Small radii of neutron stars as an indication of novel in-medium effects
aa r X i v : . [ nu c l - t h ] S e p Small radii of neutron stars as an indication of novel in-medium effects
Wei-Zhou Jiang,
1, 2, ∗ Bao-An Li, † and F. J. Fattoyev
2, 3, ‡ Department of Physics, Southeast University, Nanjing 211189, China Department of Physics and Astronomy, Texas A&M University-Commerce, Commerce, TX 75429, USA Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USA (Dated: September 8, 2018)At present, neutron star radii from both observations and model predictions remain very uncertain.Whereas different models can predict a wide range of neutron star radii, it is not possible for mostmodels to predict radii that are smaller than about 10 km, thus if such small radii are establishedin the future they will be very difficult to reconcile with model estimates. By invoking a new termin the equation of state that enhances the energy density, but leaves the pressure unchanged wesimulate the current uncertainty in the neutron star radii. This new term can be possibly dueto the exchange of the weakly interacting light U-boson with appropriate in-medium parameters,which does not compromise the success of the conventional nuclear models. The validity of this newscheme will be tested eventually by more precise measurements of neutron star radii.
PACS numbers: 26.60.-c, 14.70.Pw, 97.60.Jd
Neutron star (NS) is a unique place to test fundamen-tal forces at the extremes of matter density, gravity andmagnetic fields. Unfortunately, uncertainties in both theEquation of State (EOS) of super-dense nuclear matterand the strong-field gravity strongly interplay with eachother in determining observational properties of neutronstars, for the latest review, see, e.g., [1]. For instance,in a simple version of modified gravity where the non-Newtonian gravity exists, neutron stars could have verydifferent structures compared to predictions using Ein-stein’s General Relativity (GR) theory of gravity [2–4].The radius of a neutron star is one of the most importantobservables sensitive to the underlying nuclear EOS andgravity theories used. Currently, within GR the radiusof a canonical NS has been predicted to be roughly from11 to 15 km [5–9] depending on the EOS used. Providedthe third family of compact stars known as strange starsexist, their radii could be as small as 7 or 8 km [10–12], although these models normally predict star massesmuch smaller than the masses of observed massive neu-tron stars. Thus, the measurement of NS radii playsa very important role in resolving several issues in fun-damental physics. Unfortunately, the extraction of NSradii from observations still suffers from large systematicuncertainties [13] involved in the distance measurementsand theoretical analyses of the light spectrum [6, 14–16].Consequently, a wide range of the radius with the massaround 1.4 M ⊙ has been reported [16–24]. In particu-lar, using the thermal spectra from quiescent low-massX-ray binaries (qLMXBs) Guillot and collaborators ex-tracted NS radii of R NS = 9 . ± . M ⊙ NS the extracted ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: ff[email protected] radii are 10 . +0 . − . km [25]. It is important to note thatat the moment no consensus has been reached yet on theextracted NS radii. For instance, Bogdanov found a 3- σ lower limit of 11.1 km on the radius of the PSR J0437-4715 [26], and Poutanen et al. got a lower limit of 13km for 4U 1608-52 [27]. Whether the radii of canoni-cal neutron stars can be as low as 10 km have been aninteresting topic of hot debate during the last few years.Whereas the situation has been significantly improvedover the last few years, the systematic errors have beenhindering severely the accurate determination of the NSradii from astrophysical observations. However, if the ex-istence of NSs with small radii is firmly established, theywould pose a severe challenge to the current models ofthe nuclear EOSs. While it is not very difficult to satisfythe maximum mass constraint, to our best knowledge, nonuclear models available with best-fit parameters to datecan reproduce the small NS radius constraint. There areonly a few microscopic models or approaches that— ei-ther disregard some of the nuclear physics constraints [5]or adjust the high-density part of the EOS using variouspolytropes to match the whole density profile obtainedfrom observation [28]—can account for both constraints.From the nuclear physics standpoint, the small NS ra-dius requires certain softness of the EOS of the isospin-asymmetric nuclear matter. As one of the basic blocksof the EOS of asymmetric matter, nuclear symmetry en-ergy around 1-2 times the saturation density of nuclearmatter plays a dominating role in determining the radiiof neutron stars [6]. A softening of the symmetry en-ergy can lead to an appreciable decrease of the NS ra-dius [7]. However, with the maximum NS mass held ap-proximately at a constant, most non-relativistic and rel-ativistic models that are facilitated with soft symmetryenergies can only bring the radii of canonical NSs downto about 12-13 km [8, 9, 17, 21]. Moreover, it becomesvery difficult to further reduce the NS radius by furthersoftening the symmetry energy. In particular, one wouldthen encounter the stability problem in the NS matterEOS when the symmetry energy becomes too soft [3].To further reduce the NS radius, one could imagine toreduce the pressure of the isospin-symmetric part of theEOS in the intermediate density region. But the spacein so doing is actually limited by the saturation prop-erties of nuclear matter and the constraint on the EOSof dense nuclear matter extracted from studying nuclearcollective flow in high energy heavy-ion reactions [29].Furthermore, as we can see from the empirical relation Rp − / ( ρ B ) ≈ C ( ρ B ) between the NS radii R and thepressure p ( ρ B ) with C ( ρ B ) being a constant at a givenbaryon density ρ B [7], such a reduction is also rather in-efficient, since the isospin-symmetric EOS contributionto the total NS matter pressure p ( ρ B ) is relatively smallin the relevant density region, where this empirical re-lation holds. Moreover, even if the significant reductionwere allowed for the total pressure, the limited decreaseof the NS radius would be at the cost of a large reductionof the NS maximum mass, because the significantly re-duced pressure needs the corresponding reduction of theNS mass to balance the gravity therein, and also becausethe NS mass is the total energy integrated in a nutshellwith the reduced radius. A significant reduction in themaximum mass is certainly disfavored by the recentlydiscovered massive neutron stars of 2 M ⊙ that require astiff EOS [30, 31].Facing currently with this severe theoretical issue, weexplore in this work a new possibility to soften the EOS:adding a new term in the EOS that enhances the en-ergy density while keeping the pressure unchanged. In sodoing, the enhancement of the energy density requiressome shrinkage in NS radius without reducing signifi-cantly the NS maximum mass. Since the pressure is p = ρ B ∂ ( ǫ/ρ B ) /∂ρ B , to keep the pressure unchanged wemodify the energy density ǫ by ǫ = ǫ + C L ρ B , (1)where ǫ is the base energy density given by any model.The second term linear in density is an addition to thebase EOS with C L being the amending coefficient . Aswe shall explain, this modification to the EOS of isospin-asymmetric nuclear matter can be realized by consideringthe interaction added by a vector boson with appropriatein-medium parameters.In principle, the modification given in Eq. (1) could beadded to any nuclear model. In this work, we just demon-strate the effects using several typical relativistic mean-field (RMF) models. The RMF models under considera-tion include the SLC, SLCd [32], SL3 [9] and NL3040 [33].The SL3 and the NL3040 have similarly stiff EOSs athigh density and both give large NS maximum masses ofmore than 2.6 M ⊙ , while the SLC and the SLCd featurethe same EOS of symmetric matter within the constraintsobtained from analyzing the collective flow in relativisticheavy-ion collisions [29]. The only difference between theSLC and the SLCd is that the latter has a softer sym-metry energy. In a similar fashion, the NL3040 was alsobuilt from the original NL3 to feature a softer symmetry C L (MeV)0100200300 SLCd p ( M e V / f m ) SL3 ε (fm -4 ) FIG. 1: The relation between the matter pressure and theenergy density with RMF models SLCd and SL3. Note thatthe displacement of energy density with a given C L is thesame at a given density but not at a constant pressure. energy. The stiffness of the symmetry energy is normallymeasured by its density slope at the saturation densityof nuclear matter ρ , L = 3 ρ ( ∂E sym ( ρ B ) /∂ρ B ) ρ . Thevalue of L for the SLC and SL3 is 92.3 and 97.1 MeV,while it is 61.5 and 45.0 MeV for the SLCd and NL3040,respectively. For a comparison, it is interesting to notethat currently the most probable value of L is in therange of 40 . L .
70 MeV according to recent analy-ses of various terrestrial experiments and astrophysicalobservations, see, e.g., Refs. [34–40] and Ref. [41] for acomprehensive review. Thus, the SLC and SL3 are obvi-ously too stiff while the SLCd and NL3040 are consistentwith the existing constraints in terms of their L values.Nevertheless, they are all appropriate for the purposes ofthis study.Shown in Fig. 1 are two examples of the EOS, i.e.,pressure versus energy density, with the SLCd and SL3parameter sets. It is seen that at a constant pressure,the amending term can soften the EOS considerably,i.e., reducing the slope of the pressure with respect tothe energy density, especially with the SLCd. However,the relative effect of the amending term goes down withthe increasing density because it is just linear in densitywhile the EOS of usual nuclear models evolves generallywith the density squared. We emphasize that the EOSsoftening scheme considered here is quite different fromthe usual mechanisms mentioned in the introduction. Inparticular, typical phase transitions to matter with newdegrees of freedom normally reduce the maximum massdramatically, but often keep the NS radius more or lessthe same because the phase transitions usually occur inthe small inner core of NSs. Of course, exceptions mayexist when the new degrees of freedom, such as the ∆resonances, can emerge at a relatively low density [42].It is interesting to note that by using Lindblom’s inver-sion algorithm [43] Chen and Piekarewicz were recentlyable to obtain a softened EOS from the given small NSradii [44].We now examine effects of the amending term on theradii of neutron stars. As in Ref. [20], here we considerthe simplest model of neutron stars consisting of justneutrons, protons and electrons. Shown in Fig. 2 are themass-radius (MR) trajectories of neutron stars with theamending term within various RMF models. The amend-ing coefficient is exemplified as C L = 100, 200 and 300MeV, and the results with original models are displayedwith C L = 0. Comparing results of the original models,we see that the softening of the symmetry energy may re-duce the NS radius by as large as 1.5 km for a canonicalNS when L is reduced from 97.1 to 45 MeV. The spacefor further reducing the slope parameter L is small, andin fact, the further reduction in L has a very limited ef-fect in decreasing the NS radius. Moreover, it is seenthat even with the significant softening of the symmetryenergy within the original models the NS radius is stillfar above 9 . ± . et al. [24]. Ifsuch small radii are established, it is then interesting tosee that the amending term can indeed further reduce theNS radius. Obviously, the role of the amending term issimilar in all models: the larger the amending coefficientis, the more is the reduction of the NS radius. With thesame amending coefficient in different models, the shiftedmagnitude is also similar. Typically, the amending coef-ficient, varying up to 300 MeV, can cause a reduction ofabout 3 km in the NS radius. With the amending coeffi-cient of C L = 300 MeV, we see that the MR trajectorieswith the SLC and SLCd fall into the regime extractedby Guillot et al. [20]. We see from Fig. 2 that the NSmaximum mass is still not reached in the SLC and SLCdmodels before reaching the causality boundary, becausethe allowed nucleon density has a maximum in the con-struction of such models to meet the chiral limit [9]. Theremoval of such a limiting density may bring the NS max-imum mass closer to the causality limit. With the largeramending coefficient, the MR trajectories for the SL3and NL3040 can be very close to the upper margin of theextracted regime. Notably, we see that the MR trajec-tories are not clearly away from the causality constraint,though the amending coefficient causes the decrease ofthe NS maximum mass. Here, the moderate reduction of the NS maximum mass is just because of the softeningof the EOS, namely no excess of pressure can resist theadditional gravity arising from the increase of the energydensity. C L (MeV) 0 100 200 300 SLC
L=92.3 MeV M / M ⋅ SLCd
L=61.5 MeV
12 10 15
SL3
L=97.1 MeV
10 15
NL3040
R (km)
L=45.0 MeV
FIG. 2: (Color online) The mass-radius trajectories of neu-tron stars with RMF models: SLC, SLCd, SL3 and NL3040.In each panel, the amending coefficient C L is taken to be thevalues 0, 100, 200, 300 MeV, respectively. The slope param-eter of the symmetry energy is also labeled in each panel. In the RMF framework, the amending term in Eq. (1)can be understood as a specific in-medium effect. Similarto the analysis in Refs. [45–47], the amending term, in-corporated into the vector potential, leads to the density-dependent coupling constant of the vector ( ω ) meson g ω ( ρ B ) = ( g ω ω + 2 C L ) m ω /ρ B . (2)This relation indicates that the larger C L is responsiblefor the stronger density dependence of the coupling con-stant, and with the increase of density, the in-mediumeffect decreases with the growing ω . Fig. 3 shows thedensity-dependent coupling constant for the models con-sidered. For a comparison, the g ω ( ρ B ) obtained from theDirac-Brueckner (DB) potential of Bonn A is also shownin the figure. It is seen that the density dependence, simi-lar to the one from the DB potential, is needed to producea significant reduction of the NS radius. We can infer, in-deed, that the density dependence here is stronger thanthat from the DB potential, because the latter owns apartial cancelation of the density dependencies betweenthe scalar and vector potentials [45–47]. In addition, wesee that the in-medium effect of all cases tend to vanishat high densities. This means that the decrease of the NSradius is dominated by the modification of the amendingterm to the low-density EOS. For the dropping of the NSmaximum mass, it can then hopefully be cured by mod-ifying the high-density component of the EOS of nuclearmatter. C L (MeV)0100200300Bonn A SLC/SLCd0102030 1 2 3 4 5NL3040 ρ B / ρ g ω ( ρ ) FIG. 3: (Color online) The density-dependent vector couplingconstant incorporated from the amending term in RMF mod-els: SLC/SLCd and NL3040. The density-dependent couplingconstant obtained from the DB potential of Bonn A is alsogiven for comparison.
The DB potential is obtained by solving the two-bodycorrelations. One may expect that the many-body ratherthan just two-body correlations may generate more in-medium effects [5]. It should, however, be pointed outthat the arbitrarily strong density dependence usuallylacks observable grounds. In the present RMF models,one should evade the violation of the low-energy con-straints in finite nuclei from the in-medium effect inducedby the amending term. One possible solution is to resortto the exchange of the very light and weakly interactingboson that is undetected to date. A favorite candidate isthe U-boson, which is actually a particle beyond the stan-dard model [48, 49]. The light U-boson, first proposed byFayet [48], might be regarded as the mediator of the pu-tative fifth force [49–51]. Recently, the possibility of theMeV dark matter with the light U-boson mediation wasconsidered to account for the bright 511 keV γ emissionof positron annihilations from the galactic bulge [52–57], despite that there are a number of conventional astro-physical sources for positron annihilations, see Ref. [58]and references therein. In the past, the significant effectsof the U-boson in neutron stars were predicted [2–4], al-beit the boson should couple to baryons very weakly. Themodification of the EOS due to the U-boson in the mean-field approximation simply reads ǫ UB = 12 (cid:18) g u m u (cid:19) ρ B , (3)with g u /m u being the ratio of the U-boson coupling con-stant and its mass.By assuming a density-dependent U-boson mass of m u = g u ρ B / C L , we realize the linear density depen-dence of the amending energy density. If the U-bosonhas a very light bare mass of m u in free space, its in-medium mass can be given as m u = g u ρ B / C L + m u . (4)In general, the in-medium effect of the boson parame-ters is associated with the contribution of the intermedi-ate states. In the RMF theory, the in-medium effect isusually attributed to the nonlinearity of the meson self-interactions [59]. The in-medium effect for the bosonmay also be realized by carefully choosing the nonlinearself-interacting terms. We leave this problem for a futurestudy.Regarding the possible violation of the low-energy nu-clear constraints for finite nuclei, one can avoid it by lim-iting the weak interaction strength of the U-boson. Forinstance, if g u is 0.01, the interaction strength is abouttwo orders of magnitude weaker than the electromagneticinteraction, and such weak interaction is not able to af-fect properties of finite nuclei. In this sense, the presentscheme to invoke the U-boson does not compromise thesuccess of optional nuclear models. The interesting ques-tion is what behavior of the U-boson mass will allowsuch a weak interaction strength. Shown in Fig. 4 isthe density-dependent mass of the U-boson as a functionof density, for g u = 0 .
01. Here, as an example, the smallbare boson mass is taken to be as 0.1 MeV, which is just afree parameter. We see that for the given coefficients C L ,the in-medium mass is just within a few MeV. With theincrease of the C L , the in-medium mass of the U-bosondecreases. For C L = 300 MeV, the U-boson mass is closeto 1 MeV, which is consistent with that considered inRef. [4].We should say that the role of the vector boson in soft-ening the EOS eventually is rather pragmatic, albeit pu-tatively accounting for small NS radii. Usually, the vectorboson is a source of the pressure, while here it gives zerocontribution. From the point of view of nuclear many-body approaches, the repulsion of the vector meson canbe softened in the intermediate density region due to thecorrelation effect from the intermediate-state contribu-tions, and such a softening can be simulated in the RMFtheory by invoking the nonlinear self-interacting terms.At high densities where the intermediate-state contribu-tion is small due to the Pauli blocking the reduction ofthe in-medium effect of the vector boson will then recoverthe repulsion which may enable a stiffer EOS at high den-sities. In this sense, if we use a more complicated density-dependent term in Eq.(1) in the high density region, thecorresponding U-boson can then stiffen the high-densityEOS and the U-boson mass would be also much closerto a constant at high densities. While we use a simpleamending term in Eq.(1) to demonstrate the reductionof the NS radius, it should not represent that the corre-sponding vector U-boson would simply soften the EOS. C L (MeV)100200300400500 ρ B / ρ m u ( M e V ) FIG. 4: (Color online) The in-medium mass of the U-bosonwith various C L as a function of density. Here, g u = 0 .
01 and m u = 0 . The NS radius can attain larger reduction by furtherincreasing the amending coefficient. In our analysis herewe have invoked the special amending term that doesnot affect the pressure. Generally, other forms could alsobe optional, as long as they result in the reduction ofthe NS radius. A favorable form should reduce the low-density pressure and satisfy all constraints on the EOSof nuclear matter at saturation density. To meet thesedemands, one can consider the light scalar boson. Notethat the scalar boson, like the scalar meson, can also modify the mass of baryons. We have checked that theinclusion of such a light scalar boson can again reducethe NS radius by 0.5-1.5 km, depending on the modelused. To keep the pressure positive definite, the couplingconstant of the scalar boson should be much weaker (e.g., g s = 0 . Acknowledgement
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