A Mass measurement in a Penning trap

[kelly0303]

Hello! My question is motivated by this paper (also attached below). They are measuring the mass of a molecular ion in a Penning trap, and they are able to see a difference due to the fact that the molecule gets polarized (the motion is classical and non-relativistic). I was able to derive their result, for an induced electric dipole, using the Lagrangian:

$$
where $$ is the polarization and $$ is the electric field. If we use the fact that $$ we can see from the form of the Lagrangian, if we take the derivative with respect to $$:

$$
From this we get an effective mass of:

$$
which is consistent with their result. However, I was wondering, if we assume that the molecule is highly (or fully) polarized and not just weakly, instead of $$ we have simply $$, where $$ is the intrinsic dipole moment of the molecule. However, assuming $$ is constant, which is (very close to being) true for a fully polarized molecule, we have $$ which gives:

$$
now we can't factor out $$ anymore and thus it's not clear anymore how to count $$ towards the mass of the molecule. However, intuitively, I would expect that the higher the polarization, the higher the shift in the measured mass. What am I doing wrong, or how should I interpret the $$ term in this case? Thank you!