%================================================================ % LaTeX file with preferred layout for H1 paper drafts % use: dvips -D600 file-name %================================================================ \documentclass[12pt]{article} \usepackage{epsfig} \usepackage{wrapfig} \usepackage{subfigure} %\usepackage{lineno} \usepackage{amsmath} \usepackage{hhline} \usepackage{amssymb} \usepackage{times} \usepackage{cite} %%%%%%%%%%%% Comment the next two lines to remove the line numbering %\usepackage[]{lineno} %\linenumbers %%%%%%%%%%%% \renewcommand{\topfraction}{1.0} \renewcommand{\bottomfraction}{1.0} \renewcommand{\textfraction}{0.0} \newlength{\dinwidth} \newlength{\dinmargin} \setlength{\dinwidth}{21.0cm} \textheight23.5cm \textwidth16.0cm \setlength{\dinmargin}{\dinwidth} \setlength{\unitlength}{1mm} \addtolength{\dinmargin}{-\textwidth} \setlength{\dinmargin}{0.5\dinmargin} \oddsidemargin -1.0in \addtolength{\oddsidemargin}{\dinmargin} 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\newcommand{\dgr}{^\circ} \newcommand{\pbarnt}{\,\mbox{{\rm pb$^{-1}$}}} % % DVCS KIN % \newcommand{\tg}{\theta_{\gamma}} \newcommand{\te}{\theta_e} % % Some useful tex commands % % kinematic variable \newcommand{\qsq}{\mbox{$Q^2$}} \newcommand{\Qsq}{\mbox{$Q^2$}} \newcommand{\s}{\mbox{$s$}} \newcommand{\ttra}{\mbox{$t$}} \newcommand{\modt}{\mbox{$|t|$}} \newcommand{\eminpz}{\mbox{$E-p_z$}} \newcommand{\eminpzs}{\mbox{$\Sigma(E-p_z)$}} \newcommand{\rap}{\ensuremath{\eta^*} } \newcommand{\W}{\mbox{$W$}} \newcommand{\w}{\mbox{$W$}} \newcommand{\Q}{\mbox{$Q$}} \newcommand{\q}{\mbox{$Q$}} \newcommand{\xB}{\mbox{$x$}} % Bjorken x \newcommand{\xF}{\mbox{$x_F$}} % Feynman x \newcommand{\xg}{\mbox{$x_g$}} % x_g \newcommand{\xbj}{x} \newcommand{\xpom}{x_{\PO}} \newcommand{\y}{\mbox{$y~$}} \newcommand{\gp}{\ensuremath{\gamma^*}p } \newcommand{\gammasp}{\ensuremath{\gamma}*p } \newcommand{\gammap}{\ensuremath{\gamma}p } \newcommand{\gsp}{\ensuremath{\gamma^*}p } % ep \newcommand{\epem}{\mbox{$e^+e^-$}} \newcommand{\ep}{\mbox{$ep~$}} \newcommand{\epl}{\mbox{$e^{+}$}} \newcommand{\emi}{\mbox{$e^{-}$}} \newcommand{\epm}{\mbox{$e^{\pm}$}} %===================================================================== % elastic VM abbreviations % % VM \newcommand{\photon}{\mbox{$\gamma$}} \newcommand{\phib}{\mbox{$\varphi$}} \newcommand{\rh}{\mbox{$\rho$}} \newcommand{\rhz}{\mbox{$\rh^0$}} \newcommand{\ph}{\mbox{$\phi$}} \newcommand{\om}{\mbox{$\omega$}} \newcommand{\ome}{\mbox{$\omega$}} \newcommand{\jpsi}{\mbox{$J/\psi$}} \newcommand{\JPSI}{J/\psi} \newcommand{\ups}{\mbox{$\Upsilon$}} % b parameter \newcommand{\bsl}{\mbox{$b$}} % %===================================================================== % units % \newcommand{\cm}{\mbox{\rm cm}} \newcommand{\GeV}{\mbox{\rm GeV}} \newcommand{\gev}{\mbox{\rm GeV}} \newcommand{\GeVx}{\rm GeV} \newcommand{\gevx}{\rm GeV} \newcommand{\MeV}{\mbox{\rm MeV}} \newcommand{\mev}{\mbox{\rm MeV}} \newcommand{\MeVx}{\mbox{\rm MeV}} \newcommand{\mevx}{\mbox{\rm MeV}} \newcommand{\GeVsq}{\mbox{${\rm GeV}^2$}} \newcommand{\gevsq}{\mbox{${\rm GeV}^2$}} \newcommand{\gevsqc}{\mbox{${\rm GeV^2/c^4}$}} \newcommand{\gevcsq}{\mbox{${\rm GeV/c^2}$}} \newcommand{\mevcsq}{\mbox{${\rm MeV/c^2}$}} \newcommand{\GeVsqm}{\mbox{${\rm GeV}^{-2}$}} \newcommand{\gevsqm}{\mbox{${\rm GeV}^{-2}$}} \newcommand{\nb}{\mbox{${\rm nb}$}} \newcommand{\nbinv}{\mbox{${\rm nb^{-1}}$}} \newcommand{\pbinv}{\mbox{${\rm pb^{-1}}$}} \newcommand{\mm}{\mbox{$\cdot 10^{-2}$}} \newcommand{\mmm}{\mbox{$\cdot 10^{-3}$}} \newcommand{\mmmm}{\mbox{$\cdot 10^{-4}$}} \newcommand{\degr}{\mbox{$^{\circ}$}} % %===================================================================== % inequation symbols % %greater than or approx. symbol \def\gsim{\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex% \raise 0.55ex\hbox{$\scriptstyle >$}\,} %less than or approx. symbol \def\lsim{\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex% \raise 0.55ex\hbox{$\scriptstyle <$}\,} % % %===================================================================== %%\linenumbers \begin{titlepage} \noindent \begin{flushleft} {\tt DESY 09-109 \hfill ISSN 0418-9833} \\ {\tt July 2009} \\ \end{flushleft} \vspace{2cm} \begin{center} \begin{Large} {\bf Deeply Virtual Compton Scattering and its Beam Charge Asymmetry in \begin{boldmath}$e^\pm p$\end{boldmath} Collisions at HERA } \vspace{2cm} H1 Collaboration \end{Large} \end{center} \vspace{2cm} %========================================================================= \begin{abstract} A measurement of elastic deeply virtual Compton scattering $\gamma^* p \rightarrow \photon p$ using $e^+ p$ and $e^- p$ collision data recorded with the H1 detector at HERA is presented. The analysed data sample corresponds to an integrated luminosity of $306$~pb$^{-1}$, almost equally shared between both beam charges. The cross section is measured as a function of the virtuality $Q^2$ of the exchanged photon and the centre-of-mass energy $W$ of the $\gamma^*p$ system in the kinematic domain $6.5 < Q^2 < 80$~GeV$^2$, $30 < W < 140$~GeV and $|t| < 1$~GeV$^2$, where $t$ denotes the squared momentum transfer at the proton vertex. The cross section is determined differentially in $t$ for different $Q^2$ and $W$ values and exponential $t$-slope parameters are derived. Using $e^+ p$ and $e^- p$ data samples, a beam charge asymmetry is extracted for the first time in the low Bjorken $x$ kinematic domain. The observed asymmetry is attributed to the interference between Bethe-Heitler and deeply virtual Compton scattering processes. % Experimental results are discussed in the context of two different models, one based on generalised parton distributions and one based on the dipole approach. \end{abstract} %========================================================================= \vspace{1.5cm} \begin{center} Accepted by Phys. Lett. {\bf B} \end{center} \end{titlepage} \begin{flushleft} %-- H1AUTS Author list by names %-- Status: Wed Jul 1 09:19:31 CEST 2009 Number of authors = 244 F.D.~Aaron$^{5,49}$, %BUCH-PD 11/06 Aaron M.~Aldaya~Martin$^{11}$, %DESY-PD 10/08 Aldayamartin C.~Alexa$^{5}$, %BUCH-PD 06/06 Alexa K.~Alimujiang$^{11}$, %DESY-PD 07/08 Alimujiang V.~Andreev$^{25}$, %LPI -PD 8/88 Andreev B.~Antunovic$^{11}$, %DESY-LEFT 12/08 Antunovic S.~Backovic$^{30}$, %PODG-PD 03/2 Backovic A.~Baghdasaryan$^{38}$, %YERE-PD 09/03 Baghdasaryana E.~Barrelet$^{29}$, %PARI-PD 11/99 Barrelet W.~Bartel$^{11}$, %DESY-PD 8/88 Bartel K.~Begzsuren$^{35}$, %ULBA-PD 04/06 Begzsuren A.~Belousov$^{25}$, %LPI -PD 8/88 Belousov J.C.~Bizot$^{27}$, %ORSA-PD 8/88 Bizot V.~Boudry$^{28}$, %ECPL-PD 1/93 Boudry I.~Bozovic-Jelisavcic$^{2}$, %BEOG-PD 03/06 Bozovicjelisavcic J.~Bracinik$^{3}$, %BIRM-PD 01/2 Bracinik G.~Brandt$^{11}$, %DESY-PD 01/20 Brandt M.~Brinkmann$^{12}$, %HAM2-ST 02/09 Brinkmann V.~Brisson$^{27}$, %ORSA-PD 8/88 Brisson D.~Bruncko$^{16}$, %KOSI-PD 8/88 Bruncko A.~Bunyatyan$^{13,38}$, %MPIH-PD 12/95 Bunyatyan G.~Buschhorn$^{26}$, %MPIM-PD 8/88 Buschhorn L.~Bystritskaya$^{24}$, %ITEP-PD 05/99 Bystritskaya A.J.~Campbell$^{11}$, %DESY-PD 8/88 Campbella K.B.~Cantun~Avila$^{22}$, %MEX1-ST 04/06 Cantunavila K.~Cerny$^{32}$, %PRG2-PD 09/08 Cernyk V.~Cerny$^{16,47}$, %KOSI-PD 06/04 Cernyv V.~Chekelian$^{26}$, %MPIM-PD 01/90 Chekelian A.~Cholewa$^{11}$, %DESY-ST 11/05 Cholewa J.G.~Contreras$^{22}$, %MEX1-PD 04/97 Contreras J.A.~Coughlan$^{6}$, %RAL -PD 8/88 Coughlan G.~Cozzika$^{10}$, %SACL-PD 10/07 Cozzika J.~Cvach$^{31}$, %PRAG-PD 8/88 Cvach J.B.~Dainton$^{18}$, %LIVE-PD 8/88 Dainton K.~Daum$^{37,43}$, %WUPP-PD 06/96 Daum M.~De\'{a}k$^{11}$, %DESY-ST 08/06 Deak Y.~de~Boer$^{11}$, %DESY-LEFT 08/08 Deboer B.~Delcourt$^{27}$, %ORSA-PD 8/88 Delcourt M.~Del~Degan$^{40}$, %ZUTH-LEFT 09/08 Deldegan J.~Delvax$^{4}$, %BRUX-ST 10/06 Delvax E.A.~De~Wolf$^{4}$, %ANTW-PD 3/93 Dewolf C.~Diaconu$^{21}$, %MARS-PD 01/05 Diaconu V.~Dodonov$^{13}$, %MPIH-PD 04/98 Dodonov A.~Dossanov$^{26}$, %MPIM-ST 01/07 Dossanov A.~Dubak$^{30,46}$, %PODG-PD 10/03 Dubak G.~Eckerlin$^{11}$, %DESY-PD 8/88 Eckerlin V.~Efremenko$^{24}$, %ITEP-PD 8/88 Efremenko S.~Egli$^{36}$, %PSI -PD 01/01 Egli A.~Eliseev$^{25}$, %LPI -PD 01/06 Eliseev E.~Elsen$^{11}$, %DESY-PD 8/88 Elsen A.~Falkiewicz$^{7}$, %CRAC-LEFT 03/09 Falkiewicz L.~Favart$^{4}$, %BRUX-PD 8/88 Favart A.~Fedotov$^{24}$, %ITEP-PD 8/88 Fedotov R.~Felst$^{11}$, %DESY-PD 11/0 Felst J.~Feltesse$^{10,48}$, %SACL-PD 03/05 Feltesse J.~Ferencei$^{16}$, %KOSI-PD 01/05 Ferencei D.-J.~Fischer$^{11}$, %DESY-ST 03/08 Fischer M.~Fleischer$^{11}$, %DESY-PD 07/0 Fleischer A.~Fomenko$^{25}$, %LPI -PD 8/88 Fomenko E.~Gabathuler$^{18}$, %LIVE-PD 10/89 Gabathulere J.~Gayler$^{11}$, %DESY-PD 8/88 Gayler S.~Ghazaryan$^{38}$, %YERE-PD 8/88 Ghazaryan A.~Glazov$^{11}$, %DESY-PD 01/04 Glazov I.~Glushkov$^{39}$, %ZEUT-LEFT 11/08 Glushkov L.~Goerlich$^{7}$, %CRAC-PD 8/88 Goerlich N.~Gogitidze$^{25}$, %LPI -PD 8/88 Gogitidze M.~Gouzevitch$^{11}$, %DESY-LEFT 06/09 Gouzevitch C.~Grab$^{40}$, %ZUTH-PD 8/88 Grab T.~Greenshaw$^{18}$, %LIVE-PD 8/88 Greenshaw B.R.~Grell$^{11}$, %DESY-ST 09/04 Grell G.~Grindhammer$^{26}$, %MPIM-PD 8/88 Grindhammer S.~Habib$^{12,50}$, %HAM2-ST 12/05 Habib D.~Haidt$^{11}$, %DESY-PD 8/88 Haidt C.~Helebrant$^{11}$, %DFLC-ST 03/06 Helebrant R.C.W.~Henderson$^{17}$, %LANC-PD 8/88 Henderson E.~Hennekemper$^{15}$, %HDB2-ST 11/07 Hennekemper H.~Henschel$^{39}$, %ZEUT-PD 06/99 Henschel M.~Herbst$^{15}$, %HDB2-ST 06/08 Herbst G.~Herrera$^{23}$, %MEX2-PD 07/98 Herrera M.~Hildebrandt$^{36}$, %PSI -PD 10/99 Hildebrandtm K.H.~Hiller$^{39}$, %ZEUT-PD 8/88 Hiller D.~Hoffmann$^{21}$, %MARS-PD 10/0 Hoffmann R.~Horisberger$^{36}$, %PSI -PD 8/88 Horisberger T.~Hreus$^{4,44}$, %BRUX-PD 10/08 Hreus M.~Jacquet$^{27}$, %ORSA-PD 09/96 Jacquet M.E.~Janssen$^{11}$, %DFLC-LEFT 07/08 Janssenm X.~Janssen$^{4}$, %BRUX-PD 02/03 Janssenx L.~J\"onsson$^{20}$, %LUND-PD 8/88 Joensson A.W.~Jung$^{15}$, %HDB2-PD 06/09 Junga H.~Jung$^{11}$, %DESY-PD 07/00 Jungh M.~Kapichine$^{9}$, %JINR-PD 3/97 Kapichine J.~Katzy$^{11}$, %DESY-PD 09/1 Katzy I.R.~Kenyon$^{3}$, %BIRM-PD 8/88 Kenyon C.~Kiesling$^{26}$, %MPIM-PD 8/88 Kiesling M.~Klein$^{18}$, %LIVE-PD 8/88 Klein C.~Kleinwort$^{11}$, %DESY-PD 8/88 Kleinwort T.~Kluge$^{18}$, %LIVE-PD 05/04 Kluge A.~Knutsson$^{11}$, %DESY-PD 04/07 Knutsson R.~Kogler$^{26}$, %MPIM-ST 01/07 Kogler P.~Kostka$^{39}$, %ZEUT-PD 8/88 Kostka M.~Kraemer$^{11}$, %DESY-ST 02/06 Kraemer K.~Krastev$^{11}$, %DESY-LEFT 12/08 Krastev J.~Kretzschmar$^{18}$, %LIVE-PD 01/08 Kretzschmar A.~Kropivnitskaya$^{24}$, %ITEP-PD 06/09 Kropivnitskaya K.~Kr\"uger$^{15}$, %HDB2-PD 01/04 Kruegerk K.~Kutak$^{11}$, %DESY-PD 01/07 Kutak M.P.J.~Landon$^{19}$, %QMWC-PD 8/88 Landon W.~Lange$^{39}$, %ZEUT-PD 8/88 Lange G.~La\v{s}tovi\v{c}ka-Medin$^{30}$, %PODG-PD 06/04 Lastovickamedin P.~Laycock$^{18}$, %LIVE-PD 11/03 Laycock A.~Lebedev$^{25}$, %LPI -PD 8/88 Lebedev G.~Leibenguth$^{40}$, %ZUTH-LEFT 09/08 Leibenguth V.~Lendermann$^{15}$, %HDB2-PD 01/2 Lendermann S.~Levonian$^{11}$, %DESY-PD 8/88 Levonian G.~Li$^{27}$, %ORSA-LEFT 12/08 Li K.~Lipka$^{11}$, %DESY-PD 01/03 Lipka A.~Liptaj$^{26}$, %MPIM-LEFT 06/09 Liptaj B.~List$^{12}$, %HAM2-PD 11/99 Listb J.~List$^{11}$, %DFLC-PD 01/05 Listj N.~Loktionova$^{25}$, %LPI -PD 03/99 Loktionova R.~Lopez-Fernandez$^{23}$, %MEX2-PD 03/2 Lopezfernandez V.~Lubimov$^{24}$, %ITEP-PD 01/95 Lubimov A.~Makankine$^{9}$, %JINR-PD 11/02 Makankine E.~Malinovski$^{25}$, %LPI -PD 01/89 Malinovskie P.~Marage$^{4}$, %BRUX-PD 8/88 Marage Ll.~Marti$^{11}$, %DESY-LEFT 04/09 Marti H.-U.~Martyn$^{1}$, %AAC1-PD 8/88 Martyn S.J.~Maxfield$^{18}$, %LIVE-PD 8/88 Maxfield A.~Mehta$^{18}$, %LIVE-PD 8/88 Mehta A.B.~Meyer$^{11}$, %DESY-PD 01/00 Meyeran H.~Meyer$^{11}$, %DFLC-LEFT 11/08 Meyerhe H.~Meyer$^{37}$, %WUPP-PD 8/88 Meyerhi J.~Meyer$^{11}$, %DESY-PD 8/88 Meyerj V.~Michels$^{11}$, %DESY-LEFT 08/08 Michels S.~Mikocki$^{7}$, %CRAC-PD 8/88 Mikocki I.~Milcewicz-Mika$^{7}$, %CRAC-ST 10/02 Milcewicz F.~Moreau$^{28}$, %ECPL-PD 01/90 Moreau A.~Morozov$^{9}$, %JINR-PD 06/99 Morozova J.V.~Morris$^{6}$, %RAL -PD 8/88 Morris M.U.~Mozer$^{4}$, %BRUX-PD 06/07 Mozer M.~Mudrinic$^{2}$, %BEOG-PD 01/07 Mudrinic K.~M\"uller$^{41}$, %ZUER-PD 8/88 Muellerk P.~Mur\'\i n$^{16,44}$, %KOSI-LEFT 02/09 Murin Th.~Naumann$^{39}$, %ZEUT-PD 01/89 Naumannt P.R.~Newman$^{3}$, %BIRM-PD 10/92 Newman C.~Niebuhr$^{11}$, %DESY-PD 3/93 Niebuhr A.~Nikiforov$^{11}$, %DESY-PD 05/07 Nikiforov D.~Nikitin$^{9}$, %JINR-PD 06/08 Nikitin G.~Nowak$^{7}$, %CRAC-PD 8/88 Nowakg K.~Nowak$^{41}$, %ZUER-ST 08/05 Nowakk M.~Nozicka$^{11}$, %DESY-LEFT 09/08 Nozicka B.~Olivier$^{26}$, %MPIM-LEFT 09/08 Olivier J.E.~Olsson$^{11}$, %DESY-PD 8/88 Olsson S.~Osman$^{20}$, %LUND-PD 06/09 Osman D.~Ozerov$^{24}$, %ITEP-ST 08/98 Ozerov V.~Palichik$^{9}$, %JINR-PD 01/04 Palichik I.~Panagoulias$^{l,}$$^{11,42}$, %DESY-ST 08/04 Panagoulias M.~Pandurovic$^{2}$, %BEOG-ST 03/06 Pandurovic Th.~Papadopoulou$^{l,}$$^{11,42}$, %DESY-PD 06/04 Papadopoulou C.~Pascaud$^{27}$, %ORSA-PD 8/88 Pascaud G.D.~Patel$^{18}$, %LIVE-PD 8/88 Patel O.~Pejchal$^{32}$, %PRG2-LEFT 10/08 Pejchal E.~Perez$^{10,45}$, %SACL-PD 10/07 Perez A.~Petrukhin$^{24}$, %ITEP-ST 01/01 Petrukhin I.~Picuric$^{30}$, %PODG-PD 01/06 Picuric S.~Piec$^{39}$, %ZEUT-ST 01/06 Piec D.~Pitzl$^{11}$, %DESY-PD 8/88 Pitzl R.~Pla\v{c}akyt\.{e}$^{11}$, %DESY-PD 10/06 Placakyte B.~Pokorny$^{12}$, %HAM2-LEFT 04/09 Pokorny R.~Polifka$^{32}$, %PRG2-ST 10/06 Polifka B.~Povh$^{13}$, %MPIH-PD 8/88 Povh V.~Radescu$^{11}$, %DESY-PD 10/06 Radescu A.J.~Rahmat$^{18}$, %LIVE-ST 06/09 Rahmat N.~Raicevic$^{30}$, %PODG-PD 03/2 Raicevic A.~Raspiareza$^{26}$, %MPIM-LEFT 03/09 Raspiareza T.~Ravdandorj$^{35}$, %ULBA-PD 06/06 Ravdandorj P.~Reimer$^{31}$, %PRAG-PD 8/88 Reimer E.~Rizvi$^{19}$, %QMWC-PD 01/05 Rizvi P.~Robmann$^{41}$, %ZUER-PD 8/88 Robmann B.~Roland$^{4}$, %BRUX-LEFT 11/08 Roland R.~Roosen$^{4}$, %BRUX-PD 8/88 Roosen A.~Rostovtsev$^{24}$, %ITEP-PD 8/88 Rostovtsev M.~Rotaru$^{5}$, %BUCH-ST 02/07 Rotaru J.E.~Ruiz~Tabasco$^{22}$, %MEX1-ST 09/06 Ruiztabascojuliaelis Z.~Rurikova$^{11}$, %DESY-LEFT 09/08 Rurikova S.~Rusakov$^{25}$, %LPI -PD 8/88 Rusakov D.~\v S\'alek$^{32}$, %PRG2-ST 11/06 Salek D.P.C.~Sankey$^{6}$, %RAL -PD 8/88 Sankey M.~Sauter$^{40}$, %ZUTH-ST 10/05 Sauter E.~Sauvan$^{21}$, %MARS-PD 11/1 Sauvan S.~Schmitt$^{11}$, %DESY-PD 09/07 Schmittst L.~Schoeffel$^{10}$, %SACL-PD 12/98 Schoeffel A.~Sch\"oning$^{14}$, %HDB1-PD 04/09 Schoening H.-C.~Schultz-Coulon$^{15}$, %HDB2-PD 01/04 Schultzcoulon F.~Sefkow$^{11}$, %DFLC-PD 09/99 Sefkow R.N.~Shaw-West$^{3}$, %BIRM-ST 10/04 Shawwest L.N.~Shtarkov$^{25}$, %LPI -PD 8/88 Shtarkov S.~Shushkevich$^{26}$, %MPIM-ST 08/07 Shushkevich T.~Sloan$^{17}$, %LANC-PD 1/96 Sloan I.~Smiljanic$^{2}$, %BEOG-PD 03/06 Smiljanic Y.~Soloviev$^{25}$, %LPI -PD 8/88 Soloviev P.~Sopicki$^{7}$, %CRAC-ST 09/07 Sopicki D.~South$^{8}$, %DORT-PD 06/03 South V.~Spaskov$^{9}$, %JINR-PD 12/97 Spaskov A.~Specka$^{28}$, %ECPL-PD 3/95 Specka Z.~Staykova$^{11}$, %DESY-ST 08/06 Staykova M.~Steder$^{11}$, %DESY-PD 09/08 Steder B.~Stella$^{33}$, %ROME-PD 8/88 Stella G.~Stoicea$^{5}$, %BUCH-PD 02/08 Stoicea U.~Straumann$^{41}$, %ZUER-PD 8/88 Straumann D.~Sunar$^{4}$, %ANTW-ST 03/05 Sunar T.~Sykora$^{4}$, %ANTW-PD 01/06 Sykora V.~Tchoulakov$^{9}$, %JINR-PD 05/03 Tchoulakov G.~Thompson$^{19}$, %QMWC-PD 8/88 Thompsong P.D.~Thompson$^{3}$, %BIRM-PD 08/99 Thompsonp T.~Toll$^{12}$, %HAM2-ST 11/08 Toll F.~Tomasz$^{16}$, %KOSI-LEFT 12/08 Tomasz T.H.~Tran$^{27}$, %ORSA-ST 10/06 Tran D.~Traynor$^{19}$, %QMWC-PD 12/01 Traynor T.N.~Trinh$^{21}$, %MARS-LEFT 10/08 Trinh P.~Tru\"ol$^{41}$, %ZUER-PD 8/88 Truoel I.~Tsakov$^{34}$, %SOFI-PD 04/03 Tsakov B.~Tseepeldorj$^{35,51}$, %ULBA-PD 06/06 Tseepeldorj J.~Turnau$^{7}$, %CRAC-PD 8/88 Turnau K.~Urban$^{15}$, %HDB2-PD 06/09 Urbank A.~Valk\'arov\'a$^{32}$, %PRG2-PD 8/88 Valkarova C.~Vall\'ee$^{21}$, %MARS-PD 8/88 Vallee P.~Van~Mechelen$^{4}$, %ANTW-PD 12/98 Vanmechelen A.~Vargas Trevino$^{11}$, %DFLC-PD 02/07 Vargastrevino Y.~Vazdik$^{25}$, %LPI -PD 8/88 Vazdik S.~Vinokurova$^{11}$, %DESY-LEFT 10/08 Vinokurova V.~Volchinski$^{38}$, %YERE-PD 12/01 Volchinski M.~von~den~Driesch$^{11}$, %DESY-ST 06/08 Vondendriesch D.~Wegener$^{8}$, %DORT-PD 8/88 Wegener Ch.~Wissing$^{11}$, %DESY-PD 07/06 Wissing E.~W\"unsch$^{11}$, %DESY-PD 8/88 Wuensch J.~\v{Z}\'a\v{c}ek$^{32}$, %PRG2-PD 8/88 Zacek J.~Z\'ale\v{s}\'ak$^{31}$, %PRAG-PD 01/05 Zalesak Z.~Zhang$^{27}$, %ORSA-PD 10/92 Zhang A.~Zhokin$^{24}$, %ITEP-PD 04/99 Zhokine T.~Zimmermann$^{40}$, %ZUTH-LEFT 01/09 Zimmermannt H.~Zohrabyan$^{38}$, %YERE-PD 11/02 Zohrabyan F.~Zomer$^{27}$, %ORSA-PD 8/88 Zomer and R.~Zus$^{5}$ %BUCH-PD 07/08 Zus %-- H1 Institutes \bigskip{\it $ ^{1}$ I. Physikalisches Institut der RWTH, Aachen, Germany \\ $ ^{2}$ Vinca Institute of Nuclear Sciences, Belgrade, Serbia \\ $ ^{3}$ School of Physics and Astronomy, University of Birmingham, Birmingham, UK$^{ b}$ \\ $ ^{4}$ Inter-University Institute for High Energies ULB-VUB, Brussels; Universiteit Antwerpen, Antwerpen; Belgium$^{ c}$ \\ $ ^{5}$ National Institute for Physics and Nuclear Engineering (NIPNE) , Bucharest, Romania \\ $ ^{6}$ Rutherford Appleton Laboratory, Chilton, Didcot, UK$^{ b}$ \\ $ ^{7}$ Institute for Nuclear Physics, Cracow, Poland$^{ d}$ \\ $ ^{8}$ Institut f\"ur Physik, TU Dortmund, Dortmund, Germany$^{ a}$ \\ $ ^{9}$ Joint Institute for Nuclear Research, Dubna, Russia \\ $ ^{10}$ CEA, DSM/Irfu, CE-Saclay, Gif-sur-Yvette, France \\ $ ^{11}$ DESY, Hamburg, Germany \\ $ ^{12}$ Institut f\"ur Experimentalphysik, Universit\"at Hamburg, Hamburg, Germany$^{ a}$ \\ $ ^{13}$ Max-Planck-Institut f\"ur Kernphysik, Heidelberg, Germany \\ $ ^{14}$ Physikalisches Institut, Universit\"at Heidelberg, Heidelberg, Germany$^{ a}$ \\ $ ^{15}$ Kirchhoff-Institut f\"ur Physik, Universit\"at Heidelberg, Heidelberg, Germany$^{ a}$ \\ $ ^{16}$ Institute of Experimental Physics, Slovak Academy of Sciences, Ko\v{s}ice, Slovak Republic$^{ f}$ \\ $ ^{17}$ Department of Physics, University of Lancaster, Lancaster, UK$^{ b}$ \\ $ ^{18}$ Department of Physics, University of Liverpool, Liverpool, UK$^{ b}$ \\ $ ^{19}$ Queen Mary and Westfield College, London, UK$^{ b}$ \\ $ ^{20}$ Physics Department, University of Lund, Lund, Sweden$^{ g}$ \\ $ ^{21}$ CPPM, CNRS/IN2P3 - Univ. Mediterranee, Marseille, France \\ $ ^{22}$ Departamento de Fisica Aplicada, CINVESTAV, M\'erida, Yucat\'an, M\'exico$^{ j}$ \\ $ ^{23}$ Departamento de Fisica, CINVESTAV, M\'exico$^{ j}$ \\ $ ^{24}$ Institute for Theoretical and Experimental Physics, Moscow, Russia$^{ k}$ \\ $ ^{25}$ Lebedev Physical Institute, Moscow, Russia$^{ e}$ \\ $ ^{26}$ Max-Planck-Institut f\"ur Physik, M\"unchen, Germany \\ $ ^{27}$ LAL, Univ Paris-Sud, CNRS/IN2P3, Orsay, France \\ $ ^{28}$ LLR, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France \\ $ ^{29}$ LPNHE, Universit\'{e}s Paris VI and VII, IN2P3-CNRS, Paris, France \\ $ ^{30}$ Faculty of Science, University of Montenegro, Podgorica, Montenegro$^{ e}$ \\ $ ^{31}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic$^{ h}$ \\ $ ^{32}$ Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic$^{ h}$ \\ $ ^{33}$ Dipartimento di Fisica Universit\`a di Roma Tre and INFN Roma~3, Roma, Italy \\ $ ^{34}$ Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria$^{ e}$ \\ $ ^{35}$ Institute of Physics and Technology of the Mongolian Academy of Sciences , Ulaanbaatar, Mongolia \\ $ ^{36}$ Paul Scherrer Institut, Villigen, Switzerland \\ $ ^{37}$ Fachbereich C, Universit\"at Wuppertal, Wuppertal, Germany \\ $ ^{38}$ Yerevan Physics Institute, Yerevan, Armenia \\ $ ^{39}$ DESY, Zeuthen, Germany \\ $ ^{40}$ Institut f\"ur Teilchenphysik, ETH, Z\"urich, Switzerland$^{ i}$ \\ $ ^{41}$ Physik-Institut der Universit\"at Z\"urich, Z\"urich, Switzerland$^{ i}$ \\ \bigskip $ ^{42}$ Also at Physics Department, National Technical University, Zografou Campus, GR-15773 Athens, Greece \\ $ ^{43}$ Also at Rechenzentrum, Universit\"at Wuppertal, Wuppertal, Germany \\ $ ^{44}$ Also at University of P.J. \v{S}af\'{a}rik, Ko\v{s}ice, Slovak Republic \\ $ ^{45}$ Also at CERN, Geneva, Switzerland \\ $ ^{46}$ Also at Max-Planck-Institut f\"ur Physik, M\"unchen, Germany \\ $ ^{47}$ Also at Comenius University, Bratislava, Slovak Republic \\ $ ^{48}$ Also at DESY and University Hamburg, Helmholtz Humboldt Research Award \\ $ ^{49}$ Also at Faculty of Physics, University of Bucharest, Bucharest, Romania \\ $ ^{50}$ Supported by a scholarship of the World Laboratory Bj\"orn Wiik Research Project \\ $ ^{51}$ Also at Ulaanbaatar University, Ulaanbaatar, Mongolia \\ \bigskip $ ^a$ Supported by the Bundesministerium f\"ur 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 PBS/DESY/70/2006 \\ $ ^e$ Supported by the Deutsche Forschungsgemeinschaft \\ $ ^f$ Supported by VEGA SR grant no. 2/7062/ 27 \\ $ ^g$ Supported by the Swedish Natural Science Research Council \\ $ ^h$ Supported by the Ministry of Education of the Czech Republic under the projects LC527, INGO-1P05LA259 and MSM0021620859 \\ $ ^i$ Supported by the Swiss National Science Foundation \\ $ ^j$ Supported by CONACYT, M\'exico, grant 48778-F \\ $ ^k$ Russian Foundation for Basic Research (RFBR), grant no 1329.2008.2 \\ $ ^l$ This project is co-funded by the European Social Fund (75\%) and National Resources (25\%) - (EPEAEK II) - PYTHAGORAS II \\ } \end{flushleft} \newpage %========================================================================= \section{Introduction} %========================================================================= \label{sec:intro} Measurements of the deep-inelastic scattering (DIS) of leptons and nucleons allow the extraction of Parton Distribution Functions (PDFs). While these functions provide crucial input to perturbative Quantum Chromodynamic (QCD) calculations, they do not provide a complete picture of the partonic structure of nucleons. In particular, PDFs contain neither information on the correlations between partons nor on their transverse spatial distribution. Hard exclusive particle production, without excitation or dissociation of the nucleon, have emerged in recent years as prime candidates to address these issues~\cite{bernard,bel,buk,diehl,strik,muller,diehl3}. % Among them, deeply virtual Compton scattering (DVCS) on the proton ($\gamma^*p\to\gamma p$) is the simplest. The DVCS reaction can be regarded as the elastic scattering of the virtual photon off the proton via a colourless exchange, producing a real photon in the final state. % In the Bjorken scaling regime, corresponding to large virtuality $Q^2$ of the exchanged photon and $|t|/Q^2 \ll 1$, where $t$ is the squared momentum transfer at the proton vertex, QCD calculations assume that the exchange involves two partons in a colourless configuration, having different longitudinal and transverse momenta. % These unequal momenta, or skewing, are a consequence of the mass difference between the incoming virtual photon and the outgoing real photon and may be interpreted in the context of generalised parton distributions (GPDs) or dipole amplitudes, respectively. % In basic terms, a GPD (off-diagonal parton distribution) is the transition amplitude for removing a parton from the fast moving proton and reabsorbing it with a different momentum, thereby imparting a certain momentum transfer to the proton. % In the dipole approach the virtual photon fluctuates into a colour singlet $q {\bar q}$ pair (or dipole) of a transverse size $r \sim 1/Q$, which subsequently undergoes hard scattering with the gluons in the proton. % The $t$-dependence of the DVCS cross section carries information on the transverse momentum of partons. In the kinematic range of the HERA collider, where DVCS is accessed through the reaction $e^\pm p \rightarrow e^\pm \photon p$~\cite{dvcsh1,Adloff:2001cn,dvcsh1a,dvcszeus,dvcszeusb}, the DVCS amplitude is mainly imaginary~\cite{bel}, while the change of the amplitude with energy gives rise to a small real part. % This reaction also receives a contribution from the purely electromagnetic Bethe-Heitler (BH) process, where the photon is emitted from the electron. % The interference between DVCS and BH processes allows the extraction of the real part of the amplitude. % In addition, the real part of the DVCS amplitude can be related to its imaginary part using dispersion relations. % In the high energy limit at low momentum fraction $x$, the dispersion relations take a simple form~\cite{Hebecker:2000xs} which can therefore be used for the DVCS process to verify the consistency between measurements of the real and imaginary parts of the amplitude. This paper presents a measurement of DVCS cross sections as a function of $Q^2$ and the $\gamma^*p$ centre-of-mass energy $W$. % The single differential cross section $d\sigma / dt$ is also extracted. % The data were recorded with the H1 detector in the years $2004$ to $2007$, during which period HERA collided protons of $920$~GeV energy with $27.6$~GeV electrons and positrons. % The total integrated luminosity of the data is $306$~pb$^{-1}$. The data comprise $162$~pb$^{-1}$ recorded in $e^+p$ and $144$ pb$^{-1}$ in $e^-p$ collisions. % During this HERA~II running period, the electron\footnote{ In this paper the term ``electron'' is used generically to refer to both electrons and positrons, unless otherwise stated.} beam was longitudinally polarised, at a level of typically $35\%$. For this analysis, the periods with left-handed and right-handed beams are combined and the analysed data samples have a left-handed residual polarisation of $1\%$ and $5\%$ for $e^+p$ and $e^-p$ collisions, respectively. % Cross section measurements are carried out in the kinematic range $6.5 < Q^2 < 80$~GeV$^2$, $30 < W < 140$~GeV and $|t| <$ 1 GeV$^2$. The range in $x \simeq Q^2/W^2$ of the present measurement extends from $5 \cdot 10^{-4}$ to $10^{-2}$. % % The cross section measurements of this analysis supersede those of a previous H1 publication~\cite{dvcsh1}, in which less than half of the present HERA II data was used. It is complementary to measurements performed at lower $Q^2$ using HERA I data~\cite{dvcsh1a}. % In addition, using both beam charges, the beam charge asymmetry of the interference between the BH and DVCS processes is measured for the first time at a collider. % % %========================================================================= \section{Theoretical Framework} %========================================================================= \label{sec:theo} In this paper, cross section measurements are compared to predictions based either on GPDs or on a dipole approach. % At the present level of understanding, the pure GPD approach and dipole models, based on the proton-dipole amplitude, are not connected. However, in the low $x$ domain, dipole amplitudes could be used to provide parameterisations for GPDs at a certain scale~\cite{McDermott:2001pt}. % In this context, the DVCS process is interesting as calculations are simplified by the absence of an unknown vector meson wave function. % % The GPD model~\cite{muller} used here has been shown to describe previous data. It is based on partial wave expansions of DVCS amplitudes and is a first attempt to parametrise all GPDs over the full kinematic domain. % The dipole model~\cite{gregory}, with a limited number of parameters, describes a large panel of low $x$ measurements at HERA, from inclusive to exclusive processes. In this model, mainly using the gluon density extracted from fits to $F_2$ data, the DVCS cross section is computed using a universal dipole amplitude. % For GPD models, a direct measurement of the real part of the DVCS amplitude is an important issue, as it gives an increased sensitivity to the parameterisation of the GPDs~\cite{bel,muller}. Indeed, a calculation of the real part of the DVCS amplitude requires a parametrisation of the GPDs over the full $x$ range. % Considering the large flexibility in the parameterisation of the GPDs, this is an important quantity to qualify the correct approach with GPDs. % % In the dipole approach, as the dipole amplitude refers only to the imaginary part, the magnitude of the real part can be predicted using a dispersion relation. In high energy electron-proton collisions at HERA, DVCS and BH processes contribute to the reaction $e^\pm p \rightarrow e^\pm \photon p$. % %This reaction also receives a contribution from the purely electromagnetic BH process, where the photon is emitted from the electron. The BH cross section is precisely calculable in QED. % Since these two processes have an identical final state, they interfere. The squared photon production amplitude is then given by \begin{equation} \label {eqn:tau} \left| A \right|^2 = \left| A_{{\scriptscriptstyle BH}} \right|^2 + \left| A_{{\scriptscriptstyle DVCS}} \right|^2 + \underbrace{ A_{{\scriptscriptstyle DVCS}} \, A_{{\scriptscriptstyle BH}}^* + A_{{\scriptscriptstyle DVCS}}^* \, A_{{\scriptscriptstyle BH}}}_I, \end{equation} \noindent where $A_{\scriptscriptstyle BH}$ is the BH amplitude, $A_{\scriptscriptstyle DVCS}$ represents the DVCS amplitude and $I$ denotes the interference term. % In the leading twist approximation, the interference term can be written quite generally as a linear combination of harmonics of the azimuthal angle $\phi$. As defined in~\cite{bel}, $\phi$ is the angle between the plane containing the incoming and outgoing leptons and the plane formed by the virtual and real photons. % For an unpolarised proton beam and if only the first harmonic in $\cos\phi$ and $\sin\phi$, which are dominant at low $x$~\cite{muller}, are considered, the interference term $I$ can be written as \begin{equation} I \propto -C \, [ a_1 \cos \phi \, \mathrm{Re} A_{DVCS} + a_2 P_l \sin \phi \, \mathrm{Im} A_{DVCS} ], \end{equation} \noindent where $C = \pm 1$ is the charge of the lepton beam, $P_l$ its longitudinal polarisation and $a_1$ and $a_2$ are functions of the ratio of longitudinal to transverse virtual photon flux~\cite{bernard,bel,buk,diehl,strik,muller}. % % Cross section measurements which are integrated over $\phi$ are not sensitive to the interference term. The measurement of the cross section asymmetry with respect to the beam charge as a function of $\phi$ allows to access the interference term. % The beam charge asymmetry (BCA) of the cross section is defined as \begin{equation} A_C(\phi) = \frac{d\sigma^+/d\phi -d\sigma^-/d\phi} {d\sigma^+/d\phi + d\sigma^-/d\phi}, \label{bcadef} \end{equation} \noindent where $d\sigma^+/d\phi$ and $d\sigma^-/d\phi$ are the differential $ep \rightarrow ep\gamma$ cross sections measured in $e^{+}p$ and $e^{-}p$ collisions, respectively. Considering the low residual polarisation of the data and the theoretical expression of $a_1$ and $a_2$~\cite{bel}, $a_1 \gg a_2 P_l$ and the contribution of the $\sin \phi$ term is neglected. % Therefore, $A_C(\phi)$ can be expressed as \begin{equation} A_C(\phi) = p_1 \cos \phi = 2 A_{BH} \frac{\mathrm{Re} A_{DVCS} } { |A_{DVCS}|^2+ |A_{BH}|^2 } \cos \phi. \label{eq1} \end{equation} \noindent The term $|A_{DVCS}|^2$ can be derived directly from the DVCS cross section measurement $\sigma_{DVCS} = |A_{DVCS}^2| / (16 \pi b)$, where $b$ is the slope of the exponential $t$-dependence $e^{-b|t|}$ of the DVCS cross section. As the BH amplitude is precisely known, the measured asymmetry is directly proportional to the real part of the DVCS amplitude and the ratio between real and imaginary parts of the DVCS amplitude, $\rho=\mathrm{Re} A_{DVCS} / \mathrm{Im} A_{DVCS}$, can be extracted. % % This ratio $\rho$ can also be derived using a dispersion relation~\cite{muller,diehl2}. In the high energy limit, at low $x$ and when the $W$ dependence of the cross section is parameterised by a single term $W^{\delta(Q^2)}$, the dispersion relation can be written as~\cite{Hebecker:2000xs} \begin{equation} \rho=\mathrm{Re} A_{DVCS} / \mathrm{Im} A_{DVCS} = \tan \left( \frac{\pi \delta(Q^2)} {8} \right). \label{dispersion} \end{equation} % The ratio $\rho$ can therefore be determined directly from the energy dependence of the DVCS cross section parameterised by $\delta(Q^2)$. Comparison between the $\rho$ values calculated from the energy dependence of the DVCS amplitude and from its real part therefore provides an important consistency test of the measured BCA. %========================================================================= \section{Experimental Conditions and Monte Carlo Simulation} \label{simul} %========================================================================= A detailed description of the H1 detector can be found in~\cite{h1dect}. Here, only the detector components relevant for the present analysis are described. H1 uses a right-handed coordinate system with the $z$ axis along the beam direction, the $+z$ or ``forward'' direction being that of the outgoing proton beam. The polar angle $\theta$ is defined with respect to the $z$ axis and the pseudo-rapidity is given by $\eta=-\ln \tan \theta /2$. The SpaCal~\cite{Appuhn:1996na}, a lead scintillating fibre calorimeter, covers the backward region ($153 ^{\rm \circ} < \theta < 176 ^{\rm \circ}$). Its energy resolution for electromagnetic showers is $\sigma(E)/E \simeq 7.1\%/\sqrt{E/{\rm GeV}} \oplus 1\%$. The liquid argon (LAr) calorimeter ($4^{\rm \circ} \leq \theta \leq 154^{\rm \circ}$) is situated inside a solenoidal magnet. The energy resolution for electromagnetic showers is $\sigma(E)/E \simeq 11\%/\sqrt{E/{\rm GeV}}$ as obtained from test beam measurements~\cite{Andrieu:1994yn}. The main component of the central tracking detector is the central jet chamber CJC ($20^{\rm \circ} < \theta < 160^{\rm \circ}$) which consists of two coaxial cylindrical drift chambers with wires parallel to the beam direction. The measurement of charged particle transverse momenta is performed in the magnetic field of $1.16$~T, with a resolution of $\sigma_{P_T}/P_T = 0.002 P_T / \rm{GeV} \oplus 0.015$. % The innermost proportional chamber CIP~\cite{CIP} ($9^\circ < \theta < 171^\circ$) is used in this analysis to complement the CJC in the backward region for the reconstruction of the interaction vertex. % The forward muon detector (FMD) consists of a series of drift chambers covering the range $1.9<\eta<3.7$. Primary particles produced at larger $\eta$ can be detected indirectly in the FMD if they undergo a secondary scattering with the beam pipe or other adjacent material. Therefore, the FMD is used in this analysis to provide an additional veto against inelastic or proton dissociative events. The luminosity is determined from the rate of Bethe-Heitler processes measured using a calorimeter located close to the beam pipe at $z=-103~{\rm m}$ in the backward direction. A dedicated event trigger was set up for this analysis. It is based on topological and neural network algorithms and uses correlations between electromagnetic energy deposits of electrons or photons in both the LAr and the SpaCal \cite{roland}. The combined trigger efficiency is $98$\%. Monte Carlo (MC) simulations are used to estimate the background contributions and the corrections for the QED radiative effects and for the finite acceptance and the resolution of the detectors. Elastic DVCS events in $ep$ collisions are generated using the Monte Carlo generator MILOU~\cite{milou}, based on the cross section calculation from~\cite{ffspaper} and using a $t$-slope parameter $b=5.4$~GeV$^{-2}$, as measured in this analysis (see section~\ref{sec:exp_xsec}). The photon flux is taken from~\cite{flux}. Inelastic DVCS events in which the proton dissociates into a baryonic system $Y$ are also simulated with MILOU setting the $t$-slope $b_{inel}$ to $1.5$~GeV$^{-2}$, as determined in a dedicated study (see section~\ref{sec:pdiss}). The Monte Carlo program COMPTON~2.0~\cite{compton2} is used to simulate elastic and inelastic BH events. % In the generated MC events, no interference between DVCS and BH processes is included. Background from diffractive meson events is simulated using the DIFFVM MC generator~\cite{diffvm}. % All generated events are passed through a detailed, GEANT~\cite{Brun:1987ma} based simulation of the H1 detector and are subject to the same reconstruction and analysis chain as are the data. %========================================================================= \section{Event Selection} \label{selection} %========================================================================= % % In elastic DVCS events, the scattered electron and the photon are the only particles that are expected to give signals in the detector. The scattered proton escapes undetected through the beam pipe. % The selection of the analysis event sample requires a scattered electron and a photon identified as compact and isolated electromagnetic showers in the SpaCal and in the LAr, respectively. % The electron candidate is required to have an energy above $15$ GeV. The photon is required to have a transverse momentum $P_T$ above $2$ GeV and a polar angle between $25^{\circ}$ and $145^\circ$. Events are selected if there are either no tracks at all or a single central track which is associated with the scattered electron. In order to reject inelastic and proton dissociation events, no further energy deposit in the LAr calorimeter larger than $0.8$~GeV is allowed and no activity above the noise level should be present in the FMD. The influence of QED radiative corrections is reduced by the requirement that the longitudinal momentum balance $E - P_z$ be greater than $45$~GeV. Here, $E$ denotes the energy and $P_z$ the momentum along the beam axis of all measured final state particles. To enhance the DVCS signal with respect to the BH contribution and to ensure a large acceptance, the kinematic domain is restricted to $6.50$ are combined, in order to increase the statistical significance and to remove effects on the asymmetry of any possible $\sin \phi$ contribution from the residual lepton beam polarisation. % The systematic error on the BCA measurement mainly arises from the part of the LAr photon energy scale uncertainty which is correlated between the $e^+p$ and $e^-p$ samples, estimated to be $0.5$\%. It leads to sizeable systematic errors on the measured asymmetry for $\phi$ close to $0^\circ$ and $180^\circ$. In a first step, the interference term between DVCS and BH processes, which is not known a priori, is not included in formula~(\ref{eq:xsec_phi}). % In order to simulate the interference term, an asymmetry of the form $p_1 \cos \phi$ is added to the MC generation and passed through the full detector simulation and analysis chain to account for all acceptance and migration effects from true to reconstructed $\phi$ values. % Similarly to the data, formulae~(\ref{eq:xsec_phi}) and (\ref{bcadef}) are used to determine the reconstructed asymmetry corresponding to these MC events. % To determine the value of $p_1$, a $\chi^2$ minimisation is performed as a function of $p_1$ to adjust the reconstructed asymmetry in the MC to the measured one. % MC events generated using this $p_1$ value are then used to correct the measured asymmetry for the effect of migrations. Bin by bin correction factors are determined from the difference between the true and the reconstructed asymmetry in the MC. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results and Interpretations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Cross Sections and $t$-dependence}\label{sec:exp_xsec} The measured DVCS cross sections as a function of $W$ for $|t| < 1$~GeV$^2$ and at $Q^2 = 10 $~GeV$^2$ as well as the $Q^2$ dependence at $W = 82$~GeV are displayed in figure~\ref{fig1d} and given in table~\ref{sig1d}. They agree within errors with the previous measurements~\cite{dvcsh1,dvcsh1a,dvcszeus,dvcszeusb}. The data agree also with models based on GPDs~\cite{muller} or the dipole approach~\cite{gregory}. DVCS cross sections for $e^{+}p$ and $e^{-}p$ data are also found in good agreement with each other. % % As already discussed in~\cite{dvcsh1}, the steep rise of the cross section with $W$ is an indication of the presence of a hard underlying process. % The $W$ dependence of the cross section for three separate bins of $Q^2$ is shown in figure~\ref{fig2d}(a) and given in table~\ref{sig2d}. A fit of the function $W^\delta$ is performed in each $Q^2$ bin. Figure~\ref{fig2d}(b) shows the obtained $\delta$ values. It is observed that $\delta$ is independent of $Q^2$ within the errors. % The average value\footnote{Here and in all other places where results are given the first error is statistical and the second systematic.} $\delta = 0.63 \, \pm \, 0.08 \, \pm \, 0.14$ is in agreement with the previous measurement~\cite{dvcsh1}, as well as with the value of $\delta = 0.52 \pm 0.09$~(stat.) measured by the ZEUS Collaboration at a lower $Q^2$ of $3.2$~GeV$^2$~\cite{dvcszeusb}. Differential cross sections are measured as a function of $t$ for three values of $Q^2$ and $W$ and presented in table~\ref{sigtq}. Fits of the form $d\sigma/d|t| \sim e^{-b|t|}$, which describe the data well~\cite{dvcsh1}, are performed taking into account the statistical and correlated systematic errors. % The derived $t$-slope parameters $b(Q^2)$ and $b(W)$ are displayed in figures~\ref{figb}(a) and (b), respectively. They confirm the result obtained in a previous analysis \cite{dvcsh1} and no significant variation of $b$ with $W$ is observed. % Experimental results are compared with calculations from GPD and dipole models~\cite{muller,gregory}. % A good agreement is obtained for both $W$ and $Q^2$ dependences of the $t$-slopes. % It should be noted that in the GPD model previous data of~\cite{dvcsh1,dvcsh1a} are used to derive the $Q^2$ and $W$ dependences of $b$, while no DVCS data enter in the determination of parameters of the dipole model. % If $b$ is parametrised as \mbox{$b=b_0+2 \alpha' \ln \frac{1}{x}$}, with $x=Q^2/W^2$, the obtained $\alpha'$ value is compatible with $0$ and an upper limit on $\alpha'$ of $0.20$ GeV$^{-2}$ at $95$\% confidence level (CL) is derived. % This value is compatible with results obtained for $\jpsi$ exclusive electroproduction~\cite{jpsih1,jpsizeus}, for which the measured $\alpha'$ is below $0.17$ GeV$^{-2}$ at $95$\% CL. % An increase of the slope with decreasing $x$ (shrinkage) is therefore not observed. % Such a behaviour is expected for hard processes and confirms that perturbative QCD can be used to describe DVCS processes. Using the complete analysis sample, the value of $b$ is found to be $5.41 \, \pm \, 0.14 \, \pm \, 0.31$~GeV$^{-2}$ at $Q^2 = 10\; \mbox{GeV}^2$. % This corresponds to a total uncertainty of $6$\% on the (elastic) $t$-slope measurement for the full data sample. As in \cite{dvcsh1}, this $t$-slope value can be converted to an average impact parameter of $\sqrt{} = 0.64 \pm 0.02$~fm. It corresponds to the transverse extension of the parton density, dominated by sea quarks and gluons for an average value \mbox{$x =1.2 \cdot 10^{-3}$}, in the plane perpendicular to the direction of motion of the proton. At larger values of $x$ ($x>0.1$), a smaller value of $\sqrt{}$, dominated by the contribution of valence quarks, is estimated~\cite{diehl}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Inelastic DVCS $t$-dependence}\label{sec:pdiss} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The increased statistical precision compared to previous analyses allows a first measurement of the $t$-slope of the inelastic DVCS process. A sample of events with a signal in the FMD is selected. % It corresponds to events with the mass of the proton dissociation system $M_Y$ in the range $1.4$ to $10$~GeV, as derived from MC studies. % The contribution of inelastic DVCS events is extracted by subtracting the BH (elastic and inelastic) and elastic DVCS contributions, as estimated from the respective MC expectations. The measured differential cross section as a function of $t$ is presented in figure~\ref{fig_pdiss}. A fit of the form $d\sigma/d|t| \sim e^{-b_{inel} |t|}$ yields $b_{inel}=1.53 \pm 0.26 \pm 0.44$~GeV$^{-2}$. % In the present event sample, no indication of a dependence of $b_{inel}$ with $Q^2$ or $W$ is observed. % The obtained value for $b_{inel}$ is compatible with previous determinations for inelastic exclusive production of $\rho$, $\phi$~\cite{Adloff:1997jd} and $\jpsi$~\cite{jpsizeus}. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Beam Charge Asymmetry} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The contributions of elastic DVCS and BH processes to the analysis sample are of similar size, as can be observed in figure~\ref{figcomp}. This is a favourable situation for the beam charge asymmetry measurement, with a maximum sensitivity for the interference term. % The measured BCA integrated over the kinematic range of the analysis and corrected for detector effects, as detailed in section~\ref{sec:xsections}, is presented in figure~\ref{fig3} and table~\ref{tabbca}. % Bins in $\phi$ with a size of the order of the experimental resolution on $\phi$ are used. % The $\chi^2$ minimisation procedure leads to a $p_1$ value of \mbox{$p_1 = 0.16 \pm 0.04 \pm 0.06$}. % % The resulting function $0.16 \cos \phi$ is displayed in figure~\ref{fig3} and is seen to agree with the prediction of the GPD model for the first $\cos \phi$ harmonic~\cite{muller}. The measured asymmetry is in good agreement with the model prediction within experimental errors. % % As detailed in section~\ref{sec:theo}, from the measured BCA and the $p_1$ value determined above, together with the DVCS cross section, the ratio $\rho$ of the real to imaginary parts of the DVCS amplitude can be calculated as $\rho = 0.20 \pm 0.05 \pm 0.08$. % This is the first measurement of this ratio. % The dispersion relation of equation~(\ref{dispersion}) and our measurement of $\delta(Q^2)$ on the other hand leads to $\rho = 0.25 \pm 0.03 \pm 0.05$, % in good agreement with the direct determination. % % While in the low $x$ domain of the present measurement, the real part of the DVCS amplitude is positive, in contrast, at larger $x$ ($x \sim 0.1$) and lower $Q^2$, a smaller and negative real part was measured\footnote{The convention used in~\cite{hermes} for the definition of the $\phi$ angle is different from the one of~\cite{bel} adopted in the present paper.} by the HERMES Collaboration~\cite{hermes}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The elastic DVCS cross section $\gamma^\ast p \rightarrow \gamma p$ has been measured with the H1 detector at HERA. The~~measurement~~is~~performed~~in the kinematic~~range~~$6.5\;<\;Q^2\;<\;80$~GeV$^2$, \mbox{$30~< W<~140$~GeV} and $|t| <$~1~GeV$^2$. The analysis uses $e^{+}p$ and $e^{-}p$ data recorded from $2004$ to $2007$, corresponding to a total integrated luminosity of $306$~pb$^{-1}$, almost equally shared between both beam charges. % The $W$ dependence of the DVCS cross section is well described by a function $W^{\delta}$. No significant variation of the exponent $\delta$ as a function of $Q^2$ is observed. For the total sample a value $\delta = 0.63 \, \pm \, 0.08 \, \pm \, 0.14$ is determined. The steep rise of the cross section with $W$ indicates a hard underlying process. % % The $t$-dependence of the cross section is well described by the form $e^{-b|t|}$ with an average slope of $b = 5.41 \, \pm \, 0.14 \, \pm \, 0.31$~GeV$^{-2}$. The $t$-slopes are determined differentially in $Q^2$ and $W$ and are compatible with previous observations. % The $t$-slope is also measured for the inelastic DVCS. % The measured elastic DVCS cross section is compared to the predictions of two different models based on GPDs or on a dipole approach, respectively. Both approaches describe the data well. % The use of $e^{+}p$ and $e^{-}p$ collision data allows the measurement of the beam charge asymmetry of the interference between the BH and DVCS processes, for the first time at a collider. The ratio $\rho$ of the real to imaginary part of the DVCS amplitude is then derived, directly from the measurements of the BCA and of the DVCS cross section to be $\rho = 0.20 \pm 0.05 \pm 0.08$. This ratio can also be calculated from a dispersion relation using only the DVCS energy dependence, leading to $\rho = 0.25 \pm 0.03 \pm 0.05$. Both results are in good agreement. % The GPD model considered here~\cite{muller} correctly describes the measured BCA as well as $\rho$. % % The measurements presented here show that a combined analysis of DVCS observables, including cross section and charge asymmetry, allows the extraction of the real part of the DVCS amplitude and subsequently a novel understanding of the correlations of parton momenta in the proton. %========================================================================= \section*{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 the hospitality which they extend to the non DESY members of the collaboration. 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The first errors are statistical, the second systematic.} \label{sig1d} \end{table} \begin{table}[htbp] \centering \begin{tabular}{|c|lcc|lcc|lcc|} \cline{2-10} \multicolumn{1}{c|}{~} &\multicolumn{9}{c|}{~} \\ [-10pt] \multicolumn{1}{l|}{} & \multicolumn{9}{c|}{$\sigma_{DVCS} \; \; \left[{\rm nb}\right]$} \\ [3.0pt] % \cline{2-10} % \multicolumn{1}{c|}{~} &\multicolumn{3}{c|}{~} % &\multicolumn{3}{c|}{~} % &\multicolumn{3}{c|}{~} \\ [-10pt] \hline \multicolumn{1}{|c|}{~} &\multicolumn{3}{|c|}{~} &\multicolumn{3}{c|}{~} &\multicolumn{3}{c|}{~} \\ [-10pt] $W$ $\left[{\rm GeV}\right] $ & \multicolumn{3}{c|}{$Q^2 = 8\,{\rm GeV}^2$} & \multicolumn{3}{c|}{$Q^2 = 15.5\,{\rm GeV}^2$} & \multicolumn{3}{c|}{$Q^2 = 25\,{\rm GeV}^2$} \\ [3.0pt] \hline $45$& $3.06$& $\pm$ $0.18$& $\pm$ $0.25$& $0.98$& $\pm$ $0.07$& $\pm$ $0.08$& $0.31$& $\pm$ $0.11$& $\pm$ $0.05$ \\ $70$& $3.54$& $\pm$ $0.29$& $\pm$ $0.34$& $1.46$& $\pm$ $0.12$& $\pm$ $0.12$& $0.52$& $\pm$ $0.08$& $\pm$ $0.06$ \\ $90$& $4.93$& $\pm$ $0.39$& $\pm$ $0.52$& $1.41$& $\pm$ $0.16$& $\pm$ $0.17$& $0.81$& $\pm$ $0.13$& $\pm$ $0.09$ \\ $110$& $5.16$& $\pm$ $0.51$& $\pm$ $0.74$& $1.66$& $\pm$ $0.23$& $\pm$ $0.28$& $0.63$& $\pm$ $0.17$& $\pm$ $0.15$ \\ $130$& $5.62$& $\pm$ $1.34$& $\pm$ $1.19$& $2.00$& $\pm$ $0.37$& $\pm$ $0.47$& $0.80$& $\pm$ $0.26$& $\pm$ $0.29$ \\ \hline \hline $\delta$ & $0.61$ & $\pm$ $0.10$ & $\pm$ $0.15$ & $0.61$ & $\pm$ $0.13$ & $\pm$ $0.15$ & $0.90$ & $\pm$ $0.36$ & $\pm$ $0.27$ \\ \hline \end{tabular} \caption{The DVCS cross section $\gamma^\ast p \rightarrow \gamma p$, $\sigma_{DVCS}$, as a function of $W$ for three $Q^2$ values and for $ |t| < 1\,{\rm GeV}^2$. The values of $\delta(Q^2)$ obtained from fits of the form $W^\delta$ are given. The first errors are statistical, the second systematic.} \label{sig2d} \end{table} \begin{table}[htbp] \centering \begin{tabular}{|c|lcc|lcc|lcc|} \cline{2-10} \multicolumn{1}{c|}{~} &\multicolumn{9}{c|}{~} \\ [-10pt] \multicolumn{1}{l|}{} & \multicolumn{9}{c|}{$d\sigma_{DVCS}/d|t| \; \; \left[{\rm nb/GeV}^2\right]$} \\ [3.0pt] \cline{2-10} \multicolumn{1}{c|}{~} &\multicolumn{9}{|c|}{~} \\ [-11pt] \multicolumn{1}{c|}{~} & \multicolumn{9}{c|}{$W=82\,{\rm GeV}$} \\ [1.0pt] \hline \multicolumn{1}{|c|}{~} &\multicolumn{3}{|c|}{~} &\multicolumn{3}{c|}{~} &\multicolumn{3}{c|}{~} \\ [-12pt] $|t|$ $\left[{\rm GeV}^2\right] $ & \multicolumn{3}{c|}{$Q^2 = 8\,{\rm GeV}^2$} & \multicolumn{3}{c|}{$Q^2 = 15.5\,{\rm GeV}^2$} & \multicolumn{3}{c|}{$Q^2 = 25\,{\rm GeV}^2$} \\ [3.0pt] \hline $0.10$& $13.3$& $\pm$ $0.80$& $\pm$ $1.73$& $4.33$& $\pm$ $0.35$& $\pm$ $0.65$& $1.68$& $\pm$ $0.31$& $\pm$ $0.42$ \\ $0.30$& $4.82$& $\pm$ $0.32$& $\pm$ $0.50$& $1.24$& $\pm$ $0.13$& $\pm$ $0.16$& $0.49$& $\pm$ $0.10$& $\pm$ $0.08$ \\ $0.50$& $1.26$& $\pm$ $0.14$& $\pm$ $0.18$& $0.45$& $\pm$ $0.06$& $\pm$ $0.05$& $0.18$& $\pm$ $0.04$& $\pm$ $0.03$ \\ $0.80$& $0.21$& $\pm$ $0.03$& $\pm$ $0.04$& $0.10$& $\pm$ $0.01$& $\pm$ $0.02$& $0.05$& $\pm$ $0.01$& $\pm$ $0.01$ \\ \hline \hline $b$ [${\rm GeV}^{-2}$] & $5.87$ & $\pm$ $0.20$ & $\pm$ $0.32$ & $5.45$ & $\pm$ $0.20$ & $\pm$ $0.29$ & $5.10$ & $\pm$ $0.38$ & $\pm$ $0.37$ \\ \hline \multicolumn{10}{c}{}\\ \cline{2-10} \multicolumn{1}{c|}{~} &\multicolumn{9}{|c|}{~} \\ [-11pt] \multicolumn{1}{c|}{~} & \multicolumn{9}{c|}{$Q^2=10\,{\rm GeV}^2$} \\ [1.0pt] \hline \multicolumn{1}{|c|}{~} &\multicolumn{3}{|c|}{~} &\multicolumn{3}{c|}{~} &\multicolumn{3}{c|}{~} \\ [-12pt] $|t|$ $\left[{\rm GeV}^2\right] $ & \multicolumn{3}{c|}{$W=40\,{\rm GeV}$} & \multicolumn{3}{c|}{$W=70\,{\rm GeV}$} & \multicolumn{3}{c|}{$W=100\,{\rm GeV}$} \\ [3.0pt] \hline $0.10$& $4.77$& $\pm$ $0.50$& $\pm$ $0.49$& $7.81$& $\pm$ $0.51$& $\pm$ $0.85$& $11.0$& $\pm$ $0.85$&$\pm$ $2.23$ \\ $0.30$& $1.62$& $\pm$ $0.23$& $\pm$ $0.18$& $2.88$& $\pm$ $0.22$& $\pm$ $0.28$& $3.71$& $\pm$ $0.31$&$\pm$ $0.49$ \\ $0.50$& $0.69$& $\pm$ $0.11$& $\pm$ $0.07$& $0.91$& $\pm$ $0.10$& $\pm$ $0.10$& $1.18$& $\pm$ $0.13$&$\pm$ $0.16$ \\ $0.80$& $0.10$& $\pm$ $0.02$& $\pm$ $0.01$& $0.16$& $\pm$ $0.02$& $\pm$ $0.02$& $0.24$& $\pm$ $0.03$&$\pm$ $0.04$ \\ \hline \hline $b$ [${\rm GeV}^{-2}$] & $5.38$ & $\pm$ $0.30$ & $\pm$ $0.23$ & $5.49$ & $\pm$ $0.19$ & $\pm$ $0.26$ & $5.49$ & $\pm$ $0.20$ & $\pm$ $0.35$ \\ \hline \end{tabular} \caption{ The DVCS cross section $\gamma^* p\rightarrow \gamma p$, differential in $t$, $d\sigma_{DVCS}/dt$, for three values of $Q^2$ at $W=82\,{\rm GeV}$, and for three values of $W$ at $Q^2=10\,{\rm GeV}^2$. Results for the corresponding $t$-slope parameters $b$ are given. The first errors are statistical, the second systematic.} \label{sigtq} \end{table} \begin{table}[htbp] \centering \begin{tabular}{ll} \begin{tabular}{|r|rrc|} \hline $\phi$ [${\rm deg.}$]& \multicolumn{3}{c|}{ $A_C(\phi)$}\\ \hline $10$ &$\; 0.326 $ &$ \pm$ $ 0.086 $ &$ \pm$ $ 0.180 $ \\ $35$ &$\; 0.119 $ &$ \pm$ $ 0.076 $ &$ \pm$ $ 0.090 $ \\ $70$ &$\; -0.039 $ &$ \pm$ $ 0.080 $ &$ \pm$ $ 0.030 $ \\ $110$ &$\; 0.035 $ &$ \pm$ $ 0.092 $ &$ \pm$ $ 0.028 $ \\ $145$ &$\; -0.234 $ &$ \pm$ $ 0.079 $ &$ \pm$ $ 0.076 $ \\ $170$ &$\; -0.210 $ &$ \pm$ $ 0.075 $ &$ \pm$ $ 0.169 $ \\ \hline \end{tabular} \end{tabular} \caption{ The DVCS beam charge asymmetry $A_C(\phi)$ as a function of $\phi$ and integrated over the kinematic range $6.5 < Q^2 < 80\,{\rm GeV}^2$, $30 < W < 140\,{\rm GeV}$ and $|t| < 1\,{\rm GeV}^2$. The first errors are statistical, the second systematic.} \label{tabbca} \end{table} %========================================================================= \vfill \newpage \begin{figure}[!htbp] \vspace*{2cm} \begin{center} \includegraphics[totalheight=5.7cm]{d09-109f1a.eps}\put(-55,40){{(a)}} \includegraphics[totalheight=5.7cm]{d09-109f1b.eps}\put(-55,30){{(b)}}\\ \includegraphics[totalheight=5.7cm]{d09-109f1c.eps}\put(-55,40){{(c)}} \includegraphics[totalheight=5.7cm]{d09-109f1d.eps}\put(-55,30){{(d)}}\\ \includegraphics[totalheight=5.7cm]{d09-109f1e.eps}\put(-55,40){{(e)}} \includegraphics[totalheight=5.7cm]{d09-109f1f.eps}\put(-55,40){{(f)}} \end{center} \caption{\label{figcomp} Distributions of the energy (a) and polar angle (b) of the scattered electron, the energy (c) and polar angle (d) of the photon, % the $\phi$ azimuthal angle between the plane of incoming and outgoing lepton and the plane of virtual and real photon~\cite{bel} (e) and the proton four momentum transfer squared $|t|$ (f). The data correspond to the full $e^{\pm}p$ sample and are compared to Monte Carlo expectations for elastic DVCS, elastic and inelastic BH and inelastic DVCS. All Monte Carlo simulations are normalised according to the luminosity of the data. The open histogram shows the total prediction and the shaded band its estimated uncertainty. } \end{figure} \begin{figure}[!htbp] \begin{center} \includegraphics[totalheight=6cm]{d09-109f2a.eps}\put(-10,48){{(a)}}\\ \includegraphics[totalheight=6cm]{d09-109f2b.eps}\put(-10,48){{(b)}} \end{center} \caption{\label{fig1d} The DVCS cross section $\gamma^\ast p \rightarrow \gamma p$ as a function of $Q^2$ at $W=82$~GeV (a) and as a function of $W$ at $Q^2=10$~GeV$^2$ (b). The results from the previous H1~\cite{dvcsh1a} and ZEUS~\cite{dvcszeusb} publications based on HERA~I data are also displayed. ZEUS measurements are propagated from $W=104$~GeV to $82$~GeV using a $W$ dependence $W^{0.52}$. The inner error bars represent the statistical errors, the outer error bars the statistical and systematic errors added in quadrature. The dashed line represents the prediction of the GPD model~\protect{\cite{muller}} and the solid line the prediction of the dipole model~\protect{\cite{gregory}}. } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[!htbp] \begin{center} \includegraphics[totalheight=6cm]{d09-109f3a.eps}\put(-10,15){{(a)}}\\ \includegraphics[totalheight=6cm]{d09-109f3b.eps}\put(-10,15){{(b)}} \end{center} \caption{\label{fig2d} The DVCS cross section $\gamma^\ast p \rightarrow \gamma p$ as a function of $W$ at three values of $Q^2$ (a). The solid lines represent the results of fits of the form $W^\delta$. The fitted values of $\delta(Q^2)$ are shown in (b) together with the values obtained using HERA I data~\protect{\cite{dvcsh1a}}. The inner error bars represent the statistical errors, the outer error bars the statistical and systematic errors added in quadrature. } \end{figure} \begin{figure}[!htbp] \begin{center} \includegraphics[width=9.5cm]{d09-109f4a.eps}\put(-10,15){{(a)}}\\ \includegraphics[width=9.5cm]{d09-109f4b.eps}\put(-10,15){{(b)}} \end{center} \vspace*{-0.5cm} \caption{\label{figb} The fitted $t$-slope parameters $b(Q^2)$ are shown in (a) together with the $t$-slope parameters from the previous H1~\cite{dvcsh1a} and ZEUS~\cite{dvcszeusb} publications based on HERA I data. In (b) the fitted $t$-slope parameters $b(W)$ are shown. The inner error bars represent the statistical errors and the outer error bars the statistical and systematic errors added in quadrature. The dashed line represents the prediction of the GPD model~\protect{\cite{muller}} and the solid line the prediction of the dipole model~\protect{\cite{gregory}}. } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[!htbp] \begin{center} \includegraphics[width=9.5cm]{d09-109f5.eps}\\ \end{center} \caption{\label{fig_pdiss} The inelastic DVCS cross section differential in $t$ at $W=82$~GeV and $Q^2=10$~GeV$^2$ and for events with $1.4 \lesssim M_Y \lesssim 10$~GeV. The inner error bars represent the statistical errors, the outer error bars the statistical and systematic errors added in quadrature. } \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[!htbp] \begin{center} \includegraphics[width=10cm]{d09-109f6.eps} \end{center} \caption{ Beam charge asymmetry as a function of the angle $\phi$ as defined in~\cite{bel}, integrated over the kinematic range of the analysis. The inner error bars represent the statistical errors, the outer error bars the statistical and systematic errors added in quadrature. The function $0.16 \cos\phi$ is also shown (solid line), together with the GPD model prediction (dashed line). } \label{fig3} \end{figure} %========================================================================= \end{document}