This might be a silly question. When the mass of a spin 1 meson is measured, is there a way to check whether it is the same for all polarizations? Is such a meson mass the same for longitudinal and for transverse polarizations?
This is your second effort to get a response on this question.
As far as I know, there has not been a reason to expect that the mass of an elementary particle is a function of its polarization , We implicitly accept that an elementary particle has one rest mass, as it has one spin and one charge. You have not given a justification for the question.
Here is the abstract of one experimental study by L3
Events from the e+e− -> Z + gamma process with hard initial-state radiation collected with the L3 detector at centre-of-mass energies between 183 GeV and 209 GeV are used to measure the mass of the Z boson. Decays of the Z boson into hadrons or muon pairs are considered and the Z mass is determined to be 91.272 +/-0 0.032 (stat.) +/- 0.033 (syst:) GeV, in agreement with the value measured at the Z resonance. Alternatively,assuming this measured value of the Z mass, the method determines the LEP centre-of-mass energy, found to be 175 +/-68 (stat.)+/-68 (syst) MeV lower than the nominal value.
In the center of mass system this sample has Z polarized with the opposite polarization to the gamma. If the longitudinal mass were different than the averaged mass already studied there would be no agreement with the standard value.
Here is also a preprint with a study of the polarization of the rho meson. The mass is a given.