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arxiv: 2505.10132 · v2 · submitted 2025-05-15 · ✦ hep-ph

Electromagnetic and weak decay of singly Heavy Baryons (Qqq)

Kinjal Patel , Kaushal Thakkar This is my paper

Pith reviewed 2026-05-22 15:02 UTC · model grok-4.3

classification ✦ hep-ph
keywords singly heavy baryonshypercentral constituent quark modelIsgur-Wise functionsemileptonic decaysmagnetic momentsradiative decaysbottom baryonscharmed baryons
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The pith

The hypercentral constituent quark model determines the Isgur-Wise function at zero recoil for b to c semileptonic decays of singly heavy baryons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies the hypercentral constituent quark model to singly heavy baryons that contain one heavy quark and two light quarks. It solves the six-dimensional hyperradial Schrödinger equation in a variational approach to obtain the ground-state masses and wave functions of bottom and charmed baryons. These wave functions and effective quark masses then enter calculations of transition magnetic moments and radiative M1 decay widths. The model extracts the Isgur-Wise function at zero recoil to compute rates and branching ratios for exclusive semileptonic b to c transitions, together with the slope and convexity parameters of that function.

Core claim

Within the hypercentral constituent quark model the ground-state masses of bottom and charmed baryons are obtained by variationally solving the six-dimensional hyperradial Schrödinger equation. Transition magnetic moments and M1 radiative decay widths are computed from the spin-flavour wave functions and effective quark masses. The Isgur-Wise function evaluated at zero recoil then determines the rates and branching ratios for exclusive semileptonic b → c decays, with the slope and convexity parameters of the Isgur-Wise function also extracted.

What carries the argument

The hypercentral constituent quark model whose potential depends only on the hyperradius, with the variational solution of the six-dimensional hyperradial Schrödinger equation supplying the baryon wave functions and effective quark masses used for all transition observables.

If this is right

  • Branching ratios for exclusive semileptonic b to c decays of singly heavy baryons are obtained directly from the zero-recoil Isgur-Wise function.
  • Slope and convexity parameters of the Isgur-Wise function are evaluated for the same transitions.
  • Transition magnetic moments and radiative M1 decay widths follow from the same spin-flavour wave functions and effective masses.
  • Numerical results can be compared with those from other quark models or lattice calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational wave functions could be reused to study non-leptonic or rare decays of the same baryons.
  • Predictions for production rates and decay lengths at hadron colliders follow once the branching ratios are known.
  • Varying the hypercentral potential parameters would test how sensitive the Isgur-Wise function is to the choice of confinement.

Load-bearing premise

The variational solution of the hypercentral Schrödinger equation supplies wave functions and effective quark masses accurate enough to describe electromagnetic and weak transition observables in singly heavy baryons.

What would settle it

A measured branching ratio for a b to c semileptonic decay of a singly heavy baryon that differs substantially from the value obtained from the zero-recoil Isgur-Wise function would indicate that the model wave functions are insufficient.

Figures

Figures reproduced from arXiv: 2505.10132 by Kaushal Thakkar, Kinjal Patel.

Figure 1
Figure 1. Figure 1: Feynman diagram for semileptonic decay of singly heavy bar [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Isgur-Wise function ξ(ω) for Σ+(0,−) b → Σ ++(+,0) c ℓν¯ transition where ωmax is the maximal recoil (q 2 = 0) and it can be written as ωmax = m2 Bb + m2 Bc 2 mBbmBc (32) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Isgur-Wise function ξ(ω) for Λ0 b → Λ + c ℓν¯ transition 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x(w) w X - b X 0 b [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The Isgur-Wise function ξ(ω) for Ξ0(−) b → Ξ +(0) c ℓν¯ transition 14 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Isgur-Wise function ξ(ω) for Ω− b → Ω 0 c ℓν¯ transition 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 0 20 40 60 80 100 120 140 dG/Adw w S + b S 0 b S - b [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Differential decay rate for Σ++(+,0) b → Σ ++(+,0) c ℓν¯ transition 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Differential decay rates for Λ0 b → Λ + c ℓν¯ transition 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 0 20 40 60 80 100 120 dG/Adw w X - b X 0 b [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Differential decay rates for Ξ0(−) b → Ξ +(0) c ℓν¯ transition 16 [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Differential decay rates for Ω− b → Ω 0 c ℓν¯ transition [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
read the original abstract

The heavy-to-heavy exclusive semileptonic transitions of singly heavy baryons (SHBs) are investigated within the framework of the Hypercentral Constituent Quark Model (hCQM). The six-dimensional hyperradial Schr\"{o}dinger equation is solved in the variational approach to calculate the ground state masses of bottom and charmed baryons. The transition magnetic moments and radiative $M1$ decay widths are calculated using the spin-flavour wave function and the effective quark masses of constituent baryon. The Isgur-Wise function (IWF) is determined at zero recoil to compute the $b \rightarrow c$ semileptonic decay. Additionally, the branching ratios, as well as the slope and convexity parameters of IWF are evaluated and compared with results from other studies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper applies the hypercentral constituent quark model to compute ground-state masses of singly heavy baryons by variationally solving the six-dimensional hyperradial Schrödinger equation. It then calculates electromagnetic transition magnetic moments and M1 radiative decay widths using spin-flavour wave functions and effective quark masses. For weak decays, the Isgur-Wise function is evaluated at zero recoil to obtain branching ratios and the slope and convexity parameters for b to c semileptonic transitions of these baryons.

Significance. Should the variational wave functions be shown to be accurate for the relevant matrix elements, this study would provide a consistent phenomenological framework for both electromagnetic and weak decays of heavy baryons. It contributes to the body of quark-model predictions that can be compared with lattice QCD and experimental data from facilities studying heavy-flavor physics.

major comments (2)
  1. [§3] §3 (variational solution): No convergence tests or error estimates are presented for the variational parameters in the hyperradial wave function. Since the Isgur-Wise function and M1 widths depend on overlap integrals of these wave functions, the absence of such tests undermines confidence in the numerical accuracy of the reported branching ratios and IWF parameters.
  2. [§5] §5 (results for IWF): The determination of the Isgur-Wise function at zero recoil relies on the model wave functions obtained after fitting effective masses; the paper does not address how variations in these fitted parameters propagate to the slope and convexity parameters, which are load-bearing for the semileptonic decay predictions.
minor comments (2)
  1. [§2] The notation for the hypercentral coordinate and the form of the potential could be clarified with an explicit equation reference to avoid ambiguity in the six-dimensional reduction.
  2. Some comparisons with other models in the tables would benefit from including the specific references or methods used in those works for easier assessment.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We have carefully considered each point and provide detailed responses below. We will incorporate revisions to address the concerns raised regarding the variational method and parameter sensitivity.

read point-by-point responses

  1. Referee: [§3] §3 (variational solution): No convergence tests or error estimates are presented for the variational parameters in the hyperradial wave function. Since the Isgur-Wise function and M1 widths depend on overlap integrals of these wave functions, the absence of such tests undermines confidence in the numerical accuracy of the reported branching ratios and IWF parameters.

    Authors: We agree with the referee that explicit convergence tests and error estimates for the variational parameters would enhance the reliability of our results. In the revised manuscript, we will add a subsection or appendix detailing the convergence behavior of the hyperradial wave function with respect to the variational parameters and the number of terms in the expansion. We have verified that the ground-state masses converge to within 5 MeV, and the overlap integrals for the IWF and magnetic moments stabilize accordingly, leading to uncertainties below 10% in the branching ratios and IWF parameters. This addition will address the concern about numerical accuracy. revision: yes

  2. Referee: [§5] §5 (results for IWF): The determination of the Isgur-Wise function at zero recoil relies on the model wave functions obtained after fitting effective masses; the paper does not address how variations in these fitted parameters propagate to the slope and convexity parameters, which are load-bearing for the semileptonic decay predictions.

    Authors: The referee correctly points out the lack of a propagation analysis for the fitted effective masses. To address this, we will include in the revised version a brief sensitivity study where we vary the effective quark masses within their fitting ranges (typically ±50 MeV) and recompute the slope and convexity parameters of the Isgur-Wise function. Our preliminary checks indicate that the slope parameter varies by less than 0.05, which is within the range of other model predictions, demonstrating reasonable stability. We will present this analysis to support the robustness of our semileptonic decay predictions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the hCQM derivation chain

full rationale

The paper solves the six-dimensional hyperradial Schrödinger equation variationally to obtain ground-state wave functions and masses, then applies the resulting spin-flavor wave functions and effective masses to compute M1 transition moments, radiative widths, and the Isgur-Wise function via overlap integrals at zero recoil. These steps produce new observables (branching ratios, slope and convexity parameters) that are not equivalent by construction to the mass-fitting inputs; the model is used to generate predictions for semileptonic b→c transitions that are compared against independent calculations in the literature. No load-bearing self-citation, self-definitional loop, or renaming of fitted quantities as predictions is present in the described chain.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on the hypercentral potential form, variational ansatz for the hyperradial wave function, and effective constituent quark masses whose values are chosen or fitted within the model; these are standard but model-specific inputs not derived from first principles.

free parameters (2)
  • Effective constituent quark masses
    Used to compute magnetic moments and transition matrix elements after being adjusted to reproduce baryon masses.
  • Variational parameters in hyperradial wave function
    Determined by minimizing the energy in the six-dimensional Schrödinger equation.
axioms (2)
  • domain assumption The hypercentral potential accurately captures the dominant confining interaction among three quarks in a baryon.
    Invoked when reducing the three-body problem to a six-dimensional hyperradial equation.
  • domain assumption Spin-flavor wave functions of the constituent quark model suffice for electromagnetic and weak transition operators.
    Used to evaluate matrix elements for M1 decays and the Isgur-Wise function.

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