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Universal geometry of twoneutron halos and Borromean Efimov states close to dissociation
by Pascal Naidon
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Authors (as registered SciPost users):  Pascal Naidon 
Submission information  

Preprint Link:  https://arxiv.org/abs/2302.08716v1 (pdf) 
Date submitted:  20230220 02:55 
Submitted by:  Naidon, Pascal 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
The geometry of Borromean threebody halos, such as twoneutron halo nuclei made of a core and two neutrons, is investigated using a threebody model. This model enables to analytically derive the universal geometric properties found recently within an effectivefield theory for halos made of a core and two resonantlyinteracting particles [Phys. Rev. Lett., 128, 212501 (2022)]. It is shown that these properties not only apply to the ground threebody state, but also to all the excited (Efimov) states where the coreparticle interaction is resonant. Furthermore, a universal geometry persists away from the resonant regime between the two particles, for any state close to the threebody threshold. This universality is different from the Efimov universality which is only approximate for the ground state. It is explained by the separability of the hyperradius and hyperangles close to the threebody dissociation threshold.
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Reports on this Submission
Anonymous Report 3 on 2023620 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.08716v1, delivered 20230620, doi: 10.21468/SciPost.Report.7379
Strengths
Universality is a crucial ingredient in our attempt to patch together seemingly disparate effects in physics and create a unifying approach to understanding of these effects. In this context and in Efimov physics, universality refers to the ability of the effect to be insensitive to physics at short scales. This has led to celebrated universal scaling of Efimov exponents and energies and its discovery in atomic ultracold systems.
In this work, the author hints at another universal behavior, albeit this time a geometrical behavior with ratio of the masses with the core particle and the ratio of threebody (Efimov) energy and twobody (neutronneutron) scattering length. This is a fairly complicated universal behavior whose utility remains to be understood or its practical applicability needs to be explored.
Weaknesses
The alluring beauty of Efimov universality lies in the concept of scattering length; a shortrange collisional parameter, which determines the exponent of Efimov scaling and its all other correlated fewbody behavior. No other parameters are necessary.
This work, while scientifically justified and interesting, applies universality to the length scales of interactions interparticle distances and is informed by a ratio of radii in Eq. (15). Past work on mass ratios revealed certain magic ratios for which the original Efimov universality held.
The current universality is a novel concept, but how practically it can visualized and interpreted is not clear. Can such universalities be observed?
That the separation between hyperradius (R) and hyperangles (alpha) holds should not be a surprise.
The author refers to the applicability of this class of universality to "all states, including Efimov states". What does this mean? Do other molecular threebody states follow such universality?
What is a Borromean Efimov?
Report
This work is scientifically sound and should be considered for publication. It is not clear to this reviewer how such universality can be tested.
Anonymous Report 2 on 2023522 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2302.08716v1, delivered 20230522, doi: 10.21468/SciPost.Report.7234
Strengths
The author investigates the geometry of threebody halos consisting of a
a core and two neutrons for different values of the neutroncore
interaction. He extends previous findings for systems with one
threebody state to systems with excited states. The universal relations
for the radii are first derived analytically using the Faddeev equations
and then verified in a numerical investigation with separable potentials.
In particular the analytical derivation is very instructive and transparent
and it is confirmed by the numerical investigation.
The paper is timely, interesting, and well written.
Weaknesses
I have a few comments that should be addressed before the paper can be published (see report).
Report
The author investigates the geometry of threebody halos consisting of a
a core and two neutrons for different values of the neutroncore
interaction. He extends previous findings for systems with one
threebody state to systems with excited states. The universal relations
for the radii are first derived analytically using the Faddeev equations
and then verified in a numerical investigation with separable potentials.
In particular the analytical derivation is very instructive and transparent
and it is confirmed by the numerical investigation.
The paper is timely, interesting, and well written. However, I have a few
comments that should be addressed before the paper can be published:
1) The relations in Sec. 4 are derived under the assumption
1/a_12 << 1/a_13, 1/a_23. Thus I can see that the prediction
for beta >> 1 is universal. But I would think that the prediction for
beta << 1 does depend on the relative sizes of E and the energy scales
determined by 1/a_13, 1/a_23, and is not valid unless sqrt(E)
<< 1/a_13, 1/a_23 (which is the case considered in [24]). Only in this
case I, would consider the limit beta << 1 to be universal.
2) Is the extension to excited states not implicitly contained in [24]?
The EFT of [24] does not use in any way that the threebody halo
is in its true ground state nor is this requirement stated anywhere.
The presence of more deeply bound threebody states would not
change the calculations in [24] in any way. Thus I find the results
of the present study not that surprising. Am I missing something here?
3) I do not quite understand the claim that the universality is
independent of the Efimov effect. This might very well be true,
but I do not see how the numerical investigations show this.
All states investigated in the numerical investigation
are bound via the Efimov effect. In particular, the threebody states
only exist above a critical value of the neutroncore scattering length.
This is exactly the behavior of Efimov states. It is true that
the true ground states of the separable model theory may have large
corrections because their energies are close to the cutoff scale.
Nevertheless, their behavior for small deviations of the pair scattering
length (i.e. in the universal window) is that of Efimov states.
Requested changes
Address comments 1)3) in report.
Author: Pascal Naidon on 20230711 [id 3795]
(in reply to Report 2 on 20230522)
1) The relations in Sec. 4 are derived under the assumption 1/a_12 << 1/a_13, 1/a_23. Thus I can see that the prediction for beta >> 1 is universal. But I would think that the prediction for beta << 1 does depend on the relative sizes of E and the energy scales determined by 1/a_13, 1/a_23, and is not valid unless sqrt(E) << 1/a_13, 1/a_23 (which is the case considered in [24]). Only in this case I, would consider the limit beta << 1 to be universal.
I thank the referee for bringing up this point. The assumption for Sec. 4 was indeed incorrectly stated. The proper assumption now reads $\sqrt{2\mu_{12}E}/\hbar , 1/a_{12} << 1/a_{13}, 1/a_{23}$. This condition implies that $\sqrt{2\mu_{12}E}/\hbar << 1/a_{13}, 1/a_{23}$, as correctly stated by the referee, and that the Faddeev component $F_{12}$ is always dominant over the two other components at low momentum, which is the basis for the derivation of Sec 4. Note that this holds for any value of $\beta$, including $\beta \ll 1$ and $\beta \gg 1$. I hope this clarifies the conditions of validity.
2) Is the extension to excited states not implicitly contained in [24]? The EFT of [24] does not use in any way that the threebody halo is in its true ground state nor is this requirement stated anywhere. The presence of more deeply bound threebody states would not change the calculations in [24] in any way. Thus I find the results of the present study not that surprising. Am I missing something here?
Indeed, the results in that reference are implicitly applicable to the excited states, but there is no mention of excited nor Efimov states in that reference. It appears that the work has been understood by many to apply only to a ground state, as exemplified by the first referee's statement: "To my understanding, the analytical relations found in the work based on effective field theory [Ref. 24 in the manuscript] apply only to the ground trimer state". So it seems needed to emphasise the applicability of their formulas to excited states, and Efimov states in particular. To acknowledge that this point was implicitly contained in the original work, I added the sentence: "Although it is implicit in the work of Ref. [24], it is here emphasised that the analytical relations apply not only to a ground state but also to excited Borromean halo states".
3) I do not quite understand the claim that the universality is independent of the Efimov effect. This might very well be true, but I do not see how the numerical investigations show this. All states investigated in the numerical investigation are bound via the Efimov effect. In particular, the threebody states only exist above a critical value of the neutroncore scattering length. This is exactly the behavior of Efimov states. It is true that the true ground states of the separable model theory may have large corrections because their energies are close to the cutoff scale. Nevertheless, their behavior for small deviations of the pair scattering length (i.e. in the universal window) is that of Efimov states.
This is indeed a subtle point. It is true that the numerical investigation exhibits states which are all bound by the Efimov attraction. The main purpose of this numerical calculation is to show that the "halo universality" does apply to any Efimov state. On the other hand, their "halo universality" close their point of disssociation is governed by a repulsion at distances beyond the scattering length, i.e. beyond the range of the Efimov attraction. This repulsion always occurs, irrespective of the presence or absence of the Efimov effect. It is therefore independent of the Efimov effect. This is confirmed by the fact that none of the analytical derivations rely on the presence or absence of the Efimov effect, and do not depend on the quantity $s_0$ characterising Efimov universality. I expanded the discussion in the manuscript to make this important point clearer.
Strengths
See attached report
Weaknesses
See attached report
Report
See attached file
Requested changes
See attached file
Author: Pascal Naidon on 20230711 [id 3794]
(in reply to Report 1 on 20230513)
I would like to thank the referee for their very thorough review.
(1) The author refers in the title to Borromean Efimov states. To my understanding Efimov states are always Borromean, so this statement seems to be redundant. Is there some particular distinction for Borromean Efimov states?
In fact, Efimov states are not always Borromean. When one of the interactions supports a weakly bound twobody state, implying a positive scattering length for this interaction, the Efimov states are not Borromean, by definition of the term ‘Borromean’ as ‘bound in the absence of bound state for any subsystem’. The present study is restricted to negative scattering lengths, for which Efimov states are Borromean. It is therefore important to specify this point in the title.
(2) To my understanding, the analytical relations found in the work based on effective field theory [Ref. 24 in the manuscript] apply only to the ground trimer state. If that is the case, I would suggest to include this statement at the last sentence of the first paragraph of the Introduction.
That is a very good point. The work based on effective theory may indeed appear to be intended to describe a ground state, although it does apply to excited states as well. This is precisely a point clarified in this manuscript. I have included a sentence in the second paragraph to emphasise this point.
(3) It would help the reader if the range of the interactions Lambda was introduced in Equation (1), instead of introducing it later in the text [Section 4].
Following the referee’s suggestion, the range is now introduced after equation (1).
(4) Regarding the three pairwise interactions, the author assumes later in the text (beginning of chapter 5) that Lambda is the same both for neutronneutron and neutroncore interactions. Are the two interactions of the same form, or is there some reason why the range of interactions can be treated to be the same?
The referee is correct that the ranges of these interactions should be different in general. However, the interactions being of the same physical nature (nuclear force for nucleons, atomic forces for atoms), their ranges must be of the same order of magnitude. They are therefore taken to be equal in the numerical calculations for simplicity. This is now explained in the manuscript. As a check of the validity of this assumption, the resulting model is applied to the case of lithium11, which yields meansquare radii in agreement with experimental data.
(5) Right before Eq. (10), where the Jacobi vectors Rij,k are introduced, it would be good if there was a reference to Fig. 1.
The reference to Fig. 1 was added.
(6) How is the lowenergy expression for the Tmatrix elements derived? If the steps have been already carried out in a different publication, I suggest to cite these works.
Although this expression looks like the textbook result for the onshell Tmatrix elements, it is in fact the general offshell expression necessitated by the threebody problem. I could not find a reference deriving this expression explicitly, therefore I added the general derivation in an appendix.
\[\sqrt{2 \mu_{12} z_{12}/ \hbar^2} << a^{1}_{12}<< a^{1}_{23},a^{1}_{31}.\](7) I think there is a typo in the inline formula after Eq. (13). I think it should read,
I thank the referee for spotting the typo. In fact, this condition was not correct, as pointed out by the second referee, and it has been corrected.
(8) The author argues before Eq. (14) that the Faddeev component associated to the neutronneutron dimer is the dominant one. Is that the case due to its dependence on the inverse square root of the twobody neutronneutron energy, which is assumed to be very small?
Yes, it is precisely for this reason.
(9) Do the results in Section 4 apply only to the ground state?
The derivation of Section 4 does not require the bound state to be a ground state, so it applies to any excited bound state as well. This is now stated explicitly at the beginning of Section 4.
(10) In Section 4, the author distinguishes between two limiting cases for the matter over core meansquare radii, providing results depending solely on A, the mass ratio. How do these limits affect the geometry of the threebody system, and what are the imposed relations on the Jacobi vectors Rij,k ?
It is quite straightforward from Eq. (10) to find the constraint on Rij,k for a given limit of the ratio <r_m^2>/<r_c^2>. More generally, any geometric property in these limits will have some analytic expression depending on A, although it may look a bit more complicated. The choice of r_m and r_c just gives one of the simplest expressions.
(11) Why does the author consider only the case A=10 as a mass ratio between the core and the neutrons? Does it refer to a particular system, or is it a prototype system, and larger or smaller mass ratios lead to the same phenomenology?
Indeed, the choice A=10 was purely arbitrary. In the revised manuscript, the case A=9 is now considered, which can be applied to the halo nucleus of lithium11 and thus gives an illustration with a physical example. Other halo nuclei with different mass ratios are also discussed in the revised manuscript.
(12) In Fig. 2, at any fixed inverse neutronneutron scattering length, an infinity of trimer states appears due to the Efimov effect, as the neutroncore length is tuned to larger values. Why only five states appear in the leftmost corner of both panels? Is it due to numerical difficulties as the energy of the states becomes smaller and smaller, or does the scaling factor become very large?
It is true that there is an infinity of trimer states, but only five of them appear in the range of the plots shown in Fig. 2. The other states, which would lie outside of these plots, are thus not missing (although the region of the sixth state is admittedly very close to the boundary of the plot).
(13) Is it true that for highly excited trimers [upper panel and leftmost part of Fig. 2], the scaling factor for the dissociation thresholds becomes the same as for three identical particles?
Actually, it is not the same scaling factor as for three identical particles (~22.7) because the mass ratio is different, although it is very close (~17.6). The small variation of the scaling factor with mass ratio can be seen in the lowest curve in Fig. 6.2 of arxiv/1610.09805v3.
(14) In the caption of Fig. 3, the last sentence should refer to $a^{1}{12}$ and not $a^{1}$ ?
I thank the referee for spotting this typo. It has now been corrected.
(15) In Fig. 4, the good agreement with the analytic formula for beta go to zero, applies only for neutroncore scattering lengths such that the system is close to the threebody dissociation threshold. The author states “Therefore, it appears that the analytical formula does require a fine tuning of the coreparticle interaction”. In PRL 128, 212501 (2022) however, it seems that the formula applies only close to the threebody dissociation threshold and neutronneutron twobody resonance. Where does the fine tuning come from? It appears that in the current work, the author tests the range of applicability of the formula derived in PRL 128, 212501 (2022), providing bounds for its validity.
As shown in Fig. 4, the formula works well when the neutroncore scattering length $a_{23}$ is within 1% of the values $a_{23}^{(i)}$ but becomes inaccurate for other values of $a_{23}$. In this sense, the formula requires a rather finetuning of the coreparticle interaction. However, the refereee is correct that this could be interpreted as the mere requirement of the binding energy being small enough. Therefore, this sentence on the fine tuning of $a_{23}$ may be removed in the final version if the referee deems it confusing.
(16) It would be less confusing if the horizontal axis in Fig. 4, was just the neutronneutron scattering length. My understanding is that both the a12 scattering length as well as the binding energy E vary as one follows the horizontal dashed lines sketched in Fig. 3. Is therefore the mattertocore radii depicted with respect to beta out of convenience? So that a comparison with the analytical formulas is straightforward?
Indeed, varying beta corresponds to following the horizontal dashed lines of Fig. 3. The reason for using $\beta$ instead of $a_{12}$ is that it is indeed the variable appearing in the analytical formula, and it also allows a direct comparison between different states. The halo universality close to the threshold would be less apparent if the variable $a_{12}$ was used.
(17) Do the curves for a23 away from the threebody threshold come closer to the 2A/3 value in Fig. 4, when one considers smaller values of beta (10^7) ? Or do they already saturate as suggested by the presented values?
As far as it is possible to tell from the numerical calculations, the curves do not come close to the value 2A/3 and indeed saturate as suggested by the figure.
(18) How does the choice of separable interactions or the Gaussian form factors affect the limit of beta go to zero in the numerical calculations presented in Fig. 4? Are there small deviations if one chooses other form factors?
That is a very good question. I have not checked what happens for other form factors or nonseparable interactions, as I was primarily interested in checking the analytical limit in this work. It is possible that there is further modelindependence beyond the analytically computable results.
(19) Regarding Eqs. (23) and (24) for the hyperspherical formalism, I suggest to provide a few references regarding the method.
It is now specified that the formalism is called the "hyperspherical adiabatic expansion" and a comprehensive reference is given.
(20) I suggest to move Section 6 at the beginning, after the introduction of the model. In that regard, a neat explanation is provided for the universal relations of the radii for the ground state, along with an extension to excited states. I think it would be better to present these results directly, to make the distinction with PRL 128, 212501 (2022) more clear.
I understand the referee’s logic to have all the analytical results presented first, and finally illustrate them with numerical calculations. However, I am afraid that Section 6 would appear too abstract without first presenting the general picture obtained with the numerical calculations. I feel it is better to keep Section 6 as the section presenting a general interpretation of both the analytical formula and the numerical calculations.
(21) The results obtained in Section 6 apply close to the zero energy threebody threshold? If yes, it would be helpful for the reader to state that explicitly.
The beginning of the last paragraph has been rephrased to make this point explicit.
(22) The sentence at the end of Section 6 is a bit confusing. From the results obtained within the hyperspherical formalism, it is shown that the limiting cases of the meansquare radii apply to all states. To which lack of Efimov universality does the author refer to?
The Efimov universality is a discrete invariance of the spectrum near unitarity. The ground state trimer conforms only approximately to this invariance. If one picks a highlyexcited state, it can be scaled very precisely onto another highlyexcited state, but it would not superimpose very well when scaled onto the ground state. In contrast, the universality presented in this work is the same for the ground state and excited state, as it concerns only the shape, and not the hyperradial distribution. I have rephrased the explanation more clearly, and now refer to this universality as “halo universality” to make the distinction with “Efimov universality” more explicit.
Author: Pascal Naidon on 20230711 [id 3796]
(in reply to Report 3 on 20230620)I thank the referee for pointing out the strenghs of the work. Regarding their concerns, here are my replies.
As shown in the present work (and the previous work of Hongo and Son) the halo universality is also governed by a single parameter, the ratio $\beta$ between the threebody energy and the twobody energy.
It is important to note that the interparticle distances in 3body halo states (states close to threebody dissociation) is much larger than the length scale of interaction. This is why such sates exhibit universal properties. Efimov states (states close to twobody dissociation) are universal for the same reason: since the scattering length approaches infinity, Efimov states are much larger than the length scale of interaction and thus universal.
The observability is indeed an important point. To observe halo universality, one should be able to probe the geometric properties of a threebody system near threebody dissociation. A new section discussing the observation of halo universality in halo nuclei and ultracold atoms has been added to the manuscript. The advantage of halo nuclei is that their geometry can be probed experimentally, but their disadvantage is that they only approach the regime of halo universality. The advantage of ultracold atomic trimers is that they can be tuned to fully reach the regime of halo universality, but determining their geometry experimentally is more difficult.
In general, these two variables are not expected to be separable. There are only two cases where they become separable:  at the unitarity limit, i.e. at the twobody dissociation point, where the Efimov effect occurs.  near the threebody dissociation point, which is pointed out in the present work.
Yes, it applies to any threebody bound state as long as it is sufficiently close to threebody dissociation. The fact that it also applies to triatomic molecules close to dissociation has been emphasised in the abstract.
A Borromean Efimov state is an Efimov state that is Borromean, i.e. it is an Efimov threebody bound state in the absence of twobody bound state. The Borromean regime for Efimov states is the regime of negative scattering lengths, for which there is no twobody bound state. This is the regime studied in this work.