5.1.7. Caveats and Limitations
For general magnetic fields we expect the masses of particles to be nonlinear functions of these fields. Such an expression has been obtained, to order , for the electron . For small fields this reduces to a power series, up to logarithmic terms, in B / Bc, where Bc is some scale. For the electron Bc = me2 / e. For the hadronic case the value of B0 is uncertain. Bc = Mp2 / e = 1.7 × 1016 T is probably too large and Mp should be replaced by a quark constituent mass and e be eq; in that case Bc = (2-4) × 1015 T, depending on the quark type. This is also the range of values of 2 / eq in Eq. (5.24). The effects we have studied need fields around a few × 1014 T or an order of magnitude smaller than the lowest candidate for Bc. Eq. (5.27) bay be viewed as a power series expansion up to terms of order (B / Bc)2; as the coefficient of the quadratic term was obtained from a fit to a numerical solution, logarithms of B / Bc may be hidden in the coefficient. As in Ref. , even powers will be spin independent and the odd ones will be linear in the spin direction and may be viewed as field dependent corrections to the magnetic moment. We cannot prove, but only hope, that the coefficient of the (B / Bc)3 term, the first correction to the magnetic moment is not unusually large; should it turn out to be big and of opposite sign to the linear and quadratic terms, the conclusions of this analysis would be invalidated. These arguments, probably, apply best to the field dependence of the magnetic moments of the quarks rather than the total moment of the baryons. We may ask what is the effect on these magnetic moments due to changes in the "orbital" part of the quark wave functions. To first order we expect no effect as all the quarks are in S states and there is no orbital contribution to the total moment. The next order perturbation correction will be down by (rb / rc)4 compared to the leading effect; rb is a hadronic radius and rc is the quarks cyclotron radius. This again contributes to the (B/Bc)3 term in the expansion for the energy of a baryon.
Another limitation is due to the results of Ref.  where it is shown that fields of the order of a few × 1014 T are screened by a changes in chiral condensates. In fact, as the chiral condensate will, in large fields, point in the charged direction, the baryonic states will not have a definite charge. Whether the proton-neutron reversal takes place for fields below those that are screened by chiral condensates or vice versa is a subtle question; the approximations used in this paper and in Ref.  are not reliable to give an unambiguous answer. The treatment of the effects of magnetic fields on the strong force contributions to the baryon masses relies on the Nambu-Jona-Lasinio model and in Ref.  the variation of f with magnetic field was not taken into account. It is clear from this discussion and the one from the previous paragraph that we cannot push the results of this calculation past few × 1014 T.
The mass evolution of protons, neutrons and electrons in magnetic fields and due to electromagnetism alone, will force a proton to decay in a very intense field. Including the effects of chiral condensates diminishes the field even further. Qualitatively, it is clear that the effect enhances the electromagnetic contribution but its exact value depends on the model. This points to a novel astrophysical mechanism for creation of extra galactic positrons.