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Suppose the axion-like particle (ALP) $a$ with the lagrangian $$ L = \frac{1}{2}(\partial_{\mu} a)^2-\frac{m^2}{2} a^2 - g_{a\gamma\gamma}aF\tilde{F} $$ Here $F$ is EM field strength, and $\tilde{F}$ is the dual strength. Unlike the QCD axion case, the mass $m$ of ALP isn't related to the coupling $g_{a\gamma\gamma}$.

This coupling causes the conversion of photon pair into axion in stars. ALP blows out the energy from the star faster than photons, since it directly interacts wery weakly (or doesn't interact) with the star media.

My question is: this cooling rate can't be too large in order not to contradict the known standard star dynamics. This definitely makes the restriction on the coupling. But does this cooling rate make the bound on the ALP mass?

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Nice question. It prompted me to search a little and I found e.g. this recent paper. Indeed, the authors put limits simultaneously on ALP mass and coupling. (See page 18 and Fig.7.)

So, strictly speaking, the cooling doesn't restrict the coupling either; both parameters must be considered for a given cooling rate.

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