Holographic cold dense matter constrained by neutron stars

Quantum chromodynamics (QCD) is believed to be the basic theory of the strong interaction. However, the strong interaction is non-perturbative at low energy scale. Thus many phenomena such as hadron and glueball spectra, chiral phase transition and confinement-deconfinement phase transition, the equation of state (EoS) of the strong interaction matter under extreme conditions (high temperature, high baryon number density, rapid rotation, and strong magnetic field) can’t be explained from the first principle until now. Therefore many efforts have been paid on developing non-perturbative methods, such as lattice QCD [1, 2, 3], chiral effective theory [4], Nambu–Jona-Lasinio (NJL) model [5, 6], QCD sum rule [7, 8, 9, 10, 11, 12, 13, 14], Dyson Schwinger equations (DSE) and functional renormalization group (FRG) [15], and recently holographic QCD method [16, 17, 18, 19, 20, 21, 22, 23, 24].

Exploring the equation of state (EoS) and phase structure of quantum chromodynamics (QCD) at high density is boosted both by neutron star (NS) and heavy ion collisions. Relativistic heavy ion collision (RHIC) can produce the quark-gluon plasma (QGP) with high temperature and/or high baryon number density, and strong magnetic field. Besides that, the QGP with rapid rotation can also be produced in non-central heavy ion collisions. That is the most extreme environment created in the laboratory. It has been theoretically predicted that there exists a critical end point (CEP) for QCD at finite baryon density. The search of the CEP has been one of the most important targets for the beam energy scan program at RHIC as well as for the future accelerator facilities at Facility for Antiproton and Ion Research (FAIR) in Darmstadt and Nuclotron-based Ion Collider Facility (NICA) in Dubna.

In the nature, the early universe and the compact stars are also in extreme conditions. All these provide good platforms to investigate the non-perturbative features of strong interaction and the properties of the QCD matter.

The neutron star (NS) is a kind of compact star, which is the remnant after a massive super-giant star collapses. From August 1967 on, when the existence of NSs was confirmed by the discovery of radio pulsars, more than 2700 radio pulsars have been detected. It has been wondered for more than a half-century what’s the internal structure of NSs. Whether quark matter exists inside the core of NSs [25, 26]? Is there any exotic matter in NSs? Do strange stars exist? We can’t answer these questions yet. There are just a few of the open questions [27]. The strong interaction matter in the core of the NS is extremely dense, the baryon number density $n_{B,\mathrm{NS}}$ of which can reach as high as $5n_{S}-10n_{S}$, where $n_{S}$ is the nuclear saturation density. Although the surface of the NS is hot and the temperature can reach several hundred thousand Kelvin [28, 29], compared with the QCD energy scale $\Lambda_{\mathrm{QCD}}$ this temperature is still very low and can be neglected. Thus we can regard the matter in the NS as cold dense QCD matter.

On the experimental side, there are some observations of the NSs, including the mass measurements through the general relativistic Shapiro delay [30] and other methods [31, 32, 33, 34, 35, 36], the mass and radius measurements through the detection of thermal emission from the surface of the stars [37, 38, 39, 40], and the radius, EoS, and tidal deformability measurements through the gravitational wave signals of the binary systems [41, 42]. We will discuss these observations in detail in Sec. II.

Recent developments in the field of observational astronomy, e.g. the currently operating satellites, the Neutron star Interior Composition Explorer (NICER), and the gravitational-wave laser interferometers (Advanced LIGO, Virgo, and KAGRA) have provided tight constraints on neutron star masses and radii, which in turn, put strict constraints on the equation of state of dense matter.

The difficulties to predict the cold QCD matter from the theoretical side come from the awkward energy scale of the QCD matter in NS $\Lambda_{\mathrm{NS}}$: the baryon number density in the neutron star $n_{B,\mathrm{NS}}$ is high so that $\Lambda_{\mathrm{NS}}$ exceeds the appropriate range where the traditional nuclear physics theories validate; however it is still not high enough to adapt the perturbative method.

Although it’s very hard to directly study the EoS of cold dense QCD matter inside the NSs, we can still derive some useful information when we combine the results of the traditional nuclear theories in the low energy scale and the results of the perturbative QCD at the high energy scale with the observation data. Under the constraints from the observation data, there are two approaches to derive the allowed range of the EoS for the cold QCD matter in the whole range of $n_{B}$: the “direct interpolation approach” [43, 44] and the “causality and thermodynamic stability constraint approach” [45, 46]. In the former approach, combining the observation data, a generic family of neutron-star-matter EoSs, which interpolate between theoretical results at low and high baryon density, are obtained. In the latter one, utilizing the thermodynamic stability and causality, it is found that $ab$-$initio$ calculations in QCD at high densities, as complementary to the astrophysical observations, provide significant and nontrivial information about the EoS of matter in the cores of the neutron star.

To directly calculate the NS interior composition, from the theoretical side, normally Nambu-Jona-Lasinio (NJL) model [47, 48, 49, 50, 51, 52, 53] has been used for quark matter. The holographic QCD model [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] has been developed at finite baryon density over recent years.

There are many efforts to describe the EoS of cold dense matter inside the neutron star in the holographic frame. The main approach is to use the background action in the holographic models to describe the holographic quark matter. Then the phase equilibrium conditions can be used to connect the EoS of the holographic quark matter and the EoS calculated in effective theories adapted at low energy scale. This kind of hybrid EoSs can describe the NSs. However, the problem is that the branch that corresponds to the holographic quark matter in the mass-radius relation is unstable according to the Bardeen-Thorne-Meltzer (BTM) stable criterion stated in Refs. [70, 71]. The BTM stable criterion is stated in the following. As one moves along the $M$-$R$ curve, where $M$ is the mass of the NS as one observes from outside and $R$ is its radius, to make sure the value of the internal energy density at the center of the star increases, all configurations remain stable until one reach the first extreme value (maximum or minimum) of $M$. The fundamental mode of radial oscillation becomes unstable at the first extreme value point. Between the first extreme value point and the second one, all configurations contain one unstable radial mode. At the second extreme value point, and also at each succeeding extreme value point, the stability of one more mode of radial oscillation changes. How the $M$-$R$ curve bends as it passes through the extreme value point determines how the stability changes. One previously stable radial mode becomes unstable at the critical point if the direction of the bend is counter-clockwise (with increasing $\epsilon_{0}$). One previously unstable mode becomes stable at the critical point if the direction of the bend is clockwise.

The key point to produce the stable branch in the holographic method is to consider the holographic nuclear matter. In Refs. [72, 60, 61, 62, 63, 64, 65, 66], the Dirac-Born-Infeld (DBI) action and the Chern-Simons (CS) terms has been used to describe the baryons, where the solitons are regarded as baryons. The authors in Refs. [60, 61, 62] used the V-QCD bottom-up model action [73, 74, 75] to describe the background. In Refs. [63, 64, 65], the authors chose the Witten-Sakai-Sugimoto model [76, 20, 21] in top-down approach as the background. In Ref. [66], the authors used the hard-wall AdS/QCD model action to describe the background. They all obtained the stable branch that corresponds to the holographic nuclear matter.

In this work, different from the topological description of the baryons, we use the improved Karch-Katz-Son-Stephanov (KKSS) action [24, 77, 78, 79, 80] to introduce the contribution of the hadrons and treat it as a probe. Then we can calculate the EoS of the holographic nuclear matter. This article is organized in the following. In Sec. II, we briefly summarize the observation data of the mass and radius of the NSs. In Sec. III, we first introduce three holographic models. Then, we solve the Einstein-Maxwell-dilaton (EMD) system to derive the thermodynamics of the background and plot the $T-\mu$ phase diagram for the confinement-deconfinement phase transition in the three models. We also compare the deformed background metric in the three models with that in the $AdS_{5}$ space-time. In Sec. IV we consider the thermodynamic contribution made by the KKSS action and calculate the EoS of the holographic nuclear matter. Combined with the Hebeler-Lattimer-Pethick-Schwenk (HLPS) intermediate EoS, we then construct the hybrid EoS of the cold strong interaction matter. In Sec. V, we use the hybrid EoSs to calculate some properties of the neutron stars, such as mass-radius relation and tidal deformability. We also compare our results with the observation data. Finally, we give the conclusion and some discussions in Sec. VI.

Here we give a summary of the astrophysical observations for NSs [81, 62].

PSR J1614-2230 is a recycled millisecond pulsar with a very low inferred magnetic field, the magnetic induction intensity of which is $B=1.8\times 10^{8}\mathrm{G}$. It was discovered in a completed search of 56 unidentified mid-latitude EGRET $\gamma$-ray sources for pulsations using the Parkes telescope and the multibeam receiver in 2004 [82]. In 2010, the authors in Ref. [31] presented radio timing observations of the binary millisecond pulsar PSR J1614-2230. The detection of the general relativistic Shapiro delay [30] implied the pulsar mass is $1.97\pm 0.04M_{\odot}$, which made it the heaviest known NS at that time. In 2016, the authors in Ref. [32] analyzed 24 binary radio pulsars in the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) nine-year data set and made 14 significant measurements of the Shapiro delay. They presented the mass of PSR J1614-2230 is $1.928_{-0.017}^{+0.017}M_{\odot}$ at $68.3\%$ credibility.

PSR J0348+0432 is a mildly partially recycled binary pulsar with $P=39.1\mathrm{ms}$ and low magnetic field ($B_{\mathrm{surf}}=3.1\times 10^{9}\mathrm{G}$), in a $2.4\mathrm{hr}$ orbit with a low-mass companion, which was detected in 2012 in a $350\mathrm{MHz}$ Drift-scan Survey using the Robert C. Byrd Green Bank Telescope with the goal of finding new radio pulsars, especially millisecond pulsars that can be timed to high precision [83]. PSR J0348+0432 is only the second NS with a precisely determined mass around $2M_{\odot}$, after PSR J1614-2230. In 2013, the authors in Ref. [33] reported on radio-timing observations of the pulsar J0348+0432 and phase-resolved optical spectroscopy of its white-dwarf companion and then derived a NS mass in the range from $1.97M_{\odot}$ to $2.05M_{\odot}$ at $68.27\%$ confidence or $1.90M_{\odot}$ to $2.18M_{\odot}$ at $99.73\%$ confidence.

In 2014, 67 new radio pulsars including J0741+66 ($P=0.00288570374(45)\mathrm{s}$) were discovered in the Green Bank Northern Celestial Cap (GBNCC) survey, which surveys the entire sky visible to the Robert C. Byrd Green Bank Radio Telescope (GBT) at 350 MHz for radio pulsars and fast radio transients [84]. In 2018, the pulsar J0741+66 was renamed J0740+6620 in Ref. [85]. In 2019, by combining data from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) 12.5-yr data set with orbital phase-specific observations using the Green Bank Telescope, the authors in Ref. [34] measured the mass of the millisecond pulsars (MSP) J0740+6620 to be $2.14_{-0.09}^{+0.10}M_{\odot}$ at $68.3\%$ credibility interval or $2.14_{-0.18}^{+0.20}M_{\odot}$ at $95.4\%$ credibility interval. In 2021, the authors in Ref. [35] used the data set that consisted of combined pulse arrival-time measurements made with the $100\mathrm{m}$ Green Bank Telescope and the Canadian Hydrogen Intensity Mapping Experiment telescope. They presented the model-averaged result of the mass of PSR J0740+6620 is $2.08_{-0.07}^{+0.07}M_{\odot}$ at $68.3\%$ credible interval, which is reduced by about $2.8\%$ compared to the result given in Ref. [34]. This model-averaged value remains the largest of all other precisely measured pulsar masses determined with the Shapiro delay to date.

There is also a special class of MSPs called black widows (and redbacks), the masses of which are higher. In black-widow systems, a millisecond pulsar is accompanied by a low-mass, few $0.01M_{\odot}$ companion, which is bloated and strongly irradiated by the pulsar, leading to outflows strong enough to eclipse the pulsar signal for significant fractions of the orbit [36]. The authors in Ref. [36] present evidence that the black-widow pulsar, PSR B1957+20, has a high mass. Their best-fit pulsar mass is $2.40\pm 0.12M_{\odot}$. However, as it was stated in Ref. [81], there are many difficulties to obtain accurate measurements from these ablated companions, and the possibility of even more massive NSs than J0348+0432 is not yet as robust as the results from radio timing.

These mass measurements provide a strict restriction on the EoS of the cold QCD matter, which can’t be too “soft” and should lead to the maximum mass of NSs is about $2$ times the solar mass. Otherwise, the high masses of the above NSs cannot be explained. Many soft EoSs have already been ruled out through these mass measurements.

The measurements of the radii of the NSs are more difficult than that of the masses. In the past decade, several different techniques have been employed in the NS radius measurements. Almost all of the methods that are currently used base on the detection of thermal emission from the surface of the stars. People either measure its apparent angular size or detect the effects of the NS space-time on this emission to extract the information of the radius. These methods can be divided into two kinds: spectroscopic measurements and timing measurements. The ongoing NASA’s NS Interior Composition Explorer Mission (NICER) [86] uses the latter method (pulse profile modeling) where a large number of counts from a small number of rotation-powered pulsars which emit thermally from their polar caps are collected. Combined with the refined theoretical models that have been developed to analyze these waveforms, those data will provide the measurement of radii and EoSs of the NSs. The results have been published in Refs. [37, 38, 39, 40], in which they also use the data from the XMM-Newton telescope.

In 2019, the authors in Ref. [37] derived that for the isolated $205.53\mathrm{Hz}$ millisecond pulsar PSR J0030+0451, the estimated equatorial circumferential radius is $R_{\mathrm{e}}=13.02_{-1.06}^{+1.24}\mathrm{km}$ and the gravitational mass is $M=1.44_{-0.14}^{+0.15}M_{\odot}$ at $68\%$ credible level. In the same year, another group presented different values for PSR J0030+0451, inferred mass $M$ and equatorial radius $R_{\mathrm{eq}}$ of which are $1.34_{-0.16}^{+0.15}M_{\odot}$ and $12.71_{-1.19}^{+1.14}\mathrm{km}$, respectively, where the credible interval bounds are approximately the $16\%$ and $84\%$ quantiles in marginal posterior mass, given relative to the median [38].

In 2021, the authors in Ref. [39] found that the equatorial circumferential radius of PSR J0740+6620 is $13.7_{-1.5}^{+2.6}\mathrm{km}$ at $68\%$ credibility based on fits of rotating hot spot patterns to NS Interior Composition Explorer (NICER) and X-ray Multi-Mirror (XMM-Newton) X-ray observations. On the same day, another group constrained the equatorial radius and mass of PSR J0740+6620 to be $12.39_{-0.98}^{+1.30}\mathrm{km}$ and $2.072_{-0.066}^{+0.067}M_{\odot}$, respectively, each reported as the posterior credible interval bounded by the $16\%$ and $84\%$ quantiles [40].

There is another way to study the NSs, in which people observe the binary merger events involving NSs. On August 17, 2017, the Advanced LIGO and Advanced Virgo gravitational-wave detectors made their first observation of a binary NS inspiral [87]. The signal, GW170817, was detected. Thereafter the multi-messenger observations of this binary NS merger are finished by telescopes and observatories ranging basically over the whole spectrum of electromagnetic waves [88]. In 2018, under the hypothesis that both bodies in the GW170817 event are NSs that are described by the same equation of state and have spins within the range observed in Galactic binary NSs, the authors in Refs. [41] employed two methods to give an analysis of the radii and EoS of the NSs. From the LIGO and Virgo data alone and using equation-of-state-insensitive relations between various macroscopic properties of the NSs, they measured the two NS radii as $R_{1}=10.8_{-1.7}^{+2.0}\mathrm{km}$ for the heavier one and $R_{2}=10.7_{-1.5}^{+2.1}\mathrm{km}$ for the lighter one at the $90\%$ credible level. Using an efficient parametrization of the defining function $P(\epsilon)$ of the equation of state itself and requiring that the equation of state supports NSs with masses larger than $1.97M_{\odot}$, they further derived $R_{1}=11.9_{-1.4}^{+1.4}\mathrm{km}$ for the heavier one and $R_{2}=11.9_{-1.4}^{+1.4}\mathrm{km}$ for the lighter one at the $90\%$ credible level. They also obtained constraints on $P(\epsilon)$ at supranuclear densities, with pressure at twice nuclear saturation density measured at $3.5_{-1.7}^{+2.7}\times 10^{34}\,\mathrm{dyn}\,{\mathrm{cm}}^{-2}$ at the $90\%$ level. The tidal deformability of a $1.4M_{\odot}$-NS could be estimated to be $\Lambda_{1.4}=190_{-120}^{+390}$ at the $90\%$ level when a common but unknown EoS is imposed. This measurement value of $\Lambda_{1.4}$ is very interesting: contrary to the constraint from the mass measurement, it suggests that “soft” EoSs such as APR4 are favored over “stiff” EoSs such as H4 or MS1. So for an EoS that satisfies both the mass measurement and the tidal deformability measurement, it should be stiff but not too stiff.

First identified in data from two detectors, LIGO Livingston and Virgo on 2019 August 14, GW190814 is the signal of a compact binary coalescence with the most unequal mass ratio yet measured with gravitational waves: $q=0.112_{-0.009}^{+0.008}$ [42]. The primary component of GW190814 is conclusively a black-hole (BH) with mass $m_{1}=23.2_{-1.0}^{+1.1}M_{\odot}$. Due to the lack of measurable tidal deformations and the absence of an electromagnetic counterpart, the secondary component may be an NS or a BH. However, comparing with the maximum NS mass predicted by studies of GW170817’s remnant indicate it’s an NS by current knowledge of the NS equation of state, and by electromagnetic observations of NSs in binary systems [42]. If it’s true, this source with the mass of $2.59_{-0.09}^{+0.08}M_{\odot}$ is the heaviest NS ever observed in a double compact-object system. Then the corresponding constraints on the radius and tidal deformability of a canonical $1.4M_{\odot}$-NS are $R_{1.4}=12.9_{-0.7}^{+0.8}\mathrm{km}$ and $\Lambda_{1.4}=616_{-158}^{+273}$ [42].

Finally, we conclude all the data in this section in Tab. 1 and Tab. 2.

Firstly we introduce the framework of the Einstein-Maxwell-dilaton (EMD) system, which comes back to the gravity-dilaton coupling system at zero chemical potential. As the background action, the EMD system is widely used in the holographic QCD models. It can be used to describe the QCD confinement-deconfinement phase diagram [89, 90], the effect of the magnetic field on the QCD phase transition [91], the rotating effect on deconfinement phase transition [92], the thermodynamic properties of QCD and the location of the critical endpoint (CEP) [93], the critical behavior near the CEP [94, 95], and the spectra glueballs and corresponding EoS [96].

Including hadrons excitation, the total action of the $5$-dimensional holographic QCD model is

where $S_{b}^{s}$ is the action for the background in the string frame, and $S_{m}^{s}$ is the action that describes hadrons in the string frame.

The EMD action in the string frame is denoted by $S_{b}^{s}$:

where $s$ denotes the string frame, $\kappa_{5}^{2}=8\pi G_{5}$, $G_{5}$ is the $5$-dimensional Newton constant. $g^{s}$ is the determinant of the metric in the string frame: $g^{s}=\det\left(g_{MN}\right)$. The metric tensor in the string frame can be extracted from

where $L$ is the curvature radius of the asymptotic $AdS_{5}$ space-time. For simplicity and without loss of generality, we assume $L=1$ in the following calculations. $R^{s}$ is the Ricci curvature scalar in the string frame. The $5$-dimensional scalar field $\Phi(z)$ is the dilaton field that depends only on the coordinate $z$. $F_{MN}$ is the field strength of the $U(1)$ gauge field $A_{M}$:

The $5$-dimensional vector field $A_{M}$ is dual to the baryon number current. The function $h(\Phi)$ describes the coupling strength of $A_{M}$ in the theory, and the function $V^{s}(\Phi)$ represents the potential of the dilaton field in the string frame. $h(\Phi)$ and $V^{s}(\Phi)$ are the functions that depend only on the value of $\Phi$.