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Flavor $SU(3)$ properties of beauty tetraquark states with three different light quarks

Beauty tetraquark states $X(\bar b q'q'' \bar q )$ composed of $ \bar b s u \bar d$, $\bar b d s \bar u$, and $\bar b u d \bar s$, are unique that all the four valence quarks are different. Although the claim of existence of the first two states by D0 was not confirmed by data from LHCb, the possibility of such states still generated a lot of interests and should be pursued further. Non-observation of $X(\bar b q'q'' \bar q )$ states by LHCb may be just due to a still lower production rate than the limit of LHCb or at some different mass ranges. In this work we use light quark $SU(3)$ flavor symmetry as guideline to classify symmetry properties of beauty tetraquark states. The multiplets which contain states with three different light quarks must be one of ${\bf \bar 6}$ or ${\bf 15}$ of $SU(3)$ representations. We study possible decays of such a tetraquark state into a $B$ meson and a light pesudoscalar octet meson by constructing a leading order chiral Lagrangian, and also provide search strategies to determine whether a given tetraquark state of this type belongs to ${\bf \bar 6}$ or ${\bf 15}$. If $X(\bar b q'q''\bar q )$ belongs to ${\bf 15}$, there are new doubly charged tetraquark states $\bar b u u \bar d$ and $\bar b u u \bar s$.