A&A 638, A40 (2020) Astronomy https://doi.org/10.1051/0004-6361/202037749 & c ESO 2020 Astrophysics

Neutron star equation of state including d∗-hexaquark degrees of freedom A. Mantziris1,2, A. Pastore1, I. Vidaña3, D. P. Watts1, M. Bashkanov1, and A. M. Romero1

1 Department of Physics, University of York, Heslington, York Y010 5DD, UK e-mail: [email protected] 2 Department of Physics, Imperial College London, London SW7 2AZ, UK 3 INFN Sezione di Catania, Dipartimento di Fisica “Ettore Majorana”, Università di Catania, Via Santa Sofia 64, 95123 Catania, Italy Received 16 February 2020 / Accepted 12 April 2020

ABSTRACT

We present the extension of a previous study where, assuming a simple free bosonic gas supplemented with a relativistic mean-field model to describe the pure nucleonic part of the equation of state, we studied the consequences that the first non-trivial hexaquark d∗(2380) could have on the properties of stars. Compared to that exploratory work, we employ a standard non-linear Walecka model including additional terms that describe the interaction of the d∗(2380) di- with the other of the system through the exchange of σ- and ω- fields. Our results show that the presence of the d∗(2380) leads to maximum masses compatible ∗ ∗ with recent observations of ∼2 M millisecond pulsars if the interaction of the d (2380) is slightly repulsive or the d (2380) does not interact at all. An attractive interaction makes the equation of state too soft to be able to support a 2 M whereas an extremely repulsive one induces the collapse of the neutron star into a black hole as soon as the d∗(2380) appears. Key words. equation of state – dense matter – stars: neutron

1. Introduction that are frequently built to reproduce the properties of nuclei (Stone & Reinhard 2007). Skyrme interactions (Skyrme 1959; Neutron stars are the remnants of the gravitational collapse of Vautherin & Brink 1972; Davesne et al. 2016; Grasso 2019) massive stars during a supernova event of Type-II, Ib, or Ic. and relativistic mean-field (RMF) models (Boguta & Bodmer Their masses and radii are typically of the order of 1−2 M 1977; Serot et al. 1986; Serot & Walecka 1997) are among those 33 (M ' 2 × 10 g is the mass of the Sun) and 10−14 km, most used. Many such interactions are built to describe nuclear respectively. With central densities in the range of four to eight systems close to the isospin symmetric case and therefore pre- times the normal nuclear matter saturation density, 0 ∼ 2.7 × dictions at high isospin asymmetries should be taken with care. 14 −3 −3 10 g cm (ρ0 ∼ 0.16 fm ), neutron stars are most likely Most Skyrme interactions are by construction well behaved among the densest objects in the Universe (Shapiro & Teukolsky close to ρ0 and moderate values of the isospin asymmetry. 2008; Glendenning 2000; Haensel et al. 2007; Rezzolla et al. However, only certain combinations of the parameters of these 2018). These objects are therefore excellent laboratories to test forces are well determined experimentally. As a consequence, our present understanding of the theory of strong interacting there exists a large proliferation of different Skyrme interactions matter under extreme conditions, offering an interesting inter- that produce a similar EoS for symmetric nuclear matter, but play between the physics of dense matter and astrophysical predict a very different one for pure neutron matter. A few years observables. ago, Stone et al.(2003) performed an extensive and systematic The conditions of matter inside neutron stars are very differ- test of the capabilities of several existing Skyrme interactions to ent from those encountered on Earth and cannot be probed via provide good neutron star candidates, finding that only a few of direct measurements. Therefore, a good theoretical knowledge these forces passed the restrictive tests imposed. of the nuclear equation of state (EoS) of dense matter is required Relativistic mean-field models are based on effective to understand them. However, its determination is very challeng- Lagrangian densities where the interaction between is ing due to the wide range of densities, temperatures, and isospin described in terms of meson exchanges. The couplings of nucle- asymmetries found in these objects, and constitutes nowadays ons with are usually fixed by fitting masses and radii of one of the main problems in nuclear astrophysics. The main dif- nuclei and the properties of nuclear bulk matter, whereas those ficulties are associated to our lack of a precise knowledge of the of other baryons, like , are fixed by symmetry relations behavior of the in-medium nuclear interaction, and to the very and hypernuclear observables. Recently, Dutra et al.(2014) ana- complicated resolution of the so-called nuclear many body prob- lyzed, as in the case of Skyrme, several parametrizations of seven lem (Baldo 1999). different types of RMF models imposing constraints from sym- The nuclear EoS has been thoroughly studied by many metric nuclear matter, pure neutron matter, symmetry energy and authors using both phenomenological and microscopic many- its derivatives finding that only a very small number of these body approaches. Phenomenological approaches, either parametrizations are consistent with all the nuclear constraints nonrelativistic or relativistic, are based on effective interactions considered in that work.

Article published by EDP Sciences A40, page 1 of8 A&A 638, A40 (2020)

Microscopic approaches on the other hand are based on where we assumed the d∗(2380) as simple gas of non-interacting realistic two- and three-body forces that describe scatter- . We have therefore decided to pursue a more detailed ing data in free space and the properties of the deuteron. study which accounts for explicit interaction of d∗(2380) with These interactions are based on meson exchange (Nagels et al. the surrounding medium. To this aim, we employ a standard non- 1973, 1978; Machleidt et al. 1987; Holzenkamp et al. 1989; linear Walecka model (Dutra et al. 2014), within the framework Haidenbauer & Meißner 2005; Maessen et al. 1989; Rijken et al. of a relativistic mean field theory (RMF). Starting from a well- 1999; Stoks & Rijken 1999; Rijken 2006; Rijken & Yamamoto known nucleonic Lagrangian (Glendenning 2000; Drago et al. 2006) or, very recently, on chiral perturbation theory (Weinberg 2014a), we employ the established d∗(2380) properties to deter- 1990, 1991; Entem & Machleidt 2003; Epelbaum et al. 2005). To mine its interaction with other particles. In particular, we aim at obtain the nuclear EoS, one has to solve the complicated many- providing first constraints on the sign (attractive or repulsive) for body problem, the main difficulty of which lies in the treatment the effective interaction of such a . of the repulsive core, which dominates the short range of the The manuscript is organized in the following way. The interaction. Different microscopic many-body approaches have Lagrangian density including the d∗(2380) is shortly presented been extensively used for the study of the nuclear matter EoS. in Sect.2. Our main results regarding the appearance and e ffect These include, among others: the Brueckner–Bethe–Goldstone of d∗(2380) on neutron stars are shown and discussed in Sect.3. (Baldo 1999; Day 1967) and the Dirac–Brueckner–Hartree–Fock Finally, our concluding remarks and possible directions for (Ter Haar & Malfliet 1987a,b; Brockmann & Machleidt 1990) future work are given in Sect.4. theories, the variational method (Akmal et al. 1998), the corre- lated basis function formalism (Fabrocini & Fantoni 1993), the self-consistent Green’s function technique (Kadanoff & Baym 2. Lagrangian density 1962; Dickhoff & Van Neck 2008), and the Vlow k approach The total Lagrangian density of a system that is composed (Bogner et al. 2003). We refer to these works for specific details. of (N = n, p), the four ∆ isobar resonances To this day, the true nature of neutron stars remains an (∆ = ∆−, ∆0, ∆+, ∆++), (l = e−, µ−), scalar-isoscalar open question. Traditionally, the core of neutron stars has (σ), vector-isoscalar (ω) and vector-isovector (ρ) mesons, and been modelled as a uniform fluid of neutron-rich nuclear includes in addition the d∗(2380) di-baryon is simply given matter in equilibrium with respect to the weak interaction by β ( -stable matter). Nevertheless, due to their high density, X X X new degrees of freedom are expected to appear in addi- L = LN + L∆ + Ll + Lm + Ld∗ , (1) tion to nucleons. Examples of widely studied new degrees N ∆ l of freedom include condensates (Haensel & Proszynski 1982), condensates (Kaplan & Nelson 1986), hyper- where " # ons (Chatterjee & Vidaña 2016; Vidaña 2018), ∆ isobars τ · ρµ L = Ψ¯ iγ ∂µ − m + g σ − g γ ωµ − g γ N Ψ , (Drago et al. 2014a,b; Ribes et al. 2019), deconfined N N µ N σN ωN µ ρN µ 2 N (Glendenning 1992) and even di-baryonic matter (Faessler et al. h i ¯ µ µ µ ν 1998). The most precise and stringent neutron star con- L∆ = Ψ∆ν iγµ∂ − m∆ + gσ∆σ − gω∆γµω − gρ∆γµ I∆ · ρ Ψ∆, straint on the nuclear EoS comes from the recent determina- h i L = Ψ¯ iγ ∂µ − m Ψ , tion of the unusually high masses of the millisecond pulsars l l µ l l − 1 1 1 1 PSR J1614 2230 (Demorest et al. 2010), PSR J0348+0432 L = ∂ σ∂µσ − m2 σ2 − bm g3 σ3 − cg4 σ4 (Antoniadis et al. 2013), and PSR J0740+6620 (Cromartie et al. m 2 µ 2 σ 3 N σN 4 σN 2020). These three measurements imply that any reliable model 1 1 1 1 + m2 ω ωµ − Ω Ωµν − R Rµν + m2ρ · ρµ, for the nuclear EoS should predict maximum masses at least 2 ω µ 4 µν 4 µν 2 ρ µ larger than 2 M . This observational constraint rules out many ∗ µ µ 2 ∗ Ld∗ = (∂µ − igωd∗ ωµ)φd∗ (∂ + igωd∗ ω )φd∗ − (md∗ − gσd∗ σ) φd∗ φd∗ . of the existent EoS models with exotic degrees of freedom, (2) although their presence in the neutron star interior is, how- ν ever, energetically favorable. This has lead to problems like the ΨN and Ψ∆ are the Dirac and Rarita–Schwinger fields for the “ puzzle” (Chatterjee & Vidaña 2016) or the “∆ puzzle” and the ∆ isobar, respectively; Ψl is the Dirac ∗ (Drago et al. 2014b,a), the solutions of which are not straightfor- field; φd∗ indicates the wave function of the d (2380) conden- ward and are presently the subject of active research. sate; g represents the different baryon–meson couplings; and Recently, we studied the role of a new degree of free- τN and I∆ are isospin 1/2 and 3/2 operators. The field tensor ∗ dom, namely d (2380) (Bashkanov et al. 2019), on the nuclear of the ω and ρ mesons is denoted by Ωµν = ∂µων − ∂νωµ ∗ EoS (Vidaña et al. 2018). The d (2380) is a massive positively and Rµν = ∂µρν − ∂νρµ − gρN (ρµ × ρν), whereas the param- charged non- with integer spin (J = 3). It rep- eters b and c associated with the non-linear self-interactions resents the first known non-trivial hexaquark for which there is of the σ field guarantee that the value of the incompressibil- experimental evidence (Adlarson et al. 2011, 2013, 2014). The ity of nuclear matter is within the experimental range. The importance of such a new degree of freedom resides in the fact masses of the nucleons, ∆ isobars, leptons, and the d∗(2380) that it has the same u, d composition as and pro- are denoted by mN , m∆, ml, and md∗ . The main properties of tons and therefore does not involve any degrees of the nucleons, ∆s, and the d∗(2380) di-baryon are summarized in freedom. Moreover, it is a and as such it may condensate Table1. We note that the form of the di-baryon Lagrangian den- within the star. In our previous work we showed that despite its sity, Ld∗ , has been taken as analogous to the one that describes very large mass, the d∗(2380) can appear in the neutron star inte- the interaction of other di-baryon species in nuclear matter, 0 rior at densities similar to those predicted for the appearance of such as for instance the nonstrange ones d1(1920) and d (2060) other nucleon resonances, such as the ∆, or hyperons. That work (Faessler et al. 1997a,b) or the long-studied strangeness −2 was a first attempt to study the consequences that the presence H-particle (Glendenning & Schaffner-Bielich 1998). We also ∗ of the d (2380) could have on the properties of neutron stars note that a minus sign is explicitly placed in front of the gσd∗

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Table 1. Mass (m), spin (J), isospin (I), isospin third component (I3), where bi and qi are the baryon number and the electric charge baryon number (b), and electric charge (q) of nucleons, ∆s, and the di- of the particle i. The chemical potentials of the different particle ∗ baryon d (2380). species read q m (MeV) JII b q µ = k2 + m∗2 + g ω¯ + g I ρ¯(3), (9) 3 B FB B ωB 0 ρB 3B 0 n 939 1/2 1/2 −1/2 1 0 (B = n, p, ∆−, ∆0, ∆+, ∆++), p 939 1/2 1/2 1/2 1 1 µ ∗ = m ∗ − g ∗ σ¯ − g ∗ ω¯ , (10) ∆− 1232 3/2 3/2 −3/2 1 −1 d d σd ωd 0 q ∆0 1232 3/2 3/2 −1/2 1 0 2 2 − − µl = k + m , (l = e , µ ). (11) ∆+ 1232 3/2 3/2 1/2 1 1 Fl l ++ ∆ 1232 3/2 3/2 3/2 1 2 Let us finish this section with a short discussion of our d* 2380 3 0 0 2 1 choice of the different coupling constants. The nucleon cou- plings gσN , gωN , and gρN as well as the parameters b and c are fitted to the bulk (binding energy, density, incompress- coupling such that the effective mass of d∗(2380) is defined sim- ibility, symmetry energy) and single particle (nucleon effec- ilarly to that of the nucleon and that of the ∆ isobar. tive mass) properties of symmetric nuclear matter at saturation. In a RMF description of infinite nuclear matter, the meson These couplings and parameters are taken here as being equal fields are treated as classical fields. Meson field equations in the to those of the well-known Glendenning–Moszkowski model mean field approximation can be easily derived by applying the (Glendenning & Moszkowski 1991). In particular, we consider Euler–Lagrange equations to the Lagrangian density of Eq. (1) the parametrizations GM1 and GM3 of this model to describe the and replacing field operators by their ground-state expectation pure nucleonic part of the system. However, we should mention (3) that these parametrizations do not fulfill the currently accepted values σ → σ¯ , ωµ → ω0, ρµ → ρ¯ . They read simply 0 constraints on the slope parameter L of the symmetry energy that    Z k ∗  impose it to be between 40 and 60 MeV. The values predicted by X  FB m  2 2JB + 1 B 2  the GM1 and GM3 models are 95 and 90 MeV, respectively, sig- m σ¯ = gσB  k dk σ  2π2 q  B=N,∆  0 k2 + m∗2  nificantly above the upper limit of the accepted range. However, B despite this deficiency we still use them in the present work. 2 3 − bmN gσN (gσN σ¯ ) − cgσN (gσN σ¯ ) + gσd∗ ρd∗ (3) The coupling of the ∆ isobar with the different meson fields is poorly constrained due to the limited existence of experimental 2 X 2JB + 1 3 m ω¯ 0 = gωB k − gωd∗ ρd∗ (4) data. This leaves us with some freedom in the choice of these ω 6π2 FB B=N,∆ couplings. We consider two sets for the ∆–meson couplings. The first one is the so-called universal coupling (UC) scheme 2 (3) X 2JB + 1 3 m ρ¯ = gρB k I3B, (5) ρ 0 6π2 FB (Lavagno & Pigato 2012; Jin 1995) where all the ∆–meson cou- B=N,∆ plings are taken as being equal to those of the nucleons: where JB is the spin of the baryon B, kFB its Fermi momentum, ∗ xσ∆ = 1, xω∆ = 1, xρ∆ = 1, (12) mB = mB − gσB its effective mass, I3B the third component of ∗ ∗ g its isospin, and ρ ∗ = 2m ∗ φ ∗ φ ∗ is the density of the d (2380) i∆ d d d d where we have introduced the dimensionless couplings xi∆ = giN di-baryon. The energy density and the pressure of the system are (i = σ, ω, ρ). The second set, referred to here as the stronger obtained from the energy-momentum tensor coupling (SC) set, corresponds to the choice xσ∆ = 1.15, xω∆ = Z kF q x X (2JB + 1) B 1, ρ∆ = 1. The interested reader is referred to Drago et al. ε = k2 + m∗2k2dk (2014a,b) for more details on this second set of parameters. 2π2 B B=N,∆ 0 Being an isospin singlet, the d∗(2380) di-baryon only cou- Z kl ples with the σ and ω mesons. Unfortunately, there is currently X 1 F q 1 1 + k2 + m2k2dk + m2 σ¯ 2 + bm (g σ¯ )3 no evidence that the interaction of d∗(2380) with other particles π2 l 2 σ 3 N σ l=e−, µ− 0 is attractive or repulsive. In light of the lack of such information, in this work we explore a wide range of positive and negative 1 4 1 2 2 1 2 (3) 2 ∗ ∗ ∗ gid + c(gσσ¯ ) + mωω¯ 0 + mρ(¯ρ0 ) + md∗ ρd (6) values for the dimensionless couplings xid∗ = (i = σ, ω) in 4 2 2 giN Z k 4 order to analyze the effect of both scenarios. X (2J + 1) FB k dk P = B 6π2 q B=N,∆ 0 k2 + m∗2 B 3. Results Z kl X 1 F k4dk 1 1 ∗ − m2 σ2 − bm g σ 3 Figure1 shows the onset density of the d (2380) di-baryon as + 2 q σ ¯ N ( σ ¯ ) 3π 2 3 a function of the dimensionless couplings x ∗ and x ∗ (pan- l=e−, µ− 0 k2 + m2 σd ωd l els a and c) and the first isospin state of the ∆ isobar appear- 1 1 1  2 ing in β-stable neutron star matter, the ∆− (panels b and d). The − c(g σ¯ )4 + m2 ω¯ 2 + m2 ρ¯(3) . (7) 4 σ 2 ω 0 2 ρ 0 GM1 parametrization of the Glendenning–Moszkoski model We note that the d∗(2380) does not contribute to the total pres- has been used to describe the pure nucleonic part of the sys- sure of the system. Chemical equilibrium in the neutron star tem. Results for the UC and SC choice of parameters for the interior without trapping leads to the following relations ∆–meson couplings are shown in panels a,b and c,d, respectively. between the chemical potentials of the different species: We notice that not all sets of values of xσd∗ and xωd∗ lead to phys- ical solutions. In particular, for some sets of couplings, nega- µi = biµn − qiµe, (8) tive values of the pressure are obtained. These nonphysical cases

A40, page 3 of8 A&A 638, A40 (2020)

(a) (b)

(c) (d)

∗ − Fig. 1. Onset density (in units of ρ0) of the d (2380) di-baryon (panels a and c) and the ∆ isobar (panels b and d) in β-stable neutron star matter as a function of the dimensionless couplings xσd∗ and xωd∗ . Panels a and b: UC choice of parameters for the ∆–meson couplings while panels c and d: SC choice. In all cases the nucleonic part is described with the GM1 parametrization of the Glendenning–Moszkowski model. The d∗(2380) noninteracting case is highlighted with squares. The black line corresponds to solutions where the onset density of the d∗(2380) coincides with that of the noninteracting case. correspond to the blank regions in the four panels of the figure. rium condition for the appearance of the d∗(2380) for the cases in ∗ ∗ We observe that the onset density of the d (2380) varies signifi- which the d (2380) is subject to attraction (xσd∗ = xωd∗ = 0.25), cantly as a function of the couplings xσd∗ and xωd∗ , ranging from repulsion (xσd∗ = xωd∗ = −0.25), or does not interact at all with − ∼2ρ0 up to ∼7ρ0. However, the onset of the ∆ varies in a much the rest of the particles of the system. As seen in the plot, an smaller range, showing that in this particular scenario there is a attractive (repulsive) interaction leads to a decrease (increase) of very weak coupling between the two species. A similar conclu- the d∗(2380) chemical potential, and consequently to an earlier sion holds using the GM3 parametrization instead of the GM1 (later) fulfillment of the equilibrium condition, µd∗ = 2µn − µe, one to describe the pure nucleonic part, as is shown in Fig.2. signaling the appearance of the d∗(2380) in matter. However, However, we note that in this case the SC choice of parame- we should note that these conclusions are based on the partic- ters for the ∆–meson couplings leads to a later appearance of ular models employed in the present work, and therefore they the ∆−, although this is not correlated with the appearance of could change if other models are used or if the presence of other the d∗(2380). As a reference, we also indicate the noninteract- degrees of freedom such as hyperons or deconfined quarks are ing d∗(2380) case in both figures with solid squares. The black taken into account. line, defined by the relation xωd∗ = −0.88 xσd∗ , shown in the To better quantify the impact of the appearance of the left panel of both figures indicates the case where a proper con- d∗(2380) in the medium, in Figs.4 and5 we show the chemi- figuration of xσd∗ and xωd∗ dimensionless couplings leads to an cal composition of β-stable matter using the GM1 (Fig.4) and onset density of the d∗(2380) that is equal to that of the noninter- GM3 (Fig.5) models to describe the pure nucleonic part and the acting case. We note that below this line, any combination of the UC (upper panels) or SC (lower panels) choice of parameters for ∗ ∗ xσd∗ and xωd∗ couplings predicts an onset density of the d (2380) the ∆–meson couplings. The interaction of d (2380) is assumed that is larger than that of the free case. This is an indication that to be either attractive, with couplings xσd∗ = xωd∗ = 0.1 (left pan- ∗ the interaction of the d (2380) is repulsive for all the values of els), or repulsive, with couplings xσd∗ = xωd∗ = −0.1 (right pan- these couplings in this region of the parameter space. Similarly, els). Results for the case in which the d∗(2380) is assumed to be a above the line xωd∗ = −0.88 xσd∗ , the predicted onset density of free particle are also shown for comparison. We highlight the fact d∗(2380) is always smaller than that of the noninteracting case, that the GM1 model leads in all the cases to an earlier appear- and consequently the d∗(2380) is subject to attraction for any ance of the ∆ isobar, and also, as mentioned above, that due to its − value of the couplings xσd∗ and xωd∗ sitting in this region. This negative charge the ∆ appears at much lower densities than the is illustrated for the GM1 parametrization and the SC choice of other members of the ∆ quadruplet. We note in addition that with the ∆ couplings in Fig.3 where we show the chemical equilib- the UC choice for the ∆–meson couplings, no other ∆ resonances

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(a) (b)

(c) (d)

Fig. 2. Same as Fig.1 but with the nucleonic sector described using the GM3 model.

3000 particles since it appears in all cases at lower densities than the d∗(2380). The appearance of the d∗(2380) induces an important µ and significant reduction of the fraction of neutrons, , and d* ∆s since its baryon number is 2. In addition, as the d∗(2380) is 2500 positively charged, the lepton fraction increases in order to main- tain charge neutrality. ∗ µ Let us now analyze the effect of the d (2380) on the mass– d* radius relation of neutron stars and, in particular, on the max- imum mass. To this end, using our EoS together with that of 2000 x =x =-0.25 Douchin & Haensel(2001) for the low-density stellar crust, we 2 - σd* ωd* µn µe Free case solved the well-known Tolmann–Oppenheimer–Volkoff (TOV) Chemical potentials [MeV] x =x =0.25 equations (Tolman 1939; Oppenheimer & Volkoff 1939) which σd* ωd* describe the structure of nonrotating spherically symmetric stel- 1500 0 1 2 3 4 5 6 lar configurations in general relativity. In Figs.6 and7, we show the pressure (left panels) and the mass–radius relation ρ/ρ0 (right panels) obtained using the GM1 (Fig.6) and GM3 Fig. 3. Evolution of the chemical potentials for the appearance of the (Fig.7) models to describe the pure nucleonic part of the d∗(2380) di-baryon in β-stable matter as a function of the density of the EoS and the UC (upper panels) or SC (lower panels) choice ∗ system. Results are shown for the cases in which the d (2380) is subject of parameters for the ∆–meson couplings. The interaction of to attraction (xσd∗ = xωd∗ = 0.25) or repulsion (xσd∗ = xωd∗ = −0.25), or d∗(2380) is assumed to be either attractive, with couplings does not interact at all with the rest of the particles of the system. The (x ∗ , x ∗ ) = (0.1, 0.1), (0.2, 0.2) and (0.3, 0.3), or repulsive, GM1 parametrization together with the SC choice of the ∆ couplings σd ωd ∗ ∗ − − − − has been adopted. with couplings (xσd , xωd ) = ( 0.1, 0.1), ( 0.2, 0.2) and (−0.3, −0.3). Results for the case in which the presence of the d∗(2380) is ignored and the case in which it is assumed to be appear in the range of baryonic densities considered either for a free particle are also shown for comparison. The first thing to the GM1 model or for the GM3 one. On the contrary, when the be noticed when looking at the pressure of the system is that SC set of couplings is adopted, the ∆0 and ∆+ also appear; both it is strongly reduced once the d∗(2380) appears. This is sim- appear in the case of the GM1 model and only the former in the ply due to: (i) the reduction of the neutron and fractions case of the GM3 one when the d∗(2380) is subject to repulsion. with the appearance of the d∗(2380) with the consequent reduc- The ∆++ is absent in all cases. Finally, we also observe that the tion of their partial contributions to the pressure, and (ii) the onset density of the ∆− is not affected by the attractive or repul- fact that the d∗(2380) itself does not contributes to the pressure. sive character of the interaction of the d∗(2380) with the other We note that the pressure continues to increase slowly after the

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0 0 10 n n 10 p p -1 -1 - 10 - 10 e e

-2 -2 10 10 * * Free d Free d * * -3 Attractive d Repulsive d -3

Particle fractions 10 10 GM1+UC − GM1+UC − − * − * µ ∆ d µ ∆ d -4 -4 10 10 0 2 4 6 8 0 2 4 6 8 0 0 10 n n 10 p -1 p -1 10 - - 10 e e

-2 -2 10 10 * * Free d Free d * * -3 Attractive d Respulsive d -3

Particle fractions 10 − − 0 10 − − * 0 GM1+SC µ ∆ * + GM1+SC µ ∆ d ∆ ∆ d ∆ -4 -4 10 10 0 2 4 6 8 0 2 4 6 8 Baryon density ρ/ρ0 Baryon density ρ/ρ0

Fig. 4. Particle fraction as a function of the baryon density in units of ρ0. The nucleonic part is described with the GM1 model. Results for the UC (SC) choice of parameters for the ∆–meson couplings are shown in the upper (lower) panels. Left and right panels: results for a case in which the ∗ d (2380) feels attraction (xσd∗ = xωd∗ = 0.1) or repulsion (xσd∗ = xωd∗ = −0.1), respectively. Dashed lines show the results for the free case for comparison.

0 0 10 n n 10 p p -1 -1 10 - - 10 e e

-2 -2 10 10 * * Free d Free d * * -3 Attractive d Repulsive d -3

Particle fractions 10 GM3+UC 10 − − * GM3+UC − − * µ ∆ µ ∆ d d -4 -4 10 10 0 2 4 6 8 0 2 4 6 8 0 0 10 n n 10 p -1 p -1 10 - - 10 e e * Free d -2 -2 * 10 Respulsive d 10 * Free d GM3+SC * -3 Attractive d -3

Particle fractions 10 * 10 − − * GM3+SC − − µ ∆ d µ ∆ d 0 -4 ∆ -4 10 10 0 2 4 6 8 0 2 4 6 8 Baryon density ρ/ρ0 Baryon density ρ/ρ0 Fig. 5. Same as Fig.4 but with the nucleonic sector described using the GM3 model.

∗ appearance of the d (2380) in the system except for the cou- xωd∗ couplings that lead to a negative gradient of the pressure at plings xσd∗ = xωd∗ = −0.2 and xσd∗ = xωd∗ = −0.3 in the case some given density represent solutions in which the appearance ∗ of the GM1 model, and xσd∗ = xωd∗ = −0.3 in the case of the of the d (2380) di-baryon induces the collapse of the neutron star GM3 one. In these cases, the gradient of the pressure becomes into a black hole. negative at a given density. A negative gradient of the pressure is Looking now into the mass–radius relation, one immedi- a signal for a mechanical instability which can give rise to a pos- ately notices that only the model GM1 (with both choices for sible phase transition. However, such a possibility has not been the ∆–meson couplings) predicts a maximum mass that is com- considered in the present work. Therefore, those sets of xσd∗ and patible with the recent measurement of the mass of the pulsar

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2.5 no d* 400 free d* 2

(0.1,0.1) PSR J0740+6620 ] (-0.1,-0.1) 1.5 Sun

] GW170817

-3 200 GM1+UC PSR J0030+0451 1 GM1+UC 0.5 0 0 2 4 6 8 6 8 10 12 14 2.5 400 GM1+SC 2 (0.2,0.2) PSR J0740+6620 (-0.2,-0.2) Pressure [MeVfm GW170817 1.5 (0.3,0.3)

200 (-0.3,-0.3) PSR J0030+0451 1 Graviational Mass [M GM1+SC 0.5 0 0 1 2 3 4 5 6 7 8 6 8 10 12 14 Baryon density ρ/ρ0 Radius [km] Fig. 6. Pressure (left panels) and mass–radius relation (right panels) obtained using the GM1 model to describe the pure nucleonic part of the EoS and the UC (upper panels) or the SC (lower panels) choice of parameters for the ∆–meson couplings. The interaction of d∗(2380) is assumed to be either attractive, with couplings (xσd∗ , xωd∗ ) = (0.1, 0.1), (0.2, 0.2), and (0.3, 0.3), or repulsive, with couplings (xσd∗ , xωd∗ ) = (−0.1, −0.1), (-0.2, −0.2), and (−0.3, −0.3). The recent measurement of the mass of the pulsar PSR J0740+6620 is shown with a 68.3% credibility interval +0.10 +0.20 (2.14−0.09 M ) and 95.4% credibility interval (2.14−0.18 M ). We also show the recent gravitational wave constraint GW170817 on the radius of a 1.4 M NS (Annala et al. 2018) and the results obtained from the NICER experiments on the PSR J0030+04512 millipulsar (Riley et al. 2019).

800 2.5 no d* 600 free d* 2

(0.1,0.1) PSR J0740+6620 ] (-0.1,-0.1) 1.5 Sun ] 400 GW170817 -3 GM3+UC PSR J0030+0451 1 200 GM3+UC 0.5 0 0 2 4 6 8 6 8 10 12 14 2.5 600 GM3+SC 2 (0.2,0.2) PSR J0740+6620 (-0.2,-0.2) Pressure [MeVfm 400 GW170817 1.5 (0.3,0.3)

(-0.3,-0.3) PSR J0030+0451 1 Graviational Mass [M 200 GM3+SC 0.5 0 0 1 2 3 4 5 6 7 8 6 8 10 12 14 Baryon density ρ/ρ0 Radius [km] Fig. 7. Same as Fig.6 but with the nucleonic sector described using the GM3 model.

+0.10 PSR J0740+6620 (2.14−0.09 M (68.3% credibility interval) and 4. Conclusions +0.20 2.14−0.18 M (95.4% credibility interval) (Cromartie et al. 2020) if the d∗(2380) does not interact or is subject to slight repul- This work is the extension of a previous study (Vidaña et al. sion. An attractive interaction of the d∗(2380) leads to values of 2018) where, assuming a simple free bosonic gas supplemented the maximum mass smaller than the highest value observed until with a RMF model to describe the pure nucleonic part of the now. Too much repulsion on the other side leads, as mentioned EoS, we explored for the very first time the consequences that the above, to a negative gradient of the pressure and consequently to presence of the d∗(2380) di-baryon could have on the properties the collapse of the star. of neutron stars. Compared to that exploratory work, we have

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