Dichotomy of Baryons as Quantum Hall Droplets and Skyrmions In Compact-Star Matter
DDichotomy of Baryons as Quantum Hall Droplets and SkyrmionsIn Compact-Star Matter
Yong-Liang Ma
1, 2, ∗ and Mannque Rho † School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China International Centre for Theoretical Physics Asia-Pacific (ICTP-AP) (Beijing/Hangzhou), UCAS, Beijing 100190, China Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique, 91191, Gif-sur-Yvette, France (Dated: November 19, 2020)We discuss the “sheet structure” of compressed baryonic matter possibly present in massivecompact stars in terms of quantum Hall droplets and skyrmions for baryons in medium. Thetheoretical framework is anchored on a generalized scale symmetric hidden local symmetry thatencompasses standard nuclear effective field theory ( s EFT) and can access the density regimesrelevant to massive compact stars. It hints at a basically different, hitherto unexplored structureof the densest baryonic matter stable against collapse to black hole. Hidden scale symmetry andhidden local symmetry together in nuclear effective field theory are seen to play a potentially crucialrole in providing the hadron-quark duality in compressed baryonic matter.
I. INTRODUCTION
The structure of highly dense matter is a totally un-charted domain. Unlike at high temperature, at highdensity, it can be accessed neither by lattice QCD norby low-temperature terrestrial experiments. While, ascomprehensively reviewed recently [1, 2], finite nuclei aswell as infinite nuclear matter can be fairly accuratelyaccessed by nuclear effective field theories, pionless or pi-onful, referred to herewith as “standard nuclear effectivefield theory ( s EFT)” anchored on relevant symmetriesand invariances along the line of Weinberg’s Folk Theo-rem [3], s EFTs, as befits their premise, are expected tobreak down at some high density (and low temperature)relevant to, say, the interior of massive stars.In s EFT, the power counting in density is O ( k qF ) where k F is the Fermi momentum and high density involveshigh q . For the “normal” nuclear matter with den-sity n ≈ .
16 fm − , the expansion requires going tohigh orders in k F , at least up to q ∼ k F comes out to be O ((1 / ¯ N ) κ )where ¯ N = k F / (˜Λ − k F ) with ˜Λ being the cutoff on top ofthe Fermi sea. Expansion in κ ≥ N → ∞ . Approach-ing Fermi-liquid theory starting from s EFT for nuclear(or neutron) matter valid up to roughly ∼ n has beenformulated [2, 6]. It has also been formulated using the V lowk approach applicable to both finite nuclei and in-finite matter [7]. Both were anchored on perturbativecalculations.Given that the k F expansion must inevitably breakdown – and hence s EFT becomes no longer valid – at ∗ [email protected] † [email protected] some high density above n , a potentially promising andastute approach is to go over to the Fermi-liquid struc-ture starting from the normal nuclear matter density atwhich the Fermi-liquid structure is fairly well establishedto hold. Our strategy is to build a model, that we shallrefer to as “G n EFT,” that while capturing fully what s EFT successfully does up to n , can be extrapolatedup to a density where s EFT is presumed to break down.Such an approach developed in [8, 9] is anchored on aLagrangian that incorporates, in addition to the pionsand nucleons of s EFT, the lowest-lying vector mesons ρ and ω and the scalar meson χ standing for f (500). Wetreat the vector mesons V = ( ρ, ω ) as dynamical fieldsof hidden local symmetry (HLS) [10] and the scalar χ asa “genuine dilaton” [11], a (pseudo-)Nambu-Goldstone(NG) boson of hidden scale symmetry [12]. It is plausi-ble although not rigorously proven that these symmetriesare intrinsic in QCD but are not visible in the mater-freevacuum. We have proposed that both symmetries get un-hidden by strong nonperturbative nuclear correlations, asnuclear matter is highly compressed. We assume that theHLS is consistent with the Suzuiki theorem [13] and thelatter with an infrared (IR) fixed point with both the chi-ral and scale symmetries realized in the NG mode [11] assome high density. Both are not rigorously established.How these symmetries, hidden in free space, could ap-pear in a hitherto unexpected way in dense medium hasbeen the subject of the past efforts [8, 9] and motivatesthe aim of this paper, going beyond what has been con-sidered up to date. We approach this issue by analyzingthe structure of cold dense baryonic matter with density n > (2 − n in terms of a “baryon-quark duality” inQCD.In this paper we point out that the combined hiddenscale symmetry and HLS, suitably formulated so as to Note that the accent, as is to be clarified, is on “duality,”not on “continuity” discussed in connection with confinement-deconfinement issue. a r X i v : . [ nu c l - t h ] N ov access high density compact-star matter as in [8, 9], in-terpreted as “emergent” from strong nuclear correlations,reveals a dichotomy in the structure of baryons treatedin terms of topology in the large N c approximation anddiscuss how it could affect the equation of state (EoS)at high density relevant to the cores of massive com-pact stars. The merit of this work is that it exploitsin strongly-interacting baryonic matter a certain ubiqui-tous topological structure of highly correlated condensedmatter, thereby bringing in a possible paradigm changein nuclear dynamics. II. THE PROBLEM: DICHOTOMY
Consider the baryons made up of the quarks withnearly zero current quark masses. We will be dealing pri-marily with two flavors, u(p) and d(own). However forthe role of scale symmetry, it is essential to think in termsof 3 flavors [11] as we will explain below. For three fla-vors, all octet baryons B ( α ) , α = 1 , ...,
8, can be obtainedas solitons, i.e., skyrmions [14] from the octet mesons.This has to do with the homotopy group π ( SU (3)) (cid:39) Z and is justified in QCD at the large N c limit. Howeverthere has been one annoying puzzle in this matter: Thereis no skyrmion associated with the singlet meson η (cid:48) . Thisis because π ( U (1)) = 0. The resolution to this conun-drum was suggested in 2018 by Komargodski [15]: Thebaryon for η (cid:48) , while not a skyrmion soliton, turns out tobe also a topological object at the large N c limit but moreappropriately a fractional quantum Hall (FQH) droplet,somewhat like a pancake (or perhaps pita [16, 17]).One way of seeing how this FQH droplet comes aboutin QCD, the approach we adopt in this paper, is in termsof the “Cheshire Cat phenomenon” (CCP) formulated along time ago [18].In the CCP, a mechanism for the trade-in of topologyfor hadron-quark continuity for low energy/long wave-length nuclear processes, a baryon charge of N c -colored N f = 1 quarks, confined in, say, MIT bag, can leak outof the bag, not into a skyrmion in 3 space dimensionsas N f ≥ x ) flowing to one higher dimension making a sheetin ( x, y ) with the anomaly caused by the “confinement”boundary condition, with the resulting system given byabelian CS theory. One can think of this as a topologicalobject of η (cid:48) with a topology different from that of theskyrmions of π ’s. We denote this baryon B (0) .One important consequence of this observation is thatthe QH droplet has the spin J = N c /
2, namely 3/2for N c = 3, corresponding to the baryon resonance∆(3 / , / B (0) in the large N c limit isof course ∼ O ( N c ) but it can also receive O ( N c ) contribu- tion [15]. In the skyrmion system, there is also a baryonof the same quantum numbers (3 / , /
2) but there is nocorrection coming at O ( N c ). The first correction to thelarge N c limit comes at O (1 /N c ) arising from the rota-tional quantization. This presents a “dichotomy prob-lem” discussed in [21].One can see this dichotomy if one applies the sameargument made for the CCP for the QH droplet for B (0) to N f = 2 systems, namely the nucleon. Insteadof dropping into the “infinite hotel” in the CC mecha-nism for the N f = 2 skyrmion when the bag is shrunk tozero [18, 22, 23], there seems to be nothing that wouldprevent the quarks from undergoing the anomaly inflowinto fractional quantum Hall droplets making the CS the-ory nonabelian [19]. Why not form a sheet-structuredmatter arranged, say, in the lasagne arrays seen at highdensity in crystal lattice simulations of compact-starmatter as will be mentioned below?The pertinent question then is: What dictates the N f = 2 quarks to (A) drop in the ∞ -hotel skyrmionsor (B) instead to flow to nonabelian FQH droplets? Orcould it be into (A) and (B) in some combination? Thissharpens and generalizes the dichotomy problem raisedabove. A solution to this dichotomy problem is recentlyaddressed by Karasik [16] in terms of a “generalized” cur-rent that unifies the N f = 1 baryon – QH droplet – andthe N ≥ N f = 2systems in terms of the EoS for dense baryonic matter.We do this by “dialing” baryon density. The hope is thatthis will unravel the putative hadron-quark duality possi-bly involved in the physics of massive compact stars. Thestrategy here is to extract the conceptual insights gainedin the phenomenological development discussed in [9] forthe physics of massive compact stars, the only systemcurrently available in nature for high density n (cid:29) n atlow temperature and translate them into a scheme thatcould address, at least qualitatively, the dichotomy prob-lem. III. G n EFT LAGRANGIAN
We begin by writing the Lagrangian involved, in assimple a form as possible, that allows us to capture thebasic idea. The details look rather involved, but we be-lieve the basic idea is quite simple. We will first deal withthe mesonic sector with baryons generated as solitonsand later explicitly incorporate baryons. In developingthe basic idea in what follows, we will frequently switchback and forth between the former description and thelatter.
A. Scale-invariant hidden local symmetric (sHLS)Lagrangian
To develop the key idea involved, we incorporate the η (cid:48) field, η (cid:48) ∈ U A (1), in addition to the pseudo-scalar NGbosons π ∈ SU (2) and the vectors ρ µ ∈ SU (2) and ω ∈ U (1). The Crewther approach [11] that we adopt in thispaper necessitates the kaons on par with the dilaton. Forour purpose, however, we can ignore the strange quark– given the presence of the η (cid:48) meson – and focus on thetwo light flavors. For the reason that will become clearlater, unless otherwise specified, the ρ and ω fields willbe treated in U (2)-nonsymmetric way.We write the chiral field U as U = ξ = e iη (cid:48) /f η e iτ a π a /f (1)and the HLS fields as V ρµ = 12 g ρ ρ aµ τ a , V ωµ = 12 g ω ω µ . (2)Expressed terms of the Maurer-Cartan 1-formsˆ α µ (cid:107) , ⊥ = 12 i (cid:0) D µ ξ · ξ † ± D µ ξ † · ξ (cid:1) (3)where D µ ξ = ( ∂ µ − iV ρµ − iV ωµ ) ξ , the HLS Lagrangian weare concerned with is of the same form as the HLS La-grangian for 3 flavors [24] with the parity-anomalous ho-mogeneous Wess-Zumino (hWZ) Lagrangian composedof 3 terms (in the absence of external fields). For the SU (2) × U (1) case we are dealing with, there is no five-dimensional Wess-Zumino (WZ) term.To implement scale symmetry, we use the conformalcompensator field χ = f χ e χ/f χ which has both massdimension and scale dimension 1. The resulting scale-symmeterized Lagrangian that we denote as sHLS is L sHLS = f Φ Tr (ˆ α µ ⊥ ˆ α ⊥ µ ) + f σρ Φ Tr (cid:16) ˆ α µ (cid:107) ˆ α (cid:107) µ (cid:17) + f Φ Tr (cid:16) ˆ α µ (cid:107) (cid:17) Tr (cid:0) ˆ α (cid:107) µ (cid:1) − g ρ Tr( V µν V µν ) − g Tr( V µν )Tr( V µν ) + 12 ∂ µ χ∂ µ χ + V ( χ )+ L hW Z (4)where L hW Z – which will be specified below – is the hWZterm that conserves parity and charge conjugation butviolates intrinsic parity, Φ is defined asΦ = χ/f χ (5)and V χ is the dilaton potential, the explicit form of whichis not needed for our purpose, and V µν = ∂ µ V ν − ∂ ν V µ − i [ V µ , V ν ] (6) In this paper, we use the convention of [24] with V µ = V µρ + V µω . In (4) f = f σω − f σρ , g = 12 (cid:18) g ω − g ρ (cid:19) , (7)where f σV figures in the mass formula m V = f σV g V .Note that U (2) symmetry is recovered at the classicallevel by setting g ω = g ρ and f = 1 /g = 0 in (4).It will be found that the hWZ Lagrangian plays a cru-cial role in unifying the N f = 1 and N f ≥ L hW Z = N c π (cid:88) i =1 c i L i (8)with L = i Tr[ˆ α L ˆ α R − ˆ α R ˆ α L ] , L = i Tr[ˆ α L ˆ α R ˆ α L ˆ α R ] , L = i Tr V [ˆ α L ˆ α R − ˆ α R ˆ α L ] . (9)From the point of view of our approach, it is importantto note that the HLS we are dealing with is dynami-cally generated [10]. This means that the coefficients c i are constants that cannot be fixed by the theory, i.e.,anomaly constraints. For low density, therefore, they areto be determined by experiments . However as we willsee later, in approaching QH droplet baryons, assumed tobe feasible at high density, the coefficients will be “quan-tized” by topology. This takes place because there canbe a phase transition from a Higgs mode to a topolog-ical phase in which the HLS fields are supposed to be(Seiberg-)dual to the gluons of QCD [25]. This point willfigure importantly in the argument presented below.Note that as written the hWZ terms are scale-invariant, so do not depend on Φ. There is a possibil-ity however that these terms will pick up Φ dependencedue to certain corrections that go beyond the lowest ap-proximation in scale-symmetry breaking (referred to as“LOSS”) associated with the anomalous dimension of thegluon stress tensor. In fact it can have an impact on thecelebrated g A quenching in nuclear Gamow-Teller transi-tions [26]. We ignore such contributions in what follows.The Lagrangian as given in (4) is in the leading orderin scale-chiral expansion [12, 27]. In the sector where η (cid:48) plays no role – or negligible role – and the dilatonfield is ignored, the HLS Lagrangian is gauge-equivalent to nonlinear sigma model and one can do a systematicchiral-perturbation calculation similar to the standard The familiar example is for V µ = g ρ ρ aµ τ a , f σρ = 2 f π , givingthe KSRF relation, m ρ = 2 f π g ρ . They could be fixed by holographic QCD, but there are no knownholographic QCD models that possess possible “orange” – not tomention ultraviolet – completion that would allow approachingthe density. χ PT [24]. To theorists’ surprise, there is something un-canny with the leading O ( p ) HLS Lagrangian with thevector dominance and certain properties that are con-nected with strong-weak dualities, typically associatedwith supersymmetric gauge theories [25, 28]. It seems,at some high density, HLS could become Seiberg-dual toQCD – i.e., gluons [25] – but what is surprising is that thedual nature seems to persist even at low density [28]. Forexample, the KSRF formula for the vector meson massis exactly given to all loop orders with the leading O ( p )HLS Lagrangian (see [24] on this matter and references).As explained below, our argument that follows willbe in the mean field corresponding to the Landau Femi-liquid fixed point theory with the Lagrangian (11) givenbelow that figures in [9]. It can be improved by going be-yond the Fermi-liquid fixed point approximation in, say, V lowk RG, but we believe it to be reliable enough as isfor our purpose.
B. “Genuine dilaton” scenario (GDS)
An important issue closely tied to the EoS of massivecompact stars is the role of dilaton and indispensablythe scale symmetry in QCD. The story of scale symme-try in QCD or more generally in gauge theories has a longhistory dating from 1960’s and it remains still a highlycontroversial issue in particle physics going beyond theStandard Model (BSM). Here we will confine ourselvesto QCD for N f ≤ f (500) is a “genuine” dilaton being associated withhidden scale symmetry as argued in [11, 12]. The dis-tinctively characteristic feature of the “genuine dilaton”scenario (GDS for short) is the IR fixed point signalingthe scale invariance at which the scale and chiral symme-tries are in the NG mode admitting in addition massiveparticles, such as nucleons, vector mesons etc. It maybe that this notion of the GDS is not widely acceptedin the particle physics community working on BSM. Inour approach to dense matter it turns out, however, as re-called below, that the GDS is consistent with the generalstructure of scale symmetry that manifests as an emerg-ing symmetry at what we call “dialton-limit fixed point(DLFP)” in dense matter and considered to be relevantin the EoS for compact-star matter.
IV. BARYONIC MATTER WITHOUT η (cid:48) We first consider baryonic matter where U A (1)anomaly does not figure. In (4), we set η (cid:48) equal to zero We must admit, working outside of the field of BSM, we arefar from being fully conversant with such issues as conformalwindow etc. for large N c , see e.g., [29]. But our discussion doesnot crucially depend on these controversy issues. or properly integrated out given its massiveness in na-ture. The property of dense matter described by thetheory, G n EFT, is analyzed in some detail in [9]. Howto address many-nucleon systems directly from the La-grangian that contains meson fields only, that is in theclass of skyrmion approach, has not been worked outin a way suitable for dense matter physics. Thereforea direct exploitation of the Lagrangian (4) treated en-tirely in terms of skyrmions is not feasible at present forstudying the properties of dense baryonic matter. How-ever an astute way is to map what are established tobe “robust” topological properties of skyrmions obtainedwith (4) to a density functional-type theory – referredfrequently to as “DFT” in nuclear physics circle – by in-troducing explicitly baryon fields, and suitably couplingthem, to (4). The strategy is to suitably capture fullynon-perturbative properties associated with the topolog-ical structure involved. One possible way of how thiscan be done is discussed in detail in [9]. Needless tosay, there are several admittedly unconfirmed assump-tions made therein that require further work. Here wesummarize what one finds in the mean-field approxima-tion of G n EFT which amounts to doing Landau fixed-point theory in the way described below. Going beyondthe approximation can be formulated in what is knownas “ V lowk ” renormalization-group (RG) approach and ap-plied to compact stars in [8, 9]. A. Dilaton limit fixed point (DLFP)
To exploit the mapping of topological inputs of thesHLS Lagrangian into a mean-field approximation withG n EFT, we add nucleon coupling to the sHLS fields im-plementing both HLS and scale symmetry as L N = ¯ N ( i / D − Φ m N ) N + g A ¯ N /ˆ α ⊥ γ N + ¯ N (cid:0) g V ρ /ˆ α (cid:107) + g V Tr[/ˆ α (cid:107) ] (cid:1) N + · · · , (10)with the covariant derivative D µ = ∂ µ − iV ρµ − iV ωµ and di-mensionless parameters g A , g V ρ and g V ≡ ( g V ω − g V ρ ).The ellipsis stands for higher derivative terms that willnot be taken into account in what follows. The La-grangian concerned that we shall refer to as bs HLS is L bsHLS = L sHLS + L N . (11)With the explicit presence of the baryon field, the roleof the hWZ terms is relegated to the baryon-field cou-pling to the vector and scalar fields that takes over the ω repulsion in dense baryonic matter.We consider what happens when the density goes upand approaches the DLFP first considered by Beane andvan Kolck [30] and apply it to the model we are consider-ing in [8, 31]. To do this we assume that approaching theDLFP at high density amounts to going toward the IRfixed point `a la CT [11, 12] described above where bothchiral symmetry and scale symmetry are realized in theNG mode with the dilaton mass and pion mass going tozero in the chiral limit.Starting from the vacuum where chiral symmetry is re-alized nonlinearly, as density increases, one would like toarrive, at some point near n , at the linear realization ofchiral symmetry, say, in the form of the Gell-Mann-L´evy(linear) sigma model [32] which qualitatively capturesnuclear dynamics as the Walecka mean-field model does.This means transforming the nonlinear structure of sHLSthat is the habitat of the skyrmion structure to a formmore adapted to dense matter, namely the half-skyrmionstructure developed for the EoS of massive compact starsin [8, 9]. This feature of transformation is encoded in thehidden scale symmetry as pointed out by Yamawaki [33].This point will be further elaborated on in Section VII.To see how the bs HLS Lagrangian behaves as density isincreased, we follow Beane and van Kolck [30] and trans-form bs HLS to a linear basis, Σ = f π f χ U χ ∝ σ (cid:48) + i(cid:126)τ · (cid:126)π (cid:48) .We interpret taking the limit S ≡
Tr(Σ † Σ) → n EFT as explained in [9], it is essential to interpretthe limiting
S → (cid:104) ¯ qq (cid:105) is found to globally goto zero at a density denoted n / > n . The condensatehowever is non-zero locally, thereby supporting a chiraldensity wave in skyrmion crystal [34]. This seems to bethe case in general as observed in various models [35].As a consequence, the pion decay constant is non-zero,hence the state is in the NG mode. The same is truefor the dilaton condensate with inhomogeneity in consis-tency with the GDS. This feature resembles the “pseu-dogap” structure in condensed matter physics. As therethe issue is subtle and highly controversial (see [36] fora comprehensive discussion on this matter). In what fol-lows we interpret the limiting
S → ∗ as (cid:104) χ (cid:105) ∗ ∝ f ∗ π and (cid:104) χ (cid:105) ∗ ∝ f ∗ χ with f ∗ π ∼ f ∗ χ (cid:54) = 0. The matter in the half-skyrmion phasegoing toward the DLFP then has a resemblance to thepseudo-gap phase with fractional skyrmions present inSYK superconductivity [38].One finds that in the limit S → f π → f χ (cid:54) = 0 , g A → g V ρ → . (12) To give an idea, n / in massive compact stars comes in [9] at ∼ n . Furthermore since the ρ -meson coupling to the nucleonis given by g ρNN = g ρ ( g V ρ − , (13)one sees that ρ meson decouples – independently of the“vector manifestation (VM)” with g ρ → ω -NN coupling g ωNN = g ω ( g V ω −
1) remainsnonzero for g ω (cid:54) = 0 because g V ω − (cid:54)→ m ∗ χ and the ω -NN coupling g ωNN . In factthis balance is the well-known story of the roles of thescalar attraction and vector repulsion in nuclear physicsat normal nuclear matter density. It becomes more acuteat higher densities. 𝑛 𝑛 𝑛 𝐷𝐿𝐹𝑃 ≈ 𝑛
𝐼𝑅𝐹𝑃 𝑛 𝑉𝑀 Linear chiral symmetryNonlinear chiral symmetry
𝑇𝑟(Σ + Σ) → 0𝑓 𝜋 → 𝑓 𝜒 ≠ 0, 𝑔 𝐴 → 𝑔 𝑉𝜌 → 1, 𝑔 𝜌𝑁𝑁 → 0 FIG. 1. The proposed schematic phase structure for densityregimes: n stands for equilibrium nuclear matter density, n / for onset density of half-skyrmions, n DLFP for dilatonlimit fixed point, n IRFP for IR fixed point and n VM for vectormanifestation fixed point. The broad phase structure involved is depicted inFig. 1. Apart from the nuclear matter equilibrium den-sity n and the topology change density n / , other den-sities are not precisely pinned down. What’s given in thereview [9] does not represent precise values, hence Fig. 1should be taken at best highly schematic. B. Interplay between g ωNN and (cid:104) χ (cid:105) ∗ The nucleon in-medium mass is connected to the ω -nucleon coupling by the equations of motion for χ and ω and the in-medium property of the χ condensate, (cid:104) χ (cid:105) ∗ ,or more appropriately the in-medium dilaton decay con-stant f ∗ χ which controls the in-medium mass of the dila-ton, hence the nucleon mass, at high density. This meansthat up to the DLFP, the effective nucleon mass willremain constant proportional to the dilaton condensate (cid:104) χ (cid:105) ∗ . This is seen in Fig. 2. This (cid:104) χ (cid:105) ∗ comes out tobe equal to the scale-chiral invariant mass of the nu-cleon m that figures in the parity-doubling model forthe nucleon [39]. This then suggests that m can show up signaling the presence at a higher density of the DLFPthrough strong nuclear correlations even if it is not ex-plicit in the QCD Lagrangian.We claim that this is in accord with the GDS (“genuinedilaton” scenario) with the nucleon mass remaining mas-sive together with the non-zero dilaton decay constant. y = = / n 〈 χ 〉 * / f χ [ M e V ] FIG. 2. The ratio (cid:104) χ (cid:105) ∗ / (cid:104) χ (cid:105) where (cid:104) χ (cid:105) ∗ ∝ f ∗ χ as a func-tion of density n for varying “induced density dependence”(DD induced ) – distinct from IDD (intrinsic density depen-dence) inherited from QCD – of g ∗ V ω which is parameterizedas g ∗ V ω − g V ω − y nn ) − . The density at which theratio (cid:104) χ (cid:105) ∗ / (cid:104) χ (cid:105) becomes constant is not given by the theorybut comes out to be ∼ n in compact-star phenomenology.This density can be identified with n / , the density at whichskyrmion matter transitions to half-skyrmion matter. The remarkable interplay between the dilaton conden-sate (cid:104) χ (cid:105) ∗ and the ω -NN coupling has an important im-pact on the EoS for density n ∼ > n / at which (cid:104) χ (cid:105) ∗ flat-tens in density As noted above, the induced density de-pendence for the ρ -NN coupling ∝ ( g V ρ −
1) drops rapidlysuch that the ρ decouples from nucleons at the DLFPwhereas ( g V ω −
1) remains non-zero. How this impactson nuclear tensor forces and consequently on the symme-try energy E sym deserves to be investigated in nuclearstructure. Furthermore the vector manifestation leadsto the gauge coupling g ρ → ω coupling g ω drops only slightly. The delicate balance be-tween the attraction due to the scalar (dilaton) exchangeand the repulsion due to the ω exchange plays a crucialrole for the EoS for n ∼ > n / of massive neutron stars [8]. The flattening to a density-independent constant of (cid:104) χ (cid:105) ∗ /f χ at n / arising from an intricate interplay between g ωNN and (cid:104) χ (cid:105) ∗ in Fig. 2 is related to that of (cid:104) ¯ qq (cid:105) ∗ /f π in a skyrmion-crystal sim-ulation of HLS [40]. It is not obvious how to correctly implementthe dilaton field in the crystal simulation, so the relation betweenthe dilaton and quark condensates does not seem to come outcorrectly on skyrmion crystals. But in the genuine dilaton sce-nario incorporated in G n EFT, we believe they should be tightlyrelated – as we argued – as density approaches the IR fixed pointdensity.
C. The trace anomaly and pseudo-conformalsymmetry
A striking consequence of the interplay between the g ωNN coupling and the condensate (cid:104) χ (cid:105) ∗ at n ∼ > n / in the G n EFT framework, not shared by other modelsin the literature, is the precocious emergence of hiddenscale symmetry in nuclear interactions. The details areinvolved but the phenomenon can be clearly seen in themean-field approximation with the bs HLS Lagrangian(11).The vacuum expectation value of the trace of theenergy-momentum tensor θ µµ is given by (cid:104) θ µµ (cid:105) = 4 V ( (cid:104) χ (cid:105) ) − (cid:104) χ (cid:105) (cid:18) ∂V ( χ ) ∂χ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) χ = (cid:104) χ (cid:105) ∗ + · · · (14)where the ellipsis stands for chiral symmetry breaking(quark mass) terms. Now if one ignores the quark massterms, then given that the (cid:104) χ (cid:105) ∗ which should be iden-tified with the dilaton decay constant is independent ofdensity [9], we have ∂ (cid:104) θ µµ ( n ) (cid:105) ∂n = ∂(cid:15) ( n ) ∂n (cid:0) − v s /c (cid:1) = 0 . (15)One expects that ∂(cid:15) ( n ) ∂n (cid:54) = 0 and hence, within the rangeof density where (15) holds, say, ∼ (3 − n , we arrive atwhat is commonly associated with the “conformal soundspeed” v s /c = 1 / . (16) Since however the trace of the energy-momentum ten-sor is not zero, it should be more appropriately called“pseudo-conformal” velocity.
This prediction made in the mean field for a neutronstar of mass M (cid:39) M (cid:12) has been confirmed – moduloof course quark-mass terms – in going beyond the mean-field approximation using the V lowk RG approach [8, 41].Needless to say the quark mass terms could affect thisresult bringing in possible deviation from (16), but itseems reasonable to assume that the corrections cannotbe significant.
V. BARYONIC MATTER WITH η (cid:48) So far baryonic matter without the η (cid:48) degree of freedomis treated as density increases toward the DLFP. Thebaryons involved there are skyrmions for N f = 2. It hasbeen assumed that the U A (1) anomaly plays no role athigh density for compact-star physics.However there are at least three reasons why the η (cid:48) de-gree of freedom cannot be ignored in nuclear dynamics.First, it is known that the U A (1) anomaly plays a cru-cial role for the color-charge conservation in the CCP [42]and consequently for the flavor-singlet axial-vector cou-pling constant of the proton g A (cid:28) η (cid:48) , though massive at low density, may become lighterand become relevant at high density. Third, it has beensuggested that the FQH droplet structure of N f = 1baryon [15] can be unified in scale-symmetric HLS theorywith the skyrmion structure of N f ≥ A. From sHLS to the η (cid:48) ring In following Karasik’s arguments [16, 17], we take thesHLS Lagrangian (4) and focus on the terms that in-volve the η (cid:48) field in that Lagrangian. With the baryonsgenerated as solitons in sHLS, the parameters of the La-grangian contain only the “intrinsic density dependence”(IDD) inherited from QCD – and not the density depen-dence induced by nuclear correlations (DD induced in [9]),e.g., g V ρ and g V ω that figures in the mean-field analysisgiven in Section ?? .First we ask what one should “dial” in the parametersof G n EFT – in the spirit of the strategy used – to accesscompact stars so that the system approaches the η (cid:48) sheet.Next we ask whether high baryon density supplied bygravity makes the η (cid:48) ring “visible.”Suppose we increase density beyond n / . Recallingfrom what we have learned in the mean-field result with(11) (with the nucleon explicitly included), it seems rea-sonable to assume the ρ decouples first at some densityabove n / before reaching the DLFP. Since the gaugecoupling g ρ goes to zero in approaching the vector mani-festation fixed point n V M (say, n ∼ > n ) [8]), the mass m ρ ∼ f π g ρ goes to zero independently of whether f π → ρ decouples from the pions. The La-grangian L sHLS (4) will then reduce to what was writtendown by Karasik [16, 17] L sHLS = 12 ( ∂ µ χ ) + V ( χ ) + 12 Φ ( ∂ µ η (cid:48) ) −
14 ( ω µν ) + 12 m ω Φ ω µ ω µ − κ N c π (cid:15) µναβ ω µ ∂ ν ω α ∂ β η (cid:48) + · · · , (17)where we have written f = f η and f σρ = af π . The coef-ficient κ is related to the coefficient c in the hWZ term(8) for the HLS dynamically generated, appropriate forlow density. We assume that it will become quantized asin [16] when the η (cid:48) ring is “probed” at some high den-sity as explained below. In this operation, we assumedthat the limit
S → In [17], it has been argued that the well-known vector dominancein the presence of electroweak fields leads to the same constraint. crystal simulation, both the dilaton and chiral conden-sates – space averaged globally – go to zero while theylocally support density waves with their decay constantsremaining nonzero . As density increases further beyondthe DLFP, the condensate will vanish locally and the ki-netic energy term of η (cid:48) field gets killed and the ω mass m ω ∝ (cid:104) χ (cid:105) goes to zero. Indeed, it is explicitly shown in[48] in the crystal approach that at a density n (cid:29) n / the phase becomes homogeneous – without density waves– so that f π ∝ f χ ∝ (cid:104) χ (cid:105) = 0. We interpreted this phaseas deconfined since both chiral and scale symmetries arerestored. We will be left with the massless χ and ω fieldsand the ωη (cid:48) coupling coming from L in the hWZ term,(8).The last term of (17) can be written as L CSη (cid:48) = − κ N c π J µνα ω µ ∂ ν ω α . (18)with the topological U (1) 2-form symmetry current J µνα = 12 π (cid:15) µναβ ∂ β η (cid:48) . (19)Now a highly pertinent observation is that the chargedobjects under these symmetries get metamorphosed toinfinitely extended sheets that interpolate from η (cid:48) = 0on one side to η (cid:48) = 2 π on the other [15, 16], involving asheet η (cid:48) = π . The current is conserved because η (cid:48) in thespace of η (cid:48) configuration is a circle and π ( S ) = Z . TheLagrangian (18) corresponds to the CS field identifiedwith the ω field coupling to the baryon charge. The CSfield is a topologically non-trivial gauge field and hencegauge invariance requires that the κ be quantized κ =1. This follows from the presumed duality between thegluon fields in QCD and the HLS fields in EFT [25]. Nowgoing beyond the DLFP as the system is brought towardthe putative density at which (cid:104) χ (cid:105) → (cid:104) ¯ qq (cid:105) → f π ∼ f χ → m χ → m ω → B = 1 and J = N c / n EFT to arrive at the N f = 1 baryonfrom the N f = 2 baryons.In what’s described above, we have assumed that the ρ field decouples first before reaching the DLFP as indi-cated in Section IV A. This is what seems to take place incompact-star matter studied in [8, 9]. Instead of U (1) CSfield theory, however, one can generalize the discussionto nonabelian CS field theory from the sHLS Lagrangian(4). The Lagrangian (18) is modified to [25] L (cid:48) CSη (cid:48) = − κ N c π J µνα Tr (cid:18) V µ ∂ ν V α + 23 V µ V ν V α (cid:19) (20)where V µ = ( τ · ρ µ + ω µ ), assuming U (2) symmetry, isrestored at the high density concerned. We now identify This argument holds if we assume that the matter is in the topo-logical phase where HLS is Seiberg-dual to QCD. And also withthe vector dominance [17]. the source of the baryon number as B = ( N c /N f )Tr V and differentiating the action (cid:82) L (cid:48) CSη (cid:48) with respect to B ,we get, with κ = 1, the baryon density ρ B = 14 π (cid:15) ijk Tr( ∂ i V j ) ∂ k η (cid:48) + · · · . (21)With the configuration η (cid:48) = 0 at x = −∞ and η (cid:48) = 2 π at x = ∞ , the baryon number is gotten B = 1 by integrat-ing over x . In [25], an interpretation of this phenomenonis made in terms of an anyon excitation of the quantumHall droplet with baryon number B = 1 /N c leading to aquark described as a soliton made of hadrons, what onemight interpret as a novel manifestation of hadron-quarkduality. This exposes the η (cid:48) ring in the N f = 2 setting. Thisobservation is relevant to the possible decay of the η (cid:48) ringto a pionic sheet described in the next subsection.Above we have seen that in some density regime, onearrives at a CS theory coupled to a baryon-charge oneobject that could be identified with the η (cid:48) ring. This isdone, we suggest, by what amounts to going beyond theDLFP in the G n EFT Lagrangian . What’s interestingis to view the process in terms of the N f = 2 skyrmiongiven by the sHLS Lagrangian, namely, how the hedgehogansatz in the background of the η (cid:48) field is “deformed” asdensity goes up. It seems plausible, as suggested in [16],that high density first impacts on the EoS such that π = π = 0 , V µ = V µ = 0 , V µ = ω µ (22)and then distorts the hedgehog configuration to( σ + iπ ) / (cid:113) σ + π = e iη (cid:48) / f η . (23)This suggests that while at low density η (cid:48) in U = e iη (cid:48) /f η e iτ · π/f present in the η (cid:48) ring plays no significantrole, except perhaps, giving an O ( N c ) correction to the∆ − N mass difference which could not be significant, the η (cid:48) ring becomes important as density increases. B. Going from the η (cid:48) ring to pionic sheet We consider the density regime where the ρ mesons aredecoupled from the nucleons and the η (cid:48) ring is unstable,so decays to skyrmions. Noting that the η (cid:48) ring, i.e., L CSη (cid:48) , is embedded in the full hWZ term, we shouldlook at the hWZ term (8). Following [16], we write (inthe unitary gauge ξ R = ξ † L = ξ ) L hW Z = N c π (cid:15) µνρσ g ω ω µ × Tr (cid:34) (cid:18) κ (cid:19) ∂ ν ξξ † ∂ ρ ξξ † ∂ σ ξξ † For doing this more realistically, it may be necessary to includehigher-lying vectors and scalars as in holographic models [25].This is beyond our scheme so we won’t go into the matter further. + (cid:18) κ (cid:19) iV ν ( ∂ ρ ξ∂ σ ξ † − ∂ ρ ξ † ∂ σ ξ )+ (cid:18) κ (cid:19) i∂ ν V ρ ( ∂ σ ξξ † − ∂ σ ξ † ξ ) (cid:35) (24)where only the terms contributing to the N f = 2 com-pletion of the topological term L CSη (cid:48) are retained. Thecoefficients κ i s can be identified with c i s of the hWZ term(8) κ = c − c , κ = c + c , κ = c . (25)Under gauge transformation ω µ → ω µ − g ω ∂ µ λ , one has δS = N c π (cid:15) µνρσ ∂ µ λ Tr (cid:34) (cid:18) κ (cid:19) ∂ ν ξξ † ∂ ρ ξξ † ∂ σ ξξ † (cid:35) + iN c π (cid:15) µνρσ ∂ µ λ∂ ν Tr (cid:34) (cid:18) κ (cid:19) V ρ ( ∂ σ ξξ † − ∂ σ ξ † ξ ) (cid:35) . (26)Then the gauge invariance yields the constraints (cid:90) dφ (cid:20)(cid:18) κ (cid:19) (cid:0) ω φ + V φ (cid:1) + (cid:18) κ (cid:19) π ∂ φ π (cid:21) = const. (27)So that, on the world-sheet for the N f = 1 baryon, onehas [16] 12 κ (cid:90) dφ ( ω φ + V φ ) = 2 π, π , = 0 . (28)The η (cid:48) ring is thereby “seen.”Now suppose the η (cid:48) sheet structure, a backgroundburied in the system of N f = 2 skyrmions, is unstableand could subsequently decay into skyrmions in a dif-ferent sheet structure containing the isovector degrees offreedom ω φ + V φ = 0 , κ (cid:90) dφπ ∂ φ π = π . (29)The question is: What is the structure of the matterencoded in the condition κ (cid:82) dφπ ∂ φ π = π/ η (cid:48) decays? Could this be a sort of droplets that can bedescribed in a topological field theory, involving isovectordegrees of freedom, e.g., the π ± , the ρ vectors etc. asin the form of a nonabelian CS Lagrangian that seemsto arise in the Cheshire Cat for N f = 2 baryons? Wehave no answer to this question. Clearly isovector mesonsmust figure. This has to do with the quantization of othercoefficients than the one giving the η (cid:48) ring. The coefficients c i have been fixed in [17] by imposing vectordominance (for which the c term corresponding to the photonfield is included). As will be remarked in the last Section, VDdoes not work well in nuclear physics but it would be interestingto study their impact on dense matter. VI. UBIQUITOUS SHEET STRUCTURE OFBARYONIC MATTER
While it is not clear how the background of the η (cid:48) ring, perhaps insignificant in the dynamics of stronglyinteracting many-nucleon matter at low density, affectsthe process of going toward the DLFP – and beyond – to“expose” the η (cid:48) ring structure, it seems to be fitting tospeculate how the QH droplets structure could manifestin the sheet structure of dense matter as seen in the EoSof massive compact stars described with fair success in[8] and reviewed in [9]. A. Crystal skyrmions
We return to the skyrmion crystal simulation onwhich the G n EFT for massive compact stars is an-chored [8, 9]. As detailed there, the topological struc-ture of the skyrmions simulated on crystal is translatedinto the parameters of the G n EFT Lagrangian, whichis then treated in an RG-approach to many-nucleon in-teractions. The key role played in this procedure is thetopological feature encoded in the skyrmion structure ofhidden scale symmetry and local symmetry of sHLS. No-table there are the cusp in the symmetry energy of densematter due to the “heavy” degrees of freedom, the par-ity doubling in the baryon spectra, and a “pseudo-gap”structure of the half-skyrmion phase. These propertiesencapsulated in the RG-approach with G n EFT led to theprediction of possible precocious emergence of scale sym-metry in massive-star matter with the pseudo-conformalsound velocity of star v s /c (cid:39) / n ∼ > n .Let us explore what this skyrmion crystal structuresuggests for a possible sheet structure of dense matter.It is observed in molecular dynamics simulation of nu-clear matter expected in neutron-star crust and core-collapse supernova at a density a packing fraction of ∼ /
16 of nuclear saturation density n ∼ × g/cm that a system of “sheets” of lasagne, among a varietyof complex shapes of so-called “nuclear pasta,” could beformed and play a significant role in the EoS in low-density regime of compact star matter [44]. Involved hereare standard nuclear interactions between neutrons andprotons in addition to electromagnetic interactions.At higher densities, say, at densities ∼ (2 − n , itis seen in skyrmion crystal simulations that a stack oflasagne sheets [45] or of tubes or spaghettis [46] is en-ergetically favored over the homogenous structure. In-volved here are fractionalized skyrmions, 1/2-baryon-charged for the former and 1 /q -charged for the latterwith q odd integer. Those fractionalized skyrmions canbe considered “dual” to (constituent) quarks in the senseof baryon-quark duality in QCD. There is even an in-dication that the sheet structure of stack of lasagnescould give a consistent density profiles of finite nuclei [45].In fact there seems little doubt that an inhomogene-ity is favored in dense matter at non-asymptotic densi- ties [34, 35]. Thus it could be considered robust.The two phenomena at low and high densities in-volve basically different aspects of strong interactions,but there is a tantalizing hint that something universalis in action in both cases. We are tempted to considerthat topology is involved there. This is particularly plau-sible at high density given that the “pasta” structure, bethat lasagne or tubes (or spaghetti), is found to be strik-ingly robust. Up to date, the analysis has been madewith an ansatz for the pion field, i.e., the Atiyah-Mantonansatz for the lasagne sheet and a special ansatz allowinganalytical treatment for the tubes. The robustness musthave to do with the fact that what is crucially involved isthe topology and it is the pion field that carries the topol-ogy. What is striking is the resulting structure does notseem to depend on the presence of other massive degreesof freedom such as the vector mesons or scalar [47, 48].There, adding infinite number of higher derivative termsto the Skyrme Lagrangian is found not to modify theansatz for the tubes. It is therefore highly plausible thatthe same structure would arise from the presence of thehidden scale-local degrees of freedom of sHLS. B. Density functional theory (DFT)
Our sHLS Lagrangian could contain the unified de-scriptions of both N f = 1 droplet – η (cid:48) ring – and N f = 2skyrmions in an EoS, but we have not been able tocapture both in a unified way. That is, how are theinfinite-hotel and the FQH structures combined in theEoS and whether and how does the latter structure fig-ure in compact-star physics?A significant recent development relevant to this mat-ter is the work treating the fractional quantum Hall phe-nomenon in the Kohn’s functional density approach `a laKohn-Sham [49]. The key ingredient in this approach isthe weakly interacting composite fermions (CF) formedas bound states of electrons and (even number of) quan-tum vortices. Treated in Kohn-Sham density functionaltheory one arrives at the FQH state that captures cer-tain strongly-correlated electron interactions. The meritof this approach is that it maps the Kohn-Sham den-sity functional, a microscopic description, to the CS La-grangian, a coarse-grained macroscopic description, forthe fractional quantum Hall effect.Now the possible relevance of this development to ourproblem is as follows. First, Kohn-Sham theory more orless underlies practically all nuclear EFTs employed withsuccess in nuclear physics, as for instance, energy densityfunctional approaches to nuclear structure. Second, ourG n EFT approach belongs to this class of theories in thestrong-interaction regime. Third, the successful work-ing of the G n EFT model backed by robust topology andimplemented with intrinsic density dependence inheritedfrom QCD could very well be attributed to the powerof the Kohn-Sham density functional in baryonic matterat high density n ∼ > n . These three observations com-0bined suggest to approach the dichotomy problem in away related to what was done for FQHE.The first indication that G n EFT anchored on thetopology change could be capturing the weak CF struc-ture of [49] in FQHE is seen in the nearly non-interactingquasiparticle behavior in the chiral field configuration U in the half-skyrmion phase (see Fig. 8 in [9]). This fea-ture may be understood as follows. Due to hidden U (1)gauge symmetry in the hedgehog configuration, the half-skyrmion carries a magnetic monopole associated withthe dual ω [50, 51]. The energy of the “bare” monopolein the half-skyrmion diverges when separated, but thedivergence is tamed by interactions in the skyrmion as abound state of two half-skyrmions where the divergenceis absent. In a way analogous to what happens in theKohn-Sham theory of FQHE [49], there could intervenethe gauge interactions between the skyrmions pierced bya pair of monopoles in sHLS– as composite fermions –possibly induced by the Berry phases due to the magneticvorticies. Thus it is possible that the topological struc-ture of the FQH is buried in the bound half-skyrmionstructure at high density. A possible avenue to the prob-lem is to formulate the EoS in terms of a stack of orderedcoupled sheets of CS droplets. This problem remains tobe worked out. C. Hadron-quark continuity a.k.a. duality
In the dilaton limit where the constraints (12) setin, there are NG excitations and the nucleon mass is O ( m ) with m (cid:39) ym N , y ∼ (0 . − . M ∼ M (cid:12) has density ∼ (6 − n . A natural questionone raises is whether the core of the star contains “decon-fined” quarks either co-existing with or without baryons.In the framework of [9], the constituents of the core arefractional-baryon-charged quasiparticles. They are nei-ther baryons nor quarks. The fractional-charged phasearises without order-parameter change and hence con-sidered evolving continuously from baryonic phase witha certain topology change. In certain models having do-main walls, those fractional-charged objects can be “de-confined” on the domain wall [52]. If the sheets in theskyrmion matter discussed above are domain walls, thenit is possible that the fractional-charged objects are “de-confined” on the sheets in the sense discussed in [52].There are two significant issues raised here.One is the possible observation of an evidence forquarks in the core of massive neutron stars [53]. Veryrecently, combining the astrophysical observations andtheoretical ab initio calculations, Annal et al. con-cluded that inside the maximally massive stars therecould very well be a quark core consisting of “decon-fined quarks” [53]. Their analysis is based on the obser-vation that in the core of the massive stars, the soundvelocity approaches the conformal limit v s /c → / √ γ < .
75, 1.75 being the value close to the minimal one obtained inthe hadronic model. It turns out that in the pseudo-conformal structure of our G n EFT, the sound velocitybecomes conformal v s /c ∼ / √ γ goes near 1at n ∼ > n [54]. Thus at least at the maximum den-sity relevant for ∼ M (cid:12) stars, what could be interpretedas “deconfined quarks” can be more appropriately frac-tionally charged quasipartlcles. Are these “deconfined”objects on domain walls as in [52] or confined two half-skyrmions as mentioned in Section VI A? We have noanswers to these.The other is what is referred to as “baryon-quark conti-nuity” in [9] in the domain of density relevant to compact-star phenomenology. This is not in the domain of densityrelevant to the color-flavor locking which is to take placeat asymptotic density [55]. It seems more appropriateto say that the gauge degrees of freedom we are deal-ing with should be considered as “dual” to the gluons inQCD [25]. D. Hadron-quark continuity or deconfinement
It has recently been argued that the hadron-quark con-tinuity in the sense of [55] is ruled out on the basis ofthe existence of a nonlocal order parameter involving a(color-)vortex holonomy [56]. But such a “theorem,” per-haps holding at asymptotic density, could very well beirrelevant even at the maximum possible density observ-able in nature, whatever the maximum mass of the starstable against gravitational collapse might be. The ar-gument of [56] cannot rule out the baryon-quark dualityargument given in [9] and in this paper which is far be-low asymptotic density. The presence of the scale-chiral-invariant nucleon mass m testifies for this assertion. E. Emergence of hidden scale symmetry in nuclearmatter
The hadron-quark duality for which the hidden scalesymmetry – together with the HLS – figures cruciallyin our discussion of resolving the dichotomy, we argue,leaves a trail of other observables where its effect hasimpacts on. One prominent case is the long-standingmystery of the “quenched” axial-vector coupling constant g A in nuclear medium.We recall as the density approaches the DLFP density,the constraints (12) require that the effective g A → g eff A in Gamow-Teller transitions (most accuratelymeasurable in doubly-magic nuclei) – which is denotedas g Landau A in [26] – is predicted to be “quenched” in thepresence of emerging scale invariance from g A = 1 .
27 infree space to g eff A ≈ v s /c = 1 / n ∼ > n is a signal for anemergence of an albeit approximate scale symmetry, therehas been up to date no evident indication of the pseudo-conformal structure at low density n ∼ < n . We arehereby suggesting that g eff A ≈ β (cid:48) corrections) and g A = 1 at the DLFP is the conti-nuity of the “emergent” scale symmetry between low andhigh densities.One way of understanding this “continuity”is the wayscale symmetry is, as mentioned before, hidden in mat-ter. Regardless of whether the hidden nature of scalesymmetry is appropriate for the “genuine dilaton” ofCrewther [11] or the dilaton in the conformal win-dow [29], scale symmetry is intrinsically hidden. Thispoint is clearly illustrated in Yamawaki’s argument [33].Yamawaki starts with the SU (2) L × SU (2) R linear sigmamodel with two parameters which corresponds to theStandard Model Higgs Lagrangian, makes a series of fieldre-parameterizations and writes the SM Higgs model intwo terms, a scale invariant term and a potential termwhich breaks scale symmetry which depends on one di-mensionless parameter λ . By dialing λ → ∞ , he gets thenonlinear sigma model with the scale symmetry break-ing shoved into the NG boson field kinetic energy term,and by dialing λ → λ in terms of nuclear dynamics.For compact-star physics, it’s the density that does thedialing. As for the g A problem, it is much more sub-tle. What makes g Landau A go to 1 in nuclei is the scalesymmetry in action buried in nuclear interactions (say, λ → ∞ ), whereas what makes g A go to 1 at the dilaton-limit fixed point is the scale symmetry becoming explicit(say, λ → In both cases, given that the axial coupling to nucle-ons g A ¯ N / ˆ α ⊥ γ N in (10) is scale-invariant below the cut-off ∼ πf π at which the effective Lagrangian bs HLS ismatched to QCD and hence the coupling g A is not “fun-damentally” renormalized, it could very well be the emer-gence of approximate scale symmetry in baryonic matterthat dictates what’s happening. VII. COMMENTS AND FURTHER REMARKS
The principal proposition of this article is that theeffective low-energy Lagrangian sHLS that incorporateshidden scale and local symmetries containing, in additionto the (octet) pions, the η (cid:48) degree of freedom could con-tain both N f = 1 baryons and N f = 2 baryons throughhidden scale and local symmetries dual to the gluons inagreement with Karasik. Our argument is admittedlyfar from rigorous. What is highly non-trivial is that the G n EFT could contain both the topological structureof quantum Hall baryons and that of skyrmion baryons.How to write the ansatz for the former as one does forthe latter is unclear, but it should be feasible to do soand would allow one to see how the former comes in intothe latter to resolve the dichotomy problem.We are uncovering an interesting role that could beplayed by the scale symmetry with its dilaton and thehidden local symmetry with the vector mesons dual tothe gluons in “unifying” the two different topologicalsheet structures. The analysis made in the G n EFTframework based on sHLS indicates that in the densityregime relevant to massive compact stars, the chiral con-densate and dilaton condensate go proportional to eachother in the NG mode. In going beyond the regime ofmassive compact stars, we find the DLFP approaching,if not coinciding with, the IR fixed point with f π = f χ .How and where the density regime for the IR fixed pointis approached cannot at present be elucidated.It should be stressed that the objective in this paper isbasically different from the fundamental issue of the roleof topology and hidden symmetries in QCD and techni-color QCD. Firstly, what is evident is that what takesplace in the dense compact-star regime is drastically dif-ferent from the presently favored scenario invoking the“conformal window” in the domain of BSM with N f ∼ f π /f χ ∼ . f π /f χ ∼ η (cid:48) singularity involved in thetopological structure of baryons in [16, 17] uncovers the ω (a.k.a. Chern-Simons) mass going to zero as the fermion(“quark”) mass m → ∞ . This contrasts with how the η (cid:48) ring structure is possibly “exposed” in nuclear processes,as argued in this note, at high density as the ω mass isto go to zero with the dilaton mass m χ ∝ (cid:104) χ (cid:105) going tozero. This – what might appear to be another dichotomy– is explainable in terms of a possible (Seiberg-type) du-ality between the gluons (in the topological phase) andthe HLS mesons (in the Higgs phase). In the former thevector dominance is found to play a crucial role for the N f = 1 baryon structure [17] whereas in the latter theVD – unless a high tower of vector mesons is taken intoaccount [57] – famously fails to work for the N f = 2 (i.e.,nucleon) EM form factors. How to correlate or reconcilethese two processes appears highly challenging. Acknowledgments
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