This question is based on this blog post 'Is Dark Matter Lurking in Neutron Decays?' and a following comment I re-read recently. It has something I have seen often in popular writing, which I believe is confusing the on-shell and off-shell descriptions.
For example, the post says
The objection would be correct if the emitted W boson were a real final-state particle in the reaction. But the W is “virtual” here: it exists for a time so short that the energy budget does not get affected by its sudden appearance and disappearance. It is as if somebody stole a trillion dollars from the bank, and then put the sum back in place before any accountant could realize that the balance does not make sense.
The W boson likes to be on-mass-shell: you can rather easily materialize a W boson if you invest 80.4 GeV or more of energy in your reaction, but if you ask the W to content itself with a smaller mass you will make a deal much less often. The poor down quark, having a single MeV or so of energy to invest, has to wait an eternity to pull it off: 15 minutes are a mindbogglingly long time for a quark!
My question is, are the two views contradictory? In the first, it is argued that the energy of 80.4 GeV is “borrowed” for a short time and then put back without noticing (on-mass-shell view where relativity is temporarily violated, i.e. the equation E^2 = p^2 + m^4). In the second, there is the different view (off-mass-shell view) where relativity and energy conservation are obeyed but the particle does not have the “real” mass of 80.4 GeV.
In my understanding these are alternative viewpoints but cannot both be true (while the off-mass shell view is the one which is covariant so may be preferred). Is this correct?