Constraints for weakly interacting light bosons from existence of massive neutron stars
CConstraints for weakly interacting light bosonsfrom existence of massive neutron stars
M. I. Krivoruchenko , F. ˇSimkovic , , Amand Faessler Institute for Theoretical and Experimental Physics , B. Cheremushkinskaya 25117218 Moscow, Russia Bogoliubov Laboratory of Theoretical Physics , JINR141980 Dubna , Moscow Region , Russia Department of Nuclear Physics and Biophysics , Comenius University , Mlynsk´a dolina F1SK–842 48 Bratislava , Slovakia and Institut f¨ur Theoretische Physik , T¨ubingen Universit¨at , Auf der Morgenstelle 14D-72076 T¨ubingen, Germany
Theories beyond the standard model include a number of new particles some of which might belight and weakly coupled to ordinary matter. Such particles affect the equation of state of nuclearmatter and can shift admissible masses of neutron stars to higher values. The internal structureof neutron stars is modified provided the ratio between coupling strength and mass squared of aweakly interacting light boson is above g /µ ∼
25 GeV − . We provide limits on the couplingswith the strange sector, which cannot be achieved from laboratory experiments analysis. When thecouplings to the first family of quarks is considered the limits imposed by the neutron stars are notmore stringent than the existing laboratory ones. The observations on neutron stars give evidencethat equation of state of the β -equilibrated nuclear matter is stiffer than expected from many-bodytheory of nuclei and nuclear matter. A weakly interacting light vector boson coupled predominantlyto the second family of the quarks can produce the required stiffening. PACS numbers: 11.10.Kk, 14.70.Pw, 26.60.Kp, 97.60.Jd
Dark energy explains the accelerating expansion of theUniverse. The density of dark energy ρ D ≈ . may correspond to a fundamental scale λ D = ρ − / D ≈ . × − m [1, 2, 3, 4]. Theoretical schemes with ex-tra dimensions suggest modifications of gravity below λ D and a multitude of states with masses above 1 /λ D veryweakly coupled to members of multiplets of the standardmodel. Scales significantly below λ D represent the inter-est for supersymmetric extensions of the standard modelwhich include generally a number of new particles, suchas the leading dark matter candidate neutralino. Typi-cally, new particles are expected with masses above sev-eral hundred GeVs or even higher. However, light parti-cles may exist also, such as a neutral very weakly coupledspin-1 gauge U -boson [5] that can provide annihilation oflight dark matter and be responsible for the 511 keV lineobserved from the galactic bulge [6, 7].Deviations from the inverse-square Newton’s law areparametrized often in terms of the exchanges by hypo-thetical bosons also. Constraints on the deviations fromNewton’s gravity have been set experimentally in thesub-millimeter scale [8, 9, 10, 11, 12, 13] and down to dis-tances ∼
10 fm where effects of light bosons of extensionsof the standard model can be expected [14, 15, 16, 17, 18].Constraints on the coupling constants from unobservedmissing energy decay modes of ordinary mesons are dis-cussed in Ref. [19].Bosons with small couplings escape detection in mostlaboratory experiments. However, bosons interactingwith baryons modify the equation of state (EOS) of nu-clear matter. Their effect depends on the ratio betweenthe coupling strength and the boson mass squared, so a weakly interacting light boson (WILB) may influence thestructure of neutron stars even if its baryon couplings arevery small.The effect of a vector boson on the energy density ofnuclear matter can be evaluated by averaging the corre-sponding Yukawa potential: E I = 12 (cid:90) d x d x ρ ( x ) g π e − µr r ρ ( x ) , (1)where ρ ( x ) = ρ ( x ) ≡ ρ is the number density of ho-mogeneously distributed baryons, r = | x − x | , g is thecoupling constant with baryons, and µ is the boson mass.A simple integration gives E I = V g ρ µ , (2)where V is the normalization volume.The coherent contribution to the energy density of nu-clear matter from vector WILBs should be compared tothat from the ordinary ω -mesons. In one-boson exchangepotential (OBEP) models, the nucleon-nucleon repulsivecore at short distances r (cid:46) b = 0 . ω -meson exchanges. Respectively, the ω -meson plays a fun-damental role in nuclear matter EOS. In the mean-fieldapproximation, the contribution of ω -meson exchanges tothe energy has the form of Eq.(2), with g and µ replacedby the ω -meson coupling g ω and the mass µ ω .The N N interactions are described with g ω /µ ω =175 GeV − [20]. The relativistic mean field (RMF) model[21] gives g ω /µ ω = 196 GeV − . The compression modu-lus of nuclear matter K = 210 ÷
300 MeV is consistent a r X i v : . [ h e p - ph ] J u l with g ω /µ ω = 125 ÷
180 GeV − [22]. Stiff RMF mod-els use g ω /µ ω up to 300 GeV − [23]. If we wish to staywithin current limits and do not want to modify the inter-nal structure of neutron stars qualitatively, as describedby realistic models of nuclear matter, one has to requirethat vector WILBs fulfill constraint g µ (cid:46) g ω µ ω ≈
200 GeV − . (3)A similar reasoning applies to scalar WILBs whichhave to compete with the standard σ -meson exchange.In OBEP models, the long-range attraction between nu-cleons is attributed to σ -meson exchanges. The contri-bution of the σ -mesons to the interaction energy has theform of Eq.(2), with g and µ replaced by the σ -mesoncoupling g σ and the mass µ σ . The sign of the contribu-tion must be negative because of the attraction. Also, ρ should be replaced by the scalar density. In RMFmodels, the σ -meson mean field decreases the nucleonmass. The effect depends on the ratio g /µ also andproduces an additional decrease of the energy at fixedvolume and baryon number. The empirical values of theratio g σ /µ σ are 40 ÷
60% higher than those of the ω -meson [20, 21, 22, 23]. The internal structure of neutronstars is not modified significantly provided the couplingstrength g and mass µ of scalar WILBs fulfill constraint g µ (cid:46) g σ µ σ ≈
300 GeV − . (4)The deviations from the Newton’s gravitational poten-tial are usually parametrized in the form V ( r ) = − Gm m r (cid:16) α G e − r/λ (cid:17) . (5)The second Yukawa term can be attributed to new bosonswith Gm α G = ± g / (4 π ) and λ = 1 /µ , where + / − stands for scalar/vector bosons and m is the proton mass.On Fig. 1 we show regions in the parameter spaces( g , µ ) and ( α G , λ ) allowed for WILBs by the constraint(3). The constraint for scalar bosons is close to (3). Con-straints from other works [10, 11, 12, 13, 14, 15, 16, 17,18] are shown also.An increase of g (a decrease of µ ) of scalar WILBs in-creases the negative contribution to pressure, makes EOSof nuclear matter softer, makes neutron stars less stableagainst gravitational compression. The ratio g /µ can-not be increased significantly above the limit (4), sincethe maximum mass of the neutron star sequence cannotbe moved below masses of the observed pulsars.An increase of g (a decrease of µ ) of vector WILBs,conversely, increases the positive contribution to pres-sure, makes EOS of nuclear matter stiffer, makes neutronstars more stable against gravitational compression anddrives the maximum mass of neutron stars up.In case of vector bosons, it is less obvious what kind ofthe observables confronts to high ratios g /µ . FIG. 1: (color online) Constraints on the coupling strengthwith nucleons g / (4 π ) and the mass µ (equivalently α G and λ ) of hypothetical weakly interacting light bosons: are con-straints from Ref. [10], - from Ref. [11], - from Ref. [12], - from Ref. [13], and are constraints from low-energy n − Pb scattering [16] and [14], respectively, - from Ref.[17], - from Ref. [15], and are constraints from spec-troscopy of antiproton atoms [16], and are constraintsfrom near-forward pn scattering for vector and scalar bosons,respectively [18]. The axes are in the log scale. The internalstructure of neutron stars is not modified qualitatively pro-vided the boson coupling strengths with baryons and masseslie at g /µ <
200 GeV − beneath the highlighted area . Realistic models of nuclear matter are based on thenucleon-nucleon scattering data. They split into soft andstiff models according to the rate the pressure increaseswith the density. The soft models correspond to low max-imum masses of neutron stars ∼ . (cid:12) , while the stiffmodels give the upper limit around ∼ . (cid:12) .The problem on the softness of nuclear EOS has re-ceived new interest due the analysis of strange particleproduction in heavy-ion collisions. The data at differentbombarding energies lead to the conclusion that EOS ofnuclear matter must be soft at densities two to threetimes of the saturation density [24, 25, 26]. Data on thetransverse and elliptic flows in heavy-ion collisions sug-gest a soft EOS around the saturation, too [27].Last years observations of pulsars with high masseshave been reported. The most massive pulsars are PSRB1516+02B in the globular cluster M5 with the massof 1 . +0 . − . M (cid:12) and PSR J1748-2021B in the globu-lar cluster NGC 6440 with the mass of 2 . ± .
22 M (cid:12) [28]. The mass of rapidly rotating neutron star in thelow mass X-ray binary 4U 1636-536 is estimated to be M = 2 . ± . (cid:12) [29]. The mass and radius of theX-ray source EXO 0748-676 are constrained to M ≥ . ± .
28 M (cid:12) and R ≥ . ± . β -equilibrated nuclear matter is stiff.The controversy between the conclusions on the soft-ness of nuclear matter as derived from the laboratoryexperiments and on the stiffness of the β -equilibratednuclear matter as derived from the astrophysical obser-vations has been of interest since after the discovery ofmillisecond pulsars [31, 32] and earlier [33].Current models use to match EOS of neutron matterwith a soft EOS at the saturation density and a stiff EOSat higher densities. Such models are in the qualitativeagreement with laboratory and astrophysical data [34].High densities provide favorable conditions for the oc-currence of exotic forms of nuclear matter: pion, kaon,and dibaryon condensates, quark matter. New degrees offreedom make EOS softer, pushing the maximum massof neutron stars down. The recent astrophysical observa-tions seem to exclude the softest EOS e.g. based on theclassical Reid soft core model [35] and make it problem-atic to accommodate the exotic forms of nuclear matterwith masses and radii of the observed pulsars [30] (seehowever [36]).The in-medium masses of vector mesons depend on thedensity. Assuming µ is a function of ρ and using Eq.(2),one may evaluate the ω -meson contribution to pressure: P I = g ρ µ (cid:18) − ρµ ∂µ∂ρ (cid:19) . (6)A positive shift of the ω -meson mass decreases the pres-sure and leads to a softer EOS, whereas a negative shiftleads to a stiffer EOS. The data on the dilepton pro-duction in heavy-ion collisions do not give evidence forsignificant mass shift [37], so the observed stiffness of the β -equilibrated nuclear matter can hardly be attributedto in-medium modifications of the vector mesons.The realistic models of neutron matter discussed inRef. [34] neglect hyperon channels e.g. reactions Σ − → n + e + ¯ ν e . In RMF models [22, 38, 39], the β -equilibriumof hyperons drops the limiting mass by 0 . ÷ . (cid:12) .This result is in accord with hypernuclear data and otherrecent calculations [40, 41, 42]. The inclusion of the β -equilibrium for all baryons brings difficulties in reproduc-ing the observed masses of neutron stars.Coming back to vector WILBs, we see that their exis-tence is desirable to provide additional stiffening of the β -equilibrated nuclear matter.The Compton wavelength of WILBs is assumed to begreater that the radius of nuclei e.g. 1 /µ > R ≈ ≈ (30 MeV) − for the lead. The contribution of WILBsto the binding energy of nuclei then equals ∼ A g /R like for photons. Since g / (4 π ) is much smaller than thefine structure constant, the effect of WILBs on nucleiis negligible. Above ∼ MeV the coupling constantof WILBs is close to unity, so WILBs there are neitherweekly interacting nor light.WILBs thus do not modify observables in laboratoryexperiments on hypernuclear physics, nuclear structureand heavy-ion collisions, since their baryon couplings arevery small. The characteristic scale of the parameters ofthese particles is fixed by the upper limit (3).The mass-radius relations for non-rotating neutronstars are shown on Fig. 2 for four values of the ratio
FIG. 2: (color online) Mass of non-rotating neutron starsas a function of radius: - RMF model of hyperon materwith the compression modulus K = 300 MeV [22]; - thesame as including a flavor-singlet vector WILB coupled tobaryons with g /µ = 25 GeV − (1 / - the same as with g /µ = 50 GeV − ; - the same as with g /µ = 100 GeV − . The highlighted area within M = 1 . +0 . − . M (cid:12) shows the mass constraint from PSRB1516+02B. The neutron star sequences should cross the ro-tation speed limit curves shown for pulsar PSR B1937+21with the rotation frequency of ν = 642 Hz [51] and the neu-tron star XTE J1739-285 showing X-ray burst oscillationswith frequency of ν = 1122 Hz [52]. The mass-dependentlower bound on radii of neutron stars determined from theblackbody radiation of RX J1856.5-3754 is shown. The dot-ted straight lines z = 0 . ÷ . z = 0 .
35 measured for EXO0748-676 constrains the radii of neutron stars by
R >
12 kmand, respectively, masses [30]. g /µ = 0 , ,
50 and 100 GeV − of a flavor-singlet vec-tor WILB. At densities below ρ drip = 4 . × g / cm the matter represents an atomic lattice. WILBs donot modify properties of nuclei and the Baym-Pethick-Sutherland EOS [43], accordingly. At densities ρ drip <ρ (cid:46) ρ nucl = 2 . × g / cm , atomic lattice coexistswith neutron liquid. The matter at ρ drip < ρ (cid:46) ρ nucl isdescribed by the Baym-Bethe-Pethick EOS [44]. Above ρ nucl , nuclei dissolve and the matter is described by the β -equilibrated hyperon liquid with the compression mod-ulus K = 300 MeV [22]. WILBs contribute to the energydensity and pressure above ρ drip , as described by Eqs.(2)and (6) with ∂µ/∂ρ = 0, through the spatially extendednucleon and hyperon liquid components of the neutronstar matter. The vector WILBs give equal contributionsto the chemical potentials of the octet baryons and do notviolate the chemical β -equilibrium. [53] The inclusion ofsuch vector bosons does therefore not change compositionof the neutron star matter.The highlighted area at the upper left corner of Fig.2 excludes within general relativity the radii of neutronstars below the Schwarzschild radius. The causal limitexcludes the area R (cid:46) GM [45]. The rotation speedlimit curves are constructed using the modified Keplerianrate ν max (cid:39) M/ M (cid:12) ) / (10 km /R ) / Hz, whichaccounts for the deformation of rotating neutron starsand effects of general relativity [46].It is seen from Fig. 2 that, despite we selected EOSwith the high compression modulus, the neutron star se-quence with g /µ = 0 contradicts to the mass mea-surement of PSR B1516+02B. It gives a very low massof the neutron star from the blackbody radiation radiusconstraint also, which confronts with the lower limit of ∼ .
85 M (cid:12) for masses of protoneutron stars [49].The value of g /µ = 200 GeV − gives the maximummass slightly above 3 . (cid:12) . However, the neutron starsequence does not cross the rotation speed limits, whilethe red shift remains always below z = 0 .
35. The upperbound (3) is thus critical for the internal structure ofneutron stars. [54]The vector WILBs increase the minimum and max-imum mass limits and radii of neutron stars and areable to bring in the agreement models of hyperon mat-ter which are soft with the astrophysical observationson neutron stars which require a stiff EOS. The ratio g /µ ≈
50 GeV − might be reasonable. Such a value,however, clearly contradicts to the laboratory constraintsshown on Fig. 1 in the entire mass range µ = 10 − to10 MeV.The in-medium modification of masses of vector bosonsmodify EOS. Vector WILBs can be compared to the ω -meson where | δµ ω | /µ ω (cid:46) . δµ ∼ g /g ω µ ω δµ ω . The in-medium modification issmall provided | δµ | (cid:46) µ i.e. g /µ (cid:46) GeV − , so inthe region of interest (3) holds for the vacuum masses.The laboratory constraints shown on Fig. 1 do not ap-ply to WILBs coupled to hyperons. A vector WILB cou-pled predominantly to the second family of the quarksmakes hyperon matter EOS stiffer also. It contributesdifferently to chemical potentials of the octet baryonsand suppresses the hyperon content of the neutron starmatter due the additional repulsion. One can expect theratio g /µ should be close to or higher than that esti-mated above ( ∼
50 GeV − ). In such a scenario, nuclearmatter without hyperons can be treated as reasonable ap-proximation for the modeling structure of neutron starsin the β -equilibrium also e.g. on line with Ref. [34] wheremodels with the blocked hyperon channels are shown to be in the qualitative agreement with the laboratory andastrophysical constraints.Gauge bosons interact with the conserved currentsonly, but flavor is not conserved. A WILB coupled tothe second family of the quarks cannot be a gauge boson,so it does not arise naturally in the current theoreticalschemes. Here, we do not have a goal whatsoever to gobeyond the phenomenological analysis.Hypernuclear data restrict N Y potentials, whereas theinteraction between hyperons
Y Y is not known experi-mentally. The stiffness of the hyperon matter might alsobe attributed to the φ (1020)-meson exchange, whose cou-pling to the nonstrange baryons is suppressed accordingto the Okubo-Zweig-Iizuka rule (see, however, [50]).Summarizing, we have assumed the existence and de-rived constraints for a new boson that couples to nuclearmatter. Such a particle contributes, by its coherent forceamong nuclear constituents, to a modified EOS and af-fects the structure of neutron stars. The neutron starsexclude scalar bosons with the coupling strengths andmasses above the line on Fig. 1, whereas in a nar-row band below it and above a vector boson coupled toquarks of the second family could modify the EOS ina direction favored by the observed masses and radii ofneutron stars. The astrophysical constraints in the non-strange sector are less stringent than the most accuratelaboratory ones. They are unique, however, for scalarWILBs in the strange sector. The region of validity ofthe astrophysical constraints extends from λ ∼
10 fmto about 10 km. Detailed studies of manifestations ofnew bosons in astrophysics, physics of neutron stars, andhadron decays to energy missing channels can shed morelight on the existence of WILBs and their possible effecton the structure of neutron stars.
Acknowledgments
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