Kaons in Dense Half-Skyrmion Matter
aa r X i v : . [ h e p - ph ] D ec Kaons in Dense Half-Skyrmion Matter
Byung-Yoon Park, Joon-Il Kim, and Mannque Rho Department of Physics, Chungnam National University, Daejon 305-764, Korea Department of Physics, Florida State University, Tallahassee, FL 32306, USA Institut de Physique Th´eorique, CEA Saclay, 91191 Gif-sur-Yvette C´edex, Franceand Department of Physics, Hanyang University, 133-791 Seoul, Korea (Dated: October 30, 2018)Dense hadronic matter at low temperature is expected to be in crystal and at high density make atransition to a chirally restored but color-confined state which is a novel phase hitherto unexplored.This phase transition is predicted in both skyrmion matter in 4D and instanton matter in 5D, theformer in the form of half-skyrmions and the latter in the form of half-instantons or dyons. Wepredict that when K − ’s are embedded in this half-skyrmion or half-instanton (dyonic) matter whichmay be reached not far above the normal density, there arises an enhanced attraction from the softdilaton field figuring for the trace anomaly of QCD and the Wess-Zumino term. This attractionmay have relevance for a possible strong binding of anti-kaons in dense nuclear matter and for kaoncondensation in neutron-star matter. Such kaon property in the half-skyrmion phase is highly non-perturbarive and may not be accessible by low-order chiral perturbation theory. Relevance of thehalf-skyrmion or dyonic matter to compact stars is discussed. I. The Problem and Results — There is a compellingindication from baryonic matter simulated on crystal lat-tice that as density increases beyond the normal nuclearmatter density n , there emerges a phase with vanish-ing quark condensate symptomatic of chiral symmetryrestoration but colored quarks still confined. This hasbeen observed [1, 2] with skyrmions put in an fcc crys-tal which turn into half-skyrmions in a cc configurationas density reaches n = n / > n where the quark con-densate h ¯ qq i goes to zero but the pion decay constant f π remains non-zero, implying that hadrons are relevantdegrees of freedom there although chiral symmetry is re-stored. This has also been shown [3] to take place withinstantons in 5D that figure in holographic dual QCD(hQCD) placed in an fcc crystal that split into half-instantons in the form of dyons in a bcc crystal configu-ration. While the skyrmion matter is constructed eitherwith the pion field only or with the pion field plus thelowest lying vector mesons ρ and ω that we shall denote˜ ρ , the instanton matter arises as a solitonic matter in 5Dwhich when viewed in 4D, contains an infinite tower ofboth vector and axial vector mesons. A single baryonis found to be much better described as an instanton inhQCD [4] which is justified for both large N c and large’t Hooft constant λ = g Y M N c than the skyrmion baryonin large N c QCD. This is particularly so for quantitiescaptured in quenched lattice QCD calculations. Now ahighly pertinent question is: How does a meson behavein the medium consisting of a large number of these soli-tons?This issue was addressed in [1] for fluctuations of pionsin dense medium. What one learned from that calcula-tion was limited because the pion, being nearly a genuineGoldstone boson, is largely protected by chiral symmetryand singling out medium effect would require a high accu-racy in theory – such as, e.g., high-order 1 /N c corrections– which the skyrmion description is not capable of provid-ing. The story however is quite different with the kaons. In fact, the Callan-Klebanov model [5] that describes thehyperons as bound states of the fluctuating K − ’s with an SU (2) soliton has been amazingly successful as recentlyreviewed in [6]. Here the Wess-Zumino term that encodeschiral anomalies plays a singularly important role in pro-viding the binding of anti-kaons to an SU (2) soliton. Asimilar observation was made for the two-baryon system K − pp in [7] where it was found that there is a substantialincrease in attraction between the kaon and the nucleonswhen the latter interact at a shorter distance.In this article, we consider negatively charged kaonsfluctuating in the medium described as a dense solitonicbackground. This involves two very important issues inphysics of dense hadronic matter. One is that there is apossibility that K − can trigger strongly correlated mech-anism to compress hadronic matter to high density [8].Such a mechanism is thus far unavailable in the literatureand may very well be inaccessible in perturbation the-ory [9]. The other issue is the role of kaon condensation indense compact star matter which has ramifications on theminimum mass of black holes in the Universe and cosmo-logical natural selection [10]. It would be most appealingand of great theoretical interest to address this problemin terms of the instanton matter given in hQCD, whichhas the potential to also account for shorter-distance de-grees of freedom via an infinite tower of vector and axialvector mesons. However numerical work in this frame-work is unavailable. A skyrmion matter consisting ofpion and ˜ ρ has been studied [11] but has not yet been fullyworked out. We shall therefore take the simple Skyrmemodel implemented with two key ingredients, viz, theWess-Zumino term and the “soft dilaton” field χ s thataccounts for scale symmetry tied to spontaneously brokenchiral symmetry as precisely stated in [12]. Our task isto understand kaon fluctuations in the background givenby the skyrmion matter described by this Lagrangian.What makes this approach different from others is thatit exploits the close connection between scale symmetrybreaking encoded in the dilaton condensate which can berestored `a la Freund-Nambu [12, 13] and chiral symmetrybreaking encoded in the skyrmion matter, both linked tothe mass of light-quark hadrons. There are certain fine-tunings required with the parameters of the model toachieve a quantitative comparison with nature, which wewill eschew in this article. We shall instead focus moreon robust qualitative features. The basic approximationthat we will adopt – which will have to be ultimately jus-tified – is that the back-reaction of kaon fluctuations onthe background matter can be ignored, which is consis-tent with large N c consideration.Our result is summarized in Fig. 1.We find in the model three different regimes in prop-erties of the kaon as density increases. At low den-sity with chiral symmetry spontaneously broken with h ¯ qq i ∝ Tr U = 0 and f π = 0, the state is populated byskyrmions and the mass of the kaon propagating thereindrops at the rate controlled by chiral perturbation theoryvalid at low density. This behavior continues up to thedensity n / at which the skyrmion matter turns into ahalf-skyrmion matter characterized by Tr U = 0 so chiralsymmetry is restored but f π = 0. In this phase, the kaonmass undergoes a much more dramatic decrease until itvanishes – in the chiral limit – at the critical density n c at which Tr U = f π = 0. The region between n / and n c ,dubbed as “hadronic freedom” regime [12] – which alsoplayed an important role in explaining dilepton processesin heavy-ion collisions [14], is most likely inaccessible bylow-order chiral perturbation theory. Given the extremetruncation of the model used here, it makes little senseto attempt to pin down precisely the onset density of thehalf-skyrmion phase. The best guess would be that itfigures at a density between 1.3 and 3 times the normalnuclear density ≈ .
16 fm − , the range indicated in Fig. 1for the set of parameters chosen. II. The Lagrangian — To construct dense nuclear mat-ter into which kaons are to be embedded, we take the SU (2) f Skyrme Lagrangian given in terms of the pionfield only [15] and construct a dense baryonic matter byputting the skyrmions on a crystal. This Skyrme La-grangian can be considered as an effective low-energy La-grangian valid at large N c in which all other degrees offreedom are integrated out. In fact, it is seen in hQCDthat the Skyrme quartic term – considered in the pastas ad hoc – does arise naturally and uniquely, captur-ing physics of shorter-distance than that of the lowestvector excitations, i.e., ˜ ρ . There is however one crucialingredient that needs to be implemented to the SkyrmeLagrangian, namely, the dilaton field that accounts forthe trace anomaly of QCD. In considering dense mat-ter, it is essential that the scale-symmetry breaking en-coded in the trace anomaly be mapped to the sponta-neously breaking of chiral symmetry. This point was im-plicit already in the 1991 proposal for scaling of hadronmasses [16] but it was in [12] that the dilaton field thatfigures in the connection was clearly identified. Thereis a subtlety in distinguishing the “soft dilaton” χ s with m* K -
16 fm − is the normalnuclear matter density) in dense skyrmion matter which con-sists of three phases: (a) h ¯ qq i 6 = 0 and f ∗ π = 0, (b) h ¯ qq i = 0and f ∗ π = 0 and (c) h ¯ qq i = 0 and f ∗ π = 0. The parameters arefixed at √ ef π = m ρ = 780 MeV and dilaton mass m χ = 600MeV (upper panel) and m χ = 720 MeV (lower pannel). which we are concerned here and the “hard dilaton” χ h which is associated with the asymptotically free runningof the color gauge coupling constant reflecting scale sym-metry breaking in QCD. This is discussed in [12]. Whatis important for our purpose is that the condensate ofthe soft dilaton vanishes across the chiral restorationpoint whereas that of the hard dilaton remains non-zeroacross the critical point. The vanishing of the soft dilatoncondensate is directly connected to the vanishing of thequark condensate (in the chiral limit) and hence to chiralsymmetry as shown in the case of dileptons in heavy-ioncollisions [14]. It has not yet been rigorously establishedbut is plausible [14, 17] that the same holds in the caseof density. In this article, we will simply assume that itdoes and deal uniquely with the soft component whichwe will denote simply by χ .Extended to three flavors and implemented with thedilaton field χ , the Skyrme Lagrangian we shall use takesthe form [11, 12] L sk = f (cid:18) χf χ (cid:19) Tr( L µ L µ ) + 132 e Tr[ L µ , L ν ] + f (cid:18) χf χ (cid:19) Tr M ( U + U † − ∂ µ χ∂ µ χ + V ( χ ) (1)where V ( χ ) is the potential that encodes the traceanomaly involving the soft dilaton, L µ = U † ∂ µ U , with U the chiral field taking values in SU (3) and f χ is thevev of χ . We shall ignore the pion mass for simplic-ity, so the explicit chiral symmetry-breaking mass termis given by the mass matrix M = diag (cid:0) , , m K (cid:1) . In SU (3) f , the anomaly term, i.e., the Wess-Zumino term, S W Z = − iN C π R d xε µνλρσ Tr ( L µ L ν L λ L ρ L σ ), turns outto play a crucial role in our approach: Fluctuating Kaons in the Skyrmion Matter — In closeanalogy to the Callan-Klebanov scheme [5], we con-sider the fluctuation of kaons in the background of theskyrmion matter u , following [18], as U ( ~x, t ) = p U K ( ~x, t ) U ( ~x ) p U K ( ~x, t ) , (2) U K ( ~x, t ) = e i √ fπ KK † , U ( ~x ) = (cid:18) u ( ~x ) 00 1 (cid:19) . (3)Substituting (2) into (1) and the Wess-Zumino term, weget the kaon Lagrangian in the background matter field u ( x ) L K = (cid:18) χ f χ (cid:19) ˙ K † G ˙ K − (cid:18) χ f χ (cid:19) ∂ i K † G∂ i K − (cid:18) χ f χ (cid:19) m K K † K + 14 (cid:18) χ f χ (cid:19) (cid:0) ∂ µ K † V µ ( ~x ) K − K † V µ ( ~x ) ∂ µ K (cid:1) , + iN c f π B (cid:16) K † G ˙ K − ˙ K † GK (cid:17) . (4)where χ ( ~x ) is the classical dilaton field and V µ ( ~x ) = i ∂ µ u † ) u − ( ∂ µ u ) u † ] , (5) G ( ~x ) = 14 ( u + u † + 2) , (6) B µ ( ~x ) = 124 π ε µνλσ Tr (cid:16) u † ∂ ν u u † ∂ λ u u † ∂ σ u (cid:17) . (7)In the spirit of mean field approximation, we will take thespace average on the background matter fields u and χ and obtain L K = α ( ∂ µ K † ∂ µ K ) + iβ ( K † ˙ K − ˙ K † K ) − γK † K (8) where α = (cid:10) κ G (cid:11) , β = N c f π (cid:10) B G (cid:11) , γ = (cid:10) κ G (cid:11) m K (9)with κ = χ /f χ . Lagrangian (8) yields a dispersion rela-tion for the kaon in the skyrmion matter as α ( ω K − p K ) + 2 βω K + γ = 0 . (10)Solving this for ω K and taking the limit of p K →
0, wehave m ∗ K ≡ lim p K → ω K = − β + p β + αγα . (11)This equation will be used for evaluating the in-mediumeffective kaon mass. III. Skyrmion Crystal — The key element in Eq. (8) forthe kaonic fluctuation is the backfround u which reflectsthe “vacuum” modified by the dense skyrmion matter.The classical dilaton field tracks the quark condensateaffected by this skyrmion background u that carries in-formation on chiral symmetry of dense medium. Our ap-proach here is to describe this background u in terms ofcrystal configuration. The pertinent u has been workedout in detail in [1], from which we shall simply import theresults for this work. As shown there, skyrmions put onan fcc – which is the favored crystal configuration – makea phase transition at a density n / to a matter consist-ing of half-skrymions. With the parameters of the La-grangian picked for the model, we find n / ∼ . n butthis is quite uncertain. As noted, our guess would rangefrom slightly above n to ∼ n . This is also where theinstanton-to-half-instantons (or dyons) transition takesplace in hQCD [3]. In this paper, we will not attempt aquantitative estimate but consider the above density asa ball-park value. IV. Vector Mode — An important aspect of the half-skyrmion state is that the quark condensate h ¯ qq i – whichis proportional to Tr( u + u † ) – is zero in this phase butthe pion decay constant f π – which is proportional to h χ i in medium – could be non-zero. This means that the half-skyrmions are hadrons, not deconfined quarks. There isa resemblance to fractionized electrons in (2+1) dimen-sions in condensed matter physics [19]. In hidden localsymmetry theory [20], that h ¯ qq i = 0 and f π = 0 impliesthat f π = f s with h | A µ | π i = ip µ f π and h | V µ | s i = ip µ f s where s is the longitudinal component of the ρ meson.This corresponds to the “vector mode” conjectured byGeorgi [21] to be realized in the large N c limit of QCD.Such a mode does not exist in QCD proper with Lorentzinvariance. However one can think of it as an emergentsymmetry in the presence of medium, which dense mat-ter provides. The hadronic freedom regime mentionedabove encompasses this mode with a ≡ f s /f π ≈ g ≈ g is the hidden gauge coupling constant)near the critical point ( T c or n c ). V. Effective Kaon Mass — We now discuss the resultsgiven in Fig. 1 in some detail.The results are given for two values of dilaton mass m χ = 600 MeV and 720 MeV. As explained in [12], thedilaton mass is known neither experimentally nor the-oretically. We have taken two values which we considerreasonable for the “soft dilaton.” The former correspondsto the lowest scalar excitation seen in nature and the lat-ter to the effective scalar needed in Fermi-liquid descrip-tion of nuclear matter implementing BR scaling [22]. Asshown in [1], the density at which the half-skyrmion mat-ter appears is independent of the dilaton mass. In fact,the n / is entirely dictated by the parameters that givethe background u . For the skyrmion background, wepick √ ef π – which is the vacuum ρ mass at tree order–to be ∼
780 MeV which fixes e for the given f π ≈ n c at which the quark condensate vanishestogether with the pion decay constant and more signifi-cantly, the rate at which the kaon mass drops as densityexceeds n .What is novel in our model is that the behavior of thekaon is characterized by three different phases, not two asin conventional approaches. In the low density regime upto ∼ . n , the kaon interaction can be described by stan-dard chiral symmetry treatments. For instance, the bind-ing energy of the kaon at nuclear matter density comesout to be ∼ ∼
80) MeV for m χ = 600(720) MeV.This is what one would expect in chiral perturbationtheory (see [24] for review). Going above n , however, asthe matter enters the half-skyrmion phase with vanishingquark condensate and non-zero pion decay constant, thekaon mass starts dropping more steeply. This propertyis consistent with what is observed in the “hadronic free-dom” regime in the approach to kaon condensation thatstarts from the vector manifestation fixed point of hid-den local symmetry [25]. This form of matter – which isundoubtedly highly nonperturbative – is most likely una-menable to a chiral perturbation approach. Finally at n c at which both the quark condensate and the pion decayconstant vanish, the kaon mass vanishes. This happensat n c ≈ . . n for the dilaton mass m χ = 600(720)MeV. The reason for this is that the dilaton condensate, h χ i , vanishes with the restoration of soft scale symmetry`a la Freund-Nambu explained in [12]. This means that“kaon condensation” in symmetric nuclear matter takesplace at the point at which the scale symmetry associatedwith the soft dilaton is restored. It is not clear whetherthis takes place before, or coincides with, deconfinementwhich requires the intervention of the “hard” componentof the gluon condensate ignored in [12]. VI. Compact-Star Matter and Kaon Condensation —Thus far, the kaon is treated at the semiclassical level asa quasiparticle bound to the skyrmion matter. To under-stand what this represents in nature, we should note thatwhen one quantizes the system where a kaon is bound toa single skyrmion, the bound system gives rise to the hy-perons, Λ and Σ, as shown by Callan and Klebanov [5].This would suggest that kaons bound to a skyrmion mat-ter, when collective-quantized, would correspond to hy- peronic matter. On the other hand, one can interpretthe system in Ginzburg-Landau mean-field theory focus-ing on the kaon fluctuation and identify the vanishing ofthe effective kaon mass in medium, m ⋆K , as the signal forkaon condensation. Thus in this picture, kaon condensa-tion and hyperon condensation are physically equivalent.Now a physically interesting question is what does thehalf-skyrmion matter do to compact stars? What wehave done above is to subject the kaon to the skyrmionbackground which is given in the large N c limit. In thelarge N c limit, there is no distinction between symmetricmatter and asymmetric matter. Compact stars have neu-tron excess which typically engenders repulsion at den-sities above n , and hence the asymmetry effect needsto be taken into account. This effect will arise whenthe system is collective-quantized which gives rise to theleading 1 /N c correction to the energy of the bound sys-tem. Most of this correction could be translated into acorrection to the effective mass of the kaon. We haveno estimate of this correction but it is unlikely to bebig. What is more important is the energy that arisesfrom neutron excess, called “symmetry energy” in nu-clear physics, labeled S ( n ) in the energy per particle E of the neutron-rich system appropriate for compact stars, E ( n, x ) = E ( n, x = 0) + E sym ( n, x ) (12)with E sym ( n, x ) = S ( n ) x + · · · (13)where x = ( P − N ) / ( N + P ) with N ( P ) standing for theneutron(proton) number. The ellipsis stands for higherorder terms in | x | ≤ n c because of the elec-tron chemical potential which tends to increase as thematter density increases [26]. Now the electron chemi-cal potential is known to be entirely controlled by thesymmetry energy S in charge-neutral beta equilibriumsystems [27]. Thus collective-quantization is required toaddress at what density kaons will condense. Quantizingthe moduli space of multi-skyrmion systems (or multi-instanton systems in hQCD) has not yet been workedout fully. As mentioned above, the effect of hyperons onelectron chemical potential will be automatically takeninto account once the moduli quantization is made. Evenin the absence of detailed computations, however, we cansay at least within the framework of our model that kaonswill condense at a relatively low density, say, n K ∼ < n . VII. Conclusion — Hidden local symmetry with vec-tor manifestation [20] combined with a soft dilaton ac-counting for scale symmetry restoration `a la Freund-Nambu [12] implies that the anti-kaon mass in densemedium falls more rapidly in the half-skyrmion phasethan in the skyrmion phase in which standard chiral per-turbation approach should be applicable. It is proposedthat this could be relevant to an “Ice-9” phenomenonin kaon-nuclear systems and kaon condensation in com-pact star matter. It is also argued that the half-skyrmionphase corresponds to the “hadronic freedom” regime indensity in parallel to that in temperature that figures indilepton production in heavy-ion collisions. In this pa-per, the issue was addressed in terms of half-skyrmionsin HLS. It would be interesting to analyze the same in terms of half-instantons (or dyons) in hQCD. We hopeto address this problem in a future publication [28].
Acknowledgments — This work was supported by theWCU project of Korean Ministry of Education, Scienceand Technology (R33-2008-000-10087-0). [1] H. J. Lee, B. Y. Park, D. P. Min, M. Rho and V. Vento,“A unified approach to high density: Pion fluctua-tions in skyrmion matter,” Nucl. Phys. A (2003)427 [arXiv:hep-ph/0302019]; B. Y. Park, D. P. Min,M. Rho and V. Vento, “Atiyah-Manton approach toskyrmion matter,” Nucl. Phys. A (2002) 381[arXiv:nucl-th/0201014].[2] For a recent review, see B. Y. Park and V. Vento,“Skyrmion approach to finite density and temperature”in
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C56 (1997) 2244.[23] We take the point of view that the mesonic Lagrangianfrom which skyrmions emerge as solitonic baryons shouldfix the parameters of the theory. See M.A. Nowak, M.Rho and I. Zahed,
Chiral Nuclear Dynamics (World Sci-entific Publishing, 1996). This view is reiterated in thehQCD approach to instantonic baryons in [4].[24] A. Gal, “Overview of strangeness nuclear physics,”arXiv:0904.4009.[25] G.E. Brown, C.-H. Lee, H.J. Park, and M. Rho, “Studyof strangeness condensation by expanding about the fixedpoint of the Harada-Yamawaki vector manifestation,”Phys. Rev. Lett. (2006) 062303.[26] G. E. Brown, V. Thorsson, K. Kubodera and M. Rho,“A Novel mechanism for kaon condensation in neutronstar matter,” Phys. Lett. B291