We present preliminary results for B K from the MILC ultrafine lattices, based on a partial ensemble of 305 configurations. We use HYP-smeared improved staggered valence quarks. The analysis is done using fitting forms based on both SU(2) and SU(3) staggered chiral perturbation thery. For the SU(2) analysis, we find that the result using the NLO fit function is consistent with that from a partial NNLO fit. For the SU(3) analysis, where we have to use partially constrained fits due to the number of fit parameters, we find that our two preferred fits ("N-BB1" and "N-BB2") are also consistent, both with each other and with the results of the SU(2) fits. These results are used in companion proceedings to improve the control over the continuum extrapolation.

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The energy shift of a nucleon in a static periodic magnetic field is evaluated at second order in the external field strength in perturbation theory. It is shown that the measurement of this energy shift on the lattice allows one to determine the unknown subtraction function in the forward doubly virtual Compton scattering amplitude. The limits of applicability of the obtained formula for the energy shift are discussed.

The paper presents an extension and a refinement of our previous work on the extraction of the doubly virtual forward Compton scattering amplitude on the lattice by using the background field technique [1]. The zero frequency limit for the periodic background field is discussed, in which the well-known result is reproduced.Further, an upper limit for the magnitude of the external field is established for which the perturbative treatment is still possible. Finally, the framework is set for the evaluation of the finite-volume corrections allowing for the analysis of upcoming lattice results.