Hybrid star properties from an extended linear sigma model
aa r X i v : . [ nu c l - t h ] F e b Hybrid star properties from an extended linear sigma model
J´anos Tak´atsy, P´eter Kov´acs and Gy¨orgy Wolf
Institute for Particle and Nuclear Physics, Wigner Research Center for Physics,Hungarian Academy of Sciences, H-1525 Budapest, Hungary ∗ The equation of state provided by effective models of strongly interacting matter should complywith the restrictions imposed by current astrophysical observations on compact stars. Using theequation of state given by the (axial-)vector meson extended linear sigma model, we determine themass-radius relation and study whether these restrictions are satisfied under the assumption thatmost of the star is filled with quark matter. We study the dependence of the mass-radius relationon the parameters of the model.
I. INTRODUCTION
By the recent astrophysical observation of heavy neutron stars [1] and neutron star collisons [2] the interest forstudying dense strongly interacting matter increased substantially. In terrestial experiments ALICE at CERN, andPHENIX and STAR at RHIC [3] explored the strongly interacting matter at low density and high temperature. Inthis case, the situation is also satisfactory on the theoretical side since lattice calculations are applicable. At highdensity the theoretical investigations are not at such a satisfactory level, therefore, effective models are needed in thisregion where the existing experimental data (NA61 at CERN, BES/STAR at RHIC) are scarce and have rather badstatistics. Soon to be finished experimental facilities (NICA at JINR and CBM at FAIR) are designed to explore thisregion more precisely [4, 5].Neutron stars are on the one side a challenge requiring to understand their properties i.e mass, radius, tidaldeformability, structure etc. On the other hand, they can provide information about the dense, strongly interactingmatter in a region of the phase diagram that is inaccessible to terrestrial experiments [6–8]. Since the Tolman-Oppenheimer-Volkoff (TOV) equations [9, 10] provide a direct relation between the equation of state (EoS) of thecompact star matter and the mass-radius (M-R) relation of the compact star, these data can help to select thoseeffective models, used to describe the strongly interacting matter, whose predictions are consistent with compact starobservables. For example, the EoS must support the existence of a two-solar-mass neutron star with star radii in thepermitted radius window of 11-12.5km, cf. [11, 12].Based on the above considerations, we investigate mass-radius sequences given by the EoS obtained for the quarkphase in [13] from the N f = 2 + 1 flavor extended linear sigma model introduced in [14]. For the hadronic phase weuse for the outer crust the BPS EoS [15], for the inner crust the NV EoS [16] and for the outer core until the phasetransition we use the TNI2u [17] EoS. The model used here should be regarded as a rather crude approximation sincethe hadron-quark phase transition is introduced in a very ad hoc way. The present work has to be considered only asour first attempt to study the problem.The paper is organized as follows. In Section II we present the model and discuss how its solution was obtainedin [13], which reproduced quite well some thermodynamic quantities measured on the lattice. It can be used in thepresence of a vector meson introduced here to realize the short-range repulsive interaction between quarks in thesimplest possible way. In Section III we compare our results for the EoS and the M-R relation (star sequences), forvarious values of the unknown model parameters, such as the vector coupling g v , the quark-hadron crossover transitiondensity ¯ ρ . We draw the conclusions in Section IV and discuss possible ways to improve the treatment of the model. II. METHODS
For the EoS of the hadronic phase we use models from nuclear physics. For very low density, (outer crust, ρ < . ρ )we use the BPS EoS [15] that corresponds to a Coulomb lattice of different nuclei embedded in a gas of electrons andthat has a remarkable influence on the mass-radius relation of the star.For the region of 0 . ρ ≤ ρ < ρ (inner crust) we use the NV EoS [16] and above this until the phase transitionwe use the TNI2u or TNI3u [17, 18] EoS, which differ in their stiffness (K=250 MeV or K=300 MeV, respectively). ∗ Electronic address: [email protected]
These EoS’s even with their derivatives go over each other continuously, so there is no need of any treatment at theconnections and the exact density of the transitions are irrelevant. The model used in this paper for the quark phaseis an N f = 2 + 1 flavor (axial)vector meson extended linear sigma model (eLSM). The Lagrangian and the detaileddescription of this model, in which in addition to the full nonets of (pseudo)scalar mesons the nonets of (axial)vectormesons are also included, can be found in [13, 14]. The model contains three flavors of constituent quarks, with kineticterms and Yukawa-type interactions with the (pseudo)scalar mesons. An explicit symmetry breaking of the mesonicpotentials is realized by external fields, which results in two scalar expectation values, φ N and φ S . Compared to [13], the only modification to the model is that we include in the Lagrangian a Yukawa term − g v √ γ µ V µ Ψ, which couples the quark field Ψ T = ( u, d, s ) to the U V (1) symmetric vector field, that is V µ = √ diag( v + v √ , v + v √ , v − √ v ) µ . The vector meson field is treated at the mean-field level as in theWalecka model [19], but as a simplification we assign a nonzero expectation value only to v : v µ → v δ µ and v µ → v µ turns out to be m v = 871 . T ≈ . T = 0 MeV. We have three background fields φ N , φ S and v , and the calculation of the grand potential, Ω, is performed using a mean-field approximation, in whichfermionic fluctuations are included at one-loop order, while the mesons are treated at tree-level. Hence, the grandpotential can be written in the following form:Ω( µ q ; φ N , φ S , v ) = U mes ( φ N , φ S ) − m v v +Ω (0)vac q ¯ q ( φ N , φ S ) + Ω (0)matter q ¯ q (˜ µ q ; φ N , φ S ) , (1)where ˜ µ q = µ q − g v v is the effective chemical potential of the quarks, while µ q = µ B / µ B being the baryochemical potential. On the right hand side of the grand potential (1), the termsare (from left to right): the tree-level potential of the scalar mesons, the tree-level contribution of the vector meson,the vacuum and the matter part of the fermionic contribution at vanishing mesonic fluctuating fields. The fermionicpart is obtained by integrating out the quark fields in the partition function. The vacuum part was renormalized atthe scale M = 351 MeV. More details on the derivation can be found in [13].The background fields φ N , φ S , and v are determined from the stationary conditions ∂ Ω ∂φ N (cid:12)(cid:12)(cid:12)(cid:12) φ N = ¯ φ N = ∂ Ω ∂φ S (cid:12)(cid:12)(cid:12)(cid:12) φ S = ¯ φ S = 0 and ∂ Ω ∂v (cid:12)(cid:12)(cid:12)(cid:12) v =¯ v = 0 , (2)where the solution is indicated with a bar. Since ∂/∂v = − g v ∂/∂ ˜ µ q , the stationary condition with respect to v reads ¯ v ( φ N , φ S ) = g v m v ρ q (˜ µ q (¯ v ); φ N , φ S ) , (3)where ρ q ( x ; φ N , φ S ) = − ∂ Ω (0)matter q ¯ q ( x ; φ N , φ S ) /∂x. When solving the model, the value of g v does not effect the properties of the system at zero chemical potential,therefore, we cannot fit it to the vacuum properties, so we leave it in the range of [0 ,
3) as a free parameter, whilefor the remaining 14 parameters of the model Lagrangian we use the values given in Table IV of [13]. These valueswere determined there by calculating constituent quark masses, (pseudo)scalar curvature masses and decay widthsat T = µ q = 0 and comparing them to their experimental PDG values [20]. Parameter fitting was done using amultiparametric χ minimization procedure [21]. In addition to the vacuum quantities, the pseudocritical temperature T pc at µ q = 0 was also fitted to the corresponding lattice result [22, 23]. We mention that the model also containsthe Polyakov-loop degrees of freedom (see [13] for details), but to keep the presentation simple we omitted them from(1), as at T = 0 they do not contribute to the EoS directly. Their influence is only through the value of the modelparameters taken from [13]: since they influence the value of T pc used for parameterization.The solution of the model at g v = 0, obtained in [13], can be used to construct the solution at g v = 0 (see e.g. Ch.2.1 of [25]): one only has to interpret the solution at g v = 0 as a solution obtained at a given ˜ µ q and determine µ q atsome g v = 0 using (3). To see that the solutions ¯ φ N , S for g v = 0 can be related to the solution obtained at g v = 0 , where ¯ v = 0, consider the grand potential at g v = 0. This potential, denoted as Ω , is subject to the stationaryconditions (2) with solutions ¯ φ , S ( µ q ). It is then easy to see using (1), that the solution ¯ φ N , S ( µ q ) of (2) satisfies¯ φ N , S ( µ q + g v v ) = ¯ φ , S ( µ q ) or, changing the variable µ q to ˜ µ q the relation becomes¯ φ N , S (˜ µ q + g v v ) = ¯ φ , S (˜ µ q ) . (4) p [ G e V /f m ] ε [GeV/fm ]g v =0.0g v =1.5g v =2.5TNI3u g v =2.5 0 0.5 1 1.5 2 2.5 3 9 10 11 12 13 14 15 16 17 M [ M S un ] R [km]g v =0.0g v =1.5g v =2.5TNI3u g v =2.5causality rotation FIG. 1: The equation of state (left) and the M-R curves (right) for different values of g v using ¯ ρ = 4 ρ and Γ = 1 with TNI2u(K=250 MeV) nuclear force for different g v ’s. The g v = 2 . . . M ⊙ ( e.g. [11, 12]) is represented by the horizontal line. The different shaded regions are excluded by causality R > . GM/c and therotational constraint based on the 716 Hz pulsar J1748-2446ad, M/M ⊙ > . · − ( R/ km) [24] The value of the grand potential Ω at the extremum can be given in terms of the value of the grand potential with g v = 0, that is Ω , at its extremum. Using that the extrema of Ω (˜ µ q , φ N , φ S , v = 0) are ¯ φ and ¯ φ , one hasΩ( µ q ; ¯ φ N ( µ q ) , ¯ φ S ( µ q ) , ¯ v )= Ω (˜ µ q , φ (˜ µ q ) , φ (˜ µ q ) , v = 0) − m v ¯ v , (5)where ¯ v ≡ ¯ v ( ¯ φ N ( µ q ) , ¯ φ S ( µ q )) = g v m v ρ q (˜ µ q ; ¯ φ N ( µ q ) , ¯ φ S ( µ q ))= g v m v ρ q (˜ µ q ; ¯ φ (˜ µ q ) , ¯ φ (˜ µ q )) (6)and µ q = ˜ µ q + g v ¯ v . The pressure p and the energy density ε are calculated from the grand potential. At v = 0 theycan be expressed in terms of the pressure obtained at g v = 0: p ( µ q ) = Ω( µ q = 0; ¯ φ N (0) , ¯ φ S (0) , ¯ v (0)) − Ω( µ q ; ¯ φ N , ¯ φ S , ¯ v )= Ω (˜ µ q = 0; ¯ φ (0) , ¯ φ (0) , v = 0) − Ω (˜ µ q ; ¯ φ , ¯ φ , v = 0) + 12 m v ¯ v = p (˜ µ q ) | g v =0 + 12 m v ¯ v , (7)where ¯ v = g v m v ρ q (˜ µ q ; ¯ φ (˜ µ q ) , ¯ φ (˜ µ q )), and then ε = − p + µ q ρ q , where µ q = ˜ µ q + g v ¯ v .To connect the nuclear phase (TNI2u) and the quark phase EoS we assume a hadron-quark crossover, and use the P -interpolation [26]: P ( ρ ) = P H ( ρ ) f − ( ρ ) + P Q ( ρ ) f + ( ρ ) , (8) f ± ( ρ ) = 12 (cid:18) ± tanh (cid:18) ρ − ¯ ρ Γ (cid:19)(cid:19) (9) p [ G e V /f m ] ε [GeV/fm ] Γ =0.2 Γ =0.6 Γ =1.0TNI3u Γ =1.0 0 0.5 1 1.5 2 2.5 3 9 10 11 12 13 14 15 16 17 M [ M S un ] R [km] Γ =0.2 Γ =0.6 Γ =1.0TNI3u Γ =1.0causality rotation FIG. 2: The equation of state (left) and the M-R curves (right) using ¯ ρ = 4 ρ and g v = 2 . . ε ( ρ ) = ε H ( ρ ) f − ( ρ ) + ε Q ( ρ ) f + ( ρ ) + ∆ ε (10)∆ ε = ρ Z ρ ¯ ρ ( ε H ( ρ ′ ) − ε Q ( ρ ′ )) g ( ρ ′ ) ρ ′ dρ ′ g ( ρ ′ ) = 12Γ cosh − (cid:18) ρ ′ − ¯ ρ Γ (cid:19) . (11)where ¯ ρ is the transition density and Γ is the width of the crossover transition. Note that neither the transitiondensity - ¯ ρ - nor the order of the phase transition between the hadron and the quark phases are known, therefore, wewill investigate the dependence of the M-R curves on these parameters, as well.By solving the TOV equation [9, 10] using a fourth-order Runge-Kutta differential equation integrator with adaptivestepsize-control for a specific EoS, one can obtain the radial dependence of the energy density (and thus of the pressure)for a certain central energy density, ε . One can then determine the mass and radius of the compact star for thatcentral energy density. By changing ε , one gets a sequence of compact star masses and radii parameterized by thecentral energy density, as shown in Fig. 1 for various models. The sequence of stable compact stars ends when themaximum compact star mass is reached with increasing central energy density. III. RESULTS
As described above we use an equation of state (EoS) sewn together from 3 nuclear (BPS, NV and TNIu) and fromthe EoS for the quark phase. We consider that the low density (below ρ < ρ ) nuclear EoS’s are safe, their ingredientsare based on well known and tested nuclear physics data, and their slight modifications change only the propeties ofthe crust, and do not show up significantly in those neutron star properties which we study here. Here we investigatethe effect of the not very well determinable parameters ¯ ρ, Γ , g v on the M-r curves and on the tidal deformability. Westudy moreover, what the effect of the stiffnes of the hadronic EoS is, so we compare results using TNI3u (K=300MeV) and TNI2u (K=250 MeV) interactions.Inclusion of the repulsive interaction between quarks in the eLSM renders the EoS stiffer compared to the g v = 0case, as expected, and at lower densities one can observe the effect of the stiffnes of the hadronic phase. It is worthnoting that relatively small differences in the p ( ǫ ) lead to significant differences in the M-R curves, as we shall see inFigs. 1, 2 and 3. It can also be seen that the maximum possible compact star mass is lower for models with less stiffEoSs. The highest mass compact star has a mass of ∼ . M ⊙ with g v = 0. The effect of hadronic compressibility israther large, and the higher stiffnes brings the M-R curve closer to the observed radius window.The effect of the crossover width Γ is sizeable for either on the EoS or in the M-R sequence 2, however, it does noteffect the maximal neutron star mass and the corresponding radius. The EOS for Γ = 0 . c s >
1) around p [ G e V /f m ] ε [GeV/fm ] ρ tr =2 ρ o ρ tr =4 ρ o TNI3u ρ tr =4 ρ o M [ M S un ] R [km] ρ tr =2 ρ o ρ tr =3 ρ o ρ tr =4 ρ o TNI3u ρ tr =4 ρ o causality rotation FIG. 3: The equation of state (left) and the M-R curves (right) for different values of ¯ ρ using g v = 2 . ρ ’s. The ¯ ρ = 4 ρ case is calculated with TNI3u (K=300 MeV), as well. The shadedregions are explained in Fig.1. Λ M [M
Sun ] ρ =2 ρ ρ =3 ρ ρ =4 ρ TNI3u ρ =4 ρ Λ GW170817 Λ M [M
Sun ]g v =0.0g v =1.5g v =2.5TNI3u g v =2.5 Λ GW170817
FIG. 4: Tidal deformability parameter, Λ as a function of the neutron star mass for various crossover densities with g v =2 . , Γ = 1 (left), and various coupling strengths between the quarks and vector mesons, g v s with Γ = 1 , ¯ ρ = 4 ρ (right). Theexperimental point is from [2]. the transition point, so that solution is not physical.As it could be expected the variation of transition density ¯ ρ has a large effect on the EoS and on the M-R curve(Fig. 3), too. Since the hadronic part of the EoS is softer than the quark one, the maximal neutron star mass ishigher for smaller ¯ ρ . With decreasing ¯ ρ also the radius corresponding to the maximal mass neutron star is increasing.So in this model the parameter set: g v = 2 . , Γ = 0 − , ¯ ρ = 2 ρ − ρ and the nuclear TNI2u interaction (with K=250MeV compressibility) fulfills the M-R constraints of neutron star observations.The tidal deformability parameter Λ as a function of the neutron star mass is shown for various parameter setsin Fig. 4. Since the experimental constraint, 70 < Λ . M ⊙ <
580 is not very conclusive it does not constrain ourparameterizing. However, more precise estimate of Λ could rule out models and parameter sets.
IV. CONCLUSIONS
We employed the zero-temperature EoS to determine the mass-radius relation of compact stars. Including therepulsive interaction in the eLSM model makes the EoS stiff enough to support in some narrow range of the Yukawacoupling compact stars with masses larger than 2 M ⊙ and in the permitted radius window of 11 . . M = 2 M ⊙ .In the future, we would like to go beyond the mean-field approximation, used for the mesons in the eLSM, in away that takes into account the effect of fermions in the mesonic fluctuations. At lowest order, this can be done byexpanding to quadratic order the fermionic determinant obtained after integrating out the quark fields in the partitionfunction and performing the Gaussian integral over the mesonic fields. In order to have a physically more reliabledescription, we also plan to improve the treatment of the interaction between vector mesons and quarks employedhere. Furthermore, to reduce the uncertainty emerging from the artificially performed phase transition, we plan toderive the phase transition parameters from the model itself. V. ACKNOWLEDGMENT