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Yeah, you prove the field exists by exciting it and making it produce quanta, kind of like photons being the quanta of the EM field being excited.
Photons are excitations in the EM field, but they also carry the electromagnetic force between particles - thus giving them charge. But in order to do that photon actually needs to be created and travel from one particle to another. If Higgs works in a similar way also being a boson, one might expect it also to need to exist to do it's job. . What is the difference here?
The electromagnetic force is mediated by virtual photons. These don’t exist as free particles, such as a photon emitted by a light source, but only as an intermediate particle. Because they’re only intermediate states, virtual photons can have non-physical energies (so long as they’re within the uncertainty principle), resulting in some having an effective mass. Suffice it to say virtual photons are quite distinct from real ones! Technically, I believe you could have some of the basic features of the em force (namely attraction/repulsion by 2 point charges) with just virtual photons. Things get tricky once charges begin accelerating though, as this leads to the emission of real photons.
The short answer is, it doesn’t. The Higgs Field gives mass to fundamental particles. Existing in that field causes certain particles to have mass due to their coupling to the field. The W and Z weak gauge bosons gain mass through electroweak symmetry breaking, quarks and leptons gain mass through a different coupling. I realize this is a very unsatisfying answer as to “how” the Higgs field creates mass, but the mechanism involves some complex math (group theory and non-abelian gauge theory), so it kind of defies a simpler explanation. Regardless, it’s through interactions with the Higgs field (which can exist without any Higgs bosons around) that fundamental particles gain mass. The search for the Higgs boson was just to confirm the existence of the field, because while the field can exist without Higgs bosons present it must be possible to excite it sufficiently to create them.
Going back to your original question: these particles have almost certainly been created “naturally” in high energy collisions between particles and matter. Nature can achieve much higher energies than our particle accelerators. The highest energy particle ever observed was a cosmic ray. However, Higgs bosons are extremely short lived, with a lifetime of 10^-22 seconds. So whenever they’re created, they don’t stick around for a meaningful amount of time.
Higgs boson has mass and quite large one at that and this puts limitations on how hard is it to generate it and on how field behaves
When you don't provide enough energy to get whole Higgs boson, interactions happen via virtual particles. It's easier to grasp this idea with weak interactions and W and Z bosons
Not really, no. At some point I'm going to exceed my own expertise here since I'm not a QFT expert, but in quantum mechanics things don't firmly exist or not exist. The photons in question are "virtual particles".