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Abstract We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive$$\rho ^0$$ meson muoproduction at COMPASS using 160 GeV/cpolarised$$ \mu ^{+}$$ and$$ \mu ^{-}$$ beams impinging on a liquid hydrogen target. The measurement covers the kinematic range 5.0 GeV/$$c^2$$ $$< W<$$ 17.0 GeV/$$c^2$$ , 1.0 (GeV/c)$$^2$$ $$< Q^2<$$ 10.0 (GeV/c)$$^2$$ and 0.01 (GeV/c)$$^2$$ $$< p_{\textrm{T}}^2<$$ 0.5 (GeV/c)$$^2$$ . Here,Wdenotes the mass of the final hadronic system,$$Q^2$$ the virtuality of the exchanged photon, and$$p_{\textrm{T}}$$ the transverse momentum of the$$\rho ^0$$ meson with respect to the virtual-photon direction. The measured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitudinally polarised vector mesons ($$\gamma ^*_T \rightarrow V^{ }_L$$ ) indicate a violation ofs-channel helicity conservation. Additionally, we observe a dominant contribution of natural-parity-exchange transitions and a very small contribution of unnatural-parity-exchange transitions, which is compatible with zero within experimental uncertainties. The results provide important input for modelling Generalised Parton Distributions (GPDs). In particular, they may allow one to evaluate in a model-dependent way the role of parton helicity-flip GPDs in exclusive$$\rho ^0$$ production.more » « less