photons (light particles used in the experiment) do not interact significantly with the Higgs field due to their lack of mass at the relevant energy scales
A lecture I was listening to a few days ago while on my treadmill said that photons have no mass because they don't interact with the Higgs field (i.e., that interacting with the Higgs field is what gives mass to things). (So, similar to what you posted but a little reversed: from 'do not interact...with the Higgs field due to their lack of mass' to 'do not have mass due to not interacting with the Higgs field')
Though I sure don't have enough background for any of this to make much sense to me, I'd even been thinking of changing the lecture series because the physics series is leaving me feeling lacking in intelligence, but now I'm glad to have been sticking it out since it gives me something to say on SF posts.