Massive neutron stars with holographic multiquark cores
aa r X i v : . [ nu c l - t h ] J u l Massive neutron stars with multiquark cores
Sitthichai Pinkanjanarod
1, 2, ∗ and Piyabut Burikham † High Energy Physics Theory Group, Department of Physics,Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand (Dated: July 22, 2020)Phases of nuclear matter are crucial in determination of the physical properties of neutronstars (NS). In the core of NS, it is possible that the density and pressure become so large thatthe nuclear matter undergoes phase transition into a deconfined phase, consisting of quarks andgluons and their colour bound states. Even though the quark-gluon plasma has been observedin ultra-relativistic heavy-ion collisions[1, 2], it is still unclear whether exotic quark matter existsinside neutron stars. Recently the result from the combination of various perturbative theoreti-cal calculations with astronomical observations[3, 4] shows that (exotic) quark matter could existinside the cores of neutron stars above 2.0 solar masses ( M ⊙ ) [5] However, due to the nonperturba-tive characteristic of interactions between quarks and gluons in the deconfined phase, perturbativeQCD (pQCD) has limitation due to the possibly large coupling of the quark-gluon soup in suchdense environment. We revisit the holographic model in Ref. [6, 7] and implement the equationof states (EoS) of multiquark nuclear matter interpolating from the high-density pQCD EoS andmatching with the nuclear EoS known at low densities. It is found that the equations of the statesof multiquark nuclear phase provide the missing link between the constraints in both high and lowenergy density regions and give the mass of NS with multiquark core within the observational range.This shows evidence for the exotic multiquark core inside massive neutron stars. The NS with mul-tiquark core at the maximum mass could have masses in the range 2 . − . M ⊙ and radii 10 −
1. INTRODUCTION
In the final fate, a star collapses under its own gravitywhen the internal pressure from nuclear fuel is depleted.The quantum pressure of fermions kicks in to rescue. Ifthe mass of the star is below 0 . > . M ⊙ .Given the star is more massive than the upper mass limitof the neutron star, it is believed that it would collapseinto a black hole eventually. However, there is a possibil-ity that under extreme pressure and density, the quarkswithin hadrons would become effectively deconfined fromthe localized hadrons but still confined by gravity withinthe star. The deconfined phase of quarks could gener-ate larger pressure to sustain even more massive neutronstars or even quark stars.Even in the deconfined phase, quarks can still formbound states via the remaining Coulomb-like interactionmediated by gluons, the multiquark states. Observa-tions of multiquark candidates such as pentaquark andtetraquark have been accumulated for decades, see e.g.Ref. [10] for the latest report. It is only natural to imag-ine an abundance of multiquarks in the core of densestars where the deconfined quarks are extremely com-pressed tightly close together. Due to the nonpertur-bative nature of the strong interaction, the difficulty oflattice QCD approach when dealing with finite baryon ∗ Electronic address: [email protected], [email protected] † Electronic address: [email protected], [email protected] density, and a reliability issue of MIT bag as a tool tostudy the behaviour of the deconfined quarks and gluonsin the dense star, it is therefore interesting to use theequation of state of the deconfined nuclear matter fromthe holographic model as a complementary tool to otherconventional approaches to investigate the properties ofthe dense star [6, 7].Recent work [5] reveals a potential double-power-lawequation of states (EoS) interpolating between low andhigh density EoS calculated from the Chiral EffectiveField Theory (CET) and perturbative QCD. The em-pirical EoS gives adiabatic index and sound speed char-acteristic of the quark matter phase, showing evidenceof quark core within the NS. In this work, we revisit theholographic model investigated in Ref. [7] and match theEoS of multiquark nuclear matter with the low and highdensity EoS and demonstrate that it can interpolate wellbetween the two regions. The masses of NS with multi-quark core are consistent with current observations, al-lowing NS with
M > M ⊙ with radii around 10 − . − . M ⊙ , a potential can-didate for the object recently found by LIGO/Virgo [11].This work is organized as the following. Section 2 re-views holographic model studied in Ref. [7] and presentsthe EoS of multiquark nuclear matter. Section 3 sum-marizes the EoS from CET and piecewise polytrope usedin the interpolation and EoS of the multiquark core inthe high density region. Mass-radius diagram, mass-central density relation and thermodynamic propertiesof NS with multiquark core are explored in Section 4.Section 5 concludes our work.
2. HOLOGRAPHIC MULTIQUARK AND THEEOS
Within the framework of gauge-gravity duality fromsuperstring theories, bound states of quarks in theboundary gauge theory can be described holographicallyby strings and branes. Mesons can be expressed as astring hanging in the bulk with both ends locating atthe boundary of the AdS space[12] while baryons canbe represented by D p -brane wrapped on the S p with N c strings attached and extending to the boundary of thebulk space[13, 14]. The gauge theory from the originalAdS/CFT duality is still apart from the actual gauge the-ory described by QCD. The gauge theory from gravitydual that captures most features of QCD is the Sakai-Sugimoto (SS) model[15, 16]. In this model, hadronsnaturally exist in the confined phase however, anotherkind of bound states of quarks can also occur in the de-confined phase at the intermediate temperatures abovethe deconfinement, the multiquark states [6, 7]. See e.g.Ref. [17] for a concise review of holographic multiquarks. The configuration in the SS model consists of D4-branebackground and D8/D8 flavor branes. N c D4-branes pro-vides 4D SU( N c ) Yang-Mills gauge theory holographi-cally. On the other hand, N f D8/ N f D8 flavor branesprovide a description for confinement/deconfinementphase transition depending on the configuration of thebranes. In terms of symmetry, N f D8/ N f D8 flavorbranes poses the global symmetries U( N f ) L and U( N f ) R which can fully describe U( N f ) L × U( N f ) R chiral sym-metry breaking when the D8 and D8 are connected. Atlow energy, the classical solution of the field configura-tion on the gravity side suggests a cigar-like shape for thecompactified spatial direction of a confined background.At high temperature, the cylindrically compactified back-ground spacetime with flavor branes in parallel embed-ding is preferred, therefore the broken chiral symmetryis restored and the corresponding nuclear matter phasebecomes deconfined [18].In the deconfined phase [6], there are 3 possible config-urations as shown in Fig. 1: (i) the parallel configurationof both D8-branes and D8 representing the χ S -QGP (chi-ral symmetric quark-gluon plasma) and (ii) connectedD8-D8 without sources in the bulk representing the vac-uum with broken chiral symmetry. Another stable con-figuration (iii) is multiquark phase consisting of the con-nected D8-D8 branes with the D4-brane as the baryonvertex submerged and localized in the middle of the D8and D8. The baryon vertex can be attached with ra-dial hanging strings that represent colour charge of themultiquark configuration. L L L u c u u u T u T u T (a) (b) (c) FIG. 1: Different configurations of D8 and D8-branes inthe Sakai-Sugimoto model that are dual to the phases of(a) χ S -QGP, (b) vacuum and (c) multiquark phase. [6] Holographically, the grand canonical potential and thechemical potential of the multiquark matter are givenby [7]Ω = Z ∞ u c du (cid:20) − F f ( u )( u + u n ) (cid:21) − u √ u + n , (1) µ = Z ∞ u c du (cid:20) − F f ( u )( u + u n ) (cid:21) − n √ u + n + 13 u c p f ( u c ) + n s ( u c − u T ) (2)respectively, where u is a radial coordinate of the back-ground metric of the bulk spacetime in the SS modelin a deconfined phase at finite temperature T , f ( u ) ≡ − u T /u , u T = 16 π R T / , R ≡ πg s N c l s , l s is thestring length and g s is the string coupling. u c is the po-sition of the baryon vertex source as shown in Fig. 1. n s is the number fractions of radial strings k r in the unitof N c that represents the colour charges of a multiquarkconfiguration. n ( u ) is the baryon number density whichis a constant of the configuration given by n ( u ) = u ˆ a ′ p f ( u )( x ′ ) + u − (1 − (ˆ a ′ ) ) = const. (3)where x is the compactified coordinate transverse to theprobe D8/D8 branes with arbitrary periodicity 2 πR . Theˆ a = 2 πα ′ ˆ A/ ( R p N f ) is a rescaled version of ˆ A , the orig-inally diagonal U (1) gauge field, where α ′ is a universalRegge slope. The position u c of the vertex is determinedfrom the equilibrium condition of the D8-D4-strings con-figuration (see Appendix A of Ref. [6]). Another constantof the configuration is( x ′ ) = 1 u f ( u ) h f ( u )( u + u n ) F − i − = const. , (4)where F is a function of u c , n , T and n s , given by F = u c f ( u c ) (cid:18) u c + n − n η c f ( u c ) (cid:19) , (5)where η c ≡ (cid:16) u T u c (cid:17) + 3 n s p f ( u c ).Thermodynamic relations of multiquark states can befound in Ref. [7]. The grand potential G Ω can be writtenas dG Ω = − P dV − SdT − N dµ (6)where the state parameters P , V , S , T , and N are thepressure, volume, entropy, temperature, and the totalnumber of particles of the system respectively. Since thechange of volume is not our main concern, we define thevolume density of G Ω , S and N to be Ω, s and n , respec-tively. Therefore, we have, at a particular T and µ , P = − G Ω /V ≡ − Ω( T, µ ) . (7)Assuming that the multiquark states are spatially uni-form, we obtain n = ∂P∂µ ( T, µ ) . (8)Using the chain rule, ∂P∂n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T = ∂µ∂n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T n, (9)so that P ( n, T, n s ) = µ ( n, T, n s ) n − Z n µ ( n ′ , T, n s ) d( n ′ ) , (10)where the regulated pressure is assumed to be zero whenthere is no nuclear matter, i.e. n = 0. In the limit of small n , the baryon chemical potentialin Eqn.(2) can be approximate as µ ≃ µ source + α n − β ( n s ) n , (11)where µ source ≡ u c p f ( u c ) + n s ( u c − u T ) α ≡ Z ∞ u du u − / − f u fu ,β ( n s ) ≡ Z ∞ u du u − / q − f u fu × (cid:20) f u f u − f u (cid:18) − η f − u u (cid:19) + 1 u (cid:21) , and u is the position when x ′ ( u ) = ∞ as shown inFig. 1.By substituting Eqn.(11) into Eqn.(10), the pressurein the limit of small n can be expressed as P ≃ α n − β ( n s )4 n . (12) In the limit of large n and relatively small T , µ ≈ µ source + n / (cid:0) (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) + u c f c (cid:18) − η c f c (cid:19) n − / Γ (cid:0) − (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) (13)where the term from lower limit of integration in Eqn.(2), u c /n approaches zero as n becomes very large. Againby using Eqn.(10), we obtain P ≃ Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) ! n / . (14)Also the energy density can be found via the relation dρ = µdn and the chemical potential is given by µ = Z n η (cid:18) ∂P∂η (cid:19) dη + µ , (15)where µ ≡ µ ( n = 0). The main results from Ref. [7] aresummarized as P = an + bn ,ρ = µ n + an + b n , (16)for small n and P = kn / ,ρ = ρ c + 52 P + µ c "(cid:18) Pk (cid:19) / − n c + kn / c − k n / c (cid:18) Pk (cid:19) / , (17)for large n respectively. For n s = 0 , n c = 0 . , µ c =0 . , a = 1 , b = 0 , µ = 0 . n s = 0 . , n c = 0 . , µ c = 0 . , a =0 . , b = 180 . , µ = 0 . k = 10 − . for both cases reflecting universal behaviour at high den-sity. n c , µ c are the number density and chemical potentialwhere the EoS changes from large n to small n .
3. EOS OF THE NS
The structure of neutron star can be investigatedthrough observations and the modeling of the stronglyinteracting hadronic matter entirely from the low-densitycrust to the high-density core inside the star. However,with the absence of accurate and direct first-principlescalculation at densities above the nuclear matter satu-ration (baryon number) density n ≈ .
16 fm − , an ac-curate determination of the state of matter inside NScores is still unclear. Fortunately, recent observationsstart offering empirical constraints in both opposing lowdensity and high density limits therefore not only themodel-independent approach [5] to the problem has be-come feasible but also could provide a proper directionfor the model-building approach. At low density, there is a limitation that comesfrom the well-studied NS crust region[19] to the den-sity n CET ≡ . n , where matter occupies the hadronic-matter phase using chiral effective field theory (CET)which provides the EoS to good precision, currently bet-ter than ±
24% [20, 21].For very low density crust, EoS can be found from Ta-ble 7 of Ref. [22], it can be fit with the following functions, P ( ρ ) = κ a ρ Γ a + κ b ρ Γ b + κ c ρ Γ c , for 1 . × − < ρ < . ,P ( ρ ) = κ e ρ + α e , for ρ < . × − (18)where (Γ a , Γ b , Γ c ) = (1.41157, 1.39774, 1.43333) and( κ a , κ b , κ c ) = ( − . κ e , α e ) = (280.00, − . × − ) for the pressure anddensity expressed in the unit of MeV/fm .For slightly higher density in the range87 .
07 MeV/fm < ρc < . of Table5 of Ref. [22], the energy density and pressure of thenuclear matter can be expressed as ρ (¯ n ) c /T = a ¯ n / + b ¯ n + c ¯ n γ +1 , (19)where ¯ n = n/n and P (¯ n ) /T = 23 n a ¯ n / + n b ¯ n + γn c ¯ n γ +1 , (20)for T = 36 .
84 MeV and a = 176 . , b = − . , c = 100 . a , b , c )= (1.55468, − . − . , Γ and Γ , can be written as follows, P ( ρ ) = κ ρ Γ , for ρ ≤ ρ ≤ ρ ,P ( ρ ) = κ ρ Γ , for ρ ≤ ρ ≤ ρ ,P ( ρ ) = κ ρ Γ , for ρ ≤ ρ ≤ ρ max . (21)With mass density ρ = mn ,1. the stiff EoS (red dashed line in Fig. 2)has the exponents (Γ , Γ , Γ ) = (4 . , . , . ρ , ρ , ρ max ) = (1 . ρ s , . ρ s , . ρ s ) and( κ , κ , κ ) = (11.6687, 51.7666, 2.56345).2. the intermediate EoS (orange dashed line in Fig. 2)has the exponents (Γ , Γ , Γ ) = (4 . , . , . ρ , ρ , ρ max ) = (3 . ρ s , . ρ s , . ρ s ) and( κ , κ , κ ) = (2.89711, 1.30607, 1.07402). 3. the soft EoS (green dashed line in Fig. 2)has the exponents (Γ , Γ , Γ ) = (1 . , . , . ρ , ρ , ρ max ) = (2 . ρ s , . ρ s , . ρ s ) and( κ , κ , κ ) = (0.0321845, 2.63607, 0.572502),when both pressure and density are in GeV/fm and thedensity scale is ρ s = 0 . / fm .Since the mass and moment of inertia of the dense neu-tron star are mostly determined by the mass in the region ρ > ρ s /
2, the cutoff may be applied at ρ ( r = R ) = ρ s / R of the star. A smaller cutoff will leadto significantly larger radius with larger uncertainty butthe mass and moment of inertia relevant to rotational dy-namics will remain the same with negligible changes (only0 . − .
7% mass increase) since the density in the ex-tended region is negligibly small.
At high densities inside the NS core, baryons would betightly compressed, quarks and gluons would be so closetogether that individual quark and gluon are deconfinedfrom a single baryon and yet the interaction could stillbe sufficiently strong. The gluons and quarks become de-confined but could still form bound states of multiquark.The multiquarks can possess colour charges in the de-confined phase while keeping the star colour singlet intotality, similar to ionized gas of positive and negativeelectric charges with total neutrality. In the multiquarkmodel of Ref. [6], the colour charge is quantified by thenumber fractions of hanging radial strings n s . For ex-treme density at relatively moderate temperature, thedeconfined phase of quarks and gluons should be in themultiquark phase instead of the pure gas of quarks andgluons where perturbative QCD (pQCD) is applicable.The multiquark EoS (16),(17) are expressed in dimen-sionless form. Apart from the colour charge parameter n s , there is only one parameter we can choose to deter-mine the entire behaviour of the EoS, the energy densityscale ǫ s which will give the physical density and pressure ρǫ s , P ǫ s . After choosing ǫ s , the corresponding distancescale of the SS model is fixed by r = (cid:0) Gǫ s /c (cid:1) − / .The pQCD calculation, for the deconfined quarks andgluons of Ref. [23, 24] is also displayed for comparison inFig. 2. The results, Fig. 1 and 2, of Ref. [5] suggest that thereis a double-power-law type EoS interpolating between thehigh and low density EoS given by pQCD and CET. Onesuch candidate can be found in early work of holographicSS model [7] where the multiquark phase is shown todominate at large density and moderate temperature.There is only one parameter to be chosen in this model,the energy density scale ǫ s . By adjusting ǫ s = 23 . ● ●● ●●●● ● ● ●● ● ● ●● ● ● ● n s = ● stiff ● inter. ● softnucl.pQCD0.1 0.5 1 5 100.0010.0100.100110 Density ( GeV fm - ) P r e ss u r e ( G e V f m - ) (a) n s = 0 with ǫ s = 23 . , ρ c = 1 . . ●●● ●● ●● ● ●● ● ●● ● ● ●● ● ● ● n s = ● stiff ● inter. ● soft nucl.pQCD Density ( GeV fm - ) P r e ss u r e ( G e V f m - ) (b) n s = 0 . ǫ s = 23 . , ρ c = 0 . . FIG. 2: EoS of multiquark interpolating betweennuclear matter and extreme density regionGeV/fm to give transition density ρ c = 0 . as suggested by the turning point of EoS in Fig. 1 ofRef. [5], a good interpolating EoS of n s = 0 . n s = 0 . n s = 0 .
4. MR DIAGRAM OF NS WITH MULTIQUARKCORE
The Tolman-Oppenheimer-Volkoff equation [7–9] isused in the consideration of mass profile of the NS, dPdr = − ( ρc + P )2 8 πP r + 2 M ( r ) r ( r − M ( r )) ,dM ( r ) dr = 4 πρr , (22) where M ( r ) is the accumulated mass of the star up toradius r . In determination of the mass-radius diagramshown in Fig. 3, we use the multiquark EoS given in(16),(17) for high density region. As the density andpressure go down within the star and reach transitionpoint with either stiff, intermediate or soft EoS, the newpiecewise polytrope EoS (21) is adopted until it reachesthe low density region where the EoS given in (20), (19),and (18) will be used subsequently. From Fig. 2, wefocus our consideration to 5 scenarios: n s = 0 . n s = 0 withtransition to stiff EoS; pure n s = 0 . n s = 0multiquark interpolates worse than n s = 0 .
3. So just forcomparison, we consider only the scenario of the n s = 0case which would provide the highest possible mass whereit is continued with the stiff EoS.From Fig. 3 for NS containing multiquark core withcolour charge fractions n s = 0 . . − . M ⊙ with the radii R ∼ − . n s = 0 . ∼ . M ⊙ with radiusaround 10 km. For star containing multiquark core with n s = 0 (colourless states), the mass could be as large as2 . M ⊙ and radii around 14 km. For pure multiquark starwith no baryon crust and n s = 0 .
3, the maximum massis ∼ . M ⊙ with radius around 11 . − . M ⊙ and 10 km for n s = 0 . . M ⊙ and 7 km for n s = 0 . n s = 0multiquark core continuing to the stiff EoS, the mass andradius of multiquark core is ∼ . − . M ⊙ and 9 km.Note that this multiquark core contains both high andlow density layers governed by (17) and (16).Multiquark EoS contains two power-laws governing athigh and low density, the corresponding multiquark mat-ter is called the multiquark core and crust in Ref. [7],but to avoid confusion we instead label them with “mqh,mql” respectively in this work. Each region gives differ-ent adiabatic indices γ and sound speed c s as shown inFig. 5. Interestingly, γ ≈ .
5) for high (low) densitymultiquark respectively while c s > / c s ≃ . n s = 0 .
3, this is the value slightly above the confor-mal bound obeyed by the typical massless free quarksphase. The adiabatic index γ of the high-density multi-quark (mqh) is very close to 1 (again the conformal limitof free quarks) while the low-density multiquark (mql)has γ ≈ .
5, behaving more similar to the hadronic nu-clear matter, but with colour charges and deconfined.On the other hand, n s = 0 colourless multiquark at highdensity has γ ≃ . , c s . . n s = n s = n s = n s = mqhmqlstiffinter.softnucl.mqhmql2 4 6 8 10 12 140.00.51.01.52.02.53.0 R star ( km ) M s t a r ( M s o l a r ) n s = n s = n s = n s = - log [ρ ] M s t a r ( M s o l a r ) FIG. 3: MR diagram and mass-central density of NSand quark star. The colour label represents thecorresponding nuclear phase at the center of the star.Each point corresponds to a star with mass profileconsisting of subsequent layers of nuclear phases inorder of high to low density: multiquark,polytrope (stiff, intermediate, soft), and CET. The purehypothetical multiquark star has only multiquark layers.
5. CONCLUSIONS AND DISCUSSIONS
The holographic SS model of multiquark nuclear mat-ter has been applied to the inner core of NS at moderatetemperature. The EoS of the multiquark is interpolatedbetween the high density and the low density where theCET is applicable. The energy density scale is fixed oncethe transition density ρ c between the power laws in theempirical EoS is chosen. It is found that multiquark withcolour-charge fraction n s = 0 . . − . M ⊙ and radii10 −
14 km.At higher temperature in the order of trillion Kelvins,the population of multiquarks should become less and thedeconfined phase would consist mainly of weakly coupledquarks and gluons. Holographic models including the SSmodel predict the pressure of this phase to be higherthan the multiquark phase. In newly formed NS or ex- n s = c(cid:0)(cid:1)(cid:2)(cid:3) n s = i(cid:4)(cid:5)(cid:6)(cid:7)(cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13) n s = (cid:14)(cid:15)(cid:16)(cid:17)(cid:18) n s = (cid:19)(cid:20)(cid:21)(cid:22)(cid:23) n s = p(cid:24)(cid:25)(cid:26) m(cid:27) R !" ( k$ ) M %&’( ( M s o l a r ) FIG. 4: MR diagram of multiquark core for each curvein Figure 3. n s = )* & soft + s = ,- & stiff . s = /2 & s = :; & stiff0 2 4 6 8 100.0 <=> ?@A BCD R ( EF ) γ G s = HI & soft J s = KL & stiff M s = NO & PQSTUVW s = XY & stiff0 2 4 6 8 100.00.20.40.60.81.0 Z ( [\ ) c s FIG. 5: The adiabatic index γ = d ln Pd ln ρ and c s of themultiquark core for each scenario, the multiquark coreconsists of two regions with high and low density givenby (17) and (16).otic quark star if the core temperature could reach over atrillion Kelvin, it is possible to have this weakly-coupledquarks and gluons in the most inner core follow by multi-quark layers resulting in even larger mass of the NS mostlikely larger than 2 M ⊙ . For aged NS with lower temper-atures however, we expect only the multiquark phase toexist in the core. As density decreases with radial dis-tance, the multiquark matter undergoes phase transitioninto confined baryonic matter or even coexist in mixedphase. For all scenarios that we consider, the NS withmultiquark core could exist in a wide range of masses M > . M ⊙ with radii around 10 − . n s = 0 . M > . M ⊙ with radii around 14 km for n s = 0.There is a considerable number of observations of NSwith masses above 2 M ⊙ , e.g. Ref. [25–35]. It seems themassive NSs are abundant and our analyses suggest thatthey likely contain the multiquark cores. Acknowledgments
P.B. is supported in part by the Thailand ResearchFund (TRF), Office of Higher Education Commission(OHEC) and Chulalongkorn University under grantRSA6180002. S.P. is supported in part by the SecondCentury Fund: C2F PhD Scholarship, ChulalongkornUniversity.
Appendix A: MR diagram for zero cutoff
In the main text, we use cutoff ρ s / . − .
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