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I know that light is electromagnetic radiation (sourced by accelerating charge) and gravitational waves are gravitational radiation (sourced by accelerating mass).

Is there equivalent radiation for the strong and weak nuclear forces? If so, what is the source? If not, what properties of these forces (short range? symmetry?) prohibits radiation?

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The $W$ and $Z$ gauge fields that mediate the weak interaction are massive, more than 80 times the mass of a proton. Combined with the fact that they interact with (at least the left-handed components of) all of the matter fields in the Standard Model, this makes them very unstable, with a very short lifetime, so we don't see any macroscopic-range "radiation" composed of the $W$ and $Z$ fields.

The gauge field (gluon field) that mediates the strong interaction is "confined" as a result of its self-interaction, just like quarks are confined. In fact, quantum chromodynamics, the theory of the strong interaction by itself, has a spectrum of only massive particles, with nothing massless like a strong-force version of a photon. It is possible that glueballs could exist, which are massive particles that are nominally made of "only gluons." I'm not sure about the status of experimental searches for glueballs, but if they do exist then they are certainly massive and unstable, so they would not qualify as macroscopic-range "radiation".

Although the question is slightly different, the answers to

Do strong and weak interactions have classical force fields as their limits?

include relevant information.

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