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{\large\bf\boldmath Elastic Electroproduction of $\rho$ Mesons at HERA
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{H1 Collaboration}\\                            
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{\small \em DESY, Hamburg, Germany}\\[6mm]
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The elastic electroproduction of $\rho$ mesons is studied 
at HERA with the H1 detector for a photon virtuality in the range 
$1 < Q^2 < 60~{\rm GeV^2}$ and for a hadronic centre of mass energy in the 
range $30 < W < 140$~{\rm GeV}.
The shape of the ($\pi \pi$) mass distribution in the $\rho$ resonance region 
is measured as a function of $Q^2$. 
The full set of $\rho$ spin density matrix elements is determined,
and evidence is found for a helicity flip amplitude at the level of
$8 \pm 3 \%$ of the non-flip amplitudes.
Measurements are presented of the dependence of the cross section on $Q^2$, 
$W$ and $t$ (the four-momentum transfer squared to the proton). 
They suggest that, especially at large $Q^2$, the $\gamma^*p$ 
cross section develops a stronger $W$ dependence than that expected from the
behaviour of elastic and total hadron$-$hadron cross sections.

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{\bf References }\\
1. H1 Collaboration, C.~Adloff {\it et al}, 
DESY-99-010 / submitted to Eur.Phys.J. C
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