Predictions for the strangeness S=-3 and -4 baryon-baryon interactions in chiral effective field theory

# Predictions for the strangeness S=−3 and −4 baryon-baryon interactions in chiral effective field theory

J. Haidenbauer U.-G. Meißner Institut für Kernphysik, Jülich Center for Hadron Physics and Institute for Advanced Simulation
Forschungszentrum Jülich, D-52425 Jülich, Germany
Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics
Universität Bonn, D-53115 Bonn, Germany
###### Abstract

The leading order strangeness and baryon-baryon interactions are analyzed within chiral effective field theory. The chiral potentials consist of contact terms without derivatives and of one-pseudoscalar-meson exchanges. Assuming flavor symmetry those contact terms and the couplings of the pseudoscalar mesons to the baryons are related to the corresponding quantities of the hyperon-nucleon channels. Specifically, the values of the pertinent five low-energy constants related to the contact terms are already fixed from our preceeding study of the and systems and thus genuine predictions for the , , and interactions can be made. Strong attraction is found in some of the and channels, suggesting the possible existence of bound states.

###### 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-09/24, FZJ-IKP-TH-2009-21

, Corresponding author.

## 1 Introduction

The study of baryon-baryon systems involving strangeness has the potential of considerably deepening our understanding of strong interaction physics. Unfortunately, the experimental information on such systems is rather limited. Though there is at least a fair amount of concrete data on the and systems, the empirical information on the strangeness sector is only of qualitative nature. It consists of limits for the and reaction cross sections [1] and in the binding energy of [2], where the latter implies that the interaction can be only moderately attractive. Virtually nothing is known about the baryon-baryon interaction in the and systems. Indeed, even theoretical investigations on those systems are rather scarce [3, 4]. In 1999, the Nijmegen group presented an extension of their meson-exchange hyperon-nucleon (; ) potential [5] to interaction channels with , , and [3] invoking flavor symmetry arguments. In the actual model calculation, symmetry is broken by using the physical values of the involved baryon and meson masses and, in addition, some breaking of the baryon-baryon-meson coupling constants is allowed. The interactions predicted in this way for the and sectors turned out to be fairly strong and attractive and even suggests the existence of bound states in the and channels [3]. Such strongly attractive and baryon-baryon interactions have further interesting implications. For example, Filikin and Gal argued that then fairly light hypernuclei with should exist [6, 7]. In particular, could then already mark the onset of nuclear stability for hyperons. A strong interaction could induce a first order phase transition from neutron matter to hyperon-rich matter [8] so that, besides ordinary white dwarfs and neutron stars, a new class of compact stars, namely hyperon stars, could exist.

The baryon-baryon interaction of Fujiwara and collaborators [4] is derived in the constituent quark model. It contains the color Fermi-Breit interaction and effective-meson exchanges (pseudoscalar, scalar, vector mesons) between quarks, among others. Also here symmetry plays a key role in extending the model from the and interaction (where free parameters are fixed) to the and channels. And also here an explicit flavor-symmetry breaking is introduced, namely in the Fermi-Breit interaction. But in this approach it was found that the baryon-baryon interaction becomes step by step less attractive when going from strangeness to . In particular, no di-baryon bound states are supported, except for the deuteron.

In this context we present here a first study of the baryon-baryon interaction in the and sectors in the framework of chiral effective field theory (EFT). Outlined by Weinberg in the early 1990s [9, 10] the concepts of chiral EFT have been applied very successfully in the last decade to the (and ) interaction and to the physics of light nuclei, resulting in a high-precision description of the experimental data, see e.g. Refs. [11, 12, 13] and references therein. Recently, this framework was utilized by us also for investigating the interactions as well as the baryon-baryon interaction in the sector [14, 15]. In particular, we showed that the leading order (LO) chiral EFT successfully describes the available scattering data [14]. Also the binding energies of the light hypernuclei are predicted well within chiral EFT [16, 17].

The extension of our investigations of the baryon-baryon interaction within chiral EFT to the strangeness and sectors is straightforward. The LO potential consists of four-baryon contact terms without derivatives and of one-pseudoscalar-meson exchanges [14, 17]. Under the assumption of symmetry the interaction for the and baryon-baryon sector depends on the same (five) contact terms that enter also in the interaction. Thus, we can take over the values which were fixed in our study of the sector by a fit to the pertinent (, ) data [14]. Then the interaction in the and channels is a genuine prediction that follows from the results of Ref. [14] and symmetry.

The paper is structured as follows: In Sect. II we provide a short overview of the chiral EFT approach with emphasis on its extension to the strangeness and sectors. In Sect. III we show results for the coupled system and for the channel. Specifically, we present integrated cross sections and effective range parameters for the -waves and we compare our results to those of the two potential models mentioned above. The paper ends with a short summary.

## 2 The effective strangeness S=−3 and −4 baryon-baryon potential

We construct the chiral effective potentials for the strangeness and sector at LO using the Weinberg power counting similar to the case considered in [14]. The LO potential consists of four-baryon contact terms without derivatives and of one-pseudoscalar-meson exchanges. Details on the derivation of the LO contact terms for the octet baryon-baryon interaction can be found in Ref. [14, 17]. Here we only list the final result for the and partial wave potentials in the singlet and triplet S-waves, cf. Table 1. Also the and potentials are included in Table 1 for completeness. For convenience in the present paper we express the baryon-baryon potentials in terms of the irreducible representations, see e.g. [18] and also [19, 20], so that the contact interaction is given by

 VB1B2→B′1B′2=14(1−\boldmathσ1⋅\boldmathσ2)C1S0+14(3+% \boldmathσ1⋅\boldmathσ2)C3S1 . (1)

From the relations in Table 1 we can see that for the channel there is a one-to-one correspondence between the and sectors, i.e. the and the interactions, and also between the and sectors, i.e. the and interactions. Thus, if symmetry is fully realized then not only the potentials but even the reaction amplitudes would be the same in the corresponding sectors. Of course, the symmetry is no longer fulfilled once physical values for the baryon masses are used in the evaluation of the scattering observables, even for a symmetric interaction potential. Note that there is no such correspondence for the channel because here the role of the and representations is interchanged when going from to and from to , respectively.

The actual values of the -wave contact terms corresponding to the irreducible representations of are summarized in Table 2 for the various cut–offs (as defined more precisely below). These values were fixed by a fit to the date in our earlier study [14]. They are obtained from the low-energy constants listed in Table 2 of that paper via the relations Eqs. (2.8) - (2.13) given therein as well.

The lowest order invariant pseudoscalar-meson–baryon interaction Lagrangian with the appropriate symmetries was discussed in [14]. In the isospin basis it reads

 L = −fNNπ¯Nγμγ5\boldmathτN⋅∂μ\boldmathπ+ifΣΣπ¯% \boldmathΣγμγ5×\boldmathΣ⋅∂μ\boldmathπ (2) −fΛΣπ[¯Λγμγ5\boldmathΣ+¯\boldmathΣγμγ5Λ]⋅∂μ\boldmathπ−fΞΞπ¯Ξγμγ5\boldmathτΞ⋅∂μ% \boldmathπ −fΛNK[¯Nγμγ5Λ∂μK+¯Λγμγ5N∂μK†]−fΞΛK[¯Ξγμγ5Λ∂μKc+¯Λγμγ5Ξ∂μK†c] −fΣNK[¯\boldmathΣ⋅γμγ5∂μK†\boldmathτN+¯Nγμγ5\boldmathτ∂μK⋅\boldmathΣ]−fΞΣK[¯\boldmathΣ⋅γμγ5∂μK†c\boldmathτΞ+¯Ξγμγ5\boldmathτ∂μKc⋅\boldmathΣ] −fNNη8¯Nγμγ5N∂μη−fΛΛη8¯Λγμγ5Λ∂μη−fΣΣη8¯\boldmathΣ⋅γμγ5\boldmathΣ∂μη−fΞΞη8¯Ξγμγ5Ξ∂μη .

The interaction Lagrangian in Eq. (2) 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 [18],

 fNNπ=f,fNNη8=1√3(4α−1)f,fΛNK=−1√3(1+2α)f,fΞΞπ=−(1−2α)f,fΞΞη8=−1√3(1+2α)f,fΞΛK=1√3(4α−1)f,fΛΣπ=2√3(1−α)f,fΣΣη8=2√3(1−α)f,fΣNK=(1−2α)f,fΣΣπ=2αf,fΛΛη8=−2√3(1−α)f,fΞΣK=−f. (3)

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 value () which was already used in our study of the system [14]. The spin-space part of the one-pseudoscalar-meson-exchange potential resulting from the interaction Lagrangian Eq. (2) is in leading order similar to the static one-pion-exchange potential in Ref. [21],

 VB1B2→B′1B′2 = −fB1B′1PfB2B′2P(\boldmathσ1⋅k)(\boldmathσ2⋅k)k2+m2P , (4)

where , are the appropriate coupling constants as given in Eq. (3) 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 . The transferred momentum is defined in terms of the final and initial center-of-mass (c.m.) momenta of the baryons, and , as . To find the complete LO one-pseudoscalar-meson-exchange potential one needs to multiply the potential in Eq. (4) with the isospin factors given in Table 3.

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. (4) 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 4th and 6th column of Table 1, respectively.

Finally, for completeness we briefly comment on the used scattering equation. The calculations are done in momentum space. We solve the coupled–channel (non–relativistic) 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+iϵTνν′,Jρρ′(p,p′;√s) . (5)

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 baryons to the center-of-mass momentum. Suppressing the particle channel label, the partial wave projected potentials are given in [14]. The LS equation for the coupled channels and 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 ,

 fΛ(p′,p)=e−(p′4+p4)/Λ4 , (6)

in order to remove high-energy components of the baryon and pseudoscalar meson fields. We consider again cut-off values in the range of 550700 MeV as in Ref. [14]. This range is also similar to the one considered in the case, see, e.g. Refs. [22, 23]. The cross sections are calculated using the (LSJ basis) partial wave amplitudes, for details we refer to [24, 25].

## 3 Results and discussion

The LO chiral EFT interaction for the and baryon-baryon sector depends only on those five contact terms that enter also in the interaction. Thus, we can take over the values which were fixed in our study of the sector [14]. Then the interaction in the and channels is a genuine prediction that follows from the results of Ref. [14] and symmetry.

Results for the , , , , and scattering cross sections are shown in Fig. 1. Partial waves with total angular momentum up-to-and-including are taken into account. The shaded bands show the cut-off dependence. From that figure one observes that the and cross sections are rather large near threshold. Though the cross section for rises too, in this case it is only due to the phase space factor . There is a clear cusp effect visible in the cross section at MeV/c, i.e. at the opening of the channel. On the other hand, we do not observe any sizeable cusp effects in the cross section around MeV/c, i.e. at the opening of the channels. The latter is in line with the results reported by the Nijmegen group for their interactions, where a cusp effect in that channel is absent too. In this context we would like to remind the reader that the cusp seen in the corresponding strangeness case, namely in the cross section at the threshold, is rather pronounced in our chiral EFT interaction [14] but also in conventional meson-exchange potential models [5, 26].

Cross section results for the and channels are shown in Fig. 2, again as a function of and with shaded bands that indicate the cut-off dependence.

Results for scattering cross sections at  150 MeV/c for the various and channels are listed in Table 4. We also include predictions by other models [3, 4] for channels where pertinent results are available in the literature. Those cross sections were evaluated from the effective range parameters given in the corresponding publications.

The table makes clear again that our predictions for the cross section are indeed sizeable. The results of the Nijmegen meson-exchange potential and of the quark model of Fujiwara et al. are significantly smaller in that channel. But we would like to remark that at least one of the interactions from the Nijmegen group (NSC97b) yields values that are already fairly close to ours (with mb at  150 MeV/c). In the channel (which is a pure isospin state) the cross sections are also rather large. But here the predictions from the other interactions considered are of comparable magnitude.

It is reassuring to see that the variation of our results with the cut-off mass is not very pronounced. In fact, in general the differences in the cross sections are not more than 20% at  150 MeV/c and, thus, exhibit uncertainties very similar to those that we have observed in our analysis of the and cross sections [14]. In this context let us mention that Stoks and Rijken noticed much more pronounced differences between their six NSC97 models when going to the and sectors, as reported in [3]. Their cross sections often differ by a factor 2 or 3, or even more. But one should keep in mind that in the Nijmegen potential an breaking, in addition to the one induced by the mass differences of the pseudoscalar mesons, was introduced, cf. Ref. [5].

Results for the , , and scattering lengths and effective ranges are listed in Table 5. Also here values from the and baryon-baryon interaction potentials of Refs. [3, 4] are included.

This table reveals the reason for the sizeable cross section predicted by the chiral EFT interactions, namely a rather large scattering length in the corresponding partial wave. It is obvious that its value is strongly sensitive to cut-off variations. It even changes sign (in other words, it becomes infinite) within the considered cut-off range. This means that a virtual bound state transforms into a real bound state, where the strongest binding occurs for the cut-off MeV and leads to a binding energy of MeV. While this behaviour is interesting per se, one certainly has to stress that in such a case the predictive power of our LO calculation is rather limited. One has to wait for at least an NLO calculation, where we expect that the cut-off dependence will become much weaker so that more reliable conclusions on the possible existence of a virtual or a real bound state should be possible. The scattering lengths of the other potentials suggest also an overall attractive interaction in this partial wave though only a very moderate one.

The results for the state of the channel are fairly similar for all considered interactions. Moreover, with regard to the chiral EFT interaction there is very little cut-off dependence. The -waves in the channel belong to the same ( and , respectively) irreducible representations where in the case real () or virtual () bound states exist, cf. Table 1. Therefore, one expects that such states can also occur for . Indeed, bound states are present for both partial waves in the Nijmegen model, cf. the discussion in Sect. III.B in Ref. [3]. Their presence is reflected in the positive and fairly large singlet and triplet scattering lengths for , cf. Table 5. The chiral EFT interaction has positive scattering lengths of comparable magnitude for , for all cut-off values, and therefore bound states, too. These binding energies lie in the range of MeV (MeV) to MeV (700 MeV). In the partial wave the attraction is obviously not strong enough to form a bound state. The same is the case (but for both waves) for the quark model fss2 of Fujiwara et al. [4].

The state of the channel belongs also to the plet irreducible representation and also here the Nijmegen as well as the chiral EFT interactions produce bound states (see also [27]). In our case the binding energies lie in the range of MeV (MeV) to MeV (700 MeV). The predictions of both approaches for the scattering length are comparable. The quark model of Fujiwara et al. exhibits a different behavior for the channel, see the last column in Table 5. The small and negative scattering length signals an interaction that is only moderately attractive. The large and positive scattering length in the partial wave, produced by that potential model, is usually a sign for the presence of a bound state, though according to the authors this is not the case for this specific interaction. Nevertheless, the cross section predicted by the quark model is significantly larger than the results of our chiral EFT interaction as well as those of the Nijmegen meson-exchange potential, see, Table 4.

## 4 Summary

In this letter we have presented leading-order results for the () and interactions obtained within a chiral effective field theory approach based on Weinberg’s power counting, derived analogous to the potential presented in [14], by relating the strangeness and baryon-baryon interactions via flavor symmetry to the one in the system.

The LO chiral potential consists of two pieces: firstly, the longer-ranged one-pseudoscalar-meson exchanges and secondly, shorter-ranged four-baryon contact terms without derivatives. All the occuring five contact terms were already fixed in our study of the interaction so that genuine predictions can be made for the and baryon-baryon interactions based on chiral EFT and the assumed symmetry. The reaction amplitude is obtained by solving a regularized coupled-channel Lippmann-Schwinger equation for the LO chiral potential. We used an exponential regulator function to regularize the potential and applied cut–offs in the range between and MeV.

It will be interesting to see whether the new facilities J-PARC (Tokai, Japan) and FAIR (Darmstadt, Germany) allow access to empirical information about the interaction in the and sector. Such information could come from formation experiments of corresponding hypernuclei or from proton-proton and antiproton-proton collisions at such high energies that pairs of baryons with strangeness or can be produced. There is also a possibility to find signals for strange di-baryon states in heavy-ion collisions [28] which likewise would provide information on the corresponding baryon-baryon interaction. The chiral EFT developed here could then be used to analyze these upcoming data in a model-independent way. It would be also interesting to see how these interactions can be tested in multi-strange three-baryon systems recently explored in lattice QCD [29].

We would like to thank Andreas Nogga for comments and a careful reading of our manuscript. We acknowledge the support of the European Community-Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of the EU. Work supported in part by DFG (SFB/TR 16, “Subnuclear Structure of Matter”), and by the Helmholtz Association through funds provided to the virtual institute “Spin and strong QCD” (VH-VI-231) and by BMBF “Strong interaction studies for FAIR” (grant 06BN9006).

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