Light-Quark $SU(3)$ Flavour Splitting of Heavy-Light Constituent Diquark Masses and Doubly-Strange Diquarks from QCD Sum-Rules

QCD Laplace sum-rules are used to examine the constituent mass spectrum of $J^P\in\{0^+,1^+\}$ heavy-light [Qq] diquarks with $Q\in\{c,b\}$ and $q\in\{u,d,s\}$. As in previous sum-rule studies, the negative parity $J^P\in\{0^-, 1^-\}$ [Qq] diquark mass predictions do not stabilize, so the sum-rule analysis focuses on positive parity [Qq] diquarks. Doubly-strange $J^P=1^{+}$ [ss] diquarks are also examined, but the resulting sum rules do not stabilize. Hence there is no sum-rule evidence for $J^P=1^{+}$ [ss] diquark states, aiding the interpretation of sum-rule analyses of fully-strange tetraquark states. The SU(3) flavour splitting effects for [Qq] diquarks are obtained by calculating QCD correlation functions of $J^P\in\{0^+,1^+\}$ diquark composite operators up to next-to-leading order in perturbation theory, leading-order in the strange quark mass, and in the chiral limit for non-strange (u,d) quarks with an isospin-symmetric vacuum $<\bar nn>=<\bar uu>=<\bar dd>$. Apart from the strange quark mass parameter $m_s$, the strange quark condensate parameter $\kappa=<\bar ss>/<\bar nn>$ has an important impact on SU(3) flavour splittings. A Laplace sum-rule analysis methodology is developed for the mass difference $M_{[Qs]}-M_{[Qn]}$ between the strange and non-strange heavy-light diquarks to reduce the theoretical uncertainties from all other QCD input parameters. The mass splitting is found to decrease with increasing $\kappa$, providing an upper bound on $\kappa$ where the $M_{[Qs]}-M_{[Qn]}$ mass hierarchy reverses. In the typical QCD sum-rule range $0.56<\kappa< 0.74$, $55~MeV < M_{[cs]}-M_{[cn]} < 100~MeV$ and $75~MeV < M_{[bs]}-M_{[bn]}< 150~MeV$, with a slight tendency for larger splittings for the $J^P=1^+$ channels. These constituent mass splitting results are discussed in comparison with values used in constituent diquark models for tetraquark and pentaquark hadronic states.