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30pt \footskip 24pt \parskip 3mm plus 2mm minus 2mm \newcommand{\hdick}{\noalign{\hrule height1.4pt}} %===============================title page============================= % Uses LATEX2e % % \title{A new Measurement of Exclusive $\rho^0$ Meson Photoproduction at HERA} %======================================================================= \newcommand{\link}[1]{File: \htmladdnormallink{{\tt #1}}{../#1}} \newcommand{\spires}[2]{Spires: \htmladdnormallink{{\tt FIN #1 #2}}% {http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+#1+#2}% ~~~Cited by: \htmladdnormallink{{\tt FIN C #2}}% {http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+C+#2}} \newcommand{\spiresc}[2]{Not in Spires% ~~~Cited by: \htmladdnormallink{{\tt FIN C #2}}% {http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+C+#2}} \newcommand{\sub}[1]{\ensuremath{_{\mathrm{#1}}}} \newcommand{\super}[1]{\ensuremath{^{\mathrm{#1}}}} \newcommand{\TeV}{\ensuremath{\mathrm{GeV}}} 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\newcommand{\pb}{\ensuremath{\mathrm{pb}}} \newcommand{\fb}{\ensuremath{\mathrm{fb}}} \newcommand{\barn}{\ensuremath{\mathrm{barn}}} \newcommand{\Nel}{\ensuremath{N\sub{el}}} \newcommand{\Npd}{\ensuremath{N\sub{pd}}} \newcommand{\Nobs}{\ensuremath{N\sub{obs}}} \newcommand{\Ncor}{\ensuremath{N\sub{cor}}} \newcommand{\Ntag}{\ensuremath{N\sub{tag}}} \newcommand{\Nuntag}{\ensuremath{N\sub{untag}}} \newcommand{\ftag}{\ensuremath{f\sub{tag}}} \newcommand{\epsel}{\ensuremath{\epsilon\sub{el}}} \newcommand{\epspd}{\ensuremath{\epsilon\sub{pd}}} \newcommand{\Lumi}{\ensuremath{\mathcal{L}}} \newcommand{\Lint}{\ensuremath{\int {\mathcal{L} \der t}}} \newcommand{\MY}{\ensuremath{M\sub{Y}}} \newcommand{\Number}[1]{{\bfseries \boldmath {#1}}} \newcommand{\Nonumber}[1]{\mbox{{\bfseries \boldmath {#1 (???)}}}} \newcommand{\OurAlphaZero}{1.093 \pm 0.003 (stat.) ^{+0.008}_{-0.007} (syst.)} \newcommand{\OurAlphaPrime}{0.116 \pm 0.027 (stat.) ^{+0.036}_{-0.046} (syst.)} \newenvironment{lit}{\frenchspacing\begin{thebibliography}{19}} {\end{thebibliography}\nonfrenchspacing} \begin{document} \pagestyle{empty} \begin{titlepage} \noindent \begin{center} %{\it {\large version of \today}} \\[.3em] \begin{small} \begin{tabular}{llrr} %Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline %Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline Submitted to & & & %\epsfig{file=/h1/www/images/H1logo_bw_small.epsi \epsfig{file=H1logo_bw_small.epsi ,width=2.cm} \\[.2em] \hline \multicolumn{4}{l}{{\bf 14th International Workshop on Deep Inelastic Scattering, DIS2006, April 20-24, 2006, Tsukuba, Japan}} \\ % & Abstract: & {\bf xx-xxx} &\\ % & Parallel Session & {\bf x} &\\ \hline & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] & \multicolumn{3}{r}{H1prelim-06-011} \\[.2em] \end{tabular} \end{small} \end{center} \vspace*{2cm} \begin{center} \Large {\bfseries A New Measurement of \\ Exclusive {\boldmath $\rho^0$} Meson Photoproduction \\ at HERA\\ } \vspace*{1cm} {\Large H1 Collaboration} \end{center} \begin{abstract} \noindent Using a sample of more than 240000 events taken with the H1 detector at HERA in 2005, a new measurement of exclusive $\rho^0$ photoproduction in the kinematic range $20 < W < 90\,\GeV$ and $|t| < 3\,\GeV^2$ is performed. From the measurement of the $W$ dependence of this process in eight bins of $t$, the Pomeron trajectory $\alpha\sub{\pom}\,(t)$ can be extracted for the first time from the data of a single experiment. The result is $\alpha\sub\pom\,(t) = \OurAlphaZero + (\OurAlphaPrime)\,\GeV^{-2} \cdot t$. \end{abstract} \vspace*{1cm} \end{titlepage} \pagestyle{plain} \section{Introduction} %--------------------- The exclusive photoproduction of $\rho^0$ mesons has been studied in great detail over the last 40 years \cite{bib:rho-fixedtarget,Aston:1982hr,Derrick:1995vq,Aid:1996bs,Derrick:1996vw,Breitweg:1997ed,Breitweg:1999jy}. %\cite{Crouch:1966,Joos:1976nm,Aleksandrov:1980pt,Apsimon:1991nu, %Derrick:1995vq,Adams:1997bh,Breitweg:1997ed,Aid:1996bs,Derrick:1996vw, %Wu:2005wf,Clifft:1976be,Anderson:1976ph,Struczinski:1975ik,Bartalucci:1977cp, %Erbe:1966,Blechschmidt:1967,Ballam:1971yd,Ballam:1972eq,Egloff:1979mg, %Aston:1982hr,Atkinson:1984fg,Breitweg:1999jy}. This process shows the typical characteristics of a diffractive reaction, i.e. a weak growth of the cross section with the photon-proton centre-of-mass energy $W$ and an exponential decrease as function of the modulus of the squared momentum transfer $|t|$. This behaviour can be understood in terms of the vectormeson dominance model (VDM) %\cite{Sakurai:1960ju, Gell-Mann:1961tg, Sakurai:1969ss}, \cite{bib:vdm}, where the hadronic part of the photon wavefunction is conceived as a superposition of vector--meson states, in which the contribution from the $\rho^0$ dominates. This hadronic component of the photon may then participate in hadron--hadron interactions, notably in elastic scattering off the proton. Thus, an understanding of $\rho^0$ photoproduction sheds light on the hadronic properties of the photon, including the total photon--proton cross section, which is linked to the elastic scattering amplitude by the optical theorem. At sufficiently high centre-of-mass energies $W$, % ($W > 20\,\GeV$ for $\rho^0$ photoproduction), elastic hadron-hadron scattering is well described by the exchange of a single Regge trajectory \cite{bib:regge}, %\cite{Regge:1959mz, Regge:1960zc}, the Pomeron, while at lower values of $W$ the exchange of additional meson trajectories becomes important. The contribution of the exchange of these so-called subleading trajectories to the production cross section is often summed up by introducing an effective Regge trajectory, the Reggeon. In this high--$W$ region, where Pomeron exchange is dominant, the energy dependence of elastic $\rho^0$ photoproduction at fixed momentum transfer $t$ is in diffractive models directly linked to the Pomeron trajectory $\alpha\sub\pom\,(t)$ by $$ \frac{\der \sigma\sub{\gamma p}\,(W)}{\der t} \propto \left ( \frac{W}{W\sub{0}} \right ) ^{4 (\alpha\sub\pom\,(t) - 1)}. $$ Thus, a measurement of the $W$ dependence of elastic $\rho^0$ photoproduction in bins of $t$ is paramount to the direct measurement of the Pomeron trajectory $\alpha\sub\pom\,(t)$, which is well approximated by a linear function $\alpha\sub\pom\,(t) = \alpha\sub{\pom, 0} + \alpha'\sub\pom \cdot t$. The ZEUS collaboration has measured $\rho^0$ photoproduction cross sections \cite{Breitweg:1999jy} %\cite {Derrick:1996vw, Breitweg:1997ed, Breitweg:1999jy} at high values of $W \approx 94\,\GeV$, and has combined its data with data taken at lower energies \cite{Aston:1982hr} and with earlier HERA data from H1 \cite{Aid:1996bs} and ZEUS \cite{Derrick:1995vq,Breitweg:1997ed} in order to extract the $W$ dependence of $\rho^0$ photoproduction and thus the Pomeron trajectory. Their result is $\alpha\sub\pom\,(t) = (1.096 \pm 0.021) + (0.125 \pm 0.038)\,\GeV^{-2} \cdot t$. The intercept $\alpha\sub{\pom, 0}$ is consistent with the expectations of Donnachie and Landshoff \cite{Donnachie:1992ny} ($\alpha\sub{\pom, 0} = 1.0808$) and Cudell \cite{Cudell:1996sh} ($\alpha\sub{\pom, 0} = 1.096^{+0.012}_{-0.009}$), whereas the measured slope $\alpha'\sub\pom$ is significantly lower than the canonical value of $\alpha'\sub\pom = 0.25\,\GeV^{-2}$ extracted from other diffractive processes \cite{bib:dlslope}. In the present analysis, we measure the $W$ dependence of diffractive $\rho^0$ photoproduction over an extended $W$ range of $20 < W < 90\,\GeV$ in bins of $t$ within a single experiment. Thus we can extract the Pomeron trajectory from the data of one experiment alone, which reduces the uncertainty from different systematic errors of different experiments. The $\rho^0$ meson is identified by its decay to two charged pions, which has a branching ratio of $99\,\%$. \section{Event Selection and Triggering} %--------------------------------------- The data were taken with the H1 detector in 2005, when HERA collided $920\,\GeV$ protons with $27.6\,\GeV$ positrons. The H1 detector has been described in detail elsewhere %\cite{Abt:1996hi, Abt:1996xv, Appuhn:1996na}. \cite{bib:h1detector}. A detailed description of the analysis can be found in \cite{bib:rondiss}. Events were mainly triggered with the Fast Track Trigger (FTT) \cite{bib:ftt} at the first H1 trigger stage (trigger level 1). This trigger combines track segments, which are identified in four three-wire-groups within the central jet chamber (CJC), to track candidates within the trigger level 1 decision time of $2.3\,\us$. Two out of four possible track segments have to be identified and found to correspond to compatible track parameters (signed curvature and azimuth) in order to form a track at level 1. Tracks with transverse momenta as low as $100\,\MeV$ can be identified. %For the data period under consideration, the track finding efficiency %is approximately %\Nonumber{$90\,\%$} %for single tracks. %About %\Nonumber{$5\,\%$} %of the tracks are wrongly identified as two separate %tracks. Events were triggered if two or three tracks were found by the FTT with a total charge of zero or $\pm 1$, and if at least two of the tracks had a transverse momentum above $160\,\MeV$. In addition, a trigger signal from the inner proportional chamber (CIP) trigger, consistent with a low multiplicity interaction in the nominal interaction region, was demanded. Events with large energy deposits in the inner forward part of the Liquid Argon (LAr) calorimeter or signals in a set of scintillators (FTI2) within the forward tracker were rejected. The data were taken during a period of three months in 2005, when the operation parameters of the FTT and the CJC were stable. Correcting for the trigger prescale, the data correspond to an integrated luminosity of $569.7\,\nb^{-1}$. % Only data from data taking runs that fulfilled certain quality criteria % were considered. On one hand, the detector components pertinent to this % analysis % % (CJC, CIP, LAr, FTT, FMD, FTS, ToF and luminosity system) % had to be operational during the run. % % In addition, the gain of the jet chambers CJC1 and CJC2, which is % % constantly monitored by measuring the pulse height spectrum of hits % % caused by cosmic ray muons, had to be within % % $4.5\,\%$ ($1.7\,\%$) % % of its average value for CJC1 (CJC2). % % This condition ensures a uniform track trigger efficiency. % To ensure a uniform track trigger efficiency, % runs with unusually high or low gain of the CJC were rejected. % % Finally, three short run ranges were excluded were the % % observed event yield was particularly low; % % in all three run cases logbook entries indicated operation problems % % of the FTT. In order to identify events where the $\rho^0$ decays to $\pi^+ \pi^-$, events were selected by demanding: \begin{itemize} \item A reconstructed primary vertex within $25\,\cm$ of the nominal interaction point in $z$, \item exactly two primary vertex fitted tracks of opposite charge in the central detector, with transverse momenta above $200\,\MeV$ and polar angle within the range $20^\circ < \theta < 160^\circ$, of good quality, \item no additional tracks in the central or forward tracker, \item no scattered electron candidate in the LAr or SpaCal calorimeters, \item no calorimeter custers with energies above $500\,\MeV$ that are not associated with one of the pion tracks. A cluster is considered to be associated with a track when the distance in pseudorapidity $\eta$ and azimuthal angle $\phi$ is less than $0.175$. \end{itemize} A total of $271'684$ events passed these selection criteria, $241'672$ of these events lie in the mass range $0.6 < m\sub{\pi \pi} < 1.1\,\GeV$, where $m\sub{\pi \pi}$ is the invariant mass of the two tracks under a charged pion mass hypothesis. The four-vectors of the two decay pions are reconstructed from their measured track momenta, the $\rho^0$ four vector is the sum of the two pions momenta. The momentum transfer $t$ is then calculated from the $\rho^0$ meson's transverse momentum: $$ t = -p\sub{t, \rho}^2 $$ The $\gamma p$ centre-of-mass energy is calculated from $$ W = \sqrt{2 E\sub p (E\sub{\rho} - p\sub{z, \rho})}, $$ where $E\sub{p} = 920\,\GeV$ is the proton beam energy. \section{Monte Carlo Modelling} %----------------------------- The elastic and proton-dissociative $\rho^0$ production was modeled using the Monte Carlo generator DIFFVM \cite{List:1998jz}. The generator models vector meson production based on the equivalent photon approximation %\cite{vonWeizsacker:1934sx,Williams:1934ad,Budnev:1974de}, \cite{bib:epa}, the vector meson dominance model %\cite{Sakurai:1960ju,Gell-Mann:1961tg,Sakurai:1969ss,Bauer:1977iq}, \cite{bib:vdm}, and pomeron exchange \cite{Goulianos:1982vk}. The generator models the diffractive dissociation of the proton according to a model where the mass $\MY$ of the dissociative system is generated according to $$ \frac{\der^2 \sigma}{\der t \, \der \MY^2} \propto \frac{f\,(\MY)}{(\MY^2)^{1 + \epsilon}}, $$ with $\epsilon = 0.0808$, where $f\,(\MY)$ is a function fitted to experimental data \cite{Goulianos:1982vk} that parametrizes deviations in the low mass region $\MY < 1.9\,\GeV$ from a pure $1/(\MY^2)^{1 + \epsilon}$ behaviour. For this low mass region, the dissociating system is treated as a $N^*$ resonance that decays according to the measured branching fractions \cite{Eidelman:2004wy}. In the higher mass region, the system is modeled as a quark--diquark system, where the quark is assumed to be scattered out of the proton, and the fragmentation is performed using the Lund fragmentation as implemented in JETSET \cite{bib:jetset}. The Monte Carlo events were reweighted in the variables $t$, $W$, and $\mpipi$ in order to get the best possible description of the experimentally observed distributions. In particular, the dipion mass spectrum was reweighted from a nonrelativistic Breit--Wigner function (as implemented in DIFFVM) to a skewed relativistic Breit--Wigner function according to the model by Ross and Stodolsky \cite{Ross:1965qa}, with the skewing parameter given by $$ n\,(t) = 6.1 \cdot \exp\, (t \cdot 1.95\,\GeV^{-2}); $$ this parametrisation was derived from a fit to ZEUS data \cite{Breitweg:1997ed}. The $t$ spectrum was generated as a purely exponential spectrum, and reweighted to the form $$ \frac{\der \sigma^{\gamma p}}{\der t} \propto \exp \,\left ( a \ln \left ( 1 - b_0 t/a \right ) \right ) $$ which smoothly interpolates between an exponential decrease $\der \sigma^{\gamma p}/\der t \propto\exp(b_0 t + b_0^2 t^2/2 a)$ with $b_0 = 11.4~(6.5)\,\GeV^{-2}$ for $W_0 = 90\,\GeV$ at low values of $|t|$, and a power--law behaviour $\der \sigma^{\gamma p}/\der t \propto |t|^{-a}$ with $a=23~(8.5)$ at large $|t|$ for elastic (proton-dissociative) $\rho^0$ production. The coefficient $a$ has been determined from fits to large-$|t|$ data for the proton-dissociative data, and from fits of the form $\der \sigma^{\gamma p}/\der t \propto\exp(b_0 t + c_0 t^2)$ \cite{Breitweg:1997ed} for elastic data. The DIFFVM generator produces events with all values of $Q^2$ and $\MY$. Events were counted as signal events if they fulfilled the cuts $Q^2 < 4\,\GeV^2$ and $(\MY^2+Q^2)/(W^2+Q^2) < 0.01$, otherwise they were treated as background. \section{Cross Section Determination} %------------------------------------ \subsection{Corrected Number of $\rho^0$ Mesons} %----------------------------------------------- The events are distributed in $80$ bins\footnote{Called ``$W/t$ bins'' herafter.} in $W$ and $t$: In $t$, there are 12 bins from $|t|=0\,\GeV^2$ to $|t|=3.5\,\GeV^2$. For $|t|<0.16\,\GeV^2$ ($|t|>0.16\,\GeV^2$), there are 10 (5) $W$ bins for each $t$ bin. The $W$ range covered varies with $t$; as $|t|$ rises, the lower (upper) value of $W$ changes from $20\,\GeV$ ($69\,\GeV$) to $25\,\GeV$ ($90\,\GeV$). For each $W/t$ bin, the number of events is evaluated in dipion mass bins of $25\,\MeV$ width. This dipion mass spectrum is corrected for acceptance, reconstruction and trigger efficiency, and fitted. The overall correction factor is typically $0.25$. The correction factors that are applied are determined for each $\mpipi$ bin in each $W/t$ bin individually. It is well known that due to the large width of the $\rho$ meson there is a significant interference between resonant $\rho$ production with a subsequent decay to two pions, and nonresonant dipion production. This interference leads to a distortion of the observed dipion mass spectrum. Two models have been put forward to describe this effect: The ansatz of S\"oding \cite{Soding:1965nh} describes the distorted spectrum as sum of three terms, a relativistic Breit--Wigner function for the $\rho$, a term for the nonresonant dipion production (here taken to be constant as function of $\mpipi$), and an interference term: \begin{equation} \frac{\der N}{\der \mpipi} = \frac{ N_0 \mrho\,\Grho\,\mpipi + I (\mrho^2 - \mpipi^2)} {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2} + B, \label{eq:soeding} \end{equation} where $N_0$, $I$, and $B$ parametrize the contributions from resonant $\rho^0$ production, interference, and background, respectively. Here, $\mrho$ is the nominal $\rho^0$ mass, and $$ \Grho = \Grhon \, \left ( \frac{\mpipi^2 - 4 m_\pi^2}{\mrho^2 - 4 m_\pi^2}\right )^{\frac 3 2} \frac{\mrho}{\mpipi} $$ is the mass dependent width of the $\rho^0$, following the suggestion of Jackson \cite{Jackson:1964zd} ($\Grhon$ is the nominal $\rho^0$ width, and $m_\pi$ the charged pion mass). In the ansatz of Ross and Stodolsky \cite{Ross:1965qa}, a relativistic Breit--Wigner function is multiplied by a skewing term $(\mpipi/m\sub{\rho})^{-n}$, where $m\sub{\rho}$ is the nominal $\rho$ meson mass, and $n$ is the so-called skewing parameter: \begin{equation} \frac{\der N}{\der \mpipi} = N_0 \frac{\mrho\,\Grho\,\mpipi} {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2} \left ( \frac{\mrho}{\mpipi}\right )^n + B. \label{eq:ross-stodolsky} \end{equation} In each of the $W/t$ bins, the corrected dipion mass spectrum $\der N\super{corr}/\der \mpipi$ (in $25\,\MeV$ bins) is fitted in the range $0.6\,\GeV < \mpipi < 1.1\,\GeV$ with a skewed relativistic Breit--Wigner function as proposed by Ross and Stodolsky, with the nominal $\rho^0$ mass and width fixed to the values $m\sub{\rho^0} = 766.4\,\MeV$ and $\Grhon = 145\,\MeV$, which are the average values of the corresponding fit quantities, obtained from fits in all 80 $W/t$ bins. These values are compatible with the PDG values \cite{Eidelman:2004wy} $\mrho = 768.5\pm 1.1\,\MeV$ and $\Grhon = 150\pm 3\,\MeV$. Fits where the mass and width are left as free parameters result in compatible values for $\mrho$ and $\Grhon$ in all $W/t$ bins. The average values of \mrho\ and \Grhon\ derived from these fits are $\mrho = 766.4\,\MeV$ and $\Grhon = 145\,\MeV$. To extract the number of resonantly produced $\rho^0$ mesons in each $W/t$ bin, the integral of the unskewed Breit--Wigner function is evaluated in the region $2 m\sub{\pi} < \mpipi < \mrho+5 \Grhon$, with the value of $\mrho+5 \Grhon = 1.52\,\GeV$ taken from the PDG \cite{Eidelman:2004wy}: $$ \Ncor = N_0 \int \limits_{\mpipi = 2 m_\pi}^{\mrho+5 \Grhon} \frac{\mrho\,\Grho\,\mpipi} {(\mrho^2 - \mpipi^2)^2 + \mrho^2 \Grho^2} \der \mpipi. $$ Extracting the number of $\rho^0$ mesons from the corresponding term in the S\"oding parametrisation leads to compatible results. % a change of the number % of events of less than % \Nonumber{$2\,\% $} % in all bins with % \Nonumber{$|t|