Effect of thresholds on the width of three-body resonances

It has been recently reported an intriguing theoretical result of a narrow three-body resonance with a large available phase space. The resonance was reported in the $N\Lambda\Lambda-\Xi NN$ system near the $\Xi d$ threshold, having a very small width in spite of the open $N\Lambda\Lambda$ channel lying around 23 MeV below the $\Xi NN$ channel. We use first-order perturbation theory as a plausible argument to explain this behavior. We apply our result to realistic local interactions. Other systems involving several thresholds are likely to follow the same behavior.


I. INTRODUCTION
The coupled NΛΛ−ΞNN system in the dominant S-wave configuration has the quantum numbers (I, J P ) = ( 1 2 , 1 2 + ), since the coupling between the lower (NΛΛ) and upper (ΞNN) components of the system is via the ΛΛ − ΞN two-body channel with quantum numbers (i, j p ) = (0, 0 + ). Therefore, if one adds an additional nucleon also in S-wave, the three-body system will have the quantum numbers (I, J P ) = ( 1 2 , 1 2 + ).
The possible existence of a stable bound state of the coupled three-body system NΛΛ − ΞNN was first studied in Refs. [2][3][4] using the two-body interactions derived in a constituent quark model framework [5]. In that model, the coupled two-body subsystem ΛΛ − ΞN in the (i, j p ) = (0, 0 + ) channel (the H dibaryon channel) is bound with a binding energy of 6.4 MeV. Thus, in order to search for bound-state solutions of the three-body equations one just needs to calculate the (real) Fredholm determinant for energies below the HN threshold which indeed leds to a stable bound-state solution at about 0.5 MeV below that threshold [2].
However, the most recent analysis of the quark mass dependence of the H dibaryon in ΛΛ scattering [6,7] point to the H dibaryon being a resonance above the ΛΛ threshold. Thus, in order to study possible bound or resonant states of the NΛΛ − ΞNN system one must solve the three-body equations in the complex plane, which makes the numerical problem much harder to deal with. This was done in Ref. [1] using simple separable potentials fitted to the low-energy data of the most recent update of the ESC08 Nijmegen potential [8,9], that give account of the pivotal results of strangeness −2 physics, the NAGARA [10] and the KISO [11] events.

II. FORMALISM
The results of Ref. [1] were obtained taking the nucleon mass as the average of the proton and neutron mass, and the Ξ mass as the average of Ξ 0 and Ξ − mass. Thus, the effect of the ΛΛ − ΞN (0, 0 + ) channel is to change the three-body eigenvalue by indicating that the lower three-body channel effectively acts as a perturbation. This is somewhat intriguing since the ΛΛ − ΞN (0, 0 + ) interaction is not small (see Tables II and   III of Ref. [1]).

B. The proposed explanation
In order to provide a plausible argument to explain the above result, we will consider first-order perturbation theory taking the ΞNN channel as the main interaction and the contribution of the lower channels NΛΛ as the perturbation. The argument is very simple.
The small binding energy of the ΞNN system causes the unperturbed wave function to have a very long range in coordinate space while the perturbation is a short-range operator so that the overlap between them is quite small, which results in δE being small. Thus, we will calculate where | Ψ 0 is the (real) ΞNN wave function δV is the (complex) perturbation given in lowest order by, which is shown graphically in Fig. 1. We have used the convention that in both three-body sectors the two identical particles are labeled 2 and 3 with particle 1 being the different one.
Using Eqs. (2)(3)(4) and taking into account the identity of particles 2 and 3 one obtains, In Eq. (5) The Green's function that appears in the perturbation term (4) is given explicitly by, where p i and q i are the Jacobi momenta and η i and ν i the corresponding reduced masses of the various configurations. Since E is a positive number this function is singular and moreover it has an imaginary part. The Green's function attached to the Faddeev components of the unperturbed wave function (3), on the other hand, is given by with so that E + ∆E is a small negative number and the function in Eq. (7) is real and it has no singularity although it is sharply peaked at low momenta. In addition, the Faddeev amplitudes U N N 1;Ξ and U ΞN 3;N are also peaked at low momenta in the NN (0, 1 + ) and ΞN (1, 1 + ) channels, corresponding to the deuteron and D * bound states, respectively, which lie very close to threshold. Thus, as mentioned above, the unperturbed wave function has a long range in coordinate space while the perturbation term has the short-range characteristic of hadronic systems. Consequently, the overlap between both terms in Eq. (5) is very small, rendering δE small.
In order to show explicitly this behavior let us consider one of the terms of Eq. (5), Since in the separable model of Ref. [1] one has that, where The expressions (12) depend only on the variables q i , i.e., We show in Fig. 2(a) the wave functions W N N 1;Ξ (q) and W ΞN 2;N (q) for the two dominant channels NN (0, 1 + ) and NΞ (1, 1 + ), respectively, and in Fig. 2(b) the diagonal perturbation term δv ′ (q, q), where one can see clearly this behavior (note the logarithmic scale of the wave function).

A. The separable model
If we now apply the formalism of the previous section to the separable potential model of Ref. [1], it gives which is of the same order of magnitude as the result of the exact calculation given by Eq.
(1). This shows that the small value of δE, and consequently the very small width, can be understood as resulting from the fact that the NΛΛ channel acts effectively as a perturbation to the ΞNN channel when the resonance lies very near the ΞNN threshold. is completely general. Thus, we have also used the Malfliet-Tjon type local potentials [12] of the NN subsystem constructed in Ref. [13] and those of the ΞN subsystem constructed in Ref. [14], based in the most recent update of the Nijmegen ESCO8 potentials [9]. We show the Fredholm determinant of the ΞNN system in Fig. 3 for energies very near the Ξd threshold where as one can see the ΞNN state lies less than 0.01 MeV above the Ξd threshold, so that both the separable and local models predict the resonance very near the Ξd threshold and consequently, as we have just shown, will have a very small width.

IV. OUTLOOK
In this letter we have presented a plausible argument to explain the small width of a three-body resonance in a coupled two-channel system lying close to the upper channel in spite of being open the lower one. This is an intriguing result, since the available phase space of the decay channel is quite large, around 23 MeV. We use first-order perturbation theory to explain this behavior. We have applied our result to realistic local interactions.
Let us finally comment that the mechanism we have discussed in this work could also help in understanding the narrow width of some experimental resonances found in the heavy hadron spectra, whose assumed internal structure allow them to split into several different channels [15,16]. It has been explained in Ref. [17] how systems with an internal structure QQnn, where n stands for a light quark, could either split into (Qn) − (nQ) or (QQ) − (nn).
For Q = c or Q = b the (QQ) − (nn) threshold is lower than the (Qn) − (nQ), the mass difference augmenting when increasing the mass of the heavy quark. Such experimental behavior can be simply understood within the constituent quark model with a Cornell-like potential [17,18]. Thus, the possibility of finding meson-antimeson molecules, (Qn) − (nQ), contributing to the heavy meson spectra becomes more and more difficult when increasing the mass of the heavy flavor, due to the lowering of the mass of the (QQ) − (nn) threshold.
This would make the system dissociate immediately. In such cases, the presence of attractive meson-antimeson threshold together with the arguments we have drawn in this work, hint to a possible explanation of a narrow width of some of the XY Z states lying close to the (Qn) − (nQ) threshold as a meson-antimeson molecule. Similar arguments could be handled for the LHCb pentaquarks, what requires a careful analysis in the models used for the study of these states.