Recent Progress on Dense Nuclear Matter in Skyrmion Approaches Yong-Liang Ma, Mannque Rho

Home , Mannque Rho, Scalar meson, Skyrmion

Recent progress on dense nuclear matter in skyrmion approaches Yong-Liang Ma, Mannque Rho

To cite this version:

Yong-Liang Ma, Mannque Rho. Recent progress on dense nuclear matter in skyrmion approaches. SCIENCE CHINA Physics, Mechanics & Astronomy, 2017, 60, pp.032001. 10.1007/s11433-016-0497- 2. cea-01491871

HAL Id: cea-01491871 https://hal-cea.archives-ouvertes.fr/cea-01491871 Submitted on 17 Mar 2017

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. arXiv:1612.06600v2 [nucl-th] 19 Jan 2017 h E The 5 oe ul nteSym arnincnitn ftequadr the of consisting Lagrangian Skyrme the on built model . es atri h kreMdl5 Infi- and QCD Holographic in Matter Skyrmion 4 2 Model Skyrme the in Matter Dense Skyrmions 3 from Matter Baryonic of ABC 2 Introduction 1 htvrtedgeso reo novd h oooyi ca is topology the involved, freedom of degrees the Whatever c Contents nie Review Invited #1 cec hn rs n pigrVra elnHeidelberg Berlin Springer-Verlag and Press China Science eetpors ndnencermte nsymo approac skyrmion in matter nuclear dense on progress Recent y“krin ilb en h eei oio oe o bar for model soliton generic the meant be will “skyrmion” By . idnlclsmer arninto Lagrangian symmetry local Hidden 5.1 and ieTwro etrMsn 10 11 . . 8 tower infinite The an 9 in mesons vector . Packing 4.3 . . “top-down” . the 4.2 . from . model 7 . . holographic The . . . . 4.1 Mesons . . Vector . of Tower . . . nite . . . . . energy . . symmetry nuclear . . The matter . dense in 5 . Kaons 3.4 constant decay pion model In-medium 3.3 crystal skyrmion the from matter Nuclear 3.2 3.1 Citation: kremdl idnsmere,symo rsa,hologr crystal, skyrmion symmetries, hidden model, Skyrme solitons as stars. treated compact baryons in putting by matter th nuclear in naturally dense figures fo matter same baryonic the nuclear compressed on by to vacuum treated approach are unified properties novel hadron a medium-modified provides model Skyrme The ASnumber(s): PACS ω prahadisBSlmt...... 11 ...... limit BPS its and approach ff n h clrDilaton Scalar the and c fteLws-yn etrMesons Vector Lowest-Lying the of ect ω ogLagM,MnqeRo eetpors ndnenuclea dense on progress Recent Rho. Mannque Ma, Yong-Liang o:******** doi: eo 12 ...... meson . 23.c 16.f 12.39.Fe 21.65.-f, 12.39.Dc, 1 etro hoeia hsc n olg fPyis Jilin Physics, of College and Physics Theoretical of Center 2 ntttd hsqeTóiu,CASca,911Gif-sur 91191 Saclay, Théorique, CEA Physique de Institut σ hsc,Mcais&Astronomy & Mechanics Physics, ogLagMa Yong-Liang 2016 O CEC CHINA SCIENCE re ytepo edonly. field pion the by rried tccretagbatr n h kreqatcterm quartic Skyrme the and term current-algebra atic ( Email:[email protected] p 4 Email:[email protected] oswtotseicto ftedgeso reo involve freedom of degrees the of specification without yons 13 . ) ρ aey krin,o rsa atc ihacnso h i the on accents with lattice crystal on skyrmions, namely, , 13 1 prah nti ril,w eiwtercn rgeson progress recent the review we article, this In approach. e 1 atri krinapoce.SiCiaPy ehAstron Mech China-Phys Sci approaches. skyrmion in matter r tn.Itiscdniydpnec ID eetn h ch the reflecting (IDD) dependence density Intrinsic oting. pi ulo,BSnce,cmatstars. compact nuclei, BPS nucleon, aphic anu Rho Mannque & hsc.I hsapoc,snl ayn ayncmte a matter baryonic baryon, single approach, this In physics. nQDadsae h aepoete scnttetquark constituent as properties same the shares and QCD in esetvsadDsusos21 [1] 17 skyrmions of terms in matter) nuclear (or multi-nucle and matter nucleon single of properties the Accessing Introduction 1 Discussions and Perspectives 7 Developments Other 6 odne atrpyisadsrn hoy[] ntelimit the colors In of number [2]. large theory with string of par and on physics physics nuclear matter in condensed topic new exciting an become . mrigsmere ndnemte?....20 17 . . . 19 . . . . matter? dense . in matter . symmetries dense Emerging for . (SHLS) HLS . simplified 6.3 The model skyrmion (near-)BPS 6.2 The 6.1 im- 15 and breaking . Explicit . symmetry: symmetry Scale 14 local hidden in matter symmetry 5.4 Skyrmion local hidden in properties 5.3 Skyrmion 5.2 nvriy hncu,101,China; 130012, Changchun, University, ato es atr...... 15 ...... matter dense on pact Yet cédex, France; -Yvette 2 only N c ot 06Vl * o *:****** ***: No. *** Vol. 2016 Month ilb eerdt s“kremodel.” “Skyrme as to referred be will 3,tesymo sabaryon a is skyrmion the [3], hssihn.o link.springer.com phys.scichina.com nteLgaga.Tesoliton The Lagrangian. the in d 06 *:******, ***: 2016, , mplications accessing o:******** doi: neof ange hes nd #1 has on model [4,5]. From then on, Skyrme’s idea has been widely view the effect of the lowest-lying vector and scalar mesons accepted in the physics communityand indeed, as one can see on nucleon and nuclear matter properties by using a “chiral- in [2], has been spectacularly successful in condensed matter scalar effective theory” implementing hidden local symmetry physics and is beginning to have significant impact in nuclear (HLS) [23,24] and hidden scale symmetry. physics. Earlier achievements were reviewed in Refs. [6, 7]. In Sec. 6.1, we discuss a variant soliton model which fo- Since the 1980s, especially in the past decade, a consid- cuses on the BPS structure of heavy nuclei [25,26]. This erable progress has been made on describing dense nuclear model departs from the standard approach to skyrmion which matter by putting skyrmions on crystal lattice, the strategy is anchored on chiral Lagrangians that describe low-energy initiated by Klebanov [8]. In this review we summarize what nonperturbative strong interactions typified by the nonlin- we have learned from that approach on compressed baryonic ear sigma model. It starts from the characteristic features matter, an area of nuclear dynamics for which there are no of heavy nuclei and nuclear matter of incompressible liq- well-established theoretical tools or experimental data. The uid structure and aims, ultimately, to go to the regime where lattice QCD technique, fruitfully applied to high tempera- soft-pion dynamics are recovered so as to make a contact the ture matter, is presently powerless in addressing dense mat- premise of QCD. In this model, the empirically small value ter and effective field theory models, presently available, or of the nuclear binding energy, especially for nuclei with large phenomenological models based on energy-density function- mass number A, can be well reproduced [27,28]. In addition, als are not predictive enough beyond the density regime near it has been suggested that the BPS model can also give pre- normal nuclear matter density. It will be seen below that the dictions of the mass and radius of massive compact stars [29], regime of large density abovethat of nuclear matter n0 0.16 addressing the recent observations [30,31]. 3 ≃ #2 4 fm− is a total wilderness. The HLS , constructed up to, and including, O(p ) terms used in the simulation of skyrmion matter, is quite involved In the skyrmion approach, one starts with an effective La- with 17 terms. It turns out however that one can simplify grangian, with as many relevant degrees of freedom as relevant taken into account, and puts the resulting skyrmions considerably the full HLS Lagrangian by numerically ana- lyzing the contributions from each term of the Lagrangian to on the crystal lattice. It has become clear from the modern the skyrmion and skyrmion matter. In Sec. 6.2, it is shown development of holographic QCD for baryons (see [2] for review) that, while the pion is essential for supplying topol- that one can reduce the 17 terms down to 7 terms with an accuracy of 97%. ogy, there is no reason to suggest that the pion is enough ∼ to arrive at a realistic description of the baryon and in par- Finally translating the features obtained in the skyrmion ticular baryonic matter. In fact an infinite tower of vector crystal simulation, with focus on the special role of the topol- mesons do and must intervene. Given a realistic Lagrangian ogy change in the nuclear tensor force, into the “bare pa- ff for skyrmion structure, dense matter can then be simulated rameters” of the e ective baryonic Lagrangian that incorpo- by putting multi-skyrmions on crystal lattice and squeezing rates hidden symmetries – hidden local symmetry and scale the crystal size. The Skyrme model can therefore provide symmetry, the properties of both nuclear and dense matter us a unified framework to study the single baryon, baryonic could be addressed in a unified way. This development, not matter and medium modified hadron properties [9]. We will widely appreciated in nuclear physics community, is briefly briefly discuss the basics of nuclear matter from skyrmions discussed in Sec. 6.3. and the potential power of the skyrmion model over other nu- We finally provide our perspectives and discussions in clear effective field theories in Sec. 2 and Sec. 3. Sec. 7. Since the discovery of the AdS/CFT correspondence [10], the holographic QCD model [11–16] has provided a novel 2 ABC of Baryonic Matter from Skyrmions approach to the strong interaction processes at low energy. The original Skyrme model is a model describing baryons in In this approach, one can access not only the elementary a mesonic theory in which the chiral symmetry is realized meson and baryon dynamics [11–16] but more significantly nonlinearly with pion field only [1]. the dense baryonic matter using the topology profiles that In the nonlinear realization of chiral symmetry, the pion is difficult to access by methods that do not resort to topol- figures as the Nambu-Goldstone (NG for short) boson of the ogy [17–22]. We will review the progress in this direction in spontaneous breaking of chiral symmetry. It is expressed in Sec. 4. Special emphasis is put on the impact of the infinite the polar parameterization through tower of vector mesons on multi-skyrmion properties. a a We have learned from the long history of nuclear phe- U(x) = exp (2iπ T / fπ) , (1) nomenology that not only the pseudoscalar mesons, pions, but also the lowest-lying vector mesons, V = (ρ, ω) and the where T a is the generator of the SU(2) group satisfying a b 1 ab scalar meson (that we shall call σ) of comparable mass are Tr(T T ) = 2 δ and fπ is the pion decay constant with an crucial for understanding the nuclear forces. In Sec. 5, we re- empirical value fπ 92.4 MeV. ≃ #2Unless specified otherwise, by HLS, we will mean hidden local symmetry with the lowest-lying vector mesons V. 2 Since the unitary field U(x) satisfies U(x)U(x)† = nuclear matter about 4% lower than that simulated by the cu- U(x)†U(x) = 1, for any fixed time, say, t0, the matrix U(x, t0) bic crystal used by Klebanov. The arrangement of skyrmions defines a map from R3 to the manifold S 3 in isospin space. on an FCC crystal and the baryon number density distribution At the low-energy limit, QCD goes to the vacuum, i.e., in the crystal face are shown in Fig. 1. So far the FCC crystal is the lowest energy crystal configuration known for nuclear U( x , t0) = 1, (2) ground state. | |→∞ therefore, all the points at x are mapped onto the north z pole of S 3 and energy of| the|→∞ system is finite. We have the nontrivial map

U(x, t ) : R3 S 3, (3) 0 → y for the static configuration U(x, t0). In the language of topology, these maps constitute the third homotopy group 3 π3(S ) Z with Z being the additive group of integers ∼ 3 which accounts for the times that S is covered by the map- x ping U(x, t0), i.e., winding numbers. Because a change of the time coordinate can be regarded as a homotopy trans- Figure 1 The arrangement of skyrmions on the FCC crystal lattice (left panel) and baryon density distribution (right panel). In the left panel, the formation which cannot transit between the field configura- black dots stand for skyrmions at the lattice vertices and the red dots repre- tion in homotopically distinct classes, the winding number sent skyrmions on the face centers. is a conserved quantity in the homotopy transformation by the unitary condition of the field U(x) and condition (2). In In the FCC configuration, the skyrmion number density, skyrmion models, the conserved winding number represents which is the baryon number density with the skyrmion mat- the conserved baryon number in QCD. The baryon arises as ter regarded as baryon matter, is expressed as 1/(2L3) where a topological soliton with the topology lodged in the chiral 2L is the crystal size. The size of the crystal lattice corre- 3 field U(x). sponding to the normal nuclear density n0 = 0.17 fm− is Note that, in the construction of the skyrmion-type model, then given by L = 1.43 fm. the unitary condition of the field U(x) and the condition (2) To simulate the nuclear matter in the skyrmion crystal ap- are the essential characteristics that should be taken into ac- proach, it is convenient to decompose the skyrmion soliton count. Indeed, in the construction of the BPS-type skyrmion configuration U0 as proposed by Adam et. al. [25,26] that will be discussed later, only these two conditions are imposed. U0 = φ0 + i~τ φ,~ (5) · In the skyrmion approach to baryon-baryon interactions, φ2 + φ~2 = φ the baryon-baryon interaction at large separation is approxi- with the constraint 0 1. In this decomposition, 0 ac- mated by the product ansatz [1] of two undistorted skyrmions counts for the magnitude of the quark-antiquark condensate with a relative rotation in spin-isospin space qq¯ which in the matter-free space satisfies φ0(x ) 1 handiφ (x) 0, (a = 1, 2, 3), due to the parity→∞ invariance→ a → U00(x, x1, x2) = U0(x x1)A(α)U0(x x2)A†(α), (4) of the strong interactions. However, in the inhomogeneous − − medium, due to the interaction of the pion fluctuation with where A(α) = exp(iτ α/2) is the rotation in the isospin space the medium, the quantities φ0(x) and φa(x) measure the inho- · with rotation angle α, U0(x) is the skyrmion configuration mogeneous quark condensate in the medium [34]. satisfying the classical equation of motion of soliton, x1 and One convenient way to simulate the skyrmion matter in the x are the centers of the two skyrmions and r = x x is crystal is to introduce the unnormalized fields (φ¯0, φ¯1, φ¯2, φ¯3) 2 | 2 − 1| the distance. To get the most attractive potential, the pair of which have the Fourier series expansions as [35] #1 skyrmions should be arranged in such a way that they should ¯ mutually rotate in the isospin space by angle π about the axis φ¯0 = βabc cos(aπx/L) cos(bπy/L) cos(cπz/L), (6) perpendicular to the line joining them. One can simulate the aX,b,c nuclear matter by putting skyrmions onto the crystal lattice, and first put forward by Klebanov [8], and regardingthe skyrmion matter as baryonic matter. The density effect enters when the φ¯1 = α¯ hkl sin(hπx/L) cos(kπy/L) cos(lπz/L), (7) crystal size is changed. Subsequently, Kugler and Shtrikman Xh,k,l proposed a new crystal structure [32,33], the face-centered- φ¯2 = α¯ hkl cos(lπx/L) sin(hπy/L) cos(kπz/L), (8) cubic (FCC) crystal, which gave the minimal energy of the Xh,k,l #1 Instead of expanding with the unnormalized modes, one can also make a Fourier expansion of the normalized modes φα(α = 0, 1, 2, 3) defined by Eq. (5) [36]. 3 φ¯3 = α¯ hkl cos(kπx/L) cos(lπy/L) sin(hπz/L). (9) Xh,k,l

The ﬁelds φ¯α relate to the normalized ﬁeld φα through

φ¯α φα = , (α, β = 0, 1, 2, 3). (10) 3 ¯ 2 β=0 φβ q P By taking the Fourier coefficients β¯ andα ¯ as free param- Figure 2 The distribution of the baryon number density in the skyrmion eters and varying them, the minimum energy per particle (left panel) phase and half-skyrmion phase (right panel). at a specific crystal size renders the quasi-nucleon density- dependent. Note that for a specific crystal structure, the When the skyrmion-half-skyrmion phase transition takes Fourier coefficients β¯ andα ¯ are not independent, but are re- place, in addition to the original lattice vertices, the baryon lated. Note also that the normalization (10) does not spoil number density accumulates in the middle of the lines con- any symmetries that the unnormalized fields possess, while necting the original FCC lattice vertices. If one integrates the expansion coefficientsα ¯ hkl and β¯abc lose their meaning as the baryon number density in the region where baryon num- Fourier coefficients in the normalized fields [32,33]. ber density accumulates, each new crystal lattice has baryon Among a variety of properties revealed in the crystal ap- number one-half. At what density the transition can take proach to dense matter, the most important of all is the ex- place in dense baryonic matter depends on, among others, the istence of half-skyrmion configurations at some higher den- degrees of freedom that intervene. Since in practice a large sity. Being topological, its presence is a robust prediction number of mesons can be involved, e.g., the infinite tower of involving maximal symmetry [37]. Its does not depend on holographic QCD, and one is limited to drastic approxima- what degrees of freedom other than the pion are involved. tions, it is difficult to pinpoint the changeover density n1/2. The detailed discussions and extensive references are found Given that there is no indication for half-skymrion phase at in [2], so we invite the interested readers to consult that vol- nuclear matter density, it cannot be lower than the normal ume. What is significant is that it involves a topology change density n0. If it were much higher, say, in the regime where from skyrmions to half-skyrmions, which is responsible for the percolationtransition can take place, then the modelcould a dramatic change in the properties of the dense matter at a not be used. Thus the optimal density for n1/2 should be density n1/2 2n0, a feature which is not visible in other somewhere above n0. The present estimate based on detailed approaches in∼ the literature. The topological change is essen- analyses using a semi-realistic model that contains π, ρ, ω and ff σ indicates that n1/2 2n0. It is this density regime where tial in the given e ective field theory (EFT) for the equation ∼ of state for compressed baryonic matter relevant for massive hadron-quark continuity seems to take place as we will men- compact stars [38]. tion in Sec. 3. We will keep this as a typical value in applications to nuclear matter and denser matter. In Fig. 2 is shown the baryon number density distribution of the skyrmion (left panel) and half-skyrmion states of mat- It should be mentioned before we enter into detailed dis- ter (right panel). In the half-skyrmion phase, the vertices cussions that the skyrmion crystal picture would make sense if N were very large. Indeed if N , then the bary- where the baryon number density accumulates form the CC c c → ∞ onic matter would be a crystal. In reality Nc = 3 is not so crystal. The space average φ0 defined as “large” and normal nuclear matter is a liquid and not a crystal. One might hope to arrive at a structure compatible with 1 2L φ = 3 φ , nuclear matter by doing proper quantum-mechanical calcula- 0 3 d x 0 (11) h i (2L) Z0 tions taking into account 1/Nc corrections starting from the crystal structure but nobody succeeded to do such calcula- which is nonzero in the skyrmion matter, is found to van- tions so far. Even within the large Nc scheme, indication ish in the half-skyrmion state. This implies that the quark from holographic QCD calculations is that at high density, condensate qq¯ vanishes when the space is averaged. But a variety of different forms, such as dyonic salt (resembling this does noth implyi a phase transition since chiral symmetry half-skyrmions), layers of “popcorns,” salty or sugared etc, is not restored with nonvanishing pion decay constant, that can be present [17–22]. All these exotic configurations could is, the pion is still present. In fact there is no bona-fide or- be artifacts of the large Nc models but at present none of them der parameter for this change of state, symptomatic of topol- can be ruled out. ogy change. Although it is not a paradigmatic phase change, Given this wilderness of scenarios, what we will adopt topology change can be considered as a phase transition, so is the topology change of skyrmion-half-skyrmion phase in what follows we will use “half-skyrmion phase.” change and exploit it in both the crystal approach and con- 4 tinuum approach that takes into account the topology change are certain properties, such as compression modulus, that are in the parametersof the effective Lagrangian. We will eschew at odds with experiments in this linear model. When higher- relying on the numerical results coming from the crystal lat- dimension meson fields are incorporated, consistently with tice. the given symmetries, this approach can post-dict 2000 nuclear spectra with deviation 6 0.5MeV atthe expenseof∼ 50 ∼ 3 Dense Matter in the Skyrme Model parameters. It will be discussed below that within the spirit of mean We first discuss certain generic properties of the skyrmion field approximations, one can formulate an EFT with a La- approach in terms of the simplest model, the Skyrme model grangian that combines both chiral symmetry and scale sym- (with pions only). The Skyrme model may be considered as metry with the parameters of the Lagrangian matched to the a simplified chiral Lagrangian with all heavy degrees of free- correlators of QCD. The power of this approach will be to dom than pion integrated out and all higher derivative terms make predictions at high density that are difficult to access dropped. The resulting Lagrangian contains the current alge- without ambiguity. The phenomenological approach, suc- bra term and the quartic term, i.e., the Skyrme quartic term. cessful up to nuclear matter density, can go wild, with the In fact the quartic term is essentially the leading term in the numerous parameters at the disposal that are unconstrained next-to-leading order O(p4) chiral Lagrangian that captures at high density with experimental guidance. A typical ex- π-π scattering [39]. For simplicity, we take the parameters ample is given by the nuclear symmetry energy that strongly given by the free-space value rather than the effective val- controls the equation of state for compact-star matter. One ues fitting nucleon properties or nuclear properties used in can see this in Fig. 3. the literature. Here the purpose is to obtain a qualitative idea of what’s going on rather than fitting data. To have a bet- ter view of what we are driving at, it is useful to recall the 120

NL-RMF(18) currently popular approach anchored on phenomenological PC-RMF(3)

RHF(2)

90 Lagrangian to ﬁnite nuclei, nuclear matter and dense matter DD-RMF(2)

Gogny-HF(2)

relevant to compact stars, namely, the energy-density func- SHF(33) tional approach (EDFA for short). Related closely to what 60 (MeV)

we are doing is the relativistic mean-ﬁeld (RMF) theory, ini- sym tiated by Walecka [40]. This theory works fairly well up to E the density appropriate for nuclear matter. This can be under- 30 stood in terms of Landau Fermi-liquid theory, to which the

RMF at large density is related. 0

A typical phenomenological Lagrangian used is of the 0 1 2 3

/ form 0

µ Figure 3 Symmetry energy as a function of density. This illustrates that = ψ¯ iγµ∂ m ψ L − N one can obtain anything for the symmetry energy at density n > n0 by adjust- ing parameters of the mean-field or equivalently energy-density functional 1 µ 2 2 + ∂µ s∂ s m s without affecting the normal-matter properties. This figure is copied from 2 − s Ref. [42]. 1 µ 2 2 + ∂µ a0 ∂ a0 m a 2 a0 0 · − 1 µν 1 2 µ The power of the skyrmion crystal approach is that this ωµνω mωωµω − 4 − 2 wildness can be cleared up, thanks to the topology change 1 µν 1 2 µ that unifies baryon and nuclear structure and properties of ρµν ρ mρρµ ρ − 4 · − 2 · baryons in medium, obtaining a unique prediction. This feat µ + g ψ¯ sψ + g ψ¯τ a ψ gωψγ¯ µω ψ s a0 · 0 − is the objective of what’s described below. µ µ 1 µν gρψγ¯ µτ ρ ψ eψγ¯ µA ψ FµνF , (12) − · − − 4 where ψ = (p, n)T are the nucleon doublet, ρ is the iso-vector 3.1 Nuclear matter from the skyrmion crystal model vector meson, ω is the iso-scalar vector meson, s is the iso- scalar scalar meson, and a0 is the iso-vector scalar meson. In this and following subsections we discuss generic features Included is the photon field Aµ. This Lagrangian, what one of the skyrmion model that follow from topology involved might call “linear meson model,” can be related to chiral- with pion field. Other massive degrees of freedom that will scale Lagrangian discussed below if the iso-vector scalar figure in the dynamics will influence the detailed structure mesons are ignored. In the mean field, this Lagrangian yields without affecting the characteristics of the model. Here we a broadly successful description of nuclear matter [41]. There first look at nuclear matter. In doing this we employ the 5 Skyrme model following Ref. [9]. The Lagrangian is

2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5

1120 1120

1080 1080

1040 1040 E/B(MeV)

1000 1000 2 fπ µ 1 2

= Tr U†∂ UU†∂ U + Tr U†∂ U, U†∂ U . 960 960 4 µ 32e2 µ ν L − h i

(13) 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5

Since we are aiming at a qualitative rather than quantitative L (fm) description of nature, we will not take eﬀective parameters ﬁt to the structure of baryon and baryonic matter as was done in 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 1.0 1.0 the literature, but will take what’s determined in the vacuum.

Thus we take fπ to be the free-space pion decay constant 93 0.8 0.8 ≃ MeV. For the Skyrme parameter e, we take e = g = 5.93 with g being the value of the gauge coupling constant of the HLS, 0.6 0.6 >

the chiral eﬀective theory of vector mesons [23,24], which <

0.4 0.4 may be obtained by assuming the ρ meson mass is “heavy” #3

. The robust, though qualitative, properties extracted there- 0.2 0.2 from will then be the basis for doing more realistic calculations including the other relevant degrees of freedom. 0.0 0.0 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5

L (fm)

Figure 4 Per-skyrmion energy E/B (upper panel) and φ0 (lower panel) h i By putting the Skyrme solitons of (13) onto the FCC as a function of crystal size L. crystal so as to simulate nuclear matter, one obtains the result of Fig. 4 for the skyrmion-matter per particle (or “per- skyrmion”) energy E/B and φ0 – which is related to the On the contrary, what is highly non-trivial and of great quark condensate in QCD variablesh i through decomposition importance that can be exploited in our more realistic ap- (5) – as a function of crystal size L. The vertical line indi- proach is the observation that the quantity φ representing h 0i cates the crystal size that corresponds to the normal nuclear the quark condensate vanishes for density above normal nu- density. One notes that the system has a minimum at a density clear density. At what precise density that takes place can- considerably higher than what is normal density. This is not not be pinned down. But it is a robust observation. One surprising. It just indicates that the Skyrme Lagrangian writ- observes in Fig. 4 that when the system is squeezed from ten to the lowest order chiral counting is missing some im- L = 2.5fm to around L = 1.3fm, the skyrmion system un- portant components in nuclear interactions which give a net dergoes a phase transition from the FCC skyrmion config- repulsive force. In fact what corresponds in standard chiral uration to the CC half-skyrmion configuration. The system perturbative approaches to multi-body (i.e., three-body) re- has a minimum energy at L 0.85fm with the energy per pulsive force – of the range of an ω-exchange force that stabi- baryon E/B 957MeV. Of significance∼ is that the value of ≃ lizes nuclear matter – is absent. Also the resulting large bind- φ0 averaged over space rapidly drops as the system shrinks. ing energy, 200 MeV, is not surprising given that the typi- It reaches zero at L 1.3 fm where the system goes to a ∼ ∼ cal energy scale involved in QCD is the ΛQCD, whichis a few half-skyrmion phase mentioned above. This “phase transi- times the pion decay constant fπ 93 MeV.All these features tion” can be interpreted, once the pion fluctuations are incor- ≈ are understandable from the large Nc counting, on which the porated, as a signal for global chiral symmetry restoration skyrmion model is relying. These defects can be remedied whereas locally the system is still in the chiral symmetry bro- in refined, though involved, treatments. In fact when vec- ken phase #4. In this description there is no hint of QCD vari- tor mesons and scalar mesons are included, the situation im- ables, i.e., quarks and gluons. However the skyrmion-half- proves substantially [43–46] and with an infinite tower one skyrmion transition can be reproduced in terms of a chiral- obtains even agreement with experiments [44, 47, 48]. We quark model where the transition is captured in a delocaliza- will return to this matter below. tion of the baryon number density leading to b = 1/2 struc- #3 This assumption must break down at large density approaching the vector manifestation fixed point. #4The density at which the skyrmion matter transits to the half-skyrmion phase obtained here is different from that obtained in Ref. [9] due to different Skyrme parameters. If we took the empirically fit value e = 4.75, the skyrmion to half-skyrmion phase transition would appear above the normal nuclear density. One should not take the numerical values we are obtaining too seriously. 6 ture [49]. There is another transition in this picture corre- Then the medium-modified (time component of) pion decay sponding to quark deconfinement at higher density, which is constant is given by the longitudinal component in the low- however not in the half-skyrmion phase. energy limit

2 fπ∗ lim GL(p0, p = 0). (18) 3.2 In-medium pion decay constant ≡ − p0 0 → One of the most crucial quantities in dense medium is the Now for the calculation, we shall ignore the contributions effective in-medium pion decay constant fπ∗. It controls from the loop diagrams of the fluctuation fields. Therefore, medium modified hadron properties [9,35]. Here, as a con- there are only two types of diagrams that contribute as il- crete example, we calculate the medium-modified pion de- lustrated in Fig. 5: (i) the contact diagram and (ii) the pion cay constant fπ∗, again taking the Skyrme model. The latter exchange diagram. They are given by contains the essential feature that is shared by more realistic models given below. 2 (i) : i f 2g δab 1 φ2 , The procedure is quite simple. We take the skyrmion crys- π µν − 3 π ! tal solution as the background classical field and interpret the D E pµ pν 2 fluctuating fields on top of it as the corresponding mesons (ii) : i f 2 δab 1 φ2 . (19) − π p2 − 3 π ! in dense baryonic matter. Following Ref. [43], we write the D E skyrmion crystal solution as U(0) and introduce the fluctuat- Summing the two contributions, one obtains ing fields as 2 2 2 2 = ˇ fπ∗ = fπ 1 1 φ , (20) U u(0)Uu(0), (14) 3 0 " − − D E# where Uˇ = exp(2iτaπˇ a/ fπ) stands for the corresponding fluc- 2 where the intrinsic density dependence is brought in by the tuating field and u(0) = U(0). Since for each crystal size, 2 minimal energy solution φ0. To arrive at (20), we used the there is a solution for U(0), U(0) is dependent on the crystal relation φ2 + φ2 = 1. Equation (20) shows the direct rela- size, or equivalently, the baryon density. The decomposition 0 π tion between the medium-modified pion decay constant f ∗ in Eq. (14) guarantees that to each order of the fluctuation π and the parameter φ0 which signals the phase transition in- the chiral invariance of the model is preserved. By substitut- volving topology. h i ing the fields in Eq. (14) into the Skyrme model Lagrangian one obtains the medium-modified Lagrangian for the pion.

Consequently the parameters in the pion Lagrangian become 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 1.0 1.0 density-dependent through their dependence on U(0).

0.8 0.8

Jµ5 Jν5 Jµ5 π Jν5

0.6 0.6 */f f

0.4 0.4 (i) (ii)

0.2 0.2 Figure 5 Two types of contributions to the correlator of Eq. (15): (i) the

contact diagram and (ii) the pion exchange diagram. Shaded blobs stand for 0.0 0.0 interaction vertices in the skyrmion matter. 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5

L (fm)

To deﬁne the pion decay constant in the skyrmion matter, we write the axial-vector current correlator Figure 6 fπ∗/ fπ as a function of the crystal size L .

ab 4 ip x a b iG (p) = i d xe · 0 TJ (x)J (0) 0 (15) In Fig. 6 is plotted the crystal-size dependence of f / f . µν Z h | 5µ 5ν | i π∗ π We see that fπ∗ decreases to 0.65 fπ at the density n1/2 at which is decomposed in medium as [50] which skyrmions fractionize to∼ half-skyrmions and then stays constant for n > n . In terms of the quark condensate qq¯ , Gab(p) = δab P G (p) + P G (p) , (16) 1/2 µν Tµν T Lµν L φ corresponds to the space average of the latter. It shouldh i h i h 0i where the polarization tensors PL,T are defined as be noted that φ0 0 does not signal the restoration of chiral symmetryh sincei → the pions are still around with the non- p p i j zero pion decay constant. This means that the quark conden- P = g δi j g jν, Tµν µi − p 2 ! sate is locally non-zero in the half-skyrmion phase, implying | | pµ pν that there is chiral-density wave. This was verified in [34]. P = g P . (17) Lµν µν 2 Tµν This feature will remain unmodified when scale symmetry − − p ! − 7 is implemented. This results from the locking of scale sym- approximation in (21), one obtains the kaon dispersion rela- metry and chiral symmetry, consistent with the scale-chiral tion in the skymion matter as effective field theory [51] – discussed below – with the con- 2 2 α ω ~p + 2βωK + γ = 0, (26) sequence that f ∗ f ∗. K K π ≈ σ − where 3.3 Kaons in dense matter 1 χ 2 α = u + u† + 2 , Another highly non-trivial prediction of skyrmion crystal 4 * fχ ! + matter is the property of kaons in medium. Kaon property Nc β = B u + u† + 2 , in nuclear and denser matter has been an important issue for 16 f 2 0 π D E the equation of state of compact stars [52] since kaon con- 3 densation was predicted on the basis of chiral symmetry in 1 χ 2 γ = u + u† + 2 m . (27) 4 f K 1985 [53]. The issue is still far from being settled and is be- * χ ! + coming an urgent matter, in particular with the discovery of Focusing on S-wave kaons with ~p 0, the effective in- massive neutron stars. We will discuss here what one can say K → medium kaon energy is obtained by solving the equation about the properties of kaons in dense nuclear matter from the skyrmioncrystal point of view. Again we will use the Skyrme β + β2 + αγ model extended to three flavors including strangeness, thus m∗K lim ωK = − . (28) ≡ ~pK 0 pα involving octet NG bosons, but minimally implemented with → dilaton dynamics. Although highly simplified, the model we use exposes a non-trivial effect of the dilaton field that is ab- 600

+ sent in the usual chiral Lagrangian approaches. K with dilaton

Following [54], we take the Lagrangian w.o. dilaton

2 3 χ µ χ 2 K = DµK†D K m K†K L f ! − f ! K χ χ 300

(MeV) * K

m* -

2iN m K c µ _

, \ B DµK†K K†DµK (21) =0 − 5 f 2 − π f* =\ 0

_ + =0 where K is the kaon doublet with K = (K , K0), the last term =0 f* =\ 0 is the topological Wess-Zumino term present for three-ﬂavor f* = 0 systems, the covariant derivative is deﬁned as 0

0 1 2 3 4 5

n/n DµK = (∂µ + Γµ)K, (22) 0 Figure 7 Skyrmion matter modified Kaon mass m as a function of density where ∗K n/n0 for dilaton mass mχ = 600 MeV. See Ref. [54] for an explanation of the µ 1 µ µ figure. Γ = u†∂ u + u∂ u† , (23) 2 h i The medium-modified kaon mass m∗ (energy for the S- with u √U and the baryon number current as K ≡ wave) is shown in Fig. 7 which is borrowed from Ref. [54]. The effect of φ 0 toward n is visible for both with and µ 1 µναβ 0 1/2 B = ǫ tr u†∂νu u†∂αu u†∂βu . (24) h i → 24π2 · · without the dilaton. However the dropping of m∗K is accentu- h i ated by the presence of the dilaton. Furthermore the mass In Eq. 21, the dilaton field σ is introduced via what is called drops at a faster rate in the half-skyrmion phase. By itself, “conformal compensator” χ which has mass dimension-1 and this would have a big effect on the process of kaon conden- scale dimension-1, sation in compact star matter. It should be noted that certain

σ/ fσ baryonic short-range correlations, missing in mean-field-type χ = fσe . (25) considerations (appropriatefor Fermi-liquid fixed-point treat- This is because the dilaton field plays a specially important ments), should be included for realistic calculations. This role. How this field enters in chiral symmetry Lagrangians to could apply both to kaon condensation and to hyperon inter- join in chiral-scale symmetry will be described later. Here it actions in compact star matter. This would require an ap- suffices to say that the Lagrangian accounts for scale symme- proach drastically different from the naive application of chi- try transformation to the leading order of scale-chiral count- ral perturbation theory. We will see later such an effect could ing rule [51,55]. have a strong impact on the formation of dense kaonic matter The kaon field is affected by the background of nuclear at the hadronization phase in the cosmological evolution of matter through the skyrmion profile u. Taking the mean-field the Universe [56]. 8 3.4 The nuclear symmetry energy Perhaps the most important impact of the skyrmion-half- skyrmion (topological) phase change in the physics of dense matter is on the nuclear symmetry energy that has defied theorists in compact-star physics since many years as discussed above (See Fig. 3). It turns out that it is tightly connected to the vector manifestation [57] for the ρ meson in the skyrmion approachand it is not at all clear how that translates into standard effective field approach based on chiral symmetry. Quite remarkably it offers a compelling evidence that the topology change, apparently absent in many-body interactions, is robust and can be exploited to construct a realistic model for dense matter that cannot be accessed by other approaches. It turns out that the change from skyrmions to half-skyrmions Figure 8 Symmetry energy as a function of density calculated in Ref. [58]. plays a key role. The cusp is located at n1/2. The low density part that cannot be located The energy per particle E(n,δ)of a dense system with den- precisely is not shown as the collective quantization method used is not ap- sity n and asymmetry δ can be written as plicable in that region.

2 4 E(n,δ) = E0(n, x = 0) + δ Esym(n) + O(δ ) + , (29) ··· Of high significance is that, strange or anomalous though where δ = (N P)/A with N(P) being the neutron (proton) it may appear, the presence of the cusp, arising from the − number and A = N + P. This equation defines the symmetry skyrmion-half-skyrmion topology change involving the pion energy Esym. It turns out that the above expansion is a good field, is robust and does not depend on what other degrees one. The corrections entering at O(δ2) are estimated to be of freedom are incorporated in the dynamics. It turns out, as quite small even for δ near 1. We will ignore them here. pointed out below, that the cusp plays an essential role in the In the skyrmion crystal approach, the symmetry energy EoS of massive compact stars [59,60]. Esym can be calculated for the neutron matter (δ = 1) for large A question that is immediately raised is whether this cusp N by collective-rotating the whole system [55, 58]. After the is not just an artifact of the crystal structure. We address this collective rotation, the energy of the A-nucleus is question and show that there is neither abnormality nor mys- 1 tery in the cusp. How this cusp arises can be explained in E = AE + ITot ITot + 1 , (30) A Sol 2A terms of standard nuclear many-body interactions involving I the nuclear tensor forces coming from the exchanges of π and where E is the per-soliton energy in the nucleus and A is Sol I ρ whose properties are influenced by the topology change. the iso-spin moment of inertia of the A = N system. ITot is the Quite independently of whether it is valid or not, it is a new total isospin of the whole system which could be the same as phenomenon hitherto undiscovered in the field. the third component of the isospin I3 for pure neutron matter, i.e., ITot = A/2. For δ (N P)/(N + P) . 1 we have The treatment is much more straightforward with HLS La- ≡ − grangian with nucleon fields put in explicitly rather than gen- 1 1 (N P) 1 ITot = (N P) = (N + P) − = Aδ. (31) erated as skyrmions. To address the nuclear symmetry en- 2 − 2 (N + P) 2 ergy, the key ingredient is the nuclear tensor force, by which The energy per nucleon can then be expressed as the symmetry energy is dominated. Thus we can focus on the structure of nuclear tensor forces. With nucleons suitably ENuclei 1 2 coupled to HLS fields, the tensor force is given by the pion ENucleon = ESol + δ , (32) A ≃ 8 exchange and the ρ exchange. One can ignore higher reso- I for a large mass number A. Consequently, the symmetry en- nances for the latter. The π and ρ tensor forces have the same ergy takes the simple form spin-isospin operator and radial form, differing in the overall coefficients with opposite signs and the dependence of mass. 1 E = . (33) With the sign difference, the pion tensor gets reduced by the Sym 8 I ρ tensor in the intermediate and long range that the poten- In Ref. [58], using the Skyrme model implemented with tials act. Now since the ρ mass scales with density because scale symmetry, ESym was calculated on crystal lattice in the of the density dependence of fπ∗, whereas the pion mass re- way described above. The result is shown in Fig. 8 copied mains unscaled, the ρ tensor, becoming stronger at increasing from Ref. [58]. The symmetry energy first decreases with density, cancels the pion tensor, almost completely, near n1/2. density up to the density n1/2 at which skyrmions fraction- However at n1/2, due to the topology change, the ρ tensor ize to half-skyrmions and then increases as density increases. gets strongly suppressed because the hidden gauge coupling This is the appearance of the cusp in Fig. 8. drops rapidly due to the vector manifestation. For n > 2n , 9 0 the pion tensor takes over. This can be seen in Fig. 9 bor- 4 Skyrmion Matter in Holographic QCD and rowed from [58]. Infinite Tower of Vector Mesons

We have seen that the minimal skyrmion crystal approach using the pion-only Skyrme model, with or without a scalar

n/n =1 degree of freedom, simulates, semi-quantitatively correctly,

n/n =2

0 certain characteristic properties of nuclear matter. But there

n/n =3

0 10 are certain other properties diﬃcult to access in the standard chiral eﬀective theory approaches, such as the symmetry energy at high density, that un-mistakenly indicate massive non- 0 [MeV] 1 2 3 4

T NG bosons can be very important. In this section we explore ~ V

r [fm] the eﬀect of the vector resonances – including the inﬁnite -10 tower inspired by the holographic model of QCD – on dense skyrmion structure.

-20 As mentioned above, a powerful – and rather mysteriously efficient [63] – way of capturing the physics of vector mesons is to resort to HLS [23,24]. It is also in the HLS approachthat the possibility of the ρ mass approaching the zero-mass (in Figure 9 Tensor force as a function of density. For definiteness n1/2 = 2n0 is taken. the chiral limit) pion at high density can be treated systemati- cally [24]. When the energy scale goes up, the number of redundancies, therefore the number of gauge bosons, increases. Although for a fair description it would require a lot more The infinite number of hidden gauge vector fields that arise detailed discussion involving ingredients that are not touched from the redundancies together with the pion field in 4D can upon in this review, such as renormalization-grouptreatment, be dimensionally de-constructed to 5D Yang-Mills (YM) ac- it is perhaps proper to mention that the property of the tensor tion in curved space [64] with the 5th (holographic) direction force in the density regime up to n0 accessible experimentally representing energy scale plus the Chern-Simons term encod- is verified in the C14-dating Gamow-Teller transition [61]. ing chiral anomalies. There are different ways of explaining the same phenomena The construction of the holographic models of QCD has but they are not alternatives but in some sense equivalent in two approaches, what one might call ”top-down” and the physics. other ”bottom-up”. In the ”top-down” approach, the action of the 5D Yang-Mills is reduced from the gravity sector, Assuming that the tensor force dominates the symmetry that is the “bulk” sector of gravity/gauge (holographic) du- energy – which is supported also by various conventional nu- ality coming from string theory. Among a variety of models clear models – and given that the tensor force excites states given in the bulk sector, the so-called Sakai-Sugimoto (SS) predominantly to those peaked at E¯ 200 MeV, the sym- model [13,14] is the one which has the symmetry properties metry energy can be approximately given∼ by the closure for- closest to QCD. When the model is dimensionally reduced mula [62] to 4D àla Kaluza-Klein, it has an infinite tower of vector mesons plus pions [65,66]. This model is justified in the large 2 T 2 = V Nc and large ’t Hooft λ NcgYM limit and the chiral limit. In these limits, there are only two parameters in the model and Esym . (34) ≈ E¯ they are fixed from meson dynamics. In the ”bottom-up” approach, instead of reducing the action from the gravity sector, From Fig. 9, it is immediately seen that Esym will decrease it is constructed from QCD in five dimensional anti-de Sitter as density approaches n1/2 from below and then increase, re- space [11,12] using the duality dictionary. In these models, in producing the cusp at the transition density of the skyrmion addition to the gauge fields, the effect of the explicit breaking crystal calculation, Fig. 8. of chiral symmetry encoded in the vacuum expectation value Two comments are in order regarding the cusp structure. (VEV) is included through that of the bulk scalars. In both approaches, the generic model action has the structure First, Esym given by (34) will, in nature, be smoothed by higher-order nuclear correlations. Indeed a refined formu- DBI CS S 5 = S + S , (35) lation based on renormalization-group flow flattens, as will 5 5 DBI CS be shown below, the cusp into a changeover from a soft to where S 5 is the 5D Dirac-Born-Infeld (DBI) part and S 5 hard EoS at n1/2 via the symmetry energy. Second, the cusp is the 5D Chern-Simons (CS) part. For different models the DBI form given by (34) provides a strong support to the topology expressions of S 5 could be different. In this review, we will change taking place at 2n with a qualitative effect at that focus on the top-down holographicmodel. The application of ∼ 0 density. the bottom-up approach to the baryon structure is reviewed 10 by Pomarol and Wulzer in [2]. No application of the bottom- where up approach to nuclear and dense matter is available in the Fmn = ∂mAn ∂nAm + [Am, An] (42) literature at this moment. − a a a b 1 with Am = T Am normalized as tr(T T ) = 2 δab. The gauge field transforms 4.1 The holographic model from the “top-down” approach and its BPS limit 1 A g(A + ∂ )g− (43) m → m m Since what we are interested in is the holographic model of with g SU(2). QCD, we consider the SS model [13,14] which has the prop- ∈ erties closest to QCD. In this model, the DBI part is written The static energy coming from the action (41), known in terms of the pure burk gauge fields as as BPS action, has a well-known bound, the Bogomol’nyi bound, DBI 4 1 2 > 2 S S YM = κ d xdz tr (36) E 8π B, (44) 5 ≈ − Z 2e(z)2 F where B is the instanton number representing baryon number = λNc ff with κ 216π3 . e(z)isthee ective YM coupling that depends on the holographic direction z and is proportional to the KK 1 = 3 , 1/2 B 2 tr(FMN ∗FMN )d x dz (45) mass as MKK− . The 5D Chern-Simons term is 16π Z

1 CS Nc 4 2 i 3 1 5 = = = + in which M 1, 2, 3, z and ∗FMN 2 ǫMNABFAB is the dual S 5 2 d xdztr . (37) 24π Z AF 2A F − 10A ! field strength. Now the bound is satisfied if FMN is self-dual, We use the index m = (µ, z) with µ = 0, 1, 2, 3. The gravity i.e., enters in the z dependence of the YM coupling, giving rise FMN = ∗FMN . (46) µ to the warping of the space. = µdx + dz is the five- A A Az This meansthat the energyof the system cannot be lower than dimensional U(N ) gaugefield and = d +i is its field f F A AA the bound. In other words, a system of instanton number A strength. In the case of N f = 2 flavors, describing mass-number A nucleus has the binding energy 1 equal to zero. This can be referred to as “BPS skyrmion” as = A + A˜ . (38) A SU(2) 2 U(1) will be explained below. The resulting YM action is 4.2 Packing vector mesons in an infinite tower 4 1 2 1 2 S YM = κ d xdz tr F + F˜ , (39) − Z 2e2(z) mn 2 mn! The 5D action (36), when KK-reduced to 4D, can be written as the sum of an infinite number of vector and axial -vector and the CS term mesons possessing hidden local symmetry [14]. In leading Nc 2 Nc 2 order, the gauge coupling constant will be a constant inde- S CS = A˜ trF + A˜ F˜ . (40) 16π2 Z ∧ 96π2 Z ∧ pendent of metric warping with no coupling to the infinite tower of iso-scalar vector mesons. The skyrmion constructed In Eqs. (39) and (40) Fmn is the field strength for the SU(2) with this infinite tour will then be equivalent to the instan- gauge field and F˜mn stands for the field strength of the U(1) gauge field. ton discussed above. A highly illustrative study was made We should stress that the SS model – which is holograph- by Sutcliffe [47, 48] which showed how the vector mesons entered in the tower. ically dual to QCD in the large Nc and λ limit (and the chiral limit) – has no free parameters. To leading order in λ, that is, Start with the pion only with all the vector mesons set to O(λ), e(z) is a constant, so the 5D YM action can be taken equal to zero. This is the Skyrme model. The soliton static energy is 1.24 times the Bogomol’nyi bound (88)#5 to be in flat space. The Chern-Simons term, which comes ∼ at O(N λ0), does not contribute. Thus to leading order, i.e., c E(0) = 1.235(8π2B). (47) O(Ncλ), the static baryon – which is B = 1 is given by the instanton solution that is self-dual [19]. Next when the lowest-lying vector meson ρ is included, the Let us first look at the instanton given by the SS action in DBI soliton mass is drastically reduced to the leading order, say, O(Ncλ). At this order only the S 5 in flat space contributes and the U(1) degrees of freedom de- E(1) = 1.071(8π2B). (48) couple from SU(2) degrees of freedom. The resulting 5D Yang-Mills theory, in unit of an arbitrary mass dimension, is The next-lying axial-vector meson a1 brings this furtherdown 1 to S = trF2 d4 x dz, (41) (2) 2 − 2 Z mn E = 1.048(8π B). (49) #5This corresponds in the case of the Skyrme Lagrangian to the Faddeev bound 12π2B. 11 Since the full tower of iso-vector vector mesons brings this background-independentand hence should be independent of ( ) 2 to the bound E ∞ = 8π B, it follows that the high-lying vec- the warping. Using our energy unit, we have the BPS mass tor mesons make the theory flow to a conformal theory. That M λNc M 559 MeV [15,16,67]#6, in agreement BPS ≈ 27π KK ≈ the lowest-lying vector meson ρ does nearly all the work in with Ref. [47]. When the CS term contribution is added, we flowing to the conformality is reminiscent of the near com- 2 get MBPS CS = MBPS + Nc MKK 1038 MeV which − 15 ≈ plete saturation of the charge sum rule of the pion [13,14] shows that the contributionq from CS term to the soliton mass and nucleon [15,16,67,68] form factors. is significant. Along the same procedure applied before, we Now suppose that the same packing strategy is applied to found that, for the given , using the HLS given by Eq. (57) many-skyrmion or nuclear systems. The above result implies in the following and determiningM the low energy constants by that the nuclear binding energy, 24% of the nucleon mass ∼ the 5D YM theory (41), the results are [44,46]: with the pion field only,will reduce to near zero as the higher- lying vector mesons are packed in. This tendency is beauti- = π, ρ, ω: fully illustrated in Fig. 10. •M M (π, ρ, ω) 1162 MeV. (50) “BPS” ≈ = π, ρ: •M M (π, ρ) 577 MeV. (51) “BPS” ≈ = π: •M M (π) 673 MeV. (52) “BPS” ≈ The ω meson is seen to block the flow to the conformal fixed point. In Eqs. (50), (51) and (52), the subindex “BPS” means the parameters are calculated from the 5D YM theory (41). We next examine the effect of the tower of the infinite Figure 10 The energy per baryon, in units of the single baryon energy vector mesons from the SS model in which the warp effect vs. baryon number up to A = 4. Squares are the experimental data, circles is included. A direct calculation of the mass of the topo- are the skyrmion energies with only pions, and diamonds are the skyrmion logical object, here, the approximate instanton, yields the energies in the extended theory with pions, ρ and a1 mesons. This figure is mass [15,16,67,68,72] taken from [48]. M 1800 MeV, (53) (approx)instanton ≃

4.3 The ω meson with fπ = 92.4 MeV and λ = 17 fixed in the meson sector [13,14]. Note that the mass (53) corresponds to the mass In the SS holographic QCD model, the ω meson figures at of a skyrmion with the effect of the infinite tower of vector the next order in λ. It figures via the Chern-Simons term. mesons in a warped space and the Chern-Simons term taken In standard paradigm of nuclear physics, as stressed above, into account. the ω degree of freedom is indispensable. In chiral perturba- By dimensionally de-constructing the 5D SS model to the tion theory, its effect, at least partially, is captured in short- 4D hidden local symmetric mesonic theory, one can check range three-body and higher-body forces brought in by high- the effect of the tower of vector mesons. It was found order derivative counter terms in the chiral Lagrangian. It is that [44, 46]: also indicated in lattice simulations of nuclear forces [69]. To confront the predictions of the holographic model on = π, ρ, ω: nuclear physics with the empirical values, therefore, one •M should consider the warped space in the YM action and the M (π, ρ, ω) 1184 MeV. (54) SS ≈ Chern-Simons term. By dimensionally de-constructing the SS model and keeping only the lowest-lying vector mesons = π, ρ: ρ and ω, the ω meson is indeed found to be significant for •M describing the nuclear force [43–46,70,71]. MSS(π, ρ) 835 MeV. (55) ≈ Here we analyze how the ω influences the flow to conformality seen above with the iso-vector vector mesons. = π: To explore the effect of the U(1) degrees of freedom, one •M should include the effect of the CS term. The CS term is MSS(π) 922 MeV. (56) ≈ #6Here we used λ = 16.66, M = 948 MeV determined from our inputs. KK 12 In the above calculation, the parameters are fixed by the two The most general HLS Lagrangian responsible for the soli- empirical values, fπ and mρ, in the meson sector [13,14]. ton mass – which is O(Nc) – can be expressed as It was found that, if the parameter MKK is reduced to ∼ = + + , (57) 500 MeV, one gets 950 MeV for the soliton mass and LHLS L(2) L(4) Lanom 300 MeV for the∼∆-N mass difference, both consistent with with∼ experiments [72]. In addition, by using the truncated = 2 µ 2 µ model π, ρ, ω, one can fit the nucleon mass and the (2) = fπ Tr αˆ µαˆ + a fπ Tr αˆ µαˆ + kin, (58) M L ⊥ ⊥ k L ∆ N mass splitting if one uses fπ 62.2 MeV and mρ k − ≃ ≃ 417.5 MeV [73]. In sum, what is found is the following: where fπ is the pion decay constant, a is the parameter of the In the holographic QCD of SS, to the leading λ order, the HLS. The two 1-forms,α ˆ µ andα ˆ µ in (58) are defined by k ⊥ more iso-vector vector mesons are included, the lighter soli- 1 ton mass becomes and the sum of the contributions from the αˆ µ = (DµξR ξ† + DµξL ξ†), (59) k 2i · R · L infinite tower reduces the soliton mass to the BPS instan- 1 ton limit. The residual interaction between the skyrmions αˆ µ = (DµξR ξ† DµξL ξ†), (60) ⊥ 2i · R − · L (as quasiparticles) gets weaker and the size becomes smaller, all going in the right direction. However this tendency gets with the chiral fields ξL and ξR, which in the unitary gauge blocked at the next order in λ, namely at O(λ0), primarily by are the presence of the ω meson present in the CS term with the iπ/2 f ξ† = ξ = e π ξ with π = π τ, (61) effect of metric warping less significant. This is at odds with L R ≡ · nature. We address this problem and point at a possible res- where τ’s are the Pauli matrices. The covariant derivative olution with intervention of a scalar dilaton in the effective associated with the HLS is defined as Lagrangian. Dµξ , = (∂µ iVµ)ξ (62) R L − R,L ff 5 The E ect of the Lowest-Lying Vector where Vµ represents the gauge boson of the HLS, Mesons ρ and ω and the Scalar Dilaton σ 1 V = g ω + g ρ (63) µ 2 ω µ ρ µ In the above section, we have seen that the infinite tower of the vector meson resonances plays an important role in the and skyrmion physics. Here we focus on the nuclear matter from 0 √ + the chiral effective theory including the lowest-lying vector ρµ 2ρµ ρµ = ρµ τ = 0 . (64) resonances, ρ and ω, based on the HLS approach. ·  √2ρµ− ρµ   −  The effect of the lowest-lying vector mesons on dense   In (58), is the kinetic term of vector mesons with skyrmion matter was first studied in Ref. [74] by using a mini- Lkin mal model including vector mesons [75]. In this model, the ω 1 1 = Tr V(ρ)V(ρ),µν Tr V(ω)V(ω),µν , (65) meson couples only to the baryon density through ω Bµ (with kin 2g2 µν 2g2 µν µ L − ρ − ω Bµ being the baryon number current) representing the homogeneous Wess-Zumino term present in HLS theory. From with the field-strength tensors of the vector mesons this model, the attractive force due to the ρ meson and re- (ρ) 1 1 1 1 pulsive force arising from ω meson between nucleons in both V = ∂µ gρρν ∂ν gρρµ i gρρµ , gρρν , µν 2 ! − 2 ! − " 2 ! 2 !# skyrmion and skyrmion matter were illustrated clearly. More recently, this study was refined in HLS including the next- (ω) 1 1 Vµν = ∂µ gωων ∂ν gωωµ . (66) to-leading order terms of the chiral counting [43–46,70,71]. 2 ! − 2 ! In these calculations, the anomalous part of the effective the- The O(p4) Lagrangian in Eq. (57) is given by ory that encodes the ω contribution corresponding to the U(1) gauge field in the CS term of the SS model was fully taken µ ν µ ν (4) = y1Tr αˆ µαˆ αˆ ναˆ + y2Tr αˆ µαˆ ναˆ αˆ into account. L ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ h µ iν h µ ν i + y3Tr αˆ µαˆ αˆ ναˆ + y4Tr αˆ µαˆ ναˆ αˆ k k k k k k k k h µ νi h µ iν 4 + y5Tr αˆ µαˆ αˆ ναˆ + y6Tr αˆ µαˆ ναˆ αˆ 5.1 Hidden local symmetry Lagrangian to O(p ) ⊥ ⊥ k k ⊥ ⊥ k k h ν µi h i + y7Tr αˆ µαˆ ναˆ αˆ As will be mentioned below, there is an indication that HLS ⊥ ⊥ k k 2 h µ iν ν µ to the leading order, that is, O(p ), has a hidden power that is + y8 Tr αˆ µαˆ αˆ ναˆ + Tr αˆ µαˆ ναˆ αˆ ⊥ k ⊥ k ⊥ k ⊥ k not obvious from general EFT considerations. We comment n h µ i h io + y Tr αˆ αˆ αˆ αˆ ν , on this later. Here we will consider up to O(p4) in the power 9 µ ν h ⊥ k ⊥ k i expansion. +iz Tr V(ρ)αˆ µ αˆ ν + iz Tr V(ρ)αˆ µαˆ ν . (67) 13 4 µν 5 µν h ⊥ ⊥i h k k i (ω) Note that Vµν does not appear in the z4 and z5 terms. Finally, These results support the intuitive picture that the soliton is the anomalous parity the hWZ term, anom, is compressed due to the attractive force from the ρ meson, L while it is expanded by the repulsive force from the ω me- 3 Nc son. The potentially important contribution from a dilaton, Γ = 4 = , hWZ d x anom 2 ci i (68) L 16π 4 L particularly to nuclear and dense matter, will also figure in Z ZM Xi=1 the balance between the attraction and the repulsion. where M4 stands for the 4-dimensional Minkowski space and Putting the skyrmions given by (57) onto crystal lattice – 3 3 here FCC, one can also study the effect of the lowest-lying 1 = i Tr αˆ Lαˆ R αˆ Rαˆ L , (69a) L − vector mesons in dense skyrmion matter. Here and in what 2 = i Tr αˆ Lαˆ Rαˆ Lαˆ R , (69b) L follows, HLSmin(π, ρ, ω) will stand for the minimal model of = Tr F αˆ αˆ αˆ αˆ , (69c) L3 V L R − R L Ref. [75] – which can be obtained from (57) by switching off and taking c = c = 2/3 and c = 0 in the in the 1-form and 2-form notations with L(4) 1 − 2 3 Lanom term. The energy per skyrmion and the vacuum expectation value φ qq¯ as functions of the crystal size (density) are αˆ L = αˆ αˆ , 0 k − ⊥ plottedh ini Fig. ≡ h 11.i αˆ R = αˆ + αˆ , k ⊥ F = dV iV2. (70) V − In the Lagrangian (57) there appear numerous many parameters, fπ, a, gρ, gω, y (i = 1, , 9), z (i = 4, 5), and i ··· i ci(i = 1, 2, 3). It is very difficult to fix them all in a phenomenological way. A possible approximation is to use a recently developed holographic QCD model in which the coefficients can be completely fixed by means of a set of “master formulae” that match the four-dimensional effective (HLS) theory to the five dimensional holographic QCD (hQCD) model. In the large Nc and large λ limit, the hQCD has, as mentioned above, two parameters which can be related to the empirical values of the pion decay constant and the ρ meson mass. Here we take the SS model [13,14] and the following S empirical values:

fπ = 92.4 MeV, mω = mρ = 775.5 MeV. (71)

5.2 Skyrmion properties in hidden local symmetry In addition to the the soliton masses obtained above – in Eqs. (54), (55) and (56), one can also study other properties, Figure 11 The energy per skyrmion and the vacuum expectation value of 2 φ0 as a function of the crystal size. The vertical line indicates the crystal size i.e., the soliton winding number radius r W and energy h i corresponding to the normal nuclear density. radius r2 . The results are: p h iE p = π, ρ, ω: •M r2 0.433 fm, From Fig. 11 we can draw the following conclusions: the h iW ≈ p 2 critical density n1/2 at which the half-skyrmion phase appears r E 0.608 fm. (72) ph i ≈ in HLS(π, ρ, ω) is larger than that in HLSmin(π, ρ, ω). This is because in HLS(π, ρ, ω) the skyrmion size is smaller so = π, ρ: •M that it needs a smaller distance to have a signiﬁcant force be- 2 tween skyrmions. This conclusion is supported by the n1/2 in r W 0.247 fm, h i ≈ HLS(π, ρ) which yields a much smaller skyrmion size. From p r2 0.371 fm. (73) h iE ≈ HLS(π, ρ) one sees that the binding energy in this case is p much smaller than that in the other two cases. This is be- = π: •M cause, the model HLS(π, ρ) includes only the lowest-lying isovector vector mesons which do the most eﬃcient drive to- r2 0.309 fm, h iW ≈ ward the conformal limit. Note also that accounting for the p 2 r E 0.417 fm. (74) full hWZ terms is important for the location of n1/2. ph i ≈ 14 5.3 Skyrmion matter in hidden local symmetry

1.0

0.8 For confronting nature, the relevant observables are the in-

medium properties discussed in Sec. 3. The interesting quan- 0.6 tities are the in-medium pion decay constant fπ∗ and the eﬀec- f * / * f tive nucleon mass m∗N . What turns out to be the most remark- 0.4 able is the close relation between the two in the half-skyrmion phase. The relation remains valid even in the presence of 0.2 scale invariance that will be taken into account below.

0.0

2.5 2.0 1.5 1.0 0.5

Consider the pion decay constant. The axial correlator L (fm) (15) receives an additional contribution to Fig. 5 for the Skyrme model from the ρ exchange as depicted in Fig. 12. Figure 13 Crystal size dependence of the fπ∗ normalized by its vacuum values.

ρ In the present approach, the skyrmion properties are calcu- Jµ5 Jν5 Jµ5 π Jν5 Jµ5 Jν5 lated with the parameters fixed by meson dynamics in HLS. Hence, by taking the medium modified parameters as input, (i) (ii) (iii) one can calculate the medium-modified skyrmion (or baryon) Figure 12 Three types of contributions to the correlator of Eq. (15): (i) the properties. This allows us to calculate the skyrmion-crystal contact diagram, (ii) the pion exchange diagram, and (iii) the rho exchange matter to obtain modified baryon properties with fπ∗ plotted diagram. Shaded blobs stand for the skyrmion matter interaction vertices. in Fig. 13 as input. The in-medium nucleon mass m∗N Msol∗ so obtained is plotted in Fig. 14. This result shows that∼ the density dependence of the nucleon mass is surprisingly similar to that of fπ∗. The simple way to understand this close connection is the large Nc property of the skyrmion model The three graphs contribute which seems to hold for low as well as high density. The nucleon mass should behave as ξ f with ξ scale independent ∼ π∗ (and density independent) O( √Nc) constant.

2 ab 1 a 2 2 (i) : i f gµνδ 1 + − 1 φ 1 π 2 − 3 π − 2 pDµ pν ab h 2 2 iE 1200 (ii) : i fπ p2 δ 1 3 φπ 1

− − − 1100 2 2 4 D E ab a g fπ pµ pν 2 2

(iii) : iδ 2 2 gµν 2 1 φ 1 . (75) 1000 p mρ mρ 3 π − − h − − i

900

800 * (MeV)* sol To the order of expansion considered, the sum of the three M 700 terms is Lorentz invariant and has the same expression of fπ∗ 600

as that from the Skyrme model, Eq. (20). The crystal size 500 dependence of fπ∗ from HLS skyrmion is plotted in Fig. 13. 400 In the skyrmion phase n < n1/2, fπ∗ decreases as density 2.5 2.0 1.5 1.0 0.5 increases. This tendency is supported in observations with L (fm) deeply bound pionic nuclei. What is noteworthy and is of importance in the developmentis that this tendencystops at n . 1/2 Figure 14 Crystal size dependence of the medium modified nucleon mass. After n1/2, fπ∗ stays more or less constant nearly (modulo approximations) independent of density. The behavior of fπ∗ in HLS theory is the same as that in the Skyrme model shown in 5.4 Scale symmetry: Explicit breaking and impact on Fig. 6. This robust property of the pion decay constant, strik- dense matter ingly independent of the degrees of freedom included, is the key feature of the approach anchored on the topology change. What is strikingly clear from the results discussed above is The topology is dictated by the pion field and the ρ field does that the presence of the ω field substantially worsens the com- not carry any influence on the topology. parison of the skyrmion matter with nature. We show here 15 β′ that what’s involved is the role of the scalar degree of free- 1 µ χ + (1 c2) ∂µχ∂ χ , (78) dom [76]. In what we discussed so far, no scalar degree of − 2 ) fσ ! freedom has figured explicitly in the hidden local symmet- χ 4 χ β′ ric Lagrangian, either with the lowest-lying vector mesons V(χ) = c3 + c4 . (79) fσ !  fσ !  or with the infinite tower. The standard strategy in nuclear     physics based on effective Lagrangians, both nonlinear sigma Following [51], we set, in the chiral limit, c1 = c2 = model and hidden local symmetric model, is that the effect 1 + O(p2) which can be arrived at by setting the dilaton field of the iso-scalar scalar excitations are generated at high or- equal to zero for processes that do not involve scalar excita- der of chiral expansions. However in the approach exploit- tions. The best way to understand this relation is that there is ing HLS, it is natural to consider explicitly the scalar exci- hidden scale symmetry in Standard Higgs-type Lagrangian tations of mass 600 MeV, given that this mass scale is that yields both the scale-symmetric form and the nonlin- ∼ comparable to that of vector mesons and that tensor forces, ear sigma model form that can be reached when one dials a essential in nuclear dynamics, involve the ρ meson in addi- constant, respectively, to weak coupling limit and to strong- tion to the pion and the ω meson provide the crucial repul- coupling limit [80]. Keeping to O(p2) in the leading-order sion that stabilize the nuclear matter. First in 1991 [77] and Lagrangian, we have since then [43,54,58,78], the scalar, interpreted as a Nambu- 2 2 σ fπ χ µ 1 µ 4 Goldstone boson, say, dilaton , was introduced to describe = Tr ∂ U∂ U† + ∂ χ∂ χ + O(p ). (80) inv 4 f µ 2 µ the properties of light-quark hadrons on dense medium, as a L σ ! means of simulating a precursor signal for chiral restoration Hidden-local-symmetrizing this, we have [76] expected at high density. The basic idea there was that chi- 2 2 ral symmetry characterized by the quark condensate qq¯ is 2 χ µ 2 χ µ h i sHLS = fπ Tr aˆ µaˆ + a fπ Tr aˆ µaˆ locked to the dilaton condensate χ . It has become clear that L fσ ! ⊥ ⊥ fσ ! k k h i h i h i describing dense baryonic matter in HLS, above the normal 1 µν 1 µ 4 Tr VµνV + ∂µχ∂ χ + O(p ). (81) nuclear matter density n0, is unrealistic without incorporat- 2g2 2 − h i ing the dilaton degree of freedom. This is because there is an The dilaton potential is of course unaffected. intricate interplay between the role of ω providing repulsion In the presence of vector fields, the anomalous-parity ho- and that of dilaton providing attraction. Thus much of the mogeneous Wess-Zumino term , Eq. (68) needs to be Lan difficulty in the single baryon as well as in baryonic matter taken into account. There are three terms. For an approxi- we observed above when the ω degree of freedom is incorpo- µ mate calculation we take only one term in the form, gωµB rated is caused by what we consider as inadequate treatment where Bµ is the baryon current. It is straightforward to treat of scale symmetry broken both explicitly and spontaneously. all three terms involved therein at the cost of more parame- This comment applies to both HLS-skyrmion matter (4D) and µ 4 ters. The gωµB term considered here is O(p ), higher order holographic matter (5D). than what’s treated above. It cannot however be ignored be- One powerful way to implement scale symmetry in HLS cause it is through this term that the ω field couples to the approach was recently suggested by Crewther and Tunstall other degrees of freedom of HLS Lagrangian. Without it, the (CT) [51] #7. This approach is anchored on the possible ex- ω does not figure in the interactions. By itself, this term is of istence in QCD of an infrared fixed point at which the trace scale dimension 4, hence scale-invariant. Therefore it has no anomaly is to vanish in the chiral limit #8 coupling to the dilaton field χ. This is however not consistent We start with the leading order nonlinear sigma model La- with scale-chiral symmetry àla CT. The correct expression grangian to which scale symmetry is implemented as written should be [76] by CT. For simplicity in notation, we work in the chiral limit. β χ ′ The CT Lagrangian, in terms of the conformal compensator = c + (1 c ) . (82) hWZχ  h h  hWZ σ/ fσ L − fσ ! L field χ = fσe , is     Unlike c1,2 in (77), there is no reason why ch should be close = + + V(χ), (76) to 1. In fact there is an indication that ch = 1 at which there is L Linv Lanom 2 2 no coupling to χ is found to be violently at odds with nature. fπ χ µ 1 µ inv = c1 Tr ∂µU∂ U† + c2 ∂µχ∂ χ,(77) In [74], when skyrmions, obtained with the Lagrangians L 4 fσ ! 2 (81) and (82) with ch = 1 and V(χ) of the Coleman-Weinberg 2 2 fπ χ µ type, were put on crystal lattice to simulate dense matter, the = (1 c ) Tr ∂ U∂ U† anom 1 4 f µ contribution to the energy of the system from the hWZ term L ( − σ ! #7There is an alternative approach that involves an IR fixed point [79] in which the role of IR fixed point is given in terms of a critical number of fla- / = c vors number of colors n f N f /Nc in what corresponds to the Veneziano limit. A comparison of this approach and C-T was given in Ref. [55]. #8 Whether such an IR fixed exists in QCD with the number of flavors NF < 8 is not yet settled. This matter is discussed elsewhere in the context of applications in nuclear system and we won’t go into it. This issue, clearly important for the fundamental structure of QCD, is however not so crucial in applications to dense matter because our approach does not require the precise nature of QCD in the chiral limit, which is not honored in dense matter. 16 diverged at high density unless the pion decay constant (or 6.1 The (near-)BPS skyrmion model equivalently the ω mass) went to . This defect can be elim- In the “derivation” of the small binding energy from a flat- inated if the explicit breaking of scale∞ symmetry dampens the space YM Lagrangian discussed above, the relevant degrees strength of the hidden gauge coupling g. In [78], the problem of freedom in nuclear physics, namely, the pion and the was resolved when the factor ( χ )n with 1 < n < 3 was arbi- fσ lowest-lying vector mesons, are augmented with the infinite trarily multiplied to . This happens because∼ in medium LhWZ tower leading to the BPS structure. Now instead of adding f / f decreases monotonically at increasing density, there- σ∗ σ the infinite tower to arrive at the BPS limit, one simply posits fore the coupling gets sufficiently suppressed. This means a BPS structure in 4D. So the philosophy would then be, in- that the solution to the problem found in [78] corresponds to stead of starting from a chiral Lagrangianwith pions only and (82) with [76] building up the tower to go to heavy nuclei, why not start with the simplest structure that could work for heavy nuclei with 1 < β′ < 3, ch 0. (83) ∼ ≈ nearly vanishing binding energy and then go from there mak- It seems that this solution is consistent with the bound for the ing appropriate corrections? This is the point of view taken 2 by Adam et al. [25,26]. (For a recent review, see Ref. [83]). anomalous dimension of G β′ = γF2,IR given in [81] The parameters entering in this approach will have no direct connection with QCD proper, in a spirit totally different to γF2 ,IR 6 3. (84) what has been resorted to above. If this turns out to be a possible solution to the problem, then The justifications offered for this drastically unorthodox the analysis reported above would need to be reexamined approach are: with the potential β′ corrections in the hWZ terms including 1. The binding energy is zero at the leading order even the CS term in holographic models [43]. though highly non-linear strong interactions may be in- What we learned in this subsection is that to correctly de- volved. One might think of this as an extreme quasi- scribe baryonic matter at high density, scalar degrees of free- particle description with no residual interactions. dom are indispensable. They could perhaps be generated at high (chiral) orders in perturbative scheme. In the spirit 2. The model effectively describes a perfect fluid with its espoused in this review, they could come in as a dilaton in energy-momentumtensor in the perfect fluid form, and the broken scale symmetry. For this, it seems essential, if the static energy functional is invariant under volume- the analysis described in this subsection is correct, that how preserving diffeomorphisms. the scale symmetry is explicitly broken has to be understood. 3. While the standard quasiparticle description of bary- What we have observed above regarding the disastrous role onic matter, namely, Landau Fermi-liquid theory, can of the ω meson in dense matter in the presence of a dila- be approximately given in the mean field of a relativis- ton indicates the necessity of the dilaton coupling to the ω tic field theory, here the quasiparticle picture is to cap- in the hWZ term in 4D skyrmion matter or the Chern-Simons ture the exact result of the theory. The mean-field result term in 5D instanton matter. And this requires understanding of the relativistic field theory can be an approximation how the explicit scale symmetry breaking manifests in dense that is obtained by “averaging” the BPS theory. medium. This is an open problem to be resolved. In the BPS skyrme model of [25], the basic quantity is the 6 Other Developments SU(2) valued field U(x, t) in which topology is lodged. The Lagrangian picked is

In this section we discuss a few interesting developments that 2 4 µ 2 = λ π µ µ V(tr(U)), (85) are not directly related to what we discussed above anchored LBPS − B B − on topology change present in the description of dense mat- where λ and µ are parameters that are not related to QCD ter with skyrmions on crystal lattice. The premise is that the paramters. They will be fit to nuclear binding energies that effective Lagrangian that is taken has a connection to QCD in are focused on. The potential V(tr(U)) is a suitably chosen the sense of Weinberg’s “folk theorem” on effective field the- potential involving nonderivative terms. In [29], it is taken ories [82]. The basic assumption is that the theory has a valid in the forms contact with strong interactions at low energy, beginning with low-energy theorems based on current algebras. Thus the 1 V(tr(U)) = Vπ = tr(1 U), model I; standard nuclear physics approach to dense matter has been 2 − 2 to first start with pions only with baryons put in explicitly or V(tr(U)) = (Vπ) , model II. (86) brought out as solitons, describe well nuclear matter and then extrapolate to the regime that is outside of the range that is fit The baryon number Bµ takes the familiar expression to the theory. Here we discuss a few cases where this standard 1 µναβ = ǫ tr U†∂ UU†∂ UU†∂ U . (87) procedure is not adhered to. µ 24π2 ν α β 17 B i~τ ~rθ(r) First to see that the BPS skyrmion has zero binding en- takes the axial symmetric ansatz U(r) = e · . With this it ergy, one shows that it satisfies the Bogomoln’yi equation. is straightforward to calculate the soliton energy for the BPS The static energy given by Eq. (85) has the bound 64 √2π 2 3 2 2 3 Esol = µλN, (94) E = d x π λ 0 µ √V 2π λµ d x 0 √V 15 Z B ± ∓ Z B 2 3 > 2π λµ d x 0 √V where N = B is the winding number, i.e., baryon number. ∓ Z B Note that the soliton energy per particle then is just a con- = 2π2λµ B √V , (88) stant µλ given by the combination of the potential and the | | S3 D E constant∝ multiplying the topological term, so the energy of N √ √ baryon number system is a multiple of baryon number and where V S3 is the average value of V on the target space S 3. TheD lastE equality follows from the condition that B > 0. hence zero binding energy at the level of the soliton. The bound is saturated by the solutions of the BPS equation To go beyond the classical soliton structure, one needs to of motion from (85). perform the collective quantization. The resulting rotational spectra are given by π2λ µ √ = 0. (89) B0 ± U ~2 j( j + 1) i(i + 1) 1 1 B2 By coupling gravity to (85), one can write the action of the E = + + k2 , rot 2 − − 3 BPS model in curved space as 1 1 3 1 1 ! ! J I I I J (95) where i and i are, respectively, the angular momentum 4 1 2 4 1 ρ σ 2 J I S BPS = d x g 2 λ π g − gρσ µ V . (90) moment of inertia and the isospin moment of inertia. For Z | | − | | B B − even B nuclei, k3 = 0. The resulting binding energy including From this action, one can show that the energy-momentum the Coulomb energy and isospin violation is given in Fig. 15. tensor has the perfect fluid form The theoretical prediction with only 3 fit-parameters agrees very well with the experimental data and the Weizsacker’s T ρσ = (p + ε)uρuσ pgρσ (91) empirical formula. − with the four-velocity uρ, energy density ε and pressure p EBA 10 given by ρ ρ σ π 8 u = / gσπ , B B B 2 4p 1 ρ σ 2 ε = λ π g − gρσ + µ V, 6 2 4| | 1 BρBσ 2 p = λ π g − gρσ µ V. (92) | | B B − 4 Going further, it can be shown that the BPS skyrmion is 2 equivalent to a non-barotropic, relativistic perfect fluid in the

Eulerian formulation. A 0 50 100 150 200 The energy of the BPS system described by the Lagrangian (85) Figure 15 Binding energies per nucleon in MeV. The diamond is from the present BPS model. The triangle is from the Weizsacker’s formula. And, the 3 4 2 2 2 solid line is from experimental data. The figure is borrowed from Ref. [27]. E = d x π λ 0 + µ V(trU) (93) Z n B o has, as stated above, a Bogomoln’yi bound. In fact it has in- The model was applied also to nuclear matter and finitely many BPS solutions saturating the bound. The topol- compact-star matter. The latter is done by coupling the BPS ogy is lodged in the field U which is of the pion field type as skyrmion model to gravity using the action (90) and the re- in the usual chiral Lagrangian. In this approach, the pion field sulting energy-momentum tensor of a perfect fluid (91). The is a highly correlated nuclear collective excitation of pionic maximum mass and the radius obtained therefrom – without quantum number that supports a solitonic structure that has resorting to the EoS and the Tolman-Oppenheimer-Volkoff no fluctuation component that enters in the low-energy theo- (TOV) equation – come out to be [29] rems of strong interactions. This would naturally exclude the possibility that the pion field that enters in this description Model I: Mmax = 3.734M ; Rmax = 18.456 km. be directly related, if any, to the asymptotic pions observed • ⊙ in nature. Assuming that it is of the soltion structure similar to that of large Nc QCD model, i.e., the Skyrme model, one Model II: Mmax = 2.439M ; Rmax = 16.801 km. 18 • ⊙ 6.2 The simplified HLS (SHLS) for dense matter 7 solar

M/M 6 As mentioned above, the HLS Lagrangian (57) yields re- markable predictions for nucleon and nuclear matter. How- 5 ever, even limited to the next-to-the leading order, it involves 4 much too many parameters and while straightforward, is not 3 very illuminating. Fortunately it turns out that the Lagrangian

2 (57) can be vastly simplified. This is feasible not only in the matter-free vacuum but more significantly in dense matter as 1 described above and elaborated further below. 0 13 14 15 16 17 18 19 To simplify the model , we first analyze how each term of r (km) the O(p4) terms in the HLS sector contributes to the soliton Figure 16 Mass-radius relation calculated from the BPS model with po- mass. The results are summarized in Table. 1. From this ta- 2 tential I (dotted line) and model II (box line). This figure is borrowed from ble, we can conclude that the (p ), z4 and hWZ terms give Ref. [29]. dominant contributions, say, aboutO 97.28% of the total, to the soliton mass. The simplified Lagrangian takes the form 1 = 2 µ + 2 µ µν SHLS fπ Tr[â µaˆ ] a fπ Tr[â µaˆ ] 2 Tr[VµνV ] L ⊥ ⊥ k k − 2g µ ν The results plotted in Fig. 16 show that both the mass and + iz4 Tr Vµναˆ αˆ + hWZ. (96) h ⊥ ⊥i L the radius – which depend appreciably on the unknown po- We next study the skyrmion matter properties from tential – tend to be a bit too big in comparison with what’s model (96) and compare the result with that from full observed [30,31]. Given that the model exploits only a few HLS (57). Our results of the crystal size L dependence of the parameters and there is a plenty of room available for refine- per-skymrion energy and φ are plotted in Fig. 17. This fig- h 0i ment, one cannot consider this as a criterion to consider the ure tells us that the SHLS model indeed captures the essence model failing. of the skyrmion dynamics. Moreover, n1/2 is pushed to a somewhat larger value which makes it more consistent with A comment on the contrast of the BPS approach to the nature. current lore of nuclear physics is in order here. There are two key ingredients in the currently predominant attitude among nuclear theorists: The first is that the small binding energy

HLS( )

observed in nuclei is an interplay between a big attraction 1600 HSkyrm e characterized by scalar degrees of freedom and a big repulsion provided by vector degrees of freedom and the second, 1400 the theory has be able to accurately satisfy the conditions met

1200 both experimentally and theoretically at around nuclear mat- E/B(MeV) ter density. The BPS model is contrary to the ﬁrst in that

1000 the BPS “nucleon” is a topological blob ignorant of the in-

tricate interactions that lead to its property. As it stands, it 800 cannot account for low-energy pionic processes that are de- 1.0

scribed by low-energy theorems that are the ground to ef- 0.8 fective chiral theories. As for the second, it cannot access the accurately measured nuclear response functions to the 0.6 >

electroweak (EW) ﬁelds. For instance, while it may provide < bulk properties of nuclear ground states, it cannot account for 0.4 the important role of meson-exchange currents that are accu-

0.2 rately described by eﬀective chiral Lagrangian approaches.

One should however not dismiss this model. It may be pos- 0.0

2.5 2.0 1.5 1.0 0.5

sible to start from the BPS matter and develop a theory that L(fm ) can access those processes given by soft-pion theorems. This is in the same spirit as starting from Landau Fermi-liquid

fixed point applicable to the regime of nuclear equilibrium Figure 17 Comparision of the E/B and φ0 calculated from HLS and h i density and arriving at processes involving soft-pions. This HSkyrme model as a function of L. is exemplified by relativistic mean-field approaches that are connected to Landau Fermi-liquid theory. There pionic inter- In summary, the model (96) is verified to capture the dom- actions can be – and are – readily incorporated. inant physics of HLS (57). The crucial observation is that, as 19 Table 1 Ratio (%) of the contribution from each term of HLS to the soliton mass. (p2) y y y y y y y y y z z c c c O 1 2 3 4 5 6 7 8 9 4 5 1 2 3 73.89 1.29 0.35 0.03 0.02 1.64 0 0.07 0.49 3.04 13.24 0.22 14.21 4.54 17.88 − − − − −

with the full HLS calculation [45], the density n1/2 at which Lagrangian, bsHLS, inherits from QCD the dependence on, the transition from skyrmions to half-skyrmions takes place among others, nonperturbative QCD quantities, such as the 2 lies higher than that of normal nuclear matter, n0. The reason condensates qq¯ , G etc. injected from QCD correlators that the terms in (96) more or less fully capture the physics of into the parametersh i h of thei bsHLS. Therefore the effective La- HLS (57) is the following: The O(p2) terms encode the cur- grangian will carry information on the “vacuum.” Now the rent algebra of hadron physics, so they are dominant in low- basic assumption in the present approach is that those con- energy nuclear interactions. The ω meson – which is indis- densates scale as the density of the system described by the pensable for stability of nuclear matter – figures through the Lagrangian. This will then constitute a Lagrangian defined in hWZ terms. Amongall the O(p4) terms– apartfromthe hWZ a “sliding vacuum” which would provide the tool to access terms – the z4 term, involving the strong ρ-π-π interaction, dense nuclear matter. contributes most importantly whereas the z5 term, describing ρ-ρ-ρ interactions, is suppressed. In fact it is this simplified Lagrangian that through the effect of explicit scale symmetry This approach has been applied to various properties of breaking removes the “ω disaster” in dense skyrmion matter baryonic matter up to the density commensurate with nor- discussed in Section 5.4. mal nuclear matter, making connection with Landau Fermi- liquid fixed point theory. The application relies on Wilsonian renormalization-group strategies [86]. Early discussions on 6.3 Emerging symmetries in dense matter? this matter can be found in [87]. That symmetries may emerge in highly correlated systems is rapidly becoming a highly plausible and acceptable concept in physics. This tendency is strikingly visible not only in con- The novelty in applying the sliding-vacuum strategy to densed matter physics but also in particle physics including density regimes higher than n0 is that the intrinsic density gravity and dark matter, see e.g., [84]. Dense compact-star dependence of QCD condensates in the parameters of the La- matter we have been addressing is equally highly-correlated grangian takes into account the topology change observed in matter and it would not be an idle conjecture that certain sym- the skyrmion crystal model. The topology change that takes metries could emerge as density increases beyond the normal place at n1/2 2n0 is then encoded in how the various param- ∼ n0. This we claim is indeed the case with both hidden scale eters of the Lagrangian behaves as density goes from below symmetry and HLS in dense nuclear systems [85]. This is a to above n1/2. notion eminently novel in nuclear/hadron physics. An approach that exploits robust topological features of, but bypassing the complexity of, the skyrmion crystal method The characteristic features resulting from bsHLS with the is a continuum Lagrangian that relies on “sliding vacuum” topology change at n = n1/2 incorporated are: As density for dense matter developed first in 1991 [77] and then elabo- goes higher than n1/2, (1) the baryonic matter flows to the rated further in [86]. The starting ingredient is an effective vector manifestation at which the ρ mass m∗ 0 [57], ρ → Lagrangian constructed along the line of Weinberg’s “folk (2) the nucleon mass m∗ f ∗ f ∗ m0 , 0 and N → π ≈ σ ≈ theorem (FT)” that combines scale symmetry and chiral sym- (3) such that m∗/m∗ 0. With these taken into account ρ N → metry (scale-chiral symmetry) of QCD discussed in Section and with n1/2 2n0, the theory has been applied to both ≈ 5.4. The effective Lagrangian is built with the pseudo-NG normal nuclear matter and dense compact-star matter using boson, pion, the lowest-lying vector mesons and the dilaton, the well-established Vlowk renormalization group (RG) tech- encoding hidden scale symmetry and HLS and the baryons nique [59,88]. This approach is verified to work well for as relevant degrees of freedom. The Lagrangian is defined nuclear properties and predicts the mass and radius in fair at a scale ΛM, referred to as “matching scale,” at which the agreement with the observed massive stars [30,31]. correlators of QCD (high-energy scale) and the correlators of the effective theory, dubbed as bsHLS (low-energy scale) are optimally matched. It is an open issue as to whether such a A full description of the calculation is rather involved, in- matching is feasible. It is usually assumed that the relevant cluding the subtle issue of double decimations in doing RG ΛM can be the chiral scale 4π fπ 1 GeV. In practice in calculation in nuclear systems, etc. It suffices for our pur- nuclear applications, it is taken∼ just∼ above the vector-meson pose to give a few results to illustrate the main point of this mass scale, m 600 700 MeV. At the matching, the EFT subsection. V ∼ − 20 Figure 18 The symmetry energy predicted by the Vlowk with bsHLS with Figure 20 The sound velocity predicted in bsHLS model [85]. α ≡ n1/2 = 2n0 [59]. The eye-ball slope change is indicated by the colored (N Z)/(N + Z) where N(Z) is the neutron(proton) number. − straight lines. The lines labelled as A, B, C are empirical constraints coming from heavy-ion experiments. 7 Perspectives and Discussions First of all, it gives a unique prediction for the symmetry energy Fig. 18, weeding out the wilderness in Fig. 3. The Skyrme’s pioneering idea of getting baryons from mesonic topology change that gave the cusp in the skyrmion crystal theories was not only to unify the mesons and baryons in a is manifested here as a changeover from soft-to-hard EoS at single framework but also to treat the nucleon, elementary n1/2. Secondly, as shown in Fig. 19, the topology change particle, and the nucleus, complex system of many nucleons, increases dramatically the maximum mass from (C) 1.7 on the same footing. One is then to arrive at nucleus from ∼ M (no topology change) to (A) or (B) > 2 M (topology pion as one does to nucleon. It is now established that the ⊙ ∼ ⊙ change). The changeover resembles hadron-quarkyonic tran- skyrmion out of the pion field is a nucleon in the large Nc sition [89], suggesting a quark-hadron duality. limit of QCD in which quarks and gluons are the elementary constituents. In making these multiple connections, at 2.6 the core lies the topology. In fact, topology figures strik- 2.4 ingly in all aspects of physics, as one can gather from the (A): new-BR n =2.0n (A) 2.2 1/2 0 volume entitled “The Multifaceted Skyrmion” [2], ranging (B): new-BR n1/2=1.5n0 (B) / 2 from quarks gluons to nucleons to nuclei to condensed mat- (C): old-BR ter to string theory. Numerous startling new discoveries are 1.8 sun being made involving topology in 2 and 3 dimensional sys- 1.6 M/M tems, such as topological supeconductivity, and some daring 1.4 (C) mathematical framework is brought out to organize all visible 1.2 matters, molecules, atoms, nuclei, in complex geometry [90]. 1 While a great progress with amazing applications is being 0.8 made in condensed matter, with new experimental discover- 7 8 9 10 11 12 13 ies, the progress in nuclear physics for which Skyrme’s orig- R [km] inal idea was put forward has met with much less success. Figure 19 Mass-radius relation calculated from the bsHLS [59]. (C) is This has in part to do with that direct experimental observa- gotten with no topology change and (A) & (B) with topology change. tions are difficult to come by and theories, involving strong interactions, are harder to control. Another feature, hitherto undiscovered in the field, is the There, however, have been efforts since a decade or so, possibility that the sound velocity of massive compact stars with some notable progress in difficult fundamental problems could approach, for n > n1/2 2n0, what is conventionally of nuclear physics. The principal effort has been directed to 2 ∼2 referred to as “conformal,” vs/c = 1/3 [85]. Thisis shownin decipher what takes place at high density relevant to the in- Fig. 20. The trace of energy momentum tensor is not equal to terior of compact stars, an extremely difficult problem which zero but density-independent constant, so the half-skyrmion is currently out of reach by the only known nonperturbative state is not strictly scale-invariant but hints at the dilaton limit tool of QCD, lattice QCD, at high density. fixed point at which the dilaton condensate goes toward zero. The progress made so far comes from several different di- This is an extremely interesting possibiltiy to explore experi- rections. With the understanding of the Cheshire Cat Phe- mentally. nomenon, what is confined – color – and what is not confined 21 – baryon charge – is clarified in terms of quark-bag-skyrmion 9 H. J. Lee, B. Y. Park, D. P. Min, M. Rho and V. Vento, “A Unified duality. This then hints at the origin of the proton mass via an approach to high density: Pion fluctuations in skyrmion matter,” Nucl. intricate interplay involving topology between confinement, a Phys. A 723, 427 (2003) [hep-ph/0302019]. mathematically daunting problem, and spontaneous breaking 10 J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. 38, 1113 (1999) [Adv. Theor. of chiralsymmetry [91]. It is nowbecomingclear that the ori- Math. Phys. 2, 231 (1998)] [hep-th/9711200]. gin of mass for bosons is most likely different from that for 11 J. Erlich, E. Katz, D. T. Son and M. A. Stephanov, “QCD and a baryons. It is the topology change from skyrmions to half- holographic model of hadrons,” Phys. Rev. Lett. 95, 261602 (2005) skyrmions in dense matter at n > 2n0 that gives the signal for [hep-ph/0501128]. 12 L. Da Rold and A. Pomarol, “Chiral symmetry breaking from five di- this difference. In the density regime∼ n > n1/2 2n0, the hidden gauge symmetry associated with the∼ vector∼ mesons and mensional spaces,” Nucl. Phys. B 721, 79 (2005) [hep-ph/0501218]. the hidden scale symmetry associated with the dilaton scalar 13 T. Sakai and S. Sugimoto, “Low energy hadron physics in holographic QCD,” Prog. Theor. Phys. 113, 843 (2005) [hep-th/0412141]. intervene in nuclear dynamics in such a way that while the ρ 14 T. Sakai and S. Sugimoto, “More on a holographic dual of QCD,” Prog. mass tends toward zero, the nucleon mass goes to a chirally Theor. Phys. 114, 1083 (2005) [hep-th/0507073]. invariant mass m which is non-zero and large m O(m ). 15 D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Chiral Dynamics of Baryons 0 0 ∼ N Thus going toward possible deconfinement density ndc n0, from String Theory,” Phys. Rev. D 76, 061901 (2007) [hep-th/0701276 ≫ [HEP-TH]]. the ratio of the effective masses in medium mρ/mN 0, which is at odds with the standard “Nambu scenario” for→ the 16 D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Dynamics of baryons from string theory and vector dominance,” JHEP 0709, 063 (2007) origin of mass based on the NJL model. [arXiv:0705.2632 [hep-th]]. In the range of densities relevant to compact stars, there is 17 A. Gorsky, P. N. Kopnin and A. Krikun, “Baryon as dyonic instanton- neither indication nor need for quark-gluon degrees of free- II. Baryon mass versus chiral condensate,” Phys. Rev. D 89, no. 2, dom. In fact, the EoS described by the topology change at 026012 (2014) [arXiv:1309.3362 [hep-th]]. 18 S. Bolognesi and P. Sutcliffe, “The Sakai-Sugimoto soliton,” JHEP n / 2n has a certain similarity to the smooth hadron- 1 2 ∼ 0 quark or hadron-quarkyonic phase change that is assumed to 1401, 078 (2014) [arXiv:1309.1396 [hep-th]]. take place at (2 3)n [89]. It is also possible that the 19 V. Kaplunovsky, D. Melnikov and J. Sonnenschein, “Baryonic ∼ − 0 popcorn,” JHEP 1211, 047 (2012) doi:10.1007/JHEP11(2012)047 half-skyrmion phase – which may be dual to the quarkyonic [arXiv:1201.1331 [hep-th]]. phase – is a precursor to (emergent) scale invariance with the 20 M. Elliot-Ripley, P. Sutcliffe and M. Zamaklar, “Phases of 2 2 sound velocity vs/c = 1/3. In the spirit of the Cheshire Cat, kinky holographic nuclear matter,” JHEP 1610, 088 (2016) this seems to suggest that “deconfinement” is irrelevant in doi:10.1007/JHEP10(2016)088 [arXiv:1607.04832 [hep-th]]. compact-star physics (See [92] for an argument that there is 21 M. Rho, S. J. Sin and I. Zahed, “Dense QCD: A holographic dyonic no deconfinement in QCD.). If deconfinement should inter- salt,” Phys. Lett. B 689, 23 (2010) [arXiv:0910.3774 [hep-th]]. 22 M. Elliot-Ripley, “Salty popcorn in a homogeneous low-dimensional vene at some density, this would mean that the treatment an- toy model of holographic QCD,” arXiv:1610.09169 [hep-th]. chored on topology should break down at that density. Clar- 23 M. Bando, T. Kugo, S. Uehara, K. Yamawaki and T. Yanagida, “Is ρ ifying this issue is beyond the framework that is taken in the meson a dynamical gauge boson of hidden local symmetry?,” Phys. works reviewed here. Rev. Lett. 54, 1215 (1985). 24 M. Harada and K. Yamawaki, “Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition,” Phys. Rept. 381, 1 (2003) [hep-ph/0302103]. The work of Y.-L. M. was supported in part by National Science Founda- 25 C. Adam, J. Sanchez-Guillen and A. Wereszczynski, “A Skyrme- tion of China (NSFC) under Grant No. 11475071, 11547308 and the Seeds type proposal for baryonic matter,” Phys. Lett. B 691, 105 (2010) Funding of Jilin University. [arXiv:1001.4544 [hep-th]]. 26 C. Adam, J. Sanchez-Guillen and A. Wereszczynski, “A BPS Skyrme model and baryons at large Nc,” Phys. Rev. D 82, 085015 (2010) 1 T. H. R. Skyrme, “A Nonlinear field theory,” Proc. Roy. Soc. Lond. A [arXiv:1007.1567 [hep-th]]. 260 (1961) 127. 27 C. Adam, C. Naya, J. Sanchez-Guillen and A. Wereszczynski, “Nuclear 2 The Multifaceted Skyrmions: Second Edition (World Scientific, Singa- binding energies from a Bogomol’nyi-Prasad-Sommerfield Skyrme pore, 2016) ed. M. Rho and I. Zahed. model,” Phys. Rev. C 88, no. 5, 054313 (2013) [arXiv:1309.0820 [hep- th]]. 3 G. ’t Hooft, “A Planar diagram theory for strong interactions,” Nucl. Phys. B 72, 461 (1974). 28 C. Adam, C. Naya, J. Sanchez-Guillen and A. Wereszczynski, “Bogomolnyi-Prasad-Sommerfield Skyrme model and nuclear bind- 4 E. Witten, “Baryons in the 1/n expansion,” Nucl. Phys. B 160, 57 ing energies,” Phys. Rev. Lett. 111, no. 23, 232501 (2013) (1979). [arXiv:1312.2960 [nucl-th]]. 5 A. V. Manohar, “Equivalence of the chiral soliton and quark models in 29 C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez and large N,” Nucl. Phys. B 248, 19 (1984). A. Wereszczynski, “BPS Skyrmions as neutron stars,” Phys. Lett. B 6 I. Zahed and G. E. Brown, “The Skyrme model,” Phys. Rept. 142, 1 742, 136 (2015) [arXiv:1407.3799 [hep-th]]. (1986). 30 P. Demorest, T. Pennucci, S. Ransom, M. Roberts and J. Hessels, 7 G. Holzwarth and B. Schwesinger, “Baryons in the Skyrme model,” “Shapiro delay measurement of a two solar mass neutron star,” Nature Rept. Prog. Phys. 49, 825 (1986). 467, 1081 (2010) [arXiv:1010.5788 [astro-ph.HE]]. 8 I. R. Klebanov, “Nuclear matter In the Skyrme model,” Nucl. Phys. B 31 J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tauris, R. S. Lynch, 262, 133 (1985). 22 M. H. van Kerkwijk, M. Kramer and C. Bassa et al., “A massive 55 Y. L. Li, Y. L. Ma and M. Rho, “Chiral-scale effective theory including pulsar in a compact relativistic binary,” Science 340, 6131 (2013) a dilatonic meson,” arXiv:1609.07014 [hep-ph]. [arXiv:1304.6875 [astro-ph.HE]]. 56 Y. Akaishi and T. Yamazaki, “High-density kaonic-proton matter 32 M. Kugler and S. Shtrikman, “A new skyrmion crystal,” Phys. Lett. B (KPM) composed of Λ K p multiplets and its astrophysical con- ∗ ≡ − 208, 491 (1988). nections,” arXiv:1610.01249 [nucl-th]. 33 M. Kugler and S. Shtrikman, “Skyrmion crystals and their symme- 57 M. Harada and K. Yamawaki, “Vector manifestation of the chiral sym- tries,” Phys. Rev. D 40, 3421 (1989). metry,” Phys. Rev. Lett. 86, 757 (2001) [hep-ph/0010207]. 34 M. Harada, H. K. Lee, Y. L. Ma and M. Rho, “Inhomogeneous quark 58 H. K. Lee, B. Y. Park and M. Rho, “Half-skyrmions, tensor forces condensate in compressed Skyrmion matter,” Phys. Rev. D 91, no. 9, and symmetry energy in cold dense matter,” Phys. Rev. C 83, 025206 096011 (2015) [arXiv:1502.02508 [hep-ph]]. (2011) Erratum: [Phys. Rev. C 84, 059902 (2011)] [arXiv:1005.0255 35 H. J. Lee, B. Y. Park, M. Rho and V. Vento, “Sliding vacua in dense [nucl-th]]. skyrmion matter,” Nucl. Phys. A 726, 69 (2003) [hep-ph/0304066]. 59 H. Dong, T. T. S. Kuo, H. K. Lee, R. Machleidt and M. Rho, “Half- 36 D. Suenaga, B. R. He, Y. L. Ma and M. Harada, “Mass degeneracy of skyrmions and the equation of state for compact-star matter,” Phys. heavy-light mesons with chiral partner structure in the half-Skyrmion Rev. C 87, 054332 (2013) [arXiv:1207.0429 [nucl-th]]. phase,” Phys. Rev. D 91, no. 3, 036001 (2015) [arXiv:1412.2462 [hep- 60 W. G. Paeng, T. T. S. Kuo, H. K. Lee and M. Rho, “Scale-invariant ph]]. hidden local symmetry, topology change and dense baryonic matter,” 37 A. S. Goldhaber and N. S. Manton, “Maximal symmetry of the Skyrme Phys. Rev. C 93, no. 5, 055203 (2016) [arXiv:1508.05210 [hep-ph]]. crystal,” Phys. Lett. B 198, 231 (1987). 61 J. W. Holt, G. E. Brown, T. T. S. Kuo, J. D. Holt and R. Machleidt, 38 M. Harada, Y. L. Ma, H. K. Lee and M. Rho, “Fractionized skyrmions “Shell model description of the C-14 dating beta decay with Brown- in dense compact-star matter,” arXiv:1601.00058 [nucl-th]. Rho-scaled NN interactions,” Phys. Rev. Lett. 100, 062501 (2008). 39 D. Diakonov and V. Y. Petrov, “Nucleons as chiral solitons,” In *Shif- 62 G. E. Brown and R. Machleidt, “Strength of the rho meson coupling to man, M. (ed.): At the frontier of particle physics, vol. 1* 359-415 nucleons,” Phys. Rev. C 50, 1731 (1994). [hep-ph/0009006]. 63 Z. Komargodski, “Vector mesons and an interpretation of Seiberg 40 J. D. Walecka, “A theory of highly condensed matter,” Annals Phys. duality,” JHEP 1102, 019 (2011) doi:10.1007/JHEP02(2011)019 83, 491 (1974). [arXiv:1010.4105 [hep-th]]. 41 B. D. Serot and J. D. Walecka, “Recent progress in quantum hadrody- 64 D. T. Son and M. A. Stephanov, “QCD and dimensional deconstruc- namics,” Int. J. Mod. Phys. E 6, 515 (1997) [nucl-th/9701058]. tion,” Phys. Rev. D 69, 065020 (2004) [hep-ph/0304182]. 42 L. W. Chen, “Symmetry energy systematics and its high density behav- 65 M. Harada, S. Matsuzaki and K. Yamawaki, “Implications of holo- ior,” EPJ Web Conf. 88, 00017 (2015) [arXiv:1506.09057 [nucl-th]]. graphic QCD in ChPT with hidden local symmetry,” Phys. Rev. D 74, 43 Y. L. Ma, M. Harada, H. K. Lee, Y. Oh, B. Y. Park and M. Rho, “Dense 076004 (2006) [hep-ph/0603248]. baryonic matter in conformally-compensated hidden local symmetry: 66 M. Harada, S. Matsuzaki and K. Yamawaki, “Holographic QCD In- Vector manifestation and chiral symmetry restoration,” Phys. Rev. D tegrated back to Hidden Local Symmetry,” Phys. Rev. D 82, 076010 90, no. 3, 034015 (2014) [arXiv:1308.6476 [hep-ph]]. (2010) [arXiv:1007.4715 [hep-ph]]. 44 Y. L. Ma, Y. Oh, G. S. Yang, M. Harada, H. K. Lee, B. Y. Park and 67 D. K. Hong, M. Rho, H. U. Yee and P. Yi, “Nucleon form-factors M. Rho, “Hidden local symmetry and infinite tower of vector Mesons and hidden symmetry in holographic QCD,” Phys. Rev. D 77, 014030 for baryons,” Phys. Rev. D 86, 074025 (2012) [arXiv:1206.5460 [hep- (2008) [arXiv:0710.4615 [hep-ph]]. ph]]. 68 K. Hashimoto, T. Sakai and S. Sugimoto, “Holographic baryons: Static 45 Y. L. Ma, M. Harada, H. K. Lee, Y. Oh, B. Y. Park and M. Rho, properties and form factors from gauge/string duality,” Prog. Theor. “Dense baryonic matter in the hidden local symmetry approach: Half- Phys. 120, 1093 (2008) [arXiv:0806.3122 [hep-th]]. skyrmions and nucleon mass,” Phys. Rev. D 88, no. 1, 014016 (2013) 69 S. Aoki, K. Hashimoto and N. Iizuka, “Matrix theory for baryons: An Erratum: [Phys. Rev. D 88, no. 7, 079904 (2013)] [arXiv:1304.5638 overview of holographic QCD for nuclear physics,” Rept. Prog. Phys. [hep-ph]]. 76, 104301 (2013) [arXiv:1203.5386 [hep-th]]. 46 Y. L. Ma, G. S. Yang, Y. Oh and M. Harada, “Skyrmions with vector 70 Y. L. Ma, G. S. Yang, Y. Oh, M. Harada, H. K. Lee, B. Y. Park and mesons in the hidden local symmetry approach,” Phys. Rev. D 87, no. M. Rho, “Baryonic matter in the hidden local symmetry induced from 3, 034023 (2013) [arXiv:1209.3554 [hep-ph]]. holographic QCD models,” arXiv:1304.0289 [hep-ph]. 47 P. Sutcliffe, JHEP 1008, 019 (2010) [arXiv:1003.0023 [hep-th]]. 71 Y. L. Ma, M. Harada, H. K. Lee, Y. Oh and M. Rho, “Skyrmions, half- 48 P. Sutcliffe, “Skyrmions in a truncated BPS theory,” JHEP 1104, 045 skyrmions and nucleon mass in dense baryonic matter,” Int. J. Mod. (2011) [arXiv:1101.2402 [hep-th]]. Phys. Conf. Ser. 29, 1460238 (2014)[arXiv:1312.2290 [hep-ph]]. 49 V. Mantovani Sarti and V. Vento, “The half-skyrmion phase in a chiral- 72 H. Hata, T. Sakai, S. Sugimoto and S. Yamato, “Baryons from in- quark model,” Phys. Lett. B 728, 323 (2014) [arXiv:1309.0639 [nucl- stantons in holographic QCD,” Prog. Theor. Phys. 117, 1157 (2007) th]]. [hep-th/0701280 [HEP-TH]]. 50 J. I. Kapusta, Finite Temperature Field Theory, Cambridge University 73 M. Harada and Y. L. Ma, “Chiral partner structure of heavy baryons Press (1989). from the bound state approach with hidden local symmetry,” Phys. 51 R. J. Crewther and L. C. Tunstall, “∆I = 1/2 rule for kaon decays de- Rev. D 87, no. 5, 056007 (2013) doi:10.1103/PhysRevD.87.056007 rived from QCD infrared fixed point,” Phys. Rev. D 91, no. 3, 034016 [arXiv:1212.5079 [hep-ph]]. (2015) [arXiv:1312.3319 [hep-ph]]. 74 B. Y. Park, M. Rho and V. Vento, “Vector mesons and dense Skyrmion 52 G. E. Brown, C. H. Lee and M. Rho, “Recent developments on kaon matter,” Nucl. Phys. A 736, 129 (2004) [hep-ph/0310087]. condensation and its astrophysical implications,” Phys. Rept. 462, 1 75 U. G. Meissner, A. Rakhimov and U. T. Yakhshiev, “The Nucleon- (2008) [arXiv:0708.3137 [hep-ph]]. nucleon interaction and properties of the nucleon in a pi rho omega soli- 53 D. B. Kaplan and A. E. Nelson, “Strange goings on in dense nucleonic ton model including a dilaton field with anomalous dimension,” Phys. matter,” Phys. Lett. B 175, 57 (1986). Lett. B 473, 200 (2000) [nucl-th/9901067]. 54 B. Y. Park, J. I. Kim and M. Rho, “Kaons in dense half-skyrmion mat- 76 Y. L. Ma and M. Rho, “Scale-chiral symmetry, ω meson and dense ter,” Phys. Rev. C 81, 035203 (2010) [arXiv:0912.3213 [hep-ph]]. baryonic matter,” arXiv:1612.04079 [nucl-th]. 23 77 G. E. Brown and M. Rho, “Scaling effective Lagrangians in a dense arXiv:1611.02269 [hep-th]. medium,” Phys. Rev. Lett. 66, 2720 (1991). 85 W. G. Paeng and M. Rho, “Scale-chiral symmetry and the sound veloc- 78 B. Y. Park, M. Rho and V. Vento, “The role of the dilaton in dense ity in compact-star matter,” arXiv:1611.09975 [nucl-th]. skyrmion matter,” Nucl. Phys. A 807, 28 (2008) . 86 G. E. Brown and M. Rho, “Double decimation and sliding vacua in the 79 M. Golterman and Y. Shamir, “Low-energy effective action for pi- nuclear many body system,” Phys. Rept. 396, 1 (2004). ons and a dilatonic meson,” Phys. Rev. D 94, no. 5, 054502 (2016) 87 C. Song, “Dense nuclear matter: Landau Fermi liquid theory and chiral [arXiv:1603.04575 [hep-ph]]. Lagrangian with scaling,” Phys. Rept. 347, 289 (2001). 80 K. Yamawaki, “Hidden local symmetry and beyond,” 88 W. G. Paeng, T. T. S. Kuo, H. K. Lee and M. Rho, “Scale-invariant arXiv:1609.03715 [hep-ph]. hidden local symmetry, topology change and dense baryonic matter,” 81 T. A. Ryttov and R. Shrock, “Scheme-Independent series expansions Phys. Rev. C 93, no. 5, 055203 (2016). at an infrared zero of the beta function in asymptotically free gauge 89 K. Fukushima and T. Kojo, “The quarkyonic star,” Astrophys. J. 817, theories,” Phys. Rev. D 94, no. 12, 125005 (2016). no. 2, 180 (2016). 82 S. Weinberg, “What is quantum field theory, and what did we think it 90 M. F. Atiyah and N. S. Manton, “Complex geometry of nuclei and is?,” In *Boston 1996, Conceptual foundations of quantum field the- atoms,” arXiv:1609.02816 [hep-th]. ory* 241-251 [hep-th/9702027]. 91 M. Rho, “Probing the source of proton mass by ”unbreaking” scale- 83 C. Adam, C. Naya, J. Sanchez-Guillen, R. Vazquez and chiral symmetry,” arXiv:1610.07012 [nucl-th]. A. Wereszczynski, “The Skyrme model in the BPS limit,” 92 L. Y. Glozman, “SU(2NF ) symmetry of QCD at high temperature and arXiv:1511.05160 [hep-th]. its implications,” arXiv:1610.00275 [hep-lat]. 84 E. P. Verlinde, “Emergent gravity and the dark universe,”