  We study general properties of the mapping between 5$d$ and 4$d$
superconformal field theories (SCFTs) under both twisted circle
compactification and tuning of local relevant deformation and CB moduli. After
elucidating in generality when a 5$d$ SCFT reduces to a 4$d$ one, we identify
nearly all $\mathcal{N}=1$ 5$d$ SCFT parents of rank-2 4$d$ $\mathcal{N}=2$
SCFTs. We then use this result to map out the mass deformation trajectories
among the rank-2 theories in 4$d$. This can be done by first understanding the
mass deformations of the 5$d$ $\mathcal{N}=1$ SCFTs and then map them to 4$d$.
The former task can be easily achieved by exploiting the fact that the 5$d$
parent theories can be obtained as the strong coupling limit of Lagrangian
theories, and the latter by understanding the behavior under compactification.
Finally we identify a set of general criteria that 4$d$ moduli spaces of vacua
have to satisfy when the corresponding SCFTs are related by mass deformations
and check that all our RG-flows satisfy them. Many of the mass deformations we
find are not visible from the corresponding complex integrable systems.

High Energy Physics - Theory