The elastic electroproduction of \documentclass[12pt]{minimal}
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\begin{document}$\rho$\end{document} mesons is studied at HERA with the H1 detector for a photon virtuality in the range \documentclass[12pt]{minimal}
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\begin{document}$1 < Q^2 < 60 {\rm GeV^2}$\end{document} and for a hadronic centre of mass energy in the range \documentclass[12pt]{minimal}
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\begin{document}$30 < W < 140$\end{document} GeV. The shape of the (\documentclass[12pt]{minimal}
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\begin{document}$\pi \pi$\end{document}) mass distribution in the \documentclass[12pt]{minimal}
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\begin{document}$\rho$\end{document} resonance region is measured as a function of \documentclass[12pt]{minimal}
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\begin{document}$Q^2$\end{document}. The full set of \documentclass[12pt]{minimal}
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\begin{document}$\rho$\end{document} spin density matrix elements is determined, and evidence is found for a helicity flip amplitude at the level of \documentclass[12pt]{minimal}
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\begin{document}$8 \pm 3\%$\end{document} of the non-flip amplitudes. Measurements are presented of the dependence of the cross section on \documentclass[12pt]{minimal}
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\begin{document}$Q^2$\end{document}, W and t (the four-momentum transfer squared to the proton). They suggest that, especially at large \documentclass[12pt]{minimal}
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\begin{document}$Q^2$\end{document}, the \documentclass[12pt]{minimal}
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\begin{document}$\gamma^*p$\end{document} cross section develops a stronger W dependence than that expected from the behaviour of elastic and total hadron–hadron cross sections.