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Mass inequalities for baryons with heavy quarks

Baryons with one or more heavy quarks have been shown, in the context of a nonrelativistic description, to exhibit mass inequalities under permutations of their quarks, when spin averages are taken. These inequalities sometimes are invalidated when spin-dependent forces are taken into account. A notable instance is the inequality $2E(Mmm) > E(MMm) + E(mmm)$, where $m = m_u = m_d$, satisfied for $M = m_b$ or $M = m_c$ but not for $M = m_s$, unless care is taken to remove effects of spin-spin interactions. Thus in the quark-level analog of nuclear fusion, the reactions $Λ_b Λ_b \to Ξ_{bb}N$ and $Λ_c Λ_c \to Ξ_{cc}^{++}n$ are exothermic, releasing respectively 138 and 12 MeV, while $ΛΛ\to ΞN$ is endothermic, requiring an input of between 23 and 29 MeV. Here we explore such mass inequalities in the context of an approach, previously shown to predict masses successfully, in which contributions consist of additive constituent-quark masses, spin-spin interactions, and additional binding terms for pairs each member of which is at least as heavy as a strange quark.