Search for QCD instantoninduced processes at HERA in the high\(\pmb {Q^2}\) domain
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Abstract
Signals of QCD instantoninduced processes are searched for in neutral current deepinelastic scattering at the electronproton collider HERA in the kinematic region defined by the Bjorkenscaling variable \(x > 10^{3}\), the inelasticity \(0.2< y < 0.7\) and the photon virtuality \(150< Q^2 < 15000\) GeV\(^2\). The search is performed using H1 data corresponding to an integrated luminosity of 351 pb\(^{1}\). No evidence for the production of QCD instantoninduced events is observed. Upper limits on the cross section for instantoninduced processes between 1.5 and 6 pb, at \(95\,\,\%\) confidence level, are obtained depending on the kinematic domain in which instantons could be produced. Compared to earlier publications, the limits are improved by an order of magnitude and for the first time are challenging predictions.
Keywords
Systematic Uncertainty Kinematic Region Track Reconstruction Boost Decision Tree Hadronic Final State1 Introduction
An experimental observation of instantoninduced processes would constitute a discovery of a basic and yet novel nonperturbative QCD effect at high energies. The theory and phenomenology for the production of instantoninduced processes at HERA in neutral current (NC) electron^{1}proton collisions has been worked out by Ringwald and Schrempp [4, 6, 7, 8, 9, 10]. The size of the predicted cross section is large enough to make an experimental observation possible. The expected signal rate is, however, still small compared to that from the standard NC DIS (sDIS) process. The suppression of the sDIS background is therefore the key issue. QCD instantoninduced processes can be discriminated from sDIS by their characteristic hadronic final state signature, consisting of a large number of hadrons at high transverse energy emerging from a “fireball”like topology in the instanton rest system [4, 9, 10]. Discriminating observables, derived from simulation studies, are exploited to identify a phase space region where a difference between data and sDIS expectations would indicate a contribution from instantoninduced processes.
Upper cross section limits on instantoninduced processes have been reported by the H1 [15] and ZEUS [16] collaborations. This analysis is a continuation of the previous H1 search for QCD instantoninduced events using a seventeen times larger data sample. The search is carried out at significantly higher virtualities of the exchanged photons as suggested by theoretical considerations [11].
2 Phenomenology of QCD instantoninduced processes in NC DIS
In photongluon fusion processes, a photon splits into a quark antiquark pair in the background of an instanton or an antiinstanton field, as shown in Fig. 1. The socalled instanton subprocess \(q' + g \mathop {\rightarrow }\limits ^{(I,\bar{I})} X\) is induced by the quark or the antiquark fusing with a gluon g from the proton. The partonic system X contains \(2 \, n_f \) quarks and antiquarks, where one of the quarks (antiquarks) acts as the current quark (\(q''\)). In addition, an average number of \(\langle n_g \rangle \sim \mathcal{O}(1/\alpha _s) \sim 3\) gluons is emitted in the instanton subprocess.
The quarks and gluons emerging from the instanton subprocess are distributed isotropically in the instanton rest system defined by \(\mathbf {q'} + \mathbf {g} = 0\). Therefore one expects to find a pseudorapidity^{3} (\(\eta \)) region with a width of typically 2 units in \(\eta \), densely populated with particles of relatively high transverse momentum and isotropically distributed in azimuth, measured in the instanton rest frame. The large number of partons emitted in the instanton process leads to a high multiplicity of charged and neutral particles. Besides this band in pseudorapidity, the hadronic final state also contains a current jet emerging from the outgoing current quark \(q''\).
The instanton production cross section at HERA, \(\sigma ^{(I)}_\mathrm{HERA}\), is determined by the cross section of the instanton subprocess \(q' + g \mathop {\rightarrow }\limits ^{(I,\bar{I})} X\). The subprocess cross section is calculable in instanton perturbation theory. It involves the distributions of the size \(\rho \) of instantons and of the distance R between them. By confronting instanton perturbation theory with nonperturbative lattice simulations of the QCD vacuum, limits on the validity of instanton perturbation theory have been derived [7, 8, 11]. The perturbative and lattice calculations agree for \(\rho \lesssim 0.35 \) fm and \(R/\rho \gtrsim 1.05\). At larger \(\rho \) or smaller \(R/\rho \), the instanton perturbative cross section grows, whereas the lattice calculations suggest that the cross section is limited. There is a relation between the variables \({Q'~}\) and \({x'~}\) in momentum space and the spatial variables \(\rho \) and \(R/\rho \). Large \({Q'~}\) and \({x'~}\) values correspond to small \(\rho \) and large \(R/\rho \), respectively. The aforementioned limits can be translated into regions of the kinematical variables \({x'~}\) and \({{Q}^{\prime 2}}\), in which the perturbative calculations are expected to be valid, \({{Q}^{\prime 2}}\ge {{Q'}^2_{\min }}\simeq (30.8\times \Lambda ^{n_f}_{\overline{MS}})^2\) and \({x'}\ge {x'_{\min }}\simeq 0.35\) [12]. Here \(\Lambda ^{n_f}_{\overline{MS}}\) is the QCD scale in the \(\overline{MS}\) scheme for \(n_f\) flavours. In order to assure the dominance of planar diagrams the additional restriction \(Q^2 \ge {{Q'}^2_{\min }}\) is recommended [6, 11, 12]. The cross section depends significantly on the strong coupling^{4} \(\alpha _s\), or more precisely on \(\Lambda ^{n_f}_{\overline{MS}}\), but depends only weakly on the choice of the renormalisation scale.
The calculation of the instanton production cross section in instanton perturbation theory [6, 7, 8] is valid in the dilute instantongas approximation for approximately massless flavours, i.e. \(n_f=3\), in the HERA kinematic domain. The contribution of heavy flavours is expected to be (exponentially) suppressed [17, 18]. Thus calculations of the instanton production cross section using the QCDINS Monte Carlo generator [12] are performed for \(n_f=3\) massless flavours. It was checked that the predicted final state signature does not change significantly when heavy flavours are included in the simulation.
The analysis is performed in the kinematic region defined by \( 0.2< y < 0.7\) and \(150< Q^2< 15000~\mathrm{GeV}^2\). In this kinematic region, and additionally requiring \({{Q'}^2~}> 113\) \({\mathrm{~GeV}^2}\) and \({x'~}> 0.35\), the cross section predicted by QCDINS is \(\sigma ^{(I)}_\mathrm{HERA} = 10\pm 3 \; \mathrm{pb}\), using the QCD scale \(\Lambda _{\overline{MS}}^{(3)}= 339\pm 17 \; \mathrm{MeV}\) [19]. The quoted uncertainty of the instanton cross section \(\sigma ^{(I)}_\mathrm{HERA}\) is obtained by varying the QCD scale by one standard deviation.
The fiducial region in \({{Q}^{\prime 2}}\) and \({x'~}\) of the validity of instanton perturbation theory was derived from \(\mathrm{n}_f=0\) lattice simulations, since \(\mathrm{n}_f=3\) was not available for this purpose. The perturbative instanton calculation is made in the “dilute instanton gas” approximation, where the average distance between instantons should be large compared to the instanton size. This approximation is valid for \({x'~}\!\rightarrow \!1\), whereas the boundary \({x'~}= 0.35\) corresponds to a configuration where the distance R is similar to the instanton size \(\rho \). A further simplifying assumption is made by choosing a simple form of the fiducial region with fixed \({{Q'}^2_{\min }}\) and \({x'_{\min }}\), whereas \({{Q'}^2_{\min }}\) could be varied as a function of \({x'_{\min }}\). In summary, the kinematic region in \({{Q}^{\prime 2}}\) and \({x'~}\), where instanton perturbation theory is reliable, is, for the reasons given above, not very well defined. Thus, the theoretical uncertainty of the instanton cross section is difficult to define and could be larger than the already significant uncertainty due to the uncertainty of the QCD scale \(\Lambda _{\overline{MS}}^{(3)}\) alone. On the other hand, given that the predicted cross section is large, dedicated searches for instantoninduced processes at HERA are well motivated.
3 Experimental method
3.1 The H1 detector
A detailed description of the H1 detector can be found elsewhere [20, 21, 22, 23]. The origin of the H1 coordinate system is given by the nominal ep interaction point at \(z=0\). The direction of the proton beam defines the positive z–axis (forward direction) and the polar angle \(\theta \) and transverse momentum \(P_T\) of every particle is defined with respect to this axis. The azimuthal angle \(\phi \) defines the particle direction in the transverse plane. The detector components most relevant to this analysis are the Liquid Argon (LAr) calorimeter, which measures the positions and energies of particles over the range \(4^\circ<\theta <154^\circ \) with full azimuthal coverage, the inner tracking detectors, which measure the angles and momenta of charged particles over the range \(7^\circ<\theta <165^\circ \), and a leadfibre calorimeter (SpaCal) covering the range \(153^\circ<\theta <174^\circ \).
The LAr calorimeter consists of an electromagnetic section with lead absorbers and a hadronic section with steel absorbers. The electromagnetic and the hadronic sections are highly segmented in the transverse and the longitudinal directions. Electromagnetic shower energies are measured with a resolution of \(\delta E/E \simeq 0.11/\sqrt{E/\mathrm{GeV}} \oplus 0.01\) and hadronic energies with \(\delta E/E \simeq 0.50/\sqrt{E/\mathrm{GeV}} \oplus 0.03\) as determined using electron and pion test beam measurements [24, 25].
In the central region, \(15^{\circ }<\theta <165^{\circ }\), the central tracking detector (CTD) measures the trajectories of charged particles in two cylindrical drift chambers immersed in a uniform \(1.16\,\mathrm{T}\) solenoidal magnetic field. In addition, the CTD contains a drift chamber (COZ) to improve the zcoordinate reconstruction and a multiwire proportional chamber at inner radii (CIP) mainly used for triggering [26]. The CTD measures charged particles with a transverse momentum resolution of \(\delta (p_T)/p_T\simeq 0.002 \, p_T/\mathrm{GeV} \oplus 0.015\). The forward tracking detector (FTD) is used to supplement track reconstruction in the region \(7^{\circ }<\theta <30^{\circ }\) [27]. It improves the hadronic final state reconstruction of forward going low transverse momentum particles. The CTD tracks are linked to hits in the vertex detector, the central silicon tracker (CST) [28, 29], to provide precise spatial track reconstruction.
In the backward region the SpaCal provides an energy measurement for hadronic particles, and has a hadronic energy resolution of \(\delta E/E \simeq 0.70/\sqrt{E/\mathrm{GeV}}\oplus 0.01\) and a resolution for electromagnetic energy depositions of \(\delta E/E \simeq 0.07/\sqrt{E/\mathrm{GeV}}\oplus 0.01\) measured using test beam data [30].
The ep luminosity is determined by measuring the event rate for the Bethe–Heitler process \(ep \rightarrow ep\gamma \), where the photon is detected in the photon tagger located at \(z=103\,\mathrm{m}\). The overall normalisation is determined using a precision measurement of the QED Compton process [31] with the electron and the photon detected in the SpaCal.
3.2 Data samples
High \(Q^2\) neutral current DIS events are triggered mainly using information from the LAr calorimeter. The calorimeter has a finely segmented pointing geometry allowing the trigger to select localised energy deposits in the electromagnetic section of the calorimeter pointing to the nominal interaction vertex. For electrons with energies above 11 GeV the trigger efficiency is determined to be close to \(100\,\,\%\) [32].
This analysis is performed using the full \(e^{\pm }p\) collision data set taken in the years 2003–2007 by the H1 experiment. The data were recorded with a lepton beam of energy 27.6 GeV and a proton beam of energy 920 GeV, corresponding to a centreofmass energy \(\sqrt{s}=319\) GeV. The total integrated luminosity of the analysed data is 351 pb\(^{1}\).
3.3 Simulation of standard and instanton processes
Detailed simulations of the H1 detector response to hadronic final states have been performed for two QCD models of the sDIS (background) and for QCD instantoninduced scattering processes (signal).
The background is modelled using the RAPGAP and DJANGOH Monte Carlo programs. The RAPGAP Monte Carlo program [33] incorporates the \(\mathcal{O} (\alpha _{s})\) QCD matrix elements and models higher order parton emissions to all orders in \(\alpha _s\) using the concept of parton showers [34] based on the leadinglogarithm DGLAP equations [35, 36, 37], where QCD radiation can occur before and after the hard subprocess. An alternative treatment of the perturbative phase is implemented in DJANGOH [38] which uses the Colour Dipole Model [39] with QCD matrix element corrections as implemented in ARIADNE [40]. In both MC generators hadronisation is modelled with the LUND string fragmentation [41, 42] using the ALEPH tune [43]. QED radiation and electroweak effects are simulated using the HERACLES [44] program, which is interfaced to the RAPGAP and DJANGOH event generators. The parton density functions of the proton are taken from the CTEQ6L set [45].
QCDINS [12, 46] is a Monte Carlo package to simulate QCD instantoninduced scattering processes in DIS. The hard process generator is embedded in the HERWIG [47] program and is implemented as explained in Sect. 2. The number of flavours is set to \(n_f = 3\). Outside the allowed region defined by \({{Q'}^2_{\min }}\) and \({x'_{\min }}\) the instanton cross section is set to zero. The CTEQ5L [48] parton density functions are employed.^{5} Besides the hard instanton subprocess, subleading QCD emissions are simulated in the leadinglogarithm approximation, using the coherent branching algorithm implemented in HERWIG. The hadronisation is performed according to the Lund string fragmentation.
The generated events are passed through a detailed GEANT3 [49] based simulation of the H1 detector and subjected to the same reconstruction and analysis chains as are used for the data.
3.4 Inclusive DIS event selection
Neutral current DIS events are triggered and selected by requiring a cluster in the electromagnetic part of the LAr calorimeter. The scattered electron is identified as the isolated cluster of highest transverse momentum. A minimal electron energy of 11 GeV is required. The remaining clusters in the calorimeters and the charged tracks are attributed to the hadronic final state (HFS), which is reconstructed using an energy flow algorithm without double counting of energy [50, 51, 52]. The default electromagnetic energy calibration and alignment of the H1 detector [53] as well as the HFS calibration [32, 54] are applied. The longitudinal momentum balance is required to be within \(45\,\mathrm{~GeV~}< \sum (E  p_z) < 65\,\mathrm{~GeV}\), where the sum runs over the scattered electron and all HFS objects. Furthermore the position of the zcoordinate of the reconstructed event vertex must be within \(\pm 35\,\text{ cm }\) of the nominal interaction point.
The photon virtuality \(Q^2\), the Bjorken scaling variable x and the inelasticity of the interaction y are reconstructed from the scattered electron and the hadronic final state particles using the electronsigma method [55]. This method is the most precise one in the kinematic range of this analysis. The events are selected to cover the phase space region defined by \( 0.2< y < 0.7\), \(x >10^{3}\) and \(150< Q^2< 15000~\mathrm{GeV}^2\).
The events passing the above cuts yield the NC DIS sample which forms the basis of the subsequent analysis. It consists of about 350000 events. The simulated events are subjected to the same reconstruction and analysis chains as the real data. They reproduce well the shape and the absolute normalisation of the distributions of the energy and angle of the scattered electron as well as the kinematic variables x, \(Q^2\) and y.
3.5 Definition of the observables and the search strategy
In the following, all HFS objects are boosted to the hadronic centreofmass frame (HCM).^{6} Jets are defined by the inclusive \(k_{T}\) algorithm [56] as implemented in FastJet [57], with the massless \(P_{T}\) recombination scheme and with the distance parameter \(R_{0}= 1.35 \times R_\mathrm{cone} \). A cone radius \(R_\mathrm{cone} = 0.5\) is used. Jets are required to have transverse energy in the HCM frame \({E_{T,\mathrm jet}}\) \( > 3 \) GeV. Additional requirements on the transverse energy and pseudorapidity of the jets in the laboratory frame are imposed, \(1.0< \eta ^\mathrm{Lab}_\mathrm{Jet} < 2.5 \) and \(E_{T,\mathrm Jet}^\mathrm{Lab} > 2.5 \) GeV, in order to ensure that jets are contained within the acceptance of the LAr calorimeter and are well calibrated. The events are selected by requiring at least one jet with \({E_{T,\mathrm jet}}\) \(> 4\) GeV. The jet with the highest transverse energy is used to estimate the 4momentum \(q''\) of the current quark (see Fig. 1). \({{Q'}^2~}\) can be reconstructed from the particles associated with the current jet and the photon 4momentum, which is obtained using the measured momentum of the scattered electron. The \({{Q'}^2~}\) resolution is about \(40\,\,\%\). However, the distribution of the true over the reconstructed value exhibits large tails, since in about \(35\,\,\%\) of the cases the wrong jet is identified as the current jet. Due to the limited accuracy of the \({{Q'}^2~}\) reconstruction, the reconstructed \({{Q}^{\prime 2}}\), labelled \({{Q'}^2_{ \mathrm rec}}\), cannot be used to experimentally limit the analysis to the kinematically allowed region \({{Q}^{\prime 2}}\) \( \gtrsim \) \({{Q'}^2_{\min }}\). Details of the \({{Q'}^2~}\) reconstruction are described in [10, 58, 59].
The hadronic final state objects belonging to the current jet are not used in the definition of the following observables. A band in pseudorapidity with a width of \(\pm 1.1\) units in \(\eta \) is defined around the mean \(\bar{\eta } = \sum E_T \eta /(\sum E_T)\), where the sum includes hadronic final state objects [60]. This pseudorapidity band is referred to as the “instanton band”. The number of charged particles in the instanton band \({n_B~}\) and the total scalar transverse energy of all hadronic final state objects in the instanton band \({E_{T,B}~}\) are measured.
An approximate instanton rest frame, where all hadronic final state objects in the instanton band are distributed isotropically, is defined by \(\mathbf {q'} + \xi \mathbf {P} = 0\). The definition of \(\xi \) is given in Fig. 1. A numerical value of \(\xi = 0.076\) is used throughout this analysis [15]. In the instanton rest frame the sphericity \({\mathrm{Sph_B}}~\) and the first three normalised FoxWolfram moments are calculated [42, 61]. For spherical events \({\mathrm{Sph_B}}~\) is close to unity, while for pencillike events \({\mathrm{Sph_B}}~\) tends to zero. Furthermore, the axes \(\mathbf {i}_\mathrm{min}\) and \(\mathbf {i}_\mathrm{max}\) are found for which in the instanton rest system the summed projections of the 3momenta of all hadronic final state objects in the instanton band are minimal or maximal [9]. The relative difference between \(E_\mathrm{in} = {\sum _h \mathbf {p}_h \cdot \mathbf {i}_\mathrm{max} }\) and \(E_\mathrm{out}= {\sum _h \mathbf {p}_h \cdot \mathbf {i}_\mathrm{min} }\) is called \({\Delta _B}= (E_\mathrm{in}E_\mathrm{out})/E_\mathrm{in}\). This quantity is a measure of the transverse energy weighted azimuthal isotropy of an event. For isotropic events \({\Delta _B~}\) is small while for pencillike events \({\Delta _B~}\) is close to unity.
The reconstruction of the variable \({x'~}\) suffers from poor resolution as in the case of \({{Q'}^2_{ \mathrm rec}}\). Using two methods to calculate the invariant mass of the quark gluon system, \(W_{I}\), \(x'\) is reconstructed as \(x'_\mathrm{rec}= (x'_{1}+x'_{2})/2\), where \(x'_{i}= {{Q'}^2_{ \mathrm rec}~}/ (W^2_{I,i}+{{Q'}^2_{ \mathrm rec}~})\) with \(W^2_{I,1}=(q'_\mathrm{rec}+ \xi P)^2\) and \(W^2_{I,2}= (\sum _{h} p_{h})^2\) where the sum runs over the HFS objects in the instanton band. The \(W^2_{I,1}\) calculation is based on the scattered electron and the current jet, while the \(W^2_{I,2}\) reconstruction relies on the measurement of the hadronic final state objects in the instanton band. The \(x'_\mathrm{rec}\) resolution achieved is about \(50 \,\,\%\). As for the case of \({{Q'}^2_{ \mathrm rec}}\), the reconstructed \(x'_{rec}\) cannot be used to limit the analysis to the kinematically allowed region \({x'}\gtrsim {x'_{\min }}\). However, \(x'_\mathrm{rec}\) as well as \({{Q'}^2_{ \mathrm rec}~}\) can be used to discriminate instanton processes from the sDIS background.
Exploiting these observables, a multivariate discrimination technique is used to find the most sensitive set of observables to distinguish between signal and background [62].
3.6 Comparison of data to standard QCD predictions
Both the RAPGAP and DJANGOH simulations provide a reasonable overall description of the experimental data in the inclusive DIS and jet sample. To further improve the agreement between Monte Carlo events and data, event weights are applied to match the jet multiplicities as a function of \(Q^{2}\). The MC events are also weighted as a function of \(P_{T}\) and \(\eta \) of the most forward jet in the Breit frame [32, 54]. Furthermore, the track multiplicity distribution is weighted. The weights are obtained from the ratio of data to the reconstructed MC distributions and are applied to the events on the generator level. After these weights are applied, the simulations provide a good description of the shapes and normalisation of the data distributions. Examples of these control distributions are shown in Fig. 2: distributions of the kinematic variables x and \(Q^2\), the transverse energy of the jets \(E_{T,\mathrm jets}\), the pseudorapidity of the jets \(\eta _\mathrm{jets}\) in the hadronic centreofmass frame and the charged particle multiplicity \(n_\mathrm{ch}\).
4 Search for instantoninduced events
A multivariate discrimination technique is employed to increase the sensitivity to instanton processes. The PDERS (Probability Density Estimator with Range Search) method as implemented in the TMVA ROOT package [63] is used.^{7}
The strategy to reduce the sDIS background is based on the observables \({E_{T,\mathrm jet}}, {n_B}\), \(x'\), \({\Delta _B~}\) and \(E_\mathrm{in}\). This set of observables has been chosen since it provides the best signal to background separation [62]. Moreover, the distributions of these variables are overall well described by both Monte Carlo simulations. The distribution of the discriminator D is shown in Fig. 4. Taking into account the systematic uncertainties, the discriminator distribution is described by the sDIS Monte Carlo simulations in the background dominated region. For \(D<0.2\) predictions and data agree within systematic uncertainties. The background events are mainly concentrated at low discriminator values, while the instanton signal peaks at large values of the discriminator. At large D both data and predicted background fall off steeply.
Number of events observed in data and expected from the DJANGOH and RAPGAP simulations in the signal region
Data  DJANGOH  RAPGAP  QCDINS  

\(D>0.86\)  2430  \(2483^{+77}_{90}\)  \(2966^{+~90}_{103}\)  \(473^{+10,+152}_{12,124} \) 
The expected and observed number of events are summarised in Table 1. In the signal region, a total of 2430 events are observed in data, while DJANGOH predicts \(2483^{+77}_{90}\) and RAPGAP \(2966^{+~90}_{103}\). The uncertainties on the expected number of events include experimental systematic uncertainties and small contributions from the finite sample sizes. For the expected number of instantoninduced events the dominating uncertainty is due to \(\Lambda _{\overline{MS}}^{(3)}\).

The energy scale of the HFS is known to a precision of \(1\,\,\%\) [32, 54].

Depending on the electron polar angle the energy of the scattered electron is measured with a precision of \(0.51\,\,\%\) [64].

The precision of the electron polar angle measurement is 1 mrad [64].

Depending on the electron polar angle, the uncertainty on the electron identification efficiency ranges from 0.5 to \(2\,\,\%\) [54].

The uncertainty associated with the track reconstruction efficiency and the effect of the nuclear interactions in the detector material on the efficiency of track reconstruction are estimated to be \(0.5\,\,\%\) each [65].
The main contributions to the experimental systematic uncertainties arise from the energy scale calibration of the scattered electron ranging from \({\sim }4\,\,\%\) in the background dominated region to \({\sim }1\,\,\%\) in the signal region and from the energy scale of the HFS ranging from \({\sim }1\,\,\%\) in the background region to \({\sim }2.5\,\,\%\) in the signal region. Uncertainties connected with the track reconstruction and secondary interactions of the produced hadrons in the material surrounding the interaction region contribute to the systematic error in the signal region at a level of \({\sim }2\,\,\%\) each, and in the background dominated region by less than \(0.5\,\,\%\). In the full range of the discriminator, the uncertainties on the electron identification and on the precision of the electron polar angle are smaller than \(0.5\,\,\%\) each.
Given the observed and expected numbers of events, no evidence for QCD instantoninduced processes is observed. In the following, the data are used to set exclusion limits.
5 Exclusion limits for instantoninduced processes

The normalisation uncertainty due to the precision of the integrated luminosity measurement is \(2.3\,\,\%\) [31].

The difference between the prediction from DJANGOH and RAPGAP is assigned as model uncertainty of the background estimation, i.e. the difference between two background histograms in Fig. 4. This model uncertainty is large, 8–20 and 13–46 %, for small \(D<0.2\) and large \(D>0.85\) values of the discriminator, respectively. For intermediate values of D it amounts to \(0.38\,\,\%\).

The uncertainty of the background normalisation is \(1.1\,\,\%\). This uncertainty is estimated as \(\epsilon = (N_\mathrm{Dj}N_\mathrm{Rap})/N_\mathrm{Dj}\), where \(N_\mathrm{Dj}\) and \(N_\mathrm{Rap}\) are the total number of predicted events in the full discriminator range for the DJANGOH and RAPGAP MC simulations, respectively.

The uncertainty of the predicted signal cross section due to the uncertainty of \(\Lambda _{\overline{MS}}^{(3)}\) (Sect. 2) varies from 20 to 50 % depending on the region in \({Q'~}\) and \({x'~}\).
In order to assess the sensitivity of the instanton cross section on the kinematic variables \({x'_{\min }}\) and \({{Q'}^2_{\min }}\), limits are also determined as a function of the lower bounds \({x'_{\min }}\) and \({{Q'}^2_{\min }}\). As explained in Sect. 3.3, outside these bounds the instanton cross section is set to zero. The results are shown in Fig. 8, where the observed confidence levels, using the QCDINS predictions, are shown in the \(({x'_{\min }},{{Q'}^2_{\min }})\) plane. At \(95\,\,\%\) confidence level, parameter values \({x'_{\min }}<0.404\) are excluded at fixed \({{Q'}^2_{\min }}=113\) \({\mathrm{~GeV}^2~}\). For fixed \({x'_{\min }}=0.35\), values of \({{Q'}^2_{\min }}<195\) \({\mathrm{~GeV}^2~}\) are excluded. The exclusion regions depend somewhat on the choice of \(\Lambda ^{(3)}_{\overline{MS}}\) and its uncertainty. In order to assess these effects, the analysis was repeated for \(\Lambda ^{(3)}_{\overline{MS}}=340\pm 8\) MeV [69] instead of \(\Lambda ^{(3)}_{\overline{MS}}=339\pm 17\) MeV . For this choice, more stringent limits are obtained. For example, at fixed \({{Q'}^2_{\min }}=113\) \(\mathrm {GeV}^2\) the excluded range at \(95\,\,\%\) confidence level would change to \({x'_{\min }}<0.413\).
A less modeldependent search is presented in Fig. 9. Here, limits on the instanton cross section are determined as a function of the parameters \({x'_{\min }}\) and \({{Q'}^2_{\min }}\), using the signal shapes predicted by QCDINS. No uncertainty on the instanton cross section normalisation is included in this determination of the experimental cross section limit. The most stringent exclusion limits of order 1.5 pb are observed for large \({{Q'}^2_{\min }}\) and small \({x'_{\min }}\). For increasing \({x'_{\min }}\) the limits are getting weaker. At the nominal QCDINS setting, \({x'_{\min }}=0.35\) and \({{Q'}^2_{\min }}=113\) \({\mathrm{~GeV}^2~}\), one expects to find back an exclusion limit of 2 pb, as discussed with Fig. 7. The limit in Fig. 9, however, is observed to be somewhat better, because the theory uncertainty on the cross section normalisation is included in Fig. 7 but not in Fig. 9.
6 Conclusions
A search for QCD instantoninduced processes is presented in neutral current deepinelastic scattering at the electronproton collider HERA. The kinematic region is defined by the Bjorkenscaling variable \(x > 10^{3}\), the inelasticity \(0.2< y < 0.7\) and the photon virtuality \(150< Q^2 < 15000\) GeV\(^2\). The search is performed using H1 data corresponding to an integrated luminosity of 351 pb\(^{1}\).
Several observables of the hadronic final state of the selected events are exploited to identify a potentially instantonenriched sample. Two Monte Carlo models, RAPGAP and DJANGOH, are used to estimate the background from the standard NC DIS processes. The instantoninduced processes are modelled by the program QCDINS. In order to extract the expected instanton signal a multivariate data analysis technique is used. No evidence for QCD instantoninduced processes is observed. In the kinematic region defined by the theory cutoff parameters \({x'_{\min }}=0.35\) and \({{Q'}^2_{\min }}=113\) \({\mathrm{~GeV}^2~}\) an upper limit of 2 pb on the instanton cross section at \(95\,\,\%\) CL is determined, as compared to a median expected limit of \(3.7^{+1.6}_{1.1}(68\%)^{+3.8}_{1.7}(95\,\,\%)\) pb. Thus, the corresponding predicted instanton cross section of \(10\pm 3\) pb is excluded by the H1 data. Limits are also set in the kinematic plane defined by \({x'_{\min }}\) and \({{Q'}^2_{\min }}\). These limits may be used to assess the compatibility of theoretical assumptions such as the dilute gas approximation with H1 data, or to test theoretical predictions of instanton properties such as their size and distance distributions.
Upper cross section limits on instantoninduced processes reported previously by the H1 [15] and ZEUS [16] collaborations are above the theoretical predicted cross sections. In a domain of phase space with a lower \(Q^2\) range (\(10 \lesssim Q^2 < 100\) GeV\(^2\)), H1 reported an upper limit of 221 pb at \(95\,\,\%\) CL, about a factor five above the corresponding theoretical prediction. At high \(Q^2\) (\(Q^2 > 120\) GeV\(^2\)), the ZEUS Collaboration obtained an upper limit of 26 pb at \(95\%\) CL in comparison to a predicted cross section of 8.9 pb. In summary, compared to earlier publications, QCD instanton exclusion limits are improved by an order of magnitude and are challenging predictions based on perturbative instanton calculations with parameters derived from lattice QCD.
Footnotes
 1.
The term “electron” is used in the following to refer to both electron and positron.
 2.
\(\Delta \mathrm chirality = 2 \, n_f\), where \(\Delta \mathrm chirality = \) # \((q_{R} +\bar{q}_{R})\) # \((q_{L} +\bar{q}_{L})\), and \(n_f\) is the number of quark flavours.
 3.
The pseudorapidity of a particle is defined as \(\eta \equiv  \ln \tan ( \theta / 2) \), where \(\theta \) is the polar angle with respect to the proton direction defining the \(+z\)axis.
 4.
The qualitative behaviour for the instanton cross section is \(\sigma _{q'g}^{(I)} \sim {\left[ \frac{2 \pi }{\alpha _{s}} \right] }^{12} e^{ \frac{4 \pi }{\alpha _{s}}}\), where \(\alpha _s\) is the strong coupling.
 5.
In the phase space of this analysis the CTEQ5L and CTEQ6L gluon density distributions are almost identical.
 6.
The hadronic centreofmass frame is defined by \(\mathbf {\gamma } + \mathbf {P} = 0\), where \(\mathbf {\gamma }\) and \(\mathbf {P}\) are the 3momentum of the exchanged photon and proton, respectively.
 7.
The PDERS method has been cross checked with other methods: the neural network MLP (MultiLayer Perceptron) method and two variants of the decision tree method, BDT (Boosted Decision Trees) and BDTG (Boosted Decision Trees with Gradient Boost) [62].
Notes
Acknowledgments
We thank A. Ringwald and F. Schrempp for many helpful discussions and for help with their computer program. 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 non DESY 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.
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