Abstract
The long history of the research concerning the possible existence of bound or resonant states in light multineutron systems, essentially \(^3\)n and \(^4\)n, is reviewed. Both the experimental and the theoretical points of view have been considered, with the aim of showing a clear picture of all the different detection and calculation techniques that have been used, with particular emphasis in the issues that have been found. Finally, some aspects of the present and future research in this field are discussed.
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This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a review article of works that were already published, so there is no associated data.]
Notes
It is indeed very disappointing that this trivial kinematical fact could have been ignored not only by the authors but also passed through the referee procedure of top level journals.
The same potential was considered in [110] to illustrate an S-wave resonance-like behavior.
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Communicated by Nicolas Alamanos
A Two neutrons in a trap
A Two neutrons in a trap
In order to illustrate the kind of problems related to the extrapolation from bound states to the continuum, we present in this Appendix a detailed solution of the simplest case, in the same spirit as it was considered in Ref. [137]: two neutrons confined in a trap.
Let us consider two neutrons interacting via \(V_{nn}\) and furthermore submitted to a potential trap having, as in Refs. [105, 129, 140], a Woods–Saxon form:
The total two-body Hamiltonian is:
The particle coordinates \(r_i\) can be expressed in terms of the relative (\(\mathbf {r}=\mathbf {r}_2-\mathbf {r}_1\)) and center-of-mass (\(2\mathbf {R}=\mathbf {r}_1+\mathbf {r}_2\)) coordinates as:
Assuming \(\mathbf {R}\equiv 0\) to get rid of the center-of-mass motion, the intrinsic two-body Hamiltonian takes the form:
The numerical solution of this problem is trivial and can be obtained with any required accuracy. For illustrative purposes, we have considered a phenomenological CD MT13 nn interaction:
reproducing the nn scattering length \(a_{nn}=-18.59\) fm and effective range \(r_{nn}=2.94\) fm values, with the parameters \(V_r=1438.72\) MeV, \(\mu _r=3.11\) fm, \(V_a=509.40\) MeV, \(\mu _a=1.55\) fm, and \({\hbar ^2/ m_n}=41.44250\) Mev fm\(^2\) [83].
The binding energies for two values of the trap size (\(R=3\) and 5 fm) and fixed diffusiveness \(a=0.65\) fm are plotted in Fig. 22 as a function of the strength \(V_0\).
At first glance it seems that both series of points can be indeed linearly extrapolated (solid lines) towards a positive value (empty square at \(V_0=0\)), which could be interpreted, as in Refs. [105, 129, 140], as a 2n resonance with positive energy. Furthermore, it is worth noting that even in the absence of any interaction, the results of the linear extrapolation (dashed lines) display a similar trend and would lead to a very disturbing resonant state between two non-interacting neutrons!
One could argue that in the case of two neutrons in a \(^1\)S\(_0\) state, the extrapolated results of Fig. 22 correspond in fact to the nn virtual state with positive energy, as it is often considered. We remind, however, that the virtual state, corresponding to a pole in the negative imaginary axis \(k_0=+i\kappa \), has negative energy \(k_0^2=-\kappa ^2\). The pole trajectory in the complex k-plane and E-surface of an artificially bound nn (\(k_i\) and \(E_i\in E_I\)) into the continuum (\(k_f\) and \(E_f\in E_{II}\)) is summarized in Fig. 23. As one can see, both bound and virtual (unbound) states correspond to negative-energy values in the E-surface, though living in different Riemann sheets.
If the extrapolation analysis sketched in Fig. 22 is done carefully, in particular using much smaller binding energies, one can see that a strong curvature in the \(E(V_0)\) dependence starts at \(B\approx 0.1\) MeV until the continuum threshold \(E=0\) is reached at a critical value \(V^c_0(R)\), which increases with R. For instance, \(V^c_0\approx 0.1\) MeV for \(R=5\) fm (see the zoom displayed in Fig. 24).
After the threshold, the \(V_0\)-dependence represented in Fig. 24 changes drastically the curvature, and turns downwards toward negative values of E as can be see in Fig. 25, and as expected from the location of a virtual state in the continuum illustrated in Fig. 23. The highly non-trivial trajectory in the \((E,V_0)\) plane displayed in Fig. 25, results from the ACCC extrapolation given by equation (5). To be faithful, a set of bound state values \((V_{0}^n,E^n)\) must be computed with very high accuracy, as accurate should be the critical values of the potential strength at threshold \(V_0^c\), when the system enters the continuum. As an example of the accuracy required, the values used for the case \(R=5\) fm are listed in Table 1.
The trajectory in the \((E,V_0)\) plane is represented in the complex k-plane in the left panel of Fig. 23. It is the straight line formed by the union of \(\gamma \cup \{0\}\cup \gamma '\). The corresponding path in the E-surface is \(\varGamma \cup \{0\}\cup \varGamma '\), shown in the right panel, where \(E_i\) belongs to \(E_I\) and \(E_f\) to \(E_{II}\). The transition from bound state to continuum is at \(k=0\) and coincides with the transition \(E_I \rightarrow E_{II}\). Note that both initial (\(k_i\)) and final (\(k_f\)) states have negative energy (\(E_i\) and \(E_f\) respectively), despite the fact that \(E_f\) is in the continuum, but belong to different Riemann sheets.
From the above results it is clear that a simple polynomial extrapolation cannot describe the non-trivial evolution of a bound state into the continuum displayed in Fig. 25. This is in fact the main conclusion of this Appendix. ACCCM can do it accurately by taking into account the analytical properties related to the thresholds. However, the method requires the previous computation of several bound state values with high accuracy and very close to \(E=0\). This can be easily done in the two-body case we have considered above, although it becomes more and more involved when increasing the number of particles. This level of accuracy is not accessible to all the methods used to solve the few-nucleon problem, but it has been successfully applied to \(A=3,4,5\) in Refs. [84,85,86,87].
The case presented above corresponds to an S-wave virtual state, which is different from the realistic 3n and 4n P-wave resonances displayed in Figs. 14 and 16. They have however in common that the trajectories of the corresponding pole positions end at negative energy, Re\((E)<0\), in the third quadrant of the second Riemann sheet \(E_{II}\), as illustrated in Fig. 23. Of course, it is clear that a polynomial extrapolation of the results displayed in Fig. 22 (or Fig. 15) is forced to end at positive energy values, and is therefore unable to reproduce the realistic behavior of the 3n and 4n resonant states.
Furthermore, it is worth noting that even in the case of positive energy P-wave resonances, with Re\((E)>0\) and Im\((E)<0\) like the one represented in Fig. 17, a polynomial extrapolation disregarding the analytic threshold structure can lead to wrong results. This can be illustrated by considering the same nn interaction (10) but suitably scaled in order to generate a P-wave nn resonant state. For instance, with a scaling factor \(s=3.0\) one has a resonance at \(E=4.40-3.32\,i\) MeV, that is a state with \(\varGamma \approx 6.6\) MeV, still smaller than the realistic case of Fig. 17. To complete our results for nn S-wave, we have computed the energies of this P-wave resonant state in the trap, as a function of \(V_0\) for two different parameters of the trap size R, and performed a quadratic extrapolation towards the ‘free’ case \(V_0=0\). Results are displayed in Fig. 26. They show a significant disagreement between the extrapolated values and the physical resonance energy, indicated by a blue solid circle. We cannot exclude, as noted in Ref. [137], that for very narrow P-wave resonances the difference could be small, but the polynomial extrapolation is not reliable in a general case. This could be the case in the unphysical example (P-wave resonance in an S-wave two-Gaussian potential) considered in Ref. [129] in order to justify their extrapolation method.
The exact solution of two neutrons in a trap can be straightforwardly extended to the \(^3\)n case. The Jacobi coordinates of the 3n system are expressed in terms of the particle coordinates \(r_i\) as:
Since in the center of mass \(\mathbf {r}_1+\mathbf {r}_2+\mathbf {r}_3=0\), one has:
and equivalent expressions for the other \(r_i\).
The three-body intrinsic Hamiltonian takes the form:
with:
Since the different sets of Jacobi coordinates are related by the linear transformations:
the problem is equivalent to adding a three-body force:
which in the case of three identical particles can be easily solved with a single Faddeev equation. This calculation requires however some care due to the presence of open \(^3n\rightarrow ^2n +n\) channels for some values of the trap parameters \((V_0,R)\), but will not be developed here.
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Marqués, F.M., Carbonell, J. The quest for light multineutron systems. Eur. Phys. J. A 57, 105 (2021). https://doi.org/10.1140/epja/s10050-021-00417-8
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DOI: https://doi.org/10.1140/epja/s10050-021-00417-8