If a weak eigenstate (such as a $d'$ or a neutrino) interacts with a gravitational field do the mass eigenstates interact on their own? Weak interaction eigenstates don't correspond with mass eigenstates. I was wondering how such a superposition of mass eigenstates would interact in a gravitational field. I assumed they would do so individually however after reading as much as i could understand of this (https://arxiv.org/abs/1909.06048) paper I get the impression they set and upper limit on the mass of an electron neutrino, rather than the typical measurement of the sum of the mass eigenstates squared. In the introduction they suggest knowing this mass would be useful for modelling the early universe.
So my question is would the weak eigenstate interact with the field, with gravitational attraction proportional to the weak eigenstates "expectation mass"?
 A: Let’s answer this question by comparing with the flavor evolution of neutrinos in flight.
Neutrinos produced by a particular source, such as the Sun’s core, a nuclear reactor, or a muon beam, are created in a flavor eigenstate with a mixture of different masses.  However, as the definite-flavor neutrinos travel away from their source through the vacuum, the phase of the mass mixture evolves in a predictable way.  Likewise, the “flavor phase” of a definite-mass neutrino evolves in a predictable way as the neutrino travels.
Suppose you could produce a pulse of same-flavor neutrinos, with exactly the same energy, at one particular instant.  If you detected that pulse from very far away, you would expect it to have segregated into three bunches, with the lower-mass neutrinos arriving slightly earlier than the higher-mass neutrinos. Your detector would force these mass-segregated neutrinos to “choose” a new flavor eigenstate, so you would predict each bunch to have a different mixture of electron-, muon-, and tau-flavored neutrinos.
This isn’t the observable in current neutrino-oscillation experiments. The Sun’s core isn’t a pulsed source (thankfully), so the information we get there is that the incoherent, time-continuous, energy-continuous beam of pure electron neutrinos from the Sun is only one-third electron neutrinos by the time it reaches Earth.  Pulsed muon beams don’t have the time or energy resolution to see this effect.  It’s possible we might observe this phenomenon if a sufficiently well-behaved supernova were to occur while a neutrino detector were operating.  For a definition of “well-behaved” you’d have to dig through the literature: not too close, not too far, constraints on collapse time, possibly constraints on the stellar composition or other astronomical nitty-gritty.  There was a neutrino pulse associated with the naked-eye 1987 supernova; I believe that the total number of neutrinos detected from that pulse was eleven, which isn’t enough to do statistics.
Gravitational interactions will “force” neutrinos in a beam to “choose” a mass in the same way as a clean measurement of the flight time. But Earth’s gravity is too feeble to have this effect.  Long-baseline neutrino-beam experiments on Earth send the beam through Earth’s crust and/or mantle, because the beam travels in a straight line and Earth’s surface is round. Charged-current forward scattering of the beam forces some fraction of the neutrinos to “choose” a flavor again, and appears in analysis of those experiments as a “matter correction” to the oscillation parameters. The gravitational correction to the oscillation is much, much smaller.
