Yong-Liang Ma
email: ylma@ucas.ac.cn

Mannque Rho
email: mannque.rho@ipht.fr

What's in the core of massive neutron stars ?

When hadron-quark continuity is formulated in terms of a topology change at a density higher than twice the nuclear matter densiy n0 the core of massive compact stars can be described in terms of quasiparticles of fractional baryon charges, behaving neither like pure baryons nor deconfined quarks. Hidden symmetries, both local gauge and pseudo-conformal (or broken scale), emerge and give rise to the long-standing quenched gA in nuclear Gamow-Teller transitions at ∼ n0 and to the pseudo-conformal sound velocity v 2 pcs /c 2 ≈ 1/3 at ∼ > 3n0. These properties are confronted with the recent observations in superallowed Gamow-Teller transitions and in astrophysical observations.

Introduction.-The phase structure of the strong interactions at high densities has been investigated for several decades but still remain more or less totally uncharted. The observation of massive neutron stars and detection of gravitational waves from neutron star merger provide indirect information of nuclear matter at low temperature and high density, say, up to ten times the saturation density n 0 . So far, such phenomena can be accessed by neither terrestrial experiments nor lattice simulation.

The study of dense matter in the literature has largely relied on either phenomenological approaches anchored on density functionals or effective field theoretical models implemented with certain QCD symmetries, constructed in terms of set of relevant degrees of freedom appropriate for the cutoff chosen for the effective field theory (EFT), such as baryons and pions, and with [START_REF] Baym | New neutron star equation of state with quarkhadron crossover[END_REF][START_REF] Mclerran | Quarkyonic matter and neutron stars[END_REF] or without [START_REF] Holt | Chiral symmetry and effective field theories for hadronic, nuclear and stellar matter[END_REF] hybridization with quarks, including other massive degrees of freedom. The astrophysical observations indicate that the density probed in the interior of neutron stars could be as high as ∼ 10 times the normal nuclear matter density n 0 0.16 fm -3 and immediately raise the question as to what the interior of the star could consist of, say, baryons and/or quarks and a combination thereof. Asymptotic freedom of QCD implies that at some superrhigh density, the matter could very well be populated by deconfined quarks [START_REF] Collins | Superdense matter: Neutrons or asymptotically free quarks?[END_REF]. But the density of the interior of stars is far from the asymptotic and hence perturbative QCD cannot be reliable there. Lacking lattice QCD, high density at low temperature cannot be theoretically accessed.

In this Letter, a conceptually novel approach going beyond the standard chiral EFT (denoted as sχEFT) to higher densities n n 0 is formulated and is used to predict that the core of massive compact stars constitutes of confined quasi-fermions of fractional baryon charge, en-coding the equation of state (EOS) with the "pseudoconfomal (PC)" sound velocity (SV) v 2 pcs /c 2 ≈ 1/3. We suggest that this phenomenon, together with the "quenched g A problem" in nuclei, certain hidden symmetries, emerge in nuclear dynamics.

Emergence of Hidden Symmetries.-The key ingredients in generating higher-energy scales for going beyond the sχEFT -that contains only baryons and (pseudo-)Nambu-Goldstone (NG) bosons -are two symmetries that are invisible in the vacuum of QCD: Hidden gauge symmetry and hidden scale symmetry. Both of them could be generated at low density, at higher-order expansions in sχEFT. But they must be constrained at high densities. This is because among others the ultraviolet completion of the effective theory involved is lacking. Our approach is to exploit the possible emergence of these symmetries as density increases to the regime relevant to compact stars, say, ∼ < 10n 0 . The higher-energy degrees of freedom we will focus on are the lowest-lying vector mesons V = (ρ, ω) and the scalar f 0 (500).

For the vectors V , we adopt the strategy of hidden local symmetry (HLS) [START_REF] Bando | Is ρ meson a dynamical gauge boson of hidden local yymmetry?[END_REF] which at low density is gauge equivalent to nonlinear sigma model, the basis of sχEFT. This is for two reasons. First, at higher densities, we can assume HLS satisfies Suzuki's theorem [START_REF] Suzuki | Inevitable emergence of composite gauge bosons[END_REF] that states "when a gauge-invariant local field theory is written in terms of matter fields alone, a composite gauge boson or bosons must be inevitably formed dynamically." Secondly, the HLS fields could be (Seiberg-)dual to the gluons [START_REF] Komargodski | Vector mesons and an interpretation of Seiberg duality[END_REF][START_REF] Kan | From 3d dualities to hadron physics[END_REF][START_REF] Karasik | Skyrmions, quantum Hall droplets, and one current to rule them all[END_REF]. This first accounts for the presence of the "vector manifestation (VM)" [START_REF] Harada | Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition[END_REF] that the V meson mass goes to zero as the gauge coupling g V flows to zero at some theoretically unknown high density n vm . We will see below that n vm ∼ > 25n 0 is indicated for the emergence of the PC velocity in stars. Second, it can account for why HLS at the leading power counting works so well and indicate a Higgs phase-topological phase transition coinciding with the quark deconfinement most likely irrelevant to the compact stars we are concerned with [START_REF] Ma | Dichotomy of baryons as quantum Hall droplets and skyrmions In compact-star matter[END_REF]. Now as for the scalar f 0 (500) that we will identify arXiv:2006.14173v2 [nucl-th] 25 Oct 2020

with the dilaton χ as the NG boson of scale symmetry, we adopt the "genuine dilaton" (GD) structure proposed in [START_REF] Crewther | Genuine dilatons in gauge theories[END_REF]. The key premise of this idea is the existence of an infrared fixed point (IRFP) with the beta function β(α IR s ) = 0 for flavor number N f = 3. The characteristic feature of this IRFP which plays a crucial role in our development is that in the chiral limit both chiral and scale symmetries are in the NG mode. Given the proximity in free-space mass between the kaon and f 0 , the s(trangeness) flavor figures together with the u(p) and d(own) flavors. In dense medium, however, the effective mass of the s-quark-baryons is to drop less importantly than the light (u,d) quark baryons, thus projecting the GD idea to the two flavors ignoring strangeness as we do in what follows could be justified.

We will confine ourselves as feasible as possible to the main line in our arguments, referring for details to [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF].

We exploit two versions of the same effective Lagrangian, one purely bosonic incorporating the pseudoscalar NG bosons π ∈ SU (3), the dilaton χ and the vectors V in consistency with the symmetries concerned. We denote the resulting scale-symmetric HLS Lagrangian as χmHLS with "m" standing for the meson fields. To do nuclear physics with it, baryons are generated as skyrmions with χHLS. And the second is to introduce baryon fields explicitly into χHLS, coupled scale-hiddenlocal symmetrically to π and χ. This Lagrangian will be denoted as χbHLS with "b" standing for baryons.

Hadron-Quark Continuity and Topology Change.-Given the two Lagrangians, how does one go about doing nuclear many-body problem ?

While in principle feasible, there is, up to date, no simple way to systematically and reliably formulate nuclear many-body dynamics in terms of skyrmions with χmHLS. We will therefore resort to χbHLS. With χbHLS whose parameters are suitably fixed in medium matched to QCD correlators [START_REF] Harada | Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition[END_REF][START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF], we resort to formulating many-body nuclear problems by a Wilsonian renormalization-group (RG) type approach to arrive at Landau(-Migdal) Fermi-liquid theory. The mean field approximation with χbHLS can be identified with Landau Fermi-liquid fixed point (FLFP) theory valid in the limit 1/ N → 0 where N = k F /(Λ -k F ) (with Λ the cutoff on top of the Fermi sea) [START_REF] Shankar | Renormalization group approach to interacting fermions[END_REF][START_REF] Rho | Exchange currents from chiral Lagrangians[END_REF]. This approach can be taken as a generalization of the energy-density functional theory familiar in nuclear physics [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF]. One can go beyond the FLFP in V lowk -RG as we will do in numerical calculations.

As will be briefly summarized below, the χbHLS applied to nuclear matter -call it GenEFT -is found to describe nuclear matter at n ∼ n 0 as well as the currently successful sχEFT to typically N 4 LO. There is a however a good reason to believe that the sχEFT with the Fermi momentum k F taken as a small expansion parameter must necessarily breakdown at high densities relevant to massive compact stars. The appearance of quark degrees of freedom in one form or other is a natural candidate for the breakdown mechanism. In our approach, where and how this breakdown occurs can be determined by the skyrmion approach with the bosonic Lagrangian χmHLS. This is because topology in χmHLS can be traded in for what corresponds to hadron-to-quark continuity.

The skyrmion approach, while too daunting to handle nuclear dynamics directly and systematically, can however provide valuable and robust information on the possible topological structure involved in going beyond the normal nuclear matter density regime. At some density above n 0 -denoted as n 1/2 -the matter described in terms of skyrmions is found to transform to that of half-skyrmions. This transition has several highly correlated important impacts on the EOS for densities n > n 1/2 . The most crucial of them are that [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF] at n 1/2 (a) the quark condensate qq vanishes globally but not locally with non-vanishing pion and dilaton decay constants (f π ∼ f χ = 0), resembling the pseudo-gap phenomenon in condensed matter, (b) a parity-doubling emerges in the baryon spectrum with the chiral invariant mass m 0 (0.6 -0.9)m N and (c) the hidden gauge coupling associated with the ρ meson coupling starts to drop and flows to zero at the VM fixed point [START_REF] Harada | Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition[END_REF], hence the vector meson becoming massless. This is in line with the Suzuki theorem [START_REF] Suzuki | Inevitable emergence of composite gauge bosons[END_REF].

Our key strategy is to incorporate these effects connected with the topology change in χmHLS into the parameters of the χbHLS in formulating GenEFT.

It is interesting to note that the skyrmion to halfskyrmion transition in the skyrmion matter has an analog in condensed matter in (2+1) dimensions. For instance, the transition from the magnetic Néel ground state to the VBS (valence bond solid) quantum paramagnet phase [START_REF] Senthil | Deconfined quantum critical points[END_REF]. The half-skyrmions intervening here are deconfined, thus the transition involves "deconfined quantum critical points" but with no local order parameters for the phase transition. What takes place in dense skyrmion matter seems however quite different because the half-skyrmions are not deconfined but confined by monopoles [START_REF] Zhang | Reinterpretation of Skyrme theory: New topological structures[END_REF]. This suggests that the confined halfskyrmion complex be treated as a local baryon number-1 field with its parameters, such as the mass, coupling constants etc. drastically modified from the vacuum quantities reflecting the topology of the half-skyrmions. This allows us to work with a single Lagrangian χbHLS with the parameters controlled by topology before and after the topology change density n 1/2 . We suggest n 1/2 to be the density at which the hadron-quark change-over takes place. There is no phase transition involved.

Quenched g A in Nuclei and Continuity in Scale Symmetry.-As detailed in [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF], the GnEFT so formulated works fairly well, in fact as well as the sχEFT up to, say, N 4 LO, for baryonic matter properties for n 0 and perhaps even up to ∼ 2n 0 . With the heavy vector and scalar degrees of freedom explicitly figuring in the dynamics, the power counting rule is considerably different from that of sχEFT in which the vector and scalar excitations enter at loop orders. The mean field with χbHLS with the parameter scaling with density captured in the dilaton condensate χ and in the FLFP approximation could well be comparable to the high-order sχEFT calculation. Furthermore -and this is not recognized in nuclear circles -there is a magical element in HLS, as stressed in [START_REF] Komargodski | Vector mesons and an interpretation of Seiberg duality[END_REF][START_REF] Kan | From 3d dualities to hadron physics[END_REF], possibly connected with the Seiberg-duality between HLS and gluons of QCD.

Just to give an idea, we quote the predictions of a few thermodynamic properties of nuclear matter compared in parentheses with available empirical information: n 0 = 0.161(0.16 ± 0.01) fm -3 , B.E. = 16.7(16.0 ± 1.0) MeV, E sym (n 0 ) = 30.2(31.7 ± 3.2) MeV, E sym (2n 0 ) = 56.4(46.9 ± 10.1; 40.2 ± 12.8) MeV, L(n 0 ) = 67.8(58.9 ± 16; 58.7 ± 28.1) MeV, K 0 = 250.0(230 ± 20) MeV.

That the standard high-order sχEFT and the GnEFT with explicit hidden symmetries fare equally well at n ∼ n 0 indicate that those symmetries are buried and not apparent in the EOS at that density. This does not mean all observables are opaque to them. Totally unrecognized in the past is that the long-standing mystery of quenched g A in nuclei as seen in shell models has a deep connection with how scale symmetry hidden in QCD emerges in nuclear correlations [START_REF] Ma | The quenched gA in nuclei and emergent scale symmetry in baryonic matter[END_REF].

The superallowed matrix element of the Gamow-Teller transition from proton (neutron) to neutron (proton) on top of the Fermi sea calculated in GnEFT at the FLFP is given by the effective g

A g Landau A /g A = 1 - 1 3 Φ F π 1 -2 (1)
where g A = 1.27 is the axial coupling constant for the free neutron decay, Φ = χ * / χ 0 and F π 1 is the Landau F 1 parameter coming from the pion exchange. Φ gives the scaling in density of the dilaton condensate in medium and F π 1 is exactly given at a given density by chiral symmetry. Φ can be related to the scaling of the in-medium pion decay constant f π so is known experimentally up to ∼ n 0 . The RHS of (1) turns out to be highly insensitive to density near n 0 , so is nearly constant between 1 2 n 0 and n 0 : it comes out to be ∼ 0.8. Thus we get from medium-heavy nuclei to nuclear matter

g Landau A ≈ 1.0. (2)
This is the famous quenching of g A = 1.27 to g sm A ≈ 1.0 that has remained unexplained since mid 1970's. As discussed in [START_REF] Ma | The quenched gA in nuclei and emergent scale symmetry in baryonic matter[END_REF], there can be corrections coming from conformal anomaly that enter at higher scale-chiral order which can be "measured" in RIB experiments [START_REF] Henke | Superallowed Gamow-Teller decay of the doubly magic nucleus 100 Sn[END_REF]. But the implication will not be significantly modified.

Equation ( 1) is justified in the large N c and large N limit in what corresponds to the Landau FLFP. It's derivation exploits soft dilaton theorems [START_REF] Crewther | Genuine dilatons in gauge theories[END_REF]. As formulated it can be precisely equated to the superallowed Gamow-Teller transition matrix element given in the extreme single-particle shell-model of a doubly magic nuclei such as 100 Sn [START_REF] Henke | Superallowed Gamow-Teller decay of the doubly magic nucleus 100 Sn[END_REF]. At the matching scale to QCD, the axial weak coupling to the nucleons is scale-invariant, hence the renormalization (1) can be considered entirely due to nuclear (many-body) interactions. Thus (2) captures the influence of the scale symmetry emergent from the (nuclear) interactions which may or may not be directly connected to QCD. We can see this also by going to high density n 0 . Starting with non-linear sigma model with constituent quarks it is shown that [START_REF] Beane | The dilated chiral quark model[END_REF] after the field redefinition, letting the dilaton mass m χ go to zero with the conformal anomaly turned off -and in the chiral limit -leads to a linearized Lagrangian that satisfies various well-established sum rules, such as, among others, the Adler-Weisberger sum rule, provided the singularities that appear as m χ → 0 are suppressed by what are called "dilaon-limit fixed-point (DLFP)" constraints. The same limiting process to our χbHLS yields [START_REF] Beane | The dilated chiral quark model[END_REF]

g V → g A → 1, (3) f π → f χ = 0. ( 4
)
We denote the g A in this limit as g DL A . The DLFP gives no constraint on the ω-nucleon coupling into which we do not enter here. It is rather involved and has a crucial interplay in how the dilaton condensate becomes densityindependent in the half-skyrmion phase leading to the PC velocity [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF].

Now the important point is that ( 3) and ( 4) allow the DLFP to be identified with the IRFP [START_REF] Crewther | Genuine dilatons in gauge theories[END_REF]. In connection with the g A quenching, this provides the mechanism to make the emergent scale symmetry pervade from low density -g Landau A ≈ 1 -to high density -g DL A ≈ 1. We will argue this is also what happens in compact stars.

Equation of State of Compact Stars.-We now turn to the application of the GnEFT formalism to the structure of massive neutrons stars based on the standard TOV equation. We shall focus on the EOS of the baryonic matter, leaving out such basic issues as corrections to gravity, dark matters etc. Unless otherwise stated the role of leptons -electrons, muons, neutrinos etc -is included in the EOS. The results we present here are not new (as summarized in [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF]), but their implications and impacts on nuclear dynamics bring a totally new light and offer a paradigm change in nuclear physics. Now to go to higher densities beyond n 0 in GnEFT for compact stars, what we first need is to fix the density at which the topology change takes place. It turns out that the available astrophysics phenomenology does provide the range 2 ∼ < n 1/2 /n 0 < 4 [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF]. Here, we pick n 1/2 = 2.5n 0 for illustration.

Up to n 1/2 , the same EOS that works well at n 0 is assumed to hold. It is what comes at n 1/2 due to the topology change that plays the crucial role for the properties of compact stars. Among the various items listed in [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF], the most prominent are (a) the cusp in the symmetry energy E sym at n 1/2 , (b) the VM with m ρ → 0 at n vm ∼ > 25n 0 , (c) the approach to the DLFP -at or close to the putative IRFP -at n dl ∼ n vm and (d) the hadronquark continuity up to n vm . The cusp in E sym leads to the suppression of the ρ tensor force, the most important for the symmetry energy, and triggers the effect (b). It effectively makes the EOS transit from soft-to-hard in the EOS, thus accounting for massive ∼ > 2M stars. The effects (b) and (c) -together with the ω coupling to nucleons -make the effective mass of the confined half-skyrmions go proportional to the dilaton decay constant f χ ∼ m 0 which is independent of density. Thus f χ depends little on density in the half-skyrmion phase.

The star properties obtained in GnEFT are found to be generally consistent with presently available observations [START_REF] Ma | Towards the hadron-quark continuity via a topology change in compact stars[END_REF]. For n 1/2 = 2.5n 0 , the maximum star mass is found to be M max ∼ 2.0M with the central density ∼ 5n 0 . For a neutron star with mass 1.4M , currently highly topical in connection with gravity-wave data, we obtain the dimensionless tidal deformability Λ 1.4 ≈ 656 and the radius R 1.4 ≈ 12.8 km.

Our theory makes a basically different prediction from that of all other "standard" EFT available in the literature. And that is in the SV of stars v s and its impact on the structure of the core of massive stars. Sound Velocity and Emergent Scale Symmetry.-To see what'a going on with v s , let's look at the trace of the energy-momentum tensor θ µ µ which encodes the conformal anomaly. Suppose the dense matter is at the IRFP in the GD scheme (ignoring quark masses), then we will have θ µ µ = 0 and hence the conformal SV v 2 s /c 2 = 1/3. But our star matter must be some distance away from the IRFP even if the quark masses are ignored as m χ = 0, so θ µ µ cannot be zero. Yet a surprise in the prediction of GnEFT was that the SV came out to be the same as the conformal SV as shown in Fig. 1 (left panel) for n ∼ > 3n 0 for the assumed VM density (which we identify with the deconfinment density as stated below) n vm ∼ n dl ∼ > 25n 0 [START_REF] Paeng | Scale-invariant hidden local symmetry, topology change, and dense baryonic matter. II[END_REF]. Were the n vm much lower, say, ∼ 6n 0 , then v 2 s /c 2 would increase continuously, passing 1/3 at ∼ 3n 0 and reaching ∼ > 1/2 at ∼ 9n 0 [START_REF] Paeng | Scale-Invariant hidden local symmetry, topology change and dense baryonic matter[END_REF]. Thus the possible "deconfinement density" seems to play an important role for the SV whereas there are no appreciable differences in other star properties.

The calculations were done in the V lowk RG formalism which includes certain 1/ N corrections, going beyond the Landau FLFP approximation. How the striking result given above comes about in our EFT can be understood in an extremely simple way. In the mean field of GnEFT, the VEV of the trace of energy momentum tensor takes the simple form θ µ µ = 4V ( χ )χ ∂V (χ) ∂χ | χ= χ where V (χ) is the dilaton potential where the conformal anomaly effect is included. As noted above, χ ∝ f χ is density-independent in the range of density involved in the interior of compact stars, so

∂ ∂n θ µ µ = ∂ (n) ∂n (1-3v 2 s /c 2 ) = 0. Now if ∂ (n)
∂n = 0 (i.e., no Lee-Wick state), then we arrive at 1 -3v 2 s /c 2 = 0. Since our system is necessarily some distance away from the IRFP to account for the non-zero dilaton mass, θ µ µ cannot be zero. The v s cannot be conformal. Thus though numerically the same, we call this "pseudo-conformal"

v 2 pcs /c 2 ≈ 1/3. ( 5
)
It is of course approximate because both the quark mass terms and the deviation from the IRFP -which must be present in the EOS -are ignored. Confronting Nature.-This prediction can be confronted with a recent analysis that combines astrophysical observations and model independent theoretical ab initio calculations [START_REF] Annala | Evidence for quark-matter cores in massive neutron stars[END_REF]. Based on the observation that, in the core of the maximally massive stars, v s approaches the conformal limit v s /c → 1/ √ 3 and the polytropic index takes the value γ < 1.75 -the value close to the minimal one obtained in hadronic models -Annala et al. arrive at the conclusion that the core of the massive stars is populated by "deconfined" quarks.

It is perhaps surprising that the predicted pseudoconformal speed [START_REF] Bando | Is ρ meson a dynamical gauge boson of hidden local yymmetry?[END_REF] sets in precociously at ∼ 3n 0 and stays constant in the interior of the star. Microscopic descriptions such as the quarkyonic model [START_REF] Mclerran | Quarkyonic matter and neutron stars[END_REF] typically exhibit more complex structures at the putative hadronquark transition density. We think the simpler structure in our description is due to perhaps too drastic suppression of higher-order 1/ N terms in the half-skyrmion phase. Similarly in the right panel, one sees the polytropic index γ drops, again rapidly, below 1.75 at ∼ 3n 0 and approaches 1 at n ∼ > 6n 0 . Finally -and most importantly -we compare in Fig. 2 our prediction for P/ with the conformality band obtained by the SV interpolation method [START_REF] Annala | Evidence for quark-matter cores in massive neutron stars[END_REF]. We see that it is close to, and parallel with, the conformality band, but most significantly, it lies outside of this band.

The predicted results of GnEFT as a whole resemble the "deconfined" quark structure of [START_REF] Annala | Evidence for quark-matter cores in massive neutron stars[END_REF]. There are, however, basic differences between the two. First of all, in our theory, conformality is broken, though perhaps only slightly at high density, in the system. This could be related to the deviation of g Landau A from the experimental value of the quenching in 100 Sn observed [START_REF] Ma | The quenched gA in nuclei and emergent scale symmetry in baryonic matter[END_REF]. There can also be fluctuations around v 2 pcs /c 2 = 1/3 coming from the effects by the anomalous dimension β . This effect can be seen in Fig. 2 where the PCM prediction deviates slightly from the "would -be" conformal band. Most important of all, the confined half-skyrmion fermion in the half-skyrmion phase is not deconfined. It is a quasiparticle of fractional baryon charge, neither purely baryonic nor purely quarkonic. In fact it can be anyonic lying on a (2+1) dimensional sheet [START_REF] Ma | Dichotomy of baryons as quantum Hall droplets and skyrmions In compact-star matter[END_REF]. What it represents is a manifestation of an emergent scale symmetry pervading at low density as in g Landau A and in g DL A at high density in the vicinity of DLFP. We suggest to classify the precocious pseudo-conformal SV in the same class of emerging scale symmetry in action in nuclear processes. In fact it is in line with how conformal symmetry permeates from the unitarity limit in light nuclei [START_REF] König | Nuclear Physics around the unitarity limit[END_REF] to the symmetry energy near n 0 [START_REF] Tews | Symmetry parameter constraints from a lower bound on neutron-matter energy[END_REF] and more.

Conclusion.-We have suggested that quark-like degrees of freedom, if observed in the interior of massive neutron stars, can be interpreted as confined quasiparticles of fractional baryon charges in consistency with hadron-quark continuity. Such fractionally-charged objects are inevitable by topology at high densities [START_REF] Ma | Dichotomy of baryons as quantum Hall droplets and skyrmions In compact-star matter[END_REF]. The mechanism in action is the emergence of conformal (or scale) symmetry, coming not necessarily from QCD proper, but from strongly-correlated nuclear interactions, which could permeate, either hidden or exposed, in baryonic matter from low density to high density. In this scheme, true deconfinement is to set in as mentioned above at much higher densities, say, ∼ > 25n 0 , than relevant to compact stars when the VM fixed point and/or DLFP are reached, possibly with the phase transition from a Higgs mode to a topological mode [START_REF] Kan | From 3d dualities to hadron physics[END_REF].

FIG. 1 .

 1 FIG. 1. Density dependence of the SV of stars vs (left panel) and the polytropic index γ = d ln P /d ln (right pannel) in neutron matter.

3 ]FIG. 2 .

 32 FIG.2. Comparison of (P/ ) between the PCM velocity and the band generated with the SV interpolation method used in[START_REF] Annala | Evidence for quark-matter cores in massive neutron stars[END_REF]. The gray band is from the causality and the green band from the conformality. The red line is the PCM prediction. The dash-dotted line indicates the location of the topology change.

Acknowledgments.-We are grateful for helpful correspondence from Jim Lattimer and Aleksi Vuorinen on their recent publications. The work of YLM was supported in part by the National Science Foundation of China (NSFC) under Grant No. 11875147 and 11475071.


Jing Cui

Manuel Nieves-Cordones

Francisco Rubio

Guillaume Tcherkez
email: guillaume.tcherkez@anu.edu.au

Involvement of salicylic acid in the response to potassium deficiency revealed by metabolomics

Keywords: Potassium, deficiency, metabolomics, salicylic acid, CBL-CIPK regulation Highlights

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Potassium (K) is a major nutrient representing up to ≈5% of plant matter. K is essential for plant stomatal conductance and photosynthesis, phloem transport and cellular ion balance [START_REF] Wang | Potassium Transport and Signaling in Higher Plants[END_REF]. In many cultivated areas where K availability is naturally low, K fertilisation is required, representing a total market of $12.6 billion at the global scale [START_REF] More | Global potassium chloride market size[END_REF]. K fertilisation is of importance for high K-demanding crops such as oil palm, since some tropical soil types are relatively K-poor and waterlogging can limit further K assimilation [START_REF] Cui | Metabolic responses to potassium availability and waterlogging reshape respiration and carbon use efficiency in oil palm[END_REF].

In the past decade, intense efforts have been devoted to understand response mechanisms to low K conditions and identify biomarkers of K deficiency. It is believed that low K sensing involves CBL1/9 (CALCINEURIN B LIKE 1/9) and CIPK23 (CBL INTERACTING PROTEIN KINASE 23) via a Ca 2+ signal triggered by low K [START_REF] Ródenas | Regulation of root nutrient transporters by CIPK23: 'one kinase to rule them all[END_REF][START_REF] Tang | The CBL-CIPK calcium signaling network: unified paradigm from 20 years of discoveries[END_REF][START_REF] Tagliani | The calcineurin β-like interacting protein kinase CIPK25 regulates potassium homeostasis under low oxygen in Arabidopsis[END_REF]. CIPK23 in turn phosphorylates and activates the voltage-gated channel AKT1 [START_REF] Lan | Mechanistic analysis of AKT1 regulation by the CBL-CIPK-PP2CA interactions[END_REF][START_REF] Sánchez-Barrena | Recognition and activation of the plant AKT1 potassium channel by the kinase CIPK23[END_REF], and the transporter HAK5 [START_REF] Ródenas | Insights into the mechanisms of transport and regulation of the Arabidopsis high-affinity K + transporter[END_REF] thereby enhancing K + absorption. Other targets of CIPK23 include KUP4 (K + UPTAKE TRANSPORTER) and the vacuolar channel TPK (TWO PORES K + ) (reviewed in [START_REF] Ródenas | Regulation of root nutrient transporters by CIPK23: 'one kinase to rule them all[END_REF]). Downstream events associated with low K conditions include a reorchestration of metabolism, due to both indirect (regulatory mechanisms) and direct effects (use of K + as a cofactor) of K + scarcity. Regulatory mechanisms seem to involve the central metabolic regulator mTOR, which interacts effectively with CIPK23 [START_REF] Deng | Target of rapamycin regulates potassium uptake in Arabidopsis and potato[END_REF]. Direct effects concern typically pyruvate kinase (EC 2.7.1.40, glycolytic enzyme) and succinate thiokinase (EC 6.2.1.4, Krebs cycle enzyme), which are K +dependent such that their activity is inhibited when K + concentration is low [START_REF] Besford | Effect of potassium nutrition on some enzymes of the tomato plant[END_REF][START_REF] Evans | Effect of potassium and other univalent cations on activity of pyruvate kinase in Pisum sativum[END_REF][START_REF] Lynn | Equilibrium constants under physiological conditions for the reactions of succinyl coenzyme A synthetase and the hydrolysis of succinyl coenzyme A to coenzyme A and succinate[END_REF]. K deficiency thus activates alternative pathways such as the malate cycle (phosphoenolpyruvate carboxylase, malate dehydrogenase, malic enzyme) to synthesize pyruvate, and the aconitate-citramalate shunt to compensate for the inhibition succinate metabolism [START_REF] Cui | Metabolic responses to potassium availability and waterlogging reshape respiration and carbon use efficiency in oil palm[END_REF][START_REF] Amtmann | Effects of N, P, K and S on metabolism: new knowledge gained from multi-level analysis[END_REF][START_REF] Armengaud | Multilevel analysis of primary metabolism provides new insights into the role of potassium nutrition for glycolysis and nitrogen assimilation in Arabidopsis roots[END_REF][START_REF] Cui | Responses to K deficiency and waterlogging interact via respiratory and nitrogen metabolism[END_REF][START_REF] Cui | Plant low-K responses are partly due to Ca prevalence and the low-K biomarker putrescine does not protect from Ca side effects but acts as a metabolic regulator[END_REF][START_REF] Cui | Potassium deficiency reconfigures sugar export and induces catecholamine accumulation in oil palm leaves[END_REF]. Also, K deficiency is accompanied by putrescine accumulation, sometimes to very high levels [START_REF] Cui | What is the role of putrescine accumulated under potassium deficiency?[END_REF]. It has been proposed recently that putrescine itself plays a regulatory role since putrescine addition down-regulates further the content in pyruvate kinase [START_REF] Cui | Plant low-K responses are partly due to Ca prevalence and the low-K biomarker putrescine does not protect from Ca side effects but acts as a metabolic regulator[END_REF]. Nevertheless, drastic metabolic changes like putrescine or aconitate build-up are usually observed when plants cultivated under contrasting K conditions are compared, or when K deficiency is well-established. In other words, it is possible that early metabolic events occurring when K becomes progressively limiting have not been observed yet. Missing early events can be problematic when monitoring the K status of crops since catching first symptoms of deficiency is essential to adjust fertilization rapidly and thereby avoid consequences on yield.

Here, we carried out metabolic kinetics in Arabidopsis, using both wild-type (WT) and mutants affected in potassium absorption and signalling, with emphasis on the triple mutant hak5 akt1 cipk23 that has been characterized previously [START_REF] Ragel | The CBL-interacting protein kinase CIPK23 regulates HAK5-mediated high-affinity K + uptake in Arabidopsis roots[END_REF]. Plants were grown in hydroponics under Ksufficient conditions and then either challenged under low K or kept under K-sufficient conditions (Fig. S1). Metabolomics analyses were performed using GC-MS profiling. Both univariate and multivariate statistics were conducted to highlight metabolites associated with genotype and low K effects.

Methods

Plant material, K content and Rb uptake

Single, double and triple mutants were characterized in [START_REF] Ragel | The CBL-interacting protein kinase CIPK23 regulates HAK5-mediated high-affinity K + uptake in Arabidopsis roots[END_REF]. Sowing was done on 0.8% agar (Bacto) and plants were grown on hydroponics under short days 8:16 h day:night. K + concentration was varied with KCl in nutrient solution composition. For the metabolomic kinetics experiment was designed to follow early metabolic changes. Plants were grown at 5 mM KCl for 43 d and then kept at 5 mM or transferred to 0.01 mM for several weeks in the nutrient solution consisting of Ca(NO3)2 (1.5 mM), MgSO4 (0.35 mM), NaH2PO4 (0.2 mM) and a mixture of microelements (Na-Fe-EDTA 0.1 mM, MnSO4 1 µM, ZnSO4 1 µM, CuSO4 0.5 µM, H3BO4 37.5 µM, Na2MoO4 0.1 µM, Co(NO3)2 0.1 µM).

Four samplings were done (7, 11, 18 and 29 days after transfer) (Fig. S1). To analyse early metabolic events, only the first three samplings were used in statistics (Fig. 1). Ion transport activity of plants was checked via Rb + uptake assays carried out in a separate experiment. Rb + uptake was measured on plants grown at 10 mM KCl (with 1/5 Hoagland solution) for 30 d and transferred to 0.5 mM for 5 d, using Rb + addition to a final concentration of 0.5 mM. K and Rb content in plants was measured by ICP-OES (Perkin-Elmer).

Metabolomics

Plant leaves were snap-frozen in liquid nitrogen, frozen-dried and ground in fine powder.

Metabolomics analyses were carried out by GC-MS on water:methanol (90:10 v:v) extracts derivatized by trimethylsilylation in pyridine, using ribitol as an internal standard. Data analysis were performed using Metabolome Express (www.metabolome-express.org). Further details are provided in [START_REF] Cui | Metabolic responses to potassium availability and waterlogging reshape respiration and carbon use efficiency in oil palm[END_REF] and in Supplementary Notes S1. Statistical multivariate OPLS analyses were carried out with Simca 16 (Umetrics) as in [START_REF] Cui | Plant low-K responses are partly due to Ca prevalence and the low-K biomarker putrescine does not protect from Ca side effects but acts as a metabolic regulator[END_REF] using metabolites as X variables and genotype, time and K as Y variables.

Results

Plant growth

Thirty-five days after germination, there were little differences in plant size between genotypes (Fig. S2a) showing that high K conditions used here were K-sufficient. Four weeks after having shifted conditions to low K, observed growth was very limited, with an increase of ≈60% only in rosette diameter in the WT. As expected, growth was nearly arrested in the double hak5 akt1 and triple hak5 akt1 cipk23 mutants (Fig. S2b). The single mutant cipk23 did not show a significant decrease in rosette size compared to the WT under low K but exhibited a smaller rosette under high K, suggesting that CIPK23 is involved in the growth response to high K availability.

Potassium content and uptake

As expected, net potassium uptake capacity (estimated using Rb + uptake assays) of the double hak5 akt1 and triple hak5 akt1 cipk23 mutants was effectively 60% lower than that of the WT (Fig. S3a).

When challenged under low K, all plants were depleted in K in shoots and roots (Fig. S3b-c), with a significantly lower content in mutants compared to WT (up to 30% lower). Interestingly, there was no significant difference in uptake rate and potassium content between the two mutants, in agreement with the same rosette size under low K (Fig. S2) (unlike results found in [START_REF] Ragel | The CBL-interacting protein kinase CIPK23 regulates HAK5-mediated high-affinity K + uptake in Arabidopsis roots[END_REF]).

Metabolic signature

Metabolomics analyses led to the identification and quantification of 204 analytes. Statistics carried out by supervised multivariate analysis (OPLS) allowed facile discrimination of sample types, with respect to genotype and K conditions (Fig. 1a). In fact, the score plots showed a discrimination of K conditions along axis 1, time along axis 2 and genotype along axis 3. The statistical model was highly explicative (R² = 0.847) and robust (Q² = 0.516). A univariate analysis by ANOVA was also conducted and best discriminant metabolites are shown in a volcano plot (Fig. 1b) and a heatmap (Fig. 1c). Most genotype-significant metabolites were sugars, with hexoses and polyols increased (cluster 2, Fig. 1c) and oxidised derivatives of sugars (galacturonate, erythronate and 2-oxoglucose) decreased (cluster 1, Fig. 1c) in the hak5 akt1 cipk23 mutant. Interestingly, only three non-sugar metabolites were significant for the genotype effect: serine, cysteine and salicylate glucoside (salicylic acid 2-O-β glucoside). Metabolites associated with a genotype × K interaction effect (Fig. 1d) were mostly sugars, with three exceptions: succinate (increased in the triple mutant early on), serine and salicin (increased in the triple mutant later on). Interestingly, most discriminant metabolites associated with the K effect were similar to those significant for the genotype effect (Fig. S4a-b): typically sugars and polyols (as well as salicylate glucoside and salicin) increased while oxidised derivatives (galacturonate, erythronate) decreased under low K. In other words, the metabolic effect of the triple mutation was rather similar to that of low K availability, confirming that mutants experienced constitutive low K cellular conditions. Salicylate glucoside and salicin accumulated under low K, to a higher extent in the triple mutant (Fig. S4c). Also, salicylic glucoside was significantly higher at the first sampling time (7 d after having switched K conditions), even before biomarkers of K deficiency, aconitate and putrescine, accumulated (Fig. S4a). This resulted in putrescine being much less significant for the K effect compared to salicylate glucoside (in Fig. S4b, note that putrescine was just above the Bonferroni threshold). It is also worth noting that while salicylate glucoside increased with time under low K, other phenylpropanoids significant for the K effect (3,4,5-trihydroxybenzoate, sinapate) declined with time, suggesting rerouting of the entire phenylpropanoid pathway in favour of salicylic acid (SA) production.

Discussion

Alteration of sugar metabolism under low K

Early metabolic changes caused by low K and genotype mostly concerned sugars, with an increase in hexoses, polyols and pentoses and a decrease in metabolites coming from sugar oxidation (summary provided in Fig. S5). Also, low K conditions led to a decrease in organic acids such as citrate and fumarate, while succinate increased in the triple mutant under low K (Fig. 1c). This rapid and concerted effect on sugar and organic acids when conditions are switched from K sufficiency to K deficiency probably reflects the direct effect of low K + cellular concentration, inhibiting (i) phloem export from leaves thereby causing a general increase in sugars and myoinositol [START_REF] Cui | Potassium deficiency reconfigures sugar export and induces catecholamine accumulation in oil palm leaves[END_REF]; (ii) enzyme activities of catabolism that are K-dependent (see Introduction). It is worth noting that while citrate and fumarate were amongst most significantly decreased metabolites, aconitate was more abundant under low K, suggesting the involvement of the aconitate-citramalate shunt as observed previously in sunflower and oil palm [START_REF] Cui | Metabolic responses to potassium availability and waterlogging reshape respiration and carbon use efficiency in oil palm[END_REF][START_REF] Cui | Responses to K deficiency and waterlogging interact via respiratory and nitrogen metabolism[END_REF]. Unlike sugars, putrescine did not increase much at the beginning of the low K response and it took two weeks to observe putrescine build-up. This effect might have come from the lag time required to induce arginine decarboxylase (the source of putrescine in Arabidopsis).

Salicylic acid metabolism is part of the low K response

A striking feature of the metabolic signature observed both under low K and in the triple mutant hak5 akt1 cipk23 is the appearance of SA derivatives: salicylate glucoside and salicin. Interestingly, salicylate glucoside increased considerably early on in the triple mutant (while salicin increased significantly later). It is possible that salicylate glucoside appeared at the expense of other forms of conjugated forms of salicylic acid not detected here by GC-MS. However, since other phenylpropanoids declined it is plausible that there was a reorientation of phenylpropanoid metabolism towards SA biosynthesis. There are two major routes for SA synthesis, involving either isochorismic acid or cinnamic acid [START_REF] Dempsey | Salicylic acid biosynthesis and metabolism[END_REF]. Here, it is likely that it involved the isochorismic pathway since sinapic acid was affected by K conditions and furthermore, in sunflower proteome under K deficiency, a significant decrease in cinnamyl alcohol dehydrogenase and coumarate CoA ligase was found [START_REF] Cui | Plant low-K responses are partly due to Ca prevalence and the low-K biomarker putrescine does not protect from Ca side effects but acts as a metabolic regulator[END_REF]. Salicylate glucoside and salicin are synthesized by UDP-glucosyl transferases, transported by several sucrose transporters, stored in vacuoles and represent a pool of remobilizable SA [START_REF] Dempsey | Salicylic acid biosynthesis and metabolism[END_REF][START_REF] Fellenberg | Discovery of salicyl benzoate UDP-glycosyltransferase, a central enzyme in poplar salicinoid phenolic glycoside biosynthesis[END_REF][START_REF] Chandran | Substrate specificity of the Arabidopsis thaliana sucrose transporter AtSUC2[END_REF][START_REF] Vaca | Mechanistic differences in the uptake of salicylic acid glucose conjugates by vacuolar membrane-enriched vesicles isolated from Arabidopsis thaliana[END_REF].

The increase in the SA pool found here maybe reflects the involvement of SA signalling under low K stress. SA is generally related to pathogenesis-related signalling but has been found to participate in the response to salt stress, drought and heavy metal (in particular cadmium) [START_REF] Dempsey | Salicylic acid biosynthesis and metabolism[END_REF][START_REF] Liu | Role of salicylic acid in resistance to cadmium stress in plants[END_REF][START_REF] Janda | Magical mystery tour: Salicylic acid signalling[END_REF].

Interestingly, in sunflower proteome under K deficient, there is an increase in benzyl alcohol Obenzoyl transferase [START_REF] Cui | Plant low-K responses are partly due to Ca prevalence and the low-K biomarker putrescine does not protect from Ca side effects but acts as a metabolic regulator[END_REF], which is involved in SA metabolism and has been found to be up-regulated during the hypersensitive reaction to pathogens [START_REF] Chedgy | Functional characterization of two acyltransferases from Populus trichocarpa capable of synthesizing benzyl benzoate and salicyl benzoate, potential intermediates in salicinoid phenolic glycoside biosynthesis[END_REF]. Also, the addition of SA has been found to alleviate symptoms of K deficiency on plant morphology [START_REF] Zhu | Salicylic acid alleviates potassium deficiency by promoting nutrient absorption and accumulation in Zinnia elegans[END_REF]. A NPR1-independent pathway of SA signalling has been shown to be essential during salt and oxidative stress [START_REF] Jayakannan | The NPR1-dependent salicylic acid signalling pathway is pivotal for enhanced salt and oxidative stress tolerance in Arabidopsis[END_REF]. Here, a similar pathway is perhaps used under K deficiency.

Conclusions and future directions

Our results show that aside strong effects on sugar metabolism and catabolism, early responses to K deficiency have consequences on SA metabolism, and this is particularly visible in the triple mutant (d) Heatmap showing significant metabolites in ANOVA for the genotype × K interaction, with also a hierarchical clustering on left. In (c), "salicylate (g)" stands for salicylate glucoside as it appears in (b). In both (c) and (d), numbers along with metabolites refer to distinct derivatives in GC-MS analyses. The description of the specific effect of K is provided in Fig. S4.

  hak5 akt1 cipk23. Future studies are warranted to provide more insight on molecular mechanisms linking low K to the activation of SA metabolism and reveal specific roles of SA in the response to low K conditions. Contributions G.T. and J.C. designed the experiment; F.R. and M.N.C. provided seeds and performed mineral analyses; J.C. set up the experiments and did metabolomics analyses; G.T. did data integration and wrote the first draft of the paper; all authors contributed to the final version of the paper.

Fig. 1 .

 1 Fig. 1. Metabolomics pattern of wild-type and triple mutant hak5 akt1 cipk23 Arabidopsis rosettes: genotype effect when plants are challenged with low (0.01 mM) or high (5 mM) KCl in hydroponics after 43 days under high K. (a) Score plot of the multivariate analysis (OPLS) showing facile discrimination of samples with K conditions (axis 1), time (axis 2) and genotype (axis 3). Along axis 2, time goes from bottom (7 days after having switched K conditions) to top (29 days). (b) Volcano plot representing best biomarker metabolites associated with the genotype effect, using the loading in the OPLS analysis (x axis) against the logarithm of the p-value in univariate analysis (ANOVA; y axis). The horizontal dash-dotted line stands for the Bonferroni threshold (here, -log(p) = 3.61). (c) Heatmap showing significant metabolites in ANOVA for the genotype effect. Hierarchical clustering (Euclidean distance) is shown on left and clusters are labelled with red frames and numbers.(d) Heatmap showing significant metabolites in ANOVA for the genotype × K interaction, with also a hierarchical clustering on left. In (c), "salicylate (g)" stands for salicylate glucoside as it appears in (b). In both (c) and (d), numbers along with metabolites refer to distinct derivatives in GC-MS analyses. The description of the specific effect of K is provided in Fig.S4.

Acknowledgements

G.T. thank the Région Pays de la Loire and Angers Loire Métropole for its financial support via the grant Connect Talent Isoseed. We thank Sergio García for his skilful assistance. This work was funded by grant number 20806/PI/18 (to FR) from Fundación Séneca de la Región de Murcia, Spain, grant PID2019-106649RB-I00 from Ministerio de Ciencia e Innovación, Spain (to FR and MN-C). This work was also supported by the Red de Excelencia BIO2016-81957-REDT from Ministerio de Economía, Industria y Competitividad. All grants from Spanish ministries were co-financed by the 7 FEDER Fund. MN-C, is recipient of a Ramón y Cajal Fellowship (RyC-2017-21924) from Ministerio de Economía y Competitividad, Spain.