**5.1.8. Proton neutron mass difference by a lattice calculation**

One can take another approach, in principle exact, to calculate the mass
difference using lattice gauge theory.
We introduce the magnetic field in the lattice by multiplying a link
*U*_{µ}(*x*) with a
phase *U*_{µ}^{B}(*x*).
Fixing the field,for convenience, in the direction z we then set:

(5.35) | |

(5.36) |

where *x*_{0} is an offset for the magnetic field.

Consequently the plaquette in the x-y plane is

(5.37) | |

(5.38) |

The magnetic field is homogeneous only if *B* is is quantized as
*a*^{2} *eB* =
2*n*(*L*_{x}*L*_{y})^{-1}
This is a troublesome condition since the field is very large for
reasonable lattice size.
In the simulations we ignored this condition inducing some inhomogeneity.
The results are very preliminary: at this stage it is difficult to find
these effects with present lattice technology, but in principle it is
possible. For details of the calculations see
[224].
This method is still not efficient given the present state of the art in
simulations.