The d^*(2380) in neutron stars - a new degree of freedom?
aa r X i v : . [ nu c l - t h ] M a r The d ∗ (2380) in neutron stars - a new degree of freedom? I. Vidaña a , M. Bashkanov b , D.P. Watts b , A. Pastore c a INFN Sezione di Catania. Dipartimento di Física, Università di Catania, Via Santa Sofia 64, 95123 Catania, Italy b School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD,UK c Department of Physics, University of York, Heslington, York, Y010 5DD, UK
Abstract
Elucidating the appropriate microscopic degrees of freedom within neutron stars remains an open question which impacts nuclearphysics, particle physics and astrophysics. The recent discovery of the first non-trivial dibaryon, the d ∗ (2380), provides a newcandidate for an exotic degree of freedom in the nuclear equation of state at high matter densities. In this paper a first calculationof the role of the d ∗ (2380) in neutron stars is performed, based on a relativistic mean field description of the nucleonic degreesof freedom supplemented by a free boson gas of d ∗ (2380). The calculations indicate that the d ∗ (2380) would appear at densitiesaround three times normal nuclear matter saturation density and comprise around 20% of the matter in the center of heavy stars withhigher fractions possible in the higher densities of merger processes. The d ∗ (2380) would also reduce the maximum star mass byaround 15% and have significant influence on the fractional proton / neutron composition. New possibilities for neutron star coolingmechanisms arising from the d ∗ (2380) are also predicted. Keywords: neutron stars; EoS; hexaquarks
1. Introduction
Neutron stars are valuable laboratories to study the funda-mental properties of dense nuclear matter at low temperatures.Despite the major advances in neutron star astronomy in recentyears, significant gaps remain in our knowledge of their inter-nal structure. The appropriate microscopic degrees of freedomare only established for densities comparable to atomic nuclei,which are found in the neutron star crust or at shallow depths intypical stars. The properties of this nucleonic matter can be con-strained by precision measurements of nuclei and nuclear reac-tions. This has led to a flurry of recent theoretical [1, 2, 3, 4, 5]and experimental [6, 7, 8] nuclear physics programmes to im-prove constraints on the equation of state for pure nucleonicmatter.However, for higher densities existing in the cores of heavyneutron stars or in neutron star mergers [9] “exotic” phases be-yond simple nucleonic matter are postulated. A number of pos-sibilities have been investigated theoretically, such as pion con-densates [10], kaon condensates [11], hyperonic matter [12], ∆ isobars [13, 14], quark matter [15] and dibaryonic matter [16].Some of the proposed exotic phases were resolved and re-jected. The astronomical observation of heavy neutron starswith ∼ M ⊙ [17, 18, 19], places stringent constraints on thecontribution of exotic components. Any extra degree of free-dom, in addition to protons, neutrons and leptons tends to makethe nuclear Equation of State (EoS) softer, consequently reduc-ing the maximum stable mass for a neutron star. Apparent in-consistencies with the observed neutron star properties when ∗ Corresponding author
Email address: [email protected] (M. Bashkanov) including some of the exotic phases has led to “puzzles” e.g., the “hyperon puzzle” [12] or the “ ∆ puzzle” [14]. However,such exotic phases are generally in competition and the inclu-sion of one process may strongly influence the role of the oth-ers. To reach firm conclusions on the microscopic compositionof dense nuclear matter a comprehensive investigation of all po-tential processes appears crucial. Achieving this goal would bevery timely. Neutron star mergers, such as recently observedby LIGO with both electromagnetic and gravitational wave sig-nals, have properties which depend strongly on the EoS [19]and are now proposed as the main site of galactic heavy ele-ment production. A better understanding of matter at high den-sities is also needed for studies of black hole formation [20] andneutron star black-hole mergers [21].Dibaryonic degrees of freedom have the potential for majorimpact on the properties of the dense nuclear matter. The exis-tence of dibaryons has long been predicted by phenomenolog-ical models based on the theory of the strong force, quantumchromodynamics (QCD). Dibaryons are predicted to be colour-less QCD objects comprising 6 valence quarks and having inte-ger spin. In the quantum mechanical environment within a neu-tron star a dibaryon would essentially become a stable particleand its bosonic nature would lead to a very di ff erent physicalbehaviour than for the (fermionic) nucleons. For example, thedibaryon can form into a Bose-Einstein condensate [22]. This,as well as the small predicted size of the dibaryons (6 quarksoccupying similar size to a single nucleon [38]) indicate theycould be a new way to distribute energy in matter under com-pression.Due to the potential significance of dibaryons in understand-ing the properties and structure of neutron stars it is very im- Preprint submitted to Physics Letters B March 21, 2018 ortant to explore the influence of the first non-trivial dibaryon,the d ∗ (2380) recently discovered by the Wasa-at-Cosy collabo-ration. In this work we perform the first evaluation of the e ff ectof this specific d ∗ (2380) dibaryon on the nuclear EoS. The pa-per will firstly summarise the evidence for the d ∗ (2380) andthen present the first theoretical investigation of its influence onneutron star properties.
2. The d ∗ (2380) dibaryon Theoretical predictions of dibaryons, such as the d ∗ , have along history. Calculations started with the pioneering work ofDyson and Xuong [23] in 1964 and have developed in manystudies since then [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,36, 37, 38, 39]. Despite the use of di ff erent ansatz in the models(constituent quarks, ∆∆ molecule, compact hexaquark with a σ -meson cloud) the majority of the models indicate the d ∗ is acompact object, I ( J P ) = + ) with a size comparable to thatof a single nucleon or somewhat larger in the case of molecularpicture. Calculation of the d ∗ dibaryon properties directly fromthe QCD lagrangian is ongoing but challenging. First LatticeQCD calculations indicate very strong ∆∆ attraction where theirspins are aligned, as expected from a d ∗ [40].Experimentally, a solid candidate for the d ∗ has only emergedin recent years. Measurement of the basic double-pionic fusionreactions pn → d π π and pn → d π + π − revealed a narrowresonance-like structure in the integrated cross section [41, 42,43, 44] at a mass M ≈ Γ ≈
70 MeV. The data are consistent with a I ( J P ) = + )assignment for this resonant structure. Additional evidence hasbeen obtained in the pn → pp π π − [45], pn → pn π π [46] and ~ np → d π π [47] reactions. Partial wave analysis includingnew polarised ~ np elastic scattering data [48, 49, 50] confirmsthe existence of a resonance pole at (2380 ± − i (40 ± d ∗ (2380) is the only exotic particle which can be pro-duced copiously at modern experimental facilities. Its ba-sic properties and all major decay branches have thereforebeen determined in a short period of time [52]. All the data[48, 49, 41, 42, 43, 45, 46, 53] collected so far suggest that in88 percent of cases the d ∗ (2380) decays into ∆∆ and in 12% to pn , with high angular momentum L = d ∗ (2380) studies indicate it also hasan electromagnetic coupling [56, 57, 58, 59, 60].From measurements of pd → He ππ [61, 62], dd → He ππ [63] and heavy ion collisions [64, 65] it is established that the d ∗ (2380) exists in the nuclear environment. The visible d ∗ width is increased but consistent with the expected trivial ef-fects of Fermi motion, the (known) increase of the ∆ widthin medium [66] and also from additional open decay channelssuch as d ∗ (2380) N → NNN [61, 62, 63]. No significant changein the mass of the d ∗ (2380) was obtained in medium from theseexperiments. We will therefore use the free space d ∗ (2380)mass and width in our neutron star calculations, and modifythese within realistic limits to explore the sensitivity for neutronstar properties. The limits we use are consistent with predicted modifications for dibaryons based on chiral symmetry restora-tion calculations [39]. We hope the current work will provokefurther studies in the area of modelling d ∗ (2380) interactions inmedium. In any case the interactions between d ∗ (2380) and nu-clear matter may be expected to be weaker than for nucleons. In infinite nuclear matter, as appropriate in neutron stars, the d ∗ (2380) exists in a very di ff erent quantum mechanical envi-ronment than atomic nuclei. This can have important conse-quences such as the possible formation of a Bose-Einstein con-densate of d ∗ (2380) in the interior of neutron stars, analogousto the ones already identified for other exotic phases of bosonssuch as pions and kaons [67]. Where the sum of neutron andproton chemical potentials are equal to that of the d ∗ (2380) itbecomes energetically favorable for the system to store baryonsin form of d ∗ (2380), making the d ∗ (2380) population stable innuclear matter and producing an abrupt cut in neutron Fermi-momentum at p Fermin ∼ MeV . For the typical separations of d ∗ (2380) in neutron stars it can be assumed as a first approxi-mation to be a point-like particle. We have used this assumptionin our calculations.
3. E ff ect of the d ∗ (2380) dibaryon on neutron stars. The e ff ect of dibaryons on the EoS for nuclear matter hasbeen considered in various theoretical investigations, see e.g., Refs. [68, 69, 16, 70]. However such studies are generic anddo not study a candidate identified experimentally, as in thispresent study. To study the impact of d ∗ (2380) on the com-position, EoS and structure of neutron stars, we start from acommonly used approach based on a relativistic Lagrangian de-scribing the interaction of nucleons by means of the exchangeof σ − , ω − , and ρ − mesons plus a free Fermi gas of leptons ( e − and µ − ). In particular, we use the GM1 parametrization of theGlendenning-Moszkowski model [71]. This model predicts amaximum mass of ∼ . M ⊙ for a pure nucleonic star, a valuecompatible with the current measurements of heavy neutronstars [17, 18]. The EoS resulting from this choice of poten-tial is also compatible with the latest stringent constraints onthe radius of the 1.4 M ⊙ star, extracted from the recent mergerobservation with gravitational waves [76]. These valuable newdata did not support the validity of NN potentials with a sti ff erEoS. It would be premature to carry out a broader study includ-ing all existing NN potentials, as the d ∗ (2380) would need toalso be included consistently for the NN potential itself. Due to the isoscalar nature of the d ∗ (2380), isovector meson exchanges be-tween the d ∗ (2380) and nucleons are prohibited. Therefore π and ρ meson ex-changes, the strongest contributor to NN-forces, are excluded and higher massmeson exchanges (e.g. the η , η ’ mesons) are known to have weak couplingto nucleons. The most likely mechanism of d ∗ (2380) interaction with matterwould be expected to be from σ -meson exchange (scalar-isoscalar two-pionexchange). From consideration of the relative coupling of the σ with nucleonsand d ∗ (2380) it may be expected that even the σ − d ∗ contribution would bestrongly suppressed. The d ∗ (2380) has recently been shown to have a significant impact on thebasic NN interaction in free space, even influencing the total cross section[48, 49], e ff ects which are not included in any current NN potential. Futurecalculations would therefore need to modify the existing NN-potentials incor-porating d ∗ -mediated NN interactions as an explicit degree of freedom. Weshould also remark that the d ∗ (2380) may also influence 3N and 4N forces. d ∗ (2380) interaction in matter is thought to beweaker than for nucleons and no significant modification is ob-served in nuclear matter, we presently lack detailed constraintof its interaction with other particles. Quark models tend topredict some attraction between the d ∗ and the nuclear matter,while in molecular picture the d ∗ interaction might become re-pulsive at higher densities [72]. Therefore, in this work weinclude it just as a free gas of pointlike bosons. Clearly, theinclusion of interactions between the d ∗ (2380) and nucleons ormesons should be a focus for future work when experimentalinformation becomes available. To explore sensitivities to in-medium modifications of the d ∗ (2380) in our model, we havevaried its mass by ±
100 MeV, in an attempt to asses the e ff ectof an attractive or repulsive interaction. ρ / ρ Density 0 2 4 6 8 P a r t i c l e f r a c t i on − − − d*(2380)np - e - µ - ∆ ∆ + ∆ ++ ∆ ρ / ρ Density 0 2 4 6 8 P a r t i c l e f r a c t i on -3 -2 -1 d*(2380)np - e - µ (a)(b) Figure 1: (Color online) Particle fractions as a function of baryon density inunits of saturation density ρ . The predictions assuming m d ∗ = [2280 , m d ∗ ≡ ∆ degrees of freedom (model (a), top) and with(model (b), bottom). The vertical dash-dot line indicate the density of a heaviestpossible neutron star for a nominal d ∗ (2380) mass (see Table 1 for details). In Fig.1 model(a), we show the predicted change in thechemical composition of a neutron star when we include the d ∗ (2380) resonance as a free particle (dashed lines). The varia-tion of the predictions with an in-medium d ∗ mass modificationin the range m d ∗ [2280 , d ∗ is pre-dicted to appear at densities between 2.7 and 3.2 times normalnuclear matter saturation density ( ρ = .
16 fm − ). Note thatsince the d ∗ (2380) has baryon number 2 its appearance inducesan important and significant reduction in the neutron and protonfractions. In addition, since it is positively charged, the e − and µ − fractions increase in order to maintain charge neutrality.We also explored possible competition of the d ∗ (2380) withother postulated constituents of nuclear matter. As the d ∗ (2380)is strongly coupled to ∆∆ , in Fig. 1 (b) we show results wherewe additionally included the ∆ quartet as an explicit degree offreedom in our model. The mass of the d ∗ (2380) is around 80MeV below the pole mass of two ∆ ’s and therefore would tendto be preferentially created in nuclear matter. The one exceptionis the ∆ − which due to its favourable negative charge, appears atlower densities than the other members of the multiplet. How-ever, in general the main conclusions for the important role ofthe d ∗ (2380) remain unchanged with inclusion of the ∆ ’s and infact the predicted d ∗ fraction at higher densities is increased.The neutron star EoS and the corresponding mass-radiusrelationship (Fig. 2), obtained by solving the Tolmann–Oppenheimer–Volko ff equations [73, 74] is observed due to thebosonic character of the d ∗ (2380) and to the reduction of theneutron and proton fractions. As a consequence the mass ofthe star is reduced from about 2 . M ⊙ to values in the range1 . − . M ⊙ .The EoS for nucleonic degrees of freedom alone illustratesthe expected relationship between pressure and energy density- increasing the pressure results in a rapidly and continuouslyincreasing energy density. However, the d ∗ (2380) o ff ers newopportunities for the matter to respond to pressure increases.With the inclusion of the d ∗ a significantly di ff erent behaviouris evident. The matter is predicted to undergo a phase transitionwhich creates a much stronger correlation between pressure andenergy density and limits the maximum achievable pressure.The upper limits for the central energy desnities of stable neu-tron stars in our model are shown by the markers on the fig-ure, illustrating that such e ff ects may contribute even within themass range of stable neutron stars.The mass-radius relationship predicted by our model isshown in the right panel. The nucleonic and nucleonic +∆ pre-dictions give similar loci. However the predictions with the d ∗ (2380) show an abrupt halt to the mass-radius locus at around2 M ⊙ . The recent LIGO observation of a neutron star merger[19] allows a strict upper limit to be placed on the maximalneutron star mass of M NS < . M ⊙ [75]. It is interesting tonote that the cuto ff produced by the d ∗ (2380) phase transitionin our model is in agreement with this stringent constraint. Ourdimensionless tidal deformability parameter Λ M NS = . M ⊙ = ff predicted here. The predicted maximum stableneutron star mass, radius, density and d ∗ fraction are presentedin Table 1. The d ∗ fraction is predicted to be around 20% atthe center of the star with maximum mass and, compared to thenucleonic case, the maximum star radius is increased and the3 able 1: Maximum mass neutron star properties Model M max ⊙ Radius [km] Central density [ ρ ] d ∗ fraction at the center [%]Pure Nucleonic 2.36 11.8 5.80 0Nucl. + d ∗ (2480) 2.14 12.94 5.23 20Nucl. + d ∗ (2380) 2.05 13.06 5.03 21Nucl. + d ∗ (2280) 1.94 13.17 4.72 22 Figure 2: (Color online) Neutron star EoS (left) and mass-radius relation (right)with and without the d ∗ degree of freedom. The predictions assuming m d ∗ = [2280 , m d ∗ ≡ ff ect of ∆ degree of freedom shown asgold line with d ∗ (2380)(dashed) and without (solid). The observational massesof the pulsars PSR J1614-2230 (1 . ± . M ⊙ ) [17] and PSR J0340 + . ± . M ⊙ ) [18] as well as neutron star merger GW170817 limits from [75]and [76] are also shown. The pink markers on a left panel represent maximumachievable pressure / energy density for heavy neutron star with d ∗ degrees offreedom for the m d ∗ = central density reduced.These first results indicate the d ∗ (2380) has the possibility toplay a signifivant role in neutron stars and dense cold nuclearmatter. The interplay of the d ∗ (2380) with other postulated de-grees of freedom in high density nuclear matter should be inves-tigated in future work. The role of individual ∆ ’s in dictatingneutron star mass limits and properties appears to be dimin-ished when the the d ∗ (2380) is included in our model. Futurework also investigating the interplay with hyperons would be avaluable advance. We note that previous work [13, 14] has in-dicated that the onset of hyperons is shifted to much higher den-sities ( ∼ ρ ) when the ∆ is included. and similar e ff ects may beexpected with the d ∗ (2380) contribution. At even higher den-sities transitions to deconfined quark matter are predicted [77].It would be interesting to investigate the influence of a bosonic d ∗ (2380) in a transition to quark matter. The locattion of thepredicted [77] quark transition is also shown in Fig. 2 left.
4. The d ∗ (2380) in star cooling and mergers. By comparing the predicted particle fractions with and with-out the d ∗ (2380) (Fig. 1) it can be inferred that the majority of d ∗ (2380) are produced from pn-pairs. The majority, but notall. Since the presence of the d ∗ (2380) also increases the lepton(electron and muon) fraction, one can conclude that some of d ∗ (2380)’s would be produced in a weak process nn → d ∗ e ¯ ν e or nn → d ∗ µ ¯ ν µ . This process would be a new, unanticipatedmechanism for neutron star cooling. The anti-neutrinos pro-duced in such weak d ∗ (2380) formation would carry away en-ergy right from the core of a neutron star. Further, the con-vection of d ∗ (2380)’s within the neutron star could also pro-vide a new mechanism for star cooling. In regions of the starhaving densities below ∼ . ρ , the d ∗ (2380) would becomeunstable to d ∗ → nne + ν e decay. These neutrinos could alsocarry away energy in a dibaryon-Urca way. The main sitesof these d ∗ (2380) catalysed cooling processes would be rathernon-uniform within the star. The antineutrinos would be pro-duced from a spherical volume near the star centre, where thedensity is higher than 3 ρ , while neutrinos would be producedfrom a thin spherical surface at ∼ . ρ .In this first theoretical study we did not consider the e ff ect ofthe d ∗ (2380) spin on nuclear matter. The particle has the high-est known spin for any hadron in its ground state, J π = + . Oneshould therefore consider that the formation of the d ∗ (2380)dibaryon does not only condense two baryons into the spaceof one, but also takes two units of angular momentum from theneutron star. Due to large factor M ∗ d / m q ∼ . m q de-noting constituent quark mass), the magnetic moment of a d ∗ isexpected to be very large, µ ∗ d = . µ N [78]. The modelling ofthis e ff ect on the rotational dynamics and magnetic field mapsof neutron stars would be an important next step.Considering that the d ∗ (2380) is predicted to have a signifi-cant influence on the EoS at high densities (Fig. 2) further in-vestigation of its role in neutron star merger and neutron starblack hole mergers processes would be valuable. For example,the presence of significant d ∗ (2380) fraction could influencethe ejecta in star mergers. Although the maximum densitiesfound in stable stars are expected (in our model) to be limited toaround 5 ρ , the densities in mergers would be much higher [9],with the possibility of enhanced roles for the d ∗ (2380) in thecollision dynamics.Further, the decay of the d ∗ (2380) in delocalised ejecta mat-ter may o ff er the possibility to give additional high energygamma production mechanisms. The kinetic energies of thenucleons inferred from the gamma ray burst spectra indicatethe energies to produce the d ∗ (2380) in NN collissions is eas-ily reached [79] and the π π → γ and d ∗ → d γ wouldboth produced 100’s MeV gammas in the d* rest frame.Theinclusion of possible d ∗ (2380) contributions into gamma rayburst simulation codes would be an interesting option. Fur-ther study to assess the possibility of detecting astronomicalsignatures of d ∗ (2380) during mergers should also be investi-gated. The detection of close to monoenergetic photons fromthe d ∗ (2380) → d γ decay would be a clear signal.For nuclear matter out of equilibrium, as would be expectedin the shock waves during a merger process, exotic routes maybe followed to reach stable particle fractions. One such routeis d ∗ (2380) induced n → p conversion: nn → d ∗ e − ¯ ν e . Thecharacteristic timescale for this process to occur can be roughly4stimated with the Fermi golden rule, used to evaluate neutron,muon, taon, etc lifetimes. According to Sargent’s Law the rateof this process is proportional to excess energy Q . In our caseQ can be evaluated as a energy density di ff erence between the d ∗ (2380) and the pure nucleon EoS at the same pressure mul-tiplied by the d ∗ (2380) volume. The d ∗ (2380) appears at adensity ρ = . ρ or at a pressure of 115 MeV / f m . At thisdensity / pressure the conversion time from two neutrons to a d ∗ would be essentially infinite compare to the time scales of amerger. However a slight increase in density to 3 . ρ or pres-sure to 120 MeV / f m would decrease the conversion time be-low the free muon lifetime. And at 5 − ρ this process wouldproceed faster than the free tau decay. This is fast enough tohave an e ff ect on d ∗ (2380) production in neutron star mergershock waves, in addition to the more trivial pn → d ∗ fusion.
5. Summary and Conclusion
We have evaluated the e ff ect of the d ∗ (2380) dibaryon on thenuclear equation of state and the mass-radius relation for neu-tron stars. The calculations used a simple bosonic gas approachfor the d ∗ (2380) supplementing a nucleonic equation of statecalculated in a relativistic mean field approach. In our calcula-tions the appearance of the d ∗ (2380) dibaryons in nuclear mat-ter limits the maximum possible neutron star mass to be around2 solar masses, consistent with current observation limits andhaving an abrupt cut-o ff compatible with recent gravitationalwave observations. The results indicate the d ∗ (2380) could po-tentially be a new degree of freedom in neutron stars. Fractionsof d ∗ (2380) of around 20% are predicted in the center of heavystars, resulting in an increased maximum star radius and a re-duced central density. New neutrino and antineutrino coolingmechanisms are possible with d ∗ (2380) formation, which havepreviously not been included in neutron star modelling.These first results indicate that the d ∗ (2380) is worthy offurther investigation theoretically and experimentally to betterconstrain its role in neutron stars and star mergers.
6. Acknowledgement
We acknowledge valuable discussions with A. Gal.This work has been supported by STFC (ST / L00478X / / L005824 /
1) and by “NewCompstar”, COST ActionMP1304.
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