Spectra and nonleptonic decays of multiply heavy baryons in a finite-size quark-diquark model

Technical Summary: Spectra and Nonleptonic Decays of Multiply Heavy Baryons in a Finite-Size Quark–Diquark Model

Problem Statement
Heavy-flavor baryons containing two or three heavy quarks ($QQq$ and $QQQ$) offer a unique laboratory for studying weak transitions alongside nonperturbative Quantum Chromodynamics (QCD). While the observation of the doubly charmed baryon $\Xi_{cc}^{++}$ and the recent measurement of $\Omega_{cc}^{+}$ by LHCb have spurred interest, the spectra and nonleptonic decay properties of these systems remain less established than those of light or singly heavy baryons. A critical theoretical challenge lies in the consistent treatment of spectroscopy and weak decays. Many existing quark-diquark models treat the diquark as a pointlike object, neglecting its finite spatial extent. This approximation fails to account for how the diquark's internal structure modifies the effective quark-diquark interaction and the wave functions that determine recoil overlaps in decay amplitudes. Furthermore, quantitative predictions for nonleptonic decays, particularly $b \to c$ transitions, are less developed than those for $c \to s$ modes.

Methodology
The authors employ a nonrelativistic constituent quark-diquark model to investigate ground-state doubly and triply heavy baryons. The core methodological advancement is the generalization of the finite-size quark-diquark interaction to accommodate unequal-mass diquarks.

1. Hamiltonian and Potential: The system is described by a Hamiltonian including kinetic energy and pairwise interactions. The interaction potential is based on an improved one-gluon exchange (OGE) model (BCN-type), featuring a Cornell-like central term and a Yukawa-smeared spin-spin term.
2. Finite-Size Convolution: To account for the diquark's spatial extent, the authors utilize a convolution of the pairwise quark-quark potential with the diquark's constituent density. For unequal-mass diquarks ($m_1 \neq m_2$), the formalism is extended by using mass-weighted constituent coordinates ($\mu_1 r$ and $\mu_2 r$) and retaining distinct pairwise spin-spin couplings ($V_{13}$ and $V_{23}$) rather than doubling a single interaction. This results in a generalized convolution potential that preserves both the finite size and mass asymmetry of the diquark.
3. Spectroscopy: The radial Schrödinger equations are solved using the Numerov method to obtain baryon masses and radial wave functions.
4. Decay Formalism: Two-body nonleptonic weak decays ($B_i \to B_f M$) are calculated within the factorization approximation. The effective weak Hamiltonian includes Wilson coefficients ($C_1, C_2$) and CKM matrix elements. The decay amplitude is factorized into a meson decay constant, a baryon transition current, and a spatial overlap integral.
   • The baryon current is reduced to nonrelativistic form, separating spin-flavor coefficients ($C_V, C_A$) from the spatial overlap integral $I$.
   • The spatial overlap integral $I$ incorporates the daughter-baryon recoil via a spherical Bessel function $j_0(\kappa r)$, where $\kappa$ depends on the meson momentum and the spectator diquark mass.

Key Contributions

• Generalized Interaction: The primary theoretical contribution is the extension of the finite-size convolution potential to unequal-mass diquarks, allowing for a consistent description of systems like $QQs$ where the diquark constituents have different masses.
• Unified Framework: The study uses the same set of wave functions derived from the finite-size potential for both mass spectrum calculations and decay width evaluations, ensuring internal consistency between spectroscopy and dynamics.
• Comprehensive Decay Analysis: The paper provides a systematic evaluation of $c \to s$ and $b \to c$ transitions for a wide range of ground-state baryons, including pion and kaon emission modes.

Results

• Mass Spectra: The predicted masses for doubly and triply heavy baryons agree with existing lattice QCD results within 1.5%. Specifically, the calculated mass of $\Omega_{cc}^{+}$ ($3.746$ GeV) is consistent with recent LHCb observations ($3725.9$ MeV) with a discrepancy of only 0.5%.
• Hyperfine Splittings: The calculated hyperfine splittings ($\Delta M = M_{3/2^+} - M_{1/2^+}$) are systematically smaller than lattice QCD values (e.g., 5 MeV vs. 35 MeV for $bbc$). This suggests the effective spin-spin interaction in the current parametrization is slightly too weak or that finite-range regularization effects are significant.
• Spatial Overlaps and Recoil:
  • $b \to c$ vs. $c \to s$: The spatial overlap integrals for $b \to c$ transitions ($0.1512 - 0.2340$) are substantially smaller than for $c \to s$ transitions ($0.5672 - 0.6525$). This suppression is attributed to the larger energy release in $b \to c$ decays, which induces stronger recoil and more rapid oscillations in the overlap integral, leading to cancellations.
  • Pion vs. Kaon: For a fixed baryonic transition, the overlap integral increases when a pion is replaced by a kaon due to the reduced momentum $k$ and weaker oscillation of the Bessel function.
• Decay Widths:
  • CKM Suppression: Combined with the CKM hierarchy ($|V_{cb}|^2 \ll |V_{cs}|^2$) and the reduced spatial overlaps, $b \to c$ widths are suppressed by approximately two orders of magnitude ($10^2 - 4.7 \times 10^2$) relative to analogous $c \to s$ widths.
  • Spin Dependence: For $3/2^+ \to 1/2^+$ transitions, the vector contribution vanishes, leaving only the axial contribution. Consequently, $3/2^+ \to 3/2^+$ transitions (which include both vector and axial terms) exhibit widths roughly 3.4 to 4.0 times larger than their spin-partner $1/2^+$ counterparts.
  • Pion-to-Kaon Ratios: The ratios of pion-to-kaon emission widths generally align with the reference value determined by CKM elements and decay constants ($R_0 \approx 13.01$), with deviations explained by phase-space differences and the specific momentum dependence of the vector and axial terms.

Significance and Claims
The paper claims to provide benchmark estimates for future studies of multiply heavy baryons rather than precision predictions. The significance of the work lies in:

1. Quantifying Recoil Dynamics: It explicitly quantifies the role of recoil dynamics within a finite-size quark-diquark framework, demonstrating how the finite extent of the diquark and the mass asymmetry influence decay amplitudes.
2. Consistency: It establishes a consistent theoretical framework where spectroscopy and weak decays are treated with the same wave functions, addressing a gap in previous models that often treated diquarks as pointlike.
3. Experimental Guidance: The predicted masses and partial decay widths serve as reference points for future experimental searches, particularly for triply heavy baryons and the $\Omega_{cc}^{+}$ system.
4. Model Limitations: The authors modestly acknowledge that the absolute widths are subject to uncertainties from the nonrelativistic reduction, the spectator convention, the factorization approximation, and the lack of a quantitative uncertainty analysis. They suggest that future systematic studies should vary constituent masses and potential parameters and assess relativistic corrections and nonfactorizable contributions.