} \)), \(x_{I\!\!P}\)^{4} and \(z^\mathrm{obs}_{I\!\!P}\) are presented in Figs. 6, 7, 8, 9, 10, 11 and 12, respectively, and compared to data. Doubledifferential predictions as functions of \(z^\mathrm{obs}_{I\!\!P}\) and \(p_{\mathrm {T}}^{*,\mathrm jet1}\) for \(Q^2\) intervals, and as a function of \(z^\mathrm{obs}_{I\!\!P}\) for \(p_{\mathrm {T}}^{*,\mathrm jet1}\) intervals are presented in Figs. 13, 14, 15 and 16. Similar conclusions as for the y distribution can be drawn from these comparisons. Some variants of selected distributions are discussed in more detail in the following.
While y is an inclusive observable, the rapidity separation of the two leading jets, \(\Delta \eta ^{*} \), is directly sensitive to effects emerging from higher order radiative corrections. Also for this observable, the NNLO predictions provide an improved description of the shape for measured distributions, as can be seen in Fig. 7. Similar observations are made for all remaining distributions. This in particular for distributions in \(Q^2\), \(\langle \eta \rangle \) and \(z^\mathrm{obs}_{I\!\!P}\) (see Figs. 6, 10, 12).
NNLO predictions as a function of \(Q^2\) obtained with different scale definitions are displayed in Fig. 17. For this study we set \(\mu := \mu _F = \mu _R\). The studied scale definitions \(\mu ^2 = Q^2/4 + \langle p_{\mathrm {T}}\rangle ^2\) and \(\mu ^2 = \langle p_{\mathrm {T}}\rangle ^2\) provide similar results as the nominal scale definition of \(\mu ^2 = Q^2 + \langle p_{\mathrm {T}}\rangle ^2\), whereas the scale choice \(\mu ^2 = Q^2 \) results in higher cross sections and a steeper \(Q^2\) spectrum. The studied scale choices are covered by the scale uncertainties.
NNLO predictions for \(z^\mathrm{obs}_{I\!\!P}\) distributions obtained using different DPDFs are displayed in Fig. 18. For this observable, NNLO predictions using the H1FitB and MRW DPDFs give quite similar results and lie above most of the data. Results obtained with the H1FitA DPDF significantly overestimate the measurements in particular for higher values of \(z^\mathrm{obs}_{I\!\!P}\). Predictions obtained with ZEUSSJ and H1FitJets give lower cross sections, but the application of the H1FitJets DPDF also results in a considerably different shape of the distribution. In general, the latter two DPDFs, including dijet data in their determination, give an improved description of the data compared to the first two DPDFs. It should be noted however, that differences arising from applications of different DPDFs are not covered by the uncertainties taken from the H1FitB DPDF. This feature is most prominent at higher values of \(z^\mathrm{obs}_{I\!\!P}\).
In summary, NNLO predictions using the stated DPDFs provide an overall satisfactorily description of the data. However, none of the studied DPDFs is able to describe the shapes of the distributions of all of the \(p_{\mathrm {T}}^{*,\mathrm jet1}\) (or \(p_{\mathrm {T}}^{*,\mathrm jet}\)) measurements equally well, as can be seen from their comparisons to predictions displayed in Fig. 19.
The studied DPDFs mainly differ in their gluon component [65]. This explains the observed differences between results obtained with different DPDFs as the gluon is the most important parton inside the DPDFs. It is therefore crucial to determine the gluonic component of the DPDFs more accurately, and once this is achieved, theoretical predictions are expected to provide an improved description of the data.
Once the additional cut on \(\eta _\mathrm{lab}^\mathrm{jet}\) is imposed, the relative NNLO scale uncertainty increases significantly, i.e. up to a factor of two in some parts of the phase space. This becomes in particular distinct at high values of W, as displayed in Fig. 20 (right). In conclusion, it is observed that the phase space definition of ZEUS LRG (HERA 1) results in more stable pQCD predictions, i.e. lower scale uncertainties, while important regions of the phase space were not accessible by the experimental device and the extrapolation factors were obtained by MC simulations. Similar considerations also apply to the H1 LRG (HERA 1) measurement.
4.5 The gluon induced fraction
In order to further elucidate the dependence of the NNLO predictions on the individual parton flavors inside the DPDFs, the decomposition of the total H1 LRG (HERA 2) cross section into gluoninduced and quarkinduced channels is shown for LO , NLO and NNLO predictions in Fig. 21. It is apparent that the rise of the cross section at higher orders is predominantly driven by the gluoninduced channels.
The fractions of gluon and quarkinduced contributions to the cross sections as a function of \(z^\mathrm{obs}_{I\!\!P}\) are displayed in Fig. 22. While the fraction of the gluoninduced contribution remains unchanged for different orders in \(\alpha _s\) at low values of \(z^\mathrm{obs}_{I\!\!P}\), there is a strong increase of the gluoninduced fraction at higher values of \(z^\mathrm{obs}_{I\!\!P}\) for higher orders in \(\alpha _s\). Hence it can be deduced that future NNLO DPDFs are required to have a significantly reduced gluon component as compared to currently available NLO DPDFs.
4.6 The sensitivity to DPDFs
A detailed study on the dependence of the cross section on the DPDF is presented for the \(\langle p_{\mathrm {T}}\rangle \) distribution of the H1 LRG (HERA 2) measurement. The contributions to the cross section in each bin as a function of the DPDF parameters \(x_{I\!\!P}\) and \(z_{I\!\!P}\) is displayed in Fig. 23. At highest values of \(\langle p_{\mathrm {T}}\rangle \), only partons with comparably high values of \(x_{I\!\!P}\) and \(z_{I\!\!P}\) are contributing to the cross section, whereas the cross section at medium values of \(\langle p_{\mathrm {T}}\rangle \) is dominated by low \(x_{I\!\!P}\) and \(z_{I\!\!P}\) partons. All three bins have recognisable contributions from high values of \(z^\mathrm{obs}_{I\!\!P}\) which is a distinct feature for predictions obtained with the H1FitB DPDF.
4.7 Quantitative comparison
The calculations are repeated for different DPDFs and different scale functional forms and also in these cases, it is observed that NNLO predictions mostly give lower \(\chi ^{2}/n_\mathrm{dof} \) values than NLO predictions (not shown). In an approximation, the normalisation of the predictions is proportional to the gluon content of the DPDFs, whereas the shapes of the differential distributions are related more closely to the hard matrix elements. Therefore, these results indicate that NNLO predictions provide a better description of the data than NLO predictions, and we believe that future DPDFs determined to NNLO QCD will be able to provide an improved description of the dijet data, this also with respect to the normalisation.
5 Discussion and summary
We present the first NNLO QCD predictions for jet production in diffractive scattering. Predictions for six measurements of dijet production in diffractive deepinelastic scattering from the H1 and ZEUS collaborations were calculated and compared to data. We observe that the NNLO cross sections are significantly higher than the data and are higher than NLO calculations by about 20–40 % in the studied kinematical range.
Since no DPDFs in NNLO accuracy are available so far, only NLO DPDFs could be employed for our calculations. The discrepancy of the NNLO predictions and data is believed to be due to an overestimated gluon component of these DPDFs. Alternative DPDFs, which also considered dijet data in their determination, already result in typically lower NNLO predictions, but these still overshoot the data. Ignoring the issue of normalisation, the shapes of differential distributions are better described by NNLO than NLO predictions. This is quantified by evaluating \(\chi ^{2} \) values for the examined experimental distributions. The large amount of studied observables, which have so far not even been studied in nondiffractive DIS, prove that NNLO predictions provide an improved description in shape of the data throughout.
We believe that the normalisation difference between data and NNLO predictions can be resolved by employing DPDFs determined to NNLO accuracy and by including dijet data for their determinations. This in particular as the gluon component is most important and is only weakly constrained by the inclusive data. The NNLO dijet calculations presented here are already in a numerical format, which is suitable for such future analyses.
The comprehensive selection of all available dijet data represents the first comparison, where all these measurements are compared to predictions obtained in an identical framework. Data taken with different experimental devices, at different centerofmass energies, and using either proton spectrometers or the LRG method for the identification of the diffractive final state are investigated. All measurements are found to be mutually consistent when compared to respective predictions.
The presented NNLO predictions provide the most precise predictions for dijet production in diffractive DIS to date, and the dominant theoretical uncertainty is reduced substantially with respect to predictions in NLO. The NNLO coefficients exhibit a precision in terms of scale uncertainties, which is of a comparable size as that of presently available DPDFs, and of comparable size as that of the HERA dijet data. It is observed, that for the given kinematical range of the HERA data, higherorder corrections are of crucial importance.
Footnotes
 1.
Here, observables in the \(\gamma ^*p\) (laboratory) frame are conventionally denoted with an asterisk ‘\(^*\)’ (superscript ‘lab’).
 2.
In inclusive DDIS the invariant \(\beta ={Q^2}/{2q\cdot (pp')}\) has a similar interpretation, which can also be calculated as \(\beta = {x_{\mathrm {Bj}}}/{x_{I\!\!P}}\), with \(x_{\mathrm {Bj}} ={Q^2}/{2 p\cdot q}\).
 3.
Although, the commonly employed approach for the determination of the DPDFs is a global fit to experimental data, the DPDFs can alternatively be defined through lightcone matrix elements, and in this framework certain aspects of their behaviour at the starting scale of the evolution can be computed [47].
 4.
The numerical values of the \(x_{I\!\!P}\) distribution of the ZEUS LRG (HERA 1) measurement were provided to us by the ZEUS physics office [64].
 5.
The ZEUS LRG (HERA 1) analysis required two jets to be within \(2<\eta _\mathrm{lab}^\mathrm{jet} <2\) [62]. For better comparability and also due to technical reasons, we study an additional cut of \(1<\eta _\mathrm{lab}^\mathrm{jet} <2.5\) in analogy to the H1 FPS (HERA 2) and H1 VFPS (HERA 2) measurements.
 6.
The H1 collaboration estimated an uncertainty for \(\mu _{\text {R}}\) and \(\mu _{\text {F}}\) separately and considered the resulting uncertainties on the cross sections as half correlated and half uncorrelated.
Notes
Acknowledgements
We thank W. Slominski and M. Wing for discussions and help with the ZEUS data, and B. Pokorny for discussions on the H1 LRG data. We thank X. Chen, J. CruzMartinez, R. Gauld, A. Gehrmann–De Ridder, N. Glover, M. Höfer, I. Majer, T. Morgan, J. Pires, D. Walker and J. Whitehead for useful discussions and their many contributions to the \(\mathrm{NNLO}\small \mathrm{JET}\) code. We are grateful for the collaboration with C. Gwenlan, K. Rabbertz and M. Sutton for the interface of fastNLO and APPLgrid to \(\mathrm{NNLO}\small \mathrm{JET}\). This research was supported in part by the Swiss National Science Foundation (SNF) under contract 200020175595 and by the Research Executive Agency (REA) of the European Union under the ERC Advanced Grant MC@NNLO (340983).
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