Widom and collaborators have put together a set of ideas which they propose accounts for cold fusion effects. The number of ideas is pretty large, and all of them would have to be spot on for the theory to be correct.
But one runs into issues in the very first of the ideas.
In the first paper of the series, Widom and Larsen talk about an increase in the electron mass due to local electric fields which are present due to atomic vibrations.
If there were an increase in the electron mass, then a proton could capture aheavy electron leading to a neutron.
So there are two questions: (1) Is there a significant mass increase large enough to do the job? (2) Are the conditions associated with such a mass increase compatible with experiment?
Widom and Larsen claim that very strong electric fields are present which are strong enough to do the job. (Source: New Energy Times)
However, the mass increase formula used was derived (by Volkov) for an electron in a plane wave electromagnetic field. However, there is no plane wave electromagnetic field present under the assumptions of Widom and Larsen.
A new formula should have been derived which comes about from Coulomb coupling with whatever atomic system is under discussion. In this case, it is moving protons at the surface.
The implicit assumption being made is that the time-dependent Coulomb fields produced by the protons can be treated on the same footing as dynamical transverse photons. This is not the case.
A better way to think about it is that if you have some protons that are oscillating at the surface with frequencies that correspond to energies at the 100 meV level (which is in the references cited by Widom and Larsen), then the associated Coulombic energies available are going to be on the order of e**2/r (where r is the distance between the proton and electron) with possible energy exchanges on the order of 100 meV possible. In this case, the relevant energy scale is O(10 eV) for protons rather than O(MeV), which would be required to begin thinking about an MeV-scale mass shift.
There is a relevant literature on electrons in strong electromagnetic fields. People have considered the problem of electrons in strong electromagnetic fields at length over the years. There are a hundred papers or more in the literature.
To get a mass shift in the MeV range, you need a very intense laser beam (made up of real transverse photons). Widom and Larsen have no such intense laser beams, and very few real transverse photons.
If you do get a sufficiently intense beam to give you a mass shift in the MeV range, the electrons have kinetic energy which leads to scattering, ionization, and Bremsstrahlung radiation, all of which are potentially observable.
There is a relevant literature on electrons in more modest Coulomb fields. People have considered the problem of electrons in metals and metal hydrides for lots of years. There are hundreds of papers in the literature.
Typically what people see is simple Coulomb interaction dominating the dynamics on the atomic scale, and molecular orbital type descriptions being very useful.
Models for plasmons and surface plasmons are available in large numbers.
Models for coupled phonon modes and plasmon modes are available.
In band structure calculations, the electrons can be described by effective masses which can be larger or smaller than an electron mass in a band. However, the effective mass is due to band curvature, not to an increased dressed mass as proposed by Widom and Larsen.
No solid state physicist would claim electron mass increases at the MeV level due to any condensed matter physics physics effect.
One could go into more detail on these topics and others, and if needed.
However, the bottom line is that Widom and Larsen are wildly disconnected from reality in the very first physical statements which they make. And until these problems are addressed properly, there is no motivation to go further.
So the answer to the first issue: There can be a mass change for electrons under the conditions considered by Widom and Larsen. However, there are not MeV-level mass shifts. Widom and Larsen are simply wrong.
The answer to the second issue: If there were MeV mass shifts, there would be associated energetic phenomena (such as ionization, scattering, radiation, and so forth) due to electron kinetic energy which would be obvious.
The headache is that the longer these ideas get promoted, the more that reasonable scientists will have reason to doubt the competence of those working on cold fusion.