DESY 13-211                                            ISSN 0418-9833
December 2013

Measurement of Inclusive ep Cross Sections at High Q^{2} at \sqrt{s} = 225 and 252 GeV and of the Longitudinal Proton

Structure Function F_{L} at HERA

H1 Collaboration

Abstract

Inclusive ep double differential cross sections for neutral current deep inelastic scattering are measured with the H1 detector at HERA. The data were taken with a lepton beam energy of 27.6 GeV and two proton beam energies of E_{p} = 460 and 575 GeV corresponding to centre-of-mass energies of 225 and 252 GeV, respectively. The measurements cover the region of 6.5 × 10^{−4} ≤ x ≤ 0.65 for 35 ≤ Q^{2} ≤ 800 GeV^{2} up to y = 0.85. The measurements are used together with previously published H1 data at E_{p} = 920 GeV and lower Q^{2} data at E_{p} = 460, 575 and 920 GeV to extract the longitudinal proton structure function F_{L} in the region 1.5 ≤ Q^{2} ≤ 800 GeV^{2}.

Submitted to EPJC V. Andreev^{22}, A. Baghdasaryan^{34}, S. Baghdasaryan^{34}, K. Begzsuren^{31}, A. Belousov^{22}, P. Belov^{10}, V. Boudry^{25}, G. Brandt^{46}, M. Brinkmann^{10}, V. Brisson^{24}, D. Britzger^{10}, A. Buniatyan^{13}, A. Bylinkin^{21,43}, L. Bystritskaya^{21}, A.J. Campbell^{10}, K.B. Cantun Avila^{20}, F. Ceccopieri^{3}, K. Cerny^{28}, V. Chekelian^{23}, J.G. Contreras^{20}, J.B. Dainton^{17}, K. Daum^{33,38}, E.A. De Wolf^{3}, C. Diaconu^{19}, M. Dobre^{4}, V. Dodonov^{10}, A. Dossanov^{11,23}, A. Dubak^{23,26}, G. Eckerlin^{10}, S. Egli^{32}, E. Elsen^{10}, L. Favart^{3}, A. Fedotov^{21}, J. Feltesse^{9}, J. Ferencei^{15}, M. Fleischer^{10}, A. Fomenko^{22}, E. Gabathuler^{17}, J. Gayler^{10}, S. Ghazaryan^{10}, A. Glazov^{10}, L. Goerlich^{6}, N. Gogitidze^{22}, M. Gouzevitch^{10,39}, C. Grab^{36}, A. Grebenyuk^{10}, T. Greenshaw^{17}, G. Grindhammer^{23}, S. Habib^{10}, D. Haidt^{10}, R.C.W. Henderson^{16}, M. Herbst^{14}, M. Hildebrandt^{32}, J. Hladkỳ^{27}, D. Hoffmann^{19}, R. Horisberger^{32}, T. Hreus^{3}, F. Huber^{13}, M. Jacquet^{24}, X. Janssen^{3}, A.W. Jung^{14,47}, H. Jung^{10,3}, M. Kapichine^{8}, C. Kiesling^{23}, M. Klein^{17}, C. Kleinwort^{10}, R. Kogler^{11}, P. Kostka^{35}, J. Kretzschmar^{17}, K. Krüger^{10}, M.P.J. Landon^{18}, W. Lange^{35}, P. Laycock^{17}, A. Lebedev^{22}, S. Levonian^{10}, K. Lipka^{10,42}, B. List^{10}, J. List^{10}, B. Lobodzinski^{10}, V. Lubimov^{21,†}, E. Malinovski^{22}, H.-U. Martyn^{1}, S.J. Maxfield^{17}, A. Mehta^{17}, A.B. Meyer^{10}, H. Meyer^{33}, J. Meyer^{10}, S. Mikocki^{6}, A. Morozov^{8}, K. Müller^{37}, Th. Naumann^{35}, P.R. Newman^{2}, C. Niebuhr^{10}, G. Nowak^{6}, K. Nowak^{11}, B. Olivier^{23}, J.E. Olsson^{10}, D. Ozerov^{10}, P. Pahl^{10}, C. Pascaud^{24}, G.D. Patel^{17}, E. Perez^{9,40}, A. Petrukhin^{10}, I. Picuric^{26}, H. Pirumov^{13}, D. Pitzl^{10}, R. Plačakytė^{10,42}, B. Pokorny^{28}, R. Polifka^{28,44}, V. Radescu^{10,42}, N. Raicevic^{26}, A. Raspereza^{10}, T. Ravdandorj^{31}, P. Reimer^{27}, E. Rizvi^{18}, P. Robmann^{37}, R. Roosen^{3}, A. Rostovtsev^{21}, M. Rotaru^{4}, S. Rusakov^{22}, D.Šálek^{28}, D.P.C. Sankey^{5}, M. Sauter^{13}, E. Sauvan^{19,45}, S. Schmitt^{10}, L. Schoeffel^{9}, A. Schöning^{13}, H.-C. Schultz-Coulon^{14}, F. Sefkow^{10}, S. Shushkevich^{10}, Y. Soloviev^{10,22}, P. Sopicki^{6}, D. South^{10}, V. Spaskov^{8}, A. Specka^{25}, M. Steder^{10}, B. Stella^{29}, U. Straumann^{37}, T. Sykora^{3,28}, P.D. Thompson^{2}, D. Traynor^{18}, P. Truöl^{37}, I. Tsakov^{30}, B. Tseepeldorj^{31,41}, J. Turnau^{6}, A. Valkárová^{28}, C. Vallée^{19}, P. Van Mechelen^{3}, Y. Vazdik^{22}, D. Wegener^{7}, E. Wünsch^{10}, J.Žáček^{28}, Z. Zhang^{24}, R. Žlebčík^{28}, H. Zohrabyan^{34}, and F. Zomer^{24}

1 I. Physikalisches Institut der RWTH, Aachen, Germany 2 School of Physics and Astronomy, University of Birmingham, Birmingham, UK^{b} 3 Inter-University Institute for High Energies ULB-VUB, Brussels and Universiteit Antwerpen, Antwerpen, Belgium^{c} 4 National Institute for Physics and Nuclear Engineering (NIPNE) , Bucharest, Romania^{j} 5 STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire, UK^{b} 6 Institute for Nuclear Physics, Cracow, Poland^{d} 7 Institut für Physik, TU Dortmund, Dortmund, Germany^{a} 8 Joint Institute for Nuclear Research, Dubna, Russia 9 CEA, DSM/Irfu, CE-Saclay, Gif-sur-Yvette, France 10 DESY, Hamburg, Germany 11 Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany^{a} 12 Max-Planck-Institut für Kernphysik, Heidelberg, Germany 13 Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany^{a} 14 Kirchhoff-Institut für Physik, Universität Heidelberg, Heidelberg, Germany^{a} 15 Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic^{e} 16 Department of Physics, University of Lancaster, Lancaster, UK^{b} 17 Department of Physics, University of Liverpool, Liverpool, UK^{b} 18 School of Physics and Astronomy, Queen Mary, University of London, London, UK^{b} 19 CPPM, Aix-Marseille Univ, CNRS/IN2P3, 13288 Marseille, France 20 Departamento de Fisica Aplicada, CINVESTAV, Mérida, Yucatán, México^{h} 21 Institute for Theoretical and Experimental Physics, Moscow, Russia^{i} 22 Lebedev Physical Institute, Moscow, Russia 23 Max-Planck-Institut für Physik, München, Germany 24 LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 25 LLR, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France 26 Faculty of Science, University of Montenegro, Podgorica, Montenegro^{k} 27 Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic^{f} 28 Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic^{f} 29 Dipartimento di Fisica Università di Roma Tre and INFN Roma 3, Roma, Italy 30 Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria 31 Institute of Physics and Technology of the Mongolian Academy of Sciences, Ulaanbaatar, Mongolia 32 Paul Scherrer Institut, Villigen, Switzerland 33 Fachbereich C, Universität Wuppertal, Wuppertal, Germany 34 Yerevan Physics Institute, Yerevan, Armenia 35 DESY, Zeuthen, Germany 36 Institut für Teilchenphysik, ETH, Zürich, Switzerland^{g} 37 Physik-Institut der Universität Zürich, Zürich, Switzerland^{g}

38 Also at Rechenzentrum, Universität Wuppertal, Wuppertal, Germany 39 Also at IPNL, Université Claude Bernard Lyon 1, CNRS/IN2P3, Villeurbanne, France 40 Also at CERN, Geneva, Switzerland 41 Also at Ulaanbaatar University, Ulaanbaatar, Mongolia 42 Supported by the Initiative and Networking Fund of the Helmholtz Association (HGF) under the contract VH-NG-401 and S0-072 43 Also at Moscow Institute of Physics and Technology, Moscow, Russia 44 Also at Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 45 Also at LAPP, Université de Savoie, CNRS/IN2P3, Annecy-le-Vieux, France 46 Department of Physics, Oxford University, Oxford, UK^{b} 47 Now at Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA

† Deceased

a Supported by the Bundesministerium für Bildung und Forschung, FRG, under contract numbers 05H09GUF, 05H09VHC, 05H09VHF, 05H16PEA b Supported by the UK Science and Technology Facilities Council, and formerly by the UK Particle Physics and Astronomy Research Council c Supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT and by Interuniversity Attraction Poles Programme, Belgian Science Policy d Partially Supported by Polish Ministry of Science and Higher Education, grant DPN/N168/DESY/2009 e Supported by VEGA SR grant no. 2/7062/ 27 f Supported by the Ministry of Education of the Czech Republic under the projects LC527, INGO-LA09042 and MSM0021620859 g Supported by the Swiss National Science Foundation h Supported by CONACYT, México, grant 48778-F i Russian Foundation for Basic Research (RFBR), grant no 1329.2008.2 and Rosatom j Supported by the Romanian National Authority for Scientific Research under the contract PN 09370101 k Partially Supported by Ministry of Science of Montenegro, no. 05-1/3-3352

1 Introduction

Deep inelastic scattering (DIS) data provide high precision tests of perturbative quantum chromodynamics (QCD), and have led to a detailed and comprehensive understanding of proton structure, see [1] for a recent review. A measurement of the longitudinal proton structure function, F_{L}, provides a unique test of parton dynamics and the consistency of QCD by allowing a comparison of the gluon density obtained largely from the scaling violations of F_{2} to an observable directly sensitive to the gluon density. Previous measurements of F_{L} have been published by the H1 and ZEUS collaborations covering the kinematic region of low Bjorken x, and low to medium four-momentum transfer squared, Q^{2}, using data taken at proton beam energies E_{p} = 460, 575 and 920 GeV corresponding to centre-of-mass energies of \sqrt{s} = 225, 252 and 319 GeV respectively [2–4]. The new data presented here improve the experimental precision on F_{L} in the region 35 ≤ Q^{2} ≤ 110 GeV^{2}, and provide the first measurements of F_{L} in the region 120 ≤ Q^{2} ≤ 800 GeV^{2} and 6.5 × 10^{−4} < x < 0.032. As the extraction of F_{L} and F_{2} is repeated using all available H1 cross section measurements, the earlier measurements of F_{L} and F_{2} [2, 3] are superseded by the present analysis. Furthermore, in the determination of the systematic uncertainties of the published H1 F_{L} measurements [3] an error has been identified in the procedure of averaging several measurements at fixed Q^{2} which is corrected here.

The differential cross section for deep inelastic ep scattering can be described in terms of three proton structure functions F_{2}, F_{L} and xF_{3}, which are related to the parton distribution functions (PDFs) of the proton. The structure functions depend on the kinematic variables, x and Q^{2} only, whereas the cross section is additionally dependent on the inelasticity y related by y = Q^{2}/sx. The reduced neutral current (NC) differential cross section for e^{+}p scattering after correcting for QED radiative effects can be written as

σ̃_{NC}(x, Q^{2}, y) ≡ \frac{d^{2}σ_{NC}}{dxdQ^{2}} \frac{xQ^{4}}{2πα^{2}} \frac{1}{Y_{+}} ≡ (F_{2} − \frac{y^{2}}{Y_{+}} F_{L} − \frac{Y_{−}}{Y_{+}} xF_{3}) ,  (1)

where Y_{±} = 1 ± (1 − y)^{2} and the fine structure constant is defined as α ≡ α(Q^{2} = 0).

The cross section for virtual boson (Z/γ^{∗}) exchange is related to the F_{2} and xF_{3} structure functions in which both the longitudinal and transverse boson polarisation states contribute. The F_{L} term is related to the longitudinally polarised virtual boson exchange process. As can be seen from equation 1 the contribution of F_{L} to the cross section is significant only at high y. For Q^{2} ≲ 800 GeV^{2} the contribution of Z exchange and the influence of xF_{3} is expected to be small.

A direct measurement of F_{L} is performed by measuring the differential cross section at different values of \sqrt{s} by reducing the proton beam energy from 920 GeV, used for most of the HERA-II run period, to E_{p} = 460 and 575 GeV. The lepton beam energy was maintained at 27.6 GeV. The two sets of cross section data are combined with recently published H1 data taken at E_{p} = 920 GeV [5], and cross section measurements at lower Q^{2} taken at E_{p} = 460, 575 and 920 GeV [3], to provide a set of measurements at fixed x and Q^{2} but at different values of y. This provides an experimental separation between the F_{2} and F_{L} structure functions. Sensitivity to F_{L} is enhanced by performing the differential cross sections measurement up to high y, a kinematic region in which the scattered lepton energy is low, and consequently the background from photoproduction processes is large. The cross sections are used to extract F_{L} and the ratio R of the longitudinally to transversely polarised photon exchange cross sections. In addition a direct local extraction of the gluon density xg(x, Q^{2}) is performed.

This paper is organised as follows: in section 2 the H1 detector, trigger system and data sets are described. The simulation programs and Monte Carlo models used in the analysis are presented in section 3. In section 4 the analysis procedure is given in which the event selection and background suppression methods are discussed followed by an assessment of the systematic uncertainties of the measurements. The results are presented in section 5 and the paper is summarised in section 6.

2 H1 Apparatus, Trigger and Data Samples

2.1 The H1 Detector

A detailed description of the H1 detector can be found elsewhere [6–9]. The coordinate system of H1 is defined such that the positive z axis is in the direction of the proton beam (forward direction) and the nominal interaction point is located at z = 0. The polar angle θ is then defined with respect to this axis. 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^{∘} < θ < 154^{∘}, the inner tracking detectors, which measure the angles and momenta of charged particles over the range 7^{∘} < θ < 165^{∘}, and a lead-fibre calorimeter (SpaCal) covering the range 153^{∘} < θ < 177^{∘}.

The LAr calorimeter consists of an inner electromagnetic section with lead absorbers and an outer hadronic section with steel absorbers. The calorimeter is divided into eight wheels along the beam axis, each consisting of eight stacks arranged in an octagonal formation around the beam axis. 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 δE/E ≃ 0.11/\sqrt{E/GeV}⊕0.01 and hadronic energies with δE/E ≃ 0.50/\sqrt{E/GeV}⊕ 0.02 as determined using electron and pion test beam data [10, 11].

In the central region, 25^{∘} < θ < 155^{∘}, the central tracking detector (CTD) measures the trajectories of charged particles in two cylindrical drift chambers (CJC) immersed in a uniform 1.16 T solenoidal magnetic field. The CTD also contains a further drift chamber (COZ) between the two drift chambers to improve the z coordinate reconstruction, as well as a multiwire proportional chamber at inner radii (CIP) mainly used for triggering [12]. The CTD measures charged particle trajectories with a transverse momentum resolution of σ(p_{T})/p_{T} ≃ 0.2% p_{T} /GeV ⊕ 1.5%. The CJC also provides a measurement of the specific ionisation energy loss, dE/dx, of charged particles with a relative resolution of 6.5% for long tracks. The forward tracking detector (FTD) is used to supplement track reconstruction in the region 7^{∘} < θ < 30^{∘} [13] and to improve the hadronic final state (HFS) reconstruction of forward going low transverse momentum particles.

The CTD tracks are linked to hits in the vertex detectors: the central silicon tracker (CST) [14, 15], the forward silicon tracker (FST), and the backward silicon tracker (BST). These detectors provide precise spatial track reconstruction and therefore also improve the primary vertex reconstruction. The CST consists of two layers of double-sided silicon strip detectors surrounding the beam pipe covering an angular range of 30^{∘} < θ < 150^{∘} for tracks passing through both layers. The FST consists of five double wheels of single-sided strip detectors [16] measuring the transverse coordinates of charged particles. The BST design is very similar to the FST and consists of six double wheels of strip detectors [17].

In the backward region the SpaCal provides an energy measurement for electrons^{1} and hadronic particles, and has a resolution for electromagnetic energy depositions of δE/E ≃ 0.07/\sqrt{E/GeV}⊕ 0.01, and a hadronic energy resolution of δE/E ≃ 0.70/\sqrt{E/GeV} ⊕ 0.01 as measured using test beam data [18].

The integrated ep luminosity is determined by measuring the event rate for the Bethe-Heitler process of QED bremsstrahlung ep → epγ. The photons are detected in the photon tagger located at z = −103 m. An electron tagger is placed at z = −5.4 m adjacent to the beampipe. It is used to provide information on ep → eX events at very low Q^{2} (photoproduction) where the electron scatters through a small angle (π − θ < 5 mrad).

At HERA transverse polarisation of the lepton beam arises naturally through synchrotron radiation via the Sokolov-Ternov effect [19]. Spin rotators installed in the beamline on either side of the H1 detector allow transversely polarised leptons to be rotated into longitudinally polarised states and back again. Two independent polarimeters LPOL [20] and TPOL [21] monitor the polarisation. Only data where a TPOL or LPOL measurement is available is used. When both measurements are available they are averaged [22].

2.2 The Trigger

The H1 trigger system is a three level trigger with a first level latency of approximately 2 μs. In the following we describe only the components relevant to this analysis. NC events at high Q^{2} are triggered mainly using information from the LAr calorimeter to rapidly identify the scattered lepton. The calorimeter has a finely segmented 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 energy above 11 GeV this LAr electron trigger is determined to be 100% efficient obtained by using LAr triggers fired by the hadronic final state particles.

At high y, corresponding to lower electron energies, the backward going HFS particles can enter the SpaCal and therefore trigger the event. In addition low energy scattered electron candidates can be triggered by the Fast Track Trigger [23, 24] based on hit information provided by the CJC, and the LAr Jet Trigger [25] using energy depositions in the LAr calorimeter. These two trigger subsystems allow electron identification to be performed at the third trigger level [26]. This L3 electron trigger and the SpaCal trigger are used to extend the kinematically accessible region to high y where scattered leptons have energies as low as 3 GeV, the minimum value considered in this analysis. For electron energies of 3 GeV, the total trigger efficiency is found to vary between 91 − 97% depending on the kinematic region.

1In this paper “electron” refers generically to both electrons and positrons. Where distinction is required, the terms e^{−} and e^{+} are used.

2.3 Data Samples

The data sets used in the measurement of the reduced cross sections correspond to two short dedicated data taking periods in 2007 in which the proton beam energy was reduced to 460 GeV and 575 GeV, and the scattered lepton was detected in the LAr calorimeter. The positron beam was longitudinally polarised with polarisation P_{e} = (N_{R} − N_{L})/(N_{R} + N_{L}), where N_{R} (N_{L}) is the number of right (left) handed leptons in the beam. The integrated luminosity and longitudinal lepton beam polarisation for each data set are given in table 1. The lepton beam polarisation plays no significant role in this analysis.

E_{p} = 460 GeV  E_{p} = 575 GeV
e^{+}p L = 11.8 pb^{−1} L = 5.4 pb^{−1}
P_{e} = (−42.3 ± 0.8)% P_{e} = (−41.8 ± 0.8)%

Table 1: Integrated luminosities, L, and luminosity weighted longitudinal lepton beam polarisation, P_{e}, for the data sets presented here.

The extraction of the F_{L} structure function in section 5.2 uses the cross section measurements presented here and measurements at E_{p} = 920 GeV in which the scattered lepton is detected in the LAr calorimeter [5] and at E_{p} = 460, 575 and 920 GeV with the electron detected in the SpaCal [3]. The two detectors provide access to different kinematic regions and the corresponding measurements are referred to as the LAr and SpaCal data for each of the three values of E_{p}.

3 Simulation Programs

In order to determine acceptance corrections, DIS processes are generated at leading order (LO) QCD using the Djangoh 1.4 [28] Monte Carlo (MC) simulation program which is based on Heracles 4.6 [29] for the electroweak interaction and on Lepto 6.5.1 [30] for the hard matrix element calculation. The colour dipole model (CDM) as implemented in Ariadne [31] is used to simulate higher order QCD dynamics. The Jetset 7.410 program [32] is used to simulate the hadronisation process in the framework of the ‘string-fragmentation’ model. The parameters of this model used here are tuned to describe hadronic Z decay data [33]. The simulated events are weighted to reproduce the cross sections predicted by the NLO QCD fit H1PDF 2012 [5]. This fit includes H1 low Q^{2} NC data and high Q^{2} neutral and charged current (CC) data from HERA I, as well as inclusive NC and CC measurements from H1 at high Q^{2} based on the full HERA II integrated luminosity at E_{p} = 920 GeV [5]. In addition the Compton 22 [34] MC is used to simulate elastic and quasi-elastic QED Compton processes, and replaces the Compton processes simulation available in Djangoh. The detector response to events produced by the various generator programs is simulated in detail using a program based on Geant3 [35]. The simulation includes a detailed time dependent modelling of detector noise conditions, beam optics, polarisation and inefficient channel maps reflecting actual running conditions throughout the data taking periods. These simulated events are then subjected to the same reconstruction, calibration, alignment and analysis chain as the real data.

4 Experimental Procedure

4.1 Kinematic Reconstruction

Accurate measurements of the event kinematic quantities Q^{2}, x and y are an essential component of the analysis. Since both the scattered lepton and the hadronic final state (HFS) are observed in the detector, several kinematic reconstruction methods are available allowing for calibration and cross checks.

The primary inputs to the various methods employed are the scattered lepton’s energy E′_{e} and polar scattering angle θ_{e}, as well as the quantity Σ = ∑_{i}(E_{i} − p_{z,i}) determined from the sum over the HFS particles assuming charged particles have the pion mass, where E_{i} and p_{z,i} are the energy and longitudinal momenta respectively [36]. At high Q^{2} and low y the HFS is dominated by one or more jets. Therefore the complete HFS can be approximated by the sum of jet four-momenta corresponding to localised calorimetric energy sums above threshold. This technique allows a further suppression of “hadronic noise” in the reconstruction arising from electronic sources in the LAr calorimeter or from back-scattered low energy particles produced in secondary interactions.

The most precise kinematic reconstruction method for y ≳ 0.1 is the e-method which relies solely on E′_{e} and θ_{e} to reconstruct the kinematic variables Q^{2} and y as

Q_{e}^{2} = \frac{(E′_{e} sin θ_{e})^{2}}{1 − y_{e}} , y_{e} = 1 − \frac{E′_{e}}{E_{e}} sin^{2} (\frac{θ_{e}}{2} ) ,  (2)

and x is determined via the relation x = Q^{2}/sy. This method is used in the analysis region y > 0.19 since the resolution of the e-method degrades at low y. The method is also susceptible to large QED radiative corrections at the highest and lowest y. A cut on quantity E − P_{z} = Σ + E′_{e}(1 − cos θ_{e}) ensures that the radiative corrections are moderate.

In the Σ-method [37], y is reconstructed as Σ/(E − P_{z}) and is therefore less sensitive to QED radiative effects. The eΣ-method [38] is an optimum combination of the two and maintains good resolution throughout the kinematic range of the measurement with acceptably small QED radiative corrections. The kinematic variables are determined using

Q_{eΣ}^{2} = Q_{e}^{2} = \frac{(E′_{e} sin θ_{e})^{2}}{1 − y_{e}} , y_{eΣ} = 2E_{e} \frac{Σ}{[E − P_{z}]^{2}} ,  (3)

and x is determined as for the e-method above. The eΣ-method is employed to reconstruct the event kinematics for y ≤ 0.19 in which Σ is determined using hadronic jets defined using the longitudinally invariant k_{T} jet algorithm [39, 40].

4.2 Polar Angle Measurement and Energy Calibration

The detector calibration and alignment procedures adopted for this analysis rely on the methods discussed in detail in [5] which uses the high statistics E_{p} = 920 GeV data recorded just prior to the 460 and 575 GeV runs. The detector was not moved or opened between these run periods.

The alignment and calibration constants obtained at E_{p} = 920 GeV are verified using the same methods [5] for the data presented here.

In this analysis the scattered lepton is detected in the LAr calorimeter by searching for a compact and isolated electromagnetic energy deposition. The polar angle of the scattered lepton, θ_{e}, is determined using the position of its energy deposit (cluster) in the LAr calorimeter, and the event vertex reconstructed with tracks from charged particles. The relative alignment of the calorimeter and tracking chambers is verified using a sample of events with a well measured lepton track [41] in which the COZ chamber provides an accurate z reconstruction of the particle trajectory. The residual difference between the track and cluster polar angles in data and simulation is found to be less than 1 mrad, and this value is used as the systematic uncertainty of the scattered lepton polar angle.

An in situ energy calibration of electromagnetic energy depositions in the LAr calorimeter is performed for both data and simulation. Briefly, a sample of NC events in which the HFS is well contained in the detector is used with the Double Angle reconstruction method [42,43] to predict the scattered lepton energy (E_{DA}) which is then compared to the measured electromagnetic energy response allowing local calibration factors to be determined in a finely segmented grid in z and ϕ. The residual mismatch between E_{DA} and E′_{e} after performing the calibration step are found to vary within ≃ 0.3 − 1% depending on the geometric location of the scattered lepton within the LAr calorimeter. An additional 0.3% correlated uncertainty is considered and accounts for a possible bias in the P_{T,DA} reconstruction and is determined by varying θ_{e} and a measurement of the inclusive hadronic polar angle, γ_{h}, by the angular measurement uncertainty. This has been verified by comparing the residual global shifts between data and MC in the kinematic peak of the E′_{e} distribution.

At the lowest electron energies the calibration is validated using QED Compton interactions ep → eγp with E′_{e} of 3 − 8 GeV in which the lepton track momentum P_{track} is compared to the measured energy E′_{e} of the cluster. The simulation on average describes the data well in this low energy region. For energies below 11 GeV an additional uncorrelated uncertainty of 0.5% is included to account for a possible nonlinearity of the energy scale.

The hadronic response of the detector is calibrated by requiring a transverse momentum balance between the predicted P_{T} in the DA-method (P_{T,DA}) and the measured hadronic final state using a tight selection of well reconstructed events with a single jet. The calorimeter calibration constants are then determined in a minimisation procedure across the detector acceptance separately for HFS objects inside and outside jets and for electromagnetic and hadronic contributions to the HFS [44]. The potential bias in the P_{T,DA} reference scale of 0.3% is also included as a correlated source of uncertainty.

The mean transverse momentum balance between the hadronic final state and the scattered lepton both in data and simulation agree to within 1% precision which is taken as the uncorrelated hadronic scale uncertainty. The hadronic SpaCal calibration is performed in a similar manner and a systematic uncertainty of 5% is adopted.

4.3 Measurement Procedure

The event selection and analysis of the NC sample follows closely the procedures discussed in [5]. Inelastic ep interactions are required to have a well reconstructed interaction vertex to suppress beam induced background events. High Q^{2} neutral current events are selected by requiring each event to have a compact and isolated cluster in the electromagnetic part of the LAr calorimeter. The scattered lepton candidate is identified as the cluster of highest transverse momentum and must have an associated CTD track. For high electron energies the track condition is relaxed as detailed in 4.3.1. The analysis is restricted to the region 32 < Q_{e}^{2} < 890 GeV^{2}.

The quantity E − P_{z} summed over all final state particles (including the electron) is required by energy-momentum conservation to be approximately equal to twice the initial electron beam energy. Restricting E −P_{z} to be greater than 35 GeV considerably reduces the photoproduction background and radiative processes in which either the scattered lepton or bremsstrahlung photons escape undetected in the lepton beam direction. Topological algorithms [46] are employed to suppress non-ep and QED Compton backgrounds ep → eγp.

The photoproduction background increases rapidly with decreasing electron energy (corresponding to high y), therefore the analysis is separated into two distinct regions: the nominal analysis (y_{e} ≤ 0.38), and the high y analysis (0.38 < y_{e} < 0.9). In the high y region dedicated techniques are employed to contend with the large background. The analysis differences in each kinematic region are described below.

4.3.1 Nominal Analysis

At low y ≤ 0.38 the minimum electron energy is kinematically restricted to be above 18 GeV. The forward going hadronic final state particles can undergo interactions with material of the beam pipe leading sometimes to a bias in the reconstruction of the primary interaction vertex position. In such cases the vertex position is calculated using a stand alone reconstruction of the track associated with the electron cluster [45,46]. For the nominal analysis the photoproduction contribution is negligible, and the only sizeable background contribution arises from remaining QED Compton events which is estimated using simulation. The electron candidate track verification is supplemented by searching for hits in the CIP located on the trajectory from the interaction vertex to the electron cluster. This optimised treatment of the vertex determination and verification of the electron cluster with the tracker information improves the reliability of the vertex position determination and increases the efficiency of the procedure to be larger than 99.5%.

For the region y < 0.19 the hadronic noise has an increasing influence on Σ and on the transverse momentum balance P_{T,h}/P_{T,e} through its effect on P_{T,h} where P_{T,h}, P_{T,e} are the hadronic and scattered lepton transverse momenta respectively. The event kinematics reconstructed with the eΣ-method in which the HFS is formed from hadronic jets only, limits the noise contribution and substantially improves the P_{T,h}/P_{T,e} description by the simulation. The jets are found with the longitudinally invariant k_{T} jet algorithm [39, 40] as implemented in FastJet [47, 48] with radius parameter R = 1.0 and are required to have transverse momenta P_{T,jet} > 2 GeV. In figure 1 the quality of the simulation and its description of the E_{p} = 460 GeV and E_{p} = 575 GeV data for y_{e} < 0.19 can be seen for the distributions of the P_{T,h}/P_{T,e}, θ_{jets}, and E − P_{z} where all HFS quantities are obtained using the vector sum of jet four-momenta. The simulation provides a reasonable description of both sets of distributions. The MC simulation is normalised to the integrated luminosity of the data.

4.3.2 High y Analysis

In the high y region (0.38 < y_{e} < 0.9) the analysis is extended to low energies of the scattered electron, E′_{e} > 3 GeV. At these energies photoproduction background contributions arise from π^{0} → γγ decays, from charged hadrons being misidentified as electron candidates, and from real electrons originating predominantly from semi-leptonic decays of heavy flavour hadrons. These contributions increase rapidly with decreasing energy of the electron candidate. Therefore additional techniques are used to reduce this background.

The background from π^{0} → γγ decays leads to different electromagnetic shower profiles compared to electrons of similar energy. In addition genuine electrons have a momentum matched track associated to the cluster. Four cluster shape variables and the ratio of the candidate electron energy E′_{e} determined using cluster information, to the momentum of the associated track p_{e}, are used in a neural network multilayer perceptron [49] to discriminate signal from background. Additional information using the specific ionisation energy loss, dE/dx, of the track is also used to form a single electron discrimination variable, D_{ele}, such that a value of 1 corresponds to electrons and a value of 0 corresponds to hadrons. The neural network is trained using single particle MC simulations, and validated with samples of identified electrons and pions from J/ψ → e^{+}e^{−} and K_{s}^{0} → π^{+}π^{−} decays in data and MC [26, 27]. For the region E′_{e} < 10 GeV isolated electrons are selected by requiring D_{ele} > 0.80 which is estimated to have a pion background rejection of more than 99% and a signal selection efficiency of better than 90% [26]. For the region E′_{e} > 10 GeV the scattered electron is identified as in the nominal analysis.

The scattered lepton candidate is required to have positive charge corresponding to the beam lepton. The remaining background is estimated from the number of data events with opposite charge. This background is statistically subtracted from the positively charged sample. However, a charge asymmetry in photoproduction can arise due to the different detector response to particles compared to antiparticles [50, 51]. The charge asymmetry has been determined by measuring the ratio of wrongly charged scattered lepton candidates in e^{+}p to e^{−}p scattering at E_{p} = 920 GeV data and was found to be 1.03 ± 0.05 [5]. This is cross checked in the E_{p} = 460 and 575 GeV data using photoproduction events in which the scattered electron is detected in the electron tagger. In this sample fake scattered electron candidates passing all selection criteria are detected in the LAr calorimeter with both positively and negatively charged tracks associated to the electromagnetic cluster. The charge asymmetry is obtained by comparing the two contributions. The results obtained are consistent with the asymmetry measured in the E_{p} = 920 GeV data, however due to the lower statistical precision of the E_{p} = 460 and 575 GeV data sets, the uncertainty of the asymmetry is increased to 0.08. The asymmetry is taken into account in the subtraction procedure. The efficiency with which the lepton charge is determined is well described by simulation within 0.5% and is discussed in section 4.5.

The control of the background in the most critical region of E′_{e} < 6 GeV is demonstrated in figure 2 for both data sets. The MC simulation is normalised to the integrated luminosity of the data. In all cases the background dominated regions are well described in shape and overall normalisation, giving confidence that the background contributions can be reliably estimated from the wrong charge sample. At low E −P_{z} a peak is observed arising from QED initial state radiation (ISR) which is reasonably well described. The cut E − P_{z} > 35 GeV suppresses the influence of ISR on the measurement. The D_{ele} distribution show two populations peaking at zero and unity arising from hadrons and real electrons respectively. The peak at D_{ele} = 1 for the background indicates that there are real electrons in the remaining background sample.

The e-method has the highest precision in this region of phase space and is used to reconstruct the event kinematics. Figure 3 shows the energy spectrum and the polar angle distribution of the scattered lepton, and the E − P_{z} spectrum of the high y sample for the E_{p} = 460 and 575 GeV data before background subtraction. The background estimates are shown together with the contribution from the remaining QED Compton process. The NC simulation provides a good description of these distributions.

In the scattered lepton energy spectrum a small discontinuity at 8 GeV can be seen. This is a consequence of suppressing events in which two electron candidates are found, one with E < 8 GeV and the other with E > 8 GeV. This criterion efficiently suppresses background from the QED Compton process in the region E′_{e} < 8 GeV.

4.4 Cross Section Measurement

The simulation is used to correct the selected event samples for detector acceptance, efficiencies, migrations and QED radiation effects. The simulation provides a good description of the data and therefore is expected to give a reliable determination of the detector acceptance. The accessible kinematic ranges of the measurements depend on the resolution of the reconstructed kinematic variables. The ranges are determined by requiring the purity and stability of any measurement bin to be larger than 30% as determined from signal MC. The purity is defined as the fraction of events generated and reconstructed in a measurement bin (N^{g+r}) from the total number of events reconstructed in the bin (N^{r}). The stability is the ratio of the number of events generated and reconstructed in a bin to the number of events generated in that bin (N^{g}). The purity and stability are typically found to be above 60%. The detector acceptance, A = N^{r}/N^{g}, corrects the measured signal event yields for detector effects including resolution smearing and selection efficiency.

The measured differential cross sections σ(x, Q^{2}) are then determined using the relation

σ(x, Q^{2}) = \frac{N − B}{L · A} · C · (1 + Δ^{QED}) ,  (4)

where N and B are the selected number of data events and the estimated number of background events respectively, L is the integrated luminosity, C is the bin centre correction, and (1+Δ^{QED}) are the QED radiative corrections. These corrections are defined in [52, 53] and are calculated to first order in α using the program Heracles [29] as implemented in Djangoh [28] and verified with the numerical analysis programs Hector [54] and Eprc [55]. No weak radiative corrections are applied to the measurements.

The bin centre correction C(x, Q^{2}) is a factor obtained from NLO QCD expectation σ^{th}, using H1PDF 2012 [5], and scales the bin integrated cross section to a differential cross section at the kinematic point x, Q^{2} defined as

C(x, Q^{2}) = \frac{σ^{th}(x, Q^{2})}{∫∫_{bin} dx′dQ^{2′} σ^{th}(x′, Q^{2′})} .  (5)

The cross section measurements are finally corrected for the effects of lepton beam polarisation using the H1PDF 2012 fit to yield cross sections with P_{e} = 0. This multiplicative correction does not exceed 2.5% in the region considered.

In order to optimise the measurement for an extraction of the structure function F_{L}, the cross sections are measured in y, Q^{2} bins for y > 0.38 at E_{p} = 460 GeV, and y > 0.304 at E_{p} = 575 GeV. At E_{p} = 920 GeV the y, Q^{2} binned cross sections are published for y > 0.19 [5]. This binning is constructed specifically for a measurement of F_{L} with fine segmentation in y. The lower limits in y for each proton beam energy are chosen such that they have the same x for all three values of E_{p}. In all other cases the cross sections are measured in Q^{2}, x bins. The bin boundaries and bin centres in the Q^{2} − x plane are chosen to be the same in the overlapping region for E_{p} = 460, 575 and 920 GeV for 35 ≤ Q^{2} ≤ 800 GeV^{2}.

4.5 Systematic Uncertainties

The uncertainties on the measurement lead to systematic errors on the cross sections, which can be split into bin-to-bin correlated and uncorrelated parts. All the correlated systematic errors are found to be symmetric to a good approximation and are assumed so in the following. The total systematic error is formed by adding the individual errors in quadrature.

The size of each systematic uncertainty source and its region of applicability are given in table 2. Further details can be found elsewhere [41, 44–46, 56] in which several of the sources of uncertainty have been investigated using the E_{p} = 920 GeV LAr data. The results of similar studies performed using the E_{p} = 460 GeV and 575 GeV LAr data are compared to these earlier analyses to determine the systematic uncertainties. The influence of the systematic uncertainties on the cross section measurements are given in tables 3-4, and their origin and method of estimation are discussed below.

Electron Energy: Uncertainties arise from the particular choice of calibration samples, and the linearity correction uncertainty. These uncertainties are taken from the analysis of the 920 GeV data [5]. The uncertainty varies as a function of z_{imp} [5], the z position of the scattered electron in the calorimeter, as given in table 2. The correlated part of the uncertainty of 0.3% accounts for a possible bias in the E_{DA} reconstruction used as a reference scale in the energy calibration procedure. This results in a systematic uncertainty which is up to 2 − 3% at low y.

Hadronic Calibration: An uncorrelated uncertainty of 1% is used for the hadronic energy measurement. The uncertainty is determined by quantifying the agreement between data and simulation in the mean of the P_{T,h}/P_{T,e} distribution in different kinematic regions. The correlated part of the uncertainty accounts for a possible bias in the E_{DA} reconstruction used as a reference scale in the energy calibration. It is determined to be 0.3% and results in a correlated systematic error on the cross section which is up to 2 − 3% at low y. The resulting correlated systematic error is typically below 1% for the cross sections.

Polar Angle: A correlated 1 mrad uncertainty on the determination of the electron polar angle is considered. This contribution leads to a typical uncertainty on the reduced cross sections of less than 1%.

Source  Region  Uncertainty

Electron energy scale

z_{imp} ≤ −150 cm 0.5% unc. ⊕ 0.3% corr.
−150 < z_{imp} ≤ −60 cm  0.3% unc. ⊕ 0.3% corr.
−60 < z_{imp} ≤ +20 cm  0.5% unc. ⊕ 0.3% corr.
+20 < z_{imp} ≤ +110 cm  0.5% unc. ⊕ 0.3% corr.
z_{imp} > +110 cm  1.0% unc. ⊕ 0.3% corr.
Electron scale linearity  E′_{e} < 11 GeV  0.5%

Hadronic energy scale LAr & Tracks 1.0% unc. ⊕ 0.3% corr.

SpaCal  5.0% unc. ⊕ 0.3% corr.
Polar angle  θ_{e}  1 mrad corr.
Noise  y < 0.19  5% energy not in jets , corr.
y > 0.19  20% corr.

Trigger efficiency

high y 0.3 − 2%
nominal  0.3%

Electron track and vertex efficiency high y 1%

nominal  0.2 − 1%
Electron charge ID efficiency  high y  0.5%

Electron ID efficiency

high y z_{imp} < 20 (> 20) cm 0.5% (1%)
nominal z_{imp} < 20 (> 20) cm  0.2% (1%)
Extra background suppression  E′_{e} < 10 GeV  D_{ele} > 0.80 ± 0.04 corr.
High y background subtraction  high y  1.03 ± 0.08 corr.

QED radiative corrections

x < 0.1 , 0.1 ≤ x < 0.3 , x ≥ 0.3 0.3% , 1.0%, 2.0%
high y: y < 0.8 (y > 0.8)  1% (1.5%)

Acceptance corrections

high y 0.5%
nominal  0.2%
Luminosity  4% corr.

Table 2: Table of applied systematic uncertainties and regions of applicability. Uncertainties which are considered point-to-point correlated are labelled corr., and all other sources are considered uncorrelated. The effect of these uncertainties on the cross section measurements is given in the tables of section 5 (except for the luminosity uncertainty).

Noise Subtraction: Energy classified as noise in the LAr calorimeter is excluded from the HFS. For y < 0.19 the calorimetric energy not contained within hadronic jets is classified as noise. The uncertainty on the subtracted noise is estimated to be 5% of the noise contribution as determined from the analysis of the HERA II E_{p} = 920 GeV data [5]. For y > 0.19 the noise contribution is restricted to the sum of isolated low energy calorimetric depositions. Here the residual noise contribution is assigned an uncertainty of 20%, to accomodate differences between data and simulation.

Nominal Trigger Efficiency: The uncertainty on the trigger efficiency in the nominal analysis is determined separately for both E_{p} = 460 and 575 GeV data taking periods. Three trigger requirements are employed: the global timing, the event timing and the calorimeter energy. The inefficiency of global timing criteria to suppress out of time beam related background was continuously monitored with high precision and found to be 0.3% and is corrected for. Finally the event timing trigger requirements were also continuously monitored in the data. After rejection of local inefficient regions the overall trigger efficiency

is close to 100% and an uncertainty of 0.3% is assigned.

High y Trigger Efficiency: At low E′_{e} the LAr electron trigger is supplemented by the SpaCal trigger and by the Level 3 electron trigger based on the LAr Jet Trigger and the Fast Track Trigger. The same global timing conditions as mentioned above are used in the high y triggers. The SpaCal trigger and the LAr electron trigger together with the L3 electron trigger are independent since the SpaCal trigger is fired by the backward going hadronic final state particles. The efficiency of each of these two groups of triggers is determined using events triggered by the other group as a monitor sample. In the analysis events from both groups of triggers are used. The combined efficiency is calculated and is found to vary between 91% and 97% at E′_{e} = 3 GeV. The statistical uncertainty of the combined efficiency together with a 0.3% uncertainty arising from the global timing conditions is adopted as uncorrelated trigger uncertainty. It varies from 0.3% at high electron energies to 2% at E′_{e} = 3 GeV.

Electron Track-Vertex Efficiency: The efficiencies for reconstructing a track associated to the scattered lepton and for reconstructing the interaction vertex are determined simultaneously. The efficiency measurement follows the procedure used in the analysis of the HERA II E_{p} = 920 GeV data and checked on the E_{p} = 460 and 575 GeV data. Three algorithms are used to determine the interaction vertex. The data and MC efficiencies are compared for each contributing algorithm. The combined efficiency in the nominal analysis is found to be larger than 99.5% in the data. The residual differences between data and simulation after correction of simulation by 0.3% define the uncorrelated systematic uncertainty which is 0.2% and is considered to be uncorrelated. In the high y analysis a more stringent requirement on the quality of the track associated to the scattered lepton is applied. The efficiency was measured using electrons in the region of E′_{e} > 18 GeV and checked at low E′_{e} using a sample of QED Compton events. It is found to be 96% in data with a difference of 1% between data and simulation. This difference was corrected for and a 1% uncorrelated uncertainty is adopted.

Electron Charge Identification Efficiency: In the high y analysis the efficiency for correct charge identification of the scattered lepton is measured in the region E′_{e} > 18 GeV which is free from photoproduction background. The simulation describes the efficiency of the data with an overall difference of 0.5%, and no significant time dependence is observed. This is validated using ISR events in which the incoming beam positron has reduced energy due to QED radiation, yielding a sample of events which is free from photoproduction background but has E′_{e} below 12 GeV. The measured cross section is corrected for the overall difference by increasing the measured values by 2 × 0.5% with an uncertainty of 2 × 0.25%. The factor of two accounts for the fact that charge misidentification has a dual influence on the measurement by causing both a loss of signal events and an increase of the subtracted background [46].

Electron Identification: A calorimetric algorithm based on longitudinal and transverse shower shape quantities is used to identify electrons in the E_{p} = 460 and 575 GeV data sample. The efficiency of this selection can be estimated using a simple track based electron finder which searches for an isolated high p_{T} track associated to an electromagnetic energy deposition. The efficiency is well described by the simulation and the uncertainty of 0.2% (0.5%) is assigned in the nominal (high y) analysis at z_{imp} < 20 cm. For z_{imp} > 20 cm the uncertainty is taken to be 1% due to the lack of statistics in this region selected by the track based algorithm.

Extra Background Suppression: The uncertainty on the efficiency of the D_{ele} requirement has been studied with J/ψ → ee decays in data and is well described by the simulation. A variation of ±0.04 around the nominal D_{ele} cut value accommodates any residual difference between data and simulation. This variation leads to a cross section uncertainty of up to 2% at highest y.

High y Background Subtraction: In the high y analysis the photoproduction background asymmetry is measured in the E_{p} = 460 and 575 GeV data, and found to be consistent with the determination using the E_{p} = 920 GeV data [5,46], albeit with reduced precision. Therefore the asymmetry is taken from the analysis of the HERA II data at E_{p} = 920 GeV and the associated uncertainty is increased to 0.08. The resulting uncertainty on the measured cross sections is at most 2.7% at y = 0.85 and Q^{2} = 35 GeV^{2}.

QED Radiative Corrections: An error on the cross sections originating from the QED radiative corrections is taken into account. This is determined by comparing the predicted radiative corrections from the programs Heracles (as implemented in Djangoh), Hector, and Eprc. The radiative corrections due to the exchange of two or more photons between the lepton and the quark lines, which are not included in Djangoh, vary with the polarity of the lepton beam. This variation, estimated using Eprc, is found to be small compared to the quoted errors and is neglected [46].

Model Uncertainty of Acceptance Correction: The MC simulation is used to determine the acceptance correction to the data and relies on a specific choice of PDFs. The assigned uncertainty is listed in table 2.

Luminosity: The integrated luminosity is measured using the Bethe-Heitler process ep → epγ with an uncertainty of 4%, of which 0.5% is from the uncertainty in the theoretical calculation of this QED process.

5 Results

5.1 Double Differential Cross Sections

The reduced cross sections σ̃_{NC}(x, Q^{2}) for P_{e} = 0 are measured in the kinematic range 35 ≤ Q^{2} ≤ 800 GeV^{2} and 0.00065 ≤ x ≤ 0.65 at two different centre-of-mass energies and are referred to as the LAr data. The data are presented in tables 3 – 4 and shown in figure 4. The figure also includes previously published H1 data [3] in the Q^{2} range of the new data reported here, and are referred to as the SpaCal data. The published 920 GeV e^{+}p LAr data [5] are scaled by a normalisation factor of 1.018 [57]. This correction factor arises from an error in the Compton generator used in the determination of the integrated luminosity of the HERA-II 920 GeV data set. The new LAr data provide additional low x measurements for Q^{2} ≥ 35 GeV^{2} (from the E_{p} = 460 and 575 GeV data sets). The data are compared to the H1PDF 2012 fit [5] which provides a good description of the data.

5.2 Measurement of F F F_{L L L}

According to equation 1 it is straightforward to determine F_{L} by a linear fit as a function of y^{2}/(1 + (1 − y)^{2}) to the reduced cross section measured at given values of x and Q^{2} but at different centre-of-mass energies. An example of this procedure is shown in figure 5 for six different values of x at Q^{2} = 60 GeV^{2}. This method however does not optimally account for correlations across all measurements, and therefore an alternative procedure is applied.

The structure functions F_{L} and F_{2} are simultaneously determined from the cross section measurements at E_{p} = 460, 575 and 920 GeV using a χ^{2} minimisation technique as employed in [3]. In this approach the values of F_{L} and F_{2} at each measured x, Q^{2} point are free parameters of the fit. For Q^{2} ≤ 800 GeV^{2} the influence of the xF_{3} structure function is predicted to be small and is neglected. In addition, a set of nuisance parameters b_{j} for each correlated systematic error source j is introduced. The minimisation is performed using the new measurements presented here as well as previously published data from H1 [3, 5]. The χ^{2} function for the minimisation is

χ^{2} (F_{L,i}, F_{2,i}, b_{j}) = ∑_{i} \frac{[(F_{2,i} − f(y_{i})F_{L,i}) − ∑_{j} Γ_{i,j}b_{j} − μ_{i}]^{2}}{Δ_{i}^{2}} + ∑_{j} b_{j}^{2} ,  (6)

where f(y) = y^{2}/(1+(1−y)^{2}) and μ_{i} is the measured reduced cross section at an x, Q^{2} point i with a combined statistical and uncorrelated systematic uncertainty Δ_{i} = \sqrt{(Δ_{i,stat}^{2} + Δ_{i,syst}^{2})}. The effect of correlated error sources j on the cross section measurements is given by the systematic error matrix Γ_{i,j}. The correlations of systematic uncertainties between the SpaCal data sets at different energies are taken from [3]. The systematic uncertainties of the LAr measurements are taken to be 100% correlated among the 460 , 575 and 920 GeV data sets. There is no correlation between LAr and SpaCal measurements except for a common integrated luminosity normalisation of the LAr and SpaCal data at E_{p} = 460 and 575 GeV. For low y ≤ 0.35, the coefficient f(y) is small compared to unity and thus F_{L} can not be accurately measured. To avoid unphysical values for F_{L} in this kinematic region the χ^{2} function is modified by adding an extra prior [3]. The minimisation of the χ^{2} function with respect to these variables leads to a system of linear equations which is solved analytically. This technique is identical to the linear fit discussed above when considering a single x, Q^{2} bin and neglecting correlations between the cross section measurements.

The χ^{2} per degree of freedom is found to be 184/210. The systematic sources include normalisation uncertainties for the SpaCal and LAr data sets for E_{p} = 460, 575 and 920 GeV data which are all shifted in the minimisation procedure by less than one standard deviation with the exception of the LAr 920 GeV data which are re-normalised by +3.4%, or 1.2 standard deviations. All other sources of uncertainty including those related to calibration scales, noise subtractions, background estimates and polar angle measurements are shifted by typically less than 0.3 and never more than 0.8 standard deviations.

The measured structure functions are given in table 5 over the full range in Q^{2} from 1.5 to 800 GeV^{2}. Only measurements of F_{L} with a total uncertainty less than 0.3 for Q^{2} ≤ 25 GeV^{2}, or total uncertainty less than 0.4 for Q^{2} ≥ 35 GeV^{2} are considered. The table also includes the correlation coefficient ρ between the F_{2} and F_{L} values. In figures 6 and 7 the measured structure functions F_{2} and F_{L} are shown in the regions 2 ≤ Q^{2} ≤ 25 GeV^{2} and Q^{2} ≥ 35 GeV^{2} respectively. The new data reported here, in which the scattered electron is recorded in the LAr calorimeter, provide small additional constraints on the F_{L} measurement for 1.5 ≤ Q^{2} ≤ 25 GeV^{2} by means of correlations in the systematic uncertainties. The SpaCal and LAr data are used together for 35 ≤ Q^{2} ≤ 90 GeV^{2} . For Q^{2} ≥ 120 GeV^{2} F_{L} is determined exclusively from the LAr cross section measurements. Therefore these data supersede the previous measurements of F_{2} and F_{L} in [2,3]. For precision analyses of H1 data it is recommended to use the published tables of the reduced differential cross sections given in tables 3 to 4 and the full breakdown of systematic uncertainties instead of the derived quantities F_{2} or F_{L}.

This measurement of F_{L} and F_{2} at high y constitutes a model independent method with no assumptions made on the values of the structure functions. Within uncertainties the F_{L} structure function is observed to be positive everywhere and approximately equal to 20% of F_{2}. Also shown are the F_{L} and F_{2} measurements from the ZEUS collaboration [4] which agree with the H1 data. The ZEUS data are moved to the Q^{2} values of the H1 measurements using the H1PDF 2012 NLO QCD fit. This QCD fit is able to provide a good description of both measurements of F_{L} and F_{2} across the full Q^{2} range.

In order to reduce the experimental uncertainties the F_{L} measurements are combined at each Q^{2} value. Furthermore the highest Q^{2} bins are also averaged to achieve an approximately uniform experimental precision over the full kinematic range of the measurement. The Q^{2} averaging is performed for Q^{2} = 300 and 400 GeV^{2}, and for the Q^{2} = 500, 600 and 800 GeV^{2} values. The resulting data are given in table 6 and shown in figure 8 where the average x for each Q^{2} is provided on the upper scale of the figure. The data are compared to a suite of QCD predictions at NNLO: HERAPDF1.5 [58], CT10 [59], ABM11 [60], MSTW2008 [61], JR09 [62] and NNPDF2.3 [63]. In all cases the perturbative calculations provide a reasonable description of the data.

A similar average of F_{L} measurements over x has already been performed in [3] for Q^{2} < 45 GeV^{2}. A small problem in [3] has been identified in the averaging procedure which lead to underestimated correlated systematic uncertainties which has been corrected in the measurements reported here. Therefore the data presented in tables 6 supersedes the corresponding table from [3].

The cross section ratio R of longitudinally to transversely polarised virtual photons is related to the structure functions F_{2} and F_{L} as

R = \frac{σ_{L}}{σ_{T}} = \frac{F_{L}}{F_{2} − F_{L}} .  (7)

This ratio has previously been observed to be approximately constant for 3.5 ≤ Q^{2} ≤ 45 GeV^{2} [3].

The values of R as a function of Q^{2} are determined by minimising the χ^{2} function of equation 6 in which F_{L} is replaced by

F_{L} = \frac{R}{1 + R}F_{2}

assuming the value of R is constant as a function of x for a given Q^{2}. The minimum is found numerically in this case, using the MINUIT package [64]. The asymmetric uncertainties are determined using a MC method in which the mean squared deviation from the measured value of R is used to define the asymmetric uncertainties. The resulting value of R(Q^{2}) is shown in figure 9. The measurements are compared to the prediction of the HERAPDF1.5 NNLO for \sqrt{s} = 225 GeV and y = 0.7. The expected small variation of R in the region of x in which the data are sensitive to this quantity is also shown.

The data are found to be consistent with a constant value across the entire Q^{2} range shown. The fit is repeated by assuming that R is constant over the full Q^{2} range. This yields a value of R = 0.23±0.04 with χ^{2}/ndf =314/367 which agrees well with the value obtained previously [3] using only data up to Q^{2} = 45 GeV^{2}, and with the ZEUS data.

At NLO and NNLO QCD, analyses of DIS data constrain the gluon density through the precision measurements of F_{2} and the reduced NC cross sections through their scaling violations. Since at order α_{s} the gluon density is related to F_{L} via the approximate relation [65, 66]

xg(x, Q^{2}) ≈ 1.77\frac{3π}{2α_{S}(Q^{2})}F_{L}(x, Q^{2}) ,  (8)

the direct measure of F_{L} can be used to demonstrate its sensitivity to the gluon density by comparing the gluon obtained from the F_{L} measurements to the predicted gluon density obtained from a NLO QCD fit to DIS data.In figure 10 the gluon density extracted according to equation 8 is compared to the prediction from the gluon density determined in the NLO HERA-PDF1.5 QCD fit. The uncertainty bands of the theoretical prediction includes experimental, model and parametrisation uncertainties of the fit procedure. In order to judge on the goodness of the approximation, the gluon density as obtained by applying equation 8 to the F_{L} prediction based on the NLO HERAPDF1.5 QCD fit is also shown. A reasonable agreement between the gluon density as extracted from the direct measurement of F_{L} based on the approximate relation with the gluon derived indirectly from scaling violations is observed.

6 Conclusions

The unpolarised neutral current inclusive DIS cross section for ep interactions are measured at two centre-of-mass energies of \sqrt{s} = 225 and 252 GeV with integrated luminosities of 11.8 pb^{−1} and 5.4 pb^{−1} respectively. The measurements are performed up to the highest accessible inelasticity of y = 0.85 where the contribution of the F_{L} structure function to the reduced cross section is sizeable. The data are used together with previously published measurements at \sqrt{s} = 319 GeV (E_{p} = 920 GeV) to simultaneously extract the F_{L} and F_{2} structure functions in a model independent way. The new data extend previous measurements of F_{L} up to Q^{2} = 800 GeV^{2} and supersede previous H1 data. Predictions of different perturbative QCD calculations at NNLO are compared to data. Good agreement is observed between the measurements and the theoretical calculations. The ratio R of the longitudinally to transversely polarised virtual photon cross section is consistent with being constant over the kinematic range of the data, and is determined to be = 0.23 ± 0.04. The F_{L} measurements are used to perform a gluon density extraction based on a NLO approximation which is found to agree reasonably well with the gluon determined from scaling violations.

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 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. We would also like to thank the members of the MSTW, CT, ABM, JR, NNPDF and HERAPDF collaborations for their help in producing theoretical predictions of F_{L} shown in figure 8.

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Q^{2}  x  y  σ̃_{NC}  δ_{tot}  δ_{stat}  δ_{unc}  δ_{unc}^{E}  δ_{unc}^{h}  δ_{cor}  δ_{cor}^{E+}  δ_{cor}^{θ+}  δ_{cor}^{h+}  δ_{cor}^{N+}  δ_{cor}^{S+}  δ_{cor}^{D+}
(GeV^{2})  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)
35  8.10 × 10^{−4}  0.850  1.343  6.5  4.5  3.8  0.6  2.8  2.8  −0.3  −0.4  0.2  0.7  2.3  1.2
45  1.04 × 10^{−3}  0.850  1.173  6.3  4.7  3.4  0.4  2.4  2.4  −0.1  −0.5  0.1  0.6  2.1  1.0
45  1.18 × 10^{−3}  0.750  1.187  5.7  5.1  2.2  0.6  0.8  1.3  0.2  −0.5  0.0  0.3  0.6  1.0
60  1.39 × 10^{−3}  0.850  1.190  6.2  5.0  3.0  0.2  2.0  2.0  −0.1  −0.3  0.1  0.5  1.7  0.8
60  1.58 × 10^{−3}  0.750  1.117  4.7  4.0  2.0  0.5  0.6  1.4  −0.2  −0.6  0.0  0.2  0.6  1.0
90  2.09 × 10^{−3}  0.850  1.269  6.3  5.3  2.9  0.3  1.8  1.8  −0.2  −0.4  0.1  0.5  1.4  0.9
90  2.36 × 10^{−3}  0.750  1.193  4.6  4.1  1.9  0.3  0.5  1.1  −0.2  −0.5  0.1  0.2  0.3  0.9
90  2.73 × 10^{−3}  0.650  1.156  4.2  3.8  1.8  0.4  0.2  0.8  −0.2  −0.6  0.0  0.2  0.3  0.4
120  2.78 × 10^{−3}  0.850  1.249  6.8  6.1  2.7  0.1  1.5  1.6  0.0  −0.4  0.1  0.4  0.9  1.2
120  3.15 × 10^{−3}  0.750  1.099  5.3  4.8  1.9  0.4  0.4  0.9  −0.2  −0.4  0.0  0.2  0.4  0.6
120  3.63 × 10^{−3}  0.650  1.052  4.7  4.3  1.9  0.6  0.2  0.6  −0.2  −0.4  0.0  0.2  0.2  0.2
120  4.82 × 10^{−3}  0.490  1.041  3.3  2.7  1.8  0.6  0.1  0.8  −0.3  −0.7  0.0  0.2  0.1  0.0
150  3.47 × 10^{−3}  0.850  1.230  7.8  7.1  2.6  0.4  1.4  1.9  −0.2  −0.3  0.1  0.4  0.8  1.6
150  3.94 × 10^{−3}  0.750  1.024  6.1  5.8  1.9  0.4  0.3  0.8  −0.2  −0.4  0.0  0.2  0.2  0.6
150  4.54 × 10^{−3}  0.650  1.010  5.4  5.0  2.0  0.9  0.1  0.5  −0.2  −0.4  0.0  0.2  0.2  0.1
150  6.03 × 10^{−3}  0.490  1.060  3.2  2.5  1.8  0.5  0.1  0.7  −0.3  −0.6  0.0  0.2  0.1  0.0
150  8.00 × 10^{−3}  0.369  0.9774  3.0  2.6  1.4  0.6  0.0  0.9  −0.4  −0.8  0.0  0.1  0.0  0.0
150  1.30 × 10^{−2}  0.227  0.8384  3.8  3.3  1.5  1.2  0.0  1.1  −0.8  −0.7  0.0  0.0  0.0  0.0
150  2.00 × 10^{−2}  0.148  0.7006  5.2  4.5  2.2  1.7  1.0  1.7  −1.0  −1.0  −0.4  −0.8  0.0  0.0
200  4.63 × 10^{−3}  0.850  1.117  9.6  9.1  2.5  0.3  1.1  2.1  −0.1  −0.4  0.0  0.3  0.6  1.9
200  5.25 × 10^{−3}  0.750  1.011  8.1  7.7  1.9  0.3  0.3  1.1  0.2  −0.4  0.0  0.2  0.1  1.0
200  6.06 × 10^{−3}  0.650  0.9997  6.8  6.5  2.0  0.9  0.2  0.6  −0.1  −0.5  0.1  0.2  0.2  0.0
200  8.04 × 10^{−3}  0.490  0.9567  3.8  3.4  1.7  0.3  0.1  0.6  −0.3  −0.6  0.0  0.2  0.1  0.0
200  1.30 × 10^{−2}  0.303  0.8430  3.4  3.1  1.0  0.6  0.0  0.8  −0.4  −0.7  0.0  0.0  0.0  0.0
200  2.00 × 10^{−2}  0.197  0.6517  4.1  3.5  1.8  1.6  0.0  1.2  −1.0  −0.7  0.0  0.0  0.0  0.0
200  3.20 × 10^{−2}  0.123  0.5275  4.2  4.0  0.9  0.1  0.1  0.6  −0.1  −0.5  −0.1  0.3  0.0  0.0
200  5.00 × 10^{−2}  0.079  0.5297  4.3  4.1  1.2  0.9  0.2  0.7  −0.5  −0.4  −0.1  0.2  0.0  0.0
200  8.00 × 10^{−2}  0.049  0.4587  5.0  4.7  1.3  0.9  0.3  1.0  −0.6  −0.7  −0.2  0.2  0.0  0.0
200  1.30 × 10^{−1}  0.030  0.3610  5.6  5.1  1.9  1.5  0.1  1.6  −0.9  −0.7  −0.1  1.0  0.0  0.0
200  1.80 × 10^{−1}  0.022  0.3201  6.8  5.8  2.3  1.1  1.4  2.6  −0.7  −0.9  −0.4  −2.3  0.0  0.0
200  4.00 × 10^{−1}  0.010  0.1694  13.1  8.2  4.5  0.3  4.0  9.2  0.2  −1.1  −0.6  −9.1  0.0  0.0
250  5.79 × 10^{−3}  0.850  1.049  10.9  10.4  2.5  0.3  0.9  2.2  −0.1  −0.2  0.0  0.3  0.6  2.0
250  6.56 × 10^{−3}  0.750  1.036  9.1  8.8  1.9  0.2  0.3  1.3  −0.2  −0.4  0.1  0.2  0.2  1.1
250  7.57 × 10^{−3}  0.650  0.9480  8.0  7.7  2.0  0.9  0.1  0.5  −0.2  −0.4  0.0  0.2  0.2  0.0
250  1.00 × 10^{−2}  0.490  0.8829  4.3  3.9  1.7  0.3  0.1  0.6  −0.3  −0.5  0.0  0.2  0.1  0.0
250  1.30 × 10^{−2}  0.379  0.8281  4.0  3.7  1.3  0.4  0.0  0.6  −0.4  −0.5  0.0  0.1  0.0  0.0
250  2.00 × 10^{−2}  0.246  0.6799  4.0  3.8  1.0  0.6  0.0  0.8  −0.5  −0.7  0.0  0.0  0.0  0.0
250  3.20 × 10^{−2}  0.154  0.5817  4.4  4.1  1.3  1.0  0.2  0.9  0.4  −0.6  0.1  0.6  0.0  0.0
250  5.00 × 10^{−2}  0.098  0.5025  4.4  4.1  1.3  0.9  0.1  0.9  0.3  −0.6  0.0  0.7  0.0  0.0
250  8.00 × 10^{−2}  0.062  0.4429  4.7  4.4  1.3  1.0  0.1  1.0  0.5  −0.6  −0.1  0.7  0.0  0.0
250  1.30 × 10^{−1}  0.038  0.3750  4.9  4.4  1.4  0.8  0.1  1.6  0.3  −0.5  −0.1  1.5  0.0  0.0
250  1.80 × 10^{−1}  0.027  0.3582  5.1  4.5  2.1  1.3  0.9  1.3  0.6  −0.7  −0.3  −0.8  0.0  0.0
250  4.00 × 10^{−1}  0.012  0.1675  12.6  6.6  4.9  2.7  3.6  9.5  1.6  −1.0  −0.6  −9.3  0.0  0.0
300  6.95 × 10^{−3}  0.850  0.8700  13.8  13.3  2.4  0.2  0.8  2.5  −0.2  −0.2  0.0  0.3  0.8  2.3
300  7.88 × 10^{−3}  0.750  0.8274  11.1  10.9  2.0  0.2  0.3  1.0  −0.2  −0.3  0.0  0.2  0.3  0.9
300  9.09 × 10^{−3}  0.650  0.8411  9.8  9.6  1.9  0.3  0.1  0.5  0.1  −0.4  0.0  0.2  0.2  0.0
300  1.21 × 10^{−2}  0.490  0.9058  4.8  4.5  1.7  0.3  0.1  0.5  −0.3  −0.4  0.0  0.1  0.0  0.0
300  2.00 × 10^{−2}  0.295  0.7296  4.4  4.2  1.0  0.6  0.0  0.8  −0.6  −0.5  0.0  0.0  0.0  0.0
300  3.20 × 10^{−2}  0.185  0.6231  4.7  4.5  0.9  0.3  0.2  0.7  0.3  −0.3  0.1  0.5  0.0  0.0
300  5.00 × 10^{−2}  0.118  0.5210  4.9  4.7  1.1  0.6  0.2  0.9  0.4  −0.5  0.1  0.6  0.0  0.0
300  8.00 × 10^{−2}  0.074  0.4584  5.2  4.9  1.4  1.0  0.2  1.0  0.6  −0.6  −0.1  0.5  0.0  0.0
300  1.30 × 10^{−1}  0.045  0.3695  5.5  5.1  1.5  1.0  0.2  1.6  0.5  −0.6  0.0  1.4  0.0  0.0
300  1.80 × 10^{−1}  0.033  0.3330  5.8  5.2  2.2  1.5  0.9  1.2  0.9  −0.8  −0.3  −0.3  0.0  0.0
300  4.00 × 10^{−1}  0.015  0.1567  13.0  7.7  5.2  3.1  3.6  9.1  2.1  −1.3  −0.6  −8.8  0.0  0.0

Table 3: The NC e^{+}p reduced cross section σ̃_{NC}(x, Q^{2}) for E_{p} = 460 GeV and P_{e} = 0 with total (δ_{tot}), statistical (δ_{stat}), total uncorrelated systematic (δ_{unc}) errors and two of its contributions from the electron energy error (δ_{unc}^{E}) and the hadronic energy error (δ_{unc}^{h}). The effect of the other uncorrelated systematic errors is included in δ_{unc}. In addition the correlated systematic (δ_{cor}) and its contributions from a positive variation of one standard deviation of the electron energy error (δ_{cor}^{E^{+}}), of the polar electron angle error (δ_{cor}^{θ^{+}}), of the hadronic energy error (δ_{cor}^{h^{+}}), of the error due to noise subtraction (δ_{cor}^{N^{+}}), of the error due to background subtraction charge asymmetry (δ_{cor}^{S^{+}}) and of the error due to variation of the cut value on the electron discriminator D_{ele} (δ_{cor}^{D^{+}}) are given. The overall normalisation uncertainty of 4% is not included in the errors.

Q^{2}  x  y  σ̃_{NC}  δ_{tot}  δ_{stat}  δ_{unc}  δ_{unc}^{E}  δ_{unc}^{h}  δ_{cor}  δ_{cor}^{E+}  δ_{cor}^{θ+}  δ_{cor}^{h+}  δ_{cor}^{N+}  δ_{cor}^{S+}  δ_{cor}^{D+}
(GeV^{2})  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)
400  9.27 × 10^{−3}  0.850  1.025  13.8  13.3  2.4  0.6  0.6  2.5  −0.2  −0.3  0.0  0.2  0.6  2.4
400  1.05 × 10^{−2}  0.750  1.074  10.4  10.1  2.2  1.0  0.3  0.7  0.2  −0.3  0.1  0.2  0.1  0.6
400  1.21 × 10^{−2}  0.650  0.9263  10.0  9.8  1.9  0.2  0.1  0.4  −0.2  −0.3  0.0  0.2  0.2  0.0
400  1.61 × 10^{−2}  0.490  0.8145  5.7  5.4  1.7  0.2  0.0  0.5  −0.2  −0.5  0.0  0.1  0.0  0.0
400  3.20 × 10^{−2}  0.246  0.6305  5.2  5.0  1.0  0.6  0.0  0.8  −0.6  −0.6  0.0  0.0  0.0  0.0
400  5.00 × 10^{−2}  0.157  0.5686  5.4  5.2  1.0  0.6  0.2  0.9  0.5  −0.5  −0.1  0.4  0.0  0.0
400  8.00 × 10^{−2}  0.098  0.4493  5.8  5.7  1.0  0.4  0.1  0.7  0.4  −0.4  0.1  0.5  0.0  0.0
400  1.30 × 10^{−1}  0.061  0.4300  5.6  5.3  1.2  0.4  0.1  1.2  0.4  −0.4  −0.1  1.1  0.0  0.0
400  1.80 × 10^{−1}  0.044  0.3375  6.2  5.8  1.7  0.8  0.7  0.9  0.7  −0.6  −0.2  −0.2  0.0  0.0
400  4.00 × 10^{−1}  0.020  0.1494  13.1  8.7  4.6  1.9  3.7  8.6  1.9  −0.9  −0.7  −8.3  0.0  0.0
500  1.16 × 10^{−2}  0.850  1.002  15.0  14.6  2.4  0.1  0.4  2.2  0.2  −0.2  0.0  0.2  0.0  2.2
500  1.31 × 10^{−2}  0.750  0.7577  13.8  13.6  2.1  0.6  0.2  0.5  −0.3  −0.3  0.0  0.2  0.1  0.1
500  1.51 × 10^{−2}  0.650  0.6938  12.4  12.2  1.9  0.3  0.1  0.4  −0.2  −0.4  0.1  0.2  0.0  0.0
500  2.01 × 10^{−2}  0.490  0.7395  6.7  6.5  1.7  0.1  0.0  0.4  −0.1  −0.4  0.0  0.1  0.0  0.0
500  3.20 × 10^{−2}  0.308  0.6559  6.1  6.0  1.0  0.3  0.0  0.6  −0.3  −0.5  0.0  0.0  0.0  0.0
500  5.00 × 10^{−2}  0.197  0.6106  6.1  5.9  1.3  0.9  0.0  1.1  −0.9  −0.6  0.0  0.0  0.0  0.0
500  8.00 × 10^{−2}  0.123  0.4712  6.5  6.3  1.0  0.5  0.0  0.8  0.5  −0.4  0.0  0.4  0.0  0.0
500  1.30 × 10^{−1}  0.076  0.4112  7.7  7.5  1.4  0.6  0.1  1.2  0.6  −0.5  0.0  0.9  0.0  0.0
500  1.80 × 10^{−1}  0.055  0.3045  8.7  8.4  1.5  0.6  0.1  1.4  0.6  −0.4  −0.1  1.1  0.0  0.0
500  2.50 × 10^{−1}  0.039  0.2759  8.5  8.3  1.9  0.9  0.8  1.2  0.9  −0.6  −0.2  −0.3  0.0  0.0
500  4.00 × 10^{−1}  0.025  0.1311  13.7  11.8  4.1  1.9  3.1  5.8  1.8  −0.7  −0.7  −5.3  0.0  0.0
500  6.50 × 10^{−1}  0.015  0.01698  27.9  23.0  7.0  2.8  5.9  14.3  2.8  −1.4  −1.0  −13.9  0.0  0.0
650  1.51 × 10^{−2}  0.850  0.8058  19.6  19.4  2.7  0.5  0.5  1.1  −0.1  0.2  0.1  0.3  0.7  0.7
650  1.71 × 10^{−2}  0.750  0.9192  14.0  13.9  2.0  0.1  0.2  0.4  0.1  −0.3  0.0  0.2  0.2  0.0
650  1.97 × 10^{−2}  0.650  0.9125  12.1  12.0  1.9  0.0  0.1  0.4  −0.1  −0.4  0.0  0.1  0.1  0.0
650  2.61 × 10^{−2}  0.490  0.6085  8.0  7.8  1.8  0.5  0.0  0.5  −0.2  −0.4  0.0  0.1  0.0  0.0
650  5.00 × 10^{−2}  0.256  0.4952  7.9  7.8  1.1  0.6  0.0  0.8  −0.6  −0.6  0.0  0.0  0.0  0.0
650  8.00 × 10^{−2}  0.160  0.4515  7.9  7.8  1.0  0.4  0.2  0.8  0.5  −0.4  0.1  0.5  0.0  0.0
650  1.30 × 10^{−1}  0.098  0.3732  9.5  9.3  1.4  0.6  0.2  0.8  0.6  −0.4  −0.1  0.3  0.0  0.0
650  1.80 × 10^{−1}  0.071  0.3397  9.7  9.5  1.5  0.5  0.2  1.1  0.6  −0.5  0.0  0.9  0.0  0.0
650  2.50 × 10^{−1}  0.051  0.2520  10.3  10.1  1.7  0.7  0.5  0.9  0.7  −0.3  0.1  0.5  0.0  0.0
650  4.00 × 10^{−1}  0.032  0.1915  12.8  11.2  3.9  1.9  2.7  4.8  1.9  −0.8  −0.7  −4.3  0.0  0.0
650  6.50 × 10^{−1}  0.020  0.02382  27.6  22.4  7.8  3.7  6.4  14.0  3.5  −1.1  −1.2  −13.5  0.0  0.0
800  1.85 × 10^{−2}  0.850  0.2872  37.1  36.9  3.9  1.1  0.4  0.5  0.2  −0.3  0.0  0.2  0.3  0.1
800  2.10 × 10^{−2}  0.750  0.6634  19.2  19.0  2.3  0.2  0.2  0.3  −0.1  −0.2  0.1  0.2  0.0  0.0
800  2.42 × 10^{−2}  0.650  0.6620  16.0  15.9  2.1  0.4  0.1  0.4  −0.3  −0.2  0.0  0.1  0.0  0.0
800  3.21 × 10^{−2}  0.490  0.6172  8.8  8.6  1.8  0.6  0.0  0.4  −0.3  −0.3  0.0  0.1  0.0  0.0
800  5.00 × 10^{−2}  0.315  0.4847  9.1  9.0  1.2  0.7  0.0  0.6  −0.5  −0.3  0.0  0.0  0.0  0.0
800  8.00 × 10^{−2}  0.197  0.4527  9.3  9.1  1.3  0.8  0.0  0.7  −0.5  −0.6  0.0  0.0  0.0  0.0
800  1.30 × 10^{−1}  0.121  0.3868  10.8  10.6  1.4  0.5  0.3  0.9  0.7  −0.4  −0.2  0.4  0.0  0.0
800  1.80 × 10^{−1}  0.087  0.3642  11.0  10.9  1.5  0.2  0.1  0.8  0.4  −0.3  0.0  0.6  0.0  0.0
800  2.50 × 10^{−1}  0.063  0.2749  11.7  11.6  1.7  0.6  0.6  1.0  0.8  −0.4  −0.2  0.5  0.0  0.0
800  4.00 × 10^{−1}  0.039  0.1262  16.7  15.8  3.5  1.5  2.3  3.9  1.5  −0.4  −0.6  −3.5  0.0  0.0
800  6.50 × 10^{−1}  0.024  0.01953  31.8  28.9  7.1  2.9  5.9  11.4  2.8  −0.7  −1.0  −11.0  0.0  0.0

Table 3: continued.

Q^{2}  x  y  σ̃_{NC}  δ_{tot}  δ_{stat}  δ_{unc}  δ_{unc}^{E}  δ_{unc}^{h}  δ_{cor}  δ_{cor}^{E+}  δ_{cor}^{θ+}  δ_{cor}^{h+}  δ_{cor}^{N+}  δ_{cor}^{S+}  δ_{cor}^{D+}
(GeV^{2})  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)
35  6.50 × 10^{−4}  0.848  1.303  8.6  7.1  3.8  0.5  2.7  3.1  −0.2  −0.5  0.2  0.6  2.7  1.2
45  8.40 × 10^{−4}  0.848  1.413  7.2  6.0  3.4  0.4  2.2  2.0  −0.2  −0.4  0.1  0.5  1.6  1.1
45  9.30 × 10^{−4}  0.760  1.235  8.2  7.7  2.6  0.5  0.7  1.4  −0.2  −0.6  0.1  0.3  0.7  0.9
60  1.11 × 10^{−3}  0.848  1.259  8.0  7.1  3.2  0.3  2.0  1.8  −0.1  −0.4  0.1  0.5  1.5  0.8
60  1.24 × 10^{−3}  0.760  1.411  6.4  5.8  2.3  0.7  0.6  1.3  −0.4  −0.5  0.0  0.2  0.6  1.0
60  1.39 × 10^{−3}  0.680  1.268  7.5  7.2  2.0  0.5  0.2  1.1  −0.2  −0.6  0.0  0.2  0.5  0.7
90  1.67 × 10^{−3}  0.848  1.310  8.6  7.9  2.9  0.3  1.7  1.7  −0.2  −0.4  0.1  0.4  1.4  0.8
90  1.86 × 10^{−3}  0.760  1.326  6.9  6.5  2.1  0.3  0.5  1.1  −0.1  −0.5  0.0  0.3  0.3  0.9
90  2.09 × 10^{−3}  0.680  1.316  6.2  5.8  1.9  0.5  0.2  1.0  −0.2  −0.6  0.0  0.2  0.2  0.7
90  2.36 × 10^{−3}  0.600  1.342  6.4  6.0  2.0  0.8  0.1  0.8  −0.2  −0.8  0.0  0.2  0.1  0.0
120  2.23 × 10^{−3}  0.848  1.374  9.0  8.4  2.7  0.3  1.5  1.4  −0.2  −0.4  0.1  0.4  0.6  1.1
120  2.49 × 10^{−3}  0.760  1.173  8.0  7.7  2.0  0.6  0.4  0.8  −0.3  −0.3  0.0  0.2  0.3  0.6
120  2.78 × 10^{−3}  0.680  1.161  7.2  6.9  1.9  0.4  0.2  0.6  0.1  −0.4  0.0  0.2  0.2  0.3
120  3.15 × 10^{−3}  0.600  1.115  6.8  6.5  1.8  0.4  0.1  0.7  −0.3  −0.6  0.0  0.2  0.3  0.0
120  3.63 × 10^{−3}  0.520  1.185  6.0  5.7  1.8  0.5  0.1  0.8  −0.4  −0.7  0.0  0.2  0.1  0.0
120  4.82 × 10^{−3}  0.392  1.074  5.5  5.1  1.6  0.5  0.0  1.0  −0.3  −0.9  0.0  0.2  0.0  0.0
150  2.79 × 10^{−3}  0.848  1.291  10.8  10.3  2.6  0.3  1.2  1.8  −0.1  −0.3  0.1  0.4  0.8  1.5
150  3.11 × 10^{−3}  0.760  1.171  9.8  9.5  2.0  0.3  0.4  1.0  −0.2  −0.5  0.1  0.3  0.5  0.6
150  3.47 × 10^{−3}  0.680  1.324  7.9  7.6  2.3  1.3  0.2  0.6  −0.1  −0.5  0.0  0.2  0.3  0.2
150  3.94 × 10^{−3}  0.600  1.244  7.2  6.9  1.8  0.2  0.1  0.6  −0.2  −0.5  0.0  0.1  0.2  0.0
150  4.54 × 10^{−3}  0.520  1.041  7.1  6.8  1.8  0.3  0.1  0.7  −0.3  −0.6  0.0  0.2  0.2  0.0
150  6.03 × 10^{−3}  0.392  1.020  4.0  3.6  1.4  0.7  0.0  0.8  −0.5  −0.6  0.0  0.1  0.0  0.0
150  8.00 × 10^{−3}  0.295  0.9700  4.2  3.9  1.1  0.8  0.0  0.9  −0.5  −0.8  0.0  0.0  0.0  0.0
150  1.30 × 10^{−2}  0.182  0.8609  5.7  5.2  1.8  1.4  0.9  1.2  −0.9  −0.6  −0.2  −0.5  0.0  0.0
150  2.00 × 10^{−2}  0.118  0.7980  7.8  6.9  2.8  2.5  1.0  2.0  −1.6  −1.0  −0.3  −0.8  0.0  0.0
200  3.72 × 10^{−3}  0.848  1.296  13.3  12.9  2.6  0.1  1.2  2.0  −0.2  −0.3  0.1  0.3  0.9  1.7
200  4.15 × 10^{−3}  0.760  1.288  11.8  11.5  2.1  0.2  0.3  1.2  0.2  −0.5  0.0  0.2  0.1  1.0
200  4.63 × 10^{−3}  0.680  1.051  11.3  11.1  2.2  1.1  0.2  0.5  0.2  −0.3  0.0  0.2  0.3  0.2
200  5.25 × 10^{−3}  0.600  1.169  9.1  8.9  1.8  0.2  0.1  0.5  −0.2  −0.4  0.0  0.2  0.1  0.0
200  6.06 × 10^{−3}  0.520  1.110  8.3  8.1  1.8  0.2  0.1  0.5  −0.2  −0.4  0.0  0.2  0.1  0.0
200  8.04 × 10^{−3}  0.392  0.9625  4.8  4.6  1.3  0.4  0.0  0.8  −0.3  −0.7  0.0  0.1  0.1  0.0
200  1.30 × 10^{−2}  0.242  0.8743  4.7  4.4  1.3  1.0  0.0  1.0  −0.6  −0.7  0.0  0.0  0.0  0.0
200  2.00 × 10^{−2}  0.157  0.7573  5.1  4.9  0.9  0.3  0.0  0.8  0.2  −0.6  −0.1  0.5  0.0  0.0
200  3.20 × 10^{−2}  0.098  0.6151  5.6  5.5  1.0  0.5  0.0  0.6  −0.3  −0.5  −0.1  0.3  0.0  0.0
200  5.00 × 10^{−2}  0.063  0.5041  6.5  6.4  1.2  0.6  0.5  0.8  −0.4  −0.6  −0.2  −0.1  0.0  0.0
200  8.00 × 10^{−2}  0.039  0.4211  7.7  7.3  1.7  1.4  0.2  1.5  −0.9  −0.6  −0.1  1.1  0.0  0.0
200  1.30 × 10^{−1}  0.024  0.3857  7.6  7.2  1.9  1.5  0.2  1.5  −0.7  −0.9  −0.2  0.9  0.0  0.0
200  1.80 × 10^{−1}  0.018  0.3034  10.4  9.3  2.6  0.6  2.1  3.7  −0.5  −0.8  −0.3  −3.6  0.0  0.0
200  4.00 × 10^{−1}  0.008  0.1910  13.2  11.0  3.7  0.7  3.1  6.3  −0.5  −1.2  −0.4  −6.1  0.0  0.0
250  4.64 × 10^{−3}  0.848  0.8545  19.8  19.5  2.5  0.2  1.0  2.5  0.1  −0.3  0.1  0.3  1.4  2.0
250  5.18 × 10^{−3}  0.760  1.080  14.3  14.1  2.2  0.2  0.3  1.5  0.1  −0.4  0.1  0.2  0.0  1.4
250  5.79 × 10^{−3}  0.680  0.9481  13.9  13.7  2.3  1.2  0.1  0.5  −0.2  −0.3  0.0  0.2  0.2  0.0
250  6.56 × 10^{−3}  0.600  0.9475  11.6  11.5  1.8  0.2  0.1  0.5  −0.1  −0.4  0.0  0.2  0.1  0.0
250  7.57 × 10^{−3}  0.520  1.018  9.8  9.6  1.8  0.2  0.1  0.6  −0.2  −0.5  0.0  0.2  0.1  0.0
250  1.00 × 10^{−2}  0.392  0.9523  5.3  5.1  1.2  0.4  0.0  0.6  −0.3  −0.5  0.0  0.1  0.0  0.0
250  1.30 × 10^{−2}  0.303  0.8513  5.3  5.1  1.1  0.6  0.0  0.8  −0.6  −0.5  0.0  0.0  0.0  0.0
250  2.00 × 10^{−2}  0.197  0.7707  5.5  5.2  1.5  1.2  0.0  1.1  −0.9  −0.6  0.0  0.0  0.0  0.0
250  3.20 × 10^{−2}  0.123  0.6210  5.9  5.7  1.4  1.0  0.3  1.0  0.4  −0.6  0.0  0.6  0.0  0.0
250  5.00 × 10^{−2}  0.079  0.5412  6.1  5.9  1.4  1.0  0.1  0.9  0.5  −0.6  0.0  0.6  0.0  0.0
250  8.00 × 10^{−2}  0.049  0.4602  6.7  6.4  1.3  0.9  0.1  1.3  0.3  −0.5  −0.1  1.2  0.0  0.0
250  1.30 × 10^{−1}  0.030  0.3906  6.6  6.2  1.4  0.7  0.2  1.8  0.3  −0.5  0.0  1.8  0.0  0.0
250  1.80 × 10^{−1}  0.022  0.3514  7.8  6.9  2.5  1.5  1.5  2.5  0.7  −0.6  −0.4  −2.3  0.0  0.0
250  4.00 × 10^{−1}  0.010  0.1556  13.0  10.1  4.2  2.5  2.7  7.1  1.5  −1.0  −0.5  −6.8  0.0  0.0

Table 4: The NC e^{+}p reduced cross section σ̃_{NC}(x, Q^{2}) for E_{p} = 575 GeV and P_{e} = 0 with total (δ_{tot}), statistical (δ_{stat}), total uncorrelated systematic (δ_{unc}) errors and two of its contributions from the electron energy error (δ_{unc}^{E}) and the hadronic energy error (δ_{unc}^{h}). The effect of the other uncorrelated systematic errors is included in δ_{unc}. In addition the correlated systematic (δ_{cor}) and its contributions from a positive variation of one standard deviation of the electron energy error (δ_{cor}^{E^{+}}), of the polar electron angle error (δ_{cor}^{θ^{+}}), of the hadronic energy error (δ_{cor}^{h^{+}}), of the error due to noise subtraction (δ_{cor}^{N^{+}}), of the error due to background subtraction charge asymmetry (δ_{cor}^{S^{+}}) and of the error due to variation of the cut value on the electron discriminator D_{ele} (δ_{cor}^{D^{+}}) are given. The overall normalisation uncertainty of 4% is not included in the errors.

Q^{2}  x  y  σ̃_{NC}  δ_{tot}  δ_{stat}  δ_{unc}  δ_{unc}^{E}  δ_{unc}^{h}  δ_{cor}  δ_{cor}^{E+}  δ_{cor}^{θ+}  δ_{cor}^{h+}  δ_{cor}^{N+}  δ_{cor}^{S+}  δ_{cor}^{D+}
(GeV^{2})  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)  (%)
300  5.57 × 10^{−3}  0.848  1.208  16.0  15.7  2.5  0.6  0.8  2.1  −0.3  −0.3  0.1  0.3  0.2  2.1
300  6.22 × 10^{−3}  0.760  0.8707  18.1  18.0  2.1  0.2  0.3  1.3  −0.2  −0.4  0.1  0.2  0.4  1.1
300  6.95 × 10^{−3}  0.680  0.9694  15.0  14.8  2.1  0.8  0.1  0.5  −0.1  −0.5  0.0  0.2  0.0  0.0
300  7.88 × 10^{−3}  0.600  1.035  12.9  12.7  1.9  0.1  0.1  0.5  0.1  −0.5  0.0  0.2  0.0  0.0
300  9.09 × 10^{−3}  0.520  0.8632  12.1  12.0  1.8  0.3  0.1  0.6  −0.3  −0.5  0.0  0.1  0.0  0.0
300  1.21 × 10^{−2}  0.392  0.9079  6.2  6.0  1.3  0.4  0.0  0.6  −0.4  −0.5  0.0  0.1  0.0  0.0
300  2.00 × 10^{−2}  0.236  0.6653  6.5  6.3  1.0  0.6  0.0  0.9  −0.6  −0.7  0.0  0.0  0.0  0.0
300  3.20 × 10^{−2}  0.148  0.6171  6.7  6.6  1.0  0.5  0.1  0.8  0.3  −0.4  0.0  0.6  0.0  0.0
300  5.00 × 10^{−2}  0.094  0.5364  6.9  6.8  1.2  0.8  0.0  0.9  0.5  −0.6  −0.1  0.5  0.0  0.0
300  8.00 × 10^{−2}  0.059  0.4802  7.4  7.2  1.3  0.9  0.0  1.2  0.5  −0.6  −0.1  0.9  0.0  0.0
300  1.30 × 10^{−1}  0.036  0.3762  7.6  7.2  1.5  1.0  0.1  1.9  0.4  −0.6  −0.1  1.7  0.0  0.0
300  1.80 × 10^{−1}  0.026  0.3190  8.7  8.1  2.5  1.8  1.1  2.0  1.1  −0.9  −0.3  −1.4  0.0  0.0
300  4.00 × 10^{−1}  0.012  0.1469  15.5  12.0  4.8  2.8  3.4  8.6  1.7  −1.1  −0.6  −8.4  0.0  0.0
400  7.43 × 10^{−3}  0.848  0.8123  23.1  22.9  2.6  0.3  0.7  2.3  −0.3  −0.3  0.1  0.3  0.6  2.2
400  8.29 × 10^{−3}  0.760  0.5949  23.1  23.0  2.3  0.6  0.2  0.8  −0.1  −0.3  0.0  0.2  0.0  0.8
400  9.27 × 10^{−3}  0.680  1.013  16.1  16.0  2.0  0.4  0.2  0.4  0.2  −0.2  0.0  0.2  0.2  0.0
400  1.05 × 10^{−2}  0.600  0.8806  15.6  15.5  1.9  0.3  0.1  0.5  −0.2  −0.4  0.0  0.2  0.0  0.0
400  1.21 × 10^{−2}  0.520  0.9991  13.0  12.9  1.9  0.1  0.1  0.5  0.2  −0.4  0.0  0.1  0.0  0.0
400  1.61 × 10^{−2}  0.392  0.8791  7.1  7.0  1.2  0.3  0.0  0.6  −0.3  −0.5  0.0  0.1  0.0  0.0
400  3.20 × 10^{−2}  0.197  0.6501  7.3  7.2  1.1  0.8  0.0  1.0  −0.8  −0.6  0.0  0.0  0.0  0.0
400  5.00 × 10^{−2}  0.126  0.5099  8.0  7.9  1.1  0.6  0.1  1.0  0.6  −0.6  0.0  0.5  0.0  0.0
400  8.00 × 10^{−2}  0.079  0.4452  8.6  8.5  1.1  0.6  0.1  1.1  0.6  −0.6  −0.1  0.7  0.0  0.0
400  1.30 × 10^{−1}  0.049  0.3769  8.5  8.2  1.3  0.4  0.2  1.7  0.4  −0.4  0.0  1.6  0.0  0.0
400  1.80 × 10^{−1}  0.035  0.3421  8.9  8.6  2.0  1.0  1.1  1.6  0.9  −0.6  −0.3  −1.1  0.0  0.0
400  4.00 × 10^{−1}  0.016  0.1488  16.6  13.4  4.6  2.0  3.5  8.6  1.9  −0.9  −0.7  −8.4  0.0  0.0
500  9.29 × 10^{−3}  0.848  0.7285  27.8  27.6  2.7  0.3  0.5  2.1  0.0  −0.1  0.1  0.2  0.4  2.1
500  1.04 × 10^{−2}  0.760  0.7348  22.8  22.7  2.3  1.0  0.3  0.5  −0.2  −0.4  0.1  0.2  0.0  0.1
500  1.16 × 10^{−2}  0.680  1.177  16.2  16.1  2.0  0.1  0.1  0.4  −0.1  −0.3  0.0  0.2  0.0  0.0
500  1.31 × 10^{−2}  0.600  0.8538  17.7  17.6  1.9  0.2  0.1  0.5  −0.2  −0.4  0.0  0.1  0.2  0.0
500  1.51 × 10^{−2}  0.520  1.040  14.2  14.1  1.9  0.4  0.1  0.4  −0.3  −0.3  0.0  0.1  0.0  0.0
500  2.01 × 10^{−2}  0.392  0.7340  9.1  9.0  1.3  0.4  0.0  0.6  −0.4  −0.4  0.0  0.1  0.0  0.0
500  3.20 × 10^{−2}  0.246  0.6891  8.4  8.3  1.1  0.7  0.0  0.8  −0.7  −0.4  0.0  0.0  0.0  0.0
500  5.00 × 10^{−2}  0.157  0.5602  8.9  8.8  1.0  0.3  0.4  0.8  0.3  −0.3  0.1  0.7  0.0  0.0
500  8.00 × 10^{−2}  0.098  0.4454  9.8  9.7  1.0  0.5  0.2  0.7  0.5  −0.3  −0.1  0.3  0.0  0.0
500  1.30 × 10^{−1}  0.061  0.3831  11.5  11.4  1.3  0.5  0.2  1.4  0.5  −0.4  0.1  1.3  0.0  0.0
500  1.80 × 10^{−1}  0.044  0.3467  11.6  11.3  1.5  0.6  0.1  1.6  0.6  −0.5  −0.1  1.4  0.0  0.0
500  2.50 × 10^{−1}  0.032  0.2290  13.8  13.5  2.0  0.8  1.2  1.7  0.8  −0.4  −0.3  −1.4  0.0  0.0
500  4.00 × 10^{−1}  0.020  0.1687  18.1  16.3  4.2  1.8  3.2  6.7  1.8  −0.6  −0.7  −6.3  0.0  0.0
500  6.50 × 10^{−1}  0.012  0.02022  31.5  28.9  5.5  2.3  4.3  11.2  2.3  −0.9  −0.7  −10.9  0.0  0.0
650  1.21 × 10^{−2}  0.848  0.4914  38.7  38.5  3.0  0.4  0.4  1.0  0.2  0.0  0.0  0.2  0.5  0.8
650  1.35 × 10^{−2}  0.760  0.6986  28.3  28.2  2.2  0.3  0.2  0.6  −0.2  −0.2  0.1  0.2  0.4  0.0
650  1.51 × 10^{−2}  0.680  0.6789  25.2  25.1  2.1  0.2  0.1  0.5  0.1  −0.3  0.0  0.2  0.3  0.0
650  1.71 × 10^{−2}  0.600  0.6957  21.4  21.3  2.0  0.3  0.1  0.5  −0.2  −0.4  0.0  0.1  0.0  0.0
650  1.97 × 10^{−2}  0.520  0.4817  22.7  22.6  2.0  0.6  0.1  0.6  −0.4  −0.4  0.0  0.1  0.0  0.0
650  2.61 × 10^{−2}  0.392  0.6348  10.8  10.7  1.3  0.4  0.0  0.5  −0.2  −0.5  0.0  0.1  0.0  0.0
650  5.00 × 10^{−2}  0.205  0.4685  11.7  11.6  1.3  0.9  0.0  1.0  −0.8  −0.6  0.0  0.0  0.0  0.0
650  8.00 × 10^{−2}  0.128  0.4525  11.4  11.3  1.1  0.5  0.0  0.9  0.6  −0.4  −0.1  0.5  0.0  0.0
650  1.30 × 10^{−1}  0.079  0.3975  13.4  13.3  1.4  0.6  0.0  1.2  0.6  −0.5  0.0  0.9  0.0  0.0
650  1.80 × 10^{−1}  0.057  0.3285  14.0  13.9  1.4  0.3  0.2  1.4  0.4  −0.2  0.0  1.4  0.0  0.0
650  2.50 × 10^{−1}  0.041  0.2401  15.5  15.3  2.0  1.0  0.9  1.3  1.0  −0.6  −0.3  −0.6  0.0  0.0
650  4.00 × 10^{−1}  0.026  0.1563  20.4  18.9  4.5  2.1  3.3  6.2  2.2  −0.9  −0.9  −5.7  0.0  0.0
650  6.50 × 10^{−1}  0.016  0.02266  35.8  33.3  6.0  2.3  4.9  11.4  2.2  −0.9  −0.8  −11.1  0.0  0.0
800  1.49 × 10^{−2}  0.848  0.6679  31.9  31.8  3.1  0.9  0.4  0.5  −0.4  −0.2  0.1  0.2  0.0  0.1
800  1.66 × 10^{−2}  0.760  0.4843  38.5  38.4  2.7  0.6  0.2  0.7  0.6  −0.3  0.1  0.3  0.0  0.0
800  1.85 × 10^{−2}  0.680  0.6761  27.1  27.0  2.2  0.2  0.1  0.4  −0.2  −0.3  0.0  0.1  0.0  0.0
800  2.10 × 10^{−2}  0.600  0.6604  24.5  24.4  2.1  0.1  0.1  0.5  −0.1  −0.5  0.0  0.1  0.0  0.0
800  2.42 × 10^{−2}  0.520  0.6435  21.9  21.8  1.9  0.1  0.0  0.6  0.0  −0.6  0.0  0.1  0.0  0.0
800  3.21 × 10^{−2}  0.392  0.4923  13.6  13.5  1.4  0.5  0.0  0.4  −0.3  −0.3  0.0  0.1  0.0  0.0
800  5.00 × 10^{−2}  0.252  0.5837  12.0  11.9  1.2  0.7  0.0  0.8  −0.6  −0.6  0.0  0.0  0.0  0.0
800  8.00 × 10^{−2}  0.157  0.5522  12.0  12.0  1.1  0.4  0.1  0.8  0.6  −0.4  0.1  0.3  0.0  0.0
800  1.30 × 10^{−1}  0.097  0.2926  18.3  18.3  1.3  0.1  0.1  0.7  0.4  −0.2  −0.1  0.6  0.0  0.0
800  1.80 × 10^{−1}  0.070  0.2636  18.7  18.6  1.5  0.5  0.1  1.5  0.7  −0.5  0.0  1.2  0.0  0.0
800  2.50 × 10^{−1}  0.050  0.1811  20.5  20.4  1.9  0.8  0.8  1.3  1.1  −0.5  −0.2  −0.5  0.0  0.0
800  4.00 × 10^{−1}  0.032  0.1614  22.2  21.3  3.8  1.5  2.7  4.9  1.5  −0.4  −0.5  −4.6  0.0  0.0
800  6.50 × 10^{−1}  0.019  0.02134  43.1  40.8  6.7  2.6  5.4  12.1  2.7  −0.6  −1.1  −11.8  0.0  0.0

Table 4: continued.

Q^{2}  x  F_{L}  Δ_{stat}F_{L}  Δ_{uncor}F_{L}  Δ_{cor}F_{L}  Δ_{tot}F_{L}  F_{2}  Δ_{stat}F_{2}  Δ_{uncor}F_{2}  Δ_{cor}F_{2}  Δ_{tot}F_{2}  ρ

(GeV^{2})

1.5  0.279 × 10^{−4}  0.088  0.113  0.186  0.053  0.224  0.732  0.066  0.096  0.028  0.120  0.882
2.0  0.372 × 10^{−4}  0.110  0.069  0.131  0.062  0.160  0.843  0.028  0.051  0.032  0.066  0.855
2.0  0.415 × 10^{−4}  0.437  0.110  0.181  0.071  0.223  0.904  0.039  0.060  0.030  0.078  0.852
2.0  0.464 × 10^{−4}  0.043  0.052  0.104  0.033  0.121  0.740  0.033  0.052  0.009  0.062  0.822
2.5  0.465 × 10^{−4}  0.013  0.057  0.120  0.046  0.141  0.846  0.022  0.045  0.016  0.053  0.856
2.5  0.519 × 10^{−4}  0.103  0.062  0.129  0.042  0.149  0.897  0.023  0.045  0.016  0.053  0.860
2.5  0.580 × 10^{−4}  0.174  0.047  0.090  0.058  0.117  0.889  0.021  0.034  0.028  0.049  0.821
2.5  0.658 × 10^{−4}  0.169  0.043  0.099  0.063  0.125  0.865  0.019  0.035  0.031  0.050  0.840
2.5  0.759 × 10^{−4}  0.413  0.096  0.155  0.079  0.198  0.877  0.024  0.035  0.026  0.050  0.783
3.5  0.651 × 10^{−4}  0.130  0.065  0.135  0.052  0.158  0.973  0.025  0.050  0.022  0.060  0.846
3.5  0.727 × 10^{−4}  0.199  0.061  0.133  0.044  0.152  0.989  0.024  0.047  0.021  0.057  0.850
3.5  0.812 × 10^{−4}  0.253  0.044  0.094  0.041  0.112  0.981  0.019  0.036  0.016  0.044  0.811
3.5  0.921 × 10^{−4}  0.230  0.037  0.099  0.037  0.112  0.968  0.015  0.033  0.014  0.039  0.816
3.5  0.106 × 10^{−3}  0.155  0.049  0.123  0.046  0.141  0.934  0.015  0.032  0.010  0.037  0.797
3.5  0.141 × 10^{−3}  0.665  0.112  0.221  0.123  0.276  0.937  0.011  0.028  0.012  0.032  0.735
5.0  0.931 × 10^{−4}  0.411  0.081  0.162  0.068  0.193  1.149  0.031  0.060  0.031  0.075  0.846
5.0  0.104 × 10^{−3}  0.344  0.065  0.142  0.044  0.163  1.072  0.027  0.052  0.024  0.063  0.859
5.0  0.116 × 10^{−3}  0.258  0.048  0.108  0.049  0.128  1.127  0.021  0.042  0.018  0.050  0.828
5.0  0.131 × 10^{−3}  0.306  0.037  0.109  0.041  0.122  1.082  0.016  0.037  0.017  0.044  0.830
5.0  0.152 × 10^{−3}  0.224  0.044  0.134  0.045  0.148  1.060  0.014  0.034  0.015  0.040  0.834
5.0  0.201 × 10^{−3}  0.533  0.057  0.203  0.084  0.227  1.018  0.008  0.028  0.012  0.032  0.809
6.5  0.121 × 10^{−3}  0.435  0.096  0.179  0.077  0.218  1.215  0.037  0.066  0.027  0.080  0.853
6.5  0.135 × 10^{−3}  0.199  0.071  0.151  0.042  0.172  1.103  0.030  0.055  0.020  0.066  0.862
6.5  0.151 × 10^{−3}  0.137  0.051  0.114  0.054  0.136  1.135  0.023  0.044  0.023  0.055  0.844
6.5  0.171 × 10^{−3}  0.357  0.040  0.119  0.044  0.133  1.158  0.017  0.041  0.020  0.048  0.844
6.5  0.197 × 10^{−3}  0.318  0.044  0.145  0.053  0.161  1.147  0.014  0.038  0.019  0.044  0.855
6.5  0.262 × 10^{−3}  0.188  0.046  0.205  0.090  0.229  1.044  0.007  0.029  0.017  0.034  0.842
8.5  0.158 × 10^{−3}  0.499  0.109  0.195  0.095  0.243  1.352  0.044  0.074  0.033  0.092  0.845
8.5  0.177 × 10^{−3}  0.489  0.089  0.184  0.051  0.210  1.335  0.038  0.067  0.022  0.080  0.862
8.5  0.197 × 10^{−3}  0.271  0.057  0.123  0.058  0.147  1.196  0.027  0.048  0.021  0.059  0.841
8.5  0.224 × 10^{−3}  0.242  0.045  0.125  0.042  0.139  1.158  0.019  0.043  0.017  0.050  0.849
8.5  0.258 × 10^{−3}  −0.123  0.045  0.140  0.051  0.156  1.038  0.015  0.036  0.016  0.042  0.853
8.5  0.342 × 10^{−3}  0.167  0.045  0.216  0.089  0.238  1.095  0.007  0.030  0.017  0.035  0.846
12  0.223 × 10^{−3}  0.094  0.101  0.159  0.084  0.206  1.314  0.039  0.041  0.044  0.072  0.855
12  0.249 × 10^{−3}  0.544  0.098  0.155  0.058  0.193  1.389  0.035  0.035  0.028  0.057  0.835
12  0.278 × 10^{−3}  0.281  0.059  0.098  0.047  0.124  1.310  0.024  0.024  0.019  0.039  0.757
12  0.316 × 10^{−3}  0.248  0.050  0.100  0.038  0.118  1.258  0.019  0.022  0.015  0.033  0.733
12  0.364 × 10^{−3}  0.435  0.055  0.121  0.041  0.139  1.268  0.016  0.022  0.013  0.030  0.728
12  0.483 × 10^{−3}  0.414  0.050  0.162  0.064  0.181  1.189  0.007  0.016  0.012  0.021  0.651
15  0.279 × 10^{−3}  0.510  0.109  0.183  0.085  0.230  1.485  0.040  0.047  0.049  0.079  0.854
15  0.312 × 10^{−3}  0.148  0.088  0.150  0.052  0.181  1.370  0.032  0.035  0.027  0.054  0.834
15  0.348 × 10^{−3}  0.188  0.061  0.099  0.039  0.122  1.329  0.023  0.023  0.017  0.036  0.748
15  0.395 × 10^{−3}  0.419  0.051  0.100  0.036  0.118  1.321  0.017  0.021  0.015  0.031  0.710
15  0.455 × 10^{−3}  0.257  0.062  0.117  0.045  0.140  1.269  0.015  0.018  0.013  0.027  0.693
15  0.604 × 10^{−3}  0.066  0.054  0.157  0.066  0.179  1.180  0.007  0.014  0.012  0.019  0.620
20  0.372 × 10^{−3}  0.216  0.116  0.197  0.065  0.238  1.452  0.041  0.051  0.033  0.073  0.877
20  0.415 × 10^{−3}  0.322  0.092  0.158  0.044  0.188  1.424  0.032  0.037  0.021  0.054  0.837
20  0.464 × 10^{−3}  0.412  0.070  0.108  0.037  0.134  1.396  0.024  0.025  0.015  0.037  0.752
20  0.526 × 10^{−3}  0.358  0.052  0.103  0.037  0.121  1.354  0.018  0.021  0.015  0.032  0.708
20  0.607 × 10^{−3}  0.304  0.062  0.119  0.041  0.140  1.295  0.015  0.019  0.013  0.027  0.693
20  0.805 × 10^{−3}  0.212  0.060  0.163  0.068  0.186  1.222  0.007  0.014  0.012  0.019  0.608
25  0.493 × 10^{−3}  0.363  0.072  0.157  0.043  0.178  1.484  0.022  0.040  0.024  0.052  0.851
25  0.616 × 10^{−3}  0.284  0.043  0.089  0.031  0.103  1.382  0.013  0.021  0.014  0.028  0.698
25  0.759 × 10^{−3}  0.296  0.065  0.124  0.042  0.146  1.330  0.015  0.020  0.013  0.028  0.700
25  0.101 × 10^{−2}  0.168  0.064  0.167  0.068  0.191  1.236  0.007  0.014  0.012  0.020  0.616

Table 5: The proton structure functions F_{L} and F_{2} measured at the given values of Q^{2} and x without model assumptions. Δ_{stat}F_{L}, Δ_{uncor}F_{L}, Δ_{cor}F_{L} and Δ_{tot}F_{L} are the statistical, uncorrelated systematic, correlated systematic, and total uncertainty on F_{L} respectively. Δ_{stat}F_{2}, Δ_{uncor}F_{2}, Δ_{cor}F_{2} and Δ_{tot}F_{2} are the statistical, uncorrelated systematic and total uncertainty on F_{2}, respectively. The correlation coefficient between the F_{L} and F_{2} values ,ρ, is also given.

Q^{2}  x  F_{L}  Δ_{stat}F_{L}  Δ_{uncor}F_{L}  Δ_{cor}F_{L}  Δ_{tot}F_{L}  F_{2}  Δ_{stat}F_{2}  Δ_{uncor}F_{2}  Δ_{cor}F_{2}  Δ_{tot}F_{2}  ρ

(GeV^{2})

35  0.651 × 10^{−3}  0.453  0.124  0.214  0.091  0.264  1.612  0.043  0.058  0.030  0.078  0.889
35  0.727 × 10^{−3}  0.041  0.144  0.232  0.065  0.281  1.419  0.038  0.048  0.020  0.065  0.884
35  0.812 × 10^{−3}  0.106  0.075  0.107  0.054  0.142  1.411  0.026  0.027  0.019  0.042  0.753
35  0.921 × 10^{−3}  0.436  0.080  0.125  0.040  0.153  1.405  0.022  0.024  0.014  0.035  0.727
35  0.106 × 10^{−2}  0.196  0.072  0.130  0.042  0.155  1.325  0.017  0.021  0.012  0.030  0.698
35  0.141 × 10^{−2}  0.057  0.067  0.170  0.065  0.194  1.226  0.008  0.015  0.011  0.021  0.639
45  0.837 × 10^{−3}  0.179  0.117  0.188  0.061  0.230  1.518  0.042  0.054  0.022  0.072  0.875
45  0.934 × 10^{−3}  0.516  0.167  0.238  0.058  0.296  1.517  0.043  0.052  0.022  0.071  0.869
45  0.104 × 10^{−2}  0.366  0.084  0.107  0.054  0.146  1.430  0.029  0.027  0.016  0.042  0.731
45  0.118 × 10^{−2}  0.396  0.108  0.118  0.042  0.165  1.395  0.025  0.025  0.014  0.038  0.732
45  0.137 × 10^{−2}  0.255  0.100  0.151  0.047  0.187  1.350  0.021  0.023  0.013  0.034  0.729
45  0.181 × 10^{−2}  0.099  0.075  0.175  0.065  0.202  1.210  0.009  0.016  0.011  0.021  0.659
60  0.112 × 10^{−2}  0.282  0.146  0.179  0.051  0.237  1.446  0.058  0.058  0.021  0.084  0.851
60  0.125 × 10^{−2}  0.279  0.165  0.198  0.055  0.263  1.548  0.048  0.050  0.018  0.072  0.844
60  0.139 × 10^{−2}  0.383  0.095  0.105  0.049  0.150  1.450  0.033  0.030  0.016  0.047  0.731
60  0.158 × 10^{−2}  0.464  0.102  0.101  0.047  0.151  1.369  0.027  0.024  0.014  0.039  0.711
60  0.182 × 10^{−2}  0.159  0.230  0.320  0.047  0.397  1.288  0.028  0.033  0.012  0.045  0.818
60  0.242 × 10^{−2}  −0.044  0.094  0.185  0.069  0.218  1.186  0.011  0.016  0.011  0.023  0.683
90  0.187 × 10^{−2}  0.041  0.222  0.207  0.045  0.307  1.330  0.095  0.077  0.017  0.123  0.862
90  0.209 × 10^{−2}  0.060  0.109  0.119  0.041  0.166  1.313  0.051  0.045  0.014  0.069  0.801
90  0.237 × 10^{−2}  0.007  0.109  0.101  0.040  0.154  1.218  0.037  0.029  0.012  0.048  0.769
90  0.273 × 10^{−2}  0.447  0.143  0.135  0.048  0.202  1.325  0.031  0.027  0.013  0.043  0.717
90  0.362 × 10^{−2}  0.163  0.167  0.233  0.058  0.293  1.145  0.015  0.017  0.011  0.025  0.699
120  0.220 × 10^{−2}  0.070  0.067  0.241  0.041  0.253  1.400  0.019  0.060  0.027  0.069  0.908
120  0.250 × 10^{−2}  0.450  0.096  0.252  0.037  0.272  1.414  0.022  0.051  0.025  0.061  0.875
120  0.280 × 10^{−2}  0.136  0.094  0.103  0.036  0.144  1.299  0.022  0.025  0.021  0.039  0.711
120  0.320 × 10^{−2}  0.073  0.175  0.342  0.032  0.385  1.129  0.103  0.146  0.018  0.179  0.963
120  0.360 × 10^{−2}  0.480  0.178  0.234  0.039  0.296  1.245  0.062  0.063  0.012  0.089  0.886
120  0.480 × 10^{−2}  0.152  0.166  0.212  0.071  0.278  1.069  0.024  0.024  0.010  0.035  0.762
150  0.280 × 10^{−2}  0.123  0.083  0.270  0.046  0.286  1.357  0.024  0.065  0.025  0.074  0.931
150  0.310 × 10^{−2}  0.306  0.099  0.290  0.035  0.308  1.330  0.022  0.056  0.023  0.065  0.904
150  0.350 × 10^{−2}  0.038  0.085  0.127  0.038  0.157  1.274  0.017  0.027  0.021  0.038  0.728
150  0.390 × 10^{−2}  0.401  0.095  0.124  0.029  0.159  1.266  0.015  0.022  0.020  0.033  0.675
150  0.450 × 10^{−2}  0.554  0.114  0.153  0.038  0.195  1.209  0.014  0.023  0.020  0.034  0.668
150  0.600 × 10^{−2}  0.137  0.118  0.194  0.060  0.235  1.069  0.009  0.020  0.018  0.028  0.677
200  0.370 × 10^{−2}  −0.039  0.110  0.305  0.044  0.327  1.231  0.033  0.072  0.023  0.083  0.944
200  0.410 × 10^{−2}  −0.157  0.141  0.321  0.033  0.352  1.160  0.032  0.059  0.020  0.070  0.929
200  0.460 × 10^{−2}  0.146  0.115  0.131  0.038  0.179  1.188  0.023  0.025  0.019  0.039  0.758
200  0.520 × 10^{−2}  0.184  0.126  0.138  0.033  0.189  1.136  0.019  0.022  0.019  0.035  0.719
200  0.610 × 10^{−2}  0.253  0.141  0.169  0.033  0.223  1.107  0.016  0.021  0.018  0.032  0.690
200  0.800 × 10^{−2}  0.228  0.126  0.203  0.057  0.246  0.995  0.008  0.018  0.017  0.026  0.654
250  0.460 × 10^{−2}  0.620  0.136  0.365  0.052  0.393  1.340  0.041  0.086  0.025  0.099  0.950
250  0.520 × 10^{−2}  0.214  0.159  0.349  0.039  0.385  1.186  0.037  0.065  0.020  0.077  0.931
250  0.580 × 10^{−2}  0.243  0.130  0.142  0.038  0.196  1.176  0.026  0.027  0.019  0.041  0.770
250  0.660 × 10^{−2}  0.163  0.145  0.146  0.030  0.208  1.087  0.022  0.022  0.016  0.035  0.742
250  0.760 × 10^{−2}  0.117  0.159  0.173  0.031  0.237  0.998  0.018  0.020  0.016  0.032  0.714
250  0.100 × 10^{−1}  0.105  0.139  0.197  0.050  0.246  0.914  0.008  0.015  0.015  0.023  0.650
250  0.130 × 10^{−1}  0.140  0.228  0.280  0.095  0.374  0.842  0.008  0.014  0.014  0.021  0.650
300  0.560 × 10^{−2}  −0.038  0.161  0.316  0.041  0.357  1.138  0.048  0.075  0.021  0.091  0.942
300  0.690 × 10^{−2}  0.345  0.151  0.149  0.039  0.216  1.118  0.030  0.028  0.019  0.045  0.781
300  0.790 × 10^{−2}  0.377  0.168  0.148  0.027  0.225  1.058  0.025  0.022  0.016  0.037  0.752
300  0.910 × 10^{−2}  0.349  0.193  0.176  0.028  0.263  0.967  0.021  0.019  0.015  0.033  0.734
300  0.121 × 10^{−1}  −0.324  0.157  0.200  0.047  0.258  0.839  0.009  0.014  0.014  0.022  0.663
400  0.930 × 10^{−2}  −0.093  0.164  0.135  0.033  0.215  0.950  0.033  0.025  0.015  0.044  0.790
400  0.105 × 10^{−1}  −0.199  0.180  0.131  0.020  0.223  0.923  0.028  0.019  0.014  0.037  0.760
400  0.121 × 10^{−1}  −0.051  0.207  0.179  0.027  0.275  0.913  0.023  0.021  0.015  0.034  0.736
400  0.161 × 10^{−1}  −0.180  0.182  0.202  0.043  0.276  0.788  0.011  0.013  0.012  0.021  0.680
500  0.116 × 10^{−1}  −0.255  0.184  0.119  0.028  0.221  0.868  0.037  0.022  0.014  0.046  0.790
500  0.131 × 10^{−1}  0.340  0.207  0.143  0.019  0.252  0.946  0.033  0.022  0.015  0.042  0.755
500  0.152 × 10^{−1}  0.279  0.244  0.149  0.021  0.287  0.860  0.028  0.017  0.012  0.035  0.742
500  0.201 × 10^{−1}  0.192  0.214  0.211  0.043  0.304  0.770  0.013  0.014  0.013  0.023  0.691
650  0.151 × 10^{−1}  0.229  0.219  0.145  0.018  0.263  0.917  0.043  0.026  0.013  0.052  0.804
650  0.171 × 10^{−1}  −0.229  0.209  0.132  0.016  0.248  0.743  0.033  0.021  0.012  0.041  0.769
650  0.197 × 10^{−1}  −0.204  0.254  0.148  0.019  0.294  0.735  0.030  0.017  0.011  0.036  0.750
650  0.261 × 10^{−1}  0.651  0.244  0.201  0.036  0.318  0.739  0.016  0.014  0.012  0.024  0.698
800  0.185 × 10^{−1}  0.625  0.228  0.158  0.014  0.278  0.821  0.045  0.028  0.013  0.054  0.812
800  0.210 × 10^{−1}  0.205  0.230  0.167  0.015  0.285  0.762  0.036  0.026  0.012  0.046  0.774
800  0.242 × 10^{−1}  0.123  0.281  0.148  0.016  0.318  0.698  0.033  0.017  0.010  0.039  0.753
800  0.322 × 10^{−1}  0.276  0.253  0.202  0.031  0.325  0.642  0.016  0.014  0.010  0.023  0.714

Table 5: continued.

Q^{2}  x  F_{L}  Δ_{stat}  Δ_{uncor}  Δ_{cor}  Δ_{tot}

(GeV^{2})

1.5  0.279 × 10^{−4}  0.088  0.113  0.186  0.053  0.224
2.0  0.427 × 10^{−4}  0.127  0.039  0.074  0.044  0.095
2.5  0.588 × 10^{−4}  0.156  0.025  0.050  0.053  0.077
3.5  0.877 × 10^{−4}  0.227  0.021  0.049  0.040  0.067
5.0  0.129 × 10^{−3}  0.314  0.022  0.055  0.045  0.074
6.5  0.169 × 10^{−3}  0.264  0.023  0.058  0.050  0.080
8.5  0.224 × 10^{−3}  0.216  0.025  0.062  0.051  0.084
12  0.319 × 10^{−3}  0.324  0.026  0.051  0.044  0.072
15  0.402 × 10^{−3}  0.266  0.027  0.051  0.042  0.071
20  0.540 × 10^{−3}  0.327  0.029  0.053  0.040  0.072
25  0.687 × 10^{−3}  0.282  0.029  0.061  0.037  0.077
35  0.958 × 10^{−3}  0.213  0.035  0.059  0.040  0.080
45  0.121 × 10^{−2}  0.303  0.043  0.060  0.044  0.086
60  0.157 × 10^{−2}  0.315  0.051  0.060  0.044  0.090
90  0.243 × 10^{−2}  0.125  0.061  0.062  0.039  0.095
120  0.303 × 10^{−2}  0.198  0.054  0.077  0.029  0.098
150  0.402 × 10^{−2}  0.264  0.044  0.068  0.035  0.088
200  0.541 × 10^{−2}  0.150  0.056  0.073  0.034  0.099
250  0.736 × 10^{−2}  0.196  0.061  0.075  0.033  0.102
346  0.986 × 10^{−2}  0.039  0.059  0.057  0.029  0.087
636  0.184 × 10^{−1}  0.152  0.066  0.045  0.020  0.082

Table 6: The proton structure function F_{L}(x, Q^{2}) obtained by averaging F_{L} data from table 5 at the given values of Q^{2} and x. Δ_{stat}, Δ_{uncor}, Δ_{cor} and Δ_{tot} are the statistical, uncorrelated systematic, correlated systematic, and total uncertainty on F_{L}, respectively.

(a)

(b)

Figure 1: Distributions of P_{T,h}/P_{T,e}, θ_{jets} and E − P_{z} for (a) E_{p} = 460 GeV and (b) E_{p} = 575 GeV for y < 0.19 data (solid points) and simulation and estimated background (histograms) normalised to the integrated luminosity of the data. The estimated QED Compton background contribution is shown as shaded histogram.

(a)

(b)

Figure 2: Distributions of E − P_{z}, D_{ele}, and E′_{e}/p_{e} for the sample of events with E′_{e} < 6 GeV. The selection requirements on E−P_{z} and D_{ele} are shown as vertical lines with all other selection criteria applied. The distributions are shown for (a) E_{p} = 460 GeV and (b) E_{p} = 575 GeV for data (solid points) and simulation and estimated background (histograms) normalised to the integrated luminosity of the data. The estimated background is shown as shaded histogram and includes the photoproduction contribution estimated using wrong charge scattered lepton candidates as well as the QED Compton contribution.

(a)

(b)

Figure 3: Distributions of E′_{e}, θ_{e} and E − P_{z} for (a) E_{p} = 460 GeV and (b) E_{p} = 575 GeV for high y data (solid points) and simulation and estimated background (histograms) normalised to the integrated luminosity of the data. The estimated background is shown as shaded histogram and includes the photoproduction contribution estimated using wrong charge scattered lepton candidates and the QED Compton contribution (dashed line).

Figure 4: The reduced cross section σ̃_{NC}(x, Q^{2}) + 0.3i measured at three proton beam energies E_{p} = 460 GeV (diamonds, i=0), 575 GeV (squares, i=1) and 920 GeV (circles, i=2). The previously published H1 SpaCal data are shown by the open symbols. The solid symbols are the H1 LAr data. The new measurements reported here correspond to the filled diamonds and squares. The inner error bars represent the statistical errors, the full error bars include the statistical and systematic uncertainties added in quadrature, excluding the normalisation uncertainty. The curves represent the prediction from the H1PDF2012 NLO QCD fit.

Figure 5: The reduced cross section σ̃_{NC}(x, Q^{2}) as a function of y^{2}/(1 + (1 − y)^{2}) for six values of x at Q^{2} = 60 GeV^{2}, measured for proton beam energies of E_{p} = 920, 575 and 460 GeV. The inner error bars denote the statistical error, the outer error bars show statistical and systematic uncertainties added in quadrature. The luminosity uncertainty is not included in the error bars. The negative slopes of the linear fits (solid line) which were performed using total errors, illustrate the non-vanishing values of the structure function F_{L}(x, Q^{2}).

Figure 6: The proton structure functions F_{L}(x, Q^{2}) (solid symbols) and F_{2}(x, Q^{2}) (open symbols) measured by H1 (circles) and ZEUS (diamonds) in the region 2 ≤ Q^{2} ≤ 25 GeV^{2}. The inner error bars represent the statistical uncertainties, the full error bars include the statistical and systematic uncertainties added in quadrature, including all correlated and uncorrelated uncertainties. The curves represent the prediction from the H1PDF2012 NLO QCD fit.

Figure 7: The proton structure functions F_{L}(x, Q^{2}) (solid symbols) and F_{2}(x, Q^{2}) (open symbols) measured by H1 (circles) and ZEUS (diamonds) in the region 35 ≤ Q^{2} ≤ 800 GeV^{2}. The inner error bars represent the statistical uncertainties, the full error bars include the statistical and systematic uncertainties added in quadrature, including all correlated and uncorrelated uncertainties. The curves represent the prediction from the H1PDF2012 NLO QCD fit.

Figure 8: The proton structure function F_{L} averaged over x at different Q^{2} (solid points). The average value of x for each Q^{2} is given above each data point. The inner error bars represent the statistical uncertainties, the full error bars include the statistical and systematic uncertainties added in quadrature, including all correlated and uncorrelated uncertainties. The F_{L} measurements by ZEUS are also shown (open points). The data are compared to NNLO predictions from a selection of PDF sets as indicated.

Figure 9: The ratio R(Q^{2}) averaged over x in the region 1.5 ≤ Q^{2} ≤ 800 GeV^{2} (solid points). The error bars represent the full errors as obtained by the Monte Carlo procedure described in the text. The ZEUS data are also shown (open symbols). The ZEUS data point at Q^{2} = 45 GeV^{2} is slightly shifted for better visibility of the erros. The solid curve represents the prediction from the HERAPDF1.5 NNLO QCD fit and its uncertainty for \sqrt{s} = 225 Gev^{2} and y = 0.7. The additional dashed and dotted curves show the variations of R in the region of x where the data are sensitive to this quantity.

Figure 10: The gluon density xg(x, Q^{2}) averaged over x in the region 1.5 ≤ Q^{2} ≤ 800 GeV^{2} (solid points). The average value of x for each Q^{2} is given above each data point. The inner error bars represent the statistical uncertainties, the full error bars include the statistical and systematic uncertainties added in quadrature, including all correlated and uncorrelated uncertainties. The shaded regions represent the prediction from the HERAPDF1.5 NLO QCD fit. The dashed line corresponds to xg as obtained by applying equation 8 to the F_{L} prediction based on the HERAPDF1.5 NLO QCD fit.