Strangeness S=-2 baryon-baryon interactions using chiral effective field theory

# Strangeness S=−2 baryon-baryon interactions using chiral effective field theory

H. Polinder J. Haidenbauer U.-G. Meißner Institut für Kernphysik (Theorie), Forschungszentrum Jülich, D-52425 Jülich, Germany Helmholtz-Institut für Strahlen- und Kernphysik (Theorie), Universität Bonn, D-53115 Bonn, Germany
###### Abstract

We derive the leading order strangeness baryon-baryon interactions in chiral effective field theory. The potential consists of contact terms without derivatives and of one-pseudoscalar-meson exchanges. The contact terms and the couplings of the pseudoscalar mesons to the baryons are related via symmetry to the hyperon-nucleon channels. We show that the chiral effective field theory predictions with natural values for the low-energy constants agree with the experimental information in the sector.

###### keywords:
Hyperon-hyperon interaction, Hyperon-nucleon interaction, Effective field theory, Chiral Lagrangian
###### Pacs:
13.75.Ev, 12.39.Fe, 21.30.-x, 21.80.+a

HISKP-TH-07/15, FZJ-IKP-TH-2007-18

, Corresponding author. ,

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As of today, theoretical investigations of the baryon-baryon interaction in the strangeness sector were performed within the meson-exchange picture [1, 2, 3, 4] as well as in the constituent quark model [5, 6, 7, 8, 9, 10, 11, 12]. In both approaches the assumption of symmetry is an essential prerequisite. It allows to connect the doubly strange cascade-nucleon () and hyperon-hyperon () interactions () to the hyperon-nucleon () and nucleon-nucleon () interactions, i.e. to systems where a wealth of experimental information is available, which can then be used to constrain the parameters inherent to those approaches.

Indeed, the experimental knowledge on the and interactions themselves is quite poor. Until the beginning of this century the only information available came from doubly strange hypernuclei and, moreover, only three candidates for such hypernuclei were reported [13, 14, 15]. The binding energies derived from these events indicated a strongly attractive interaction. However, more recently a new candidate for with a much lower binding energy was identified [16], the so-called Nagara event, suggesting that the interaction should be only moderately attractive. This conjecture is also in line with evidence provided by the latest searches for the dibaryon, a bound state in the sector proposed by Jaffe back in 1977 [17], whose existence is now considered to be practically ruled out [18]. (See also the theoretical works [19, 20] concerning the attraction in the system.) Very recently doubly strange baryon-baryon scattering data at lower energies, below GeV, were deduced for the first time [21, 22]. An upper limit of mb at confidence level was provided for elastic scattering, and for the cross section at MeV a value of mb was reported [22].

Over the last decade a new powerful tool for understanding hadronic interactions has emerged, namely chiral effective field theory (EFT). This approach, which was pioneered by Weinberg in the early nineties, incorporates explicitly the scales and symmetries of Quantum Chromodynamics. An important advantage of EFT is that there is an underlying power counting that allows to improve calculations systematically by going to higher orders in a perturbative expansion and, at the same time, it allows to estimate theoretical uncertainties. In addition, it is possible to derive two- and corresponding three-baryon forces in a consistent way. The concepts of chiral EFT have been applied in the last decade to the interaction and to the physics of light nuclei, resulting in a high-precision description of the experimental data, see e.g. Refs. [23, 24] and references therein. Recently we utilized the chiral EFT framework for investigating the interaction. In particular, we showed that the leading order (LO) chiral EFT successfully describes the available scattering data [25]. Also the binding energies of the light hypernuclei are predicted well within chiral EFT [26, 27].

The and interactions have not been studied using chiral EFT so far. In this letter we report on the first chiral EFT investigation of the sector, starting with a LO calculation. For this purpose we extend the recently constructed LO chiral EFT potential of the interaction [25]. We employ relations to connect the doubly strange with the singly strange channels and we confront the LO chiral EFT predictions with the (poor) experimental knowledge.

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We construct the chiral effective potentials for the sector at LO using the Weinberg power counting similar to the case considered in [25]. The LO potential consists of four-baryon contact terms without derivatives and of one-pseudoscalar-meson exchanges. The LO invariant contact terms for the octet baryon-baryon interactions that are Hermitian and invariant under Lorentz transformations were discussed in [25, 27] and we refer the reader to these works for details. The pertinent Lagrangians read

 L1 = C1i⟨¯Ba¯Bb(\mathchar0\relaxiB)b(\mathchar0\relaxiB)a⟩ ,L2=C2i⟨¯Ba(\mathchar0\relaxiB)a¯Bb(\mathchar0\relaxiB)b⟩ , L3 = C3i⟨¯Ba(\mathchar0\relaxiB)a⟩⟨¯Bb(\mathchar0\relaxiB)b⟩ . (1)

Here, the labels and are the Dirac indices of the particles, the label denotes the five elements of the Clifford algebra, is the usual irreducible octet representation of given by

 B = ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝\mathchar6\relax0√2+\mathchar3\relax√6\mathchar6\relax+p\mathchar6\relax−−\mathchar6\relax0√2+\mathchar3\relax√6n−\mathchar4\relax−\mathchar4\relax0−2\mathchar3\relax√6⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ , (2)

and the brackets denote taking the trace in the three-dimensional flavor space. The Clifford algebra elements are here actually diagonal -matrices.

The and partial wave potentials derived from the above Lagrangians are given in Table 1 for the singlet S-waves and in Table 2 for the triplet S-waves. The coefficients and are linear combinations of the low-energy coefficients ’s in Eq. (1) and refer to the central and spin-spin parts of the potential, see e.g. Ref. [25]. The and potentials are listed also in Tables 1 and 2 for completeness. Using the Clebsch-Gordan coefficients one can express the baryon-baryon potentials in terms of the irreducible representations, see e.g. [28] and also [29]. The last columns of Tables 1 and 2 show the content of the various potentials.

Contrary to the Nijmegen meson-exchange models [1, 2] or the constituent quark model [12], say, the chiral EFT for the channels is not completely fixed by imposing symmetry. The symmetry connects only five of the six LO contact terms to those appearing in the and interactions. Thus, one contact term remains undetermined. In the present study those five LO contact terms are taken over from Ref. [25], where they were fixed by a fit to the data. The additional contact term, which occurs in the singlet isospin-zero doubly strange and channels, needs to be determined, in principle, from empirical information on the baryon-baryon interaction. For the additional contact term we have chosen the projection on the singlet S-wave: . All other LO and contact potentials are then fixed by symmetry.

The lowest order invariant pseudoscalar-meson–baryon interaction Lagrangian with the appropriate symmetries was discussed in [25]. In the isospin basis it reads

 L = −fNNπ¯Nγμγ5\boldmathτN⋅∂μ\boldmathπ+if\mathchar6\relax\mathchar6\relaxπ¯\boldmath\mathchar6\relaxγμγ5×\boldmath\mathchar6\relax⋅∂μ\boldmathπ (3) −f\mathchar3\relax\mathchar6\relaxπ[¯\mathchar3\relaxγμγ5\boldmath\mathchar6\relax+¯% \boldmath\mathchar6\relaxγμγ5\mathchar3\relax]⋅∂μ\boldmathπ−f\mathchar4\relax\mathchar4\relaxπ¯\mathchar4\relaxγμγ5\boldmathτ\mathchar4\relax⋅∂μ\boldmathπ −f\mathchar3\relaxNK[¯Nγμγ5\mathchar3\relax∂μK+¯\mathchar3\relaxγμγ5N∂μK†] −f\mathchar4\relax\mathchar3\relaxK[¯\mathchar4\relaxγμγ5\mathchar3\relax∂μKc+¯\mathchar3\relaxγμγ5\mathchar4\relax∂μK†c] −f\mathchar6\relaxNK[¯\boldmath\mathchar6\relax⋅γμγ5∂μK†\boldmathτN+¯Nγμγ5\boldmathτ∂μK⋅\boldmath\mathchar6\relax] −f\mathchar4\relax\mathchar6\relaxK[¯% \boldmath\mathchar6\relax⋅γμγ5∂μK†c\boldmathτ\mathchar4\relax+¯\mathchar4\relaxγμγ5\boldmathτ∂μKc⋅\boldmath\mathchar6\relax]−fNNη8¯Nγμγ5N∂μη −f\mathchar3\relax\mathchar3\relaxη8¯\mathchar3\relaxγμγ5\mathchar3\relax∂μη−f\mathchar6\relax\mathchar6\relaxη8¯\boldmath\mathchar6\relax⋅γμγ5\boldmath\mathchar6\relax∂μη−f\mathchar4\relax\mathchar4\relaxη8¯\mathchar4\relaxγμγ5\mathchar4\relax∂μη .

The interaction Lagrangian in Eq. (3) is invariant under transformations if the various coupling constants fulfill specific relations which can be expressed in terms of the coupling constant and the -ratio as [28],

 fNNπ=f,fNNη8=1√3(4α−1)f,f\mathchar3\relaxNK=−1√3(1+2α)f,f\mathchar4\relax\mathchar4\relaxπ=−(1−2α)f,f\mathchar4\relax\mathchar4\relaxη8=−1√3(1+2α)f,f\mathchar4\relax\mathchar3\relaxK=1√3(4α−1)f,f\mathchar3\relax\mathchar6\relaxπ=2√3(1−α)f,f\mathchar6\relax\mathchar6\relaxη8=2√3(1−α)f,f\mathchar6\relaxNK=(1−2α)f,f\mathchar6\relax\mathchar6\relaxπ=2αf,f\mathchar3\relax\mathchar3\relaxη8=−2√3(1−α)f,f\mathchar4\relax\mathchar6\relaxK=−f. (4)

Here , is the axial-vector strength, , which is measured in neutron –decay and is the weak pion decay constant, MeV. For the -ratio we adopt here the SU(6) value () which was already used in our study of the system [25]. The spin-space part of the one-pseudoscalar-meson-exchange potential resulting from the interaction Lagrangian Eq. (3) is in leading order similar to the static one-pion-exchange potential in [31],

 VB1B2→B′1B′2 = −fB1B′1PfB2B′2P(\boldmathσ1⋅k)(\boldmathσ2⋅k)k2+m2P , (5)

where , are the appropriate coupling constants as given in Eq. (4) and is the actual mass of the exchanged pseudoscalar meson. With regard to the meson we identified its coupling with the octet value, i.e. the one for . We defined the transferred and average momentum, and , in terms of the final and initial center-of-mass (c.m.) momenta of the baryons, and , as and . To find the complete LO one-pseudoscalar-meson-exchange potential one needs to multiply the potential in Eq. (5) with the isospin factors given in Table 3.

We want to remark that for the and interactions couplings between channels with non-identical and with identical particles occur which requires special attention [30]. We follow the treatment of the flavor-exchange potentials as done by the Nijmegen group. Then the proper anti-symmetrization of the states is achieved by multiplying specific transitions with factors that are included in Table 3, see Refs. [1, 2]. In Table 3 is the flavor-exchange operator having the values for even-L singlet and odd-L triplet partial waves (antisymmetric in spin-space), and for odd-L singlet and even-L triplet partial waves (symmetric in spin-space). We note that for , for example, -exchange contributes only to spin-space antisymmetric i.e. flavor symmetric partial waves, i.e. ,, etc.

The symmetry of the one-pseudoscalar-meson exchanges is broken by the masses of the pseudoscalar mesons. This is taken into account explicitly in Eq. (5) by taking the appropriate values for . In case one would consider identical pseudoscalar-meson masses, the corresponding potential obeys the relations as shown in the last column of Tables 1 and 2. This can easily be checked by assuming equal masses and adding the contributions of all one-pseudoscalar-meson exchanges for each channel – using Eqs. (4), (5) and Table 3 – and compare the result with the last column of Tables 1 and 2.

Finally, for completeness we briefly comment on the used scattering equation. The calculations are done in momentum space. We solve the coupled channels (nonrelativistic) Lippmann-Schwinger (LS) equation,

 Tν′′ν′,Jρ′′ρ′(p′′,p′;√s) = Vν′′ν′,Jρ′′ρ′(p′′,p′)+∑ρ,ν∫∞0dpp2(2π)3Vν′′ν,Jρ′′ρ(p′′,p)2μνq2ν−p2+iηTνν′,Jρρ′(p,p′;√s) . (6)

The label indicates the particle channels and the label the partial wave. is the pertinent reduced mass. The on-shell momentum in the intermediate state, , is defined by . Relativistic kinematics is used for relating the laboratory energy of the hyperons to the c.m. momentum. Suppressing the particle channels label, the partial wave projected potentials are given in [25]. The LS equation for the and systems is solved in the particle basis, in order to incorporate the correct physical thresholds. The potential in the LS equation is cut off with the regulator function ,

 f\mathchar3\relax(p′,p)=e−(p′4+p4)/\mathchar3\relax4 , (7)

in order to remove high-energy components of the baryon and pseudoscalar meson fields. The cross sections are calculated using the (LSJ basis) partial wave amplitudes, for details we refer to [32, 33].

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The LO chiral EFT interaction for the baryon-baryon sector depends, in principle, on six LO contact terms. Five of those contact terms enter also in the interaction. Thus, for those we can take over the values which were fixed in our study of the sector [25]. Indeed only the and channels that contain the isospin-zero interaction depend on the sixth contact term, which is not fixed yet. The interaction in the other channels are genuine predictions that follow from the results of Ref. [25] and symmetry.

The experimental knowledge on the doubly strange baryon-baryon interaction is quite poor, but since the observation of the Nagara event [16] it is generally accepted that the interaction is only moderately attractive. Also, very recently an inelastic cross section has been deduced at a laboratory momentum of 500 MeV and, in addition, an upper limit with 90% confidence level has been provided for the elastic cross section for laboratory momenta in the range of 200-800 MeV, see Ref. [22].

Since one contact terms is not yet fixed, we investigate whether those mentioned experimental scattering cross sections of the baryon-baryon interaction in the sector constrain this additional contact term, . For this purpose we evaluate the relevant doubly strange baryon-baryon cross sections and study their dependence on this additional contact term. This is done for a fixed cut-off value, namely = 600 MeV.

We restricted the variations of to a range less then twice the natural value, which is equal to for this partial-wave projected contact term [24]. Since the interaction is expected to be only moderately attractive, as mentioned, we considered only such variations of where the absolute value of the resulting scattering length was less than fm. For the same reason, we excluded regions which led to bound states or near-threshold resonances in the system, which are very unlikely to exist in view of the available experimental information. Based on these considerations the additional contact term was varied in the range 2.0,…,-0.05 times the natural value. The corresponding results are depicted by the bands in Fig. 1. (Note that we show only the purely hadronic cross sections. The Coulomb interaction is not taken into account in the present study.) From Figs. 1 (b) and (c) we conclude that the chiral EFT results are consistent with the recently deduced scattering cross sections in the elastic and inelastic channels. But it is obvious that these data do not allow to constrain the value of more quantitatively. Note that in Figs. 1 (a) and (c) one clearly sees the opening of the inelastic and channels, respectively.

The range of the scattering length corresponding to the variations in is fm. It is interesting to compare those values with the ones of the Nijmegen ESC04 model [2] and the model of Fujiwara et al. [12]. These are the only baryon-baryon interactions for which a direct comparison of the binding energy of the Nagara event, i.e. of the hypernucleus, with corresponding predictions based on those models is available. Specifically, Fujiwara et al. used their and interactions in a Faddeev calculation where they considered the nucleus as an -particle and two baryons having strangeness . The two- separation energy, defined as , obtained for their model is close to the experimental number of MeV [16]. The recent Nijmegen ESC04 model also reproduces the two- separation energy correctly [2].

The scattering length given for the Nijmegen ESC04d interaction is fm [34]. The value for the interaction based on the constituent quark model of Fujiwara et al. is fm, see [12]. The range of values predicted by the chiral EFT is close to the one of the Nijmegen models. Although the Nijmegen and the quark model have different scattering lengths, both give a good reproduction of the experimentally observed two- separation energy, as mentioned. Obviously, from the scattering length alone one can not draw any conclusions on the magnitude of the two- separation energy. Thus, the only reliable way to determine the two- separation energy corresponding to the chiral EFT interaction consists in a concrete calculation of doubly-strange hypernuclei. This has not been done so far. Clearly, such a calculation might help to further constrain the size of .

In order to study the cut-off dependence of the chiral EFT predictions, we first perform a reference calculation with 0 and with the cut-off value of =600 MeV. In subsequent calculations for other cut-off values we then vary in such a way that the scattering length remains practically unchanged. We considered cut-off values in the range of 550,…,700 MeV like we did in Ref. [25]. This range is also similar to the one considered in the case, see, e.g. Refs. [35, 36]. Results for the , , and scattering lengths are listed in Table 4 together with the values for the additional contact term.

Cross sections for those reactions and some more channels are presented in Fig. 2.