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As our recent quark model calculation~\cite{Noh:2023fdy} suggests a strongpossibility of a compact $T_{cc}$ that closely reproduces experimental mass, wehave a strong incentive to extend our work to investigate the possible compactconfiguration of a pentaquark $udcc\bar{s}$, which is related to the structureof the doubly charmed tetraquark $T_{cc}$. Since the introduction of a completeset of 3-dimensional harmonic oscillator bases to a spatial wave function insolving a quark model-based Hamiltonian with variational method leads to a moreaccurate value of the mass, it seems natural that future studies of thepentaquark should be treated with the same elaborate technical approach. Toattain such precision for the ground state energy, we utilize a complete set of3-dimensional harmonic oscillator base up to 6th quanta. Before carrying outthis process, one important thing that has to be taken into account is to findout the color $\otimes$ spin states of the pentaquark for the evaluation ofcolor and spin interaction most essential to the quark model configuration. Toeasily identify the suitable configuration, we make a systematic analysis of$SU(6)_{CS}$ irreducible representation of the pentaquark, from which we findthat there is a correspondence between the color $\otimes$ spin states obtainedfrom their coupling scheme and the multiplet of the $SU(6)_{CS}$ irreduciblerepresentation of the pentaquark. We find that the energy of the pentaquarkconfiguration is +18.5 MeV above the lowest threshold for decay into $\Xi_{cc}$and $K$, suggesting that this configuration is not stable against its decay.Nonetheless, while we used a Gaussian hyperfine potential, it was recentlyfound that a Yukawa form leads to a stronger attraction for the $T_{cc}$configuration. Therefore it is important to study the same configuration usingthe latter potential.

The doubly-charmed pentaquark in a quark model with a complete set of harmonic oscillator bases