Proton Mass, Topology Change and Tensor Forces in Compressed Baryonic Matter Mannque Rho1 1Institut de Physique Th´eorique, CEA Saclay, 91191 Gif-sur-Yvette c´edex, France & Department of Physics, Hanyang University, Seoul 133-791, Korea (Dated: December 11, 2013) ThisisasummaryofthetalksIgaveatKoreanPhysicalSocietymeeting(April26,2012,Daejeon, Korea) and the 4th Asian Triangle Heavy Ion Conference (ATHIC) (November 14, 2012, Pusan, Korea). They are based on the series of work done at Hanyang University in the World Class University III Program underthe theme of “From Dense Matter to Compact Stars.” The program wasconceivedandexecutedtounderstandhighlycompressedbaryonicmatterinanticipationofthe forthcomingRIBmachine“RAON”whichisinconstructionintheInstituteforBasicScience(IBS) 3 inKorea. Theproblemstreatedranged from theorigin of theprotonmass, topological structureof 1 barynic matter, chiral symmetry and conformal symmetry to the EoS of nuclear matter and dense 0 neutron-richmatterand to themaximum mass of neutron stars. The resultsobtained arenew and 2 intriguing and could have an impact on the novel structure of dense matter to be probed in the n accelerators “RAON,”FAIR etc. and in compact stars. a J PACSnumbers: 1 1. INTRODUCTION a “mass without mass.”[1]. ] h Inthestandardlore,theprotonmass(andthemassof t “light-quarkmesons,” say,the ρ meson to be specific), if - The landscape of hadronic phases has been fairly ex- l oneassumesthattheupanddownquarkmassesarezero c tensivelyexploredathightemperaturesthankstolattice (called the chirallimit), is saidto be entirely “generated u QCD on theory side and to RHIC and now LHC on ex- n perimentalside, butit is a totally barrenunchartedfield dynamically.” Phrasedin terms of symmetries, the mass [ is attributed to the spontaneous breaking of chiral sym- in the directionofhighdensity atlowtemperature. Lat- 1 tice QCD cannot access the density regime relevant to metry (SBCS for short). The SBCS is characterized by that the quark condensate hq¯qi, the vacuum expectation v the interior of compact stars, at present the only source 6 available for high density, and there are no theoretical value(VEV)ofthebilinearquarkfields,isnonvanishing, 6 tools thathavebeen confirmedreliable,giventhe lackof hq¯qi0 6= 0. If this were the entire story, then one would 0 have, by ‘dialling’ the quark condensate, that experiments available at high density. The phase struc- 0 tureofcompressedbaryonicmatterbeyondnuclearmat- . m (hq¯qi)→0 as hq¯qi→0. (1.1) 1 terdensityismoreorlessunknownandposesachallenge N 0 in hadron/nuclear physics. 3 But QCD does not say that this is the entire story. In In this note I would like to discuss a line of work done 1 fact,itispossibletohaveamasstermintheprotonthat recently to go from nuclear matter density to densities : does not vanish when the quark condensate goes to zero v relevant to the interior of compact stars at zero tem- without violating chiral symmetry in the chiral limit. It i X perature. The merit of the work is that it is a unified iseasytoundersandthisifonerecallstheSU(2)×SU(2) approach to baryons and mesons in and out of medium r Gell-Mann-L´evylinearsigmamodelwiththedoubletnu- a anchored on one effective Lagrangian with the symme- cleons, the triplet pions and the scalar σ meson [2]. In tries (assumed to be) embodied in QCD, valid up to the this model, the nucleon and the scalar σ acquire masses density at which deconfinement sets in. The principal entirelybytheVEVofthesigmafieldhσi 6=0whilethe 0 theme will be that the origin of the proton mass plays a pionremainsmasslessbyNambu-Goldstonetheorem. On keyroleintheEoSforcompactstarsandcanbeexplored the otherhand, asnotedbyDeTar andKunihiro[3], one in the forthcoming terrestrial accelerators such as RIB can have the nucleon mass in the form machines (e.g., “KoRIA” or RAON in Korea, FRIB of MSU/Michigan...),FAIRofDarmstadt/Germany,NICA m =m +m¯(hq¯qi) (1.2) N 0 atDubna/Russiaetc. andthespaceobservatoriesinop- eration and in project. I suggest this as a direction for such that m¯ → 0 as the condensate is dialled to zero the coming era in hadron/nuclear physics in Korea. provided one introduces parity doublet to the nucleon. The problem is that while quark masses could be ex- Then the nucleon mass does not vanish if m does not. 0 plained by the discovery of the Higgs or Higgs-like bo- Here m is a chirally-invariantmass term. This model is 0 son, the bulk, say, 98%, of the mass of proton whose referred to as baryon parity-doublet model. constituents are nearly zero mass quarks and massless Now what about the mesons? In the framework I am gluons, remains unexplained. This is because unlike adopting, the situation is different for mesons. When molecules, atoms and nuclei, the mass of the proton is the chiral symmetry is restored, the σ mass has to join 2 the pion mass, hence must go to zero in the chiral limit. to date no convincing evidence, either positive or nega- Whenvectormesonsaresuitablyintroducedaccordingto tive, has come up. No experiments so far succeeded to hidden local symmetry which is gauge equivalent to the single out unambiguously the order parameter that in- (nonlinear)sigmamodel[4],themassofthevectormeson dicates what is happening. Let me briefly explain this ρ also has to go to zero due to what is known as “vector without going into details of the experiments performed manifestation”(VMforshort)ofhiddenlocalsymmetry. andthenumeroustheoreticalinterpretationsputforward Itmaybethatthemassesofotherlight-quarkmesonsbe- ‘explaining’ the observations. longing to certain flavor symmetries all go to zero in the The currently popular idea for unravelling vacuum sense of “mended symmetries” `a la Weinberg [5]. There structure of chiral symmetry is to ‘measure’ the mass is thus an apparent difference in this picture between ofa hadronin a medium, at hightemperature and/orat baryon and meson masses in the way chiral restoration highdensity. Butthequestionis: Whatisthemeaningof is reached. I will discuss later that this difference can “in-medium” mass? Since the detector is outside of the be avoided if one resorts to the constituent quark pic- mediumwherethetemperatureanddensityhavevacuum ture. I should note in this connection that by artificially values while the quantity of a hadronone wants to mea- unbreaking chiral symmetry in a dynamical lattice sim- sureistheoneinsideamedium,thereisnounambiguous ulation [6], Glozman et al. find that both mesons and meaning of mass inside hot or dense medium when mea- baryonsremainmassiveafterchiralsymmetryispresum- sured on the detector located outside. In the past, what ably restored [7]. In fact, in baryons, m0 is found to be one did was to detect weakly interacting particles that large. carrywith as little disturbance as possible the snap-shot The underlying theme in my discussion will be that a pictureoftheinteractionsthattakeplaceinthemedium. substantialm0 isindicatedintheequationofstate(EoS) Thusonemeasuredtheinvariantmassoftheleptonpairs of nuclear matter and dense neutron-rich(compact-star) l+l− where l=e,µ arising from the decay of in-medium matter. It figures in nuclear dynamics in a highly intri- vector mesons, typically the ρ meson. The hope was to cate way. It is important to note that such a large m0 see the invariant masses corresponding to the ρ meson raisesafundamentalquestioninphysics: Wheredoesthe mass sliding in medium with densitiy or temperature re- protonmassassociatedwithm0 comefromifitisuncon- flecting the in-medium “vacuum” property of the quark nected to the spontaneous breaking of chiral symmetry? condensate. These were studied in heavy-ion collisions probing high temperature and in the electroproduction of ρ mesions in nuclei probing density effects. It turned 2. UNBREAKING SYMMETRIES out that the results were inconclusive. In [8], I give my arguments chiefly drawn from the references given in [8] thatthoseexperimentsdidnotsucceedtosingleout,and If hadron masses arise by the spontaneous breaking take snapshot of, the order parameter hq¯qi as intended. of chiral symmetry by the vacuum characterized by the condensate hq¯qi 6= 0, then one should be able to tweak That the vector meson mass sliding with density the vacuum such that the broken symmetry is restored or with temperature measured in dilepton productions and the fate of hadron masses is revealed. It is gener- could not give a good snapshot of the vacuum structure ally believed that when a hadronic system is heated to reflecting chiralsymmetry was already pointed out right high enough temperature or compressed to high enough after the scaling relation – called “BR scaling” [9] – was density, the quark condensate will go to zero or to near proposed. See [10] for a footnote remark on this point. zero. This is confirmed by lattice QCD in the chiral There are several reasons why such measurements limit for high temperature. The situation with the den- could have failed to exhibit the phenomenon looked for. sity, however,is totally unclear because there is no QCD The most trivial – and unquestionably disappointing – calculation for dense matter: Lattice method cannot ac- possibility I did not touch on in [8] could be that as one cess high-density matter. So it is not known rigorously approachesthe chiralrestorationpoint at high tempera- whetherthequarkcondensatedoesindeedvanishatsome tureand/orhighdensity,hadronscouldsimplybreakinto high density or put another way, whether chiral symme- pieces and cannot be described by local fields. The class try is actually restored at such high density presumably ofpictureswherepercolationtakesplacebeforethetran- present in the interior of compact stars. One would like sition could perhaps be put in this category. Although to find this out by experiments. this possibilitycannotbe dismissed–andI havenothing Numerous efforts have been made – and will continue further to say thereon, let me focus on the alternative tobemade–toverifywhetherthepresumedlinkbetween possibility that hadronic degrees of freedom can be de- the mass and the quark condensate can be established. scribed in terms of local hadronic fields up to the point One obvious experimental way is to try to “unbreak chi- where the phase transition takes place. ral symmetry” by heating hadronic matter to high tem- If one assumes that local field theory makes sense up perature or compressing it to high density by heavy ion to the phase change, then one can convince oneself, us- collisions with the objective to make the quark conden- inghiddenlocalsymmetry,thatcertainpropertiesofEM sate go to zero. So far there have been many (elaborate) interactions that are known to hold in the (matter-free) experiments to do just this, but I have to say that up vacuum do not apply to hot and/or dense matter. Of 3 particular importance is that the “vector dominance” – out the role of chiral symmetry in nuclear dynamics by which holds well in matter-free space and is simply as- ‘measuring’ the property of a (light-quark) hadron in a sumed by most workers in the field to hold in medium process that takes place in a density regime near that – does not hold as the temperature or the density ap- of nuclear matter. The presence of large widths etc. as proachesthe critical. The consequence discussedin [8] is observed in the experiments and much discussed in the that the dileptons become “nearly blind” to the part of literature should come as no surprise; it merely reflects the mass, i.e., m¯(hq¯qi), that encodes information on the that strong interactions are undeed taking place with a quark condensate as it approaches zero. largenumber ofchannels openin heatbathand/orcom- As far as I know, the connection of the in-medium pressed matter into which excitations of the ρ quantum masstothequarkcondensatehq¯qiispreciselygivenonly numbers can decay. ‘Seeing’ the ρ meson signalling the in hidden local symmetry theory and even there (in the effect of (2.1), even if present in such an environment, chiral limit) only in the vicinity of the VM fixed point would be like seeing a needle in a haystack. with vanishing condensate. It has been established [4] I will mention below that something analogous hap- thatashq¯qi→0,thepolemass–notjusttheparametric pens with the nuclear symmetry energy. mass – of the vector meson ρ goes as m∗ g∗ hq¯qi∗ ρ → ∝ (2.1) 3. TOPOLOGY CHANGE m g hq¯qi ρ where g is the hidden gauge coupling. This comes about I will now focus on density effects and consider dense because of the flow to the VM fixed point in HLS the- baryonic matter. It will be described in terms of ory (VM/HLS for short) that is forced upon by match- skyrmionsthatariseastopologicalsolitonsfromaneffec- ing with QCD. I expect this to more or less hold near tive Lagrangian that has the symmetries assumed to be the chiral transition point even if the chiral limit is not presentinQCD.Therewillbe noneedtoputinbaryons assumed. In the close vicinity of the VM fixed point, by hand. As stated in Introduction, I will stick to a sin- the gauge coupling approaches zero so the width will gle effective Lagrangian, a generalized nonlinear sigma also get suppressed. Thus the ρ meson should become modelthatcontainsvectormesons`alahiddenlocalsym- a sharper resonance near the critical point. The irony metry(HLS)andascalarmeson,thedilatonχassociated here is, however,that the photon becomes ‘nearly blind’ with spontaneously broken scale symmetry (SBSS).#2 to such dropping-mass vector mesons by the same VM Let me call this dHLS Lagrangian, “d” standing for the mechanism [8]. dilaton. I will come back in the next section to the role Now most of the experimental measurements per- of a scalar meson of vacuum mass ∼ 600−700 MeV in formed so far involve temperatures or densities remote nuclearinteractions. Itwillplaya crucialroleonhadron from the critical point. In this case, the meaning of in- masses sliding with density in effective Lagrangians. mediummassbecomes evenmoreblurred. Inthe hidden The power of the skyrmion approach is that one can gauge theory framework that I am adopting, there is no describe mesons, elementary baryons and multibaryon reasonforthe directlink (2.1) betweenthe massandthe matter, all with one effective Lagrangian. This allows condensate to hold far away from the VM fixed point. one to do as consistent a treatment as feasible, avoiding For instance in dense matter near nuclear matter den- arbitrarymixingofvariousdifferentmodelsingoingfrom sity,the“effectivemass”oftheρmesoncanberelatedto oneregimeofdensitytoanotherregimeashasbeendone the Landau-Migdal Fermi-liquid parameter F , which is 1 inthepast. Imustadmitthatthereisofcourseapriceto afixed-pointparameterintheeffectivefieldtheoryofnu- payforsucha“unifiedapproach”: Giventhe constraints clearmatter[11]. Anaptandnon-trivialexampleofsuch and nonlinearity inherent in the Lagrangianpicked, it is a relation is found in the anomalous orbital gyromag- technically difficult to do fully reliable quantum calcula- netic ratio δg of heavy nuclei [11]. This relation means l tions. I will try – and to some extent, succeed – to fi- that the so-called density-dependent “ρ mass” contains, nessethisdifficultybyresortingtowhatNatureindicates, among others, quasiparticle interactions near the Fermi in particular in fixing parameters of the model. I will sea,aquantitythatcannotbesimplyanddirectlylinked be generally thinking in terms of the dHLS Lagrangian to the order parameter of chiral symmetry.#1 Stated L(π,ρ,ω,χ)#3. Howeverwheneverqualitatively reliable, more to the point, there is no way that one can single I will discuss with the simpler Skyrme Lagrangian (con- tainingthequadraticcurrentalgebratermandthequar- tic Skyrme term) [13] implemented with the dilaton χ #1 It may very well be that the Landau parameters – or in that matter,othermundane-lookingnucleareffectssuchascollisional broadening discussed in the literature – have something ulti- mately to do with chiral symmetry of QCD but arguing that #2 WhatismeantbydilatonisexplainedinSection5. they areconsistent withor reflecting chiral symmetrywouldbe #3 AworkisinprogresswithaLagrangianthatcontainsaninfinite senseless. It’s as devoid of meaning as saying “chiral dynamics tower of vector mesons coming from gravity-gauge dual QCD explainnuclei.” models[12]. 4 (call it dSkyrme). The dSkyrme can be considered as a physics[18]aswellasahighlytopicalissueincondensed dHLS from which the vector mesons are integrated out. matter physics [19]. What might be happening in the It should be qualitatively reliable at densities far away hadronic case I am dealing with is much less clear be- from the VM fixed point where the hidden gauge sym- cause we do not know what quantum (1/N ) corrections c metry is indispensable and the vector mesons cannot be will do. What I will do is to exploit the possibility that integrated out.. topologymaybecapturedbyasmoothchangeinbound- One natural way to describe many-baryon systems in ary conditions in Hilbert space#5 which in the present thepresentframeworkistoputmultiskyrmionsonFCC case, corresponds to a change of parameters in the La- crystalandreducethecrystalsizetosimulatedensemat- grangian. The parameter change will then reflect the ter [14]. In the large N consideration on which the vacuum change in medium as density exceeds n . c 1/2 skyrmion Lagrangian relies, it is justified to consider dense matter in a crystal form [15]. But nuclear mat- ter is most likely not in a crystal form, the deviation 4. PARAMETER CHANGES FROM NUCLEAR from crystal being effects higher order in 1/N . I will, SYMMETRY ENERGY c however, argue that what we can reliably deduce from the crystal calculation is the topology involved rather Since we do not know how to fully quantize skyrmion than specific dynamical contents, and it can be applied matter, the strategy is to extract density-dependent pa- to lower density even if the crystal structure may not be rametersoftheeffectiveLagrangiancapturingthe topol- a good dynamical description there. ogy change from the skyrmion crystal calculation and Ithasbeenestablishedthatatcertaindensity,itisen- then do field theoretic many-body calculations with the ergetically more favorablethat a skyrmion on FCC frac- Lagrangiansodefined. This hasbeendoneby lookingat tionize into two half-skrymions in CC or BCC [16, 17]. the nuclear “symmetry energy factor” E [21], sym What happens is that by squeezing the crystal size (in- creasing density), one induces a “phase change”#4 from E(n,δ)=E0(n)+Esym(n)δ2+··· (4.1) skyrmion matter to half-skyrmion matter at a density where E is the energy per baryon of the system, E denoted n [14]. This phase change – which is generic 0 1/2 is the symmetric part of the energy with δ = 0 and independently of the specific meson degrees of freedom δ =(N −Z)/(N +Z) with N(Z) is the number of neu- involved apart from the pions – engenders the change of trons (protons). One cancalculate the symmetry energy the quark condensate defined on the average in the unit factorE bycollective-quantizingtheskyrmioncrystal cell from Σ ≡ hq¯qi| 6= 0 in the skyrmion phase to sym unit representing pure neutron matter [22]. One obtains Σ = 0 in the half-skyrmion phase. Locally the conden- sate is not equal to zero in the half-skyrmion phase, so 1 there is a modulated scalar density distribution. In this Esym(n)≈ (4.2) 8Ω (n) cell phase, the pion decay constant does not vanish. There- fore, pions are propagating and so are other mesons and where Ω is the moment of inertial of the single cell of cell baryons. This means that although the Σ is zero on the the crystal given by the integral over the cell with the average, chiral symmetry is not actually restored, and crystal configuration U (r) for the chiral field with the 0 the confinement persists. There must therefore be an lowest energy for a given density. orderparameterforchiralsymmetry thatisofhigherdi- The result is shown in Fig. 1 for two different masses mensionfieldoperators. Thismustbeamedium-induced for the dilaton appropriate for the system #6. The ki- operator. One can also think of this phase as “quarky- netic energy contribution – suppressed for large N – is c onic.” not included in (4.2)#7. As I will show later, there are One may question whether this transition is real and other correlation terms, higher order in 1/N , that turn c not just an artifact of the crystal structure which may out to become significant in medium, so this cannot be not be realistic at not so high density. I have no simple compareddirectlywithnature. Whatissignificantisthe answer to this question since I do not know how to com- qualitative feature of the cusp at the transition density putequantumcorrectionshigherorderin1/N . However n . c 1/2 I suggest that given that what is involved is a topology changeinvolvingdifferentsymmetriesasinotherareasof physics, the approach could well be reliable, unaffected qualitatively by higher order 1/Nc corrections. In fact, #5 Anexamplethat illustratestopology change intermsofchange topology change is currently a deep issue in quantum in boundary conditions is the chiral bag model [20]. There the topological charge representing baryon charge can be continu- ouslychanged bychangeinthebagboundaryconditions. #6 Inrelativisticmeanfieldcalculations ofnucleiandnuclearmat- ter, the scalar meson mass usually taken is ∼ 600 MeV. This #4 I put this in parenthesis to indicate that it is not clear how to corresponds to the vacuum mass ∼ 750 MeV which drops to interpret the phenomenon in terms of the standard Ginzburg- ∼600MeVbyscalingatnuclearmatterdensity. Landau-Wilson paradigm for phase transitions. I will however #7 Becauseofthestrongtensorforces,thekineticenergycontribu- looselyusethisterminologyforthechangeover involved. tioncouldinrealitybestronglysuppressed[23]. 5 coupling drops rapidly toward zero as density in- creases. Thisisbecausebothg →0duetotheVM and (1−g )→0 due to the DLFP [25]. V The net result is that the old scaling in the Lagrangian – called BR scaling [9] – is replaced by a new scaling at n = n [21, 22, 26], that I will call “BLPR,” stand- 1/2 ing for Brown, Lee, Park and Rho involved in various aspects of the scaling relation. One of the principal con- sequences of BLPR is that the ρ-tensor force is strongly suppressed for n ≥ n , while the pion tensor remains 1/2 more or less intact, thereby leaving only the π tensor for n >∼2n0. This is depicted in Fig. 2, lower panel, for the case with n ∼ 1.3n taken as an example#8. We see 1/2 0 FIG. 1: Symmetry energy factor predicted by the skyrmion crystal at n1/2 =1.3n0. Note thecusp structureat the tran- 20 n/n0=1 sition density. n/n0=2 n/n0=3 10 To understand the cusp structure, it turns out to be V] most informative to look at the behavior of the tensor Me forcesasdensityincreases. Thisisbecausethesymmetry T~ V[ energyisdominatedbythe tensorforces(foranupdated 0 1 2 3 4 account, see [23]). Approximately, the symmetry energy factor E from the tensor forces in standard nuclear r [fm] sym theory can be written as [24] -10 12 E ∼hV i≈ hV2(r)i (4.3) sym sym E¯ T where E¯ ≈200 MeV is the averageenergy typical of the 20 tensorforceexcitationandV istheradialpartofthenet n/n0=1 T n/n0=2 tensor force. This will be identified with the mean-field 10 n/n0=3 result of the Lagrangian with density-dependent param- eters deduced from the topology change. V] e M forDceesscbreibtwedeenintwthoenudcHleLoSnsm(iondeplrinadcioppleteodb,tathineedtefnrosomr T~ V[ 0 1 2 3 4 skyrmions) are given by the exchange of a pion and a ρ r [fm] -10 meson. They come in with opposite signs, so that their effects tend to cancel. The property of their net effect depends sensitively on the scaling of the masses of the -20 mesonsandthenucleoninvolvedaswellasonthemeson- nucleon coupling constants in medium. The main qual- FIG. 2: Sum of π and ρ tensor forces in units of MeV for itative results from the skyrmion crystal combined with densities n/n0 =1, 2 and 3 with the “old scaling” (upper the VM/HLS and the dilaton-limit fixed point (DLFP panel) and with the “new scaling,” (lower panel) with m0 ≈ for short) of dHLS discoveredin [25] can be summarized 0.8mN. The topology change is put at n1/2 =1.3n0. by the modifications in the scaling of parameters in the Lagrangianfromtheonesproposedin1991[9]. Theyare, for n>∼n1/2, from Fig. 2 that the net tensor strength at the interme- diate and long ranges decreases due to the cancellation • The effective mass of the nucleon as a soliton in medium goes like m∗ ∼ f∗ and the effective pion N π decayconstantf∗dropsslowlyinthehalf-skyrmion π phase. Therefore the nucleon mass stays more or less un-scaling after n , consistent with a large #8 Theexactlocationofn1/2isnotdeterminedfromthecrystalcal- 1/2 culation performedsofar. It willrequirean involved numerical m0 in (1.2). work with the dHLS Lagrangian. With the simplified dSkyrme Lagrangian, it comes, for reasonable values of (scaling) param- • The effective coupling of the ρ meson to nucleons eters, to 1.3 <∼n1/2/n0 <∼2.0. This is the range supported by gρN =g(1−gV) where gV is the “induced” vector nuclearphenomenology [21]. 6 between the π and ρ tensors up to the density n , af- sionsonthismatterandforpreviousreferences.) Onecan 1/2 ter which the ρ tensor starts getting suppressed and the associate this dilaton, i.e., soft glue, with the low-mass pion tensor quickly taking over with its full strength. It scalar needed in nuclei and nuclear matter. I note here is easy to see that the cusp structure of the skyrmion ananalogyto what is being done for the 125 GeV boson result is reproduced qualitatively by the formula (4.3) discovered at LHC, that is, to describe it as a Higgs- with the BLPR scaling. This supports the identification like dilaton, with scale symmetry broken spontaneously of the scaling parameters with the topology change at by a weak external conformal symmetry breaking. The the mean-field level. We will see that higher order nu- pseudo-Goldstone nature of the boson is to account for clear correlations do smooth the cusp structure, but the the low mass of the discovered boson. main feature will remainin a more realistic treatmentof Currently a highly topical issue, the QCD structure EoS. of the low-mass scalar is poorly known. It is most Itshouldbenotedthatthechangeinthetensorforces, likelya complicatedmixture ofglue balland(qq¯)n (with affecting importantly the symmetry energy of compact n = 1,2,···) configurations. For our purpose, we will stars, could also have a precursor effect on the structure notneedtospecify itsdetailedstructure. Wewillsimply ofasymmetricnuclei,suchasforinstanceshellevolution, take it to be the dilaton χ associated with spontaneous to be studied in RIB accelerators. breaking of scale symmetry. Its condensate hχi, locked to hq¯qi, will go to zero when chiral symmetry is restored (in the chiral limit). The trace anomaly associated with 5. DILATON IN NUCLEAR MATTER theSBSSwillthengotozeroatthechiraltransition. Af- ter chiral restoration there will still remain the explicit breakingduetothetraceanomalytiedtotheasymptotic Innuclear physics,itis essentialthatthere be a scalar freedom. This piece is chirally invariant and is presum- degree of freedom with a vacuum mass of ∼ 600−700 ably responsible for non-zero m . MeV. It plays a key role in phenomenological as well 0 In constructing the dHLS Lagrangian consistent with as one-boson-exchange potentials and also in relativis- SBSS, the χ field enters as a “conformal compensator tic mean field approaches to nuclei and nuclear matter. field.” Except for the explicit chiral symmetry breaking Roughly, the scalar field provides the attraction to bind term (i.e., pion mass term), the procedure is straightfor- nucleons in nuclei with the stabilization provided by a ward. The condensatehχi–signallingSBSS–tracksthe vector repulsion. In nature there is no sharp and well- vacuumchangeasdensityisincreasedandmakesthepa- defined scalar in the vacuum with the mass needed. The rameters of the Lagrangian dHLS change as the density meson f (600) is listed in the particle data booklet, but 0 of the system goes across n . its structure in terms of QCD is not understood and 1/2 Inthelowdensityregimesuchasinnucleiandnuclear remains controversial despite intensive works on it and matter, relativistic mean field models require that the other scalars. In chiral perturbation theory, the scalar χ field be chiral-singlet or dominated by a chiral-singlet “resonance” in π−π interactions can be approximately component. However as one approaches the density at reproducedbyhigh-orderloopcalculations. Alsotheone- which chiral phase transition is to take place, the scalar scalar-boson(σ)exchangeintheOBEPNNpotentialcan should change over to the σ – which is the fourth com- be approximately described by irreducible two-pion ex- ponentof the chiralfour vector– of the Gell-Mann-L´evy changes(involving∆resonanceintermediatestates,form sigma model or its generalization to the parity-doublet factors etc.). Both are perturbative in nature. sigma model. How one can go from low density to high Whatweneedforourpurposeis,however,alocalfield density in the framework of dHLS is discussed in terms forthescalarexcitationtobetreatednon-perturbatively, of the dilaton limit fixed point (DLFP) in [25]. The that is, in the mean-field approximation. One way to changeoverpresumably involves the scalar meson under- introduce such a scalar field in the context of scaling goingasortoflevel-crossingatsomedensity. Itmayalso parameters in effective Lagrangians was first suggested involve the role of 1/N corrections at varying densities. in 1991 [9]. What has been done in the work I am re- c Both are very difficult questions to address at present. porting here follows essentially that approach but in a Whether the picture we have adopted is viable or not much more refined and improved form. The basic idea remains to be checked by experiments. is to link chiral symmetry to scale (or conformal) sym- metry in such a way that spontaneous breaking of scale symmetry (SBSS) triggers spontaneous breaking of chi- ral symmetry (SBCS). Scale symmetry cannot sponta- 6. EOS OF COMPACT STARS neously break unless there is explicit breaking, and it is the trace anomaly that provides the natural source for The change of the symmetry energy caused by the theexplicitbreaking. Itispreciselythisobservationthat modified tensor forces at a density slightly above that suggestedtheseparationinto“softglue”and“hardglue” ofnuclearmatter hasanimportantconsequenceonEoS: of the gluon condensate and the linking of the soft com- the EoS which is soft below n , gets hardened above 1/2 ponent to the dilaton condensate hχi which in turn is n [21,26]. ThisispreciselythefeatureofEoSrequired 1/2 locked to the quark condensatehq¯qi. (See [27] for discus- forexplainingmassivecompactstarswithM ∼2M⊙ be- 7 ing observed. I would suggest that this feature could be 70 checkedinheavy-ionexperiments thatwillprobe baryon 60 denLseittiemsenb>∼rie2flny0.describe, following [21], how the ap- 50 ((AB)):: nneeww--BBRR nn11//22==21..05nn00 (A) 40 (C): no new-BR proach given above fares in confronting compact stars. The idea here is to implement the BLPR in an effective eV] 30 (B) M field theory (EFT) for nuclear matter. Since the dHLS A [ 20 Lagrangian with the parameters fixed from the crystal E/0 10 calculation is a tree-level Lagrangian, one needs to do 0 quantum calculations with this Lagrangian to confront -10 nature. In doing this, the “double-decimation” strategy can be adopted [21, 28]. -20 (C) 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1. The first decimation consists of obtaining via RG n/n0 equation the V in free space by decimating to lowk 180 the scale Λ that describes nucleon-nucleon in- lowk 160 teractions up to lab momentum to ∼ 300 MeV. It solid square: new-BR n1/2=2.0n0 is best for our problem to do this in terms of the 140 open square: new-BR n1/2=1.5n0 generalized HLS Lagrangian with the parameters 120 with the intrinsic density dependence given by the V] BLPR. E [Mesym 1 0800 expt-Li1 eexxpptt--LTis2ang 2. Theseconddecimationistodonuclearmany-body 60 calculationwiththisV totheFermi-momentum lowk 40 scale Λ . There are a variety of ways of doing fermi 20 thisstep. TheyallamountessentiallytodoingLan- dau Fermi-liquid fixed point theory and arrive at 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 nuclearmatterattheequilibriumdensityn0 ∼0.16 n/n0 fm−3. This is an EFT well-justified up to density nearn0. Thisstepfixesthescalingpropertiesofthe FIG.3: EoS forsymmetricnuclearmatter(upperpanel)and parameters in the Lagrangian up to near n = n0. symmetry energy (lower panel) with comparison with exper- Thesamescalingisassumedupton1/2 providedit imental fits, for n1/2/n0 =1.5 and 2.0. is not too high above n . 0 3. The last step is to smoothly extrapolate with the the dilepton experiments and the anomalous orbital gy- formalism to high densities and calculate the EoS romagnetic ratio δg of heavy nuclei where one could not l forcompactstars. Indoingthis,onecanadoptchi- naively associate what is observed with a signal for chi- ralperturbationstrategyandincluden-bodyforces ral symmetry, here what is predicted from the topology – if one wishes – for n > 2, suitably introducing change, i.e., the cusp, becomes nearly invisible in the scaling parameters for the n-body forces. In [21], background of many-body correlations. We clearly need the topology change is incorporated in a smooth a cleverer idea to unearth the effect we would like to ex- manner in terms of the changes in intrinsic scaling pose. at n . It however ignores other degrees of free- 1/2 The equations of state so obtained predict the star domthatmightenter,suchasstrangenessthatcan properties depicted in Fig. 4. Thanks to the stiffening be manifested in terms of kaon condensation (or at n1/2, one gets a massive star M ∼2.4M⊙. The maxi- equivalently hyperons) and strange quarks etc. mum density reached for such a star is ∼5n . 0 Thecalculatedresultsthatcomefromtheaboveproce- dures[21]aregiveninFig.3. Theupperpanelshowsthe EoSforsymmetricnuclearmatterandthelowerpanelthe 7. STRANGENESS DEGREE OF FREEDOM symmetry energy factor E . One notes that without sym BLPR,nuclearmatter wouldsaturateattoo high a den- If the central density of the massive compact star sity with much too large a binding energy. For the sym- reaches ∼ 5n as is found, then it is possible that kaons 0 metry energy, although the tree-order cusp is smoothed will condense or equivalently hyperons will appear.#9 by higher-order nuclear correlations, it leaves a distinc- tive imprint in the change of its slope at n : it is soft 1/2 below n and becomes stiffer above n . 1/2 1/2 Here is yet another case where it is dangerous to jump #9 It has been suggested in a unified approach with an effective to a conclusion based on what is apparent. Similarly to Lagrangianthathyperonscanappearonlywhenkaonscondense, 8 8. COMMENTS ON THE ORIGIN OF HADRON 2.4 MASSES 2.2 To conclude, I would like to make two comments. (A) 2 (B) 1. One of the most important observations made in Msun (A): new-BR n1/2=2.0n0 the work reported here is that in applying BLPR M/ 1.8 (B): new-BR n1/2=1.5n0 scaling to compact stars in [21], one has no free- domtoletthenucleonmassdropappreciablybelow 1.6 ∼ 0.8mN for n >∼n1/2 with the other parameters held fixed. Lower nucleon mass would bring too 1.4 strong a repulsion and make the symmetry energy – and the EoS in general – go haywire. The ques- 10 10.5 11 11.5 12 12.5 tion then is: Why can one not fiddle with other R [km] scalingparametersofthe Lagrangianso asto com- 2.6 pensate the effect of the dropping nucleon mass? 2.4 For instance if the ω-NNcoupling in the dHLS La- 2.2 grangian is arbitrarily allowed to drop, then one 2 may suitably soften the repulsion due to ω ex- changes to compensate the repulsion from the de- Msun 1.8 creased nucleon mass and keep the EoS within the M/ 1.6 rangegivenbyheavy-ionexperiments. Howeveran (A) (A): new-BR n1/2=2.0n0 1.4 (B) (B): new-BR n1/2=1.5n0 approximate one-loop renormalization-group anal- 1.2 ysis made so far with the dHLS Lagrangian indi- cates that the ω-NN coupling does not scale [32], 1 which means that at least at one-loop order, the 0.8 2 3 4 5 6 7 coupling does not drop. This indicates that the ncenter/n0 U(2) symmetry, fairly good in the (matter-free) vacuum for the ρ and ω, could be breaking down FIG. 4: Mass-radius trajectories (upper panel) and central in medium, given that the ρ coupling does flow to densities (n ) (lower panel) of neutron stars calculated the VM fixed point. Higher-loop RG analysis may center forn1/2 =2.0(A)and1.5n0 (B).Themaximumneutron-star benecessarytoconfirmthisresult. Whatwouldbe mass and its radius for these two cases are respectively (2.39 the most exciting could be that the symmetry en- M⊙, 10.90 km) and (2.38 M⊙, 10.89 km). ergyincompactstarsresultingfromthepresenceof half-skyrmionstructureispointingtoasubstantial m which carries the crucial imprint of the origin 0 of the proton mass. The appearance of the strangeness degrees of freedom will affect the EoS: It will soften it. It was observed in 2. Within the frameworkofdHLS,I havearguedthat [30]thattheonsetofthehalf-skyrmionmatterthatleads while mostofthe nucleonmassneednotfollowthe to the stiffening of the EoS induced a propitious drop quark condensate (with a significant m0), meson of the mass of the kaon propagating in dense medium. masses most likely do. This conclusion would be What this means in compact-star matter is that kaons invalidated if the constituent quark model which will condense more rapidly with the kaon mass drop- holds fairly well in the vacuum and which gets a pingfaster. Sotherearetwoopposingphenomenataking strong theoretical support on the basis of large Nc place at n , one stiffening the EoS and the other soft- considerations [33], held in dense matter. In this 1/2 ening. Since stiffening the EoS in the nucleon sector is case, one could construct a parity-doublet model known to push the kaon condensation to higher density, for the constituent quark instead of for the nu- it is not at all clear what the net effect will be. This cleonwithalargechirallyinvariantmassm0Q with matter remains to be resolved. I should mention that the mass formula for the constituent quark of the condensed kaons do not necessarily imply that a mas- form mQ =moQ+m¯Q(hq¯qi). Then the mass ratio sive star of ∼ 2M⊙ cannot be formed. In fact, the state mM/mB ≃ 2/3 (where M and B stand, respec- of condensed kaons can play the role of a doorway state tively, for light-quark mesons and baryons) which to strange quark matter that can accommodate massive holds fairly well in matter-free space would hold stars as suggested in [26, 31]. equally well in the vicinity of the chiral restora- tion. This would be consistent with Glozman et al’s observation that both baryons and mesons have rather large masses in the chirally restored whichwouldmeanthattheyrepresentthesamephysics. See[29]. phase [7]. However naive consideration paralleling 9 the nucleon parity-doubler would not work since supported by the Korean Ministry of Education, Sci- therewillbeproblemswithbothmesonandbaryon enceandTechnology(R33-2008-000-10087-0))withHyun spectra due to the parity doubling of the con- Kyu Lee, Kyungmin Kim and Won-Gi Paeng and with stituentquarks. Asubtler approachwillbe needed the participation of Masayasu Harada, Tom Kuo, Yong- to make that picture work, if at all#10. Further- Liang Ma, Yongseok Oh, Byung-Yoon Park and Chihiro morethisscenariowouldseriouslyrevampboththe Sasaki. Noneofthemshouldofcoursebeheldresponsible “old” and “new” (BLPR) scalings. for whatever errors I may be committing in my note. 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