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 Q^2 60 GeV^2 and for a hadronic centre of mass energy in the range 30 W 140 GeV. The shape of the pipi 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 +- 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.