Which assumption of the no-hair theorem does the Einstein-Maxwell-Dilaton-Axion (EMDA) black hole violate? I know that generally a black hole should have no hair. For example, as proved by Bekenstein in this paper, we can not couple as massive scalar/vector etc. to a black hole which is stationary.
However, I am aware that there is a solution called  Einstein-Maxwell-Dilaton-Axion (EMDA) black hole, as introduced in this paper this paper. I wonder what assumption of the "no hair" theorem does this EDMA solution violate so it can end up with a black hole with scalar fields hair.
 A: Specifically the cited Bekenstein's 1972 no scalar hair theorem is evaded because it is for free scalar fields, while in EMDA theory dilaton and axion fields are coupled to Maxwell field and to each other. Moreover, while the EMDA black hole has parameters called dilaton and axion charges, they do not represent a true hair, since variation of these parameters results in variation of asymptotic values of dilaton and axion fields at infinity. Hence these are not true independent degrees of freedom of the black hole itself but more like properties of the world around it. This type of behavior is referred to as “secondary hair” (while electric charge for example, would be primary hair, independent degree of freedom of black hole).
For more details about various no scalar hair theorems and various ways those theorems could be evaded see the paper:

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*Herdeiro, C. A., & Radu, E. (2015). Asymptotically flat black holes with scalar hair: a review. International Journal of Modern Physics D, 24(09), 1542014, doi:10.1142/S0218271815420146, arXiv:1504.08209.

