%================================================================ % LaTeX file with prefered layout for H1 paper drafts % use: dvips -D600 file-name %================================================================ %%%%%%%%%%%%% LATEX HEADER %%%%%%%%%%%%% DO NOT DELET NOT CHANGE ANYTHING IN THE HEADER %%%%%%%%%%%%% UNLESS CLEARLY RECOMMENDED \documentclass[12pt]{article} \usepackage{epsfig} \usepackage{amsmath} \usepackage{hhline} \usepackage{amssymb} \usepackage{times} \usepackage{cite} %%% I add rotating pakidge for figures: \usepackage{rotating} %%%%%%%%%%%%%%%%%%new color%%%%%%%%%%%%%%%% \usepackage{color} \definecolor{darkgreen}{rgb}{0,0.5,0} \definecolor{DeepPink3}{rgb}{0.803922,0.062745,0.462745} %%%%%%%%%%%% 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} 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\newcommand{\dstar}{\ensuremath{D^*}} \newcommand{\dstarp}{\ensuremath{D^{*+}}} \newcommand{\dstarm}{\ensuremath{D^{*-}}} \newcommand{\dstarpm}{\ensuremath{D^{*\pm}}} \newcommand{\zDs}{\ensuremath{z(\dstar )}} \newcommand{\Wgp}{\ensuremath{W_{\gamma p}}} \newcommand{\ptds}{\ensuremath{p_t(\dstar )}} \newcommand{\etads}{\ensuremath{\eta(\dstar )}} \newcommand{\ptj}{\ensuremath{p_t(\mbox{jet})}} \newcommand{\ptjn}[1]{\ensuremath{p_t(\mbox{jet$_{#1}$})}} \newcommand{\etaj}{\ensuremath{\eta(\mbox{jet})}} \newcommand{\detadsj}{\ensuremath{\eta(\dstar )\, \mbox{-}\, \etaj}} % Journal macro \def\Journal#1#2#3#4{{#1} {\bf #2} (#3) #4} \def\NCA{\em Nuovo Cimento} \def\NIM{\em Nucl. Instrum. Methods} \def\NIMA{{\em Nucl. Instrum. Methods} {\bf A}} \def\NPB{{\em Nucl. Phys.} {\bf B}} \def\PLB{{\em Phys. Lett.} {\bf B}} \def\PRL{\em Phys. Rev. Lett.} \def\PRD{{\em Phys. Rev.} {\bf D}} \def\ZPC{{\em Z. Phys.} {\bf C}} \def\EJC{{\em Eur. Phys. J.} {\bf C}} \def\CPC{\em Comp. Phys. Commun.} \begin{titlepage} \noindent %Date: \today \\ %Version: 2.1 \\ %Editors: Anna Kropivnitskaya (kropiv@mail.desy.de), %Andrei Rostovtsev (rostov@itep.ru) \\ %Referees: Eddi de Wolf (Eddi.DeWolf@ua.ac.be), %Tim Greenshaw (green@liverpool.ac.uk) \\ %\\ %Date of final reading: December 5th, 2008; 9:00am, Spokesman's office DESY $08-172$ \hspace{10 cm} ISSN $0418-9833$\\ February $2009$ \noindent %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \vspace{2cm} \begin{center} \begin{Large} {\bf Inclusive Photoproduction of \boldmath{$\bf\rho^0$}, \boldmath{$\bf K^{*0}$} and \boldmath{$\bf \phi$} Mesons at HERA \\}%(version of \today )} \vspace{2cm} H1 Collaboration \end{Large} \end{center} \vspace{2cm} \begin{abstract} \hspace{-0.55 cm}Inclusive non-diffractive photoproduction of $\rho(770)^0$, $K^*(892)^0$ and $\phi(1020)$ mesons is investigated with the H1 detector in $ep$ collisions at HERA. The corresponding average $\gamma p$ centre-of-mass energy is $210$ GeV. The mesons are measured in the transverse momentum range $0.50.4$~GeV. The photon virtuality $Q^2$ is smaller than $0.01$~GeV$^2$, due to the positron tagger acceptance. The photon energy is determined from the difference between the positron beam energy and the energy measured in the positron tagger. In order to reduce the non-$ep$ background and to ensure good reconstruction of the event kinematics, the following criteria are applied: \begin{itemize} \item{Events are selected if the reconstructed $\gamma p$ centre-of-mass energy lies within the interval $1742$ GeV is detected in the photon tagger. This suppresses the background arising from random coincidences of Bethe-Heitler events in the positron tagger with beam-gas interactions in the main H1 detector.} \item{Events are selected if the $z$ coordinate of the event vertex, reconstructed using the CJCs, lies within $35$ cm of the mean position for $ep$ interactions.} \end{itemize} Background from elastic and diffractive events is suppressed by the above trigger requirements. To further reduce the contribution of diffractive processes, the presence of an energy deposit of at least $500$ MeV is required in the forward region of the LAr, defined by $2.03<\eta_{lab}<3.26$. Monte Carlo studies show that, with this requirement, less than $1\%$ of the final event sample consists of diffractive events with $X_\pom<0.05$, where $X_\pom=M_X^2/W^2$ and $M_X$ is the invariant mass of the diffractive system. In total, about $1.8\times10^6$ events satisfy the above selection criteria. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Selection of \boldmath{$\bf\rho(770)^0$}, \boldmath{$\bf K^*(892)^0$} and \boldmath{$\bf \phi(1020)$} Mesons} %%%%%%%%%%%%%%%%%%%%%% %\subsubsection{Signal extraction and fitting procedures} The mesons are identified using the $\rho(770)^0\to\pi^+\pi^-$, $K^*(892)^0\to K^+\pi^-$ or $\overline{K}^*(892)^0\to K^-\pi^+$ and $\phi(1020)\to K^+K^-$ decays\footnote{ In the following, the notation $K^{*0}$ is used to refer to both the $K^{*0}$ and $\overline{K}^{*0}$ mesons unless explicitly stated otherwise.}. Charged tracks reconstructed in the CJCs with $p_T>0.15~$GeV and pseudorapidity $|\eta_{lab}|<1.5$ are considered as charged pion or kaon candidates. Since most of the charged particles in $ep$ collisions are pions, no attempt to identify pions is made, while identification criteria for charged kaons are applied for the extraction of the $K^{*0}$ and $\phi$ signals. This is done by measuring the momentum-dependent specific energy loss $dE/dx$ in the CJCs. This method gives a significant improvement in the signal-to-background ratio for low $p_T$ mesons, $p_T<1.5$~GeV, where the $dE/dx$ resolution allows good particle identification. For high $p_T$ mesons, $p_T>1.5$~GeV, the $dE/dx$ method is inefficient and therefore particle identification % AS criteria are not applied. Such tracks are considered as both pion and kaon candidates and their four-momenta are determined from the track measurements using the corresponding mass hypothesis~\cite{pdg}. Vector meson candidates are reconstructed from these four-momenta. The kinematic range for the reconstructed neutral mesons is restricted to $|y_{lab}|<1$ and $p_T>0.5$~GeV. To extract the $\rho^0$, $K^{*0}$ and $\phi$ signals, the distributions of respective invariant masses of their decay products, $m_{\pi^+\pi^-}$, $m_{K^\pm\pi^\mp}$ and $m_{K^+ K^-}$, are fitted using a function composed of three parts: \begin{equation} F(m)=B(m)+\sum R(m)+\sum S(m). \label{invfit} \end{equation} The terms correspond to contributions from the combinatorial background, $B(m)$, %% AS from reflections which result from decays other than the signal under consideration, $R(m)$, and from the relevant signal, $S(m)$, respectively. The combinatorial background function is taken to be: $$ B(m) = (a_0+a_1m+a_2m^2+a_3m^3)\cdot B^0(m)\, , $$ where $a_0$, $a_1$, $a_2$ and $a_3$ are free parameters, and $B^0(m)$ is the invariant mass distribution of the like-sign charged particle combinations: $\pi^\pm\pi^\pm$ for the $\rho^0$ and $K^\pm\pi^\pm$ for the $K^{*0}$. %Monte Carlo studies show that it is the best parametrisation. The shape of the combinatorial background for $\phi$ is described by the following function: $$ B(m) = b_1(m^2-4m_K^2)^{b_2} e^{-b_3 m}\, , $$ where $b_1$, $b_2$ and $b_3$ are free parameters and $m_K$ is the kaon mass. The second term, $\sum R(m)$, in~(\ref{invfit}) represents the sum of the reflections; %in the invariant mass spectra originating from other hadronic decays. for example, charged particles from the decay $K^{*0}\to K^\pm \pi^\mp$ with the kaon misidentified as a charged pion will give rise to structure in the $m_{\pi^+\pi^-}$ spectrum and must be taken into account as a separate contribution. In addition, there are two other contributions to the $m_{\pi^+\pi^-}$ spectrum in the mass region of interest. These arise from the decays $\omega(782)\rightarrow \pi^+ \pi^-$ and $\omega(782)\rightarrow \pi^+ \pi^-\pi^0$ in which the $\pi^0$ is not observed. % %% AS For the $\omega(782)$ meson, the production rate relative to that of the $\rho^0$ is varied within the range $1.0\pm0.2$, which is consistent with measurements of the $\omega(782)/\rho^0$ ratio in hadronic collisions~\cite{reflection_ISR} and in $Z^0$ boson decays~\cite{reflection_LEP}. The $\omega(782)$ branching ratios are taken from ~\cite{pdg}. %% The five major reflections in the $m_{K^\pm\pi^\mp}$ spectrum are due to: the decay $\rho^0\rightarrow \pi^+ \pi^-$ with the $\pi^+$ or $\pi^-$ misidentified as a charged kaon; the decays $\omega(782)\rightarrow \pi^+ \pi^-$ and $\omega(782)\rightarrow \pi^+ \pi^-\pi^0$ with the $\pi^0$ not observed and with one of the $\pi^+$ or $\pi^-$ mesons misidentified as a charged kaon; the decay $\phi\rightarrow K^+K^-$ with one of the kaons misidentified as a charged pion and a self-reflection from the $K^{*0}$, where the pion and kaon are interchanged. % For the $m_{K^+K^-}$ spectrum, there are no reflections from known resonances in the invariant mass region of interest. % Therefore, the shapes of the reflections are taken from Monte Carlo calculations. The contribution of the reflections from the $\rho^0$, $K^{*0}$ and $\phi$ mesons is tied to the production rates determined in this analysis and is therefore iteratively calculated. The function $S(m)$ used to describe the signal in~(\ref{invfit}) is a convolution of a relativistic Breit-Wigner function $BW(m)$ and a detector resolution function $r(m,m')$. %which accounts for the uncertainty in the momentum measurement using CJCs. The relativistic Breit-Wigner function \begin{equation} BW(m)=A_0\,\frac{m_0\,m\,\Gamma(m)}{(m^2-m_{0}^{2})^2+m_{0}^{2}\,\Gamma^{2}(m)}\;, \label{relbw} \end{equation} is used with $$\Gamma(m)=\Gamma_{0}\left(\frac{q}{q_0}\right)^{2l+1}\frac{m_0}{m}\;,$$ where $A_0$ is a normalisation factor, $\Gamma_{0}$ is the resonance width, $l=1$ for vector mesons, $m_0$ is the resonance mass, $q$ is the momentum of the decay products in the rest frame of the parent meson, and $q_0$ is their momentum at $m=m_0$. The cross sections cited in this paper assume that the meson signal is defined as the integral of the relativistic Breit-Wigner function~(\ref{relbw}) in the region $\pm2.5\Gamma_0$ around the mass $m_0$. Monte Carlo studies show that a non-relativistic Breit-Wigner function with width $\Gamma_{res}$ provides a good description of the detector resolution function: \begin{equation} r(m,m') = \frac{1}{2\pi}\,\frac{\Gamma_{res}}{(m-m')^2+(\Gamma_{res}/2)^2}\;. \label{resol} \end{equation} For the $K^{*0}$ analysis, the resolution parameter is determined from Monte Carlo with $\Gamma_{res}=12$~MeV. %The mass resolution is about four times smaller than It is small compared to the width of the $K^{*0}$ meson ($50.3\pm0.6$~MeV)~\cite{pdg}, leading only to a small change in the shape of the resonance. For the $\phi$, $\Gamma_{res}$ is comparable to the width of the $\phi$ meson ($\Gamma_0=4.26\pm0.05$~MeV)~\cite{pdg}. As a result, the shape of the $\phi$ signal is significantly changed, and hence the detector resolution $\Gamma_{res}$ is taken as a free parameter in the fit. It is found to vary from $3.4$~MeV to $6.0$~MeV, increasing with the $p_T$ of the $\phi$ meson. For the $\rho^0$ meson, the detector resolution is significantly smaller than its width. However, BEC between the $\rho^0$ decay pions and other pions in the event strongly distort the $\rho^0$ line shape. %While the phase space effect is much smaller. The BEC plays an important role in broadening the $\rho^0$ mass peak and in shifting it towards lower masses. Similar effects are observed in $pp$ and heavy-ion collisions at RHIC~\cite{STAR} and in $e^+e^-$ collisions at LEP using $Z^0$ decays~\cite{reflection_LEP}. It is therefore important to check that the Monte Carlo model used for the extraction of the cross sections describes the di-pion spectra in the data. The data spectra and the Monte Carlo simulations with and without BEC are shown in figure~\ref{bec}. The Monte Carlo model with BEC is in a good agreement with the data in the region of the $\rho^0$ resonance, whereas the model without BEC fails to describe the di-pion mass spectrum. The results of fitting the function~(\ref{invfit}) to the $m_{\pi^+\pi^-}$ data in the mass range from $0.45$ to $1.7~$GeV with the contributions due to the combinatorial background and the reflections are shown in figure~\ref{fig:signal}a$)$, and after combinatorial background subtraction in figure~\ref{fig:signal}b$)$. In this mass range, the signal from the $K^0_S$, $f_0(980)$ and $f_2(1270)$ mesons is taken into account. The $K^0_S$ signal is fitted using a Gaussian centred on the nominal mass and with fixed width. The relativistic Breit-Wigner function given in equation~(\ref{relbw}) is used for the $f_0(980)$ and $f_2(1270)$ mesons. In the fit, the resonance masses $m_0$ %% AS and the yields %% are free parameters. The $\rho^0$ and $f_2(1270)$ widths are fixed to the Particle Data Group~\cite{pdg} values and the $f_0(980)$ width is fixed to $70$~MeV. %% AS Due to the small signal and the non-trivial background behaviour, which lead to large uncertainties, cross sections for $f_0(980)$ and $f_2(1270)$ meson production are not measured here. %% The $K^{*0}$ signal is measured under the assumption that there is no difference between the particle and antiparticle production rates, and the signal obtained from the $m_{K^\pm \pi^\mp}$ spectrum is divided by $2$ to determine the $K^{*0}$ rate in the following. The result of fitting the function~(\ref{invfit}) to the $m_{K^\pm\pi^\mp}$ data in the mass range from $0.7$ to $1.2~$GeV with the contributions due to the combinatorial background and the reflections is shown in figure~\ref{fig:signal}c$)$. In the fit, the $K^{*0}$ width is fixed to the nominal value while the mass parameter is left free. The result for the $K^{*0}$ mass is compatible with the world average~\cite{pdg}. %%%%%%%%%%%%%%%%%%%%% The result of fitting function~(\ref{invfit}) to the $m_{K^+K^-}$ data in the mass range from $0.99$ to $1.06~$GeV, together with the background contribution, is shown in figure~\ref{fig:signal}d$)$. In the fit, the $\phi$ width, $\Gamma_0$, is fixed to the nominal value while the mass is left a free parameter and is found to be compatible with the world average~\cite{pdg}. % %%%%%%%%%%%%%%%%%%% \subsection{Cross Section Determination and Systematic Errors} The invariant differential cross section for $\rho^0$, $K^{*0}$ and $\phi$ meson production is measured in the $y_{lab}$ region from $-1$ to $1$ in seven bins in transverse momentum from $0.5$ to $7$~GeV. It is calculated according to: $$ \frac{1}{\pi}\,\frac{d^2\sigma^{\gamma p}}{dp_T^2\,dy_{lab}} = \frac{N} {\pi \cdot {\cal L}\cdot BR \cdot \Phi_\gamma\cdot \epsilon \cdot \Delta p_T^2\cdot \Delta y_{lab}} \, , $$ where $N$ is the number of mesons from the fit in each bin. The corresponding bin widths are $\Delta y_{lab}=2$ and $\Delta p_T^2 = 2p_T^{bin}\Delta p_T$. Bin centre corrections based on equation~(\ref{pawer}) are applied to define the value of $p_T^{bin}$ at which the differential cross section is measured. ${\cal L}$ denotes the integrated luminosity and $\epsilon$ the efficiency. The branching fractions $BR$ are taken from~\cite{pdg} and are equal to $1$, $0.67$ and $0.49$ for $\rho^0\to \pi^+\pi^-$, $K^{*0}\to K^\pm\pi^\mp$ and $\phi\to K^+K^-$, respectively. The photon flux $\Phi_\gamma = 0.0127$ is calculated using the Weizs\"acker-Williams approximation~\cite{Weizs}. The single differential cross section for $\rho^0$, $K^{*0}$ and $\phi$ meson production for $p_T>0.5$~GeV is measured in four bins in rapidity from $-1$ to $1$ according to: $$ \frac{d\sigma^{\gamma p}}{dy_{lab}} = \frac{N} {{\cal L}\cdot BR \cdot \Phi_\gamma\cdot \epsilon \cdot \Delta y_{lab}} \, . $$ Here, the bin width is $\Delta y_{lab}=0.5$. The fit procedure described in the previous section is repeated to determine the number of mesons, $N$, in each measurement bin, calculated as an integral over the signal function~(\ref{relbw}) within $\pm2.5\Gamma_0$ around the meson mass. Similarly, the total visible cross section for $\rho^0$, $K^{*0}$ and $\phi$ meson production is measured from the number of mesons fitted in the range $|y_{lab}|<1$ and $p_T>0.5$~GeV. The efficiency is given by $\epsilon=\epsilon_{rec}\cdot {\cal A}_{etag}\cdot {\cal A}_3\cdot\epsilon_{trig}$. The reconstruction efficiency for the mesons, $\epsilon_{rec}$, includes the geometric acceptance and the efficiency for track reconstruction. It is calculated using Monte Carlo data and is at least $45\%$ at low $p_T$ and rises to about $90\%$ with increasing $p_T$. % AS For the acceptance determination, the Monte Carlo generators are reweighted to model the observed $p_T$-dependences. % The average acceptance of the positron tagger, ${\cal A}_{etag}$, is about $50\%$, as determined in~\cite{H1total}. The trigger acceptance, ${\cal A}_3$, arises from the requirement that at least three tracks are reconstructed in the CJCs with $p_T>0.4$ GeV. It is determined from Monte Carlo simulations with PYTHIA and PHOJET and varies from $50\%$ to $95\%$. The trigger efficiency, $\epsilon_{trig}$, is calculated from the data using monitor triggers. It is about $90\%$. % AS The efficiencies and acceptances as calculated from the PYTHIA and PHOJET simulation are found to be consistent. Small residual differences, attributed to different track multiplicity predictions, are taken into account in the systematic uncertainties of the measurement. % The statistical error varies from $7$ to $15\%$ for the $\rho^0$, $10$ to $18\%$ for the $K^{*0}$ and $13$ to $24\%$ for the $\phi$ meson cross sections. The systematic errors arise from the uncertainties in the track reconstruction efficiency ($4\%$) and the trigger efficiency (up to $6\%$), the variation of the $f_0(980)$ width by $\pm30$~MeV in the $\rho^0$ fit (up to $7\%$), %the resolution width variation in $\phi(1020)$ meson ($3.5-27\%$), the uncertainties in the $dE/dx$ kaon identification procedure ($6\%$ for the $K^{*0}$ and $12\%$ for the $\phi$) and the luminosity calculation ($2\%$), the variation of the background shape ($5\%$) and the variation of the assumptions about the normalisation of the contributions from the reflections ($4\%$ for the $\rho^0$ and up to $15\%$ for the $K^{*0}$). The total systematic error varies from $10$ to $12\%$ for the $\rho^0$, $11$ to $21\%$ for the $K^{*0}$ and $10$ to $17\%$ for the $\phi$ meson cross sections. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Results and Discussion} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The inclusive non-diffractive photoproduction cross sections for $\rho^0(770)$, $K^{*0}(892)$ and $\phi(1020)$ mesons in the kinematic region $Q^2<0.01$ GeV$^2$, $1740.5$ GeV and $|y_{lab}|<1$ are found to be: $$ \phantom{} \sigma_{vis}^{\gamma p}(\gamma p \rightarrow \rho^0 X) \phantom{0}\, = 25600 \pm 1800 \pm 2700 \text{ nb;} $$ $$ \sigma_{vis}^{\gamma p}(\gamma p \rightarrow K^{*0} X) = \phantom{0}6260 \pm \phantom{0}350 \pm \phantom{0}860 \text{ nb;} $$ $$ \, \sigma_{vis}^{\gamma p}(\gamma p \rightarrow \phi X) \phantom{00}= \phantom{0}2400 \pm \phantom{0}180 \pm \phantom{0}340 \text{ nb.} $$ The first error is statistical and the second systematic. %Here, and in the following, Note that the $K^{*0}$ cross section is the sum of the particle and antiparticle contributions divided by $2$. The differential cross sections for the photoproduction of $\rho^0$, $K^{*0}$, and $\phi$ mesons are presented in tables~$1$~and~$2$ and in figure~\ref{fig:sigma}. Within the rapidity range of this measurement, the resonance production rates are constant as a function of rapidity, within errors. The transverse momentum spectra of the $\rho^0$, $K^{*0}$ and $\phi$ mesons can be parametrised by function~(\ref{pawer}), where $d\sigma/dy_{lab}$ in equation~(\ref{power2}) corresponds to the average value of the cross section over central rapidities, $\langle d\sigma/dy_{lab}\rangle_{|y_{lab}|<1}$. In the fit, the value of the power $n$ is fixed to be $6.7$, as derived previously from measurements of charged particle spectra by the H1 collaboration~\cite{H1pions} which gave $n=6.7\pm0.3$. The power law distribution, with this value of $n$, describes %the spectra of charged particles~\cite{H1pions}, $K^0_S$ meson, $\Lambda^0$ baryon~\cite{HERA_L0} and $D^{*\pm}$ meson production~\cite{HERA_charmed} at HERA, as is shown in figure~\ref{fig:allxsec}. A similar shape of the transverse momentum distribution, but with different values of the parameters $n$ and $E_{T_0}$, was reported for charged particles produced in hadronic collisions~\cite{UA1charged}. %???????????????? The results of the fits of the data to function~(\ref{pawer}) are shown in figure~\ref{fig:sigma}a$)$. In table~$3$, the parameters of the fit and the average transverse kinetic energy $\langle E_T^{kin}\rangle$, the average transverse energy $\langle E_T\rangle=\langle E_T^{kin}\rangle +m_0$ and the average transverse momentum $\langle p_T\rangle = \sqrt{\langle E_T\rangle^2-m_0^2}$ derived from~(\ref{pawer}) are presented. The errors include the experimental uncertainty on the value of $n$. Also given are the $\langle p_T\rangle$ values measured at RHIC in $pp$ and Au-Au collisions~\cite{STAR} . % and to PYTHIA predictions for photoproduction. It is interesting to observe that the resonances with different masses, lifetimes and strangeness content are produced with about the same value of the average transverse kinetic energy $\langle E_T^{kin}\rangle$. This observation supports the thermodynamic picture of hadronic interactions~\cite{therm}, in which the primary hadrons are thermalised during the interaction. The values of $\langle p_T\rangle$ for $\rho^0$, $K^{*0}$ and $\phi$ mesons are similar in $\gamma p$ and $pp$ collisions with about the same centre-of-mass energy $\sqrt{s} \approx 200$~GeV, while these values are all higher in Au-Au collisions. The PYTHIA and PHOJET models do not describe the shape of the measured $p_T$ spectra. Moreover, contrary to the data, the Monte Carlo $p_T$ spectra are not described by the power law function~(\ref{pawer}). These observations are illustrated in figures~\ref{fig:sigma}c$)$ and~\ref{fig:sigma}d$)$. The measurements in the visible kinematic range of the $\rho^0$, $K^{*0}$ and $\phi$ mesons, $p_T>0.5$~GeV and $|y_{lab}|<1$, are extrapolated to the full $p_T$ range using the parametrisation~(\ref{pawer}) to determine the total inclusive non-diffractive photoproduction cross sections. The extrapolation factors are of order two. In the rapidity interval $|y_{lab}|<1$ and integrated over the full $p_T$ range the following cross section ratios are obtained: $$ \,R(K^{*0}/\rho^0) =0.221\pm0.036\, ;$$ $$\quad\, R(\phi/\rho^0) =0.078\pm0.013\, ; $$ $$\;\, R(\phi/K^{*0}) =0.354\pm0.060\, .$$ The errors are given by the statistical and systematic errors added in quadrature. PYTHIA and PHOJET, with the strangeness suppression factor $\lambda_s = 0.286$~\cite{BEC}, predict the ratios $0.200$, $0.055$ and $0.277$, respectively, which are similar to the measured values, but are all somewhat lower than these. In table~$4$, $R(\phi/K^{*0})$ is compared to the corresponding ratios measured by STAR in $pp$ and Au-Au collisions~\cite{STAR} at $\sqrt{s_{NN}}=200$ GeV. %The H1 result is in good agreement with the STAR $pp$ ratio at a similar energy scale. Although the rapidity ranges at the H1 and RHIC experiments differ\footnote{The difference in rapidity between the laboratory frame and the $\gamma p$ frame is about two units at H1.}, the resulting ratios for $pp$ and $\gamma p$ interactions are very close. %Some tendency for $\phi$ meson production to be more abundant in Au-Au %collisions is observed. %Some tendency to produce $\phi$ mesons with a higher relative rate than $K^{*0}$ %mesons is observed in Au-Au collisions. However, the corresponding result in Au-Au collisions is observed to be higher. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conclusions} First measurements of the inclusive non-diffractive photoproduction of $\rho(770)^0$, $K^*(892)^0$ and $\phi(1020)$ mesons at HERA are presented. The differential cross sections for the production of these resonances as a function of transverse momentum are described by a power law distribution while the single differential cross sections as a function of rapidity are observed to be flat in the visible range. Despite their different masses, lifetimes and strangeness content, these resonances are produced with about the same value of the average transverse kinetic energy. This observation supports a thermodynamic picture of hadronic interactions. The description of the shape of the $\rho^0$ resonance produced in $\gamma p$ collisions at HERA is improved by taking Bose-Einstein correlations into account. A similar effect is observed in $pp$ and heavy-ion collisions at RHIC and in $e^+e^-$ annihilation at LEP, using $Z^0$ decays. The cross section ratios $R(K^{*0}/\rho^0)$, $R(\phi/\rho^0)$ and $R(\phi/K^{*0})$ are determined, and $R(\phi/K^{*0})$ is compared to results obtained in $pp$ and heavy-ion collisions by the STAR experiment at RHIC. The ratio $R(\phi/K^{*0})$ measured in $\gamma p$ interactions is in agreement with the $pp$ results, while this ratio is observed to be smaller than the result obtained in Au-Au collisions. %Some tendency for $\phi$ meson production to be more abundant in Au-Au %collisions is observed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 support and for the hospitality which they extend to the non-DESY members of the collaboration. %\\(You can of course thank people external to H1 for discussions and help when they contributed to clarifying the physics analysis, provided software programs, theory calculations etc.) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{References} \begin{thebibliography}{99} \bibitem{string} B.~Andersson {\it et al.}, %``Parton fragmentation and string dynamics," Phys. Rept. {\bf 97} (1983) 31. \bibitem{cluster} G.C.~Fox and S.~Wolfram, %``A model for parton showers in QCD," Nucl. Phys. 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The first error is statistical and the second systematic. For each bin in $p_T$ the range as well as the bin-centred value $p_T^{bin}$ are given. % } \end{table} %%=============================== \begin{table} \begin{center} \begin{tabular}{|c|c|c|c|} %\multicolumn{4}{l}{$\rho(770)^0$}\\ \hline & \multicolumn{3}{|c|}{$ d\sigma/dy_{lab}$~[$\mu$b]} \\ \hline $y_{lab}$&$\rho^0$& $(K^{*0}+\overline{K}^{*0})/2$& $\phi$\\ \hline $[-1.0,-0.5]$ & $11.0\pm1.0\pm1.2$ & $3.36\pm0.35\pm0.72$ & $1.44\pm0.25\pm0.22$ \\ \hline $[-0.5,0.0]$ & $13.1\pm1.1\pm1.4$ & $2.52\pm0.27\pm0.36$ & $1.08\pm0.12\pm0.16$ \\ \hline $[0.0,0.5]$ & $10.4\pm1.5\pm1.1$ & $3.07\pm0.30\pm0.44$ & $1.44\pm0.13\pm0.22$ \\ \hline $[0.5,1.0]$ & $14.6\pm1.3\pm1.5$ & $4.28\pm0.44\pm0.79$ & $1.61\pm0.33\pm0.25$ \\ \hline \end{tabular} \end{center} \caption{ Inclusive non-diffractive photoproduction single differential cross sections $d\sigma/dy_{lab}$ for $\rho(770)^0$, $K^*(892)^0$ and $\phi(1020)$ mesons in the transverse momentum range $p_T>0.5$ GeV in bins of $y_{lab}$. The first error is statistical and the second systematic.} \end{table} %\newpage \clearpage %\vspace*{.0cm} \begin{table} \begin{center} \begin{tabular}{|l|r|c|c|c|} \hline \multicolumn{2}{|c|}{} &$\rho^0$& $(K^{*0}+\overline{K}^{*0})/2$& $\phi$\\ \hline $\gamma p$ (H1) &$\langle d\sigma/dy_{lab}\rangle_{|y_{lab}|<1}$ [$\mu$b] &$23.6\pm2.7$ & $\phantom{.}5.22\pm0.60\phantom{.0}$&$\phantom{.}1.85\pm0.23\phantom{.0}$ \\ \cline{2-5} &$T$ [GeV] &$\phantom{0}0.151\pm0.011$ &$0.166\pm0.012$&$0.170\pm0.012$ \\ \cline{2-5} % $T^{PYTHIA}$ [GeV]&$0.136$ &$0.140$&$0.149$ \\ % \hline &$\langle E_T\rangle$ [GeV]&$\phantom{0}1.062\pm0.018 $ &$1.205\pm0.020$&$1.333\pm0.022$ \\ \cline{2-5} &$\langle E_T^{kin}\rangle$ [GeV]&$\phantom{0}0.287\pm0.018$ &$0.313\pm0.020$&$0.314\pm0.022$ \\ \cline{2-5} &$\langle p_T\rangle$ [GeV]&$\phantom{0}0.726\pm0.027$ &$0.810\pm0.030$&$0.860\pm0.035$ \\ \hline \hline $pp$ (STAR) &$\langle p_T\rangle_{pp}$ [GeV]&$\phantom{0}0.616\pm0.062$ &$0.81\pm0.14$& $0.82\pm0.03$\\ \hline Au-Au (STAR) &$\langle p_T\rangle_{AuAu}$ [GeV]&$\phantom{00}0.83\pm0.10\phantom{0}$ &$1.08\pm0.14$&$0.97\pm0.02$ \\ \hline \end{tabular} \end{center} \caption{The parameters $\langle d\sigma/dy_{lab}\rangle_{|y_{lab}|<1}$ and $T=E_{T_0}/n$ for $\rho^0$, $K^{*0}$ and $\phi$ mesons from a fit of function~(\ref{pawer}) to the differential cross sections. The average transverse energy $\langle E_T\rangle$, kinetic energy $\langle E_T^{kin}\rangle$ and momentum $\langle p_T\rangle$ are also presented. The errors correspond to the quadratically summed statistical and systematic errors. %Results from $pp$ and Au-Au collisions obtained by the STAR experiment~\cite{STAR} are also shown. %The temperature $T$ is also compared to the value $T^{PYTHIA}$ obtained from PYTHIA simulations. Also shown are measurements in $pp$ and Au-Au interactions at nucleon-nucleon centre-of-mass energy $\sqrt{s_{NN}}=200$ GeV~\cite{STAR} at central rapidities. } \end{table} %=============================== \begin{table} \begin{center} \begin{tabular}{|l|l|c|} \hline Experiment &Measurement & $R(\phi/K^{*0})$\\ \hline H1&$\gamma p$, $\langle W\rangle=210$~GeV, $|y_{lab}|<1$ & $0.354\pm0.060$\\ \hline STAR&$pp$, $\sqrt{s}=200$~GeV, $|y|<0.5$ & $0.35\pm0.05$\\ \cline{2-3} &Au-Au, $\sqrt{s_{NN}}=200$~GeV, $|y|<0.5$ & $0.63\pm0.15$\\ \hline \end{tabular} \end{center} \caption{The ratio $R(\phi/K^{*0})$ of the total cross-sections for $\phi$ and $K^{*0}$ production obtained in $\gamma p$ collisions (H1) at $\langle W\rangle=210$~GeV. The errors correspond to the quadratically summed statistical and systematic errors. Also shown are measurements in $pp$ and Au-Au interactions at nucleon-nucleon centre-of-mass energy $\sqrt{s_{NN}}=200$ GeV~\cite{STAR} at central rapidities. } \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \clearpage \vspace*{4.0cm} \begin{figure}[ht] \center \setlength{\unitlength}{1cm} \hspace*{1.5cm} \begin{picture}(15.0,10.0) \epsfig{file=d08-172f1.eps, bburx=350,bbury=570,bbllx=20,bblly=300, width=10.0cm} \end{picture} \caption{ %The like-sign di-pion invariant mass spectrum $(a)$ and The unlike-sign di-pion mass spectrum after subtracting the like-sign contribution, normalised to the total number of entries. The solid and dashed curves correspond to the PYTHIA simulation with and without Bose-Einstein correlations (BEC), respectively. } \label{bec} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \clearpage \vspace*{4.0cm} \begin{figure}[ht] \center \setlength{\unitlength}{1cm} \hspace*{1.5cm} \begin{picture}(15.0,10.0) \epsfig{file=d08-172f2.eps, bburx=530,bbury=600,bbllx=50,bblly=80, clip=,width=14.0cm,angle=0} \end{picture} \caption{The invariant mass spectra for $\pi^+\pi^-$ in $a)$ and $b$), for $K^\pm\pi^\mp$ in $c$) and for $K^+K^-$ in~$d$). The full curves show the result of the fit; the dashed curves correspond to the contribution of the combinatorial background $B(m)$. In~$b)$, the data and the fit $F(m)$ are shown after subtraction of the combinatorial background $B(m)$; the dotted and dash-dotted curves show the contributions from $\omega$ and $K^*$ reflections, respectively. In~$c)$, the dotted curve corresponds to the contribution of the reflections and the dash-dotted curve corresponds to the contribution of the $K^*(892)$ signal. In~$d)$, the dotted curve corresponds to the contribution of the $\phi(1020)$ signal. } \label{fig:signal} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \clearpage \begin{figure}[ht] \center \setlength{\unitlength}{1cm} %\hspace*{1.5cm} %\vspace*{4.0cm} \begin{picture}(17.0,16.0) \epsfig{file=d08-172f3.eps, bburx=520,bbury=580,bbllx=0,bblly=40, % bburx=520,bbury=580,bbllx=0,bblly=230, clip=,width=15.0cm} \end{picture} \caption{The inclusive differential non-diffractive cross sections for $\rho^0(770)$, $K^{*0}(892)$ and $\phi(1020)$ mesons measured in~$a)$ as a function of transverse momentum for $|y_{lab}|<1$ and in~$b)$ as a function of rapidity for $p_T>0.5$ GeV. The curves on the figure~$a)$ correspond to the power law, equation~(\ref{pawer}), with $n=6.7$. The ratios of data to Monte Carlo predictions ``Data/MC'' are shown for the PYTHIA (full points) and PHOJET (empty points) simulations as a function of transverse momentum for $|y_{lab}|<1$ in~$c)$ and as a function of rapidity for $p_T>0.5$ GeV in~$d)$. Statistical and systematic errors are added in quadrature.} \label{fig:sigma} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \clearpage \begin{figure}[ht] \center \setlength{\unitlength}{1cm} %\hspace*{1.5cm} \vspace*{4.0cm} \begin{picture}(15.0,10.0) \epsfig{file=d08-172f4.eps, bburx=530,bbury=580,bbllx=120,bblly=220, clip=,width=12.0cm} \end{picture} \caption{ The inclusive invariant differential cross sections as a function of transverse momentum. The curves show the results of fits to the power law, equation~(\ref{pawer}). Statistical and systematic errors are added in quadrature. } \label{fig:allxsec} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 \end{document}