Sub-percent calculations of an increasing number of physical observables are within the reach of modern Lattice QCD. In order to achieve such precision, we must include the (typically) O(1%) corrections from isospin-breaking contributions in our calculations. These corrections include disconnected diagrams, which can be prohibitively expensive to resolve using standard techniques. In this talk, I will discuss ongoing efforts to calculate several disconnected topologies relevant at O(alpha) to e.g. Kl2 decays, on RBC-UKQCD physical-point domain-wall ensembles, following on from exploratory calculations [1]. We make use of the ‘split-even’ estimator [1, 2], which can improve statistical errors by an order of magnitude or more, and explore the use of a distance-splitting technique [1] to take advantage of the dominant short-distance behaviour of some topologies. These techniques are applicable beyond disconnected diagrams, and might improve statistical errors in many contractions involving noisy loop estimators. I will explore the potential to make great improvements on the statistical error of our previous result for rare K+->Pi+l+l- decays [3] using these techniques.

[1] Harris, T., Gülpers, V., Portelli, A., Richings, J. Efficiently unquenching QCD+QED at O(alpha) PoS LATTICE2022 (2023). https://arxiv.org/abs/1903.10447
[2] Giusti, L., Harris, T., Nada, A. et al. Frequency-splitting estimators of single-propagator traces. Eur. Phys. J. C 79, 586 (2019). https://doi.org/10.1140/epjc/s10052-019-7049-0
[3] Boyle, P. A., Erben, F., Flynn, J. M., Gülpers, V., Hill, R. C., Hodgson, R., Jüttner, A., Ó hÓgáin, F., Portelli, A., Sachrajda, C. T. Simulating rare kaon decays K+->Pi+l+l- using domain wall lattice QCD with physical light quark masses. Phys. Rev. D 107 (2023) L011503. https://link.aps.org/doi/10.1103/PhysRevD.107.L011503