Elastic Electroproduction of rho Mesons at HERA

The elastic electroproduction of rho mesons is studied at {HERA} with the H1 detector for a photon virtuality in the range 1 {\textless} Q{\textasciicircum}2 {\textless} 60 {GeV{\textasciicircum}2} and for a hadronic centre of mass energy in the range 30 {\textless} W {\textless} 140 {GeV.} The shape of the pipi mass distribution in the rho resonance region is measured as a function of Q{\textasciicircum}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 +- 3 \% of the non-flip amplitudes. Measurements are presented of the dependence of the cross section on Q{\textasciicircum}2, W and t (the four-momentum transfer squared to the proton). They suggest that, especially at large Q{\textasciicircum}2, the gamma{\textasciicircum}*p cross section develops a stronger W dependence than that expected from the behaviour of elastic and total hadron-hadron cross sections.