Light-quark <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> flavor 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 ${J}^{P}\ensuremath{\in}{{0}^{+},{1}^{+}}$ (scalar, axial vector) heavy-light $[Qq]$ diquarks with $Q\ensuremath{\in}{c,b}$ (charm, bottom) and $q\ensuremath{\in}{u,d,s}$ (up, down, strange). As in previous sum-rule studies, the negative parity ${J}^{P}\ensuremath{\in}{{0}^{\ensuremath{-}},{1}^{\ensuremath{-}}}$ (pseudoscalar, vector) $[Qq]$ diquark mass predictions do not stabilize, so the sum-rule analysis focuses on positive parity $[Qq]$ diquarks. Doubly strange ${J}^{P}={1}^{+}$ (axial vector) $[ss]$ 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 $SU(3)$ flavor splitting effects for $[Qq]$ diquarks are obtained by calculating QCD correlation functions of ${J}^{P}\ensuremath{\in}{{0}^{+},{1}^{+}}$ 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 nonstrange ($u$, $d$) quarks with an isospin-symmetric vacuum $⟨\overline{n}n⟩=⟨\overline{u}u⟩=⟨\overline{d}d⟩$. Apart from the strange quark mass parameter ${m}_{s}$, the strange quark condensate parameter $\ensuremath{\kappa}=⟨\overline{s}s⟩/⟨\overline{n}n⟩$ has an important impact on $SU(3)$ flavor splittings. A Laplace sum-rule analysis methodology is developed for the mass difference ${M}_{[Qs]}\ensuremath{-}{M}_{[Qn]}$ between the strange and nonstrange heavy-light diquarks to reduce the theoretical uncertainties from all other QCD input parameters. The mass splitting is found to decrease with increasing $\ensuremath{\kappa}$, providing an upper bound on $\ensuremath{\kappa}$ where the ${M}_{[Qs]}\ensuremath{-}{M}_{[Qn]}$ mass hierarchy reverses. In the typical QCD sum-rule range $0.56<\ensuremath{\kappa}<0.74$, $55\text{ }\text{ }\mathrm{MeV}\ensuremath{\lesssim}{M}_{[cs]}\ensuremath{-}{M}_{[cn]}\ensuremath{\lesssim}100\text{ }\text{ }\mathrm{MeV}$ and $75\text{ }\text{ }\mathrm{MeV}\ensuremath{\lesssim}{M}_{[bs]}\ensuremath{-}{M}_{[bn]}\ensuremath{\lesssim}\phantom{\rule{0ex}{0ex}}150\text{ }\text{ }\mathrm{MeV}$, with a slight tendency for larger splittings for the ${J}^{P}={1}^{+}$ axial-vector channels. These constituent mass splitting results are discussed in comparison with values used in constituent diquark models for tetraquark and pentaquark hadronic states.

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