Hybrid Stars in a Strong Magnetic Field
aa r X i v : . [ a s t r o - ph . H E ] N ov EPJ manuscript No. (will be inserted by the editor)
Hybrid Stars in a Strong Magnetic Field
V. Dexheimer , R. Negreiros , and S. Schramm UFSC, Florianopolis, Brazil Gettysburg College, Gettusburg, PA, USA FIAS, Johann Wolfgang Goethe University, Frankfurt, DE Instituto de Fisica, UFF, Av. Gal. Milton Tavares de Souza s/n. Gragoata, Niteroi, 24210-346, BrazilReceived: date / Revised version: date
Abstract.
We study the effects of high magnetic fields on the particle population and equation of stateof hybrid stars using an extended hadronic and quark SU(3) non-linear realization of the sigma model.In this model the degrees of freedom change naturally from hadrons to quarks as the density and/ortemperature increases. The effects of high magnetic fields and anomalous magnetic moment are visible inthe macroscopic properties of the star, such as mass, adiabatic index, moment of inertia, and cooling curves.Moreover, at the same time that the magnetic fields become high enough to modify those properties, theymake the star anisotropic.
PACS.
Magnetars are compact stars that have surface magneticfields up to 10 − G [1,2]. Such fields can be esti-mated from observations of the star’s period and periodderivative. According to Virial theorem estimates, neu-tron stars could have a central magnetic field as largeas 10 or 10 G. An accurate calculation of this limitis complicated, since all energies to be weighed againstthe magnetic energy (the one from matter and the grav-itational one) also depend on the magnetic field (due tothe nonlinear nature of General Relativity). For this rea-son the limit might be different for different equations ofstate (EOS’s). Furthermore, due to the asymmetry intro-duced by the magnetic field in the z-direction, the pres-sure becomes different in the parallel and perpendiculardirections. In reality, depending on the magnitude of themagnetic field, the parallel pressure becomes much smallerthan the perpendicular one, and in extreme cases, can golocally to zero. In Ref. [3], as well as in our calculations,this limit was found to be around 10 - 10 G. Beyondthis value strong instabilities can occur, which indicatesthat further investigation is necessary.References [4,5] solve the Einstein and Maxwell equa-tions self-consistently for non-rotating and rotating stars.They study axisymmetric and poloidal magnetic field con-figurations and find B ∼ . . × G as limits forthe star central magnetic field. This range arises from theuse of different hadronic EOS’s, from a simple politropic
Send offprint requests to : to a hyperonic relativistic one. Although such EOS do nottake into account magnetic field modifications, their for-malism provides reliable results for star masses and radiibecause it takes into consideration different pressures indifferent directions of the star. It is important to noticethat Ref. [6] states that the use of different current func-tions and symmetries, other than the axial one, togetherwith effects of the magnetic field on the EOS may alterthese limits. Other simulations including magnetic fieldeffects can be found in [7,8,9,10].It has also been shown by refs. [11,12] that using wellaccepted hadronic EOS’s together with those based onfirst principles QCD calculations, one finds a phase tran-sition to deconfined matter already at a few times satura-tion density inside compact stars. Such a phase transitioncan be a sharp first order transition or might exhibit amixed phase, depending on the surface tension betweenthe phases. In this work we study the effect of the latter.There have also been works [13,14,15,16,17,18,19,20,21] showing that the inclusion of sophisticated quark cou-pling terms (like vector couplings, effective 6-quark inter-actions and quark-quark pairing) to the NJL and PNJLmodels, can lead to the appearance of a crossover chi-ral symmetry restoration/deconfinement phase transitionat high density and small temperatures. In this case, theQCD phase diagram becomes more complicated due tothe presence of a second critical point. Unfortunately, thevalues of the coupling constants which determine the pre-cise nature of the phase transition in this regime are notpredictable from first principles QCD calculations at this V. Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field point. Thus, the determination of the nature of the phasetransition in this limit is still an active area of research.One important question we will address is whetherthe effects of strong magnetic fields in the deconfinementphase transition are strong enough to be observed. Otherstudies along this line can be found in [22,23,24,25]. Inorder to do so in a realistic way, we adopt as our micro-scopic model a chiral approach that includes hadronic aswell as quark degrees of freedom in a unified description[26,27,28]. Here, besides the analysis of the mass-radiusdiagram, we will extend our study to the adiabatic in-dex and moment of inertia of the stars. Furthermore, wewill, for the first time, investigate the thermal evolution ofhighly magnetized hybrid stars that include modificationsin the EOS, which will help us to assess the effect of ahigh magnetic field on the cooling of the star. Finally, wewill use two different approximations to take into accountthe pure magnetic field contribution to the energy densityand pressure of the system. With this, we can model anEOS that is in principle anisotropic in an isotropic wayand try to determine how reliable those approximationsare.
Chiral sigma models are effective quantum relativistic mod-els that describe hadrons interacting via meson exchangeand, most importantly, are constructed from symmetry re-lations. They are constructed in a chirally invariant man-ner as the particle masses originate from interactions withthe medium and, therefore, go to zero at high densityand/or temperature. Adopting the nonlinear realizationof the sigma model gives a significant improvement to thewidely used linear sigma model [29,30] as it is in betteragreement with nuclear physics results [31,32].The Lagrangian density of the SU(3) non-linear sigmamodel in the mean field approximation constrained furtherby astrophysics data can be found in Ref. [33]. A recentextension of this model also includes quarks as dynamicaldegrees of freedom [26]. In this version, the degrees of free-dom change due to the effective masses of the baryons andquarks. Their masses are generated by the scalar mesons( σ , isovector δ , strange ζ ), except for a small explicit massterm M and the term containing the field Φ . This fieldserves as an effective order parameter for the deconfine-ment transition and is modeled in such a way as to repro-duce the behavior of the Polyakov loop at zero chemicalpotential, as determined by lattice QCD calculations [34].The baryon and quark effective masses are given by M ∗ B = g Bσ σ + g Bδ τ δ + g Bζ ζ + M B + g BΦ Φ , (1) M ∗ q = g qσ σ + g qδ τ δ + g qζ ζ + M q + g qΦ (1 − Φ ) , (2)where the values for the coupling constants can be foundin Ref. [26]. With the increase of density/temperature,the σ field (non-strange chiral condensate) decreases itsvalue, causing the effective masses of the particles (inthe absence of Φ ) to decrease towards chiral symmetry µ B (MeV) × × × × × B * ( M e V ) B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 1. (Color online) Effective magnetic field as a function ofbaryon chemical potential shown for different central magneticfields. restoration. The field Φ assumes non-zero values with theincrease of temperature/density and, due to its presencein the baryons effective mass (Eq. (1)), suppresses theirpresence. On the other hand, the presence of the Φ fieldin the effective mass of the quarks, included with a nega-tive sign (Eq. (2)), ensures that they will not be presentat low temperatures/densities. The value for the g Φ cou-pling constants is intrinsically related to the strength ofthe phase transition and only high values reproduce a firstorder phase transition at high densities and low tempera-tures.Both phase transitions, chiral symmetry restoration(breaking) and deconfinement (confinement) happen atthe same density/temperature. Such a fact comes fromthe correlation of these quantities in the effective massesof the particles. The potential for Φ reads U = ( a T + a µ B + a T µ B ) Φ (3)+ a T log (1 − Φ + 8 Φ − Φ ) . This potential was modified from its original form in thePNJL model [35,36] by adding terms that depend on thebaryon chemical potential. This allows us to reproduce thephase structure over the whole range of chemical poten-tials and temperatures, as suggested in lattice QCD stud-ies [37,38], i.e. a cross over at small chemical potentialand, at higher chemical potential, a first-order transitionline that stops at a second-order critical end-point. Here,by using the modified µ B dependent form for the Φ poten-tial U , combined with a high value for the coupling of the Φ field for the hadrons and quarks, we obtain a first orderphase transition at high densities, as it has been conjec-tured in other hybrid star calculations [39,40,41,42,43].We stress again that this behavior is only a model as-sumption and, for instance in a Polyakov-extended quark-meson model [44] or in the quarkyonic picture [45] onemight expect a smooth deconfinement crossover in thisregime. . Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field 3 µ B (MeV) Φ B=0B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 2. (Color online) Deconfinement order parameter as afunction of baryon chemical potential imposing local chargeneutrality shown for different central magnetic fields. Dottedlines include AMM.
In order to have a more complete description of hybridstars, we include magnetic fields that, while being in the z-direction, are not constant. The effective magnetic dipolefield B ∗ increases with baryon chemical potential goingfrom a surface value of B surf = 10 G (when µ B ≃ B c at high baryon chemical po-tential B ∗ ( µ B ) = B surf + B c [1 − e b ( µB − a ] , (4)with a = 2 . b = − . × − and µ B given in MeV. Theuse of the formula above generates no discontinuity in theeffective magnetic field or in the increase of the effectivemagnetic field at the phase transition. Such an unphys-ical discontinuity would be present if we had chosen theeffective magnetic field to be a function of baryon density,as is normally the case. The constants a , b and the formof B ∗ are chosen to reproduce (in the absence of quarks)the effective magnetic field curve as a function of densityfrom Refs. [23,46,24]. As can be seen in Fig. 1, even withthe use of extremely high central magnetic fields, the val-ues for the effective magnetic fields only become extremeat very high baryon chemical potentials. In practice, onlyabout 70% of B c can be reached inside the star. Still, thehighest central magnetic field considered in this work isabove the limit established by hydrostatic stability fromrefs. [4,5] and is only shown to illustrate the influence ofan extreme high magnetic field in the EOS.The magnetic field in the z-direction forces the eigen-states in the x and y directions of the charged particles tobe quantized into Landau levels νE ∗ i νs = s k z i + (cid:18)q m ∗ i + 2 ν | q i | B ∗ − s i κ i B ∗ (cid:19) , (5)where k i is the fermi momentum, q i the charge and s i thespin of each baryon or quark. The last term comes from the ε (MeV/fm ) P ( M e V / f m ) B=0B=1.4x10 GB=7.2x10 GB=1.4x10 G Fig. 3. (Color online) Equation of State (pressure of matter asa function of energy density of matter) assuming global chargeneutrality for different central magnetic fields. Dotted lines in-clude AMM. anomalous magnetic moment (AMM) of the particle thatsplits the energy levels with respect to the alignment/anti-alignment of the spin with the magnetic field. The AMMalso modifies the energy levels of the uncharged particles E ∗ i s = q k i + (cid:0) m ∗ i − s i κ i B ∗ (cid:1) . (6)The constants κ i determine the tensorial coupling strengthof baryons with the electromagnetic field tensor and havevalues κ p = 1 . κ n = − . κ Λ = − . κ + Σ = 1 . κ Σ = 1 . κ − Σ = − . κ Ξ = − . κ − Ξ = 0 .
06. The signof κ i determines the preferred orientation of the spin withthe magnetic field. The sum over the Landau levels ν runsup to a maximum value, beyond which the momentum ofthe particles in the z-direction would be imaginary ν max = E ∗ i s + s i κ i B ∗ − m ∗ i | q i | B ∗ . (7)We choose to include in our calculations the AMM ef-fect for the hadrons only, since the coupling strength ofthe particles κ i depends on the corresponding magneticmoment, that up to now is not fully understood for thequarks. Furthermore, it is stated in Ref. [47], that quarksin the constituent quark model have no anomalous mag-netic moment. For calculations including AMM effects forthe quarks see refs. [22,48,49,50]. The AMM for the elec-trons is also not taken into account as its effect is negligi-bly small. Note that the AMM removes the spin degener-acy of the particles and their energy levels are further splitin two levels each, which increases even more the particlechemical potentials.As can be seen in Fig. 2, a phase transition which is offirst order at zero temperature is delayed with the inclu-sion of high magnetic field. This effect is due to a strongerstiffening of the hadronic part of the EOS (as the mag-netic field increases). The delay of the phase transition isfurther increased when the anomalous magnetic moment V. Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field µ B (MeV) -3 -2 -1 ρ i (f m - ) pn Λ e µ uds Fig. 4. (Color online) Particle densities as a function of baryonchemical potential assuming global charge neutrality for B=0.Quark densities are divided by 3. is considered, as it only affects the hadronic phase, ren-dering its EOS stiffer (due an increase in the chemicalpotential of the baryons). If global charge neutrality is as-sumed, the location of the mixed phase does not changesubstantially, except for the highest magnetic field consid-ered with AMM, when the entire mixed phase is pushedto slightly higher chemical potentials.The same effect of confinement and/or chiral symme-try enhancement in the presence of high magnetic fieldswas already predicted by other works. In refs. [51,52,53]this was shown for high density and small or zero temper-ature, whereas in refs. [54,55,53,56] this was shown forsmall or zero density and high temperatures. Together,these features show the importance of the study of the in-fluence of high magnetic fields and deconfinement/chiralsymmetry restoration in compact stars as a part of agreater picture that forms the whole QCD phase diagramand contains heavy-ion experiments in the other extreme.The EOS is shown in Fig. 3 for the case in which globalcharge neutrality is allowed and a mixed phase appears.We do not show the pure quark phase region, since inthis model it is present only at very high densities thatcannot be reached in the interior of stars. It is importantto notice that at very low densities, the hadronic EOSof non-interacting matter gets softer in the presence ofhigh magnetic field, in agreement with refs. [57,6] andother papers. In our case this effect is smaller because themagnetic field is only large at high densities. Note thatonly the energy density and pressure of matter are shownin this figure and no direct contribution from the magneticfield is included. In this way we can see the direct effectsof the Landau quantization and the AMM in the model.Fig. 4 shows the baryon density of fermions. In themixed phase the hadrons disappear as the quarks smoothlyappear. The hyperons, despite being included in the cal-culation, are suppressed by the appearance of the quarkphase. Only a very small amount of Λ baryons appearright before the phase transition. The density of electronsand muons is significant in the hadronic phase but not µ B (MeV) -3 -2 -1 ρ i (f m - ) e µ pn Λ uds Fig. 5. (Color online) Particle densities as a function of baryonchemical potential assuming global charge neutrality for B c =7 . × G, and including AMM. Quark densities are dividedby 3. in the quark phase. The reason for this behavior is that,because the down and strange quarks are also negativelycharged, there is no necessity for the presence of electronsto maintain charge neutrality, and only a small amount ofleptons remains to ensure beta equilibrium. The strangequarks appear after the other quarks, and do not makesubstantial changes in the system.Fig. 5 shows the change in population when a centralmagnetic field of 7 . × G with AMM is considered.The wiggles in the charge particle densities come from theLandau levels, more precisely when the Fermi energy ofthe particles crosses the discrete threshold of a Landaulevel. The charged particles are enhanced (as their chemi-cal potentials increase with B ), which is especially visibleby the amount of electrons in the quark phase. The re-maining hyperons are even more suppressed due to theincrease in the proton density. All these effects are furtherenhanced when higher central magnetic fields are consid-ered, like B c = 1 . × G. A key point of this investigation is to determine whetherthe changes due to the presence of magnetic field in theEOS are strong enough to affect observable properties ofthe stars. To answer this question, we are going to analyzethe adiabatic index, moment of inertia, mass-radius dia-gram and thermal evolution for different central magneticfields.
We begin with the adiabatic index (Fig. 6). The disconti-nuities in the B = 0 curve show the appearance of the Λ ’sand the beginning of the mixed phase. The extra peaksin the finite magnetic field with AMM curves show the . Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field 5 ρ B (fm -3 ) Γ B=0B c =7.2x10 GB c =1.4x10 G Fig. 6. (Color online) Adiabatic index as a function of baryondensity for different central magnetic fields including AMM.
Landau level thresholds that also can be seen in the pop-ulation plot. Note that the Landau level thresholds appearin the hadronic phase as well as in the mixed phase. Aspointed out in Ref. [58], these rapid changes in the pres-sure can cause instabilities in the star that might causestar-quakes, glitches and giant flares.
In Fig. 7 we show the moment of inertia as function of ra-dius for several of the investigated stellar sequences, tak-ing into account the AMM. We see that for the maximumcentral magnetic field considered (1 . × G), there isan increase of about 10% from the unmagnetized case (forthe maximum mass star). For B c = 7 . × G this in-crease is of about ∼ As a first approach to the problem, the possible hybrid starmasses and radii are calculated by solving the Tolman-Oppenheimer-Volkoff equations for spherical isotropic staticstars [59,60] using the EOS of matter. In Fig. 8, besidesour EOS for the core, a separate EOS was used for thecrust [61]. The maximum mass supported against grav-ity is higher for higher magnetic fields and even higherwhen the AMM is included. It is important to note thatin this model pure quark matter only appears at very highdensities, that correspond to the unstable branch of themass-radius diagram. Thus, only hadronic and “mixed”matter appear in the star. Similar results for stars con-taining only hadronic and “mixed matter” were also found
13 13.5 14 14.5
R (km) I ( g c m ) B c = 0B c = 1.4x10 GB c = 7.2x10 GB c = 1.4x10 G Fig. 7. (Color online) Moment of inertia as a function of radiusfor different central magnetic fields including AMM. in a calculation using the Brueckner-Hartree-Fock modelfor the hadronic phase and the Dyson-Schwinger approachfor the quarks [62].So far, the magnetic field energy density and pressurecontributions were not considered and only the influence ofthe magnetic field on the energy levels of the particles wastaken into account. The problem in including the magneticpressure is that it has different values in the directionsparallel and perpendicular to the field, as can be seen inthe electromagnetic energy-momentum tensor T µν = 14 π B ∗ B ∗ B ∗
00 0 0 − B ∗ , (8)where the first term is related to the energy density, theother three terms are related to the pressure in the x , y and z directions and the 4 π comes from the choice of Gaus-sian natural units. This problem was pointed out in manypapers such as refs. [63,64,49,50,65,3,66,25]. In these pa-pers, there is also a term coming from the magnetic dipoleinteraction, which is linear in B and is therefore sublead-ing at extremely high magnetic fields.A consistent inclusion of the macroscopic magneticpressure requires a 2D calculation. As mentioned before,such calculations exist [4,5]; however, most calculationsperformed using realistic EOS’s for the macroscopic prop-erties of magnetized stars assume isotropy and considerthe pure magnetic pressure term to be either positive ornegative in all directions. It was pointed out by refs. [23,67] that considering the magnetic pressure to be nega-tive in all directions constrains the maximum values thatcan be used for the magnetic field to lower values. In ourcase, B c = 7 . × G would already cause an insta-bility at very high densities. A third option, suggestedby Ref. [68] uses, as a monopole approximation of theenergy-momentum tensor, the average between the threepressures and adds + B / π for the magnetic pressurein all directions. We show the difference caused by eitherchoosing the pure magnetic pressure to be positive in all V. Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field
R (Km) M / M s un B=0B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 8. (Color online) Mass-radius diagram shown for differentcentral magnetic fields including AMM. The dashed and dottedlines have the extra magnetic pressure term of B ∗ / π and B ∗ / π , respectively, added to the total pressure. directions or using the average pressure in the mass-radiusdiagram. Evidently, the first case is unphysical (for ex-tremilily strong magnetic fields) since the magnetic pres-sure dominates over any other contribution. We speculatethat the correct value would be close to the second op-tion that uses the average of the pressures in differentdirections, but any more precise statement requires theuse of axisymmetric equations, as was already pointed byRef. [25]. Work along this line using our hybrid matterEOS is ongoing and will allow for a more exact estimateof the increase of the star mass based on the magnitudeof the magnetic field of the magnetar. We now turn our attention to the thermal evolution ofhybrid stars, whose microscopic composition is given bythe model described in this paper. The cooling of compactstars is given by the thermal balance and thermal energytransport equation ( G = c = 1) [69] ∂ ( le φ ) ∂m = − ρ p − m/r (cid:18) ǫ ν e φ + c v ∂ ( T e φ ) ∂t (cid:19) , (9) ∂ ( T e φ ) ∂m = − ( le φ )16 π r κρ p − m/r . (10)In Eqs. (9) − (10) the structure of the star is given bythe variables r , ρ ( r ), e φ and m ( r ), that represent the ra-dial distance, the energy density, the metric function, andthe stellar mass, respectively. The thermal variables aregiven by the temperature T ( r, t ), luminosity l ( r, t ), neu-trino emissivity ǫ ν ( r, T ), thermal conductivity κ ( r, T ), andspecific heat c v ( r, T ). The boundary conditions of (9) and(10) are determined by the luminosity at the stellar centerand at the surface. The luminosity vanishes at the stellarcenter since there is no heat flux there. The surface tem-perature (luminosity) is defined by the crust temperature -1 Age (years) T ∞ ( K ) B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 9. (Color online) Cooling curves for a 1.1 M ⊙ mass starin the presence of different central magnetic fields includingAMM. T ∞ denotes the redshifted temperature observed at in-finity. The observed data consists of circles denoting spin-downages and squares denoting kinematic ages [76,79]. The dashedand dotted lines include the extra magnetic pressure term of B ∗ / π and B ∗ / π , respectively. All curves overlap. and the properties of the stellar surface (surface gravity,magnetic field and etc.) [70,71,72,73].For the hadronic phase, we have considered the fol-lowing neutrino emission processes: direct Urca, modifiedUrca and bremsstrahlung processes; whereas for the quarkphase, the processes taken into account are the quark di-rect Urca, quark modified Urca, and quark bremsstrahlungprocesses. Details about the emissivities of such processescan be found in refs. [74,75]. The specific heat of thehadrons is the usual specific heat of fermions, as describedin Ref. [76]. As for the quarks, we use the expression for thespecific heat calculated in Ref. [75]. The thermal conduc-tivities for the hadronic and quark matter were calculated,respectively in refs. [77,78], and in this work we follow theresults presented in these references.We have calculated the cooling for magnetized hybridstars (with anomalous magnetic moment of the hadronsincluded) for three different masses: 1.1, 1.4 and 1.93 M ⊙ .For each hybrid star mass we considered three values forthe central magnetic field (1 . × , . × , . × G). In addition to that we have also considered the effectof including the pure magnetic field contribution in theEoS.The cooling of 1.1 M ⊙ stars is shown in Fig. 9. Onecan see that in this case the star exhibits a slow cool-ing, which agrees relatively well with the observed data.Furthermore, we can also conclude that in this case themagnetic field has no substantial effect on the thermalevolution of the object, and neither does the inclusion ofthe pure magnetic contribution in the EoS.The situation is similar for stars with higher masses,as can be seen by the full lines in Figs. 10 and 11, whichshow the thermal evolution of stars with 1.4 and 1.93 M ⊙ ,respectively. We see that in this case the modifications in-troduced by the magnetic field in the composition are not . Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field 7 -1 Age (years) T ∞ ( K ) B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 10. (Color online) Cooling curves for a 1.4 M ⊙ mass star.Otherwise same notation as Fig. 9. The dashed and dotted linescurves overlap. enough to alter the cooling significantly (as was the casefor lower mass stars). This picture changes, however, if oneintroduces the effect of the magnetic pressure. As shownby the dotted lines in Figs. 10 and 11, the extra B ∗ / π leads to the slow cooling of a star that would otherwise ex-hibit fast cooling. This stems from the fact that adding anextra source of pressure stiffens the EoS. Hence, stars withhigher masses possess smaller central densities and, there-fore, smaller proton fractions. The small proton fractionwill hinder the otherwise present direct Urca process, thusleading to a slow cooling. The results for the inclusion ofthe magnetic pressure of B ∗ / π (dashed lines) are qual-itatively the same. In this case, however, the stiffening ofthe EoS is more mild, and thus the effect is only appre-ciable for moderate masses, as seen in Fig. 10, where thethermal evolution is the same as for the magnetic pressureof B ∗ / π . For the 1.93 M ⊙ star we see that the stiffeningof the EoS is not enough to hinder the direct Urca pro-cess completely, and cooling of hybrid stars with magneticpressure of B ∗ / π is only slightly slower.Note that in Figs 9, 10 and 11 we have assumed aneffective magnetic field that changes with density withinthe star but does not change in time. In a realistic sce-nario it is expected that the magnetic field decreases as afunction of time as explained in Ref. [80,81,82]. In thesereferences the authors also conclude that the inclusion ofthe magnetic field directly in the cooling simulation re-produces stars that remain warmer for a longer time, inbetter agreement with observation. On the other hand,such studies, do not include the effect of magnetic fieldsin the EOS, and therefore are complemented by our study. To model magnetars, we use an effective model that in-cludes hadronic and quarks degrees of freedom. As thedensity increases, the order parameters signal the decon-finement and chiral symmetry phase transitions, which -1 Age (years) T ∞ ( K ) B c =1.4x10 GB c =7.2x10 GB c =1.4x10 G Fig. 11. (Color online) Cooling curves for a 1.93 M ⊙ massstar. Otherwise same notation as Fig. 9. tend to take place closer to the the center of the star forhigher magnetic fields. The stiffness of the equation ofstate (EOS), and the consequent star masses, depend onthe chosen central magnetic field. It is clear that a higher B c allows for more massive stars, but the quantitativeanalysis of how more massive the star might be requiresthe use of a 2D solution of Einstein’s equations which takesinto account the breaking of spherical symmetry by themagnetic field. This was shown by the dramatic changein the star masses when different assumptions were usedfor the inclusion of the pure magnetic field contribution tothe EOS. As a consequence, only the lower magnetic fieldconfiguration with B c = 1 . × gives reliable results,as it has a pressure anisotropy ( p z − p x,y ) / ( p z + p x,y ) of ∼ B ∗ that increases with baryon chemical potential,and therefore avoids a discontinuity in the phase transitionregion. This allows B ∗ to increase smoothly from a sur-face value up to a chosen central one ( B c ). Furthermore,our investigation indicates that the composition changesintroduced by the magnetic field are not high enough toalter significantly the thermal evolution of hybrid stars.We have found, however, that by including the magneticpressure ( B ∗ / π or B ∗ / π ), the consequent stiffeningof the EoS allows for stars with higher masses (1.4 – 1.93 M ⊙ ) to exhibit slower cooling, which is in better agree-ment with the observed data. This is an important result,which indicates the importance of correctly introducingthe macroscopic pressure into the structure of the star. Weare cautious in interpreting these results, since the appro-priate treatment of both the structure, and the thermalevolution of stars with such high magnetic fields requires a V. Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field full two-dimensional study [83]. We will save this investi-gation for a future work, as it seems clear to us that the ef-fect of the magnetic pressure on the structure and thermalevolution are substantial. Although calculations which ad-dress this problem using model EOS’s have already beenperformed, we believe that a well-motivated microscopicEOS that also includes the effects of the star’s magneticfield is important for fully understanding the behavior ofmagnetars.Subsequently, we would like to expand our study tofinite temperature, in order to have a complete phasediagram (temperature as a function of baryon chemicalpotential) for high magnetic fields. A phase diagram for B = 0 was already constructed using the extended non-linear realization of the SU(3) sigma model in Ref. [26].Such a complete analysis is important, since it connectsthe physics at high temperature and small densities suchas those created in heavy ion collisions, with the physicsinside neutron stars. A finite- B phase diagram was alreadyconstructed in Ref. [22] but we believe that we can givefurther insight into the matter with our model, since italso includes chiral symmetry restoration and allows for asmooth chiral and deconfinement crossover transition atsmall densities. It would also be interesting to study dif-ferent parametrizations of the model which would give asmooth crossover in the low temperature regime and studythe influence of the magnetic field in that case. Acknowledgments
We are grateful to Constanca Providencia, Milva Orsaria,Fridolin Weber, Debora Menezes and Aurora Perez Mar-tinez for the fruitful discussions on the subject of strongmagnetic fields. R. N. acknowledges financial support fromthe LOEWE program HIC for FAIR. V. D. acknowledgesfinancial support from CNPq-Brazil.
References
1. B. Paczynski, Acta Astron. , 145 (1992)2. A.I. Ibrahim, C. Markwardt, J. Swank, S. Ransom,M. Roberts et al., AIP Conf.Proc. , 294 (2004)3. E.J. Ferrer, V. de la Incera, J.P. Keith, I. Portillo, P.P.Springsteen, Phys. Rev. C82 , 065802 (2010)4. M. Bocquet, S. Bonazzola, E. Gourgoulhon, J. Novak, As-tron.Astrophys. , 757 (1995)5. C.Y. Cardall, M. Prakash, J.M. Lattimer, Astrophys.J. , 322 (2001)6. A.E. Broderick, M. Prakash, J.M. Lattimer, Phys. Lett.
B531 , 167 (2002)7. V.C.A. Ferraro, apsrev4-1 , 407 (1954)8. S. Lander, D. Jones, Mon.Not.Roy.Astron.Soc. , 482(2012)9. L. Rezzolla, F.K. Lamb, S.L. Shapiro, Astrophys.J. ,L141 (2000)10. K. Kiuchi, K. Kotake, Mon.Not.Roy.Astron.Soc. , 1327(2008)11. A. Kurkela, P. Romatschke, A. Vuorinen, Phys.Rev.
D81 ,105021 (2010) 12. A. Kurkela, P. Romatschke, A. Vuorinen, B. Wu (2010),
13. T. Hatsuda, M. Tachibana, N. Yamamoto, G. Baym, Phys.Rev. Lett. , 122001 (2006)14. G. Baym, T. Hatsuda, M. Tachibana, N. Yamamoto,J.Phys.G G35 , 104021 (2008)15. P. Powell, G. Baym,
The axial anomaly and three-flavorNJL model with confinement: constructing the QCD phasediagram , in
APS Meeting Abstracts (2012), p. 1000116. N.M. Bratovic, T. Hatsuda, W. Weise (2012),
17. O. Lourenco, M. Dutra, A. Delfino, M. Malheiro,Phys.Rev.
D84 , 125034 (2011)18. O. Lourenco, M. Dutra, T. Frederico, A. Delfino, M. Mal-heiro, Phys.Rev.
D85 , 097504 (2012)19. N. Yamamoto, M. Tachibana, T. Hatsuda, G. Baym,Phys.Rev.
D76 , 074001 (2007)20. N. Yamamoto, JHEP , 060 (2008)21. H. Abuki, G. Baym, T. Hatsuda, N. Yamamoto, Phys.Rev.
D81 , 125010 (2010)22. S. Chakrabarty, Phys. Rev.
D54 , 1306 (1996)23. D. Bandyopadhyay, S. Chakrabarty, S. Pal, Phys. Rev.Lett. , 2176 (1997)24. A. Rabhi, H. Pais, P.K. Panda, C. Providencia, J. Phys. G36 , 115204 (2009)25. L. Paulucci, E.J. Ferrer, V. de la Incera, J.E. Horvath,Phys. Rev.
D83 , 043009 (2011)26. V.A. Dexheimer, S. Schramm, Phys. Rev.
C81 , 045201(2010)27. S. Schramm, R. Negreiros, J. Steinheimer, T. Schurhoff,V. Dexheimer, Acta Phys.Polon.
B43 , 749 (2012)28. R. Negreiros, V. Dexheimer, S. Schramm, Phys.Rev.
C85 ,035805 (2012)29. P. Papazoglou, S. Schramm, J. Schaffner-Bielich,H. Stoecker, W. Greiner, Phys. Rev.
C57 , 2576 (1998)30. J.T. Lenaghan, D.H. Rischke, J. Schaffner-Bielich, Phys.Rev.
D62 , 085008 (2000)31. P. Papazoglou et al., Phys. Rev.
C59 , 411 (1999)32. L. Bonanno, A. Drago, Phys. Rev.
C79 , 045801 (2009)33. V. Dexheimer, S. Schramm, Astrophys. J. , 943 (2008)34. K. Fukushima, Phys. Lett.
B591 , 277 (2004)35. C. Ratti, M.A. Thaler, W. Weise, Phys.Rev.
D73 , 014019(2006)36. S. Roessner, C. Ratti, W. Weise, Phys.Rev.
D75 , 034007(2007)37. Z. Fodor, S. Katz, JHEP , 050 (2004)38. Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, K. Szabo, Nature , 675 (2006), hep-lat/0611014
39. W. Lin, B.A. Li, J. Xu, C.M. Ko, D.H. Wen, Phys.Rev.
C83 , 045802 (2011)40. C. Lenzi, A. Schneider, C. Providencia, R. Marinho,Phys.Rev.
C82 , 015809 (2010)41. B. Agrawal, Phys.Rev.
D81 , 023009 (2010)42. H. Chen, M. Baldo, G. Burgio, H.J. Schulze, Phys.Rev.
D84 , 105023 (2011)43. T. Endo, Phys.Rev.
C83 , 068801 (2011)44. T.K. Herbst, J.M. Pawlowski, B.J. Schaefer, Phys.Lett.
B696 , 58 (2011)45. L. McLerran, R.D. Pisarski, Nucl.Phys.
A796 , 83 (2007)46. G.J. Mao, A. Iwamoto, Z.X. Li, Chin. J. Astron. Astro-phys. , 359 (2003)47. S. Weinberg, Phys.Rev.Lett. , 1181 (1990)48. I.S. Suh, G. Mathews, F. Weber, Phys.Rev.D (2001). Dexheimer et al.: Hybrid Stars in a Strong Magnetic Field 949. A. Perez Martinez, H. Perez Rojas, H.J. Mosquera Cuesta,M. Boligan, M.G. Orsaria, Int. J. Mod. Phys. D14 , 1959(2005)50. R.G. Felipe, A.P. Martinez, H.P. Rojas, M. Orsaria, Phys.Rev.
C77 , 015807 (2008)51. D. Menezes, M. Benghi Pinto, S. Avancini, A. Perez Mar-tinez, C. Providencia, Phys.Rev.
C79 , 035807 (2009)52. D. Menezes, M. Benghi Pinto, S. Avancini, C. Providencia,Phys.Rev.
C80 , 065805 (2009)53. S.S. Avancini, D.P. Menezes, C. Providˆencia, Phys. Rev.C (6), 065805 (2011)54. M. D’Elia, S. Mukherjee, F. Sanfilippo, Phys.Rev. D82 ,051501 (2010)55. A.J. Mizher, M. Chernodub, E.S. Fraga, Phys.Rev.
D82 ,105016 (2010)56. R. Gatto, M. Ruggieri, Phys.Rev.
D83 , 034016 (2011)57. A. Broderick, M. Prakash, J. Lattimer, Astrophys.J.(2000)58. C.Y. Ryu, T. Maruyama, T. Kajino, G.J. Mathews, M.K.Cheoun, Phys.Rev.
C85 , 045803 (2012)59. R.C. Tolman, Phys.Rev. , 364 (1939)60. J. Oppenheimer, G. Volkoff, Phys.Rev. , 374 (1939)61. G. Baym, C. Pethick, P. Sutherland, Astrophys.J. , 299(1971)62. H. Chen, M. Baldo, G. Burgio, H.J. Schulze, Phys.Rev. D84 , 105023 (2011)63. M. Chaichian, S. Masood, C. Montonen, A. Perez Mar-tinez, H. Perez Rojas, Phys.Rev.Lett. , 5261 (2000)64. A.P. Martinez, H. Rojas, H.J. Mosquera Cuesta,Eur.Phys.J. C29 , 111 (2003)65. A. Perez Martinez, H. Perez Rojas, H. Mosquera Cuesta,Int. J. Mod. Phys.
D17 , 2107 (2008)66. M. Orsaria, I.F. Ranea-Sandoval, H. Vucetich, Astrophys.J. , 41 (2011)67. M. Sinha, B. Mukhopadhyay (2010),
68. I. Bednarek, A. Brzezina, R. Manka, M. Zastawny-Kubica,Nucl.Phys.
A716 , 245 (2003)69. F. Weber,
Pulsars as astrophysical laboratories for nuclearand particle physics , 1st edn. (Institute of Physics, Bristol,1999)70. E.H. Gudmundsson, C.J. Pethick, R.I. Epstein, Astrophys.J. Lett. , L19 (1982)71. E.H. Gudmundsson, C.J. Pethick, R.I. Epstein, Astrophys.J. , 286 (1983)72. D. Page, U. Geppert, F. Weber, Nucl.Phys.
A777 , 497(2006)73. D. Blaschke, T. Klahn, D. Voskresensky, Astrophys.J. ,406 (2000)74. D. Yakovlev, A. Kaminker, O.Y. Gnedin, P. Haensel,Phys.Rept. , 1 (2001)75. N. Iwamoto, Annals of Physics , 1 (1982)76. D. Page, J.M. Lattimer, M. Prakash, A.W. Steiner, Astro-phys.J.Suppl. , 623 (2004)77. E. Flowers, N. Itoh, Astrophys. J. , 750 (1981)78. P. Haensel, Nuclear Physics B Proceedings Supplements , 23 (1991)79. D. Page, J.M. Lattimer, M. Prakash, A.W. Steiner, Astro-phys.J. , 1131 (2009)80. D.N. Aguilera, J.A. Pons, J.A. Miralles, Astrophys.J. ,L167 (2008)81. D.N. Aguilera, J.A. Pons, J.A. Miralles, Astron.Astrophys. , 255 (2008)82. Y. Xie, S.N. Zhang (2011),
83. R. Negreiros, S. Schramm, F. Weber, Phys.Rev.