Electroweak sphaleron in a magnetic field

Using lattice simulations we calculate the rate of baryon number violating processes, the sphaleron rate, in the Standard Model with an external (hyper)magnetic field for temperatures across the electroweak crossover, focusing on the broken phase. Additionally, we compute the Higgs expectation value and the pseudocritical temperature. The electroweak crossover shifts to lower temperatures with an increasing external magnetic field, bringing the onset of the suppression of the baryon number violation with it. When the hypermagnetic field reaches the magitude BY approximate to 2T2 the crossover temperature is reduced from 160 to 145 GeV. In the broken phase for small magnetic fields the rate behaves quadratically as a function of the magnetic flux. For stronger magnetic fields the rate reaches a linear regime which lasts until the field gets strong enough to restore the electroweak symmetry where the symmetric phase rate is reached.