On the possibility of rho-meson condensation in neutron stars
Ritam Mallick, Stefan Schramm, Veronica Dexheimer, Abhijit Bhattacharyya
aa r X i v : . [ a s t r o - ph . H E ] A ug On the possibility of rho-meson condensation in neutron stars
Ritam Mallick ∗ and Stefan Schramm † Frankfurt Institute for Advanced Studies,60438 Frankfurt am Main, Germany
Veronica Dexheimer ‡ Department of Physics, Kent State University, Kent, Ohio 44242, USA
Abhijit Bhattacharyya § Department of Physics, University of Calcutta, 700009 Kolkata, India (Dated: September 27, 2018)
Abstract
The possibility of meson condensation in stars and in heavy-ion collisions has been discussed inthe past. Here, we study whether rho meson condensation ( ρ − ) can occur in very dense matterand determine the effect of strong magnetic fields on this condensation. We find that rho mesoncondensates can appear in the core of the neutron star assuming a rho mass which is reduceddue to in-medium effects. We find that the magnetic field has a non-negligible effect in triggeringcondensation. PACS numbers: 26.60.Kp, 52.35.Tc, 97.10.CvKeywords: dense matter, stars: neutron, stars : magnetic field, equation of state ≤
30 GeV per nucleon and the beamenergy scan at RHIC aim to study the properties of matter not only at high temperaturebut also at comparatively high densities.Increasing the density of the nuclear matter it is expected that the spontaneously brokenchiral symmetry, characterized by a large quark condensate, is at least partially restored.However, a clear observable signature of this effect is still not well established. From thetheory side, an early conjecture suggested by Brown and Rho [1], argued that hadrons (exceptfor the pseudo-Goldstone bosons like pions) experience a mass reduction in nuclear matter,which is proportional to the in-medium quark condensate. More recent calculations, usingQCD sum rules [2], quark-meson models [3] and hadronic models of vacuum polarization [4]also suggest such mass reduction. However, a full understanding of the restoration of chiralsymmetry in dense matter remains an open and much-discussed topic.More direct evidence of a potential mass reduction of vector mesons, the enhancement inthe production of low invariant-mass dileptons in heavy-ion collisions has been investigatedextensively, following such observations for an invariant mass around 400 −
500 MeV, asmeasured by the CERES experiment at the CERN-SPS [5]. Various model studies rangingfrom simple thermal models to more detailed transport calculations, predicted enhanceddilepton production [6–9]. One of the conjectures which explains such enhanced dileptonproduction in HIC, suggested the reduction of the in-medium masses of the vector mesons.The theoretical results were consistent with experimental data. However, later dynamicalstudies hinted that the earlier studies overestimated the effect [10]. The enhancement indilepton production due to vector meson mass reduction also differed from experiment toexperiment (consistent at SPS energies but not with DLS results [11]). Overall the situationwith respect to dilepton enhancement is still not unambiguously resolved.More recent experiments of γ − A [12] reactions provided s clearer signature of an in-medium ω mass reduction. In the experiment by the TAPS collaboration [12] the modi-fication of the ω in nuclei was measured in photo-production experiments, where its mass2as found to be m ∗ ω = 722 ± . n . Similar numbers were also found in 12 GeVphoton-nuclear reaction by Naruki et al. [13]. The ω meson mass and quark-condensatewere studied in more recent NJL model calculations [14], where they approximately recov-ered the B-R scaling law using constraints from the TAPS experiment. However, for the ρ meson the picture is still not clear.At the other extreme QCD exhibits exciting physics in a strong background magneticfield, since quarks are electrically charged and can be strongly affected by the field [15]. Itchanges the chiral symmetry breaking by increasing the quark-condensate [16, 17]. However,the strength of the magnetic field which typically can influence QCD effects is about eB ∼ m π ∼ × G, where e is the charge of the electron, B is the magnetic field and m π denotes the mass of pion. Quarks and antiquarks can form condensates in the presence ofstrong magnetic fields. For the quark-antiquark bound state with light quarks, i.e. for the ρ meson, strong magnetic fields can enhance the formation of a condensate.As will be discussed further below, the key idea lies in the fact that a sufficiently strongmagnetic field can trigger an instability in the vacuum leading to condensates in specificquark channels, especially with respect to the charged vector mesons generating non-zero ρ condensates [15, 17]. Huge magnetic fields are generated at the periphery of heavy-ioncollisions, where the large nuclear charges and momenta magnify the magnetic field [18]. AtLHC energies of 3 . − G [19]. However, such a field only exists for a very short time and is restricted to a smallvolume.On the other hand, the extreme conditions discussed above can occur naturally in neutronstars (NS), where the nuclear density is several times the nuclear saturation density ( n )and the magnetic field can be as high as 10 − G in their cores [20, 21]. The highdensity might reduce the ρ meson mass due to in-medium effects and with the aid of strongmagnetic field, a condensate could subsequently form.The possibility of pion and/or kaon condensation in neutron stars has been investigatedin numerous studies in the past. Since it was first suggested by Migdal [22, 23], pioncondensation and its impact on NS physics became the focus of intense discussion. Therewere calculations, both supporting and challenging a condensed state in the dense coresof compact stars. Similarly, for kaon ( K − ) condensation [24], it was argued that as theelectron chemical potential is an increasing function of density, whereas the effective (anti-3kaon mass decreases with density, condensation sets in at some density. Depending on thekaon-nucleon sigma term and the specific model, kaon condensation might occur already atabout 3 − M ⊙ puts severe constraints on the stiffness of theequation of state (EoS) of the core region of stellar matter. Therefore, a potential onset ofa condensate is important since (zero-momentum) condensates soften the equation of statethereby reducing maximum star masses.In this Letter, we study the possibility of rho meson ( ρ − ) condensation occurring insidea NS and look into the conditions which might make rho condensation possible. Thereare various aspects that increase the possibility of condensation in a neutron star environ-ment. Firstly, in the core of the star extreme densities exist, which could strongly amplifya potential density-dependence of the meson mass. Secondly, as the neutron star is in acharge-neutral state, condensation of negatively charged particles set in when their massdrops below the lepton Fermi energy, in striking contrast to the situation in a heavy-ioncollision, where there is no such effect. Finally, there is a possibility of strong extendedmagnetic fields in at least certain classes of neutron stars like magnetars, which can gener-ate a substantial effect for a charged spin-1 particle.We use a standard relativistic mean field approach to study the possibility of rho-mesoncondensation. Within this approach the nucleons interact through meson exchange rep-resented by their mean field values. The scalar meson ( σ ) provides the attractive force,whereas the vector mesons ( ω, ρ ) generate repulsion. In order to investigate whether the rhomeson will condense in a neutron star we assume a simple relation for the medium-dependent ρ mass. Certainly, in more extensive calculations a fuller study of density- (or field-) de-pendent masses of all involved hadrons should be performed. Here, in order to determinewhether such a condensation could take place in principle, for simplicity we adopt a linear4ependence of the rho mass on the scalar field m ∗ ρ = m ρ − gσ (1)where g is a dimensionless constant and m ρ = 776 MeV is the vacuum mass of the rhomeson. This term effectively introduces a coupling between the vector and scalar fields.In the presence of a background magnetic field of strength B , the energy level E n,S x of aparticle of mass m , charge e , and spin s in the Landau level n is given by [15, 19, 28–30] E n,S x = q p + m + (2 n − g B S x + 1) eB (2)where S x is the spin projection onto the magnetic field axis, and g B is gyromagnetic ratio ofthe particle which is taken to be 2 for the ρ meson [31–33]. Therefore, the energy squaredof the lowest state for a charged rho meson corresponding to p = 0 , n = 0 and S x = 1 is m ρ − ∗ = m ρ − − eB. (3)The upper equations point to the fact that in principle, if the in-medium effect or mag-netic field is high enough, the effective mass squared of the ρ meson can be negative andcondensation can take place. However, for neutron stars the situation is less restrictive asmentioned above, since the mass of the ρ − only has to fall below the electron chemical poten-tial, so that electrons can be replaced by negative rho mesons for generating a charge-neutralsystem. In this case, we obtain a neutron star with rho meson condensation.To have a better idea about if and how the rho meson condensation takes place in aneutron star, we have to adopt a specific relativistic mean field description including electronsto ensure charge neutrality of the matter. We choose the GM3 model for our calculationas its equation of state (EoS) has frequently been used in the literature to describe nuclearmatter in NSs [34]. This EoS can generate 2 solar mass NSs, which is an important criterion.We solve the mean field equations at finite density including the additional term Eq. (1).As this term does not affect isospin symmetric matter the isospin-independent quantitiesof saturated matter for the GM3 parameters do not change. However, the GM3 value ofthe asymmetry energy ( a sym = 32 . g from Eq. (1), we accordingly adjust the g Nρ coupling of the nucleon to the rho meson in order to maintain the original value of a sym .The results for stellar matter calculations are shown in Fig 1. We plot the variation ofthe electron chemical potential and the effective mass of the rho meson as a function of5 µ e , m ρ * ( M e V ) µ e , g=0, B=0m ρ , g=0, B=0 µ e , g=0, B=/ 0m ρ , g=0, B=/ 0 µ e , g=-4, B=0m ρ , g=-4, B=0 (a) µ e , m ρ * ( M e V ) µ e , g=-4, B=/ 0m ρ , g=-4, B=/ 0 µ e , g=-6, B=/ 0m ρ , g=-6, B=/ 0 (b) FIG. 1. (Color online) The electron chemical potential (solid lines) and effective ρ − mass (dashedlines) are plotted as functions of normalized density. The crossing points mark the density at whichthe condensation appears. normalized baryon density. The dashed lines represent the rho meson effective mass andthe solid lines depict the electron chemical potential. In Fig. 1a the black lines shows therespective values without any in-medium effect or magnetic field ( g = 0, B = 0). In thiscase, we find that there is no crossing point between the lines, which means the rho meson6oes not condense in this environment for any reasonable density value. The inclusion of themagnetic field ( B = 0) shifts the curves but is still not able to generate a crossing (red lines),which points to the fact that the magnetic field alone cannot give rise to condensation in acompact star. In order to investigate the maximum size of the effect with some reasonableassumptions, we considered a magnetic field of a strength of 7 . × G (5 × MeV ). Inorder to produce rho condensation solely due to a strong magnetic field, much higher valuesfor the magnetic field is required (3 − × G). However, studies suggest that such highmagnetic fields are difficult, if not impossible, to realize in a NS; however, they may locallyappear for a short time during heavy ion collisions. For the crossing to take place, the rhomass modification due to in-medium effect must be included. Taking this effect into accountwe find that the blue lines cross each other at about 7 . g = −
4. We have checkedthe parameter setting with nuclear matter constraints, and all are well within the acceptedvalues. Recent measurements of pulsar masses suggest that the central energy density ofneutron stars might lie around 5 − B and g the condensate setsin at approximately 6 . g , i.e. assuming a larger rho mass modification due to in-medium effects, the condensatesets in at smaller densities. For g = −
6, the condensate appears around 4 . . ρ − condensate. The maximum mass achieved is about 2 . M ⊙ . The red and blue curvesshow the sequences where rho meson condensation takes place. The solid lines in the curverepresent stars which do not contain rho condensate, whereas the dotted points of the curveindicate stars that have some amount of ρ − condensate in their cores. For these curves the7 e+15 2e+15 3e+15 4e+15 ε c (gm/cm )00.511.522.5 M / M o g=0, B=0g=-4, B=/ 0g=-6, B=/0 FIG. 2. (Color online) Sequence of stars for the GM3 EoS. The mass is shown as function of thecentral energy density. When both g and B are set to zero there is no rho condensate inside thestars and the maximum mass achieved is 2 .
05 solar mass. When g, B = 0 the condensate appears.The dotted lines indicate stars that contain some amount of rho condensate. magnetic field is kept constant at B = 7 . × G. We find that as we increase g , thecondensate appears much earlier i.e. more and more stars of the sequence have a condensateregion in their cores. At this point we have determined the onset of condensation butso far have not performed a full calculation of the star structure including a condensate.However, assuming a drastic softening of the equation of state, the biggest possible shift ofthe maximum mass is given by the star mass when condensation starts to set in, which, forthis specific model, yields a shift downward by about 0 . M ⊙ for a coupling of g = − ∗ mallick@fias.uni-frankfurt.de † [email protected] ‡ [email protected] § [email protected][1] Brown, G. E., & Rho, M., Phys. Rev. Lett. 66, 2720 (1991)[2] Hatsuda, T., & Lee, S. H., Phys. Rev. C 46, R34 (1992)[3] Saito, K., & Thomas, A. W., Phys. lett. B 327, 9 (1994)[4] Jean, H. C., Piekarewicz, J., & Williams, A. G., Phys. Rev. C 49, 1981 (1994)[5] Agakichiev, G. et al., Phys. Rev. Lett. 75, 1272 (1995)[6] Cassing, W., Ehehalt, W., & ko, C. M., Phys. lett. B 363, 35 (1995)[7] Srivastava, M. K., Sinha, B., & Gale, C., Phys. Rev. C 53, R567 (1996)[8] Koch, V., & Song, C., Phys. Rev. C 54, 1903 (1996)[9] Bratkovskaya, E. L., & Cassing, W., Nucl. Phys. A 619, 413 (1997)[10] Cassing, W., Bratkovskaya, E. L., Rapp, R., & Wambach, J., Phys. Rev. C 57, 916 (1998)[11] Bratkovskaya, E. L., Ko, C. M., Phys. Lett. B 445, 265 (1999)[12] Trnka, D. et al., Phys. Rev. Lett. 94, 192303 (2005)[13] Naruki, M. et al., Phys. Rev. Lett 96, 092301 (2006)[14] Huguet, R., Caillon, J. C., & Labarsouque, J., Phys. Rev. C 75, 048201 (2007)[15] Chernodub, M. N., Lect.Notes Phys. 871, 143 (2013)
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