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Hyperons and Condensed Kaons in Compact Stars Hyun Kyu Lee1 and Mannque Rho2 1Department of Physics, Hanyang University, Seoul 133-791, Korea 2Institut de Physique Théorique, CEA Saclay, 91191 Gif-sur-Yvette cédex, France & Department of Physics, Hanyang University, Seoul 133-791, Korea (Dated: December 11, 2013) UsingtheCallan-Klebanovboundstatemodelforhyperonssimulatedoncrystallatticetodescribe strangebaryonicmatter,wearguethatto (1)inthelargeN countingtowhichthetheoryisrobust, c O hyperonscan figureonlywhen–orafter–kaonscondensein compact-starmatter. Wealso discuss how the skyrmion-half-skyrmion topological transition affects the equation of state (EoS) of dense baryonic matter. The observations made in this note open wide the issue of how to theoretically access theEoS of compact stars. 3 1 PACSnumbers: 0 2 Introduction: Strangeness can figure in compact stars ing is chiefly via the Wess-Zumino term – related to the n in a very crucial way. The appearance of hyperons [1] Weinberg-Tomozawa term in chiral Lagrangian – which a and/or condensed kaons [2] at a density above that of distinguishes kaon (i.e., K+) and anti-kaon (i.e. K−), J nuclear matter can soften the EoS for dense matter, in- pushing the K+ mass up and the K− mass down. For 1 fluencing the maximum star mass and the cooling of the the given“heavy-meson”ansatz for the kaon fluctuation, star formed in supernova explosion. There has been a thismodeldoesnotappeartohaveflavorSU(3)symme- ] h long-standingdebateastowhatappearsfirstinthegrav- try. However it turns out that one can recover both chi- t itational compression of compact-star matter, hyperons ralSU(3)×SU(3)symmetryandthe(Isgur-Wise)heavy - l orkaoncondensation. Ifhyperonsappearfirst,thenthey quark symmetry by tuning the symmetry breaking, i.e., c u willsubstantiallylowertheelectronchemicalpotentialso the kaon mass. An important point to note here that is n thattheelectronscannolongertransformtokaonsinthe highlyrelevantfor,andwillbeexploitedin,whatfollows [ way discussed in [3]. This will then banish kaon conden- is that in the limit m → 0, the bound-state descrip- K sation from the scenario. The alternative possibility – tion of hyperons smoothly approaches the rigid rotator 1 v which we think is more likely – that kaon condensation description [7]. On the other hand as mK → ∞, the 7 cantakeplacebeforetheappearanceofhyperonshasnot heavy-quark symmetry for the s quark emerges [8]. We 6 been exploredup to date. It is the purpose of this paper will exploit the former feature in applying the model to 0 to suggest that hyperons can appear not before but only kaon fluctuation in dense matter. 0 when or after kaons condense. The CK model wasfound to be fairly successfulin de- . 1 Our approach will be anchored on a multi-skyrmion scribing the non-exotic hyperons (with strangeness S = 0 description of dense hadronic matter [4]. The merit of −1,−2,−3). It has the merit to account for the O(N0) 3 our approach is that all the relevant degrees of freedom, strangeness fluctuation that is absent in the rigid rotor 1 i.e.,baryonsandmesons,intheenergyregimeconcerned, : model [9]. That one can single out the strangeness fluc- v canbetreatedonthesamefootinginaunifiedwayusing tuationis significantsincethe flavor-independentO(N0) i a single effective Lagrangian that models QCD for both X correction, i.e., the Casimir energy, that is difficult to elementary and many-body processes. The approach is calculate reliably can be eliminated from the considera- r a quite involved,however,in concepts as wellas in numer- tion. The bound state model can be made to work even ics, so the results obtained therefrom can be rigorously better phenomenologically by incorporating vector me- justified only in the limit of large N . c son degrees of freedom in the SU(2) sector à la hidden The main ingredients of our approach are (1) the localsymmetry(HLS)[10]. Moreovertherecentdevelop- Callan-Klebanov (CK) bound-state picture for hyperons ment inapplied string theorypoints to that a morereal- formed with (anti-)kaons bound to SU(2) skyrmions [5] istic description of baryonic systems could be obtained and(2)multiskyrmionsputoncrystallatticetosimulate with the incorporation of the infinite tower of hidden dense baryonic matter [4]. In the framework so defined, local vector mesons inherent in the gauge-gravity dual- our main result stated above will turn out to be very ity [11, 12]. This is manifested in the striking role that simple to establish. We will first establish it to O(1) in the tower plays in the vector dominance in the meson the N counting and then discuss what may happen at c and baryon form factors [11, 12]. Now once the vector O(1/N ). c mesonsarepresentintheSU(2)sector,thehomogeneous Callan-Klebanov Bound State Model for Hyperons: In Wess-ZuminotermsinheritedfromtheChern-Simonsac- the CK model, (anti-)kaons, taken to be “heavy,” are tioninthe5Dholographicdualactionplaysanextremely bound to the SU(2) skyrmion constructed with the importantrolein capturing the physicsof short-distance Skyrme Lagrangian [6] consisting of the current algebra stronginteractions. There the roleofthe ω mesoninthe term and the four-derivative Skyrme term. The bind- baryon structure is found to be crucial [13]. 2 In calculating baryon properties in the skyrmion To capture most, if not all, of the scalar degree of free- model, one is resorting to the large N expansion of domassociatedwiththeCasimireffectmentionedabove, c the baryon mass m = M + M + M + ··· where adilatonfieldχisintroducedviaQCDtraceanomalyto 1 0 −1 the subscripts stand for the power n in the Nn depen- betreatedinmeanfieldaswassuggestedfordensematter c dence. The first is the leading O(N ) soliton contribu- in [16, 17]. c tion and the third is the O(1/N ) hyperfine splitting ob- Putting on crystal lattice the HLS Lagrangian imple- c tained by collective-quantization. Both are well defined mentedwiththeχfieldisyettobeworkedout. Thiswill in the framework with a given Lagrangian. The second yieldamorereliableresultthanavailableuptodate,par- term is less straightforward. There are two possible con- ticularly for the observables sensitive to higher orders in tributions to the O(N0) term, M = M0 + Ms. The 1/N . Since we are working to O(N0), however, we can c 0 0 0 c c first is the flavor-independent Casimir energy which is work with the simplified Lagrangian of [14] where the presentinboththe CKmodelandthe rigidrotormodel. vector mesons are integrated out. This can be justified Thesecondisthe kaonicfluctuationthatdepends onthe formoderatedensityawayfromtheVM(vectormanifes- strangenessS. This termis presentinthe SK modelbut tation) density of HLS. absent in the rigid rotor model. It is this term that will In[14], inclose analogyto the CK hyperon,kaonfluc- play a key role in our discussion that follows. tuation U in the background of multiskyrmions simu- K The Casimir energy is highly model-dependent and lated on FCC crystal U was considered by taking the 0 hence is difficult to compute reliably. For this reason, ansatz for the chiral field it is mostly ignored in the literature. We claim that ig- noring this term canbe a seriousdefect in dense matter. U(~x,t)=pUK(~x,t)U0(~x)pUK(~x,t). (3) Approximate loop calculations using chiral perturbation There the focus was on the properties of kaons in the theorygiveM0 <∼−(1/3)M1[15]inmatter-freespace. In medium,this contributioncanbeassociatedwiththe at- tractionbroughtin by a scalarisoscalarfield (commonly 600 (b) denoted σ) that binds nucleons and hence is indispens- mK*+ with dilaton able in nuclear processes. w.o. dilaton In the CK model, the mass formula for hyperon with V) strangeness S is of the form, e * (MmK 300 m|S| =Msol+M00+|S|ωK +O(1/Nc)+··· (1) <q_q>=\0 mK*- f* =\ 0 _ whereω is the energyofthe (anti)kaonbound– chiefly <q_q>=0 <qq>=0 K f* =\ 0 f* = 0 by the Wess-Zumino term – to the soliton. We will be 0 interestedinthemassdifferencebetweenthelowest-lying 0 1 2 3 4 5 hyperon – that we shall denote Y – and the nucleon. n/n0 The first two terms of (1) are flavor-singlet and hence FIG.1: m∗K± vs. n/n0 in denseskyrmion matter consisting are cancelled out in the difference. So taking |S|=1 for ofthreephases: (I)skyrmionphase,(II)half-skyrmionphase the lowest-lying hyperon, we obtain the simple result and (III) chiral symmetry restored phase. The parameters are fixed at √2ef = 780 MeV and dilaton mass m = 720 π χ ∆m≡mY −mN =ωK +O(1/Nc). (2) MeV with n1/2 ∼1.3n0. As will be explained later, the O(1/N ) corrections that c givenbackgroundU alreadycalculatedinpreviousstud- 0 leadtothehyperon-multipletstructurearehighlymodel- ies [19]. A striking qualitative feature found in [19] was dependent and cannot be calculated reliably at present. theskyrmion-half-skyrmiontransitionthattakesplaceat We will therefore limit ourselves to O(N0) to which the c n ∼(1.3−2.0)n , a range of density that is compati- treatment is robust. 1/2 0 blewithnuclearphenomenology[20]. While the location The CK Skyrmions on Crystal: We are interested in of n was found to be insensitive to the dilaton mass, 1/2 studying kaonic fluctuations in a baryonic medium at thebehaviorofthein-mediumkaondependedsensitively high density. For this we put K−s (that we shall simply on the dilaton mass, reflecting the importance of O(N0) c referto as kaons– omitting anti– forshortunless other- effects in medium. In Fig. 1 is reproduced the scaling wisenoted)oncrystallatticefollowingtheapproachfirst of the kaon mass as a function of density for the dilaton discussedin[14]. Thelatticesizethendefinesthematter mass in the vacuum m ≈720 MeV #1. Note that there χ density. This is the only method we know of for describ- isanabruptdropinthe K− massatn , probablycon- 1/2 ing in-medium kaons in a unified framework that treats nected to the abrupt change in the nuclear tensor forces mesons and baryons (both strange and non-strange) in multibaryonicmatterwithasingleLagrangianwithsym- metries(i.e.,chiralsymmetry,scalesymmetry)consistent with QCD. The vector mesons ρ, ω can be incorporated #1 Thisisroughlythevacuum massfortheσ thatwouldyieldthe – naturally and readily – by hidden local symmetry [18]. in-mediummassneededfornuclearmatterinmean-fieldmodels. 3 that affects the symmetry energy in the EoS of compact is spontaneously broken. The structure of such strange star matter seen in [20]. hadronic matter is an open problem that has not been Let us now turn the focus from kaon properties to hy- addressed up to today. peron properties in medium described as CK skyrmion Hyperfine Effects: Our discussion so far has been lim- matter. Forthiswewillignorethepossibleback-reaction ited to O(N0). Let us see what one can say about c of the kaons on the background of skyrmions that is O(1/N ) corrections. These corrections are responsible c suppressed by 1/N and consider K−’s embedded in an c for the hyperon multiplet structure as well as the split- SU(2)skyrmionmatter. The ansatz we takeis the same ting between even-parity and odd-parity states#2. Since asin(3). TheskyrmionmatterwithbaryonnumberB is we do not know how to collective-quantize the skyrmion described by the U simulated on FCC crystal. As with 0 matter with baryon number A ≫ 1, we look at the the CK hyperon for B = 1, the difference in energy per system as a collection of A quasiparticles (i.e., “quasi- baryon between the hyperon matter with |S| = A with hyperons”). The hyperfine energy for the lowest-lying K−’sboundtotheskyrmionmatterwithbaryonnumber quasi-hyperon Λ of spin 1/2, isospin 0 and strangeness B = A ≫ 1 and the non-strange skyrmion matter (with −1 is [5, 10] B =A and |S|=0) then will be given to O(N0) by c 3 E∗ −E∗ =ω∗ +O(N−1) (4) E∗ (Λ)= (c∗2−1) (7) Y N K c −1 8Ω∗ where the asterisk represents medium-dependence. As where the asterisk stands for in-medium quantity, Ω∗ is in the case of B = 1, the soliton (O(N )) and Casimir c the moment of inertia of the skyrmion rotator and c∗ is (O(N0))energieswillcanceloutinthedifference,leaving c the coefficient multiplying the effective spin operator of onlythestrangeness-dependentterm,i.e.,thein-medium S =−1. Taking into account the hyperfine term for the kaon mass, to O(N0). The mass shift occurs as depicted c nucleon, we get in Fig. 1 [14]. The O(N−1) term will capture the hy- c perfine effect as well as other medium-dependent terms. 3 Forthisweneedtocollective-quantizetheskyrmionmat- EY∗ −EN∗ =ωK∗ + 8Ω∗(c∗2−1). (8) ter with baryon number A. The result will be strongly model-dependent such as the degrees of freedom taken What this says is that which comes first, hyperons or into account in the Lagrangian and quantum loop cor- kaon condensation, will depend on whether c∗2 > 1 or rections of the Casimir type. At present we do not know c∗2 <1. As stressedbyCallanandKlebanov,this quan- how to collective-quantize such multi-skyrmion systems tityishighlymodel-dependent,andcannotbecalculated in general. We will mention later what one finds for with reliability; one is unable to determine it even for the simple case where the skyrmion matter is a pure hyperons in the vacuum [5]. If one uses the mass for- neutron matter. In this case an approximate collective- mula to determine c in matter-free space, it comes out quantization is feasible [21]. What is of significance is to be c ∼ 0.7. If this held in medium, then (8) would that the treatment given here is robust to the NLO in imply that hyperons appear before kaons condense. On N . the other hand, as mentioned, the CK model smoothly c Appearance of Strangeness in Compact-Star Matter: approaches the SU(3) rotor model as mK → 0. In this To see the implication of what we found above in limit c∗ →1. In medium, the kaon mass drops, accentu- compact-starmatter,werecallthataccordingtothesce- ated by the topology change, as density increases,which nario of [3], (s-wave) kaons condense in dense neutron- means that in-medium effects will become more impor- richmatter whenthe electronchemicalpotential µe that tantatO(1/Nc)andmedium-dependentloopcorrections increases with increasing baryonic density is equal to or canoverwhelmtreecontributions. Allthatcanbesaidat greater than the dropping in-medium kaon mass presentisthatc∗ islikelytobecloseto1athighdensity. Symmetry Energy: As done in [21] following Kle- µ ≥m∗ . (5) e K banov’s suggestion [22], we can approximately calculate how the symmetry energy is modified in the pres- On the other hand, hyperons can appear when ence of strangeness by collectively-quantizing the pure µ ≥E∗ −E∗. (6) A-neutron crystal system (for A → ∞) with the max- e Y N imum isospin I = |I | = A/2. In [21], the symmetry 3 FromEq.(4), it follows that to NLO in Nc, the hyperons energy of pure neutron matter in the absence of hyper- will figure at the same density as the threshold density ons (or equivalently kaon condensation)was found to be at which s-wave kaons condense. This is our principal E = 1 δ2 where δ = (N − P)/(N + P) and Ω˜ is result announced above. sym 8Ω˜ This result could mean either that hyperons and condensed kaons play the same role in compact-star matter or hyperons appear only when kaons condense. For the #2 An important odd-parity hyperon that is often considered for latter,itwouldbeessentialtofigureouthowhyperonsin- kaon-nuclear physics is Λ(1405). This is correctly given by the teractinkaon-condensedmatterwhereisospinsymmetry hyperfinecorrection. 4 the moment of inertia of the unit cell of the crystal lat- Without calculating these higher order corrections with tice. Doingthesamecalculationwiththeansatz(3)that confidence,itwouldbedifficulttoexcludethepossibility includes kaon fluctuation, we obtain of the reversed appearance of the strangeness degrees of freedom. 1 Ehyp ≈ (1−d∗)δ2 (9) As illustrated in Figure 1, the kaon mass in the sym 8Ω˜ skyrmion matter drops continuously up to near nuclear matter density which is more or less in accordance with where d∗ is what corresponds to c∗ in (8) with U in (3) 0 chiral perturbation theory (ChPT) predictions. But the replacedbypureneutronmatterandΩ˜ isthemomentof steep drop at n is not seen in ChPT. This drop may 1/2 inertia of the soliton U . 0 have some connection with the Akaishi-Yamazaki “con- A few remarks are in order here. traction effect” needed to form dense kaonic nuclei but NotefirstthatatO(N−1)atwhichE figures,there c sym not present in conventional nuclear interactions [24]. It aretermsindependentofδ butdependentonstrangeness is also interesting to note that with the topology change that could contribute to the hyperfine effects. These translated into parameter changes in effective nuclear terms as well as the d∗ cannot be computed accurately. many-body formulations as discussed in [20], the drop- Second, Ω˜ is precisely what was computed in [21] and ping of the kaon mass after the density n is close to 1/2 hence will have the cusp structure at n . However as 1/2 what one gets for kaon fluctuation in compact stars cal- shownin[20],thecuspwillbesmoothedbynuclearcorre- culated from the VM fixed point [25]. lations . Third, for d∗ >0, the presence of hyperons will Thetopologychangeatn withthedropofthekaon make the symmetry energy softer – as expected – and in 1/2 masswillspeedupkaoncondensationandsoftentheEoS. the limit m∗ → 0, we expect d∗ → 1, so the symmetry K On the other hand, in the absence of kaon condensation energy will vanish. (or hyperons in our description), it is found to stiffen Note also that Eq.(9) will be modified once kaons are the EoS at n [20]. To accommodate both the soft condensed,because then isospinis spontaneouslybroken 1/2 EoS found in heavy-ion experiments at low density and and there will be a term linear in δ to the symmetry the stiff EoS needed at higher density to account for the energydiscussedin[23]. Thismeansthatthedescription recently measured 1.97 M neutron star, the two mech- intermsofhyperons–withoutcondensedkaons–canno ⊙ anisms must intervene in a delicate balancing act. This, longer be correct. webelieve,throwsopenwidetheissueofEoSatdensities Comments: The main conclusion reached in this pa- relevant to compact stars. per is that to O(N0), the appearance of strangeness in c Acknowledgments: The work reported here was par- compact star matter can be equivalently described ei- tially supported by the WCU project of Korean Min- ther in terms of hyperons or in terms of kaons. 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