Triply Heavy Ω Baryons with JETHAD: A High-Energy Viewpoint

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents a comprehensive study on the construction of fragmentation functions (FFs) for triply heavy baryons and their phenomenological applications at future colliders. The work is logically structured, methodologically sophisticated, and represents a significant contribution to the field. The results are intriguing and certainly warrant publication. I have a couple of minor points that I would like to clarify with the authors, which could further strengthen the manuscript.

(1) The parameter \langle \bar{q}_T^2 \rangle is clearly pivotal in shaping the initial FFs. The authors have chosen values of 60 GeV^2, 70 Ge^2, and 90 GeV^2 for the 3Q, 4Q, and 5Q systems, respectively, indicating a correlation with the system's complexity. Could the authors elaborate on whether a more fundamental pattern (linear or other functional dependence) governs this choice? A brief discussion on the physical reasoning behind this scaling would be highly informative for the reader.

(2)The mass of the \Omega_{bbb} is taken as 3 \times m_b, which provides a reasonable and computationally convenient benchmark. However, as the authors are well aware, this approach does not account for the binding energy, which is non-negligible. Given the large mass of the b quark, its kinetic energy is small (resulting in weak repulsive forces), such that the binding energy is generally non-negligible in quark models [J. Phys. G 39, 105001 (2012].  Admittedly, this binding energy is still small relative to the total mass of \Omega_{bbb}​ overall. I am uncertain about the extent to which this affects the authors' results. Here, I provide the theoretical mass prediction of \Omega_{bbb}​ from [Phys. Rev. D 112, no.7, 074007 (2025)] for the authors' reference.

Author Response

We are grateful to the Referee for the positive assessment of our work and for the insightful comments, which helped us improve the clarity and completeness of the manuscript. We address the two points raised below.

For the sake of readability, all new or modified portions of the text in response to the Referee’s comments are highlighted in red color in the revised version of the manuscript.

Comment (1):

The parameter \langle \bar{q}_T^2 \rangle is clearly pivotal in shaping the initial FFs. The authors have chosen values of 60 GeV^2, 70 Ge^2, and 90 GeV^2 for the 3Q, 4Q, and 5Q systems, respectively, indicating a correlation with the system's complexity. Could the authors elaborate on whether a more fundamental pattern (linear or other functional dependence) governs this choice? A brief discussion on the physical reasoning behind this scaling would be highly informative for the reader.

Reply:

We thank the Referee for raising this important point and for recognizing the central role of the parameter $\langle \bar{q}_T^2 \rangle$ in shaping the initial fragmentation functions. Our choice of the values 60 GeV², 70 GeV², and 90 GeV² for the 3Q, 4Q, and 5Q systems should not be interpreted as merely heuristic. Rather, these numbers reflect a physically motivated trend that emerges when combining phenomenological constraints with basic features of heavy-flavor fragmentation.

First, the selected values ensure that the resulting FFs peak at sufficiently large $z$ (typically $\langle z \rangle > 0.4$), in agreement with well-established behavior observed in single-heavy fragmentation and in previous determinations for tetraquarks and pentaquarks. At the same time, they preserve a reasonable balance between the heavy-quark and gluon channels, preventing an unphysical dominance of one over the other. This requirement acts as a phenomenological anchor for fixing $\langle \bar{q}_T^2 \rangle$ across systems with different numbers of heavy constituents.

Second, a deeper physical rationale underlies the observed scaling with the system’s complexity. As established in early studies of heavy-flavor fragmentation, heavy-quark FFs peak at large $z$ because binding effects scale with the heavy-quark mass. Extending this argument to multiquark systems, an increase in the number of heavy constituents tends to shift the fragmentation peak further, reflecting a more rigid momentum distribution within the hadronic bound state. This behavior aligns with equal-velocity arguments, where the relation $\langle z \rangle \approx 1 - \Lambda_q/m_Q$ (derived for heavy–light mesons) provides intuitive guidance on how binding effects become progressively less influential as the heavy content grows. Although such expressions cannot be applied literally to fully heavy systems, the qualitative trend remains consistent.

To better emphasize this conceptual reasoning, we have added two dedicated paragraph in Section 2.3 of the revised manuscript. We also acknowledge the importance of quantifying the systematic uncertainty associated with variations of $\langle \bar{q}_T^2 \rangle$. A dedicated study is beyond the scope of the present work, but we are planning a future update of our fragmentation-function sets where this nonperturbative source will be explicitly assessed and included.

Comment (2):

The mass of the $\Omega_{bbb}$ is taken as $3 \times m_b$, which provides a reasonable and computationally convenient benchmark. However, as the authors are well aware, this approach does not account for the binding energy, which is non-negligible. [...] I am uncertain about the extent to which this affects the authors’ results. Here, I provide the theoretical mass prediction of $\Omega_{bbb}$ from [Phys. Rev. D 112, no.7, 074007 (2025)] for the authors’ reference.

Reply:

We thank the Referee for this pertinent observation and for providing a useful reference on the mass prediction of the $\Omega_{3b}$ baryon.

Following the suggestion, we have added a brief discussion on the role of the binding energy and its implications for the $\Omega_{3b}$ mass. This new paragraph appears immediately after the sentence where the baryon mass is fixed to $3m_b$. We now cite both the quark model study of Ref. [J. Phys. G 39, 105001 (2012)] and the recent theoretical prediction of Ref. [Phys. Rev. D 112, no.7, 074007 (2025)], kindly indicated by the Referee.

As stated in the added text, we acknowledge that the binding energy is non-negligible for triply heavy baryons, but we emphasize that the associated uncertainty has a subleading impact on our phenomenological observables. For the sake of consistency and simplicity, and since the dominant theoretical uncertainties originate from the modeling of the initial FFs and their evolution, we retain the $3m_b$ approximation in our computations.

Further remarks:

We added four new references [90,386,431,432], to better support the related discussions.

We hope that the improved version of the manuscript meets the expectations of the Referee.

Reviewer 2 Report

Comments and Suggestions for Authors

Please, read the attached paper.

Comments for author File: Comments.pdf

Author Response

We thank the Referee for her/his careful reading and for the positive and encouraging evaluation of our manuscript.
We are grateful for the constructive remarks and suggestions, which helped us to improve the quality and clarity of the paper.

We have carefully addressed all the indicated points and list our detailed responses below:

1. Paragraph 2.1: Done.

2. Paragraph 2.3: Done. We replaced “$c$” with “$Q$” and added a clarifying sentence (in red) right below Eq.~(11), explicitly stating the meaning of $C_F$.

3. Paragraph 3.1: Done. We added spaces before all references as requested.

4. Paragraph 3.2: Done. 

- Subscripts have been fixed: $p_{2}^B \to P_2^B$, $p_b \to P_b$.

– $P_{5b}$ was replaced with $\Omega_{3b}$.

– Concerning Eq. (35), the logarithm of the product between the two transverse momenta (and not the ratio) is indeed correct. Its dimensional structure matches that of the logarithmic derivative involving the emission functions in the same equation (note that the second function appears as a complex conjugate).

5. Paragraph 4.3: Done. We corrected both the LaTeX typo (“andeqrefLOJEF”) and the repetition in the sentence on rapidity cuts.

6. Further remarks: We added four new references [90,386,431,432], to better support the related discussions.

We hope that the improved version of the manuscript meets the expectations of the Referee.

Reviewer 3 Report

Comments and Suggestions for Authors

Referee's report on the paper "Triply Heavy baryons with JETRAD.." by F.Celiberto.

In my view, this is one of the most impressive reviews I have recently seen on this topic. It contains an impressive number of citations (approximately 500) and covers a tremendous amount of information related to the description of such baryons in HEP. I did not find any problems with the way the material is presented and could not detect any inaccuracies. This paper will represent a valuable reference on this topic for many years to come.

In this particular case, I recommend publishing this paper without modifications.

 

Author Response

We sincerely thank the Referee for her/his careful reading, for the encouraging and generous comments, and for recommending the publication of our manuscript without modifications.

We are truly grateful for the recognition of the effort devoted to the bibliographic and technical aspects of this work.
The Referee's appreciation reinforces our commitment to developing high-quality and comprehensive references for the heavy-flavor and exotic-hadron community.