1 DESY 15-037                                            ISSN 0418-9833
2 March 2015

3 Combination of Differential D^{∗±} D^{∗±} D^{∗±} Cross-Section Measurements 4 in Deep-Inelastic eeep p p Scattering at HERA

5

6 Abstract

7

8 H1 and ZEUS have published single-differential cross sections for inclusive D^{∗±} -meson 9 production in deep-inelastic ep scattering at HERA from their respective final data sets. 10 These cross sections are combined in the common visible phase-space region of photon 11 virtuality Q^{2} > 5 GeV^{2}, electron inelasticity 0.02 < y < 0.7 and the D^{∗±} meson’s trans-12 verse momentum p_{T}(D^{∗}) > 1.5 GeV and pseudorapidity |η(D^{∗})| < 1.5. The combination 13 procedure takes into account all correlations, yielding significantly reduced experimental 14 uncertainties. Double-differential cross sections d^{2}σ/dQ^{2}dyare combined with earlier D^{∗±} 15 data, extending the kinematic range down to Q^{2} > 1.5 GeV^{2}. Perturbative next-to-leading-16 order QCD predictions are compared to the results.

17 Submitted to JHEP

18 The H1 and ZEUS Collaborations

19 H. Abramowicz^{49,a1}, I. Abt^{36}, L. Adamczyk^{22}, M. Adamus^{58}, V. Andreev^{34}, S. Antonelli^{6}, 20 V. Aushev^{25,26,b21}, Y. Aushev^{26,a2,b21}, A. Baghdasaryan^{60}, K. Begzsuren^{55}, O. Behnke^{17}, U. Behrens^{17}, 21 A. Belousov^{34}, A. Bertolin^{40}, I. Bloch^{62}, E.G. Boos^{2}, K. Borras^{17}, V. Boudry^{42}, G. Brandt^{39}, 22 V. Brisson^{38}, D. Britzger^{17}, I. Brock^{7}, N.H. Brook^{30}, R. Brugnera^{41}, A. Bruni^{5}, A. Buniatyan^{4}, 23 P.J. Bussey^{15}, A. Bylinkin^{33,a3}, L. Bystritskaya^{33}, A. Caldwell^{36}, A.J. Campbell^{17}, K.B. Cantun Avila^{32}, 24 M. Capua^{10}, C.D. Catterall^{37}, F. Ceccopieri^{3}, K. Cerny^{45}, V. Chekelian^{36}, J. Chwastowski^{21}, 25 J. Ciborowski^{57,a4}, R. Ciesielski^{17,a5}, J.G. Contreras^{32}, A.M. Cooper-Sarkar^{39}, M. Corradi^{5}, 26 F. Corriveau^{46}, J. Cvach^{44}, J.B. Dainton^{28}, K. Daum^{59,a6}, R.K. Dementiev^{35}, R.C.E. Devenish^{39}, 27 C. Diaconu^{31}, M. Dobre^{8}, V. Dodonov^{17}, G. Dolinska^{17}, S. Dusini^{40}, G. Eckerlin^{17}, S. Egli^{56}, 28 E. Elsen^{17}, L. Favart^{3}, A. Fedotov^{33}, J. Feltesse^{14}, J. Ferencei^{20}, J. Figiel^{21}, M. Fleischer^{17}, 29 A. Fomenko^{34}, B. Foster^{16,a7}, E. Gabathuler^{28}, G. Gach^{22,a8}, E. Gallo^{16,17}, A. Garfagnini^{41}, J. Gayler^{17}, 30 A. Geiser^{17}, S. Ghazaryan^{17}, A. Gizhko^{17}, L.K. Gladilin^{35}, L. Goerlich^{21}, N. Gogitidze^{34}, 31 Yu.A. Golubkov^{35}, M. Gouzevitch^{17,a9}, C. Grab^{63}, A. Grebenyuk^{3}, J. Grebenyuk^{17}, T. Greenshaw^{28}, 32 I. Gregor^{17}, G. Grindhammer^{36}, G. Grzelak^{57}, O. Gueta^{49}, M. Guzik^{22}, D. Haidt^{17}, W. Hain^{17}, 33 R.C.W. Henderson^{27}, J. Hladkỳ^{44}, D. Hochman^{47}, D. Hoffmann^{31}, R. Hori^{54}, R. Horisberger^{56}, 34 T. Hreus^{3}, F. Huber^{18}, Z.A. Ibrahim^{24}, Y. Iga^{50}, M. Ishitsuka^{51}, A. Iudin^{26,a2}, M. Jacquet^{38}, X. Janssen^{3}, 35 F. Januschek^{17,a10}, N.Z. Jomhari^{24}, A.W. Jung^{19,a11}, H. Jung^{17,3}, I. Kadenko^{26}, S. Kananov^{49}, 36 M. Kapichine^{13}, U. Karshon^{47}, M. Kaur^{9}, P. Kaur^{9,b22}, C. Kiesling^{36}, D. Kisielewska^{22}, R. Klanner^{16}, 37 M. Klein^{28}, U. Klein^{17,a12}, C. Kleinwort^{17}, R. Kogler^{16}, N. Kondrashova^{26,a13}, O. Kononenko^{26}, 38 Ie. Korol^{17}, I.A. Korzhavina^{35}, P. Kostka^{28}, A. Kotański^{23}, U. Kötz^{17}, N. Kovalchuk^{16}, H. Kowalski^{17}, 39 J. Kretzschmar^{28}, K. Krüger^{17}, B. Krupa^{21}, O. Kuprash^{17}, M. Kuze^{51}, M.P.J. Landon^{29}, W. Lange^{62}, 40 P. Laycock^{28}, A. Lebedev^{34}, B.B. Levchenko^{35}, S. Levonian^{17}, A. Levy^{49}, V. Libov^{17}, S. Limentani^{41}, 41 K. Lipka^{17,b23}, M. Lisovyi^{17}, B. List^{17}, J. List^{17}, E. Lobodzinska^{17}, B. Lobodzinski^{36}, B. Löhr^{17}, 42 E. Lohrmann^{16}, A. Longhin^{40,a14}, D. Lontkovskyi^{17}, O.Yu. Lukina^{35}, I. Makarenko^{17}, E. Malinovski^{34}, 43 J. Malka^{17}, H.-U. Martyn^{1}, S.J. Maxfield^{28}, A. Mehta^{28}, S. Mergelmeyer^{7}, A.B. Meyer^{17}, H. Meyer^{59}, 44 J. Meyer^{17}, S. Mikocki^{21}, F. Mohamad Idris^{24,a15}, A. Morozov^{13}, N. Muhammad Nasir^{24}, K. Müller^{64}, 45 V. Myronenko^{17,b24}, K. Nagano^{54}, Th. Naumann^{62}, P.R. Newman^{4}, C. Niebuhr^{17}, T. Nobe^{51}, 46 D. Notz^{17,†}, G. Nowak^{21}, R.J. Nowak^{57}, J.E. Olsson^{17}, Yu. Onishchuk^{26}, D. Ozerov^{17}, P. Pahl^{17}, 47 C. Pascaud^{38}, G.D. Patel^{28}, E. Paul^{7}, E. Perez^{14,a16}, W. Perlański^{57,a17}, A. Petrukhin^{17}, I. Picuric^{43}, 48 H. Pirumov^{17}, D. Pitzl^{17}, R. Plačakytė^{17,b23}, B. Pokorny^{45}, N.S. Pokrovskiy^{2}, R. Polifka^{45,a18}, 49 M. Przybycień^{22}, V. Radescu^{17,b23}, N. Raicevic^{43}, T. Ravdandorj^{55}, P. Reimer^{44}, E. Rizvi^{29}, 50 P. Robmann^{64}, P. Roloff^{17,a16}, R. Roosen^{3}, A. Rostovtsev^{33}, M. Rotaru^{8}, I. Rubinsky^{17}, S. Rusakov^{34}, 51 M. Ruspa^{53}, D. Šálek^{45}, D.P.C. Sankey^{11}, M. Sauter^{18}, E. Sauvan^{31,a19}, D.H. Saxon^{15}, M. Schioppa^{10}, 52 W.B. Schmidke^{36,a20}, S. Schmitt^{17}, U. Schneekloth^{17}, L. Schoeffel^{14}, A. Schöning^{18}, 53 T. Schörner-Sadenius^{17}, F. Sefkow^{17}, L.M. Shcheglova^{35}, R. Shevchenko^{26,a2}, O. Shkola^{26,a21}, 54 S. Shushkevich^{17}, Yu. Shyrma^{25}, I. Singh^{9,b25}, I.O. Skillicorn^{15}, W. S�omiński^{23,b26}, A. Solano^{52}, 55 Y. Soloviev^{17,34}, P. Sopicki^{21}, D. South^{17}, V. Spaskov^{13}, A. Specka^{42}, L. Stanco^{40}, M. Steder^{17}, 56 N. Stefaniuk^{17}, A. Stern^{49}, P. Stopa^{21}, U. Straumann^{64}, T. Sykora^{3,45}, J. Sztuk-Dambietz^{16,a10}, 57 D. Szuba^{16}, J. Szuba^{17}, E. Tassi^{10}, P.D. Thompson^{4}, K. Tokushuku^{54,a22}, J. Tomaszewska^{57,a23}, 58 D. Traynor^{29}, A. Trofymov^{26,a13}, P. Truöl^{64}, I. Tsakov^{48}, B. Tseepeldorj^{55,a24}, T. Tsurugai^{61}, 59 M. Turcato^{16,a10}, O. Turkot^{17,b24}, J. Turnau^{21}, T. Tymieniecka^{58}, A. Valkárová^{45}, C. Vallée^{31}, 60 P. Van Mechelen^{3}, Y. Vazdik^{34}, A. Verbytskyi^{36}, O. Viazlo^{26}, R. Walczak^{39}, W.A.T. Wan Abdullah^{24}, 61 D. Wegener^{12}, K. Wichmann^{17,b24}, M. Wing^{30,a25}, G. Wolf^{17}, E. Wünsch^{17}, S. Yamada^{54}, 62 Y. Yamazaki^{54,a26}, J. Žáček^{45}, N. Zakharchuk^{26,a13}, A.F. Żarnecki^{57}, L. Zawiejski^{21}, O. Zenaiev^{17}, 63 Z. Zhang^{38}, B.O. Zhautykov^{2}, N. Zhmak^{25,b21}, R. Žlebčík^{45}, H. Zohrabyan^{60}, F. Zomer^{38} and 64 D.S. Zotkin^{35}

65 ^{1} I. Physikalisches Institut der RWTH, Aachen, Germany 66 ^{2} Institute of Physics and Technology of Ministry of Education and Science of Kazakhstan, Almaty, Kazakhstan 67 ^{3} Inter-University Institute for High Energies ULB-VUB, Brussels and Universiteit Antwerpen, Antwerpen, 68 Belgium^{b1} 69 ^{4} School of Physics and Astronomy, University of Birmingham, Birmingham, UK^{b2} 70 ^{5} INFN Bologna, Bologna, Italy^{b3} 71 ^{6} University and INFN Bologna, Bologna, Italy^{b3} 72 ^{7} Physikalisches Institut der Universität Bonn, Bonn, Germany^{b4} 73 ^{8} National Institute for Physics and Nuclear Engineering (NIPNE) , Bucharest, Romania^{b5} 74 ^{9} Panjab University, Department of Physics, Chandigarh, India 75 ^{10} Calabria University, Physics Department and INFN, Cosenza, Italy^{b3} 76 ^{11} STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire, UK^{b2} 77 ^{12} Institut für Physik, TU Dortmund, Dortmund, Germany^{b6} 78 ^{13} Joint Institute for Nuclear Research, Dubna, Russia 79 ^{14} CEA, DSM/Irfu, CE-Saclay, Gif-sur-Yvette, France 80 ^{15} School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom^{b2} 81 ^{16} Institut für Experimentalphysik, Universität Hamburg, Hamburg, Germany^{b6,b7} 82 ^{17} Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany 83 ^{18} Physikalisches Institut, Universität Heidelberg, Heidelberg, Germany^{b6} 84 ^{19} Kirchhoff-Institut für Physik, Universität Heidelberg, Heidelberg, Germany^{b6} 85 ^{20} Institute of Experimental Physics, Slovak Academy of Sciences, Košice, Slovak Republic^{b8} 86 ^{21} The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Krakow, 87 Poland^{b9,b15} 88 ^{22} AGH-University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow, 89 Poland^{b9} 90 ^{23} Department of Physics, Jagellonian University, Krakow, Poland 91 ^{24} National Centre for Particle Physics, Universiti Malaya, 50603 Kuala Lumpur, Malaysia^{b10} 92 ^{25} Institute for Nuclear Research, National Academy of Sciences, Kyiv, Ukraine 93 ^{26} Department of Nuclear Physics, National Taras Shevchenko University of Kyiv, Kyiv, Ukraine 94 ^{27} Department of Physics, University of Lancaster, Lancaster, UK^{b2} 95 ^{28} Department of Physics, University of Liverpool, Liverpool, UK^{b2} 96 ^{29} School of Physics and Astronomy, Queen Mary, University of London, London, UK^{b2} 97 ^{30} Physics and Astronomy Department, University College London, London, United Kingdom^{b2} 98 ^{31} Aix Marseille Université, CNRS/IN2P3, CPPM UMR 7346, 13288 Marseille, France 99 ^{32} Departamento de Fisica Aplicada, CINVESTAV, Mérida, Yucatán, México^{b11} 100 ^{33} Institute for Theoretical and Experimental Physics, Moscow, Russia^{b12} 101 ^{34} Lebedev Physical Institute, Moscow, Russia 102 ^{35} Lomonosov Moscow State University, Skobeltsyn Institute of Nuclear Physics, Moscow, Russia^{b13} 103 ^{36} Max-Planck-Institut für Physik, München, Germany 104 ^{37} Department of Physics, York University, Ontario, Canada M3J 1P3^{b14} 105 ^{38} LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 106 ^{39} Department of Physics, University of Oxford, Oxford, United Kingdom^{b2} 107 ^{40} INFN Padova, Padova, Italy^{b3} 108 ^{41} Dipartimento di Fisica e Astronomia dell’ Università and INFN, Padova, Italy^{b3} 109 ^{42} LLR, Ecole Polytechnique, CNRS/IN2P3, Palaiseau, France 110 ^{43} Faculty of Science, University of Montenegro, Podgorica, Montenegro^{b16} 111 ^{44} Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic^{b17} 112 ^{45} Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic^{b17} 113 ^{46} Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8^{b14} 114 ^{47} Department of Particle Physics and Astrophysics, Weizmann Institute, Rehovot, Israel 115 ^{48} Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria 116 ^{49} Raymond and Beverly Sackler Faculty of Exact Sciences, School of Physics, Tel Aviv University, Tel Aviv, 117 Israel^{b18} 118 ^{50} Polytechnic University, Tokyo, Japan^{b19} 119 ^{51} Department of Physics, Tokyo Institute of Technology, Tokyo, Japan^{b19} 120 ^{52} Università di Torino and INFN, Torino, Italy^{b3} 121 ^{53} Università del Piemonte Orientale, Novara, and INFN, Torino, Italy^{b3} 122 ^{54} Institute of Particle and Nuclear Studies, KEK, Tsukuba, Japan^{b19} 123 ^{55} Institute of Physics and Technology of the Mongolian Academy of Sciences, Ulaanbaatar, Mongolia 124 ^{56} Paul Scherrer Institut, Villigen, Switzerland 125 ^{57} Faculty of Physics, University of Warsaw, Warsaw, Poland 126 ^{58} National Centre for Nuclear Research, Warsaw, Poland 127 ^{59} Fachbereich C, Universität Wuppertal, Wuppertal, Germany 128 ^{60} Yerevan Physics Institute, Yerevan, Armenia 129 ^{61} Meiji Gakuin University, Faculty of General Education, Yokohama, Japan^{b19} 130 ^{62} Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany 131 ^{63} Institut für Teilchenphysik, ETH, Zürich, Switzerland^{b20} 132 ^{64} Physik-Institut der Universität Zürich, Zürich, Switzerland^{b20} 133 ^{†} Deceased

134 ^{a1} Also at Max Planck Institute for Physics, Munich, Germany, External Scientific Member 135 ^{a2} Member of National Technical University of Ukraine, Kyiv Polytechnic Institute, Kyiv, Ukraine 136 ^{a3} Also at Moscow Institute of Physics and Technology, Moscow, Russia 137 ^{a4} Also at �ódź University, Poland 138 ^{a5} Now at Rockefeller University, New York, NY 10065, USA 139 ^{a6} Also at Rechenzentrum, Universität Wuppertal, Wuppertal, Germany 140 ^{a7} Alexander von Humboldt Professor; also at DESY and University of Oxford 141 ^{a8} Now at School of Physics and Astronomy, University of Birmingham, UK 142 ^{a9} Also at IPNL, Université Claude Bernard Lyon 1, CNRS/IN2P3, Villeurbanne, France 143 ^{a10} Now at European X-ray Free-Electron Laser facility GmbH, Hamburg, Germany 144 ^{a11} Now at Fermilab, Chicago, United States 145 ^{a12} Now at University of Liverpool, United Kingdom 146 ^{a13} Now at DESY ATLAS group 147 ^{a14} Now at LNF, Frascati, Italy 148 ^{a15} Also at Agensi Nuklear Malaysia, 43000 Kajang, Bangi, Malaysia 149 ^{a16} Now at CERN, Geneva, Switzerland 150 ^{a17} Member of �ódź University, Poland 151 ^{a18} Also at Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 152 ^{a19} Also at LAPP, Université de Savoie, CNRS/IN2P3, Annecy-le-Vieux, France 153 ^{a20} Now at BNL, USA 154 ^{a21} Member of National University of Kyiv - Mohyla Academy, Kyiv, Ukraine 155 ^{a22} Also at University of Tokyo, Japan 156 ^{a23} Now at Polish Air Force Academy in Deblin 157 ^{a24} Also at Ulaanbaatar University, Ulaanbaatar, Mongolia 158 ^{a25} Also at Universität Hamburg and supported by DESY and the Alexander von Humboldt Foundation 159 ^{a26} Now at Kobe University, Japan 160 ^{b1} Supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT and by Interuniversity Attraction Poles 161 Programme, Belgian Science Policy 162 ^{b2} Supported by the UK Science and Technology Facilities Council, and formerly by the UK Particle Physics and 163 Astronomy Research Council 164 ^{b3} Supported by the Italian National Institute for Nuclear Physics (INFN) 165 ^{b4} Supported by the German Federal Ministry for Education and Research (BMBF), under contract No. 05 166 H09PDF 167 ^{b5} Supported by the Romanian National Authority for Scientific Research under the contract PN 09370101 168 ^{b6} Supported by the Bundesministerium für Bildung und Forschung, FRG, under contract numbers 05H09GUF, 169 05H09VHC, 05H09VHF, 05H16PEA 170 ^{b7} Supported by the SFB 676 of the Deutsche Forschungsgemeinschaft (DFG) 171 ^{b8} Supported by VEGA SR grant no. 2/7062/ 27 172 ^{b9} Supported by the National Science Centre under contract No. DEC-2012/06/M/ST2/00428 173 ^{b10} Supported by HIR grant UM.C/625/1/HIR/149 and UMRG grants RU006-2013, RP012A-13AFR and 174 RP012B-13AFR from Universiti Malaya, and ERGS grant ER004-2012A from the Ministry of Education, 175 Malaysia 176 ^{b11} Supported by CONACYT, México, grant 48778-F 177 ^{b12} Russian Foundation for Basic Research (RFBR), grant no 1329.2008.2 and Rosatom 178 ^{b13} Supported by RF Presidential grant N 3042.2014.2 for the Leading Scientific Schools and by the Russian 179 Ministry of Education and Science through its grant for Scientific Research on High Energy Physics 180 ^{b14} Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) 181 ^{b15} Partially Supported by Polish Ministry of Science and Higher Education, grant DPN/N168/DESY/2009 182 ^{b16} Partially Supported by Ministry of Science of Montenegro, no. 05-1/3-3352 183 ^{b17} Supported by the Ministry of Education of the Czech Republic under the project INGO-LG14033 184 ^{b18} Supported by the Israel Science Foundation 185 ^{b19} Supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and its 186 grants for Scientific Research 187 ^{b20} Supported by the Swiss National Science Foundation 188 ^{b21} Supported by DESY, Germany 189 ^{b22} Also funded by Max Planck Institute for Physics, Munich, Germany, now at Sant Longowal Institute of 190 Engineering and Technology, Longowal, Punjab, India 191 ^{b23} Supported by the Initiative and Networking Fund of the Helmholtz Association (HGF) under the contract 192 VH-NG-401 and S0-072 193 ^{b24} Supported by the Alexander von Humboldt Foundation 194 ^{b25} Also funded by Max Planck Institute for Physics, Munich, Germany, now at Sri Guru Granth Sahib World 195 University, Fatehgarh Sahib, India 196 ^{b26} Partially supported by the Polish National Science Centre projects DEC-2011/01/B/ST2/03643 and 197 DEC-2011/03/B/ST2/00220

198 1 Introduction

199 Measurements of open charm production in deep-inelastic electron^{1}–proton scattering (DIS) at 200 HERA provide important input for stringent tests of the theory of strong interactions, quantum 201 chromodynamics (QCD). Previous measurements [1–20] have demonstrated that charm quarks 202 are predominantly produced by the boson–gluon-fusion process, γg → c\overline{c}, whereby charm pro-203 duction becomes sensitive to the gluon distribution in the proton. Measurements have been 204 obtained both from the HERA-I (1992–2000) and HERA-II (2003–2007) data-taking periods.

205 Different techniques have been applied at HERA to measure open-charm production in DIS. 206 The full reconstruction of D or D^{∗±} mesons [1–6, 10–12, 15, 16, 18, 20], the long lifetime of 207 heavy flavoured hadrons [7–9, 12, 14, 17, 19] or their semi-leptonic decays [13] are exploited. 208 After extrapolation from the visible to the full phase space, most of these data have already 209 been combined [21] at the level of the reduced cross-sections and have provided a consistent 210 determination of the charm contribution to the proton structure functions, a measurement of the 211 charm-quark mass m_{c}(m_{c}) and improved predictions for W - and Z-production cross sections at 212 the LHC. However, the extrapolation procedure requires theoretical assumptions, which lead to 213 theoretical uncertainties comparable in size to the experimental uncertainties [21]. Moreover, 214 this combination was restricted to inclusive DIS variables only, such as the photon virtuality, 215 Q^{2}, and the inelasticity, y. Alternatively, the measured cross sections can be combined directly 216 in the visible phase space. In this case, dependences on the theoretical input are minimised 217 and the charm production mechanism can be explored in terms of other variables. Such a 218 combination, however, is possible only for data with the same final state, covering a common 219 visible phase space. The recently published differential cross-section measurements by H1 220 [15,18] and ZEUS [20] for inclusive D^{∗±} -meson production fulfil this requirement. The analysis 221 of fully reconstructed D^{∗±} mesons also offers the best signal-to-background ratio and small 222 statistical uncertainties.

223 In this paper, visible D^{∗±} -production cross sections [6, 15, 18, 20] at the centre-of-mass en-224 ergy \sqrt{s} = 318 GeV are combined such that one consistent HERA data set is obtained and 225 compared directly to differential next-to-leading-order (NLO) QCD predictions. The combina-226 tion is based on the procedure described elsewhere [21–24], accounting for all correlations in 227 the uncertainties. This yields a significant reduction of the overall uncertainties of the measure-228 ments.

229 The paper is organised as follows. In Section 2 the theoretical framework is briefly intro-230 duced that is used for applying phase-space corrections to the input data sets prior to combina-231 tion and for providing NLO QCD predictions to be compared to the data. The data samples used 232 for the combination are detailed in Section 3 and the combination procedure is described in Sec-233 tion 4. The combined single- and double-differential cross sections are presented in Section 5 234 together with a comparison of NLO QCD predictions to the data.

235 2 Theoretical predictions

236 The massive fixed-flavour-number scheme (FFNS) [25] is used for theoretical predictions, since 237 it is the only scheme for which fully differential NLO calculations [26] are available. The cross-238 section predictions for D^{∗±} production presented in this paper are obtained using the HVQDIS 239 program [26] which provides NLO QCD (O(α_{s}^{2})) calculations in the 3-flavour FFNS for charm 240 and beauty production in DIS. These predictions are used both for small phase-space corrections 241 of the data, due to slightly different binning schemes and kinematic cuts, and for comparison to 242 data.

1In this paper, ‘electron’ is used to denote both electron and positron if not stated otherwise.

243 The following parameters are used in the calculations and are varied within certain limits to 244 estimate the uncertainties associated with the predictions:

245 • The renormalisation and factorisation scales are taken as μ_{r} = μ_{f} = \sqrt{Q^{2}+ 4m_{c}^{2}}. The 246 scales are varied simultaneously up or down by a factor of two for the phase-space cor-247 rections where only the shape of the differential cross sections is relevant. For absolute 248 predictions, the scales are changed independently to 0.5 and 2 times their nominal value.

249 • The pole mass of the charm quark is set to m_{c} = 1.50 ± 0.15 GeV. This variation also 250 affects the values of the renormalisation and factorisation scales.

251 • For the strong coupling constant the value α_{s}^{n_{f} =3} (M_{Z}) = 0.105 ± 0.002 is chosen [21] 252 which corresponds to α_{s}^{n_{f} =5} (M_{Z}) = 0.116 ± 0.002.

253 • The proton parton density functions (PDFs) are described by a series of FFNS variants 254 of the HERAPDF1.0 set [24] at NLO determined within the HERAFitter [27] framework, 255 similar to those used in the charm combination paper [21]. Charm measurements were 256 not included in the determination of these PDF sets. For all parameter settings used here, 257 the corresponding PDF set is used. By default, the scales for the charm contribution to the 258 inclusive data in the PDF determination were chosen to be consistent with the factorisa-259 tion scale used in HVQDIS, while the renormalisation scale in HVQDIS was decoupled 260 from the scale used in the PDF extraction, except in the cases where the factorisation 261 and renormalisation scales were varied simultaneously. As a cross check, the renormal-262 isation scales for both heavy- and light-quark contributions are varied simultaneously in 263 HVQDIS and in the PDF determination, keeping the factorisation scales fixed. The re-264 sult lies well within the quoted uncertainties. The cross sections are also evaluated with 265 3-flavour NLO versions of the ABM [28] and MSTW [29] PDF sets. The differences are 266 found to be negligible compared to those from varying other parameters.

267 The NLO calculation performed by the HVQDIS program yields differential cross sections 268 for charm quarks. These predictions are converted to D^{∗±} -meson cross sections by applying 269 the fragmentation model described in a previous publication [21]. This model is based on the 270 fragmentation function of Kartvelishvili et al. [30] which provides a probability density func-271 tion for the fraction of the charm-quark momentum transferred to the D^{∗±} meson. The function 272 is controlled by a single fragmentation parameter, α_{K}. Different values of α_{K} [21] are used 273 for different regions of the invariant mass, ŝ, of the photon–parton centre-of-mass system. The 274 boundary ŝ_{1} = 70 ± 40 GeV^{2} between the first two regions is one of the parameter variations. 275 The boundary ŝ_{2} = 324 GeV^{2} between the second and third region remains fixed. The model 276 also implements a transverse-fragmentation component by assigning to the D^{∗±} meson a trans-277 verse momentum, k_{T}, with respect to the charm-quark direction [21]. The following parameters 278 are used in the calculations together with the corresponding variations for estimating the uncer-279 tainties of the NLO predictions related to fragmentation:

280 • The fragmentation parameter α_{K} α_{K} α_{K}, the bin boundary ŝ_{1} ŝ_{1} ŝ_{1} and the average k_{T} k_{T} k_{T}are varied 281 according to a prescription described elsewhere [21].

282 • The fraction of charm quarks hadronising into D^{∗+} D^{∗+} D^{∗+} mesons is set to 283 f(c → D^{∗+} ) = f(c̄→ D^{∗−}) = 0.2287 ± 0.0056 [31].

284 The small beauty contribution to the D^{∗±} signal needs a detailed treatment of the B hadron 285 decay to D^{∗±} mesons and is therefore obtained from NLO QCD predictions for beauty hadrons 286 convoluted with decay tables to D^{∗±} mesons and decay kinematics obtained from EvtGen [32]. 287 The parameters for the calculations and the uncertainties are:

288 • The renormalisation and factorisation scales μ_{r} = μ_{f} = \sqrt{Q^{2}+ 4m_{b}^{2}} are varied in the 289 same way as described above for charm. The variations are applied simultaneously for 290 the calculation of the charm and beauty cross-section uncertainties.

291 • The pole mass of the beauty quark is set to m_{b} = 4.75 ± 0.25 GeV.

292 • The fragmentation model for beauty quarks is based on the Peterson et al. [33] parametri-293 sation usind ϵ_{b} = 0.0035 ± 0.0020 [34].

294 • The fraction of beauty hadrons decaying into D^{∗±} D^{∗±} D^{∗±} mesons is set to 295 f(b → D^{∗±}) = 0.173 ± 0.020 [35].

296 • The proton structure is described by the same PDF set (3-flavour scheme) used for the 297 charm cross-section predictions.

298 The total theoretical uncertainties are obtained by adding all individual contributions in 299 quadrature.

300 3 Data samples for cross-section combinations

301 The H1 [36–38] and ZEUS [39] detectors were general purpose instruments which consisted 302 of tracking systems surrounded by electromagnetic and hadronic calorimeters and muon detec-303 tors. The most important detector components for the measurements combined in this paper are 304 the central tracking detectors (CTD) operated inside solenoidal magnetic fields of 1.16 T (H1) 305 and 1.43 T (ZEUS) and the electromagnetic sections of the calorimeters. The CTD of H1 [37] 306 (ZEUS [40]) measured charged particle trajectories in the polar angular range^{2} of 15^{∘} < Θ < 307 165(164)^{∘}. In both detectors the CTDs were complemented with high-resolution silicon ver-308 tex detectors: a system of three silicon detectors for H1, consisting of the Backward Silicon 309 Tracker [41], the Central Silicon Tracker [42] and the Forward Silicon Tracker [43], and the 310 Micro Vertex Detector [44] for ZEUS. For charged particles passing through all active layers 311 of the silicon vertex detectors and CTDs, transverse-momentum resolutions of σ(p_{T})/p_{T} ≃ 312 0.002p_{T}/ ⊕ 0.015 (H1) and σ(p_{T})/p_{T} ≃ 0.0029p_{T}/ ⊕ 0.0081 ⊕ 0.0012/p_{T} (ZEUS), with p_{T} 313 in units of GeV, have been achieved.

314 Each of the central tracking detectors was enclosed by a set of calorimeters comprising an 315 inner electromagnetic and an outer hadronic section. The H1 calorimeter system consisted

2In both experiments a right-handed coordinate system is employed with the Z axis pointing in the nominal proton-beam direction, referred to as “forward direction”, and the X axis pointing towards the centre of HERA.

The origin of the coordinate system is defined by the nominal interaction point in the case of H1 and by the centre

of the CTD in the case of ZEUS.

Data set  Kinematic range
Q^{2}  y  p_{T}(D^{∗}) η(D^{∗})  L
( GeV^{2})  ( GeV)  (pb^{-1})
I H1 D^{∗±} HERA-II (medium Q^{2}) [18] 5 : 100 0.02 : 0.70 > 1.5 −1.5 : 1.5 348
II H1 D^{∗±} HERA-II (high Q^{2}) [15] 100 : 1000 0.02 : 0.70 > 1.5 −1.5 : 1.5 351
III ZEUS D^{∗±} HERA-II  [20] 5 : 1000 0.02 : 0.70 1.5 : 20.0 −1.5 : 1.5 363
IV ZEUS D^{∗±} 98-00  [6] 1.5 : 1000 0.02 : 0.70 1.5 : 15.0 −1.5 : 1.5  82

Table 1: Data sets used in the combination. For each data set the respective kinematic range and the integrated luminosity, L, are given.

316 of the Liquid Argon calorimeter (LAr) [45] and the backward lead–scintillator calorimeter 317 (SpaCal) [38] while the ZEUS detector was equipped with a compensating uranium–scintillator 318 calorimeter (CAL) [46]. Most important for the analyses combined in this paper is the electro-319 magnetic part of the calorimeters which is used to identify and measure the scattered electron. 320 Electromagnetic energy resolutions σ(E)/E of 0.11/\sqrt{E} (LAr) [47], 0.07/\sqrt{E} (SpaCal) [48] 321 and 0.18/\sqrt{E} (CAL), with E in units of GeV, were achieved.

322 The Bethe–Heitler process, ep → eγp, is used by both experiments to determine the lumi-323 nosity. Photons originating from this reaction were detected by photon taggers at about 100 m 324 downstream of the electron beam line. The integrated luminosities are known with a precision of 325 3.2% for the H1 measurements [15,18] and of about 2% for the ZEUS measurements [6,20,49].

326 Combinations are made for single- and double-differential cross sections. In Table 1 the 327 datasets^{3} used for these combinations are listed together with their visible phase-space regions 328 and integrated luminosities. The datasets I–III are used to determine single-differential com-329 bined cross sections as a function of the D^{∗±} meson’s transverse momentum, p_{T}(D^{∗}), pseu-330 dorapidity, η(D^{∗}), and inelasticity, z(D^{∗}) = (E(D^{∗}) − p_{Z}(D^{∗}))/(2E_{e}y), measured in the lab-331 oratory frame, and of Q^{2} and y. The variables E(D^{∗}), p_{Z}(D^{∗}) and E_{e} denote the energy 332 of the D^{∗±} meson, the Z component of the momentum of the D^{∗±} meson and the incoming 333 electron energy, respectively. Owing to beam-line modifications related to the HERA-II high-334 luminosity running [50] the visible phase space of these cross sections at HERA-II is restricted 335 to Q^{2} > 5 GeV^{2}, which prevents a combination with earlier D^{∗±} cross-section measurements 336 for which the phase space extends down to Q^{2} = 1.5 GeV^{2}.

337 In the case of the double-differential cross section, d^{2}σ/dydQ^{2}, the kinematic range can be 338 extended to lower Q^{2} using HERA-I measurements [4, 6, 10]. In order to minimise the use of 339 correction factors derived from theoretical calculations, the binning scheme of such measure-340 ments has to be similar to that used for the HERA-II data. One of the HERA-I measurements, 341 set IV of Table 1, satisfies this requirement and is therefore included in the combination of 342 this double-differential cross section. The visible phase spaces of the combined single- and 343 double-differential cross sections are summarised in Table 2.

344 The measurements to be combined for the single- and double-differential cross sections are 345 already corrected to the Born level with a running fine-structure constant α and include both 346 the charm and beauty contributions to D^{∗±} production. The total expected beauty contribution 347 is small, varying from ∼ 1% at the lowest Q^{2} to ∼ 7% at the highest Q^{2}. The cross sections

3Of the two sets of measurements in [18], that compatible with the above cuts is chosen.

single  double
Range in  differential cross section
Q^{2}  (GeV^{2}) 5 − 1000  1.5 − 1000
y  0.02 − 0.7
p_{T}(D^{∗}) (GeV)  > 1.5
|η(D^{∗})|  < 1.5

Table 2: Visible phase space of the combined cross sections.

348 measured previously [6, 15, 18] are here corrected to the PDG value [35] of the D^{0} branching 349 ratio.

350 3.1 Treatment of data sets for single-differential cross sections

351 In order to make the input data sets compatible with the phase space quoted in Table 2 and with 352 each other, the following corrections are applied before the combination:

• The H1 collaboration has published measurements of D^{∗±} cross sections separately for 5 GeV^{2} < Q^{2} < 100 GeV^{2} (set I) and for 100 GeV^{2} < Q^{2} < 1000 GeV^{2} (set II). Due to the limited statistics at high Q^{2}, a coarser binning in p_{T}(D^{∗}), η(D^{∗}), z(D^{∗}) and y was used in set II compared to set I. Therefore the cross section in a bin i of a given observable integrated in the range 5 GeV^{2} < Q^{2} < 1000 GeV^{2} is calculated according to

σ_{i}(5 < Q^{2}/GeV^{2} < 1000) = σ_{i}(5 < Q^{2}/GeV^{2} < 100)  (1)

+ σ_{i}^{NLO}(100 < Q^{2}/GeV^{2} < 1000) · \frac{σ_{int}(100 < Q^{2}/GeV^{2} < 1000)}{σ_{int}^{NLO}(100 < Q^{2}/GeV^{2} < 1000)} .

353 Here σ_{int} denotes the integrated visible cross section and σ^{NLO} stands for the NLO pre-354 dictions obtained from HVQDIS. In this calculation both the experimental uncertainties 355 of the visible cross section at high Q^{2} and the theoretical uncertainties as described in 356 Section 2 are included. The contribution from the region 100 GeV^{2} < Q^{2} < 1000 GeV^{2} 357 to the full Q^{2} range amounts to 4% on average and reaches up to 50% at highest p_{T}(D^{∗}).

358 • The bin boundaries used for the differential cross section as a function of Q^{2} differ be-359 tween sets I, II and set III. At low Q^{2} this is solved by combining the cross-section 360 measurements of the first two bins of set I into a single bin. For Q^{2} > 100 GeV^{2} no 361 consistent binning scheme could be defined directly from the single-differential cross-362 section measurements. However, the measurements of the double-differential cross sec-363 tions d^{2}σ/dQ^{2}dy have been performed in a common binning scheme. By integrating 364 these cross sections in y, single-differential cross sections in Q^{2} are obtained also for 365 Q^{2} > 100 GeV^{2} which can be used directly in the combination.

366 • The cross-section measurements in set III are restricted to p_{T}(D^{∗}) < 20 GeV while there 367 is no such limitation in the phase space of the combination. Therefore these cross sections 368 are corrected for the contribution from the range p_{T}(D^{∗}) > 20 GeV using HVQDIS. This 369 correction is found to be less than 0.1%.

370 3.2 Treatment of data sets for double-differential cross sections

371 Since the restriction to the same phase space in Q^{2} does not apply for the combination of the 372 double-differential cross sections in Q^{2} and y, the HERA-I measurement, set IV, is also included 373 in the combination. This allows an extension of the kinematic range down to Q^{2} > 1.5 GeV^{2}. 374 The p_{T}(D^{∗}) ranges of the measurements of sets III and IV are corrected in the same way as for 375 the single-differential cross sections.

376 To make the binning scheme of the HERA-I measurement compatible with that used for 377 the HERA-II datasets, the binning for all datasets is revised. Cross sections in the new bins 378 are obtained from the original bins using the shape of the HVQDIS predictions as described in 379 Section 2. The new binning is given in Section 5 (Table 9). Bins are kept only if they satisfied 380 both of the following criteria:

381 • The predicted fraction of the cross section of the original bin contained in the kinematical 382 overlap region in Q^{2} and y between the original and corrected bins is greater than 50% 383 (in most bins it is greater than 90%).

384 • The theoretical uncertainty from the correction procedure is obtained by evaluating all 385 uncertainties discussed in Section 2 and adding them in quadrature. The ratio of the 386 theoretical uncertainty to the uncorrelated experimental uncertainty is required to be less 387 than 30%, where the theoretical uncertainty is obtained by evaluating all uncertainties 388 discussed in Section 2 and adding them in quadrature.

389 This procedure ensures that the effect of the theoretical uncertainties on the combined data 390 points is small. Most of the HERA-II bins are left unmodified; all of them satisfied the criteria 391 and are kept. Out of the 31 original HERA-I bins, 26 bins satisfy the criteria and are kept. The 392 data points removed from the combination mainly correspond to the low-y region where larger 393 bins were used for the HERA-I data.

394 4 Combination method

The combination of the data sets uses the χ^{2} minimisation method developed for the combination of inclusive DIS cross sections [22,24], as implemented in the HERAverager program [51]. For an individual dataset e the contribution to the χ^{2} function is defined as

χ_{exp,e}^{2}(m^{i},b_{j}) = ∑_{i} \frac{( m^{i}− ∑_{j} γ_{j}^{i,e}m^{i}b_{j} − μ^{i,e}) ^{2}}{(δ_{i,e,stat}μ^{i,e})^{2}+ (δ_{i,e,uncor}m^{i})^{2}} .  (2)

Here μ^{i,e} is the measured value of the cross section in bin i and γ_{j}^{i,e}, δ_{i,e,stat}and δ_{i,e,uncor}are the relative correlated systematic, relative statistical and relative uncorrelated systematic uncertainties, respectively, from the original measurements. The quantities m^{i} express the values of the expected combined cross section for each bin i and the quantities b_{j} express the shifts of the correlated systematic-uncertainty sources j, in units of the standard deviation. Several data sets providing a number of measurements (index e) are represented by a total χ^{2} function, which is built from the sum of the χ_{exp,e}^{2} functions of all data sets

χ_{tot}^{2}(m^{i},b_{j}) = ∑_{e} χ_{exp,e}^{2}(m^{i},b_{j}) + ∑_{j} b_{j}^{2}.  (3)

395 The combined cross sections m^{i} are obtained by the minimisation of χ_{tot}^{2}with respect to m^{i} and

396 b_{j}.

397 The averaging procedure also provides the covariance matrix of the m^{i} and the uncertain-398 ties of the b^{j} at the minimum. The b_{j} at the minimum and their uncertainties are referred to 399 as “shift” and “reduction”, respectively. The covariances V of the m^{i} are given in the form 400 V = V_{uncor}+∑_{k}V_{sys}^{k} [23]. The matrix V_{uncor}is diagonal. Its diagonal elements correspond to the 401 covariances obtained in a weighted average performed in the absence of any correlated system-402 atic uncertainties. The covariance matrix contributions V_{sys}^{k} correspond to correlated systematic 403 uncertainties on the averaged cross sections, such that the elements of a matrix V_{sys}^{k} are obtained 404 as (V_{sys}^{k})_{ij} = δ_{i}^{sys,k}δ_{j}^{sys,k}, given a vector δ^{sys,k} of systematic uncertainties. It is worth noting that, 405 in this representation of the covariance matrix, the number of correlated systematic sources is 406 identical to the number of correlated systematic sources in the input data sets.

407 In the present analysis, the correlated and uncorrelated systematic uncertainties are predom-408 inantly of multiplicative nature, i.e. they change proportionally to the central values. In equa-409 tion (2) the multiplicative nature of these uncertainties is taken into account by multiplying the 410 relative errors γ_{j}^{i,e} and δ_{i,e,uncor}by the cross-section expectation m^{i}. In charm analyses the statis-411 tical uncertainty is mainly background dominated. Therefore it is treated as being independent 412 of m^{i}. For the minimisation of χ_{tot}^{2}an iterative procedure is used as described elsewhere [23].

413 The 55 systematic uncertainties obtained from the original publications were examined for 414 their correlations. Within each data set, most of the systematic uncertainties are found to be 415 point-to-point correlated, and are thus treated as fully correlated in the combination. In to-416 tal there are 23 correlated experimental systematic sources and 5 theory-related uncertainty 417 sources. A few are found to be uncorrelated and added in quadrature. For the combination 418 of single-differential cross sections the uncorrelated uncertainties also include a theory-related 419 uncertainty from the corrections discussed in Section 3, which varies between 0 and 10% of the 420 total uncertainty and is added in quadrature. Asymmetric systematic uncertainties were sym-421 metrised to the larger deviation before performing the combination. Except for the branching-422 ratio uncertainty, which was treated as correlated, all experimental systematic uncertainties were 423 treated as independent between the H1 and ZEUS data sets. Since the distributions in p_{T}(D^{∗}), 424 η(D^{∗}), z(D^{∗}), Q^{2} and y are not statistically independent, each distribution is combined sepa-425 rately.

426 5 Combined cross sections

427 The results of combining the HERA-II measurements [15, 18, 20] as a function of p_{T}(D^{∗}), 428 η(D^{∗}), z(D^{∗}), Q^{2} and y are given in Tables 3 – 7, together with their uncorrelated and correlated 429 uncertainties^{4}. The total uncertainties are obtained by adding the uncorrelated and correlated 430 uncertainties in quadrature.

431 The individual data sets and the results of the combination are shown in Figures 1 – 5. The 432 consistency of the data sets as well as the reduction of the uncertainties are illustrated further by 433 the insets at the bottom of Figures 1 and 4. The combinations in the different variables have a 434 χ^{2} probability varying between 15% and 87%, i.e. the data sets are consistent. The systematic 435 shift between the two input data sets is covered by the respective correlated uncertainties. The 436 shifts and reductions of the correlated uncertainties are given in Table 8. The improvement of 437 the total correlated uncertainty is due to small reductions of many sources. While the effective 438 doubling of the statistics of the combined result reduces the uncorrelated uncertainties, the 439 correlated uncertainties of the combined cross sections are reduced through cross-calibration 440 effects between the two experiments. Typically, both effects contribute about equally to the 441 reduction of the total uncertainty.

4A detailed breakdown of correlated uncertainties can be found on https://www.desy.de/h1zeus/heavy-flavour/dstarcombination/results/cs/.

442 The combined cross sections as a function of p_{T}(D^{∗}), η(D^{∗}), z(D^{∗}), Q^{2} and y are compared 443 to NLO predictions^{5} in Figures 6 – 10. In general, the predictions describe the data well. 444 The data reach an overall precision of about 5% over a large fraction of the measured phase 445 space, while the typical theoretical uncertainty ranges from 30% at low Q^{2} to 10% at high Q^{2}. 446 The data points in the different distributions are statistically and systematically correlated. No 447 attempt is made in this paper to quantify the correlations between bins taken from two different 448 distributions. Thus quantitative comparisons of theory to data can only be made for individual 449 distributions.

450 In order to study the impact of the current theoretical uncertainties in more detail, the effect 451 of some variations on the predictions is shown separately in Figure 11, compared to the same 452 data as in Figures 6, 8 and 10. Only the variations with the largest impact on the respective 453 distribution are shown in each case.

454 The NLO prediction as a function of p_{T}(D^{∗}) (Figure 11, top) describes the data better by 455 either

456 • setting the charm-quark pole mass to 1.35 GeV or

457 • reducing the renormalisation scale by a factor 2 or

458 • increasing the factorisation scale by a factor 2.

459 Simultaneous variation of both scales in the same direction would largely compensate and 460 would therefore have a much smaller effect.

461 The prediction for the z(D^{∗}) distribution (Figure 11, bottom left) describes the shape of the 462 data better if the fragmentation parameters are adjusted such that the boundary between the two 463 lowest fragmentation regions [21] is varied from the default of 70 GeV^{2} to its lower boundary 464 of 30 GeV^{2}.

465 The preference for a reduced renormalisation scale already observed for p_{T}(D^{∗}) is con-466 firmed by the z(D^{∗}) distribution (Figure 11, bottom right). However, the shape of the z(D^{∗}) 467 distribution rather prefers variations of the charm mass and the factorisation scale in the oppo-468 site direction to those found for the p_{T}(D^{∗}) distribution. The distributions of the other kinematic 469 variables do not provide additional information to these findings [52].

470 As stated before, within the large uncertainties indicated by the theory bands in Figures 1 – 471 5, all distributions are reasonably well described. However, the above study shows that the dif-472 ferent contributions to these uncertainties do not only affect the normalisation, but also change 473 the shape of different distributions in different ways. It is therefore not obvious that a variant of 474 the prediction thet gives a good description of the distribution in one variable will also give a 475 good description of the distribution in another.

5The NLO QCD prediction for the beauty contribution to D^{∗±} production, calculated as described in Section 2, can be found on https://www.desy.de/h1zeus/heavy-flavour/dstarcombination/results/beauty/.

476 Based on the above study, a ‘customised’ calculation is performed to investigate the possi-477 bility of obtaining an improved description of the data in all variables at the same time, both 478 in shape and normalisation, within the theoretical uncertainties quoted in Section 2. For this 479 calculation:

480 • The renormalisation scale is reduced by a factor 2, with the factorisation scale unchanged.

481 • The change of the fragmentation parameter ŝ_{1} = 30 GeV^{2} is applied.

482 • At this stage, the resulting distributions are still found to underestimate the data normal-483 isation. As the renormalisation and factorisation scales are recommended to differ by at 484 most a factor of two [53], the only significant remaining handle is the charm-quark pole 485 mass. This mass is set to 1.4 GeV, a value which is also compatible with the partially 486 overlapping data used for a previous dedicated study [21] of the charm-quark mass.

487 • All other parameters, which have a much smaller impact [52] than those discussed above, 488 are left at their central settings as described in Section 2.

489 The result of this customised calculation is indicated as a dotted line in Figures 6 – 10. 490 A reasonable agreement with data is achieved simultaneously in all variables. This a posteri-491 ori adjustment of theory parameters is not a prediction, but it can be taken as a hint in which 492 direction theoretical and phenomenological developments could proceed. The strong improve-493 ment of the description of the data relative to the central prediction through the customisation 494 of the renormalisation scale is in line with the expectation that higher-order calculations will 495 be helpful to obtain a more stringent statement concerning the agreement of perturbative QCD 496 predictions with the data. The improvement from the customisation of one of the fragmentation 497 parameters and the still not fully satisfactory description of the z(D^{∗}) distribution indicate that 498 further dedicated experimental and theoretical studies of the fragmentation treatment might be 499 helpful.

500 In general, the precise single-differential distributions resulting from this combination, in 501 particular those as a function of p_{T}(D^{∗}), η(D^{∗}) and z(D^{∗}), are sensitive to theoretical and 502 phenomenological parameters in a way which complements the sensitivity of more inclusive 503 variables like Q^{2} and y.

504 The combined double-differential cross sections with the uncorrelated, correlated and total 505 uncertainties^{6} as a function of Q^{2} and y are given in Table 9. The total uncertainty is obtained 506 by adding the uncorrelated and correlated uncertainties in quadrature. The individual data sets 507 as well as the results of the combination are shown in Figure ??. Including data set IV slightly 508 reduces the overall cross-section normalisation with respect to the combination of sets I–III 509 only. The pull distribution of the combination is shown in Figure 13. The combination has a χ^{2} 510 probability of 84%, i.e. all data sets are consistent. The shifts and reductions of the correlated 511 uncertainties are given in Table 8.

6A detailed breakdown of correlated uncertainties can be found on https://www.desy.de/h1zeus/ heavy-flavour/dstarcombination/results/cs/combined_q2y.txt.

512 These combined cross sections are compared to NLO predictions^{7} in Figure 14. The cus-513 tomised calculation is also shown. In general the predictions describe the data well. The data 514 have a precision of about 5–10% over a large fraction of the measured phase space, while the 515 estimated theoretical uncertainty ranges from 30% at low Q^{2} to 10% at high Q^{2}. As well as the 516 single-differential distributions, these double-differential distributions give extra input to test 517 further theory improvements.

518 6 Conclusions

519 Measurements of D^{∗±} -production cross sections in deep-inelastic ep scattering by the H1 and 520 ZEUS experiments are combined at the level of visible cross sections, accounting for their sys-521 tematic correlations. The data sets were found to be consistent and the combined data have 522 significantly reduced uncertainties. In contrast to the earlier charm combination at the level of 523 reduced cross sections, the present combination does not have significant theory-related uncer-524 tainties and in addition distributions of kinematic variables of the D^{∗±} mesons are obtained. 525 The combined data are compared to NLO QCD predictions. The predictions describe the data 526 well within their uncertainties. Higher order calculations would be helpful to reduce the theory 527 uncertainty to a level more comparable with the data precision. Further improvements in the 528 treatment of heavy-quark fragmentation would also be desirable.

529 Acknowledgements

530 We are grateful to the HERA machine group whose outstanding efforts have made these experi-531 ments possible. We appreciate the contributions to the construction, maintenance and operation 532 of the H1 and ZEUS detectors of many people who are not listed as authors. We thank our 533 funding agencies for financial support, the DESY technical staff for continuous assistance and 534 the DESY directorate for their support and for the hospitality they extended to the non-DESY 535 members of the collaborations. We would like to give credit to all partners contributing to the 536 EGI computing infrastructure for their support.

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602 [44] A. Polini et al., Nucl. Instrum. Meth. A581, (2007) 656. 603 [45] B. Andrieu et al. [H1 Calorimeter Group], Nucl. Instrum. Meth. A336, (1993) 460. 604 [46] M. Derrick et al., Nucl. Instrum. Meth. A309, (1991) 77; 605 A. Andresen et al.[ZEUS Calorimeter Group and ZEUS Collaborations], Nucl. Instrum. 606 Meth. A309, (1991) 101; 607 A. Caldwell et al., Nucl. Instrum. Meth. A321, (1992) 356; 608 A. Bernstein et al.[ZEUS Barrel Calorimeter Group Collaboration], Nucl. Instrum. Meth. 609 A336, (1993) 23. 610 [47] B. Andrieu et al. [H1 Calorimeter Group], Nucl. Instrum. Meth. A336, (1993) 499. 611 [48] T. Nicholls et al. [H1 SpaCal Group], Nucl. Instrum. Meth. A374, (1996) 149.

612 [49] L. Adamczyk et al., Nucl. Instrum. Meth. A744, (2014) 80 [arXiv:1306.1391]. 613 [50] “The HERA luminosity upgrade”, U. Schneekloth (ed.) (DESY), July 1998, 614 DESY-HERA-98-05.

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616 [52] O. Zenaiev, Charm Production and QCD Analysis at HERA and LHC, PhD thesis, 617 Hamburg University, Hamburg, 2015. 618 [53] O. Behnke et al., Benchmark cross sections for heavy flavour production, in J. Baines et 619 al., Heavy Quarks (Working group 3), Summary report for the HERA-LHC workshop 620 proceedings, [hep-ph/0601164].

p_{T}(D^{∗})  \frac{dσ}{dp_{T}(D^{∗})} δ_{uncor} δ_{cor} δ_{tot}
( GeV)  (nb/ GeV) (%) (%) (%)
1.50 : 1.88  2.35  6.4 4.7 8.0
1.88 : 2.28  2.22  4.9 4.2 6.4
2.28 : 2.68  1.98  3.7 4.0 5.5
2.68 : 3.08  1.55  3.5 3.7 5.1
3.08 : 3.50  1.20  3.7 3.5 5.1
3.50 : 4.00 9.29 × 10^{−1}  3.2 3.4 4.7
4.00 : 4.75 6.14 × 10^{−1}  3.0 3.5 4.6
4.75 : 6.00 3.19 × 10^{−1}  3.1 3.3 4.5
6.00 : 8.00 1.15 × 10^{−1}  3.8 3.7 5.3
8.00 : 11.00 3.32 × 10^{−2}  5.4 3.7 6.5
11.00 : 20.00 3.80 × 10^{−3} 10.4 6.4 12.2

Table 3: The combined differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of p_{T}(D^{∗}), with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

Figure 13: The pull distribution for the combination of the double-differential D^{∗±} cross sections.

18

η(D^{∗})  \frac{dσ}{dη(D^{∗})} δ_{uncor} δ_{cor} δ_{tot}
(nb) (%) (%) (%)
−1.50 : −1.25 1.36 5.8 4.3 7.2
−1.25 : −1.00 1.52 4.6 4.0 6.1
−1.00 : −0.75 1.59 4.6 4.0 6.1
−0.75 : −0.50 1.79 3.8 3.5 5.2
−0.50 : −0.25 1.83 3.8 3.3 5.1
−0.25 : 0.00  1.89 3.8 3.7 5.3
0.00 : 0.25  1.86 4.0 3.4 5.2
0.25 : 0.50  1.88 4.0 3.6 5.4
0.50 : 0.75  1.91 4.1 3.5 5.4
0.75 : 1.00  1.92 4.3 4.0 5.9
1.00 : 1.25  2.08 4.7 4.0 6.1
1.25 : 1.50  1.81 6.3 4.8 7.9

Table 4: The combined differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of η(D^{∗}), with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

z(D^{∗})  \frac{dσ}{dz(D^{∗})} δ_{uncor} δ_{cor} δ_{tot}
(nb) (%) (%) (%)
0.00 : 0.10 3.28 9.5 5.9 11.2
0.10 : 0.20 7.35 4.8 6.3 7.9
0.20 : 0.32 8.61 3.5 4.6 5.7
0.32 : 0.45 8.92 2.7 3.9 4.7
0.45 : 0.57 8.83 1.8 4.0 4.3
0.57 : 0.80 4.78 2.4 5.1 5.6
0.80 : 1.00 0.63 8.1 10.2 13.0

Table 5: The combined differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of z(D^{∗}), with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

Q^{2}  \frac{dσ}{dQ^{2}}  δ_{uncor} δ_{cor} δ_{tot}
( GeV^{2}) (nb/ GeV^{2}) (%) (%) (%)
5 : 8 4.74 × 10^{−1} 4.0 5.0 6.4
8 : 10 2.96 × 10^{−1}  4.3 3.8 5.8
10 : 13 2.12 × 10^{−1}  3.8 4.0 5.6
13 : 19 1.24 × 10^{−1}  3.2 3.8 5.0
19 : 28 7.26 × 10^{−2}  3.5 3.6 5.0
28 : 40 3.97 × 10^{−2}  3.7 4.0 5.5
40 : 60 1.64 × 10^{−2}  4.4 4.7 6.4
60 : 100 7.45 × 10^{−3}  5.2 3.9 6.5
100 : 158 2.08 × 10^{−3}  7.2 5.3 9.0
158 : 251 8.82 × 10^{−4}  7.6 5.0 9.1
251 : 1000 7.50 × 10^{−5} 12.0 6.7 13.3

Table 6: The combined differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of Q^{2}, with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

y  \frac{dσ}{dy} δ_{uncor} δ_{cor} δ_{tot}
(nb) (%) (%) (%)
0.02 : 0.05 12.1 5.8 9.1 10.8
0.05 : 0.09 18.8 3.9 4.6 6.0
0.09 : 0.13 17.0 3.4 4.3 5.5
0.13 : 0.18 13.4 3.7 4.2 5.6
0.18 : 0.26 11.2 3.4 3.7 5.0
0.26 : 0.36 7.65 3.7 4.2 5.6
0.36 : 0.50 4.78 4.0 5.3 6.6
0.50 : 0.70 2.65 5.6 6.4 8.5

Table 7: The combined differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of y, with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

Data set Name  \frac{dσ}{dQ^{2}}  \frac{dσ}{dy}  \frac{dσ}{dp_{T}(D^{∗})} \frac{dσ}{dη(D^{∗})}  \frac{dσ}{dz(D^{∗})}  \frac{d^{2}σ}{dQ^{2}dy}
sh red sh red sh red sh red sh red sh red
I,II  H1 CJC efficiency  0.8 0.9 0.3 0.9 0.5 0.9 0.5 0.9 0.4 0.9 0.6 0.8
I,II  H1 luminosity  0.5 0.9 0.4 0.9 0.6 0.9 0.6 0.9 0.4 0.9 0.1 0.9
I,II  H1 MC PDF  0.1 1.0 0.1 1.0 0.2 1.0 0.2 1.0 0.1 1.0 0.0 1.0
I,II  H1 electron energy  0.2 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.7 0.9 0.0 0.8
I,II  H1 electron polar angle  0.2 1.0 0.1 1.0 0.1 1.0 0.2 1.0 0.2 1.0 0.3 0.9
I,II  H1 hadronic energy scale  0.1 1.0 0.2 0.9 0.0 1.0 0.0 1.0 −1.0 0.7 0.0 1.0
II  H1 fragmentation threshold at high Q^{2} 0.0 1.0  0.0 1.0
I,II  H1 alternative MC model  0.4 0.9 0.4 0.9 0.1 1.0 0.0 1.0 −1.0 0.8 1.2 0.7
I,II  H1 alternative MC fragmentation  0.0 1.0 0.0 1.0 0.0 1.0 −0.1 1.0 0.2 1.0 0.3 0.9
I,II  H1 fragmentation threshold  0.0 1.0 −0.4 0.9 0.2 1.0 0.0 1.0 0.6 0.9 0.2 0.8
I  H1 high Q^{2} uncertainty  0.1 1.0 0.0 0.9 0.1 1.0 0.1 1.0
III  ZEUS hadronic energy scale  0.0 1.0 −0.1 0.8 0.0 1.0 0.0 1.0 −0.9 0.9 −0.5 0.7
III  ZEUS electron energy scale  0.1 0.9 0.2 0.9 0.0 1.0 0.2 1.0 0.0 1.0 0.4 0.7
III  ZEUS p_{T}(π_{s}) correction  −0.1 1.0 −0.1 1.0 −0.1 1.0 −0.3 1.0 0.0 1.0 −0.7 0.9
III  ZEUS M(Kπ) window variation  −0.3 0.8 −0.7 0.8 0.4 0.6 −0.3 0.7 0.5 0.8 −0.7 0.9
III  ZEUS tracking efficiency  −0.2 0.9 −0.4 0.9 −0.4 0.9 −0.2 0.9 −0.2 0.9 −0.7 1.0
III  ZEUS b MC normalisation  0.0 1.0 0.0 1.0 0.0 1.0 0.0 1.0 0.1 1.0 0.0 1.0
III  ZEUS PHP MC normalisation  0.0 1.0 −0.1 1.0 0.0 1.0 −0.1 1.0 0.1 1.0 −0.3 1.0
III  ZEUS diffractive MC normalisation  0.0 1.0 0.1 0.9 0.2 1.0 0.0 1.0 0.0 1.0 0.7 0.9
III  ZEUS MC reweighting (p_{T}(D^{∗}) and Q^{2}) 0.3 0.9 0.0 1.0 −0.1 1.0 0.0 1.0 0.0 1.0 0.6 0.9
III  ZEUS MC reweighting (η(D^{∗}))  0.0 1.0 0.0 0.8 −0.2 1.0 −0.3 1.0 −0.2 1.0 0.4 0.8
III  ZEUS luminosity (HERA-II)  −0.2 1.0 −0.1 1.0 −0.2 1.0 −0.2 1.0 −0.1 1.0 −0.7 0.9
IV  ZEUS luminosity (98-00)  0.8 0.9
I-IV Theory m_{c} variation  0.0 1.0
I-IV Theory μ_{r}, μ_{f} variation  0.0 1.0
I-IV Theory α_{s} variation  0.0 1.0
I-IV Theory longitudunal frag. variation  0.1 1.0
I-IV Theory transverse frag. variation  0.0 1.0

Table 8: Sources of point-to-point correlated uncertainties. For each source the affected data sets are given, together with the shift (sh) and reduction factor (red) in the combination obtained after the first iteration. For sources which do not affect the combination of a given differential cross section, no shifts and reductions are quoted.

Q^{2}  y  \frac{d^{2}σ}{dQ^{2}dy}  δ_{uncor} δ_{cor} δ_{tot}
( GeV^{2})  (nb/ GeV^{2}) (%) (%) (%)
1.5 : 3.5 0.02 : 0.09  4.76  12.9 2.5 13.2
0.09 : 0.16  5.50  11.3 2.6 11.5
0.16 : 0.32  3.00  12.0 2.6 12.3
0.32 : 0.70 9.21 × 10^{−1} 20.5 2.5 20.7
3.5 : 5.5 0.02 : 0.09  2.22  11.3 2.8 11.6
0.09 : 0.16  1.98  7.9 2.7 8.3
0.16 : 0.32  1.09  20.2 2.7 20.4
0.32 : 0.70 3.47 × 10^{−1} 14.6 2.6 14.8
5.5 : 9 0.02 : 0.05  1.06  12.3 4.4 13.1
0.05 : 0.09  1.46  7.8 4.1 8.8
0.09 : 0.16  1.32  5.4 4.3 6.9
0.16 : 0.32 7.73 × 10^{−1}  4.9 3.9 6.3
0.32 : 0.70 2.51 × 10^{−1}  5.6 4.2 7.0
9 : 14 0.02 : 0.05 5.20 × 10^{−1} 13.0 6.6 14.6
0.05 : 0.09 7.68 × 10^{−1}  6.6 3.9 7.7
0.09 : 0.16 5.69 × 10^{−1}  4.6 2.8 5.4
0.16 : 0.32 4.12 × 10^{−1}  4.6 3.1 5.6
0.32 : 0.70 1.51 × 10^{−1}  5.6 4.0 6.9
14 : 23 0.02 : 0.05 2.29 × 10^{−1} 11.4 6.3 13.0
0.05 : 0.09 3.78 × 10^{−1}  6.5 4.1 7.7
0.09 : 0.16 2.90 × 10^{−1}  4.8 3.3 5.8
0.16 : 0.32 1.86 × 10^{−1}  5.0 3.4 6.0
0.32 : 0.70 6.92 × 10^{−2}  6.2 4.4 7.7
23 : 45 0.02 : 0.05 6.91 × 10^{−2} 14.8 8.2 16.7
0.05 : 0.09 1.23 × 10^{−1}  5.9 3.6 6.9
0.09 : 0.16 1.14 × 10^{−1}  4.4 3.0 5.3
0.16 : 0.32 7.42 × 10^{−2}  4.3 3.0 5.2
0.32 : 0.70 3.21 × 10^{−2}  5.2 3.7 6.4
45 : 100 0.02 : 0.05 6.16 × 10^{−3} 33.5 11.1 35.3
0.05 : 0.09 2.70 × 10^{−2} 11.0 4.4 11.8
0.09 : 0.16 2.05 × 10^{−2}  8.0 3.7 8.8
0.16 : 0.32 1.99 × 10^{−2}  5.4 3.2 6.3
0.32 : 0.70 7.84 × 10^{−3}  6.9 4.0 7.9
100 : 158 0.02 : 0.32 4.12 × 10^{−3}  8.2 4.1 9.2
0.32 : 0.70 2.18 × 10^{−3} 11.1 4.1 11.9
158 : 251 0.02 : 0.30 1.79 × 10^{−3} 10.2 4.4 11.1
0.30 : 0.70 9.28 × 10^{−4} 11.6 4.6 12.5
251 : 1000 0.02 : 0.26 1.31 × 10^{−4} 14.5 4.7 15.3
0.26 : 0.70 1.18 × 10^{−4} 12.7 5.0 13.6

Table 9: The combined double-differential D^{∗±} -production cross section in the phase space given in Table 2 as a function of Q^{2} and y, with its uncorrelated (δ_{uncor}), correlated (δ_{cor}) and total (δ_{tot}) uncertainties.

Figure 1: Differential D^{∗±} -production cross section as a function of p_{T}(D^{∗}). The open triangles and squares are the cross sections before combination, shown with a small horizontal offset for better visibility. The filled points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. The histogram indicates the binning used to calculate the cross sections. The bottom part shows the ratio of these cross sections with respect to the central value of the combined cross sections.

Figure 2: Differential D^{∗±} -production cross section as a function of η(D^{∗}). The open triangles and squares are the cross sections before combination, shown with a small horizontal offset for better visibility. The filled points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. The histogram indicates the binning used to calculate the cross sections.

Figure 3: Differential D^{∗±} -production cross section as a function of z(D^{∗}). The open triangles and squares are the cross sections before combination, shown with a small horizontal offset for better visibility. The filled points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. The histogram indicates the binning used to calculate the cross sections.

Figure 4: Differential D^{∗±} -production cross section as a function of Q^{2}. The open triangles and squares are the cross sections before combination, shown with a small horizontal offset for better visibility. The filled points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. The histogram indicates the binning used to calculate the cross sections. The bottom part shows the ratio of these cross sections with respect to the central value of the combined cross sections.

Figure 5: Differential D^{∗±} -production cross section as a function of y. The open triangles and squares are the cross sections before combination, shown with a small horizontal offset for better visibility. The filled points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. The histogram indicates the binning used to calculate the cross sections.

Figure 6: Differential D^{∗±} -production cross section as a function of p_{T}(D^{∗}). The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown. The bottom part shows the ratio of these cross sections with respect to the central value of the combined cross sections.

Figure 7: Differential D^{∗±} -production cross section as a function of η(D^{∗}). The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown.

Figure 8: Differential D^{∗±} -production cross section as a function of z(D^{∗}). The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown.

Figure 9: Differential D^{∗±} -production cross section as a function of Q^{2}. The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown. The bottom part shows the ratio of these cross sections with respect to the central value of the combined cross sections.

Figure 10: Differential D^{∗±} -production cross section as a function of y. The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown.

Figure 11: Differential D^{∗±} -production cross section (ratio to data) as a function of p_{T}(D^{∗}) (top) and z(D^{∗}) (bottom) compared to NLO predictions with different variations: charm-quark mass m_{c}, renormalisation scale μ_{r}, factorisation scale μ_{f} and fragmentation bin boundary ŝ_{1}.

Figure 14: Double-differential D^{∗±} -production cross section as a function of Q^{2} and y. The data points are the combined cross sections. The inner error bars indicate the uncorrelated part of the uncertainties. The outer error bars represent the total uncertainties. Also shown are the NLO predictions from HVQDIS (including the beauty contribution) and their uncertainty band. A customised NLO calculation (dotted line, see text) is also shown.