**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
[37].
For small fields this reduces to a power series, up to logarithmic terms,
in *B* / *B*_{c}, where *B*_{c} is some
scale. For the electron *B*_{c} =
*m*_{e}^{2} / *e*.
For the hadronic case the value of *B*_{0} is uncertain.
*B*_{c} = *M*_{p}^{2} /
*e* = 1.7 × 10^{16} T is probably too large and
*M*_{p}
should be replaced by a quark constituent mass and *e* be
*e*_{q}; in
that case *B*_{c} = (2-4) × 10^{15} T,
depending on the quark type. This is also the range of values of
^{2} /
*e*_{q} in
Eq. (5.24). The effects we have studied need fields around a few
× 10^{14} T or an order of magnitude smaller than the lowest
candidate for *B*_{c}. Eq. (5.27) bay be viewed as a power
series expansion up to terms of order (*B* /
*B*_{c})^{2}; as the coefficient of the
quadratic term was obtained from a fit to a numerical solution,
logarithms of *B* / *B*_{c} may be hidden in the
coefficient. As in
Ref. [37],
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* /
*B*_{c})^{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
(*r*_{b} / *r*_{c})^{4} compared to
the leading effect; *r*_{b} is a hadronic radius and
*r*_{c} is the quarks
cyclotron radius. This again contributes to the
(*B*/*B*_{c})^{3} term in
the expansion for the energy of a baryon.

Another limitation is due to the results of Ref.
[217]
where it is shown that fields of the order of a few ×
10^{14} 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.
[217]
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.
[217]
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 × 10^{14} T.

Experimental Consequences

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.