ΛΛNΞNN S wave resonance

  • We use an existing model of the ΛΛNΞNN three-body system based on two-body separable interactions to study the (I,JP)=(1/2,1/2+) three-body channel. For the ΛΛ, ΞN, and ΛΛΞN amplitudes, we have constructed separable potentials based on the most recent results of the HAL QCD Collaboration. They are characterized by the existence of a resonance just below or above the ΞN threshold in the H-dibaryon channel, (i,jp)=(0,0+). A three-body resonance appears 2.3 MeV above the Ξd threshold. We show that if the ΛΛΞN H-dibaryon channel is not considered, the ΛΛNΞNN S wave resonance disappears. Thus, the possible existence of a ΛΛNΞNN resonance would be sensitive to the ΛΛΞN interaction. The existence or nonexistence of this resonance could be evidenced by measuring, for example, the Ξd cross section.
  • The ΛΛΞN system in a pure S wave configuration has quantum numbers (i,jp)=(0,0+), such that by adding one more nucleon, the ΛΛNΞNN system necessarily has quantum numbers (I,JP)=(1/2,1/2+). In a series of works based on the chiral constituent quark model [1-3], this system was studied under the assumption that the H dibaryon [4] has the lower limit mass determined by the E373 experiment at KEK [5] from the observation of a 6ΛΛHe double hypernucleus. Despite significant experimental and theoretical efforts, the existence of the H dibaryon remains inconclusive, see Ref. [6] for a recent update. The experimental evidence disfavors large binding energies [7], as predicted in Ref. [4], and the high statistics study of Υ decays at Belle [8] found no indication of an H dibaryon with a mass near the ΛΛ threshold. Recently, the HAL QCD Collaboration [9] has published a Nf=2+1 study of coupled channel (ΛΛ and ΞN) baryon-baryon interactions with near-physical quark masses, namely mπ=146 MeV, concluding that the H dibaryon could be a ΛΛ resonance just below or above the ΞN threshold. Similar results were obtained in a low-energy effective field theory study of the H dibaryon in ΛΛ scattering [10].

    The HAL QCD results are being used as input for the study of strangeness - 2 baryon-baryon interactions, as recently done in relativistic chiral effective field theory studies [11]. The HAL QCD ΞN interactions have also recently been used to study the possible existence of ΞNN bound states, see Ref. [12], with negative results for the (I,JP)=(1/2,1/2+) channel. For the NN interaction, they used the AV8 potential [13]. As the coupling between ΛΛ and ΞN was found to be weak in Ref. [9], they used an effective single-channel ΞN potential, in which the coupling to ΛΛ in 11S0 was renormalized into a single Gaussian form chosen to reproduce the ΞN phase shift obtained with channel coupling. The three-body ΞNN problem is solved in the real axis by means of a variational method with Gaussian bases, the Gaussian Expansion Method [14,15]. The full coupling between the ΞNN and ΛΛN channels was not explicitly considered. A similar calculation based on the Nijmegen ESC08c potentials [16-18] was presented in Ref. [19], also with negative results for the (I,JP)=(1/2,1/2+) channel, see Fig. 2(a) of Ref. [19].

    Unlike the calculation in Ref. [12], we developed a model in Ref. [20] of the ΛΛNΞNN three-body system, which allowed us to look for possible three-body resonances. Using separable two-body interactions fitted to the low-energy data of the Nijmegen S wave baryon-baryon amplitudes [16-18], we found a resonance just below the Ξd threshold with a very small width of only 0.09 MeV. (It is worth to note that the results for the ΞNN system with maximal isospin have been independently reproduced within the integral Faddeev equation formalism [21] in agreement with high accuracy.) Qualitatively similar results have been obtained and are described in Ref. [12]; although as stressed in this manuscript, they are numerically different due to a different NN potential and a different treatment of the ESC08c Nijmegen S wave baryon-baryon interactions. Such dependencies on the models and parametrizations of the two-body interactions result in this three-body system being ideally suited for testing different models for two-body interactions.

    However, contrary to the recent results of the HAL QCD Collaboration, the Nijmegen baryon-baryon interactions gave no indication of either a bound state or a resonance in the ΛΛΞN (0,0+) two-body channel, the H-dibaryon channel. It is thus interesting to see if the existence of a resonance just below or above the ΞN threshold, as it has been found by the HAL QCD Collaboration [9] and low-energy effective field theory studies [10], may affect the position of the three-body S wave (1/2,1/2+) ΛΛNΞNN resonance found in Ref. [20]. For this purpose, we have now constructed separable potential models of the ΛΛ, ΞN, and ΛΛΞN amplitudes reproducing the behavior of the HAL QCD collaboration results [9]. We have also performed a full-fledged coupled-channel study of the ΛΛNΞNN three-body system.

    We use rank-one separable potentials for all uncoupled two-body channels, that is, for all channels except the ΛΛΞN (0,0+) interaction. They are as follows,

    Vρi=gρiλgρi,

    (1)

    such that the two-body t-matrices are

    tρi=gρiτρigρi,

    (2)

    with

    τρi=λ1λgρi|G0(i)|gρi,

    (3)

    where G0(i)=1/(EKi+iϵ) and Ki is the kinetic energy operator of channel i. We use Yamaguchi form factors [22] for the separable potentials of Eq. (1), i.e.,

    gρi(p)=1α2+p2.

    (4)

    For the case of the coupled (0,0+) ΛΛΞN channel, we use a rank-two separable potential of the form [23]

    Vρσij=gρiλijgσj,

    (5)

    such that

    tρσij=gρ1τρσijgσj,

    (6)

    with

    τΛΛΛΛ11=λ213GΞNλ11(1λ33GΞN)λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),τΞNΞN33=λ213GΛΛλ33(1λ11GΛΛ)λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),τΛΛΞN13=τΞNΛΛ31=λ13λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),

    (7)

    and

    GΛΛ=gΛΛ1|G0(1)|gΛΛ1,GΞN=gΞN3|G0(3)|gΞN3.

    (8)

    In this case, we also use Yamaguchi-type form factors as

    gΛΛ1(p)=1α2+p2,gΞN3(p)=1β2+p2.

    (9)

    The parameters of the ΛΛΞN model based on the latest HAL QCD potentials are given in Table 1. In Figs. 1(a), (b), and (c) we show the predictions for the ΛΛ and ΞN phase shifts as well as the inelasticity, which are rather similar to those of model t/a=12 of the HAL QCD Collaboration presented in Fig. 4 of Ref. [9]. The corresponding parameters of the uncoupled ΞN models are given in Table 2. Note that our results have been obtained by taking the nucleon mass to be the average of the proton and neutron masses and the Ξ mass as the average of the Ξ0 and Ξ masses. Thus, the ΞN and ΞNN thresholds are 25.6 MeV above the ΛΛ and ΛΛN thresholds, respectively. However, this threshold is 32 MeV for the HAL QCD results [9], as they use the values obtained from their lattice QCD study for the baryon masses. Therefore, to compare our phase shifts with those of Ref. [9], one should keep in mind that the energy scale of Ref. [9] corresponds to those of Fig. 1 multiplied by 1.25. The models of the NN and ΛN subsystems are the same as in Ref. [20].

    Table 1

    Table 1.  Parameters α and β (in fm1), λ11, λ33, and λ13 (in fm2) of the separable-potential model of the coupled (i,jp)=(0,0+) ΛΛΞN two-body system.
    α β λ11 λ33 λ13
    1.3465 1.1460 0.1390 0.3171 0.0977
    DownLoad: CSV
    Show Table

    Figure 1

    Figure 1.  (a) ΛΛ scattering phase shifts, (b) ΛΛ inelasticity, and (c) NΞ scattering phase shifts in the (i,jp)=(0,0+) channel.

    Table 2

    Table 2.  Parameters α (in fm1) and λ (in fm2) of the separable-potential model of the uncoupled (i,jp) ΞN two-body channels.
    Channel α λ
    (0,1+) 1.41 -0.117
    (1,0+) 7.333 22.97
    (1,1+) 0.803 -0.016
    DownLoad: CSV
    Show Table

    The coupled ΛΛNΞNN three-body system presents the special characteristic that each three-body component consists of two identical fermions and a third one that is different. The homogeneous integral equations of this system appropriate for the search of bound and resonant states were derived in Ref. [3] using a graphical method. Using the new separable models presented in Tables 1 and 2 based on the HAL QCD interactions, we do not find any bound state below the Ξd threshold, which is in agreement with the results of Ref. [12]. Therefore, we investigate the possible existence of a resonance above the Ξd threshold by calculating the Ξd scattering amplitude.

    We adopt the same convention as in Refs. [3,20], i.e., particles 2 and 3 are identical and particle 1 is different in each three-body component. After the reduction for identical particles, the inhomogeneous integral equations appropriate for Ξd elastic scattering take the following form

    1|T1|ϕ0=21|tΛΛ1|11|3G0(3)3|T3|ϕ0+1|tΛΛNΞ13|33|1G0(1)1|U1|ϕ01|tΛΛNΞ13|32|3G0(3)3|U3|ϕ0,3|T3|ϕ0=3|tNΛ3|32|3G0(3)3|T3|ϕ0+3|tNΛ3|33|1G0(1)1|T1|ϕ0,

    1|U1|ϕ0=21|tNN1|11|3G0(3)3|U3|ϕ0,3|U3|ϕ0=23|tNΞ3|ϕ03|tNΞ3|32|3G0(3)3|U3|ϕ0+3|tNΞ3|33|1G0(1)1|U1|ϕ0+23|tNΞΛΛ31|11|3G0(3)3|T3|ϕ0,

    (10)

    where |ϕ0 is the initial state consisting of the deuteron wave function times a Ξ plane wave.

    Substituting Eqs. (2) and (6) into the integral Eq. (10) and introducing the transformations i|Ti|ϕ0=i|gαiii|Xi|ϕ0 and i|Ui|ϕ0=i|gβiii|Yi|ϕ0, one obtains the inhomogeneous one-dimensional integral equations

    1|X1|ϕ0=2τΛΛ1gΛΛ1|11|3G0(3)3|gNΛ33|X3|ϕ0+τΛΛNΞ13gNΞ3|33|1G0(1)1|gNN11|Y1|ϕ0τΛΛNΞ13gNΞ3|32|3G0(3)3|gNΞ33|Y3|ϕ0,3|X3|ϕ0=τNΛ3gNΛ3|32|3G0(3)3|gNΛ33|X3|ϕ0+τNΛ3gNΛ3|33|1G0(1)1|gΛΛ11|X1|ϕ0,1|Y1|ϕ0=2τNN1gNN1|11|3G0(3)3|gNΞ33|Y3|ϕ0,

    3|Y3|ϕ0=2τNΞ3gNΞ3|ϕ0τNΞ3gNΞ3|32|3G0(3)3|gNΞ33|Y3|ϕ0+τNΞ3gNΞ3|33|1G0(1)1|gNN1|Y1|ϕ0+2τNΞΛΛ31gΛΛ1|11|3G0(3)3|gNΛ33|X3|ϕ0.

    (11)

    If one neglects the inhomogeneous terms in Eqs. (10) and (11), they become identical to Eqs. (14) and (15) of Ref. [20]. The Ξd scattering amplitude normalized as in the Argand diagram is given by

    F=πq0νϕ0|U3|ϕ0,

    (12)

    where q0 and ν, respectively, are the Ξd on-shell momentum and the reduced mass. We solve the integral Eqs. (11) using the standard method [24], where the momentum variables are rotated into the complex plane as qiqieiϕ and the results are checked to be independent of the rotation angle ϕ. If the resonance lies below the Ξd threshold, as it was the case in Ref. [20], the contour rotation method allows to simultaneously consider both the momentum variables and the energy variable as complex, such that one can determine the position of the pole in the complex plane. However, if the resonance lies above the Ξd threshold, as in the present case, the contour rotation method works only if one takes the momentum variables as complex but leaves the energy variable real, such that one cannot determine the position of the pole in the complex plane.

    We show, in Fig. 2, the Argand diagram of the Ξd system between 0 and 10 MeV above the Ξd threshold, where one sees the typical counterclockwise behavior of a resonant amplitude. If one neglects the (i,jp)=(0,0+) channel, the counterclockwise behavior disappears, which shows that the H-dibaryon channel (i,jp)=(0,0+) is basic for the existence of the three-body ΛΛNΞNN S wave resonance.

    Figure 2

    Figure 2.  Argand diagram of the Ξd system between 0 and 10 MeV above the Ξd threshold. Several relevant energies, in MeV, are indicated.

    As already mentioned in the introduction, the HAL QCD ΞN interactions have recently been used to study the possible existence of ΞNN bound states in Ref. [12] with negative results for the (I,JP)=(1/2,1/2+) channel, which is in agreement with our findings despite using a different NN interaction and method.

    We have finally evaluated the Ξd elastic cross-section as a function of energy, where we have included not only the (I,JP)=(1/2,1/2+) Ξd amplitude but also the (1/2,3/2+) amplitude, which is very small. The result is shown in Fig. 3. As one can see, the resonance shows up as a change of slope of the cross section at an energy around 2.3 MeV, i.e., close to the Ξd breakup threshold S=2mN+mΞ. The bump in the cross section would become larger for a stronger (i,jp)=(0,0+) transition potential, and as noted above, it would disappear if the (i,jp)=(0,0+) channel is not considered or the two-body resonance in the H-dibaryon channel does not exist. The Ξd cross section would allow for discrimination among the different models for the strangeness 2 two-baryon interactions. It could be studied through the quasifree Ξ production in the (K,K+) reaction on a deuteron target [25,26].

    Figure 3

    Figure 3.  Ξd elastic cross section.

    Let us finally note that if one drops the coupling to the ΛΛN channel, Fig. 2 changes by about 10 % while maintaining its shape and rotating slightly to the right; similarly, in Fig. 3 the cross-section at E=2.3 MeV changes from 55 mb to 58 mb.

    It is interesting to compare this resonance with the nucleon-nucleon 1D2 Hoshizaki resonance [27], which has a mass close to S=mN+mΔ, since it arises due to the process, NNπNN, which is driven by the pion-nucleon Δ resonance [28]. The resonance we are studying here is driven by the ΛΛΞN H-dibaryon resonance, which appears either just below or above the NΞ threshold [9], such that it has a mass, mH=mN+mΞ. Following the comparison with the Hoshizaki state, one expects that the ΛΛNΞNN resonance will have a mass close to S=mN+mH=2mN+mΞ, which is precisely the Ξd threshold and in agreement with Figs. 2 and 3.

    In brief, we have shown that the possible existence of a ΛΛNΞNN resonance would be highly sensitive to the ΛΛΞN interaction. In particular, by using a separable potential based on the most recent results of the HAL QCD Collaboration, characterized by the existence of a resonance just below or above the ΞN threshold in the H-dibaryon channel, (i,jp)=(0,0+), a three-body resonance appears at 2.3 MeV above the Ξd threshold. A theoretical and experimental effort to constrain the ΛΛΞN interaction is the basic requirement for progress in our investigation of the strangeness 2 sector.

    [1] H. Garcilazo and A. Valcarce, Phys. Rev. Lett., 110: 012503 (2013); A. Gal, Phys. Rev. Lett., 110: 179201 (2013); H. Garcilazo and A. Valcarce, Phys. Rev. Lett., 110: 179202 (2013)
    [2] H. Garcilazo, A. Valcarce, and T. Fernández-Caramés, J. Phys. G, 41: 095103 (2014) doi: 10.1088/0954-3899/41/9/095103
    [3] H. Garcilazo, A. Valcarce, and T. Fernández-Caramés, J. Phys. G, 42: 025103 (2015) doi: 10.1088/0954-3899/42/2/025103
    [4] R. L. Jaffe, Phys. Rev. Lett., 38: 195 (1977) doi: 10.1103/PhysRevLett.38.195
    [5] H. Takahashi et al., Phys. Rev. Lett., 87: 212502 (2001) doi: 10.1103/PhysRevLett.87.212502
    [6] A. Francis, J. R. Green, P. M. Junnarkar et al., Phys. Rev. D, 99: 074505 (2019) doi: 10.1103/PhysRevD.99.074505
    [7] A. Gal, E. V. Hungerford, and D. J. Millener, Rev. Mod. Phys., 88: 035004 (2016) doi: 10.1103/RevModPhys.88.035004
    [8] B. H. Kim et al. (Belle Collaboration), Phys. Rev. Lett., 110: 222002 (2013) doi: 10.1103/PhysRevLett.110.222002
    [9] K. Sasaki, S. Aoki, T. Doi et al. (HAL QCD Collaboration), Nucl. Phys. A, 998: 121737 (2020) doi: 10.1016/j.nuclphysa.2020.121737
    [10] Y. Yamaguchi and T. Hyodo, Phys. Rev. C, 94: 065207 (2016) doi: 10.1103/PhysRevC.94.065207
    [11] K. -W. Li, T. Hyodo, and L. -S. Geng, Phys. Rev. C, 98: 065203 (2018) doi: 10.1103/PhysRevC.98.065203
    [12] E. Hiyama, K. Sasaki, T. Miyamoto et al., Phys. Rev. Lett., 124: 092501 (2020) doi: 10.1103/PhysRevLett.124.092501
    [13] R. B. Wiringa, R. A. Smith, and T. L. Ainsworth, Phys. Rev. C, 29: 1207 (1984) doi: 10.1103/PhysRevC.29.1207
    [14] E. Hiyama, Y. Kino, and M. Kamimura, Prog. Theor. Nucl. Phys., 51: 223 (2003) doi: 10.1016/S0146-6410(03)90015-9
    [15] E. Hiyama, PTEP, 2012: 01A204 (2012)
    [16] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, Phys. Rev. C, 99: 044002 (2019) doi: 10.1103/PhysRevC.99.044002
    [17] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, Phys. Rev. C, 99: 044003 (2019) doi: 10.1103/PhysRevC.99.044003
    [18] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, arXiv: 1504.02634
    [19] H. Garcilazo and A. Valcarce, Phys. Rev. C, 93: 064003 (2016) doi: 10.1103/PhysRevC.93.064003
    [20] H. Garcilazo, Phys. Rev. C, 93: 024001 (2016) doi: 10.1103/PhysRevC.93.024001
    [21] I. Filikhin, V. M. Suslov, and B. Vlahovic, Math. Model. Geom., 5: 1 (2017)
    [22] Y. Yamaguchi, Phys. Rev., 95: 1628 (1954) doi: 10.1103/PhysRev.95.1628
    [23] S. B. Carr, I. R. Afnan, and B. F. Gibson, Nucl. Phys. A, 625: 143 (1997) doi: 10.1016/S0375-9474(97)00480-6
    [24] B. C. Pearce and I. R. Afnan, Phys. Rev. C, 30: 2022 (1984) doi: 10.1103/PhysRevC.30.2022
    [25] T. Tamagawa et al., Nucl. Phys. A, 691: 234c (2001) doi: 10.1016/S0375-9474(01)01035-1
    [26] Y. Yamamoto, T. Tamagawa, T. Fukuda et al., Prog. Theor. Phys., 106: 363 (2001) doi: 10.1143/PTP.106.363
    [27] N. Hoshizaki, Phys. Rev. C, 45: R1424 (1992) doi: 10.1103/PhysRevC.45.R1424
    [28] A. Gal and H. Garcilazo, Nucl. Phys. A, 928: 73 (2014) doi: 10.1016/j.nuclphysa.2014.02.019
  • [1] H. Garcilazo and A. Valcarce, Phys. Rev. Lett., 110: 012503 (2013); A. Gal, Phys. Rev. Lett., 110: 179201 (2013); H. Garcilazo and A. Valcarce, Phys. Rev. Lett., 110: 179202 (2013)
    [2] H. Garcilazo, A. Valcarce, and T. Fernández-Caramés, J. Phys. G, 41: 095103 (2014) doi: 10.1088/0954-3899/41/9/095103
    [3] H. Garcilazo, A. Valcarce, and T. Fernández-Caramés, J. Phys. G, 42: 025103 (2015) doi: 10.1088/0954-3899/42/2/025103
    [4] R. L. Jaffe, Phys. Rev. Lett., 38: 195 (1977) doi: 10.1103/PhysRevLett.38.195
    [5] H. Takahashi et al., Phys. Rev. Lett., 87: 212502 (2001) doi: 10.1103/PhysRevLett.87.212502
    [6] A. Francis, J. R. Green, P. M. Junnarkar et al., Phys. Rev. D, 99: 074505 (2019) doi: 10.1103/PhysRevD.99.074505
    [7] A. Gal, E. V. Hungerford, and D. J. Millener, Rev. Mod. Phys., 88: 035004 (2016) doi: 10.1103/RevModPhys.88.035004
    [8] B. H. Kim et al. (Belle Collaboration), Phys. Rev. Lett., 110: 222002 (2013) doi: 10.1103/PhysRevLett.110.222002
    [9] K. Sasaki, S. Aoki, T. Doi et al. (HAL QCD Collaboration), Nucl. Phys. A, 998: 121737 (2020) doi: 10.1016/j.nuclphysa.2020.121737
    [10] Y. Yamaguchi and T. Hyodo, Phys. Rev. C, 94: 065207 (2016) doi: 10.1103/PhysRevC.94.065207
    [11] K. -W. Li, T. Hyodo, and L. -S. Geng, Phys. Rev. C, 98: 065203 (2018) doi: 10.1103/PhysRevC.98.065203
    [12] E. Hiyama, K. Sasaki, T. Miyamoto et al., Phys. Rev. Lett., 124: 092501 (2020) doi: 10.1103/PhysRevLett.124.092501
    [13] R. B. Wiringa, R. A. Smith, and T. L. Ainsworth, Phys. Rev. C, 29: 1207 (1984) doi: 10.1103/PhysRevC.29.1207
    [14] E. Hiyama, Y. Kino, and M. Kamimura, Prog. Theor. Nucl. Phys., 51: 223 (2003) doi: 10.1016/S0146-6410(03)90015-9
    [15] E. Hiyama, PTEP, 2012: 01A204 (2012)
    [16] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, Phys. Rev. C, 99: 044002 (2019) doi: 10.1103/PhysRevC.99.044002
    [17] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, Phys. Rev. C, 99: 044003 (2019) doi: 10.1103/PhysRevC.99.044003
    [18] M. M. Nagels, Th. A. Rijken, and Y. Yamamoto, arXiv: 1504.02634
    [19] H. Garcilazo and A. Valcarce, Phys. Rev. C, 93: 064003 (2016) doi: 10.1103/PhysRevC.93.064003
    [20] H. Garcilazo, Phys. Rev. C, 93: 024001 (2016) doi: 10.1103/PhysRevC.93.024001
    [21] I. Filikhin, V. M. Suslov, and B. Vlahovic, Math. Model. Geom., 5: 1 (2017)
    [22] Y. Yamaguchi, Phys. Rev., 95: 1628 (1954) doi: 10.1103/PhysRev.95.1628
    [23] S. B. Carr, I. R. Afnan, and B. F. Gibson, Nucl. Phys. A, 625: 143 (1997) doi: 10.1016/S0375-9474(97)00480-6
    [24] B. C. Pearce and I. R. Afnan, Phys. Rev. C, 30: 2022 (1984) doi: 10.1103/PhysRevC.30.2022
    [25] T. Tamagawa et al., Nucl. Phys. A, 691: 234c (2001) doi: 10.1016/S0375-9474(01)01035-1
    [26] Y. Yamamoto, T. Tamagawa, T. Fukuda et al., Prog. Theor. Phys., 106: 363 (2001) doi: 10.1143/PTP.106.363
    [27] N. Hoshizaki, Phys. Rev. C, 45: R1424 (1992) doi: 10.1103/PhysRevC.45.R1424
    [28] A. Gal and H. Garcilazo, Nucl. Phys. A, 928: 73 (2014) doi: 10.1016/j.nuclphysa.2014.02.019
  • 加载中

Cited by

1. Wu, T.-W., Luo, S.-Q., Liu, M.-Z. et al. Tribaryons with lattice QCD and one-boson exchange potentials[J]. Physical Review D, 2023, 108(9): L091506. doi: 10.1103/PhysRevD.108.L091506
2. Meher, G., Raha, U. Investigation of Ξ-nn (S= - 2) hypernucleus in low-energy pionless halo effective theory[J]. European Physical Journal: Special Topics, 2021, 230(2): 579-601. doi: 10.1140/epjs/s11734-021-00007-1
3. Garcilazo, H., Valcarce, A. Doubly charmed multibaryon systems[J]. European Physical Journal C, 2020, 80(8): 720. doi: 10.1140/epjc/s10052-020-8320-0

Figures(3) / Tables(2)

Get Citation
H. Garcilazo and A. Valcarce. ΛΛNΞNN S wave resonance[J]. Chinese Physics C. doi: 10.1088/1674-1137/abab8c
H. Garcilazo and A. Valcarce. ΛΛNΞNN S wave resonance[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abab8c shu
Milestone
Received: 2020-06-11
Article Metric

Article Views(1630)
PDF Downloads(42)
Cited by(3)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Email This Article

Title:
Email:

ΛΛNΞNN S wave resonance

    Corresponding author: H. Garcilazo, humberto@esfm.ipn.mx
    Corresponding author: A. Valcarce, valcarce@usal.es
  • 1. Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Edificio 9, 07738 México D.F., Mexico
  • 2. Departamento de Física Fundamental, Universidad de Salamanca, 37008 Salamanca, Spain

Abstract: We use an existing model of the ΛΛNΞNN three-body system based on two-body separable interactions to study the (I,JP)=(1/2,1/2+) three-body channel. For the ΛΛ, ΞN, and ΛΛΞN amplitudes, we have constructed separable potentials based on the most recent results of the HAL QCD Collaboration. They are characterized by the existence of a resonance just below or above the ΞN threshold in the H-dibaryon channel, (i,jp)=(0,0+). A three-body resonance appears 2.3 MeV above the Ξd threshold. We show that if the ΛΛΞN H-dibaryon channel is not considered, the ΛΛNΞNN S wave resonance disappears. Thus, the possible existence of a ΛΛNΞNN resonance would be sensitive to the ΛΛΞN interaction. The existence or nonexistence of this resonance could be evidenced by measuring, for example, the Ξd cross section.

    HTML

  • The ΛΛΞN system in a pure S wave configuration has quantum numbers (i,jp)=(0,0+), such that by adding one more nucleon, the ΛΛNΞNN system necessarily has quantum numbers (I,JP)=(1/2,1/2+). In a series of works based on the chiral constituent quark model [1-3], this system was studied under the assumption that the H dibaryon [4] has the lower limit mass determined by the E373 experiment at KEK [5] from the observation of a 6ΛΛHe double hypernucleus. Despite significant experimental and theoretical efforts, the existence of the H dibaryon remains inconclusive, see Ref. [6] for a recent update. The experimental evidence disfavors large binding energies [7], as predicted in Ref. [4], and the high statistics study of Υ decays at Belle [8] found no indication of an H dibaryon with a mass near the ΛΛ threshold. Recently, the HAL QCD Collaboration [9] has published a Nf=2+1 study of coupled channel (ΛΛ and ΞN) baryon-baryon interactions with near-physical quark masses, namely mπ=146 MeV, concluding that the H dibaryon could be a ΛΛ resonance just below or above the ΞN threshold. Similar results were obtained in a low-energy effective field theory study of the H dibaryon in ΛΛ scattering [10].

    The HAL QCD results are being used as input for the study of strangeness - 2 baryon-baryon interactions, as recently done in relativistic chiral effective field theory studies [11]. The HAL QCD ΞN interactions have also recently been used to study the possible existence of ΞNN bound states, see Ref. [12], with negative results for the (I,JP)=(1/2,1/2+) channel. For the NN interaction, they used the AV8 potential [13]. As the coupling between ΛΛ and ΞN was found to be weak in Ref. [9], they used an effective single-channel ΞN potential, in which the coupling to ΛΛ in 11S0 was renormalized into a single Gaussian form chosen to reproduce the ΞN phase shift obtained with channel coupling. The three-body ΞNN problem is solved in the real axis by means of a variational method with Gaussian bases, the Gaussian Expansion Method [14,15]. The full coupling between the ΞNN and ΛΛN channels was not explicitly considered. A similar calculation based on the Nijmegen ESC08c potentials [16-18] was presented in Ref. [19], also with negative results for the (I,JP)=(1/2,1/2+) channel, see Fig. 2(a) of Ref. [19].

    Unlike the calculation in Ref. [12], we developed a model in Ref. [20] of the ΛΛNΞNN three-body system, which allowed us to look for possible three-body resonances. Using separable two-body interactions fitted to the low-energy data of the Nijmegen S wave baryon-baryon amplitudes [16-18], we found a resonance just below the Ξd threshold with a very small width of only 0.09 MeV. (It is worth to note that the results for the ΞNN system with maximal isospin have been independently reproduced within the integral Faddeev equation formalism [21] in agreement with high accuracy.) Qualitatively similar results have been obtained and are described in Ref. [12]; although as stressed in this manuscript, they are numerically different due to a different NN potential and a different treatment of the ESC08c Nijmegen S wave baryon-baryon interactions. Such dependencies on the models and parametrizations of the two-body interactions result in this three-body system being ideally suited for testing different models for two-body interactions.

    However, contrary to the recent results of the HAL QCD Collaboration, the Nijmegen baryon-baryon interactions gave no indication of either a bound state or a resonance in the ΛΛΞN (0,0+) two-body channel, the H-dibaryon channel. It is thus interesting to see if the existence of a resonance just below or above the ΞN threshold, as it has been found by the HAL QCD Collaboration [9] and low-energy effective field theory studies [10], may affect the position of the three-body S wave (1/2,1/2+) ΛΛNΞNN resonance found in Ref. [20]. For this purpose, we have now constructed separable potential models of the ΛΛ, ΞN, and ΛΛΞN amplitudes reproducing the behavior of the HAL QCD collaboration results [9]. We have also performed a full-fledged coupled-channel study of the ΛΛNΞNN three-body system.

    We use rank-one separable potentials for all uncoupled two-body channels, that is, for all channels except the ΛΛΞN (0,0+) interaction. They are as follows,

    Vρi=gρiλgρi,

    (1)

    such that the two-body t-matrices are

    tρi=gρiτρigρi,

    (2)

    with

    τρi=λ1λgρi|G0(i)|gρi,

    (3)

    where G0(i)=1/(EKi+iϵ) and Ki is the kinetic energy operator of channel i. We use Yamaguchi form factors [22] for the separable potentials of Eq. (1), i.e.,

    gρi(p)=1α2+p2.

    (4)

    For the case of the coupled (0,0+) ΛΛΞN channel, we use a rank-two separable potential of the form [23]

    Vρσij=gρiλijgσj,

    (5)

    such that

    tρσij=gρ1τρσijgσj,

    (6)

    with

    τΛΛΛΛ11=λ213GΞNλ11(1λ33GΞN)λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),τΞNΞN33=λ213GΛΛλ33(1λ11GΛΛ)λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),τΛΛΞN13=τΞNΛΛ31=λ13λ213GΛΛGΞN(1λ11GΛΛ)(1λ33GΞN),

    (7)

    and

    GΛΛ=gΛΛ1|G0(1)|gΛΛ1,GΞN=gΞN3|G0(3)|gΞN3.

    (8)

    In this case, we also use Yamaguchi-type form factors as

    gΛΛ1(p)=1α2+p2,gΞN3(p)=1β2+p2.

    (9)

    The parameters of the ΛΛΞN model based on the latest HAL QCD potentials are given in Table 1. In Figs. 1(a), (b), and (c) we show the predictions for the ΛΛ and ΞN phase shifts as well as the inelasticity, which are rather similar to those of model t/a=12 of the HAL QCD Collaboration presented in Fig. 4 of Ref. [9]. The corresponding parameters of the uncoupled ΞN models are given in Table 2. Note that our results have been obtained by taking the nucleon mass to be the average of the proton and neutron masses and the Ξ mass as the average of the Ξ0 and Ξ masses. Thus, the ΞN and ΞNN thresholds are 25.6 MeV above the ΛΛ and ΛΛN thresholds, respectively. However, this threshold is 32 MeV for the HAL QCD results [9], as they use the values obtained from their lattice QCD study for the baryon masses. Therefore, to compare our phase shifts with those of Ref. [9], one should keep in mind that the energy scale of Ref. [9] corresponds to those of Fig. 1 multiplied by 1.25. The models of the NN and ΛN subsystems are the same as in Ref. [20].

    α β λ11 λ33 λ13
    1.3465 1.1460 0.1390 0.3171 0.0977

    Table 1.  Parameters α and β (in fm1), λ11, λ33, and λ13 (in fm2) of the separable-potential model of the coupled (i,jp)=(0,0+) ΛΛΞN two-body system.

    Figure 1.  (a) ΛΛ scattering phase shifts, (b) ΛΛ inelasticity, and (c) NΞ scattering phase shifts in the (i,jp)=(0,0+) channel.

    Channel α λ
    (0,1+) 1.41 -0.117
    (1,0+) 7.333 22.97
    (1,1+) 0.803 -0.016

    Table 2.  Parameters α (in fm1) and λ (in fm2) of the separable-potential model of the uncoupled (i,jp) ΞN two-body channels.

    The coupled ΛΛNΞNN three-body system presents the special characteristic that each three-body component consists of two identical fermions and a third one that is different. The homogeneous integral equations of this system appropriate for the search of bound and resonant states were derived in Ref. [3] using a graphical method. Using the new separable models presented in Tables 1 and 2 based on the HAL QCD interactions, we do not find any bound state below the Ξd threshold, which is in agreement with the results of Ref. [12]. Therefore, we investigate the possible existence of a resonance above the Ξd threshold by calculating the Ξd scattering amplitude.

    We adopt the same convention as in Refs. [3,20], i.e., particles 2 and 3 are identical and particle 1 is different in each three-body component. After the reduction for identical particles, the inhomogeneous integral equations appropriate for Ξd elastic scattering take the following form

    1|T1|ϕ0=21|tΛΛ1|11|3G0(3)3|T3|ϕ0+1|tΛΛNΞ13|33|1G0(1)1|U1|ϕ01|tΛΛNΞ13|32|3G0(3)3|U3|ϕ0,3|T3|ϕ0=3|tNΛ3|32|3G0(3)3|T3|ϕ0+3|tNΛ3|33|1G0(1)1|T1|ϕ0,

    1|U1|ϕ0=21|tNN1|11|3G0(3)3|U3|ϕ0,3|U3|ϕ0=23|tNΞ3|ϕ03|tNΞ3|32|3G0(3)3|U3|ϕ0+3|tNΞ3|33|1G0(1)1|U1|ϕ0+23|tNΞΛΛ31|11|3G0(3)3|T3|ϕ0,

    (10)

    where |ϕ0 is the initial state consisting of the deuteron wave function times a Ξ plane wave.

    Substituting Eqs. (2) and (6) into the integral Eq. (10) and introducing the transformations i|Ti|ϕ0=i|gαiii|Xi|ϕ0 and i|Ui|ϕ0=i|gβiii|Yi|ϕ0, one obtains the inhomogeneous one-dimensional integral equations

    1|X1|ϕ0=2τΛΛ1gΛΛ1|11|3G0(3)3|gNΛ33|X3|ϕ0+τΛΛNΞ13gNΞ3|33|1G0(1)1|gNN11|Y1|ϕ0τΛΛNΞ13gNΞ3|32|3G0(3)3|gNΞ33|Y3|ϕ0,3|X3|ϕ0=τNΛ3gNΛ3|32|3G0(3)3|gNΛ33|X3|ϕ0+τNΛ3gNΛ3|33|1G0(1)1|gΛΛ11|X1|ϕ0,1|Y1|ϕ0=2τNN1gNN1|11|3G0(3)3|gNΞ33|Y3|ϕ0,

    3|Y3|ϕ0=2τNΞ3gNΞ3|ϕ0τNΞ3gNΞ3|32|3G0(3)3|gNΞ33|Y3|ϕ0+τNΞ3gNΞ3|33|1G0(1)1|gNN1|Y1|ϕ0+2τNΞΛΛ31gΛΛ1|11|3G0(3)3|gNΛ33|X3|ϕ0.

    (11)

    If one neglects the inhomogeneous terms in Eqs. (10) and (11), they become identical to Eqs. (14) and (15) of Ref. [20]. The Ξd scattering amplitude normalized as in the Argand diagram is given by

    F=πq0νϕ0|U3|ϕ0,

    (12)

    where q0 and ν, respectively, are the Ξd on-shell momentum and the reduced mass. We solve the integral Eqs. (11) using the standard method [24], where the momentum variables are rotated into the complex plane as qiqieiϕ and the results are checked to be independent of the rotation angle ϕ. If the resonance lies below the Ξd threshold, as it was the case in Ref. [20], the contour rotation method allows to simultaneously consider both the momentum variables and the energy variable as complex, such that one can determine the position of the pole in the complex plane. However, if the resonance lies above the Ξd threshold, as in the present case, the contour rotation method works only if one takes the momentum variables as complex but leaves the energy variable real, such that one cannot determine the position of the pole in the complex plane.

    We show, in Fig. 2, the Argand diagram of the Ξd system between 0 and 10 MeV above the Ξd threshold, where one sees the typical counterclockwise behavior of a resonant amplitude. If one neglects the (i,jp)=(0,0+) channel, the counterclockwise behavior disappears, which shows that the H-dibaryon channel (i,jp)=(0,0+) is basic for the existence of the three-body ΛΛNΞNN S wave resonance.

    Figure 2.  Argand diagram of the Ξd system between 0 and 10 MeV above the Ξd threshold. Several relevant energies, in MeV, are indicated.

    As already mentioned in the introduction, the HAL QCD ΞN interactions have recently been used to study the possible existence of ΞNN bound states in Ref. [12] with negative results for the (I,JP)=(1/2,1/2+) channel, which is in agreement with our findings despite using a different NN interaction and method.

    We have finally evaluated the Ξd elastic cross-section as a function of energy, where we have included not only the (I,JP)=(1/2,1/2+) Ξd amplitude but also the (1/2,3/2+) amplitude, which is very small. The result is shown in Fig. 3. As one can see, the resonance shows up as a change of slope of the cross section at an energy around 2.3 MeV, i.e., close to the Ξd breakup threshold S=2mN+mΞ. The bump in the cross section would become larger for a stronger (i,jp)=(0,0+) transition potential, and as noted above, it would disappear if the (i,jp)=(0,0+) channel is not considered or the two-body resonance in the H-dibaryon channel does not exist. The Ξd cross section would allow for discrimination among the different models for the strangeness 2 two-baryon interactions. It could be studied through the quasifree Ξ production in the (K,K+) reaction on a deuteron target [25,26].

    Figure 3.  Ξd elastic cross section.

    Let us finally note that if one drops the coupling to the ΛΛN channel, Fig. 2 changes by about 10 % while maintaining its shape and rotating slightly to the right; similarly, in Fig. 3 the cross-section at E=2.3 MeV changes from 55 mb to 58 mb.

    It is interesting to compare this resonance with the nucleon-nucleon 1D2 Hoshizaki resonance [27], which has a mass close to S=mN+mΔ, since it arises due to the process, NNπNN, which is driven by the pion-nucleon Δ resonance [28]. The resonance we are studying here is driven by the ΛΛΞN H-dibaryon resonance, which appears either just below or above the NΞ threshold [9], such that it has a mass, mH=mN+mΞ. Following the comparison with the Hoshizaki state, one expects that the ΛΛNΞNN resonance will have a mass close to S=mN+mH=2mN+mΞ, which is precisely the Ξd threshold and in agreement with Figs. 2 and 3.

    In brief, we have shown that the possible existence of a ΛΛNΞNN resonance would be highly sensitive to the ΛΛΞN interaction. In particular, by using a separable potential based on the most recent results of the HAL QCD Collaboration, characterized by the existence of a resonance just below or above the ΞN threshold in the H-dibaryon channel, (i,jp)=(0,0+), a three-body resonance appears at 2.3 MeV above the Ξd threshold. A theoretical and experimental effort to constrain the ΛΛΞN interaction is the basic requirement for progress in our investigation of the strangeness 2 sector.

Reference (28)

目录

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return