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We study the quasi-hydrodynamics of a system with a softly broken $U(1)$global symmetry using effective field theory (EFT) and holographic methods. Inthe gravity side, we consider a holographic Proca model in the limit of smallbulk mass, which is responsible for a controllable explicit breaking of the$U(1)$ global symmetry in the boundary field theory. We perform a holographicSchwinger-Keldysh analysis, which allows us to derive the form of the boundaryeffective action in presence of dissipation. We compare our results with thepreviously proposed EFT and hydrodynamic theories, and we confirm theirvalidity by computing the low-energy quasi-normal modes spectrum analyticallyand numerically. Additionally, we derive the broken holographic Ward identityfor the $U(1)$ current, and discuss the recently proposed novel transportcoefficients for systems with explicitly broken symmetries. The setupconsidered is expected to serve as a toy model for more realistic situationswhere quasi-hydrodynamics is at work, such as axial charge relaxation in QCD,spin relaxation in relativistic systems, electric field relaxation inmagneto-hydrodynamics, or momentum relaxation in condensed matter systems.

U(1) quasi-hydrodynamics: Schwinger-Keldysh effective field theory and holography