Do gauge bosons really exist or are they only a mathematical model? Have we ever detected them?
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Photons, which are gauge bosons, are absorbed by the retina and cause impulses in the optic nerve. In a very dark room, the eye can detect small numbers of photons. Researchers argue about how few, but you don't have to have a classical electromagnetic wave to excite retinal cells.
Direct perception of photons through one of the five senses tells me that they exist, although questions of "existence" are more about philosophy than physics. (Does Fock space "exist"? If so, where?) I choose to believe that many things "exist" that I cannot perceive with my senses. People once did not even believe in atoms; but I do, even though I haven't seen one with my eyes.
We can't perceive W and Z bosons, or gluons, through our senses the way can perceive photons. But the Standard Model makes accurate predictions so it makes sense to assume that they exist as much as anything else exists.