Vector-boson fusion Higgs pair production at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">N</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msup><mml:mi>LO</mml:mi></mml:mrow></mml:math>

We calculate the next-to-next-to-next-to-leading order (${\mathrm{N}}^{3}\mathrm{LO}$) QCD corrections to vector-boson fusion (VBF) Higgs pair production in the limit in which there is no partonic exchange between the two protons. We show that the inclusive cross section receives negligible corrections at this order, while the scale variation uncertainties are reduced by a factor four. We present differential distributions for the transverse momentum and rapidity of the final state Higgs bosons, and show that there is almost no kinematic dependence to the third order corrections. Finally we study the impact of deviations from the Standard Model in the trilinear Higgs coupling, and show that the structure of the higher order corrections does not depend on the self-coupling. These results are implemented in the latest release of the proVBFH-incl program.

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