Manifestation of Hidden Symmetries in Baryonic Matter: From Finite Nuclei to Neutron Stars
MManifestation of Hidden Symmetries in Baryonic Matter:From Finite Nuclei to Neutron Stars
Mannque Rho ∗ and Yong-Liang Ma
2, 3, † Universit´e Paris-Saclay, CNRS, CEA, Institut de Physique Th´eorique, 91191, Gif-sur-Yvette, France School of Fundamental Physics and Mathematical Sciences,Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou, China (Dated: January 19, 2021)When hadron-quark continuity is formulated in terms of a topology change at a density higherthan twice the nuclear matter density ( n ), the core of massive compact stars can be describedin terms of quasiparticles of fractional baryon charges, behaving neither like pure baryons norlike deconfined quarks. Hidden symmetries, both local gauge and pseudo-conformal (or brokenscale), emerge and give rise both to the long-standing “effective g ∗ A ≈
1” in nuclear Gamow-Tellertransitions at ∼ < n and to the pseudo-conformal sound velocity v pcs /c ≈ / ∼ > n . It issuggested that what has been referred to, since a long time, as “quenched g A ” in light nuclei reflectswhat leads to the dilaton-limit g DL A = 1 at near the (putative) infrared fixed point of scale invariance.These properties are confronted with the recent observations in Gamow-Teller transitions and inastrophysical observations. I. INTRODUCTION
While the structure of nuclear matter at density n = n ≈ .
16 fm − is fairly well understood, the phase struc-ture of the strong interactions at high densities, investi-gated for several decades, still remains largely uncharted.Recent precision measurements of massive ∼ > M (cid:12) neu-tron stars and detection of gravitational waves from starmergers provide indirect information of nuclear matter atlow temperature and at high density, say, up to ten timesthe saturation density n . So far, such phenomena canbe accessed by neither terrestrial experiments nor latticesimulation, the only model-independent tool known instrong-interaction physics.The study of dense matter in the literature has largelyrelied on either largely phenomenological approaches an-chored on density functionals or effective field theoreticalmodels implemented with certain QCD symmetries, con-structed in terms of set of relevant degrees of freedom ap-propriate for the cutoff chosen for the effective field the-ory (EFT), such as baryons and pions, and with [1, 2] orwithout [3, 4] hybridization with quarks, including othermassive degrees of freedom. The astrophysical observa-tions indicate that the density probed in the interior ofneutron stars could be as high as ∼
10 times the nor-mal nuclear matter density n and immediately raise thequestion as to what the interior of the star could consistof, say, baryons and/or quarks and a combination thereof.Asymptotic freedom of QCD implies that at some super-high density, the matter could very well be populated bydeconfined quarks [5]. But the density of the interior ofstars is far from the asymptotic and hence perturbativeQCD cannot be reliable there. Now the question that im- ∗ [email protected] † [email protected] mediately arises is whether and how the relevant degreesof freedom of QCD, namely, the gluons and quarks inter-vene in the high density regime relevant to the interiorof massive compact stars. Given the apparently differ-ent degrees of freedom from hadrons at low density andthe quark-gluon degrees of freedom presumably figuringat high density, does addressing massive star physics in-volve one or more phase transitions?In this review, we describe a conceptually novel ap-proach going beyond the standard chiral EFT (denotedas sChEFT) to higher densities n (cid:29) n formulated byus since some years and exploit it to not only post-dict– with success – the experimentally studied propertiesof baryonic matter at density near n but also predicthigh density properties of massive compact stars. Giventhat at low densities, the relevant degrees of freedomare hadrons while at some high densities, they must bequarks and gluons of QCD, it seems reasonable to pre-sume that the proper approach would involve going fromhadronic theory to quark-gluon theory. At low densi-ties it is clear that the appropriate theory is an EFTanchored on chiral symmetry, i.e., χ EFT, but what isthe appropriate theory at high density which cannot bedirectly accessed by QCD proper? It must be necessar-ily an EFT but not knowing what the UV completion ofan EFT is, there is no obvious way to formulate a semi-microscopic theory involving quarks and gluons. Cur-rently resorted to are the class of linear sigma modelssuch as Nambu-Jona-Lasinio-type theories or bag modelswith or without coupling to hadrons such as pions, vec-tor mesons etc. with arbitrary constants phenomenolog-ically adjusted with no link to QCD. The strategy thereis then to hybrid to certain extent arbitrarily a hadronicdescription to the putative quark/gluonic description ata suitable density regime where the changeover of degreesof freedom is presumed to take place.The merit of the approach that we rely on is that wewill have a single unified, admittedly perhaps oversim- a r X i v : . [ nu c l - t h ] J a n plified , effective Lagrangian formulated in a way that en-compasses from low density to high density, involvingonly manifestly “macroscopic” degrees of freedom, butcapturing the continuity to “microscopic” quarks-gluondegrees of freedom. How to effectuate the change of de-grees of freedom will be formulated in terms of a possibletopology change at a density denoted n / encoded in thebehavior of the parameters of the EFT Lagrangian as onemoves from below to above the changeover density n / . II. HIDDEN SYMMETRIES AND NEWDEGREES OF FREEDOM
We start by specifying the degrees of freedom that fig-ure in our formulation of the theory. For consistency withthe scale symmetry discussed below, we need the num-ber of flavors to be 3 with nearly “massless” u(p), d(own)and s(trange) quarks. We will however confine ourselvesto the u and d quarks in most of what follows. In whatway, the s flavor figures will be explained below.Since we will be dealing with an effective field the-ory (EFT), the degrees of freedom figuring in the theorywill depend on the energy/momentum (or length) scaleinvolved. Specifically in terms of density at zero tem-perature with which we will be concerned here, if we areinterested in nuclear interactions in the vicinity of nuclearequilibrium density, say, n ∼ < n , then the relevant en-ergy scale can be slightly less than the mass of the lowest-lying vector-meson, ρ , i.e., so Λ ∼ (400 − π inaddition to the nucleons N T = ( np ). The nucleon hasmass higher than the scale Λ, but must figure for nuclearphysics either directly or as a skrymion. This impliesthat it makes sense only if nuclear dynamics involved are“soft” relative to the cutoff, in the sense the pion mass m π is soft. The EFT that comprises of pions and nucle-ons, suitably anchored on chiral symmetry of QCD, is the standard chiral EFT (denoted sChEFT) which has beenestablished, when formulated in ab inito approach ex-ploiting powerful numerical techniques, to be fairly suc-cessful in nuclear physics at low energy and at densitynear n . The well-defined chiral power expansion withsystematic renormalization of higher-order terms allowsone to account for scalar and vector-channel excitationsmore massive than the pion. This sChEFT, currentlyfeasible in practice up to N n LO for n ∼ < ∼ <
700 MeV.Where this breakdown takes place is not known preciselybut it is considered to be very likely to be of ∼ > n in-volving the possible change of degrees of freedom fromhadrons to effective quarks/gluons. In our approach thischangeover will be identified with “hadron-quark conti-nuity” without deconfinement .The key point in the approach is that in going above n toward compact-star densities ∼ (5 − n , two hiddensymmetries play a crucial role in capturing the hadron- quark continuity. The symmetries involved are hiddenlocal symmetry (HLS for short) for V µ and hidden scalesymmetry for the dilaton that we denote by χ corre-sponding to f (500) . This means that the appropriatecutoff of the EFT is set above the lowest vector and scalarmass. In some cases, a higher tower of vector mesonscould figure as in holographic models but we will havethem integrated out in this review except when they areneeded as will be discussed later. There are also othermore subtle symmetries such as parity-doubling, pseu-dogap symmetry etc. which will figure associated withthe two main hidden symmetries. They will be explainedwhen needed. A. Hidden local symmetry (HLS)
We first address in this section how to bring in thehidden local symmetry into the chiral Lagrangian that isthe basis for sChEFT. The hidden scale symmetry willbe treated in the next section.In the chiral SU (2) L × SU (2) R symmetry sponta-neously broken to SU (2) V = L + R , there is a redundancyin the chiral field UU = e iπ/f π = ξ † L ξ R = ξ † L h ( x ) h ( x ) † ξ R , (1) h ( x ) ∈ SU (2) V . Gauging this redundancy in a particular way picked by[6, 7] leads to what is referred to in the literature as“hidden local symmetry (HLS)” . “Gauging” the redun-dancy by itself is of course empty but endowing it witha kinetic energy term makes the vector field propagateand hence become gauge field. The basic assumption isthat the local (gauge) symmetry is dynamically gener-ated. As such there is nothing that implies that thatsymmetry is intrinsic in QCD, and hence could very wellbe a gauge symmetry emergent from the strong dynamicsas in strongly correlated condensed systems. Accordingto Suzuki [8], a gauge-invariant local field theory writtenin matter fields alone inevitably gives rise to compositegauge bosons (CGB). This theorem would then implythat the HLS as constructed in the way stated abovemust, in the chiral limit (i.e., m u = m d = 0), have whatis called in [7] as the “vector manifestation” (VM) fixedpoint at which the vector meson mass goes to zero.There are two important observations to make here –to be elaborated in greater details later – on the notionof the vector mesons as composite gauge fields. In the literature, it is denoted as σ which transforms nonlinearlyunder scale transformation. We use χ that transforms linearlyas will be explained. Extensive references particularly relevant to this review are foundin [7]. The argument involves subtle notions of global, local and gaugesymmetries juxtaposed with physical symmetries anchored onNoether’s theorem [9].
The first is the role of V µ channels in nuclear dynam-ics which is encoded in sChEFT at high orders in chiralperturbation series. Now the HLS Lagrangian is gauge-equivalent to non-linear sigma model which is the basis ofsChEFT. This means that `a la Suzuki theorem, sChEFT,correctly treated, should give rise to the VM fixed-pointstructure. In other words, there must exist at some highdensity (or temperature) a phase where the m V = 0. The second observation is that the HLS fields describedas CGB fields could become “Seiberg-dual” [10] at highdensity (or temperature) to the gluons of QCD. Thiswould mean that HLS is actually encoded – but hid-den – in QCD proper and emerges un-hidden at highdensity (or temperature). How this feature manifests inexperiments is quite subtle and not obvious because ofthe topological structure that involves an η (cid:48) singularityat high density, possibly associated with quantum Halldroplet baryons [11–15]. B. Hidden scale symmetry
1. “Genuine dilaton scheme”
In QCD, scale symmetry is broken by quantum (a.k.a.trace) anomaly and quark masses . In our approach weadopt the “genuine dilaton” scheme [16] for scale symme-try. In this scheme the scalar f (500) is considered to beas light as the kaons K , and classed as a scalar pseudo-Nambu-Goldstone boson (of spontaneously broken scalesymmetry) – called “dilaton” from here on – put on parwith the kaons in the pseudo-scalar pseudo-NG bosons.The genuine dilaton scheme requires taking the flavorsymmetry SU (3). Unless otherwise noted, however, wewill continue our discussion with 2 flavors. Later η (cid:48) willenter accounting for the 3rd flavor. Together with thevector mesons V µ , the mass scales involved are given inFig. 1 copied from [16]. Combined with the three-flavorchiral symmetry, the dilaton gives rise to “chiral-scalesymmetry” captured at low energy by chiral-scale per-turbation theory χ PT σ . As indicated, in the matter-freevacuum, there is a net scale separation between the NGsector and the massive sector denoted as non-NG sector.But in nuclear medium at high density, the scale separa-tion between the NG sector and the vector mesons in theNG sector becomes blurred. Also the η (cid:48) is most likely to Such a phase should be visible at high temperature in heavy-ion dilepton processes. If Nature confirms the absence of sucha phase it would then rule out the composite gauge structureof the vector mesons. We will argue later that up to date apossibly crucial phase structure associated with an η (cid:48) singularity– a topological object involving Chern-Simons topological field –has been ignored at near chiral phase transition. In what follows, quark masses will not be explicitly shown in thediscussions although they are implicitly included in numericalresults given in the review. How the quark masses figure is givenin the “genuine dilaton scheme.” join the NG sector at high density. These features willbe elaborated in what follows.
FIG. 1. Scale separation for chiral-scale perturbation theory(denoted χ PT σ ) in GDS copied from [16]
We should mention that this genuine dilaton scheme(
GDS for short) – which has been extended to BSM(beyond the Standard Model) involving dilatonic Higgsphysics – seems to be fundamentally different from thecurrently popular scenario involving the conformal win-dow for N f ∼ GDS with HLSfor treating nuclear interactions in the density regimeencompassing from finite nuclei to dense compact-starmatter density. It will be shown that this
GDS – withits associated soft (NG boson) theorems – is particularlypredictive and strikingly consistent with what takes placein nuclear interactions both at low and high densities .The idea is to extend the chiral perturbation strat-egy established in HLS [7] to a combined “chiral-scalesymmetry” scheme that incorporates the power countingdue to scale symmetry. In [16], neither vector bosons norbaryons were explicitly addressed. The scheme combin-ing these degrees of freedom essential for nuclear physics– that we dub ψd HLS with ψ standing for baryon and d for dilaton – was first formulated in [17]. What is distinctive of this
GDS is that in the chirallimit , there is an infrared (IR) fixed point α IRs – atwhich the QCD β function vanishes – as indicated inFig. 2 in the χ PT σ flow. It is contrasted in the figurewith the β function flow for N f = 3 denoted χ PT . TheIR fixed point is characterized by that conformal invari-ance is realized in NG mode with f π ∼ f σ (cid:54) = 0 withmassive matter fields accommodated. The striking dif-ference of this scheme from that based on conformal win-dow adopted by many authors in the field of BSM (see,e.g., [18] and references given therein) where the IR fixedpoint involved, that is, conformal symmetry, is realizedin Wigner mode with f π = f σ = 0. In its present form, it looks much too complicated with explodingparameters although the basic idea involved is extremely simple.It could surely be made much simpler and more readily manage-able for applications in nuclear physics, a work for the future. Unless otherwise stated, we will always be dealing with the chirallimit.
FIG. 2. β function flows for the N f = 3 QCD ( χ PT ) and forthe genuine dilaton flow ( χ PT σ ): Copied from [16].
2. Chiral-scale symmetry
We briefly explain how to combine scale symmetry tohidden local gauge symmetry explained above so as toperform chiral-scale perturbation expansion. Details canbe found in [16, 17, 19]. For this, it is convenient touse the “conformal compensator field” χ related to thedilaton field σ by χ = f χ e σ/f χ . (2)In this form, χ – with mass dimension 1 – transformslinearly under scale transformation whereas σ transformsnonlinearly. From hereon when we refer to dilaton, weshall mean χ instead of σ .Given that HLS is generated dynamically from thenon-linear sigma model, the chiral-scale power countingof HLS Lagrangian is straightforward. What is neededis the power counting in scale symmetry. Among variouspower counting schemes that have been made in the lit-erature (e.g., [20] in the large N f context), we adopt the GDS . The “small” expansion parameter in scale symme-try, apart from the power of derivatives O ( ∂ ) ∼ O ( p ),is the deviation of the β function from the IR fixed point β ( α IRs ). Expanding the β function near the IR fixedpoint, β ( α s ) = (cid:15)β (cid:48) + O ( (cid:15) ) where (cid:15) = α s − α IRs and β (cid:48) = ∂β ( α s ) /∂α s | α s = α IR is the anomalous dimension ofthe gluon stress tensor tr G µν . It is clear from Fig. 2 that β (cid:48) >
0, so the chiral-scale power counting is (cid:15) ∼ O ( p ) . (3)In accordance of chiral-scale symmetry, the dilaton massscales as the pion mass does m χ ∼ O ( (cid:15) ) ∼ O ( p ) . (4)Now the chiral-scale HLS [17] (denoted d HLS) can beorganized in the same way as with nonlinear sigma modelwithout the vector fields [16]. Take for example theleading chiral order (LO) HLS with scale dimension d s =2 which is made scale-invariant by multiplying by (cid:0) χf χ (cid:1) .Denote it L HLS:LO . The
GDS is to take into account theanomalous dimension β (cid:48) calculated by Callan-Symansikrenormalization group equation to incorporate the scaleanomaly effect L LO = q ( c, β (cid:48) ) L d HLS:LO (5) where q ( c, β (cid:48) ) = (cid:16) c + (1 − c ) (cid:0) χf χ (cid:1) β (cid:48) (cid:17) . (6)Here c is a constant not given by symmetry alone. Notethat the anomalous dimension effect is present only when c (cid:54) = 1. The first term has scale dimension d s = 4, hencescale-invariant but the second term has d > β (cid:48) cannot be zero in this scheme.All chiral-scale leading order terms can be given both d s = 4 and d > c J fordifferent J . How to go to chiral N n LO terms for n >
3. Soft Theorems
It should be stressed that a striking feature of thisGDS– which will play a key role in what’s discussed below– is that approaching the IR fixed point “soft theorems”apply to both scalar and pseudo-scalar NG bosons.
Bothplay a very powerful role to the developments made belowin nuclear processes. In the
GDS , the soft theorems areclosely parallel between chiral symmetry and scale sym-metry. Among various soft theorems, the Goldberger-Treiman relation for the pion f π g πNN = g A m N has thecounterpart for the dilaton . f χ g χNN = m N (7)and the dilaton counterpart for the Gell-Mann-Oakes-Renner (GMOR) relation for the pion m π f π = − ¯ m (cid:104) ¯ qq (cid:105) is m χ f χ = (cid:15)β (cid:48) α IR (cid:104) G µν (cid:105) . (8)There should of course be nonzero quark mass term inthis expression for the dilaton which is ignored in thechiral limit. Note that it is β (cid:48) (cid:15) that plays here the role of“explicit” symmetry breaking due to quantum anomalyeven in the absence of the quark masses.In the matter-free vacuum, these soft theorems are ver-ified to hold very well for the pion, e.g., to high orders inchiral perturbation theory and also on lattice QCD. Theyare also verified to hold fairly well in chiral perturbationtheory in nuclear matter (see, e.g., [21] for the GMOR re-lation). Disappointingly there are up-to-date no higher-order calculations that give support to soft-dilaton theo-rems in both the vacuum and in medium although argu-ments are given in [16] that suggest that they will work We announce in advance that the pionic and dilatonicGoldberger-Treiman relations applied to nuclear systems will befound to lead to g A = 1 in dense matter at what will be called“dilaton-limit fixed point.” at least as well as soft-kaon theorems in the vacuum. Itwill be shown however that combined with soft-pion the-orems, soft-dilaton theorems do resolve a long-standingpuzzle of quenched g A in nuclear Gamow-Teller transi-tions [22]. This, together with what takes place in highlydense compact-star matter, will be the key contributionof this review.
4. Chiral-scale HLS and “LOSS” approximation
In applying the chiral-scale HLS Lagrangian describedabove to nuclear processes at normal nuclear systems at n ∼ n and highly compressed matter at n ∼ (5 − n relevant compact stars, we will employ two versions ofthe same effective Lagrangian, one purely bosonic incor-porating the pseudo-scalar NG bosons π ∈ SU (3), thedilaton χ and the vectors V in consistency with the sym-metries concerned. We are denoting the resulting scale-symmetric HLS Lagrangian as d HLS with d standing fordilaton. To do nuclear physics with it, baryons will begenerated as skyrmions with the d HLS Lagrangian. Andthe second is to introduce baryon fields explicitly intochiral-scale HLS, coupled scale-hidden-local symmetri-cally to π and χ . This Lagrangian will be denoted as ψd HLS with ψ standing for baryons.At this point we should underline an approximationdubbed “leading-order scale symmetry (LOSS)” approx-imation which has figured in applications to compact-star properties in our approach [19]. The LOSS approx-imation would be valid if β (cid:48) (cid:28) d s = 4 terms. All scale-symmetry explicit breaking canthen be put in the Coleman-Weinberg-type dilaton po-tential. However we will see that in some cases β (cid:48) (cid:28) d > β (cid:48) (cid:54)(cid:28)
1, particu-larly in nuclear Gamow-Teller transitions.
III. HADRON-QUARK CONTINUITY
Given the two Lagrangians d HLS and ψd HLS, howdoes one go about doing nuclear many-body problem ?While in principle feasible, there is, up to date, no sim-ple way to systematically and reliably formulate nuclearmany-body dynamics in terms of skyrmions with d HLS.We will therefore resort to ψd HLS. With the ψd HLS La-grangian whose parameters are suitably fixed in mediummatched to QCD correlators [7, 19], we have formulateda Wilsonian renormalization-group (RG)-type approachto arrive at Landau(-Migdal) Fermi-liquid theory alongthe line developed in condensed matter physics [23]. The This terminology may very well be a misnomer. mean field approximation has been identified with Lan-dau Fermi-liquid fixed point (FLFP) theory, valid in thelimit 1 / ¯ N → N = k F / (Λ − k F ) (with Λ the cut-off on top of the Fermi sea) [24, 25]. This approach canbe taken as a generalization of the energy-density func-tional theory familiar in nuclear physics [19]. One can gobeyond the FLFP in V lowk -RG by accounting for selected1 / ¯ N corrections as we have done numerical calculations.As will be briefly summarized below (see [19] for de-tails), the ψd HLS applied to nuclear matter — call it Gn EFT — is found to describe nuclear matter at n ∼ n as well as currently successful standard χ EFT to typicallyN n LO for n ≤
4. There is however a good reason to be-lieve, although not rigorously proven, that the standardChEFT with the Fermi momentum k F taken as a smallexpansion parameter must inevitably breakdown at highdensities relevant to massive compact stars, say, ∼ > n .Quark degrees of freedom in one form or other could bea natural candidate for the breakdown mechanism. Ourapproach has the merit that where and how this break-down occurs can can be inferred from the skyrmion ap-proach with the d HLS Lagrangian. The strategy is to ex-ploit topology encoded in the dilaton-HLS Lagrangian toaccess the putative hadron-to-quark continuity in QCD.The way we approach this is distinctively novel in thefield and could surely be further refined but the resultsobtained thus far are promising as we will show. Thereare certain predictions that have not been made in otherapproaches.
A. Topology change
We shall eschew going into details that are abundantlyavailable elsewhere [19] and focus on the key aspects thatare new without trying to be precise.Given that topology is in the pion field in skyrmionapproaches to compressed baryonic matter, the originalSkyrme model (with the Skyrme quartic term) – imple-mented with the dilaton scalar – suffices for the purposeat hand. To differentiate it from other skyrmions suchas with the d HLS Lagrangian, used in our approach ,or with holographic QCD models, we will specificallyrefer to the Skyrme model implemented with the dila-ton as skyrmion dπ . Otherwise “skyrmion” will refer tothe generic topological baryon with various different La-grangians.The skyrmion approach to baryonic matter, while stillfar too daunting to handle nuclear dynamics directly andsystematically , can however provide valuable and ro-bust information on the possible topological structure in-volved in going beyond the normal nuclear matter density Even nucleon-nucleon potentials in the Skyrme model with thepion are still to be worked out satisfactorily (with some suc-cess) after many decades since the model was revived in early1980’s [26] regime. One can exploit its topological structure in densematter by putting skyrmions on crystal lattice. By re-ducing the lattice size L , one can compress the skyrmionmatter. It has been found that at some density denoted n / (corresponding to L / in the crystal)) the skyrmionmatter simulated on, say, FCC lattice is found to turn toa half-skyrmion matter. This transition, being topologi-cal, is considered to be robust. At what density this tran-sition takes place depends on what’s in the Lagrangianand hence cannot be determined by theory. We will de-termine it by phenomenology below.Now the most important feature of this transition isthat the chiral condensate Σ ≡ (cid:104) ¯ qq (cid:105) , the order parame-ter of chiral symmetry, which is nonzero in the skyrmionmatter for n < n / , goes, when space averaged, goes tozero ( ¯Σ →
0) for n ≥ n / while nonzero locally sup-porting chiral wave. This implies that chiral symmetryremains spontaneously broken with nonzero pion decayconstant f π (cid:54) = 0 in n ∼ > n / although ¯Σ = 0. This resem-bles the pseudogap structure in superconductivity wherethe order parameter is zero but the gap is not. In factthere are other phenomena in strongly correlated con-densed matter systems with such pseudogap structure.The skyrmion-half-skyrmion transition in dense mat-ter has an analog in condensed matter in (2+1) dimen-sions, for instance, the transition from the magnetic N´eelground state to the VBS (valence bond solid) quantumparamagnet phase [27]. There the half-skyrmions in-tervening in the phase transitions are deconfined withno local order parameters. What takes place in denseskyrmion matter seems however quite different becausethe half-skyrmions remain confined by monopoles [28].This suggests that the confined half-skyrmion complexbe treated as a local baryon number-1 field with its pa-rameters, such as the mass, coupling constants etc. dras-tically modified reflecting the topology change from thevacuum. This observation will be seen to play a key rolein “embedding” the topological inputs in Gn EFT. Thetopology change from below to above n / is to capture interms of hadronic variables the hadron-quark change-overinvolving no phase transition, i.e., no deconfinement .Our key strategy is to incorporate these topology ef-fects embedded in d HLS into the parameters of the ψd bHLS in formulating Gn EFT.
B. From nuclear symmetry energy to HLS gaugecoupling
One of the most remarkable impacts of the topologychange at n / is its effect on the nuclear symmetry en-ergy E sym in the equation of state (EoS) of baryonicmatter and on the HLS gauge coupling constant, spe-cially g ρ , as the vacuum is modified by density. We canwork with the skyrmions with d HLS but the argumentcan be made with the skyrmion dπ .Consider the symmetry energy E sym in the energy per baryon of nuclear matter E ( n, ζ ) = E ( n,
0) + E sym ( n ) ζ + O ( ζ ) + · · · (9)where ζ = ( N − Z ) / ( N + P ) with N ( Z ) being the neutron(proton) number. In the skyrmion dπ , while one cannotcalculate the energy per baryon reliably, E sym can becalculated in the large N c limit by collectively rotatingthe whole skyrmion matter. For A = N + Z system with N (cid:29) Z , it is found [29] E sym ≈ (8 λ I ) − (10)where λ I is the isospin moment of inertia. There aretwo contributions, one from the leading current algebra( O ( p )) term and the other from the Skyrme quartic( O ( p )) term. Both turn out to be equally important.The predicted E sym [29] is schematically given in Fig. 3. FIG. 3. Schematic cusp structure of E sym at n / that ap-pears both in the skyrmion dπ crystal simulation and in theclosure approximation with the tensor forces. What is notable here is the cusp at what correspondsto the skyrmion-to-half-skyrmion transition density n / .Approaching from below the density n / at which thecusp is located, the symmetry energy decreases and itincreases after n / almost linearly in density. It is easyto understand how this cusp appears and it has a crucialimpact on theEoS of dense matter. It arises from theinterplay between the two terms with ¯Σ = 0 figuringimportantly. The cusp cannot be present without thequartic term.Now the question is: What could this mean in termsof the effective Lagrangian ψd HLS? The answer lies inthat the Skyrme quartic term represents heavy degreesof freedom, inherited from the lowest-lying ρ meson andhigher towers of isosvector vector mesons integrated outin the Lagrangian. It suffices to limit to the lowest ρ meson and ask how it can contribute to the symmetryenergy. In standard EFT approaches, one can addressit in terms of nuclear potentials given in the ψd HLS La-grangian. It is well-known that the symmetry energy canbe fairly well approximated by the closure approximationof the two-nucleon diagrams with iterated tensor forces, E sym ≈ κ (cid:104) ( V T ) (cid:105) ∆ E , (11)where κ > E ≈ (200 − V T connects dominantly from the groundstate of the matter. It is also very well known in nu-clear theory that the net tensor force gets contributionsfrom one-pion and one- ρ exchanges with opposite signs,so the dominant one-pion exchange tensor force, attrac-tive in the range of two-nucleon interactions, is weak-ened by the interference of the two terms. In mediumthe pion mass stays more or less unaffected by densitybut the ρ mass decreases with density with the decreas-ing dilaton condensate (which is related to the pion de-cay constant) – `a la Brown-Rho scaling which holds in ψd HLS – so the cancellation in the tensor force reducingthe net strength would make the symmetry energy ulti-mately vanish at some density above n . This would bestrongly at odds with what is known in nuclear physics at ∼ n and slightly above via heavy-ion experiments. This(potential) disaster can be avoided if at some density thegauge coupling g ρ drops faster than the pion decay con-stant does. By imposing this condition at n / , one canmake the ρ tensor drastically suppressed so as to let thepion tensor force take over and make the cusp structuredevelop at n / . This reproduces qualitatively the samecusp in Fig. 3. Indeed the HLS coupling g ρ going tozero so that m ρ ∝ g ρ → ψd HLS. What’s given in Fig. 3 corresponds roughly to mean-field approximations valid near n / . The cusp singular-ity structure must be an artifact of the approximationinvolved in both the skyrmion crystal and the closureapproximation, involving large N c and large ¯ N limits.Such a cusp structure must be smoothed by fluctuationson top of the mean field. In [30], corrections to the meanfield treated in terms of 1 / ¯ N corrections `a la V lowk RG doindeed smooth the “singularity” but significantly leaveunmodified the soft-to-hard cross-over at n . . This isshown in Fig. 4.We will return to this cusp structure in Sec. VI re-garding the structure of E sym inferred from the PREX-IIexperiment in Pb [31]. We mention as a side remark that this scenario invalidates thenotion made in some heavy-ion circles working on dilepton pro-cesses that chiral restoration at high temperature implies degen-eracy in mass between ρ and a , not the mass going to zero (inchiral limiet). FIG. 4. E sym in V lowk RG in Gn EFT for n / ∼ n copiedfrom [30]. The theoretical points given in V lowk RG are con-nected by blue line below n / and by red above n / . Thedotted lines indicate experimental constraints. C. Soft-to-hard crossover in EoS
The cusp structure in E sym discussed above have acrucial impact on the EoS of massive compact stars. Itdescribes softening of the EoS going toward n / frombelow and hardening going above n / . This feature,which will be elaborated in what follows, is consistentwith what is observed in the tidal polarizability (TP) Λfrom gravity waves and also with the massive ∼ > M (cid:12) stars. The former requires a soft EoS below n / and thelatter a hard EoS above n / , both of which are naturallyoffered by the cusp structure. This cross-over behavioris the main reasoning for identifying the skyrmion-half-skyrmion transition with hadron-quark continuity.
D. Scale-invariant quasiparticles
Another striking feature in the skyrmion-half-skyrmiontransition is that at n / the confined half-skyrmioncomplex behaves like non-interacting fermion, i.e., “freequasiparticles.”Let us write the chiral field as U ( (cid:126)x ) = φ ( x, y, z ) + iφ jπ ( x, y, z ) τ j (12)and the fields placed in the lattice size L as φ η,L ( (cid:126)x ) for η = 0 , π and normalize them with respect to their maxi-mum values denoted φ η,L,max for given L . The propertiesof these fields are plotted for L > L / – skyrmion phase– and L ≤ L / – half-skyrmion phase – in Fig. 5 [19].Note the remarkable feature that the field configurationsin the half-skyrmion phase are density-independent instrong contrast with the skyrmion phase. This transition FIG. 5. The field configurations φ and φ π as a functionof t = x/L along the y = z = 0 line. The maximum valuesfor η = 0 , π are φ , L, max = φ π, L, max = 1. The left andright panels correspond respectively to the skyrmion phaseand half-skrymion phase. The half-skyrmion phase sets inwhen L = L / ∼ < . is interpreted to signal the onset of pseudo-conformal in-variance in baryonic matter to be observed in the pseudo-conformal sound velocity in massive compact stars dis-cussed in Sec. V. IV. Gn EFT
We are now adequately equipped to specify the struc-ture of Gn EFT built from ψd HLS Lagrangian.We will work with ψd HLS defined in the “vacuum”sliding with density. The parameters of the Lagrangianwill therefore depend on density. With the input takinginto account the topology change at n / , the parameterswill change qualitatively as density goes from below toabove n / .There are several ways by which the dependence canoccur.First the conformal compensator field will pick up theVEV, χ → (cid:104) χ (cid:105) n + χ as density changes, so the parame-ters of the Lagrangian term coupled to χ will depend ondensity via (cid:104) χ (cid:105) n .Next the matching of correlators between ψd HLS andQCD [7] at a given density will introduce density depen-dence, such as the NG boson decay constants and cou-pling constants g V , g χNN etc., a most important quantityof which being the ρ coupling g ρ in medium which flowsto g ρ = 0 as density goes to the VM fixed point.The density dependence on parameters will be denotedby superscript ∗ , e.g., m ∗ for masses, g ∗ for couplingconstants, (cid:104) ( χ, ¯ qq ) (cid:105) ∗ for comdensates etc.Now given the ψd HLS endowed with scaling parame-ters encoding the topology change at n / , we adopt toapproach nuclear dynamics in RG-implemented Fermi- liquid as announced in Introduction We will discussfirst Landau Fermi-liquid (FL) fixed point approach and,whenever available, the V lowk RG approach that goes be-yond the FL fixed point. The details of calculations givenin [19] will be skipped. We go immediately into con-fronting Nature.
A. Dilaton-limit fixed point
The first application of Gn EFT is to go to high densityin the mean-field approximation with ψd HLS.As noted, the mean-field approximation with ψd HLS isequivalent to Fermi-liquid fixed point approach valid inthe large N c and large ¯ N approximation [25, 32]. Assuggested in Sect III D, this approximation should bemore appropriate at high density. For simplicity we takethe LOSS approximation for scale symmetry described inSec. II B 4.Following Beane and van Kolck [33], we do the fieldre-parametrization Z = U χf π /f χ = s + i(cid:126)τ · (cid:126)π in ψd HLSin LOSS and take the limit Tr( ZZ † ) →
0. Two qualita-tively different terms appear from the manipulation: oneregular and the other singular in the limit. The singularpart is of the form L sing = (1 − g A ) A (1 / Tr (cid:0) ZZ † ) (cid:1) + ( f π /f χ − B (cid:0) / Tr( ZZ † ) (cid:1) . (13)The first (second) term is with (without) the nucleonsinvolved. The requirement that there be no singular-ity leads to the dilaton-limit fixed point (DLFP) “con-straints” g A → g DLA = 1 (14)and f π → f χ (cid:54) = 0 . (15)We have denoted the g A arrived at in the dilaton-limitfixed point as g DL A to be distinguished from the g LA –Eq. (27) – arrived at the Landau Fermi-liquid fixed point.We note that these “constraints” are more or less (ifnot exactly) the same as what are in the genuine dila-ton properties approaching the IR fixed point. There aremore constraints associated with the DLFP, among whichhighly relevant to the EoS at high density is the “emer-gence” of parity doubling in the nucleon structure [19].We will come to this later as the dilaton condensatein the half-skyrmion phase (cid:104) χ (cid:105) ∗ converges to the chiral-invariant mass m in the parity-doubling model. It is notknown whether and how this mass vanishes ultimately.But in our approach it can be present near the IR fixed We could in principle formulate chiral-scale-HLS perturbationtheory along the line of nuclear χ PT. This has not been formu-lated yet. point and more likely in the density regime relevant tocompact stars.In what follows we will simply assume that the DLFPis equivalent to the IR fixed point of GDS.
If we make the assumption that soft theorems are alsoapplicable near the DLFP, which seems justified, then wecan write, following the discussion in Sec. II B 3, m ∗ N ≈ f ∗ χ g ∗ χNN ≈ f ∗ π g ∗ πNN /g ∗ A (16)from which we have g ∗ A ≈ (cid:16) f ∗ π f ∗ χ (cid:17)(cid:16) g ∗ πNN g ∗ χNN (cid:17) . (17)This relation is to hold for both low and high density. Atlow density this relation could be verified in chiral-scaleperturbation theory. At high density n > n / , we expectfrom (14) and (15) that g ∗ πNN → g ∗ χNN , what could beconsidered as a prediction of this theory. B. Gn EFT in finite nuclei and nuclear matter
As discussed in great detail in [19], the parametersof ψd HLS Lagrangian for Gn EFT, primarily controlledby the topology change encoding putative hadron-quarkcontinuity, are different from below to above n / . Phe-nomenology fixes the range 2 ∼ < n / /n ∼ <
4. As a typicalvalue, we will use n / ≈ n . The recent developmentsuggests n / could be higher as we will comment later.Extended to a higher cutoff scale with heavier degreesof freedom than sChEFT, Gn EFT should, by construc-tion, go beyond the scale applicable to sChEFT for bary-onic matter properties valid at ∼ n [3]. With the heavyvector and scalar degrees of freedom explicitly figuring inthe dynamics, the power counting rule, however, is con-siderably different from that of sChEFT in which the vec-tor and scalar excitations enter at loop orders. Instead ofdoing the standard EFT based on power counting involv-ing the vector mesons and the dilaton, we opt to approachthe problem in Wilsonian RG-type strategy along theline worked out for strongly correlated condensed Fermi-liquid systems [23]. At the mean-field level, this proce-dure gives the Fermi-liquid fixed point theory [25] andgoing beyond the mean field in V lowk RG, it has been ver-ified that all thermodynamics properties of nuclear mat-ter come out as well as in sChEFT treated up to N LO–but with much fewer number of fit parameters [19]. Thatthe extremely simple Gn EFT calculation works as wellas the high chiral-order sChEFT calculation may be re-flecting the “magic” of HLS (Seiberg-)dual to the gluonof QCD as recently noted by several authors [10, 13].We stress here that the Gn EFT applied here to nuclearmatter properties at ∼ n is in the LOSS approxima-tion described in Sec. II B 4. It has not been yet checkedhow the leading order (LO) chiral-scale Lagrangian with-out the LOSS approximation – with the possible role of β (cid:48) dependence – affects the results . This point is per-tinent in connection with the quenched g A problem to be treated next where hidden scale symmetry gets “un-hidden” together with a possible influence of the anoma-lous dimension β (cid:48) . Also note that the result that thestandard high-order sChEFT and the Gn EFT with ex-plicit hidden symmetries fare equally well at n ∼ n indi-cate that those symmetries are buried and not apparentin the EOS at that density. This does not mean all ob-servables in nuclear processes at low density are opaqueto them. Indeed totally unrecognized in the past is thatthe long-standing mystery of quenched g A in light nu-clei as seen in shell models has a deep connection withhow scale symmetry hidden in QCD emerges in nuclearcorrelations [22]. In fact this quantity exhibits the mul-tifarious ways hidden scale symmetry manifests in theselow-energy nuclear processes.
1. Quenching of g A in nuclei It has been a long-standing mystery that simple shell-model calculations of the superallowed Gamow-Teller(GT) transition in light nuclei required an effective GTcoupling constant g ∗ A ≈
1, quenched from the free-spacevalue 1.267, to explain the observed decay rates [34].There have been a variety of explanations of the mys-tery debated in the literature since 1970’s but none withconvincing arguments. Here we offer a mechanism whichseems to resolve the half-a-century mystery: It involvesan up-to-date unrecognized working of hidden scale sym-metry in nuclear interactions [22]. This mechanism hasa remarkable impact not only on the structure of densecompressed baryonic matter in compact stars but also onhow scale (or conformal symmetry) is realized in nuclearmedium.We first give precise definitions for addressing nuclearGamow-Teller transitions both with and without neu-trinos. What has been referred to in the literature as“quenched g A ” in (supperallowed) GT transitions in lightnuclei treated in simple shell models [34] is a misnomer.Most, if not all, of the so-called “quenching” involved inbringing the g A effective in nuclei – denoted genericallyas g ∗ A – have little to do with a “genuine quenching” ofthe axial-vector coupling constant in the weak current.In fact g ∗ A ≈ g DL A = 1at high density discussed above, Eq. (14).In order to clear up the above confusion and zero-inon the issue concerned, we write the leading chiral-scale-order axial current A ± µ in ψd HLS in medium as A ± µ = g A q ssb ¯ ψτ ± γ µ γ ψ + · · · (18)where q ssb = c A + (1 − c A )Φ ∗ β (cid:48) , Φ ∗ = f ∗ χ /f χ (19)where q ssb represents the quantum-anomaly-inducedscale-symmetry-breaking (ssb). Note that in the GDS ,0 q ssb when (cid:54) = 1 is a generically density-dependent mul-tiplicative factor on the scale-invariant axial current g A ¯ ψτ ± γ µ γ ψ . As mentioned in Sec. II B 2, c A is an un-known constant. The LOSS approximation correspondsto taking q ssb ≈
1. For what follows, it is important tonote that what amounts to the true coupling-constantquenching of g A – in the sense of renormalization inher-ited from QCD – is ¯ g A ≡ g A d ssb . (20)This is the quantity that should be identified as the bona-fide quenched g A in nuclear medium and that we aim toextract from GT transitions in nuclei.First we will calculate g ∗ A for the LOSS with d ssb = 1.We would like to see how the scale invariance of the axialcurrent impacts on the nuclear GT matrix element wherenuclear correlations are treated in the LOSS approxima-tion.In the large N c and large ¯ N limit an astute way to ap-proach is to apply the Goldberger-Treiman relation in nu-clear medium as suggested in Sec. II B 3. This, as workedout in [25] for the axial current (18) with q ssb = 1, is ofthe form [32, 35] ( g eff A /g A ) d ssb =1 = g LA /g A = (Φ ∗ ) − ( m LN /m N ) + O (1 /N c , / ¯ N ) (21)where the superscript L represents density-dependentFermi-liquid fixed point quantities. g LA depends on den-sity via Φ ∗ and m LN . It should be stressed that g eff A is theeffective axial-vector coupling constant for the quasipar-ticle on the Fermi surface which is not to be associatedwith a true quenched coupling constant as has been donein the literature. What exactly it corresponds to will beexplained below.In the RG approach, the Landau mass is taken as afixed point quantity [23]. That the axial coupling con-stant is also taken as a fixed point quantity is somethingnew and unfamiliar in the field. What it implies will beexplained.There are other quantities in nuclear electroweak re-sponses where the same arguments based on soft theo-rems in the large N c and large ¯ N work very well.We now explain what g LA signifies and how it is con-nected to what is computed in shell-model calculations.We are concerned only with the superallowed Gamow-Teller (GT) transition which involves zero momentumtransfer. We assume that the matrix element is not ac-cidentally suppressed. Now in the Wilsonian-type RGequation for strongly correlated fermion (nucleon here) As is well-known, calculating the g πNN coupling in the skyrmionmodel is highly involved, so what was done in Adkins, Nappiand Witten [36] was to use the Goldberger-Treiman relation toarrive at that coupling. We are using the same strategy to thein-medium Goldberger-Treiman relation exploiting the scale in-variance of the Skyrme quartic term. (See [35]). systems, decimated all the way to the top of the Fermisea, the GT transition from, say, a quasi-proton to aquasi-neutron on top of the Fermi sea is given by the GTmatrix element M d ssb =1 GT = g A q L ( τ − σ ) fi (22)where all nuclear correlations are captured entirely inwhat we denote as q L standing for Landau Fermi-liquidfixed point.Before discussing the prediction of this approach, letus explain in what way this matrix element is relatedto shell-model calculations. To be specific, we considerthe doubly magic nucleus Sn with 50 neutrons and 50protons. We focus on this nucleus because it has whatmay be up-to-date the most accurate data on the super-allowed GT transition. This process allows to exploit the“extreme single-particle shell model (ESPSM)” descrip-tion [37, 38] which provides a precise mapping of theshell-model result to the Fermi-liquid fixed point result(22). In the ESPSM description, the GT process involvesthe decay of a proton in a completely filled shell g / to aneutron in an empty shell g / . We write the GT matrixelement in the ESPSM description in the form M espsm GT = g A q ESPSM ( τ − σ ) fi (23)where q ESPSM is the ratio of the full
GT matrix elementgiven in principle by nature (that is to be hopefully cap-tured in an accurate experiment) to the accurately calcu-lable ESPSM matrix element for the GT transition of theaxial-current with q ssb = 1. We are taking the RIKENexperiment [38] to provide the presently available empir-ical value of q ESPSM . Our proposal is that the Landaufixed-point description is equivalent to the ESPSM de-scription q L = q ESPSM /q ssb . (24)Now to access q L , we first note that the Fermi-liquidfixed point prediction (21) requires the Landau mass forthe nucleon. Unlike in the Landau Fermi-liquid theory forelectrons in which the fixed-point two-body interactionsare all local, the presence of Nambu-Goldstone bosons re-quires a nontrivial input, The pion exchange contributionto the Landau(-Migdal) interaction (for nuclear physics)brings in a non-local term. This term is of O (1 / ¯ N ), but itis extremely important due to the special role of soft the-orems in low-energy nuclear dynamics as stressed above.The result for q L , obtained a long time ago [25, 32], turnsout to be surprisingly – and deceptively – simple, q L = (cid:16) −
13 Φ ∗ ˜ F π (cid:17) − . (25)In Gn EFT the dilaton decay constant f ∗ χ is locked to thepion decay constant f ∗ π . The latter has been measuredin deeply bound pionic nuclear experiments [41], so isknown as function of density. The pionic Landau param-eter ˜ F π is given by the Fock diagram, so is calculablefor given densities. Thus both Φ ∗ and ˜ F π are accurately1known for any density in the vicinity of n . Furthermorethe product Φ ∗ ˜ F π turns out to be surprisingly indepen-dent of density, so (25) applies to light as well as heavynuclei. At n ≈ n , we have q L ≈ . − . . (26)This gives g LA = g A q L ≈ . . (27)There are caveats in this result, among which the numer-ical value of (26) depends on the accuracy of the LandauFermi-liquid fixed point approximation (the large N c andlarge ¯ N limit), the “precise” value of density, the couplingconstants used etc., so it would be unwise to take (27)too literally. Ultimately this formula could be checked bya high-order chiral-scale expansion as is done with somesoft theorems in nuclear matter discussed in Sec. II B 3.One can however have confidence in this result in thatthe same soft theorems applied to nuclear EM responsefunctions at low energy appear to work extremely well.An example is the anomalous orbital gyromagnetic ratioof the proton δg pL [25] δg l = 49 [(1 / Φ ∗ ) − −
12 ˜ F π ] τ . (28)This involves the same quantities as in g LA , Φ ∗ and ˜ F π and follows from the same soft theorems in the LOSS ap-proximation as in (27). The prediction for the Pb nucleus δg pL = 0 . τ is in good agreement with the presentlyavailable experiment ( δg pl ) exp = 0 . ± .
03 [25]. Wetake this as an indication of the reliability of (25).Now what does g LA ≈ q ssb = 1 should correspond to the FLFP matrix el-ement. Of course at the present status of computationalpower, such a full scale shell-model calculation is notdoable for the double-closed-shell nucleus Sn. Howeverin light nuclei, one expects q ssb ≈ ∗ as well asthe c coefficients could be so far from the vacuum values.What Eq. (24) implies is that high-quality calculationssuch as quantum Monte Carlo method should reliablyexplain the Gamow-Teller transitions without quenchingof g A from the free-space value. We suggest that thisexpectation is supported by the recent (powerful) quan-tum Monte Carlo calculations for β decay and electroncapture in A = 3 −
10 [40] that can explain the measuredexperimental values at ∼
2% uncertainty with the un-quenched g A = 1 .
276 and without multi-body exchangecurrents. This should be contrasted with the presently available result insCHEFT [3] which disagrees with the empirical value by morethan factor of 3.
It is worth noting that the result (27), predicted al-ready in early 1990, has a totally new aspect to it.At the risk of redundancy, let us recall that Eq. (27),justified in the large N c and large ¯ N limits, correspondsto the Landau FLFP that exploits soft-dilaton theo-rems [16]. As formulated it can be precisely equatedto the superallowed Gamow-Teller transition matrix el-ement given in the extreme single-particle shell-modeldoubly magic nuclei, e.g., Sn. At the matching scaleto QCD, the axial weak coupling to the nucleons is scale-invariant, hence the renormalization leading to (27) canbe considered entirely due to nuclear (many-body) in-teractions taking place below the matching scale . Thus(27) captures the influence of the putative scale symme-try emergent from the (nuclear) interactions which may(or may not) be intrinsically connected to QCD. We cansee this also by going to high density (cid:29) n . Startingwith non-linear sigma model with constituent quarks, asshown by [33], letting the dilaton mass m χ go to zero withthe conformal anomaly turned off — and in the chirallimit — leads to a linearized Lagrangian that satisfies var-ious well-established sum rules based on soft theorems,provided the singularities that appear as m χ →
2. Evidence for d ssb (cid:54) = 1 ? Up to density in the vicinity of n , there is no indica-tion for corrections to the LOSS approximation q ssb ≈ c coefficients be ≈ β (cid:48) (cid:28) ∗ (cid:54) = 1. In fact treating dense matter in termsof skyrmions on crystal lattice is found to give a sensibleresult [42] only if β (cid:48) ∼ (2 −
3) and c hWZ ≈ c hWZ is the c coefficient in the q ssb accompanying the homo-geneous Wess-Zumino term in the HLS Lagrangian [7].Otherwise the system becomes catastrophically unstableunless the ω meson mass goes to ∞ . This would of coursebe a nonsense. It may just be an indication for anoma-lous behavior in the high density regime which is notunderstood at all, but it seems totally unnatural.Given what appears to be an accurate result for the su-perallowed Gamow-Teller transition for the doubly magicnucleus Sn, one can raise the question as to what onecan learn about the correction to the LOSS, namely, thedeviation from q ssb = 1. Now to extract q ssb from theRIKEN experiment, we write the Sn matrix elementexpressed in terms of the ESPSM M GT:riken = g A q ESPSMriken ( τ − σ ) fi .. (29)The RIKEN experiment leads to q ESPSMriken = 0 . − . . (30)2From the equivalence relation (24), one can get q RIKENssb ≈ . − . . (31)The difference (1 − q RIKENssb ) represents the anomaly-induced correction to the LOSS. This is substantial.Next given the experimental (RIKEN) information on q ssb , (31), one might hope to get an idea on β (cid:48) . Since c A is not known, one cannot zero-in on a unique answer. Letus however see what one gets if one just takes β (cid:48) ≈ . c A ≈ .
15 compatible with what’s indicated for thehWZ term in the skyrmion crystal model [42]. That gives q ssb ≈ .
62 which is in the ball-park of the RIKEN valuefor q ssb , (31).There is a caveat here. The skyrmion crystal simula-tion involves high density for which Φ ∗ (cid:28) ∗ ∼ <
1. There is no reason whythe same value of the c coefficient makes sense.It would therefor be desirable to measure q ESPSM X fordifferent double-magic systems X in addition to recon-firming the RIKEN data. Certain forbidden axial tran-sitions treated in the next subsection offer a promisingpossibility.
3. Forbidden axial transitions
The deviation from the LOSS approximation encodingthe scale symmetry breaking (19), if firmly confirmed,implies that the g ∗ A ≈ g A applicable to all axial-vector processes in nu-clear medium. For example, it cannot be applied to suchprocesses as neutrinoless double β decays – relevant forgoing BSM – where momentum transfers can be of ∼ all processesin nuclei. This issue is recently raised in the analysisof effective g A , denoted ¯ g A , in the β decay spectrum-shape function involving leptonic phase-space factors andnuclear matrix elements in forbidden non-unique betadecays [43] – referred to as COBRA in what follows.The nuclear operators involved there are non-relativisticmomentum-dependent impulse approximation terms. Inprinciple there can be n -body exchange-current correc-tions with n ≥
2. However the corrections to the single-particle (impulse) approximation are typically of m -thorder with m ∼ > χ EFT and could be – justifiably– ignored. Unlike in the superallowed GT transition, nei-ther soft theorems nor the Fermi-liquid renormalization-group strategy can be exploited in this work. Hence (27)is not relevant to the spectrum-shape function discussedin [43]. In defining ¯ g A , nuclear correlations are takeninto account, so what is relevant is the possible the scale-anomaly correction (19) in the analysis of [43].The ¯ g A gotten from the RIKEN data corresponding tothe “genuinely quenched g A ” can be written as¯ g RIKEN A = g A q RIKENssb ≈ (0 . − . . (32) This is to be compared with the analysis of the spectrum-shape factor of Cd β decay [43], listed in the increasingaverage χ -square for three nuclear models used to calcu-late the forbidden non-unique decay process [43]¯ g COBRA A = g A q COBRAspectshape = 0 . ± . , . ± . , . ± . . (33)Given the range of uncertainties involved both in theexperimental data and in the theoretical models usedin [43], what can be identified as the anomaly-inducedquenching in the COBRA spectrum-shape result (33)is more or less consistent with the same effect in theRIKEN’s result (32). However ¯ g A ∼ > ∗ is very close to Φ ∗ ( n ) and β (cid:48) < β decay processes give an indication on scale symmetry andthe anomalous dimension β (cid:48) reflecting on the working ofscale symmetry in strong interactions. At present thereis no trustful information on β (cid:48) for N f ∼ V. EQUATION OF STATE FOR COMPACTSTARS
We now turn to the application of the Gn EFT formal-ism to the structure of massive neutron stars based onthe standard TOV equation. We shall focus on the EoSof the baryonic matter, leaving out such basic issues ascorrections to gravity, dark matters etc. Unless otherwisestated the role of leptons — electrons, muons, neutrinosetc — is included in the EOS. The results we present hereare not new (as summarized in [19]), but their implica-tions and impacts on nuclear dynamics offer a paradigmchange in nuclear physics.We will argue that what we developed above appliesto massive compact stars [19].Up to this point, we have not fixed the topology changedensity, apart from that it should be in the range con-nected to the possible “hadron-quark transition.” To goto higher densities beyond n in Gn EFT for compactstars, what we need is to fix the density at which thetopology change takes place. It is given neither by reli-able theory nor by terrestrial experiments. It turns outthat the available astrophysics phenomenology does pro-vide the range 2 ∼ < n / /n < n / = 2 . n for illustration.Up to n / , the same EoS that works well at n isassumed to hold. It is what comes at n / due to thetopology change that plays the crucial role for the prop-erties of compact stars. Among the various items listedin [19], the most prominent are (a) the cusp in the sym-metry energy E sym at n / , (b) the VM with m ρ → n vm ∼ > n , (c) the approach to the DLFP — at or3close to the putative IRFP — at n dl ∼ n vm and (d) thehadron-quark continuity up to n vm . The cusp in E sym leads to the suppression of the ρ tensor force, the mostimportant for the symmetry energy, and triggers the ef-fect (b). It effectively makes the EoS transit from soft-to-hard in the EoS, thus accounting for massive ∼ > M (cid:12) stars. The effects (b) and (c) — together with the ω coupling to nucleons — make the effective mass of theconfined half-skyrmions go proportional to the dilatondecay constant f χ ∼ m which is independent of density.Thus f χ depends little on density in the half-skyrmionphase. The star properties obtained in Gn EFT are found tobe generally consistent with presently available observa-tions [19]. There are none that are at odds with the ob-servations – both terrestrial and astrophysical – withinthe range of error bars quoted. For n / = 2 . n , themaximum star mass is found to be M max ∼ . M (cid:12) with the central density ∼ n . For a neutron star withmass 1 . M (cid:12) , currently highly topical in connection withgravity-wave data, we obtain the dimensionless tidal de-formability Λ . ≈
650 and the radius R . ≈ . v s and its impact on thestructure of the core of massive stars, both of which arehighly controversial in the community and await verdictsfrom experimenters. A. Emergence of pseudo-conformality
As noted above in connection with the Gamow-Tellercoupling constant in nuclei, there is scale symmetry thatis hidden in nuclear Gamow-Teller matrix elements butbecomes “visible” at high density. Here we will arguethat that symmetry manifests in the sound velocity ofmassive stars v s /c ≈ /
3. This resembles the conformalsound velocity v conf : s /c = 1 / (cid:104) θ µµ (cid:105) (cid:54) = 0. This is because thecompressed matter is some (cid:15) distance away from the IRfixed point, e.g., m χ ∝ (cid:15) (cid:54) = 0. However it comes out thatin the range of density relevant to compact stars, (cid:104) θ µµ (cid:105) is albeit approximately density-independent ∂∂n (cid:104) θ µµ (cid:105) ≈ . (34)One can see this in the mean-field approximation of ψd HLS (or in the large N c and ¯ N limit or Landau Fermi-liquid fixed point approximation of Gn EFT). The traceof the energy momentum tensor comes out to be (cid:104) θ µµ (cid:105) = 4 V ( (cid:104) χ (cid:105) ) − (cid:104) χ (cid:105) ∂V ( χ ) ∂χ | χ = (cid:104) χ (cid:105) . (35) For simplicity we have taken the LOSS approximation,so the dilaton potential V ( χ ) contains all the conformalanomaly effects including quark mass terms. B. Sound velocity Now we recall that approaching the IR fixed-point, (cid:104) χ (cid:105) ∼ f χ (cid:54) = 0 and with the parity-doubling f χ ∝ m which is independent of density. Thus ∂∂n (cid:104) θ µµ (cid:105) = ∂(cid:15) ( n ) ∂n (1 − v s ) = 0 (36)where v s = ∂P ( n ) ∂n ( ∂(cid:15)∂n ) − is the sound velocity and (cid:15) and P are respectively the energy density and the pressure.We assume that there is no Lee-Wick-type anomalousnuclear state at the density involved, so ∂(cid:15) ( n ) ∂n (cid:54) = 0 whichis consistent with Nature. Therefore we have v pc : s /c ≈ / . (37)We call this pseudo-conformal (PC) sound velocity.The result (37) was confirmed in the V lowk RG goingbeyond the Fermi-liquid fixed point with 1 / ¯ N correctionsincluded [19, 30]. The result v pc : s obtained in the V lowk RG with n / set at 2 . n is given in Fig 6. / n v s / c FIG. 6. Density dependence of the sound velocity v s in neu-tron matter for n / = 2 . n . The strength of the “spike” at ∼ n / depends on n / and could exceed the causality limitfor n / ∼ > n . What this result shows is that the pseudo-conformalitysets in slightly above n / . At the point of topologychange (a.k.a hadron-quark crossover), there is strongfluctuation that leads to a strong increase of the soundvelocity (which approaches the causality limit v s = c if n / is taken at ∼ n [19]) followed by rapid convergence We will comment below how the LOSS approximation could bein tension as in the case of quenched g A . This and next subsections are based on results to be publishedelsewhere [44].
C. Core of massive stars / n γ FIG. 7. Density dependence of the polytropic index γ = d ln P /d ln (cid:15) in neutron matter for n / = 2 . n This prediction (37) on the PCV can be confrontedwith a recent analysis that combines astrophysical ob-servations and model independent theoretical ab initio calculations [45]. Based on the observation that, in thecore of the maximally massive stars, v s approaches theconformal limit v s /c → / √ γ ≡ d (ln P ) /d (ln (cid:15) ) < .
75 — the valueclose to the minimal one obtained in hadronic models —Annala et al. [45] arrive at the conclusion that the core ofthe massive stars is populated by “deconfined” quarks.It is perhaps oversimplified and surprising that the pre-dicted pseudo-conformal speed (37) sets in precociously at ∼ n and stays constant in the interior of the star, butour description does qualitatively jive with the observa-tion of [45]. As already mentioned, microscopic descrip-tions such as the quarkyonic model [2] typically exhibitmore complex structures at the putative hadron-quarktransition density. We think the simpler structure in ourdescription is due to the suppression of higher-order 1 / ¯ N terms in the half-skyrmion phase. The global structureshould however be robust. Similarly in Fig. 7, one seesthe polytropic index γ drops, again rapidly, below 1.75at ∼ n and approaches 1 at n ∼ > n .Finally — and most importantly — we compare inFig. 8 our prediction for P/(cid:15) with the conformality bandobtained by the SV interpolation method [45]. We seethat it is close to, and parallel with, the conformalityband, but most significantly, it lies outside of this band.
200 400 600 800 100050100150200250300350400 ϵ [ MeV / fm ] P [ M e V / f m ] FIG. 8. Comparison of (
P/(cid:15) ) between the PCM velocity andthe band generated with the SV (sound velocity) interpolationmethod used in [45]. The gray band is from the causality andthe green band from the conformality. The red line is thePCM prediction. The dash-dotted line indicates the locationof the topology change.
The predicted results of Gn EFT as a whole resemblethe “deconfined” quark structure of [45]. There are, how-ever, basic differences between the two. First of all, inour theory, conformality is broken, though perhaps onlyslightly at high density, in the system. This could be re-lated to the deviation of g LA from the experimental valueof the quenching in Sn observed [22]. There can alsobe fluctuations around v pcs /c = 1 / β (cid:48) . This effect canbe seen in Fig. 8 where the PCM prediction deviates onlyslightly from the “would -be” conformal band. Most im-portant of all, the confined half-skyrmion fermion in thehalf-skyrmion phase is not deconfined. It is a quasipar-ticle of fractional baryon charge, neither purely baryonicnor purely quarkonic. In fact it can be anyonic lying ona (2+1) dimensional sheet [15]. What it represents is amanifestation of an emergent scale symmetry pervadingat low density as in g LA and in g DLA at high density inthe vicinity of DLFP a.k.a IR fixed-point. We suggest toclassify the precocious pseudo-conformal SV in the sameclass of emerging scale symmetry in action in nuclear pro-cesses. In fact it is in line with how conformal symmetrypermeates from the unitarity limit in light nuclei [46] tothe symmetry energy near n [47] and more. VI. TENSIONS
There are a few cases of tension between the predic-tions in Gn EFT and presently available astrophysical ob-servables. At present none of the tensions are fatal but5some of them are on the verge of being one. We discussa few here and suggest possible solutions.
A. Going from R in Pb to massive compactstars?
The answer to the question posed above is negative inthe theory formulated in this review. We discuss this asa possible tension of our approach with the structure ofmassive compact stars.It has been a current lore in nuclear theory largely ac-cepted by workers in the field that “precision data” infinite nuclei and nuclear matter should give constraints on the EoS for massive neutron stars. This lore is high-lighted by the recent astute and elegant attempt to havethe neutron skin thickness of
Pb, specifically the up-dated PREX-II, provide a stringent laboratory constrainton the density dependence of E sym and make statementon the neutron star matter involving much higher densitythan n [31]. The most interesting quantity obtained in[31] is the derivative of the symmetry energy at a densityin the vicinity of n inferred from R and is found togive L = 3 n ∂∂n E sym ( n ) | n = n = (109 . ± .
4) MeV . (38)Using this data as the first “rung in a density ladder” togo to compact-star properties, it is argued that the EoSis “stiff” at the densities relevant for atomic nuclei .Indeed this L value is considerably higher than what’sobtained in Gn EFT which comes out to be L ≈ E sym (2 n ) ≈
56 MeVwhich is compatible with a presently available empiri-cal information (46 . ± .
8) MeV [48]. It is the topologychange involving no phase transition that admits a “soft”EoS below n / and “hard” EoS above. As discussed be-low, the cusp structure, soft going to n / and hard go-ing above n / , could lower Λ . below ∼
650 Mev with R . <
13 km with n / ∼ > n . No phase transition(s)mentioned in [31] is (are) required. This observation con-traries the lore accepted in some nuclear circles that whattakes place at massive star density need be constrainedby what takes place at the extensively studied regime atnormal matter density. B. Confronting gravity wave
1. Tidal polarizability
As noted before, the predicted upper bound for thetidal polarizability Λ . is ∼
650 with R . ≈ . n cent ≈ . n in Gn EFT. This is consistent withthe present situation. However if future measurementslower it, say, to ∼
400 or lower as some standard ChEFTseems to manage to bring down, then it would imply that we would have to reassess what we have done [30] – inFig. 4 – in accounting for higher-order 1 / ¯ N correctionsin the V lowk RG treatments in approaching n / frombelow. As shown in Fig. 9, the symmetry energy at themean field level can become more attractive as densityincreases from slightly below to n / in the cusp of E sym .The central density relevant to Λ . is located below n / – i.e., “soft” regime – and a fine-tuning which is avoidedin the treatment may be needed. But unlike in the caseof [31] where the cusp-type changeover is absent, it posesno insurmountable obstacle without phase transitions.
2. Maximum star mass M max In the present approach, the range of density for thetopology change allowed is 2 ∼ < n / /n ∼ <
4. The upperbound of n / , 4 n , supports the maximum star mass M max (cid:39) . M (cid:12) . A somewhat higher M max could be ar-rived at by increasing n / beyond the upper bound, buta mass as high as M max ∼ > . M (cid:12) would put our theoryin tension with the pressure measured in heavy-ion colli-sions as noted in [19, 30]. It has recently been suggestedthat within a 2 σ confidence level the maximum mass is M max = 2 . +0 . − . M (cid:12) and the GW190814 object of M max ∼ > . M (cid:12) could be a black hole [49]. This wouldlift the tension in our approach. It seems very difficult, ifnot impossible, within the present framework to accom-modate such a star of mass ∼ > . M (cid:12) without a drasticrevamping in taking into account 1 / ¯ N corrections. C. Duality between HLS and QCD gluons
The cusp structure of E sym in the skyrmion crys-tal simulation results from the fractionalization of theskyrmion with baryon charge B = 1 into 2 half-skyrmions with baryon charge B = 1 /
2. The change-overfrom skyrmions to half-skyrmions involving no Ginzburg-Landau-Wilsonian phase transitions is a pseudogap-likephenomenon, highly exotic in nuclear physics. In factthere can be a variety of interesting fractionalized quasi-particles, such as 1 /n with n = 3 , ... , in the skyrmionstructure [50] or even more intriguing sheet structuresin the half-skyrmion phase [51]. Also discussed in thetheory literature are possible domain walls with de-confined quasiparticles associated with the η (cid:48) ring sin-gularity involving hidden symmetries as mentioned inSec. II [13, 52]. These phenomena could play an impor-tant role at high densities relevant to compact stars [15].Some of the topological structure imported into Gn EFTmay be invalidated by them.One of the most startling novel arguments raised inthe current development of hidden symmetries in stronginteractions is that the hidden symmetries of the sorttreated in this review, local gauge and perhaps also scale,are “indispensable” for accessing phase transitions, be6they chiral or Higgs-topology or deconfinement [13, 14,52].
VII. FURTHER REMARKS
We have suggested that quark-like degrees of freedom,if observed in the interior of massive neutron stars, canbe interpreted as confined quasi-particles of fractionalbaryon charges in consistency with hadron-quark conti-nuity. Such fractionally-charged objects seem inevitableby topology at high densities [15]. The mechanism inaction is the emergence of conformal (or scale) sym-metry, coming not necessarily from QCD proper, butfrom strongly-correlated dynamical nuclear interactions,which could permeate, either hidden or exposed, in bary-onic matter from low density to high density. In thisscheme, true deconfinement is to set in as mentionedabove at much higher densities, say, ∼ > n , than rel- evant to compact stars when the VM fixed point and/orDLFP are reached, possibly with the phase transitionfrom a Higgs mode to a topological mode [14].Together with the multitude of puzzling observations,we are inevitably led to the conclusion that the densestcompressed matter stable against collapse to black holes,massive neutron stars, requires a lot more than what hasbeen debated in the literature and presents a totally un-charted domain of research. Acknowledgments
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