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Particle representation for NNˉK system

by B. Vlahovic, I. Filikhin, and R. Ya. Kezarashvili

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Igor Filikhin
Submission information
Preprint Link: scipost_201910_00029v1  (pdf)
Date accepted: Jan. 28, 2020
Date submitted: Oct. 16, 2019, 2 a.m.
Submitted by: Filikhin, Igor
Submitted to: SciPost Physics Proceedings
Proceedings issue: 24th European Few Body Conference (EFB2019)
Ontological classification
Academic field: Physics
Specialties:
  • Nuclear Physics - Theory
  • Quantum Physics
Approaches: Theoretical, Computational

Abstract

In the framework of the Faddeev equations in configuration space we perform an analysis of quasi-bound state of the NNˉK system within a particle model. In our approach, the system NNˉK(sNN=0) (NNˉK(sNN=1)) is described as a superposition of ppK and pnˉK0 (nnˉK0 and pnK) states, which is possible due to a particle transition. The relation of the particle model to the theory of a two-state quantum system is addressed and discussed taking into account the possibilities of deep and shallow NNˉK(sNN=0) quasi-bound states.

Published as SciPost Phys. Proc. 3, 044 (2020)


Reports on this Submission

Report #1 by Anonymous (Referee 1) on 2020-1-13 (Contributed Report)

  • Cite as: Anonymous, Report on arXiv:scipost_201910_00029v1, delivered 2020-01-13, doi: 10.21468/SciPost.Report.1447

Report

The proceedings contribution "Particle representation for NN¯K system" by Vlahovic, Filikhin and Kezarashvili
provides a succinct summary of coupled-channels calculations in this system. It particularly focuses on the ppK- and npK0 channels, and provides an analytical discussion of how coupling between these two channels affects the theoretical understanding of the three-body problem (including extension of the Faddeev formalism to coupled channels). The paper has a clear introduction and justification, and is overall well written when it comes to general physics discussions.

While I think I understand the motivation of this work, I have trouble understanding some of the details. First, I think I see how the coupled channels approach can be introduced for the Faddeev formalism, but the physical role of the A-matrix in Eq. (2) is unclear to me. On the one hand, the mixing between different isospin states should be directly dictated by the laws of angular momentum/isospin (eg some Clebsch-Gordan coefficients for three, rather than 2, states). I presume similar decompositions exist for the case of three spins, for instance, and I would assume the same should hold for isospin systems.

The discussion around the freedom on the alfa/beta parameters, and the discussion of their impact, is given in Section 3 and in the context of Figure 2. Having done that, the authors immediately settle for \alpha=1/\sqrt{2}. I don't think I understand the motivation to choose this value (which in Fig 2 provides the same energies for the two systems). I suggest a more elaborate discussion is provided, and in particular I would also like to know whether the choice V_12=V_21=0 is fixed across the paper (or only at this stage).

Figure 3: the authors state in Table 2 that epsilon_2 is "unbound". If I understand correctly, epsilon_2 corresponds to the dashed line in the "coupled" panel (right), which indicates that it is bound (negative energy) but above the other state. Should the dashed line be above 0?

Finally, and in the context of the isospin splitting, is there a simple reason to see why the ppK or nnK systems are significantly more bound than the npK states? np pairs have a bound states but pp and nn don't, so I would have expected the kaon to produce further attraction in the np channel than in the nn/pp one. I don't think I found a justification of this behavior in the paper.
  • validity: good
  • significance: good
  • originality: good
  • clarity: high
  • formatting: good
  • grammar: reasonable

Author:  Igor Filikhin  on 2020-01-24  [id 718]

(in reply to Report 1 on 2020-01-13)
Category:
answer to question
correction

Dear Referee,
We thank you for carefully reading our manuscript and making insightful comments that will enhance the clarity of our work. We greatly value your time and energy in helping us improve our manuscript. We are humbled by your assessment that our fellow researchers will find our work relevant and interesting. We completely agree with your comments and have addressed your concerns individually below.
Sincerely,
B. Vlahovic, I. Filikhin, R. Ya. Kezarashvili

Reviewer
The proceedings contribution "Particle representation for NN¯K system" by Vlahovic, Filikhin and Kezarashvili
provides a succinct summary of coupled-channels calculations in this system. It particularly focuses on the ppK- and npK0 channels, and provides an analytical discussion of how coupling between these two channels affects the theoretical understanding of the three-body problem (including extension of the Faddeev formalism to coupled channels). The paper has a clear introduction and justification, and is overall well written when it comes to general physics discussions.
Our Response
Thank you for the favorable assessment of our manuscript.
Reviewer
While I think I understand the motivation of this work, I have trouble understanding some of the details. First, I think I see how the coupled channels approach can be introduced for the Faddeev formalism, but the physical role of the A-matrix in Eq. (2) is unclear to me. On the one hand, the mixing between different isospin states should be directly dictated by the laws of angular momentum/isospin (eg some Clebsch-Gordan coefficients for three, rather than 2, states). I presume similar decompositions exist for the case of three spins, for instance, and I would assume the same should hold for isospin systems.
Our Response
Thank you very much for this comment. We added after Eq. (5), the second line, the following explanation in the revised manuscript:

Thus, the matrix A describes the coupling ppK/npˉK0 via non-diagonal elements V12 and V21.
The isospin states of ppK and npˉK0 are the same and the possible transformation pK nˉK0 can be described as a coupling
between the systems.

Reviewer
The discussion around the freedom on the alfa/beta parameters, and the discussion of their impact, is given in Section 3 and in the context of Figure 2. Having done that, the authors immediately settle for \alpha=1/\sqrt{2}. I don't think I understand the motivation to choose this value (which in Fig 2 provides the same energies for the two systems). I suggest a more elaborate discussion is provided, and in particular I would also like to know whether the choice V_12=V_21=0 is fixed across the paper (or only at this stage).
Our Response
Thank you for this this insightful and interesting comment. The choice V_12=V_21=0 is not fixed across the paper, it is only at this stage for the results shown in Figure 2. For more clarity we wrote (page 6, second paragraph, line 5):

If the coupling, defined by the potentials V12 and V21, is ignored, the systems transit as one to another when the coupling
constant α increases from 0 to 1. In Fig. ??? we present the results of calculations for the binding energies of the ppK and npˉK0 versus
α2 for the case when V12 and V21 are omitted in Eq. (???).
Important is that the "ppK" cluster can be described as having two levels with the same energy for the α=12.
Taking into account that the transition pK/nˉK0 makes the same probability for the pK and nˉK0 states of NˉK system, we assume the same probability for ppK and npˉK0 states. According the two-level system theory, two levels of the ppK/npˉK0 system
are crossing when the coupling is absent. When α2=12 and V12=V21=0, the Hamiltonians corresponding to the ppK and npˉK0 states are the same. It is initial point of the theory.
Switching on the coupling, i.e. considering V12 and V21 as non-zero terms in Eq. (???), has to lead to "repulsion of the levels" or anti-crossing of the levels.
Thus, the coupling constant has to be chosen as 12. It means that the probability of the ppK and npˉK0 states is equal in the "ppK" cluster. The energy in this point is different from one obtained in the framework of the "traditional" isospin NNˉK model \cite{Kezerashvili2016}. However, the same coupling constant was used in Ref. \cite{R16}, where an isospin model was actually employed based on a "charge isospin basis". This choice for α creates a correspondence between the isospin and particle models.


Reviewer
Figure 3: the authors state in Table 2 that epsilon_2 is "unbound". If I understand correctly, epsilon_2 corresponds to the dashed line in the "coupled" panel (right), which indicates that it is bound (negative energy) but above the other state. Should the dashed line be above 0?
Our Response
Many thanks for this comment. The dashed line is above the two-body threshold, not shown in the figure, which corresponds the breakup of the system on two fragments and it is in the negative region of energies. In the revised manuscript we replace Figure 3 by the new one, where explicitly is shown the two-body threshold p+(pK^-) by the dotted line., as well as we added in the caption for Figure 3:

The two-body threshold p+(pK) is shown by the dotted line.

Reviewer
Finally, and in the context of the isospin splitting, is there a simple reason to see why the ppK or nnK systems are significantly more bound than the npK states? np pairs have a bound states but pp and nn don't, so I would have expected the kaon to produce further attraction in the np channel than in the nn/pp one. I don't think I found a justification of this behavior in the paper.
Our Response
Thank you for this this insightful and interesting comment. For the clarification of this situation we added the following text (page 7, last paragraph):

It is interesting to mention that nnˉK0 system is significantly more bound than the npK due to two strong nˉK0 interactions which are specularly associated with the Lambda(1405) state, while npK is formed by the week nK and strong pK interactions. The attractive contribution of the spin triplet np potential %(adding the energy to NN) pair
cannot compensate the weak contribution of the nK potential.

Attachment:

RK_SciPost_particle_rep_VFK_081.pdf

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