arXiv:1904.09913v5 [hep-ph] 26 Jun 2020 ∗ in.Termse n unu parameters quantum and masses composi- Their three-quark conventional or tions. The quark-antiquark have physicists. of hadrons con- interest multiquark attracted and/or tents numbers hadrons quantum unusual (QCD), with Chromodynamics Quantum and model n rm20,ie rmfis bevto fteexotic the of observation Start- first from [1]. Jaffe i.e. meson R. 2003, by of from context suggested ing the model in four-quark explanation the its scalar lowest four- found the which inside of A hierarchy multiplet, four mass investigations a gluons. of was intensive states valence built quark triggered contain be reason or to quarks main expected valence is more Un- hadrons and physics. exotic cal- particle or be of can usual methods and standard scheme this using of culated predictions with accord in .Jff 7.H osdrdsxqak(iayn states (dibaryon) by six-quark light considered also only He of obtained built [7]. was Jaffe consequences R. far-reaching with the of components [2–6]. progress essential physics theoretical two hadrons as form exotic well date, col- as to information years, achieved in- energy experimental past the high during valuable of of lected Now, one branches growing became physics. rapidly pentaquarks and and teresting tetra of tions ebudadsal.UigteMTqakbgmodel quark-bag MIT the Using stable. and bound be ursadforming and quarks elgteog ob on rrsnn.Aogsix- Among resonant. representations or two bound these from be states to quark enough light be o group vor rmtesnltadottrpeettosof representations octet only and dibaryons singlet that the found from and representations its alyzed ltprilshv eosis ttesm ie l of all there- particles, time, merely strange same containing are the structures dibaryons fore At octet spins. and zero singlet have particles glet orsodn author Corresponding Introduction: nte neetn eutaotmliur hadrons multiquark about result interesting Another X 37)tertcladeprmna investiga- experimental and theoretical (3872) SU ymti,del on eta clrhxqakS= S hexaquark scalar neutral bound deeply symmetric, fteQDsmrl ehd u rdcin o t asare mass its for predictions Our ca method. a t rule of as sum constant coupling QCD considered and the be mass of the may calculate far, we work so present detection experimental from oda bnacsti ruetmyb vdd hnahypot a Then an evaded. nucleosynthesis be primordial may argument before this out abundances mordial frozen was DM that and h omlgclD bnac,eitneo hssaemyc may analysis. state thorough this further of requires existence which abundance, nuclei, DM cosmological the f ti ovninlyage htDr atr(M a non-b a has (DM) Matter Dark that argued conventionally is It 1 3.B obnn h oo n pnof spin and color the combining By (3). eateto hsc,Uiest fTha,NrhKaregar North Tehran, of University Physics, of Department m e h clrHexaquark Scalar The 2 eateto hsc,Doˇgu¸s Acibadem-Kadi Physics, University, of Department S u 3 rmfis aso h quark-parton the of days first From nttt o hsclPolm,Bk tt nvriy Az University, State Baku Problems, Physical for Institute , SU 1239 = d and , cs 6 clrpn ru,Jff an- Jaffe group, ”colorspin” (6) − +42 4 28 eateto hsc,KceiUiest,430Imt Tu Izmit, 41380 University, Kocaeli Physics, of Department s ursta eogt fla- to belong that quarks e ( MeV u .Azizi, K. and m s 2 e) lhuhteevle fms ol rdc therm produce would mass of values these Although MeV). 128 = d urscannot quarks SU ,2, 1, SU f J f ∗ 3 may (3) uuddss 3 sin- (3) C P .S Agaev, S. S. are addt oDr Matter? Dark to Candidate a : o nlss aepeitdeitneo H-dibaryon, a six-quark of neutral and existence flavor-singlet of predicted i.e., Jaffe analysis, for tt ihisospin-spin-parities with state is8 e eo h 2 the below MeV MeV 2150 80 mass lies with structure six-quark double-strange sec fabudsaelig4 e eo h 2 the below MeV 40 ex- lying confirmed state also bound [16] a Ref. of in istence Calculations model. bag iis2 limits ssal gis togdcy.I a ea through decay H- of can lifetime It mean that dibaryon, means decays. which strong interactions, weak against stable is fJff 1,1] nfact, In QCD 16]. result the original [15, the from Jaffe with consistent extracted of is mass 12– rules [10, sum H-dibaryon’s two-point well The as invoked were 14]. models quark other energy h orce I a oe e to led model H- of bag framework MIT the the corrected in calculate studies the calculations these to of thus, results controversial: used usual, are As were [8–16]. physics mass dibaryon’s particle meth- of and models ods various which to in investigations, subject retical a state, loosely-bound mass is transformations. the weak limit with this H-dibaryon The obeying mesons. conventional of uemethod rule oaayeΛ analyze To hrlmdlteatos[]found [9] authors the model chiral a hc sjs bv h 2 the above just is which hehl.Teltiesmltospromdi e.[17] that Ref. conclusion in to performed led simulations lattice The threshold. h tblt odtoso h ulu n extracted and nucleus account the into of took authors m conditions the stability paper this the In MeV. 1880 xsec fabudsaeHdbro,adpeitdits predicted and energy H-dibaryon, binding bound-state a of existence eueahxqakisedo i-ur tt,adde- and state, six-quark a of instead hexaquark a use we iaynwsepoe nRf 2] nwihisms was mass its which H- in QCD about [20], holographic estimated Ref. in the explored of was context dibaryon the In respectively. H h rgnlwr 7 a olwdb ueostheo- numerous by followed was [7] work original The h eaur ecp o rgnlppr,hereafter papers, original for (except S hexaquark The uuddss 3 ≈ n .Sundu H. and . 80MV h ae atc tde confirmed studies lattice later The MeV. 1850 0 esaa i-ur atceSb means by S particle six-quark scalar he hc u oisfaue a escaped has features its to due which , ol o infiatyipc pri- impact significantly not could d − m τ nrdc osaiiyo h oxygen the of stability to ontradict ddt o ayncD.I the In DM. baryonic a for ndidate hetical S 2 ¨ y 42 sabl Turkey Istanbul, k¨oy, 34722 ≈ ∼ . roi aue u fw assume we if but nature, aryonic v. ern13557 Iran 14395-547, Tehran Ave., 1180 = e n ihna cuayo h sum the of accuracy an within and GeV 4 − 14 au Azerbaijan Baku, –1148 10 ≈ 0 geswt h euto h quark- the of result the with agrees 20% neato n siaeΛ binding ΛΛ estimate and interaction Λ m − 74 SU H 10 4 . − +40 e 1]ad(19 and [18] MeV 6 ,i osdrbylne hnthat than longer considerably is s, 1 = 26 3 ao-ige,highly flavor-singlet, (3) m e ( MeV m H . rkey GeV. 7 Λ m a eo h 2 the below was m 20MVtrsodand threshold MeV 2230 = Λ H m hehl 8,weesin whereas [8], threshold s rmRf 1]vre in varies [15] Ref. from I 5MeV) 95 = ( J P 0(0 = ) m m H H ally ± uuddss m 10MeV 1130 = 0 e [19], MeV 10) 20MeV 2240 = N + threshold .This ). bound m Λ . 2 note it by S) was searched for by KTeV, Belle, and BaBar turbative methods to explore hadrons. As starting point, collaborations in exclusive S Λpπ− and inclusive the method uses the correlation function Υ(1S) and Υ(2S) decays, in processes→ Υ(2S, 3S) SΛΛ → [21–24]. All these experiments could not find evidence for Π(p)= i d4xeipx 0 J(x)J †(0) 0 , (1) the hexaquark S near the threshold 2mΛ and were able Z h |T { }| i only to impose limits on its mass m the latest being S and extract from its analysis sum rules to compute spec- m < 2.05 GeV. S troscopic parameters of the hexaquark. The main ingre- Recent activities around the S is inspired by renewed dient of this analysis is the interpolating current J(x) suggestions to consider it as a possible candidate to which we choose it in the following form dark matter [25–29]. In accordance with this scenario if m 2(m + m ) = 1877.6 MeV the hexaquark is ab- a b S p e J(x) = ǫabc uT (x)Cγ d(x) uT (x)Cγ s(x) solutely≤ stable, because all possible decay channels of a 5 5 T c free S is kinematically forbidden. For the mS obeying the d (x)Cγ s(x) , (2) × 5 inequality mS
M 2 [1.3, 1.6] GeV2, s [2.5, 2.9] GeV2, (10) ∈ 0 ∈ which obey all aforementioned restrictions. 1.3
1.2 2.9 mSHGeVL 1.1 2.8 1.0 1.3 2.7 2 s0 HGeV L 1.4 2.6 1.0 M2HGeV2L 1.5 2.5 2.9 1.6 PC 0.5 2.8 FIG. 2: The mass of the hexaquark S as a function of the 0.0 Borel and continuum threshold parameters. The mass of the 1.3 2.7 s HGeV2L 0 strange quark is ms = 95 MeV. 1.4 2.6 M2HGeV2L 1.5 Discussion and Conclusions: We have considered 2.5 1.6 the spin-0, parity-even, highly symmetric S-hexaquark of uuddss with Q = 0, B = 2 and S = 2. Us- 2 − FIG. 1: Dependence of the pole contribution on M and s0. ing the technique of QCD sum rule, we have found The (red) surface PC = 0.2 is also shown. that for the chosen interpolating current the intervals M 2 [1.3, 1.6] GeV2, s [2.5, 2.9] GeV2 for the aux- ∈ 0 ∈ In Fig. 1 we plot the pole contribution, when M 2 and iliary parameters fulfill the requirements of the method 2 s0 are varying within limits (10): at M = 1.3 the pole discussed above. For these intervals, we could able to contribution is 0.9, whereas at M 2 =1.6 it becomes equal reach [0.9 0.2] pole contributions to the sum rules, and − +40 to 0.2. The predictions for the mass mS is pictured in Fig. have extracted mS = 1180−26 MeV (ms = 95 MeV) and 2 +42 2, where a mild dependence on the parameters M and mS = 1239−28 MeV (ms = 128 MeV) for the mass of the s0 is seen. The results for the spectroscopic parameters S-hexaquark. This range of the mass implies that the of the hexaquark S read: at the scale µ = 2 GeV (for hexaquarke S is an absolutely stable particle. ms = 95 MeV) It is worth noting that in the context of the sum rule method the scalar six-quark particle was investigated in +40 +0.03 −6 7 mS = 1180− MeV, fS =8.56− 10 GeV , (11) Refs. [15, 16]. In Ref. [15], the authors explored differ- 26 0.26 × ent interpolating currents and carried out calculations by and at the scale µ = 1 GeV (for ms = 128 MeV) taking into account nonperturbative terms up to qq¯ 4 5 h i and ms qq¯ orders. Using ms = 0.2 GeV, and an ap- m = 1239+42 MeV, f =9.18+0.03 10−6 GeV7. (12) h i S −28 S −0.22 × proximation qq¯ = ss¯ , for the mass of the hexaquark they found mh =2i .4h GeV.i But, because of uncertainties e e S Let us note that in computation of mS and fS the of calculations, the authors could not determine whether Borel parameter has been varied within the limits M 2 the mass of this particle lies above or below the ΛΛ 2 e ∈ [1.34, 1.63] GeV . e threshold, i.e. whether it is stable or not. In Ref. [16] the Theoretical uncertainties in the sum rule predictions two-point correlation function was found by employing (11) and (12) appear mainly due to working windows for the hexaquark a molecular type current. The authors 2 2 for the auxiliary parameters M and s0, and the scale took into account only terms proportional to qq¯ and Λ. The ambiguities connected with various vacuum con- qq¯ 4, and neglected the gluon and mixed condensates.h i densates are numerically small. We do not include into hPredictioni in Ref. [16] was made using the strange quark errors of the mass and coupling corrections generated by mass ms = 150 MeV; it was found that the mass of the different choices of the scale µ, but keep (mS, fS) and hexaquark is m 2.19 GeV which corresponds to a S ≈ (mS, fS) as two sets of parameters. Let us note that bound state 40 MeV below the ΛΛ threshold. variation of the mass ∆mS(µ) 60 MeV is comparable The accuracy of calculations performed in the present withe othere errors and does not≈ exceed a few percent of work considerably exceeds the accuracy of previous in- mS. It is not difficult to check also self-consistent charac- vestigations. In this paper we have included into anal- ter of obtained results. Indeed, estimating √s m we ysis not only the quark, but also the gluon and mixed 0 − S 5 condensates. In computations we took into account non- information provided by the SuperCDMS (see, Fig. 13 of −36 2 perturbative terms up to dimension thirty, and for the [38]) allows us to estimate σDMN 10 cm . The limit ∼ −30 2 mass of the strange quark used its contemporary value on DM-proton cross-section σDMp 0.6 10 cm was +9 ≤ × ms = 95−3 MeV. Therefore, our result for the mass of S- extracted also in Ref. [39], in which the authors relied on hexaquark differs from estimations of Refs. [15, 16], but XDC data [40]. Results of other experiments devoted to is in accord with output of the chiral model [9]. Our pre- DM-nucleon scattering can be found in Refs. [37–39]. diction for mS (and mS) almost coincides with one made The theoretical investigation of the hexaquark-nucleon recently in Ref. [32], in which it was obtained by model- scattering cross-section σ should be performed in the e NS ing the S-hexaquark as a bound state of scalar diquarks context of QCD, i.e., in a framework of the Standard and using recent progress in theoretical and experimental Model (SM). In this aspect, this task differs considerably physics of multiquark hadrons. from the situation in theories, where DM particle and There are a lot of constraints on the mass and radius a mediator of DM-nucleon interaction are introduced at of the hexaquark as a candidate to the DM. They stem expense of various extensions of SM [41]. Because the from analyses of the different production modes of S- hexaquark is the particle composed of the conventional hexaquark in high energy hadron and e−e+ collisions, quarks, a mediator of S N interaction also should be- from its interactions with ordinary baryons, and from long to ordinary hadron− spectroscopy. It should be neu- analysis of cosmological processes [27, 28]. As we have tral flavor-singlet scalar particle with a mass < 1 GeV: noted above, accelerator experiments could not observe There are few candidates to play a role of such media- the hexaquark S near the threshold 2mΛ and only im- tor. The scalar mesons σ [in new classification f0(500) ], posed limits on its mass mS < 2.05 GeV. There are some f0(980), and a scalar glueball G may couple to both S and reasons that make problematic detection of S-hexaquark baryons, and carry this interaction. The σ and f0(980) at colliders. In fact, the hexaquark is neutral and com- are singlets from the lowest nonet of scalar mesons. Dif- pact particle and its mass is close to the mass of neutron, ficulties in interpretation of these mesons as qq states which makes difficult to separate relevant signals from inspired suggestion about their diquark-antidiquark na- ones generated by neutron. Additionally, as a flavor- ture [1]. Within this paradigm problems with low masses, singlet particle, the S-hexaquark presumably does not and a mass hierarchy inside the light nonet seem found couple to a hadronic content of photon, pion and other their solutions. The current status of relevant theoretical flavor-nonsinglet mesons, or such interactions are very studies can be found in Refs. [42–44]. The scalar glue- weak. It is possible that due to these features it escapes ball G, in accordance with various estimations, has the detection in hadron collisions. Strategies for discover- mass & 1 GeV, but due to mixing with quark component ing a stable hexaquark in various hadronic processes and would appear as a part of σ and f0(980) mesons [45]. relevant problems were discussed in Ref. [27]. In accor- Here, for the sake of concreteness, we analyze only σ- dance with estimates of this work, within existing exper- exchange processes. Then, to evaluate the cross-section imental datasets should be a few hundred events with σNS one has to compute strong couplings corresponding S or its antiparticle S. For creation of the S-hexaquark to vertices NNσ and SSσ: This is necessary to evaluate e−e+ collisions may be more promising than hadronic σNS using one of QCD approaches. The NNσ coupling, processes, because flavor-singlet multi-gluon states copi- actually, is known, and was calculated in the context of ously produced in these collisions eventually may trans- the sum rule method in Refs. [46, 47]. In these arti- form to S particles [28]. In any case, the constraint cles the meson σ was treated either as mixed qq + G mS < 2.05 GeV extracted from collider processes does or pure qq states. The coupling NNσ, where σ is the not contradict to existence of a stable particle with the diquark-antidiquark exotic meson or admixture of four- mass m 1.2 GeV. S ≈ quark and glueball components, as well as SSσ were not As a particle carrying the baryon number B = 2, the S- calculated using available methods of QCD. It is clear, hexaquark interacts with other baryons and nuclei, and that the hexaquark S is self-interacting particle, which these processes might be utilized to detect it. Param- takes place through the same σ-exchange mechanism. In eters of such interactions have to be calculated using other words, the S-hexaquark as the DM particle belongs perturbative or nonperturbative methods of QCD, and to class of self-interacting DM models. The cross-section confronted with available experimental data. There are σSS for elastic S S interaction can be computed us- some active experiments designed for direct detection of ing the strong coupling− of the vertex SSσ. It is worth DM scattering on target materials of detectors [36–38]. noting that σNS was evaluated in Ref. [25] by model- Recently the XENON and SuperCDMS collaborations re- ing S N interaction via one-boson exchange Yukawa − ported about constraints on light DM-nucleon scattering potential. The result obtained there put on σNS the −3 −30 2 cross-section σDMN extracted from their studies [37, 38]. limit σNS . 10 mb or . 10 cm . At high veloc- In accordance with Ref. [37], for DM particles with the ities of the interacting particles v c , naive estimates ≈ mass 1 GeV and a spin-independent DM N interac- for σNS and σSS led to constraints σNS 0.25σNN ∼ −−38 2 ≥ ≈ tion the cross-section σDMN is σDMN 10 cm ; and 5 mb and σSS 0.25σNS 1.25 mb [28], respectively. −32 2 −33≈ 2 ≥ ≈ σDMp 10 cm and σDMn 10 cm if this inter- The restrictions on σDMp for spin-dependent interactions ≈ ≈ −27 2 −24 2 action is spin dependent (see, Fig. 5 in Ref. [37] ). The 10 cm < σDMp < 10 cm were obtained in Ref. 6
[48]. A strong theoretical analysis, in accordance with is the stable particle, and its self-interaction and elas- the scheme outlined above, is required to make a reli- tic scattering on ordinary matter does not reduce the able prediction for σNS compatible or not with existing total mass of DM in some location, which is crucial to experimental limits. describe aforementioned phenomena. Of course, interac- Besides direct detection experiments, there are outer tion of S with baryon and photon fluids may alter matter space cosmic ray (CR) experiments, results of which can power spectrum and the cosmic microwave background, be utilized to explore the Dark Matter through its decays but they are not strong enough to generate significant ef- to SM particles. Such investigations may be helpful also fects [32]. Contrary, the S-baryon (i.e., DM-galactic gas) for studying matter-antimatter asymmetry in the Uni- interactions may produce a DM disk embedded within verse. The finished satellite-based PAMELA and ongo- the spherical galactic halo, lead to co-rotation of DM ing AMS-02 experiments provided information useful for with the gas and forming DM density structure similar these purposes. The PAMELA was constructed to detect to that of the gas [28]. Parameters of the DM disk in a galactic CRs, and mainly their positron and antiproton galaxy, it thickness, for example, depend on the mass of components [49]. Measurements revealed a rise in the DM and the cross-section σDMN [57]. positron-electron ratio e+/(e− + e+) at energies above Another constraint on mS is connected with stability 10 GeV. The abundance of positrons in CRs was con- of conventional nuclei. The reason is that very small firmed by AMS-02 up to energies 350 GeV [50] and later mass of the hexaquark may contradict to stability of ex- till 500 GeV. The similar enhancement was observed in isting nuclei, because for small mS nucleons inside nu- antiproton to proton ratio p/p, as well [51, 52]. A stan- clei would bind to the S state faster than what is al- dard scenario implies that antiparticles are produced due lowed by observed stability of these nuclei. This process ∗ to inelastic interactions of CR nuclei with particles of the runs through double-weak production of the off-shell Λ interstellar gas. But rates of such processes are small, baryons by a pair of nucleons pn, pp or nn. Because our therefore deviation of these ratios from expected small estimate for the mass of the hexaquark is mS 1.2 GeV, ∗ ≈ values may be interpreted in favor of DM decaying to SM the main sources of the virtual Λ baryons are the weak ∗ ∗ particles. Collected data on antiparticle anomaly in the decays p Λ π+, n Λ π0, and an internal conversion → → ∗ Universe provide valuable information to verify different (udd) (uds). Generated by this way virtual Λ s after- → ∗ ∗ DM models, and impose constraints on DM-SM inter- wards through the strong-interaction process Λ Λ S → actions [41]. Alternatively, the observed abundance of form the hexaquark S. high-energy positrons in CRs may be connected with ac- The matrix element of the reaction NN SX can → celerating effects of nearby sources, such as a pulsar, su- be written as a product of the amplitude for the nucle- ∗ pernova remnants [53–55]. This mechanism implies some ons’ double-weak transitions to a pair of virtual Λ s, and ∗ anisotropy in detected antiparticle fluxes, whereas data matrix element for creation of the S from the Λ s [26] are consistent with their isotropic distributions: this is (NN SX) (NN Λ∗Λ∗X) (Λ∗Λ∗ S). significant obstacle in attempts to explain antiparticle M → ≈M → M → excess in CRs using local sources. (13)
The hexaquark S produces leptons in the SS annihila- ′ tion and inelastic interactions with particles of the inter- Then lifetime of the nucleus decaying to and the hexaquark is N N stellar gas. Productions of various groups of σ, f0(980), π, and K mesons would be main channels at low energy 3yr annihilations. At high energies these light mesons should τ( ′SX) , (14) N → N ≃ (Λ∗Λ∗ S) 2 be accompanied by numerous baryons. Dominant decay |M → | modes of the mesons σ, f0(980), π, and K, and rele- which should be confronted with the Super-Kamiokande vant branching ratios are well known [35], which can be (SK) limit for the oxygen nuclei employed to estimate production rates of electrons and positrons to account for discussed effects. 16 ′ 26 τ( O8 SX) & 10 yr. (15) Important restrictions on the mass of the hexaquark → N arise from observed cosmological abundance of the DM, Equation (14) is rather rough estimate for τ, which is and DM to ordinary matter ratio in the Universe. The seen from treatment, for instance, of the matrix ele- concept of DM is necessary (excluding models with modi- ment (Λ N) 2 used to derive it [26]). This ma- fied theory of gravity) to account for observed astrophys- trix element|M → was calculated| there in the harmonic oscil- ical effects, such as rotation curves of galaxies, gravita- lator model, and was also inferred from phenomenological tional lensing, and other phenomena. Direct evidence for analysis: obtained results differ from each other by ap- the existence of DM came from analysis of bullet clusters proximately 10 times. The prediction (14) was made by [56]. All these phenomena can be explained within the employing an average value of (Λ N) 2. concept of DM provided DM particles are stable. The The situation with (Λ∗Λ∗ |MS) 2→is even| worst than |M → | ∗ ∗ hexaquark with mass mS 1.2 GeV reproduces observed in the previous case. The matrix element (Λ Λ ≈ 2 ∗ ∗ |M → DM abundance and DM/matter ratio, while a larger mS S) describes the strong process Λ Λ S and plays a | → 16 gives a smaller relic abundance [32]. The S-hexaquark crucial role in theoretical estimations of the O8 lifetime. 7
In Ref. [26] the (Λ∗Λ∗ S) was calculated as overlap also contains phenomenological contributions to repro- integral of the finalM hexaquark→ and initial Λ∗Λ∗ baryons’s duce nucleon-nucleon elastic scattering. The new wave wave functions, the latter being factored into wave func- functions, of course, present a more detailed picture of a tions of the two Λ∗ baryons, and a wave function of two nucleus, but are nonrelativistic quantities and do not take nucleons inside nucleus. The Λ∗ baryon and hexaquark into account dynamical effects of quark-gluon interac- S wave functions were written down using the Isgur-Karl tions in nuclei. The updated results for (Λ∗Λ∗ S) 2 (IK) nonrelativistic harmonic oscillator quark model [58] are presented in Fig. 5 of Ref. [32] as|M a function→ of the| and its generalization to six-quark system [26]. These ratio rS /rN . Unfortunately, the authors did not show functions depend on parameters α = 1/ r2 , the region rS/rN = 0.11 0.2, in which the restriction B(S) B(S) − ∗ ∗ 2 −25 qh i on the matrix element (Λ Λ S) . 10 may be where r2 and r2 are mean charge radii of the Λ∗ |M → | h B i h S i satisfied. baryon and hexaquark S, respectively. The features of As it has been just emphasized above, (Λ∗Λ∗ the two nucleons inside nucleus were modeled in Ref. [26] S) 2 critically depends on a behavior of the relevant|M wave→ using Brueckner-Bethe-Goldstone (BBG) wave function. functions| at small inter-nucleon distances r . 1 fm. At A wide class of the two-nucleon wave functions including these distances nucleons are not the suitable degrees of the BBG, the Miller-Spencer wave functions, and ones freedom, and quark-gluon content of the nucleons be- extracted from results of Ref. [59], was employed in Ref. comes essential to describe correctly processes inside nu- [32] to study the stability of the oxygen nucleus. clei. Neither the Isgur-Karl type wave functions of the The IK wave functions used to model the Λ∗ baryons Λ∗ baryon and S-hexaquark, nor the two-nucleon wave and the hexaquark S suffer from serious drawbacks. functions discussed till now contain detailed information −1 Thus, the parameter αB = 0.406 fm necessary to on relativistic and nonperturbative interactions of quarks 1 + 3 + and gluons at distances r . 1 fm and high densities. describe the mass splitting of lowest lying 2 and 2 baryons corresponds to radius of the proton 0.49 fm, Therefore, the predictions for (Λ∗Λ∗ S) 2 made in |M → | whereas to reproduce the experimental value 0.86 fm one Refs. [26, 32] cannot be considered as credible ones and −1 needs αB = 0.221 fm . In other words, the IK func- used to confirm or exclude existence of the hexaquark S. tions could not explain simultaneously the mass splitting Only after thorough exploration of aforementioned prob- of positive parity baryons and the proton radius. Usage lems, we can get serious estimate for (Λ∗Λ∗ S) 2 ∗ |M → | of such nonrelativistic wave function to describe the Λ and see whether the requirement rS << rN –necessary 16 baryon, moreover an attempt to generalize and apply it for stability of O8 in the present picture–survives or to the relativistic multiquark system like the hexaquark not. Then we will be able to answer the question moved is, at least, questionable. Existing problems of the IK to the title of the present article as well. model were emphasized already in the original paper [26] and repeated in Ref. [32], nevertheless in both of them Appendix: Details of calculations and the correlation these wave functions were employed to carry out numer- OPE 2 ical computations. function Π (M ,s0) Calculations demonstrate that the matrix element (Λ∗Λ∗ S) 2 is highly sensitive to the choice of the In this appendix we present some details of calcula- |M → | tions, which are necessary to derive the sum rules for the parameters αB and αS, and is suppressed if the hex- mass and coupling of the hexaquark. It is evident that a aquark S has a small radius rS/rN 1, where rN is the nucleon radius. The radius of the≪ hexaquark was esti- main problem is calculation of the QCD side of the sum rules ΠOPE(p). Using explicit expressions for the light mated in Ref. [29] as 0.1 rS 0.3 fm which implies ≤ ≤ quarks propagators and inserting relevant ones into Eq. rS /rN =0.11 0.34. Then, for example, for rS 0.13 fm and, as a result,− for the ratio r /r 6.6≈ the BBG (6), we get the Fourier integrals of following types: N S ≈ wave function with the core radius c 0.4 fm, and 2 2 m 2 l −x Λ −1 ≃ ∗ ∗ [1, xµ, xµxν , ...](x ) ln αB = 0.406 fm satisfies the constraint (Λ Λ 4 ipx 4 2 −25 |M → T [l,m] = d xe h i . S) . 10 necessary to evade the SK bound (see, Fig. Z (x2)n 1 in| Ref. [26]). At the same parameters of the hexaquark, −1 (A.1) but for αB = 0.221 fm the core radius in the BBG model should be larger to achieve required suppression For simplicity, let us consider the case l = 0 and m = 0. of the matrix element (Λ∗Λ∗ S) 2. After a Wick rotation in the Euclidean space for T [0, 0] |M → | ≡ To update predictions for (Λ∗Λ∗ S) 2, the au- T , one finds thors in Ref. [32] used the new|M two-nucleon→ wave| func- e−ipE xE tions. The latter were extracted by utilizing an in- n 4 T = i( 1) d xE 2 n . (A.2) formation on the two-nucleon point density inside nu- − − Z (xE ) clei ρNN (r) [59]. The original calculations of ρNN (r) By applying the Schwinger parametrization were performed by employing the nonrelativistic Hamil- ∞ tonian, where the phenomenological NN potential in- 1 1 n−1 −tA n = dt t e , A> 0, (A.3) cludes electromagnetic and one-pion-exchange terms, and A Γ(n) Z0 8 it is not difficult to recast T into the form Afterwards we apply the continuum subtraction proce- ∞ dure using the replacement 2 n 1 n−1 4 −ipE xE −txE T = i( 1) dt t d xE e e . − − Γ(n) Z0 Z By performing the resultant Gaussian integral over four- x, we obtain s0 N 1 2 M 2 dse−s/M sN−1, N> 0, (A.8) 2 ∞ π 2 → Γ(N) Z0 T = i( 1)n dt tn−3e−pE /4t. (A.4) − − Γ(n) Z0 The next step is to apply the Borel transformation with 2 respect to pE to suppress contributions of the higher res- onances and continuum states. To this end, we utilize where s0 is the continuum threshold parameter. Then the formula for the Borel transformed and subtracted integral T , we get 2 1 1 e−pE /4t = δ , (A.5) BM M 2 − 4t which leads to 0 2 3−n s 2 π2 ∞ 1 1 n π 4 −s/M N−1 T = i( 1)n dt tn−3δ . M T = i( 1) dse s , M 2 B − − Γ(n)Γ(n 3) Z0 B − − Γ(n) Z0 M − 4t − (A.6) (A.9) By carrying out the integration over t, we immediately Calculations of the other terms T [l,m] in QCD side of get the sum rule can be performed in a similar manner. In the general case of T [l,m], for continuum subtraction one − π2 M 2 n 3 should use more complicated formulas, full list of which T = i( 1)n . (A.7) can be found in Ref. [60] BM − − Γ(n) 4
OPE 2 As a result, for Π (M ,s0) we get
0 30 s 2 OPE 2 1 7 −s/M (i) 2 Π (M ,s0)= dss e + Π (M , s0), (A.10) 147 41252π10 Z · 0 Xi=3
where the first term is the perturbative contribution. 9
For the nonperturbative (3) (10) dimensional terms we get: − 0 s 2 (3) 7ms ss 5 −s/M Π = 3 9 h2 i8 dss e , 3 4 5 π Z0 0 2 2 s 2 (4) gs G 5 −s/M Π = h 11 2 i 10 dss e , 2 4 5 π Z0 · m Π(5) = s [2 qg σGq 2 3341152π8 h s i 0 0 s · 2 s 2 4 2 −s/M 4 2 −s/M dss 12783 2340 ln s/Λ e + sgsσGs dss 346 + 120 ln s/Λ e , × Z0 − h i Z0 0 s 2 (6) 1 2 2 2 4 −s/M Π = 3 7 6 2 qq + ss + 12 2 ss qq + qq dss e , 2 3 4 5π h i h i h ih i h i Z0 · 0 0 2 2 s 2 s 2 (7) ms gs G 3 2 −s/M 3 2 −s/M Π = 3h 11 8 i 5 qq dss 311 + 60 ln s/Λ e + ss dss 865 36 ln s/Λ e , 3 4 π h i Z0 − h i Z0 − 0 s 2 (8) 1 2 2 2 4 2 2 2 3 −s/M Π = 4 15 10 1429 gsG + 21576π m0 181 qq +5 ss + 342 qq ss dss e , −2 3 4 π h i h i h i h ih i Z0 · 0 s 2 (9) ms 2 2 2 2 2 −s/M Π = 4 12 8 gs G m0 dss 77464 qq + 13922 ss (27320 qq 748 ss ) ln s/Λ e 2 3 4 π h i Z0 − h i h i− h i− h i · s0 m 2 s 12 ss 2 qq + 14 ss qq 2 + 24 qq 3 dss2e−s/M , −2 3344π4 − h i h i h ih i h i Z 0 · 2 2 s0 4 s0 g G 2 m 2 Π(10) = h s i 386 qq 2 + 34 ss 2 + 636 ss qq dss2e−s/M 0 27 qq 2 + ss 2 dss2e−s/M . 3349π6 h i h i h ih i Z − 3 49π6 h i h i Z 0 · 0 (A.11)
(i) 2 In Π (M , s0) we have assumed uu = dd and denoted both of them as qq . We do not provide explicit expressions of the terms (i) > (10). h i h i h i
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