  We study the quantum quench in two coupled Tomonaga-Luttinger Liquids (TLLs),
from the off-critical to the critical regime, relying on the conformal field
theory approach and the known solutions for single TLLs. We exploit the
factorization of the initial state in terms of a massive and massless modes
which emerges in the low energy limit, and we encode the non-equilibrium
dynamics in a proper rescaling of the time. In this way, we compute several
correlation functions, which at leading order factorize into multipoint
functions evaluated at different times for the two modes. Depending on the
observable, the contribution from the massive or from the massless mode can be
the dominant one, giving rise to exponential or power-law decay in time,
respectively. Our results find a direct application in all the quench problems
where, in the scaling limit, there are two independent massless fields: these
include the Hubbard model, the Gaudin-Yang gas, and tunnel-coupled tubes in
cold atoms experiments.
