Measurement of \({\varvec{D^{*}}}\) production in diffractive deep inelastic scattering at HERA
Abstract
Measurements of \(D^{*}(2010)\) meson production in diffractive deep inelastic scattering \((5<Q^{2}<100\,\mathrm{GeV}^{2})\) are presented which are based on HERA data recorded at a centreofmass energy \(\sqrt{s} = 319\,\mathrm{GeV}\) with an integrated luminosity of 287 pb\(^{1}\). The reaction \(ep \rightarrow eXY\) is studied, where the system X, containing at least one \(D^{*}(2010)\) meson, is separated from a leading lowmass proton dissociative system Y by a large rapidity gap. The kinematics of \(D^{*}\) candidates are reconstructed in the \(D^{*}\rightarrow K \pi \pi \) decay channel. The measured cross sections compare favourably with nexttoleading order QCD predictions, where charm quarks are produced via bosongluon fusion. The charm quarks are then independently fragmented to the \(D^{*}\) mesons. The calculations rely on the collinear factorisation theorem and are based on diffractive parton densities previously obtained by H1 from fits to inclusive diffractive cross sections. The data are further used to determine the diffractive to inclusive \(D^{*}\) production ratio in deep inelastic scattering.
1 Introduction
In the framework of Regge theory of soft hadronic interactions, the energy dependence of total hadron–hadron scattering cross sections is described only after taking into account a specific type of effective exchange with vacuum quantum numbers [1]. Although it is used in various contexts, such an exchange is often referred to as a ‘pomeron’ (\({I\!\!P}\)) [2]. Pomeron exchange is a tool to describe diffractive processes, which are characterised by large gaps, devoid of activity, in the rapidity distribution of final state particles.
Diffractive processes in electron–proton^{1} deep inelastic scattering were observed already in the very early part of the HERA experimental program [3, 4] and lead to revived interest in this class of soft peripheral hadronic interactions [5]. In reactions of the type \(ep \rightarrow eXY\) they are characterised by a large gap in rapidity between the systems X and Y. The system X can be considered as resulting from a diffractive dissociation of the virtual photon, while the system Y consists of the initial state proton or its low mass hadronic excitation, scattered at a small momentum transfer squared t relative to the initial state proton.
Perturbative quantum chromodynamics (pQCD) calculations are applicable in deep inelastic scattering even though the partonic structure of the proton is a priori unknown. In order to overcome this difficulty, the collinear factorisation theorem [6] is used, where the calculation of deep inelastic scattering (DIS) cross sections is described by a processdependent partonic hard scattering part convoluted with a universal set of parton distribution functions of the proton (PDF). Collinear factorisation, therefore, opens the possibility to extract PDFs from one process and use them to predict cross sections for another process. For the PDF extraction the validity of the DGLAP evolution equations [7, 8, 9] is assumed.
A similar strategy can also be adapted to diffractive deep inelastic scattering (DDIS), where collinear factorisation is expected to be valid as well [10]. Assuming in addition the validity of proton vertex factorisation [11], diffractive processes are described by the exchange of collective colourless partonic states, such as the pomeron. Diffractive parton distribution functions (DPDFs) are extracted from diffractive data [12, 13]. Similarly to the normal PDFs, the DPDFs are expected to evolve as a function of the scale as predicted by the DGLAP equations. QCD analyses of diffractive data show that gluons constitute the main contribution to the DPDFs [12, 13]. To date, analyses of HERA data support the validity of the collinear factorisation theorem in DDIS as evidenced by experimental results on inclusive production [12, 13], dijet production [13, 14, 15, 16, 17, 18, 19] and \(D^{*}\) production [20, 21, 22, 23, 24].
2 Kinematics of diffractive deep inelastic scattering
3 Monte Carlo models and fixed order QCD calculations
The diffractive and nondiffractive processes are modelled with the RAPGAP Monte Carlo event generator [25]. The generated Monte Carlo events are subjected to a detailed H1 detector response simulation based on GEANT3 [26]. These simulated samples are passed through the same analysis chain as used for data and are used to correct the data for detector effects.
Predictions for \(D^{*}\) cross sections in nexttoleadingorder (NLO) QCD precision are obtained from the HVQDIS [33, 34] program adapted for diffraction. The calculation relies on collinear factorisation using H1 2006 DPDF Fit B NLO parton density functions involving gluons and light quarks in the quark singlet (fixedflavournumber scheme). Massive charm quarks are produced via \(\gamma ^{*}\)gluon fusion with the QCD scale parameter set to \(\Lambda _{5} = 0.228~\mathrm{GeV}\), which corresponds to a 2loop \(\alpha _{s}(M_{Z})=0.118\), as was used in the DPDF extraction. The charm quarks are fragmented independently into \(D^{*}\) mesons with \(f(c\rightarrow D^{*}) = 0.235 \pm 0.007\) [35] in the \(\gamma ^{*}p\) rest frame using the Kartvelishvili parameterisation with parameters suited for use with HVQDIS [31]. The factorisation and renormalisation scales are set to \(\mu _{r} = \mu _{f} = \sqrt{Q^{2} + 4m_{c}^{2}}\) with the value \(m_{c} = 1.5\mathrm{~GeV}\) for the charm pole mass. The uncertainties arising from the choice of scales are estimated by simultaneously varying them by factors of 0.5 and 2. The uncertainty introduced in the calculation caused by the uncertainty of \(m_c\) is evaluated by varying \(m_{c}\) to 1.3 and 1.7 GeV. The Kartvelishvili parameters are varied within their uncertainties [31]. The DPDF uncertainties are estimated by propagating the eigenvector decomposition of the errors of the DPDF parameterisation. The individual sources of uncertainties are added in quadrature separated for up and down variations of the cross sections. The contribution of Bhadrons due to beauty fragmentation to the diffractive \(D^{*}\) cross section is neglected; it is expected to be less than 3% for nondiffractive DIS (see [36]) and even smaller for the diffractive production.
The HVQDIS calculation is performed also in the nondiffractive mode using the CT10F3 proton PDF set [37]. It is used for comparisons of predictions with measurements of the diffractive to inclusive cross section ratio (Sect. 5.2). The calculation is done following the one used for comparison with inclusive \(D^{*}\) data [38]. The uncertainties from the choice of scales and \(m_{c}\) as well as the fragmentation uncertainty are evaluated in the same manner as for the diffractive calculation. The uncertainty of the CT10F3 PDF set is not considered for this analysis but is expected to be small in comparison to the DPDF uncertainties.
4 Experimental technique
4.1 The H1 detector
A detailed description of the H1 detector can be found elsewhere [39]. Here, a brief account of the detector components most relevant to the present analysis is given. The H1 coordinate system is defined such that the origin is at the nominal ep interaction point and the polar angle \(\theta = 0\) and the positive z axis correspond to the direction of the outgoing proton beam. The region \(\theta < 90^\circ \), which has positive pseudorapidity \(\eta =  \ln \tan \theta /2\), is referred to as the ‘forward’ hemisphere.
The ep interaction point in H1 is surrounded by the central tracking system, which includes silicon strip detectors [40] as well as two large concentric drift chambers. These chambers cover a region in polar angle \(20^{\circ }< \theta < 160^{\circ }\) and provide a resolution of \(\sigma (P_{T})/P_{T} = 0.006P_{T}/\mathrm{~GeV} \oplus 0.02\). They also provide triggering information [41, 42]. The forward tracking detector, a set of drift chambers with sense wires oriented perpendicular to the z axis, extends the acceptance of the tracking system down to \(7^{\circ }\) in polar angle. The central tracking detectors are surrounded by a finely segmented liquid argon (LAr) sampling calorimeter covering \(1.5< \eta < 3.4\). Its resolution is \(\sigma (E)/E = 0.11/\sqrt{E/\mathrm{~GeV}} \oplus 0.01\) in its electromagnetic part and \(\sigma (E)/E = 0.50/\sqrt{E/\mathrm{~GeV}} \oplus 0.02\) for hadrons, as measured in test beams [43, 44].
The central tracker and LAr calorimeter are placed inside a large superconducting solenoid, which provides a uniform magnetic field of 1.16 T. The backward region \(4< \eta < 1.4\) is covered by a leadscintillating fiber calorimeter (SpaCal [45]) with electromagnetic and hadronic sections. In the present analysis the energy and angle of the scattered electron is measured in the electromagnetic section of the SpaCal. It has an energy resolution for electrons \(\sigma (E)/E = 0.07 / \sqrt{E/\mathrm{~GeV}} \oplus 0.01 \) as measured in test beams [46].
Information from the central tracker and the LAr and SpaCal calorimeters is combined in an energy flow reconstruction algorithm which yields a list of hadronic final state objects [47, 48]. For these objects a calibration is applied ensuring the relative agreement of hadronic energy scale between the data and simulations at 1% accuracy [49].
In the forward region the H1 detector is equipped with drift chambers comprising the forward muon detector (FMD, \(1.9< \eta < 3.7\)). The forward tagger system (FTS) is a set of scintillators surrounding the beam pipe at several locations along the proton beamline, downstream of the H1 main detector. The FTS station at \(28 \mathrm{~m}\) covering the range \(6.0< \eta < 7.5\) is used in this analysis. Both FMD and FTS are sensitive to the very forward energy flow and improve the selection of large rapidity gap events.
The luminosity is measured via the Bethe–Heitler bremsstrahlung process \(ep \rightarrow ep\gamma \), with the final state photon detected by a photon detector located close to the beam pipe at position \(z=103~\mathrm{m}\). The precision of the integrated luminosity determination is improved in a dedicated analysis of the QED Compton process [50].
4.2 Event selection
The analysis is based on data collected by H1 in the 2005–2006 electron and the 2006–2007 positron running periods with \(\sqrt{s}=319~\mathrm{GeV}\), where the proton and lepton beam energies are 920 and 27.6 GeV, respectively. The corresponding integrated luminosity is 287 pb\(^{1}\). The events are triggered on the basis of a scattered electron signal in the SpaCal calorimeter together with at least one track above the 900 MeV transverse momentum threshold in the drift chambers of the central tracker.
4.2.1 Diffractive DIS selection and reconstruction of kinematics
The activity in the FTS and the FMD is required not to exceed the noise levels monitored with an event sample triggered independently of detector activity. Noise effects are also propagated into the simulation of the detector response in a similar manner. The diffractive event selection requirements ensure that the analysed sample is dominated by \(ep \rightarrow eXp\) processes at small t with an intact proton in the final state, often called proton elastic processes. A small admixture of events is present with leading neutrons or low \(M_{Y}\) baryon excitations, referred to as proton dissociation contributions (PD). The values of \(M_Y\) and t are not reconstructed explicitly. However, as the diffractive selection rejects events at large \(M_{Y}\) or large \(\left t \right \), the measurement is corrected to the \(M_{Y} < 1.6 \mathrm{~GeV}\) and \(\left t \right < 1 \mathrm{~GeV}^{2}\) range.
4.2.2 \(D^{*}\) selection
The mass difference \(\Delta m = m(K^{\mp } \pi ^{\pm } \pi ^{\pm }_{slow})m(K^{\mp } \pi ^{\pm })\) is used to determine the \(D^{*}\) signal. It is expected to peak near \(\Delta m=0.145~\mathrm{GeV}\) [53]. The wrong charge combinations \(K^{\pm } \pi ^{\pm } \pi ^{\mp }_{slow}\) selected with otherwise unchanged criteria do not contribute to the signal, they are, however, utilised to constrain the shape of the combinatorial background. The right and wrong charge \(\Delta m\) distributions are fitted simultaneously by means of an unbinned extended likelihood fit using RooFit [54] and ROOT [55]. The Crystal Ball [56] and Granet [57] probability distribution functions are used for modelling the signal and background, respectively. The fit to the total sample of selected \(D^{*}\) candidates is shown for the right and wrong charge combinations in Fig. 2. The fit to the total number of \(D^{*}\) mesons in the data yields \(N(D^{*}) = 1169 \pm 58\). The observed width is dominated by experimental effects.
The fits are repeated in bins of reconstructed kinematic quantities. Figure 3 shows the \(D^{*}\) yields determined as a function of the variables \(Q^2\), y, \(\text {log}_{10}(x_{{I\!\!P}})\), \(z_{{I\!\!P}}^{obs}\), \(p_{t,D^{*}}\) and \(\eta _{D^{*}}\). The \(N(D^{*})\) distributions are well described by the reweighted simulation. The fraction of proton dissociation processes adjusted globally (as given by Eq. 4) is largely independent of the kinematics. The reggeon contribution is generally small and reaches 5% at large \(x_{I\!\!P}\). The nondiffractive background contribution is below 1% level and is not shown.
4.3 Cross section measurement

\(N_{i}^\mathrm{data}\) is the number of \(D^{*}\) mesons from the fit passing the experimental cuts in the data.

\(N_{i}^\mathrm{sim, bgr}\) is the number of \(D^{*}\) mesons from the fit to simulated events passing the experimental cuts while being generated outside the phase space (Table 1) of the measurement.

\(A_{i}\) is the acceptance correction factor accounting for effects related to the transition from the hadron level to the detector level determined from MC simulations.

\(\mathcal {L}_\mathrm{int}\) is the integrated luminosity of the data.

\(B_{r}\) is the branching ratio of the golden decay channel.

\(\varepsilon _\mathrm{trigg}\) is the trigger efficiency.

\(C^\mathrm{{QED}}_\mathrm{{corr},i}\) are corrections for QED radiation defined as \(\sigma ^\mathrm{{QEDoff}}/\sigma ^\mathrm{{QED}{\text {}}\mathrm{on}}\) as obtained from Monte Carlo generated events, where \(\sigma ^\mathrm{\mathrm{QED}{\text {}}\mathrm{off}}\) (\(\sigma ^\mathrm{{QED}{\text {}}\mathrm{on}}\)) is the binintegrated cross section predicted by RAPGAP with QED radiation turned off (turned on) as described in Sect. 3.

\(\Delta _{i}^{x}\) is the bin width of the ith bin of x.
Definition of the phase space of the cross section measurement
DIS phase space  
\( 5< Q^{2} < 100~\mathrm{GeV}^{2}\)  
\(0.02< y < 0.65\)  
\(D^{*}\) kinematics  
\(p_{t,D^{*}} > 1.5~\mathrm{GeV}\)  
\(1.5< \eta _{D^{*}} < 1.5 \)  
Diffractive phase space  
\(x_{{I\!\!P}} < 0.03\)  
\(M_{Y} < 1.6~\mathrm{GeV}\)  
\(t < 1~\mathrm{GeV}^{2} \) 
Bin averaged hadron level \(D^{*}\) production cross sections in diffractive DIS as a function of y, \(Q^{2}, \mathrm{log}_{10}(x_{{I\!\!P}})\), \(z_{{I\!\!P}}^{obs}\), \(p_{t,D^{*}}\) and \(\eta _{D^{*}}\) together with statistical (\(\delta _\mathrm{stat}\)), systematic (\(\delta _\mathrm{syst}\)) and total (\(\delta _\mathrm{tot}\)) uncertainties. The total uncertainties are obtained as the statistical and systematic uncertainties added in quadrature
y  \(\mathrm{d}\sigma /\mathrm{d}y\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb})\) 

\(0.02 \div 0.09\)  770  120  110  160 
\(0.09 \div 0.18\)  870  110  100  150 
\(0.18 \div 0.26\)  660  98  117  152 
\(0.26 \div 0.36\)  558  78  58  97 
\(0.36 \div 0.50\)  282  55  41  68 
\(0.50 \div 0.65\)  197  52  51  73 
\(Q^{2}\,(\mathrm{GeV}^{2})\)  \(\mathrm{d}\sigma /\mathrm{d}Q^{2}\,\,(\mathrm{pb} / \mathrm{GeV}^{2})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb} / \mathrm{GeV}^{2})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb} / \mathrm{GeV}^{2})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb} / \mathrm{GeV}^{2})\) 

\(5 \div 8\)  29.6  3.7  5.0  6.2 
\(8 \div 13\)  14.8  1.9  1.9  2.7 
\(13 \div 19\)  9.0  1.2  0.9  1.5 
\(19.0 \div 27.5\)  4.81  0.79  0.48  0.92 
\(27.5 \div 40.0\)  1.63  0.45  0.52  0.69 
\(40 \div 60\)  0.95  0.25  0.17  0.30 
\(60 \div 100\)  0.30  0.11  0.07  0.14 
\(\mathrm{log}_{10}(x_{{I\!\!P}})\)  \(\mathrm{d}\sigma /\mathrm{d}\mathrm{log}_{10}(x_{{I\!\!P}})\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb})\) 

\(3.00 \div 2.70\)  59  17  22  27 
\(2.70 \div 2.41\)  147  22  32  39 
\(2.41 \div 2.11\)  172  24  47  53 
\(2.11 \div 1.82\)  223  29  27  40 
\(1.82 \div 1.52\)  464  53  79  96 
\(z_{{I\!\!P}}\)  \(\mathrm{d}\sigma /\mathrm{d}z_{{I\!\!P}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb})\) 

\(0.0 \div 0.1\)  470  120  70  140 
\(0.1 \div 0.3\)  652  71  98  121 
\(0.3 \div 0.6\)  211  29  28  40 
\(0.6 \div 1.0\)  174  19  13  23 
\(p_{t,D^{*}}\,(\mathrm{GeV})\)  \(\mathrm{d}\sigma /\mathrm{d}p_{t,D^{*}}\,\,(\mathrm{pb} / \mathrm{GeV})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb} / \mathrm{GeV})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb} / \mathrm{GeV})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb} / \mathrm{GeV})\) 

\(1.50 \div 2.28\)  180  24  22  33 
\(2.28 \div 3.08\)  120  12  14  19 
\(3.08 \div 4.75\)  45.6  4.4  3.5  5.7 
\(4.75 \div 8.00\)  4.8  1.0  0.6  1.2 
\(\eta _{D^{*}}\)  \(\mathrm{d}\sigma /\mathrm{d}\eta _{D^{*}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{stat}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{syst}}\,\,(\mathrm{pb})\)  \(\delta _{\mathrm{tot}}\,\,(\mathrm{pb})\) 

\(1.5 \div 1.0\)  129  18  16  24 
\(1.0 \div 0.5\)  119  16  15  22 
\(0.5 \div 0.0\)  119  15  12  20 
\(0.0 \div 0.5\)  103  15  14  21 
\(0.5 \div 1.0\)  58  15  11  19 
\(1.0 \div 1.5\)  91  18  12  22 
4.4 Systematic uncertainties

The energy scale (polar angle) of the scattered lepton is known to the 1% (1 mrad) level resulting in a 0.5% (1.5%) uncertainty.

The relative energy scale of the hadronic final state is known with a precision of 1% resulting in a 0.06% uncertainty.

Changing the function \(f_\mathrm{{corr}}\) (Eq. 7) to the constant 1.16 results in 2.7% uncertainty.

There is a certain ambiguity in describing the tails of the \(\Delta m\) signal distribution. Choosing a modified Crystal Ball function with an extra Gaussian component for the fits to the \(D^{*}\) signal has 3.8% effect.

The normalisation of the proton dissociative contribution (Eq. 4) introduces an uncertainty of 7.1%.

In a dedicated study [58], using forward proton tagging data, a 10% uncertainty on the large rapidity gap selection inefficiency is determined, which translates to a 2.4% uncertainty.

The shapes of the generated spectra of \(Q^{2}\), y, \(x_{{I\!\!P}}\), \(p_{t,D^{*}}\) and t are varied independently with the help of multiplicative weights of \(\mathrm{e}^{{}^{+0.007 }_{0.013}\,Q^{2}/\mathrm{GeV}^{2}}\), \(y^{^{+0.9}_{1.1}}\), \((x_{{I\!\!P}})^{^{+0.13}_{0.16}}\), \(\mathrm{e}^{^{+0.06}_{0.15}{\,}p_{t,D^{*}}/\mathrm{GeV}}\) and \(\mathrm{e}^{^{+0.8}_{0.9}{\,}t/\mathrm{GeV}^{2}}\) resulting in variations of the fitted differential distributions compatible with the data control distributions (Fig. 3). The reweighting is an approach to assess the uncertainties on the data correction procedure stemming from the Monte Carlo model. The resulting uncertainties are 0.5, 0.9, 0.4, 3.7 and 1.1%, respectively.

The integrated luminosity is known to 2.7% and the golden channel branching ratio to 1.1%.

The uncertainty on the trigger efficiency (98% on average) is covered by a 2% variation.

The impact of the restriction to the \(D^{0}\) mass window in terms of \(N(D^{*})\) yield loss caused by the choice of the 80 MeV value is evaluated. A systematic uncertainty of 2% covers the observed difference between data and simulation.

The reflections contribute about 3% to the fitted \(N(D^{*})\). The branching fractions of \(D^{*}\) decaying to reflections are not precisely reproduced in the simulation. The integrated cross section increases by about 1.2% if recent branching ratios of reflections are used [53].

The track reconstruction efficiency is known with 1% uncertainty resulting in 3% per \(D^{*}\).

The contribution of nondiffractive processes is suppressed by the diffractive selection to a level of less than 1%. A conservative uncertainty of 1% is assigned.
5 Results
In the first part of this section the measured integrated and differential cross sections for \(D^{*}\) production in diffractive deep inelastic scattering are presented. Theoretical predictions based on nexttoleading order QCD calculations are compared with the data. In the second part ratios of diffractive to nondiffractive \(D^{*}\) production cross sections are extracted and confronted with theoretical predictions as well as with previous results from HERA.
5.1 Diffractive \(D^{*}\) production cross sections
The values of diffractive fraction for \(D^{*}\) production cross sections together with statistical (\(\delta _\mathrm{stat}\)), systematic (\(\delta _\mathrm{syst}\)) and total uncertainties (\(\delta _\mathrm{tot}\)) as a function of y, \(Q^{2}\), \(p_{t,D^{*}}\) and \(\eta _{D^{*}}\). The total uncertainties are obtained as the statistical and systematic uncertainties added in quadrature
y  \(R_{D}\,\,(\%)\)  \(\delta _{\text {stat}}\,\,(\%)\)  \(\delta _{\text {syst}}\,\,(\%)\)  \(\delta _{\text {tot}}\,\,(\%)\) 

\(0.02 \div 0.09\)  5.3  0.8  0.8  1.1 
\(0.09 \div 0.18\)  6.2  0.8  0.8  1.0 
\(0.18 \div 0.26\)  6.0  0.9  1.2  1.5 
\(0.26 \div 0.36\)  8.2  1.2  1.0  1.6 
\(0.36 \div 0.50\)  6.7  1.3  1.1  1.7 
\(0.50 \div 0.65\)  8.5  2.4  2.3  3.3 
\(Q^{2}\,(\text {GeV}^{2})\)  \(R_{D}\,\,(\%)\)  \(\delta _{\text {stat}}\,\,(\%)\)  \(\delta _{\text {syst}}\,\,(\%)\)  \(\delta _{\text {tot}}\,\,(\%)\) 

\(5 \div 8\)  6.7  0.9  1.2  1.5 
\(8 \div 13\)  6.5  0.9  0.9  1.2 
\(13 \div 19\)  7.4  1.0  0.9  1.3 
\(19.0 \div 27.5\)  7.2  1.2  0.8  1.5 
\(27.5 \div 40\)  4.4  1.2  1.5  1.9 
\(40 \div 60\)  6.2  1.7  1.2  2.1 
\(60 \div 100\)  4.2  1.6  1.1  2.0 
\(p_{t,D^{*}}\,(\text {GeV})\)  \(R_{D}\,\,(\%)\)  \(\delta _{\text {stat}}\,\,(\%)\)  \(\delta _{\text {syst}}\,\,(\%)\)  \(\delta _{\text {tot}}\,\,(\%)\) 

\(1.5 \div 2.28\)  8.4  1.2  1.1  1.6 
\(2.28 \div 3.08\)  7.3  0.8  0.9  1.2 
\(3.08 \div 4.75\)  5.8  0.6  0.5  0.8 
\(4.75 \div 8.00\)  3.1  0.7  0.4  0.8 
\(\eta _{D^{*}}\)  \(R_{D}\,\,(\%)\)  \(\delta _{\text {stat}}\,\,(\%)\)  \(\delta _{\text {syst}}\,\,(\%)\)  \(\delta _{\text {tot}}\,\,(\%)\) 

\(1.5 \div 1\)  10.6  1.5  1.4  2.1 
\(1.0 \div 0.5\)  7.8  1.1  1.0  1.5 
\(0.5 \div 0.0\)  7.5  1.0  0.8  1.3 
\(0.0 \div 0.5\)  6.2  0.9  0.9  1.3 
\(0.5 \div 1.0\)  3.3  0.9  0.7  1.1 
\(1.0 \div 1.5\)  4.9  1.0  0.7  1.2 
The measured bin averaged singledifferential cross sections as a function of y, \(Q^{2}\), \(\text {log}_{10}(x_{{I\!\!P}})\), \(z_{{I\!\!P}}^{obs}\), \(p_{t,D^{*}}\) and \(\eta _{D^{*}}\) are given in Table 2 and are shown in Figs. 4, 5 and 6 together with the NLO predictions. In order to compare the shapes between data and theory the ratios of data to NLO calculations are also shown.
Figure 4 shows that the shape of the measured \(\mathrm{d}\sigma /\mathrm{d}y\) is well described by the theory. The measured \(\mathrm{d}\sigma /\mathrm{d}Q^{2}\) might indicate a slightly harder dependence in the data, however, within the large uncertainties the shape is in agreement with the theory. The shape of the \(\mathrm{d}\sigma /\mathrm{dlog}_{10}(x_{{I\!\!P}})\) shown in Fig. 5 is satisfactorily described by the prediction given the large relative uncertainties at low \(x_{{I\!\!P}}\) values. The shape of \(\mathrm{d}\sigma /\mathrm{d}z_{{I\!\!P}}^{obs}\) shown in Fig. 5 is not described as well by the prediction, however the experimental uncertainties at low \(z_{{I\!\!P}}^{obs}\) are sizeable. The shapes of \(\mathrm{d}\sigma /\mathrm{d} p_{t,D^{*}}\) and \(\mathrm{d}\sigma /\mathrm{d}\eta _{D^{*}}\) are well described by the theory (see Fig. 6). For \(\eta _{D^{*}} > 1\), however, the theory predicts a value which underestimates the data by about \(50\%\) with a large uncertainty. There is an indication of a similar effect in the corresponding nondiffractive \(D^{*}\) cross section measurement [38].
5.2 Diffractive fractions
The \(D^{*}\) and DIS selection criteria given in Table 1 are close to those used in the corresponding nondiffractive analysis [38]. The nondiffractive \(D^{*}\) differential cross sections thus can be used to calculate the diffractive fraction, \(R_{D} = {\sigma ^{\mathrm{diff}}_{{D^{*}}}} / {{\sigma ^{\text {nondiff}}_{{D^{*}}}}}\), in the phase space defined in Table 1.
The nondiffractive cross sections [38], originally given for \(0.02< y < 0.7\), are interpolated to \(0.02< y < 0.65\) using small corrections calculated with HVQDIS. The correction factors reduce the nondiffractive cross sections by about 1.5–3.5% differentially in \(Q^2\), \(p_{t,D^{*}}\) and \(\eta _{D^{*}}\) with typical uncertainties of \(0.2\%\). The uncertainties of both the diffractive and nondiffractive cross sections are accounted for in the \(R_{D}\) measurement. Integrated over the whole phase space the results are \(R_{D} = 6.6 \pm 0.5 \mathrm{(stat)} \text { }^{+0.9}_{0.8} \mathrm{(syst)}\%\) for the data and \(R_{D}^\mathrm{\, theory} = 6.0 {}^{+1.0}_{0.7} \mathrm{(scale)} {}^{+0.5}_{0.4} \mathrm{(}m_c\mathrm{)} {}^{+0.7}_{0.8} \mathrm{(DPDF)} {}^{+0.02}_{0.04} \mathrm{(frag)}\%\) for the theoretical prediction. The uncertainties of the theoretical predictions are obtained from simultaneous variations of \(m_c\), fragmentation parameters and the factorisation and renormalisation scales. The DPDF uncertainty is also propagated to the prediction.
In Fig. 8 the diffractive fraction, integrated over the full phase space, is compared with previous measurements performed at HERA both in the DIS regime [20, 21, 22] and in photoproduction [24]. The average value of \(R_{D}\) measured in this article is in agreement with the previous results and within the sizeable experimental uncertainties is observed to be largely independent of the varying phase space constraints in \(x_{{I\!\!P}}\), \(Q^2\) and \(p_{t,D^{*}}\). In particular, the ratios observed in DIS and in photoproduction are compatible with each other.
6 Conclusions
Integrated and differential cross sections of \(D^{*}(2010)\) production in diffractive deep inelastic scattering are measured. The analysis is based on a data sample taken by the H1 experiment at the HERA collider corresponding to an integrated luminosity of 287 pb\(^{1}\). The measured cross sections are compared with theoretical predictions in next to leading order QCD. Compared to the previous measurement in a similar kinematic domain the precision is improved by a factor of two. The new measurements are well described by the predictions within the large theoretical uncertainties which are dominated by variations of scales and the charm quark mass. This supports the validity of collinear factorisation in diffraction.
Measurements of diffractive fractions of \(D^{*}\) production cross section in deep inelastic scattering are also presented, using nondiffractive cross sections published earlier by H1. The fractions are in agreement with theoretical predictions in nexttoleading order QCD. Although the value of the diffractive fraction is found to decrease at high \(p_{t,D^{*}}\) and at high \(\eta _{D^{*}}\) due to limitations of the diffractive phase space, it is observed to be largely independent of other details of the phase space definition. This is confirmed by comparisons to previous measurements of the diffractive fraction.
Footnotes
 1.
The term electron is referring to both \(e^{}\) and \(e^{+}\) unless stated otherwise.
 2.
A detailed analysis of the systematic uncertainties is available http://wwwh1.desy.de/publications/H1publication.short_list.html.
Notes
Acknowledgements
We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. We thank the engineers and technicians for their work in constructing and maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the nonDESY members of the collaboration. We would like to give credit to all partners contributing to the EGI computing infrastructure for their support for the H1 Collaboration. We express our thanks to all those involved in securing not only the H1 data but also the software and working environment for long term use allowing the unique H1 dataset to continue to be explored in the coming years. The transfer from experiment specific to central resources with long term support, including both storage and batch systems has also been crucial to this enterprise. We therefore also acknowledge the role played by DESYIT and all people involved during this transition and their future role in the years to come.
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