Formation of Lower Mass-gap Black Hole–Neutron Star Binary Mergers through Super-Eddington Stable Mass Transfer

Super-Eddington accretion of neutron stars (NSs) has been suggested both observationally and theoretically. In this paper, we propose that NSs in close-orbit binary systems with companions of helium (He) stars, most of which systems form after the common-envelope phase, could experience super-Eddington stable Case BB/BC mass transfer (MT), and can sometimes occur accretion-induced collapse (AIC), resulting in the formation of lower mass-gap black holes (mgBHs). Our detailed binary evolution simulations reveal that AIC events tend to happen if the primaries NS have an initial mass ≳ 1 . 7 𝑀 ⊙ with a critical accretion rate of ≳ 300 times the Eddington limit. These mgBHs would have a mass nearly equal to or slightly higher than the NS maximum mass. The remnant mgBH–NS binaries after the core collapses of He stars are potential progenitors of gravitational-wave (GW) source. Multimessenger observation between GW and kilonova signals from a population of high-mass binary NS and mgBH–NS mergers formed through super-Eddington stable MT are helpful in constraining the maximum mass and equation of state of NSs. S230529ay, a mgBH–NS merger candidate recently detected in the fourth observing run of the LIGO-Virgo-KAGRA Collaboration, could possibly originate from this formation scenario.


INTRODUCTION
On the one hand, the X-ray and radio observations of Galactic pulsars revealed a likely NS maximum mass of ∼ 2 − 2.3  ⊙ (e.g., Antoniadis et al. 2013;Alsing et al. 2018;Romani et al. 2022), consistent with the maximum mass inferred by the observations of NSs in GW binaries (e.g., Margalit & Metzger 2017;Landry & Read 2021;Zhu et al. 2022a;Abbott et al. 2023a).On the other hand, the measurements of the mass distribution of BHs in Galactic X-ray binaries suggested a lower boundary close to ∼ 5  ⊙ (Bailyn et al. 1998;Özel et al. 2010;Farr et al. 2011).This led to the conjecture of the presence of the mass gap between the heaviest NSs and lightest BHs, which was thought to be potentially caused by the observational bias (Kreidberg et al. 2012), natal kick (Mandel et al. 2021), or the intrinsic physics mechanism of core-collapse supernova (SN) explosions, e.g., the rapid model (Fryer et al. 2012).However, recent electromagnetic (EM) observations discovered a few compact objects with mass plausibly lying in the range of ∼ 2.5 − 5  ⊙ , indi-★ E-mail: jin-ping.zhu@monash.edu† E-mail:yingqin2013@hotmail.com cating that the putative mass gap could be at least partly populated.For example, gravitational microlensing identified eight compactobject candidates with masses within the mass gap (Wyrzykowski & Mandel 2020).Thompson et al. (2019) reported that the unseen companion of giant star 2MASS J05215658+4359220 could have a mass of 3.3 +2.8 −0.7  ⊙ , which could be a noninteracting mgBH.Using radial velocity measurements, Rivinius et al. (2020) suggested HR 6819, a star in a hierarchical triple system, might be accompanied by an unseen companion, which could be a nonaccreting BH with a mass of ≥ 4.2  ⊙ .van der Meĳ et al. ( 2021) calculated a mass of 4U 1700-37 of 2.54  ⊙ , which could either be a NS or a BH in the mass gap.Andrews et al. (2022) used Gaia data release to estimate the masses of the dark companions in wide-orbit binaries, which span a range of 1.35 − 2.7  ⊙ , partially intersecting with the mass gap.
Currently, it remains uncertain that whether X-ray binaries and GW sources follow a similar evolutionary pathway (e.g., Belczynski et al. 2021;Fishbach & Kalogera 2022).It is also expected that GW searches could discover a number of mgBHs in the merging systems of binary BH-NS and binary BH (BBH).During the third observing run (O3) of the LIGO-Virgo-KAGRA (LVK) Collaboration, a few GW candidates that potentially contain a mass-gap compact object were indeed detected.GW190814 was reported to be a merger between a 23.2 +1.1 −1.0  ⊙ BH and a mass-gap compact object with a mass of 2.59 +0.08  −0.09  ⊙ (Abbott et al. 2020b).The LVK Collaboration also detected a similar, but marginal event namely GW200210_092254 in O3 (Abbott et al. 2023b), where the component masses were inferred to be 24.1 +7.5  −4.6  ⊙ and 2.83 +0.47 −0.42  ⊙ , respectively.The posterior distributions for the primary source mass of the lowsignificance BH-NS candidate GW190426_152155 and the BH-NS merger GW200115, i.e., 5.7 +3.9  −2.3  ⊙ and 5.9 +1.4 −2.1  ⊙ , partly lie in the mass gap (Abbott et al. 2021a,b).However, by applying alternative astrophysically motivated priors, the primary mass of GW200115 would be more tightly constrained to be 7.0 +0.4  −0.4  ⊙ (Mandel & Smith 2021), which could be completely higher than the mass gap.Furthermore, the secondary component of GW191113_071753 has a mass of 5.9 +4.4  −1.3  ⊙ with 13% probability of being a mgBH (Abbott et al. 2023b).Despite the detection of a few GW candidates containing mass-gap compact objects in O3, the population properties of merging compact binaries using GWs in GWTC-2 and GWTC-3 still indicated a relative dearth of events with masses in the mass gap (Farah et al. 2022;Zhu et al. 2022a;van Son et al. 2022;Olejak et al. 2022;Ye & Fishbach 2022;Biscoveanu et al. 2023;Abbott et al. 2023a).
Systematic investigations of BH-NS systems formed through isolated binary evolution have been recently explored (e.g., Román-Garza et al. 2021;Hu et al. 2022;Xing et al. 2023;Wang et al. 2024).Binary population synthesis showed that the rapid SN model does not form any mgBHs (e.g., Belczynski et al. 2012;Giacobbo & Mapelli 2018), while the delayed and stochastic models (Mandel & Müller 2020) suggested that ∼ 30 − 80% of BH-NS mergers and ∼ 20 − 40% of BBH mergers are expected to host at least one mgBH (e.g., Shao & Li 2021;Drozda et al. 2022).Both scenarios appear to be inconsistent with the observational results from GWTC-3.On the other hand, NSs can occur AIC to mgBHs via accretion when NSs grow to the point of exceeding the maximum mass allowed by their equation of state (EoS).AICs of accreting NSs are mostly predicted to take place in intermediate/low-mass X-ray binaries with companions of degenerated hydrogen/He dwarf stars (MacFadyen et al. 2005;Dermer & Atoyan 2006;Giacomazzo & Perna 2012;Gao et al. 2022;Chen et al. 2023).Furthermore, based on the standard scenario of the formation of a binary NS (BNS) system (e.g., Bhattacharya & van den Heuvel 1991;Tauris et al. 2017;Vigna-Gómez et al. 2018), an NS orbiting a main-sequence star needs to undergo common-envelope phase to become a close-orbit NS-He star system when the secondary star expands to a giant star.The NS could accrete materials from the hydrogen envelope during the common-envelope phase and from the He star during the subsequent Case BB/BC MT phase to increase its mass.As predicted by several simulations (e.g., MacLeod & Ramirez-Ruiz 2015;Esteban et al. 2023), since the common-envelope stages are months-long with accretion rates up to  ≲ 0.1  ⊙ if the super-Eddington accretion for the NS is permissible, the mass transfer (MT) during the commonenvelope phase is usually limited during this phase.The discovery that a number of ultra-luminous X-ray sources (ULXs) are pulsating NSs unambiguously suggested that a fraction of NSs in binary systems can accrete at a super-Eddington accretion rate (see Kaaret et al. 2017, for a review).Population synthesis simulations by Shao et al. (2019) suggested that a significant fraction of NS ULXs in a Milky Way-like galaxy could contain a He star companion.Most recently, Zhou et al. (2023) for the first time identified a He donor star in NGC 247 ULX-1.Thus, some NSs could experience super-Eddington accretion in an NS-He star binary during the stable MT stage.The MT rates could range from a few 10 −5  ⊙ yr −1 to a few 10 −4  ⊙ yr −1 with durations of a few 10 4 yr.Thus, AIC of an NS to a mgBH could happen if the NS reaches its maximum mass after accreting sufficient materials.In this paper, we present a new formation scenario in which an accreting NS during the stable Case BB/BC MT with super-Eddington accretion could undergo AIC leading to the formation of an mgBH.The final remnant system could be an ideal GW source of an mgBH-NS binary merger if the binary system survives the second SN.

Super-Eddington Accretion of NSs
Traditionally, the accretion rate of compact objects is thought to be limited by the Eddington limit, i.e., with the Eddington luminosity of where  acc is the accretor's mass,  is the opacity of the accreting material,  is the gravitational constant, and  is the speed of light.
Hereafter, the conventional notation   = /10  is adopted in cgs units.The radiation efficiency of accretion  is set to be  = 0.1 for an NS accretor and  = 1− √︃ 1 − ( BH / BH,i ) 2 /9 if  BH < √ 6 BH,i for a BH accretor (Podsiadlowski et al. 2003), where  BH is the BH mass, and  BH,i is the initial BH mass which is equal to the NS maximum mass  NS,max .In our work,  NS,max ∼ 2.2  ⊙ is defined following the constraints by the observations of Galactic pulsars and GW binaries (e.g., Margalit & Metzger 2017;Romani et al. 2022;Zhu et al. 2022a;Abbott et al. 2023a).
The discoveries of NS ULXs demonstrated that NSs can accrete at super-Eddington rate.Some of ULXs, e.g., M82 X-2 (Bachetti et al. 2014), NGC 7793 P13 (Fürst et al. 2016), and NGC 1313 X-2 (Sathyaprakash et al. 2019) have an apparent luminosity of a few 10 40 erg s −1 implying that the accretion rate could be ≳ 100  Edd .Israel et al. (2017) reported that NGC 5907 ULX-1 and some other extreme ULXs that might harbor NSs can even have a maximum Xray luminosity of a few 10 41 erg s −1 corresponding to ∼ 1000  Edd if the emission is isotropic.
For a standard gas-pressure-dominated thin disk with super-Eddington accretion, the location at which the accretion disk just possesses a local luminosity reaching the Eddington limit and has mass loss is defined as the spherization radius (Shakura & Sunyaev 1973), i.e., where  is the accretion rate of the disk.Furthermore, the NS magnetic field can interact with the accretion disk.The disk around a magnetized NS is disrupted at the magnetospheric radius (Frank et al. 2002), which is given by where  A is the Alfvén radius,  is the dimensionless coefficient (Ghosh & Lamb 1979;Wang 1996;Long et al. 2005;Kulkarni & Romanova 2013),  in is the accretion rate at  mag ,  NS is the NS mass,  =  NS  3 NS is the magnetic moment of the NS with  NS and  NS being the magnetic field strength and radius of the NS, respectively.One can determine a critical accretion rate of  cr ∝  4/9 which provides a limit on  in by equating  mag with  sph .Thus, the NS accretion with a higher super-Eddington critical rate requires a larger NS magnetic field.If  mag ≳  sph , for a subcritical disk, we have  in = ; otherwise, for a supercritical disk, then The model of Shakura & Sunyaev (1973) is based on the assumption of a standard geometrically thin disk.Most recently, Chashkina et al. (2017) proposed that the NS accretion disk could be geometrically thick and radiation-pressure-dominated in its inner part, leading to a larger magnetosphere size and, hence, a larger critical accretion rate of (5) For a higher accretion rate, the inner disk would then become advection-dominated and the critical rate can be further enhanced to (Chashkina et al. 2019) Therefore, one value of  in can be determined by two different magnetic field strengths.For example, for a 1.4  ⊙ NS with an accretion rate of 100  Edd , the magnetic field strength can be ∼ 5.6 × 10 12 G if the disk is radiation-dominated by Equation ( 5) or ∼ 1.1 × 10 11 G if the disk is advection-dominated by Equation ( 6), respectively.Here,  = 0.1 and an NS radius of  NS = 12.5 km are adopted (e.g., Miller et al. 2019;Landry & Read 2021).Since we will discuss the influence of magnetic field decay on accretion rates in Section 3 and two different magnetic field strengths will give consistent results, for simplicity, we assume that the inner disk of the super-Eddington accretion disk during the Case BB/BC MT stage could always be advection-dominated and, hence, we can use Equation ( 6) to estimate critical accretion rate for specific NS magnetic field.The accretion rate of mgBHs formed after AICs of NSs is assumed to be limited by the Eddington limit, i.e.,  cr =  Edd , because the magnetic field effect stops operating.The mass accretion rate of the compact object, including NS and BH, can be expressed as (7)

Physics Implemented in MESA
We adopt the release version mesa-r15140 of the Modules for Experiments in Stellar Astrophysics (MESA) stellar evolution code (Paxton et al. 2011(Paxton et al. , 2013(Paxton et al. , 2015(Paxton et al. , 2018(Paxton et al. , 2019;;Jermyn et al. 2023) to evolve secondary He stars, which are assumed to be tidally locked by primary NS companions (treated as point masses).The zero-age He main-sequence stars are created following the same method with Qin et al. (2018Qin et al. ( , 2023)); Bavera et al. (2020Bavera et al. ( , 2021)) 2024), and then are relaxed to reach the thermal equilibrium.Convection is implemented based on the Ledoux criterion, and mixing-length theory (Böhm-Vitense 1958) with a mixing length of  MLT = 1.93 is adopted.Convective mixing is treated as a top step decay process with an overshoot parameter  ov = 0.1 p , where  p is the pressure scale height at the Ledoux boundary limit.We also consider semiconvection (Langer et al. 1983) with an efficiency parameter  SC = 1.0 in our model.The network of approx12.net is chosen for nucleosynthesis.
For stellar winds, we use the "Dutch" scheme for both RGB and AGB phase, as well as the cool and hot wind.We adopt the default RGB_to_AGB_to_wind_switch = 1d-4, as well as cool_wind_full_on_T = 0.8d4 and hot_wind_full_on_T = 1.2d4.The standard "Dutch" scheme was calibrated by multiplying with a Dutch scaling factor of 0.667 to match the recently updated modeling of He star wind mass loss (Higgins et al. 2021) is adopted.Angular momentum transport and rotational mixing diffusive processes (Heger et al. 2000;Heger & Langer 2000), including the effects of Eddington-Sweet circulations, and the Goldreich-Schubert-Fricke instability, as well as secular and dynamical shear mixing, are incorporated.We adopt diffusive element mixing from these processes with an efficiency parameter of   = 1/30 (Chaboyer & Zahn 1992;Heger & Langer 2000).Additionally, the traditional Spruit-Tayler (Spruit 2002) dynamo-induced angular momentum transport is implemented in our models.
We model MT following the Kolb scheme (Kolb & Ritter 1990), and the implicit MT method (Paxton et al. 2015) is adopted.The dynamical tides are applied to massive stars with radiative envelopes, and the corresponding timescale for orbital synchronization is calculated following the prescription in Hurley et al. (2002).We adopt the updated tidal torque coefficient  2 provided in Qin et al. (2018).

PROPERTIES OF MGBH-NS FORMED THROUGH SUPER-EDDINGTON ACCRETION
We choose three different initial masses for the primary NS, including  NS 1,i = 1.4,1.7, and 2.0  ⊙ .If the primary NS mass exceeds  NS,max during the process of super-Eddington accretion, the AIC to mgBH can occur.The initial He star mass  He 2,i is in a range of 2.5 − 8  ⊙ .We follow the evolution of these He stars from the zero-age He main-sequence until carbon depletion occurred in their cores.Then, we estimate the baryonic mass of the remnant NS and BH formed through core collapse based on the delayed SN mechanism rather than the rapid SN mechanism (Fryer et al. 2012), prompted by recent plausible discoveries on compact objects in the lower mass gap (e.g., Wyrzykowski & Mandel 2020;Thompson et al. 2019;Abbott et al. 2020b).The baryonic mass of the remnant NS formed through electron-capture SN is set to 1.38  ⊙ following Fryer et al. (2012) if the pre-SN carbon-oxygen core mass is within the range between 1.37 − 1.43  ⊙ (Tauris et al. 2015).We also take into account neutrino loss as in Zevin et al. (2020).For the binary systems which could have MT, we cover the initial orbital periods  orb,i of 0.04 − 40 d.Three critical super-Eddington accretion rates of  cr = 100  Edd , 300  Edd and 500  Edd are considered.For a 1.4  ⊙ NS, these three rates correspond to ∼ 3.6×10 −6  −1 −0.47  ⊙ yr −1 , ∼ 1.1×10 −5  −1 −0.47  ⊙ yr −1 and ∼ 1.8×10 −5  −1 −0.47  ⊙ yr −1 , which are always much lower than the MT rate of Case BB/BC (i.e., from a few 10 −5  ⊙ yr −1 to a few 10 −4  ⊙ yr −1 ; see Figure 1).By setting  = 0.1 and  NS = 12.5 km (e.g., Miller et al. 2019;Landry & Read 2021) in Equation ( 6), NSs are required to have magnetic field of ∼ 1.1 × 10 11 G, ∼ 1.3 × 10 12 G, and ∼ 4.0 × 10 12 G, respectively.Besides defined parameters for evolving binary systems introduced in Section 2.2, we set the grids of  NS 1, ,  He 2,i ,  orb,i , metallicity , and  cr as initial input parameters to simulate the final primary mass  1,f and second-born NS/BH mass  2,f through MESA.When we consider the influence of accretion-induced magnetic field on the accreted mass and, hence, the final primary mass, we transfer the critical accretion rates to magnetic field strengths by Equation ( 6) and evolve them with accreted masses.
We firstly ignore the possible decay of NS magnetic field and, hence, the accreting NSs can always have a constant critical super-Eddington accretion rate, since it is currently still unclear which physical effect, such as spindown-induced flux expulsion, Ohmic evolution of the crustal field, and diamagnetic screening of the field by accreted plasma, dominates the decay of magnetic field (e.g., Bhattacharya 2002).It is worth noting that the observation of a large bubble nebula surrounding NGC 1313 X-2 suggested that this ULX pulsar could have maintained a super-Eddington phase for more than 1 Myr, indicating that the magnetic field of this source has not been suppressed during the accreting phase (Sathyaprakash et al. 2019).Furthermore, despite the experience of extensive MT, some longhistory accreting-powered pulsars could still have a strong magnetic field.For instance, the NS in low-mass X-ray binary 4U 1626-67 is almost older than 100 Myr, but its magnetic field is still as strong as a few 10 12 G (Verbunt et al. 1990).Therefore, some NSs could accrete a large abundance of materials without losing their strong magnetic field.
Figure 2 displays the parameter space that allows the formation of mgBH-NS binaries in a solar-metallicity environment, where  ⊙ = 0.0142 (Asplund et al. 2009) is employed in this work.We find that mgBH-NS binaries can hardly be formed via super-Eddington accretion from binary systems composed of ≲ 1.4  ⊙ NSs and He stars.When the primary NSs with  NS 1,i ∼ 1.7  ⊙ have an accretion rate of ≳ 300  Edd or the NSs with  NS 1,i ∼ 2.0  ⊙ have an accretion rate of ∼ 100  Edd , AICs can happen only if the He star companions have an initial mass of  He 2,i ∼ 3 − 5  ⊙ and the binary systems have extremely close orbits with  orb,i ≲ 0.2 d.These AICs majorly occur duration the Case BB MT, and the second-born NSs could have a mass of  NS 2 ∼ 1.1 − 1.5  ⊙ (see the left panel of Figure 3).Furthermore, all NSs with a mass range of ≳ 2.0  ⊙ and an accretion rate of ≳ 300  Edd that undergo the Case BB MT, as well as some undergoing the Case BC MT, can easily experience AIC to BHs.If the secondary is a ∼ 3  ⊙ He star, the formation of mgBHs can be allowed for binary systems with an initial orbital period reaching  orb,i ∼ 2 d.In binary systems with  orb,i ≲ 0.1 d, the primary NSs can finally collapse, even if the secondary He stars are as massive as  He 2,i ∼ 6  ⊙ .We note that the NS could undergo AIC if the initial binary system is composed of a 1.7  ⊙ NS and a 2.5  ⊙ He star that end up as a white dwarf (WD) with a critical accretion rate of 300  Edd (see the right-middle panel in Figure 2) while AICs in the nearby MESA grid systems are prohibited, because the duration of the MT rate of this system above the rate is much longer compared to those of nearby MESA grid systems, as shown in Figure 1.
Figure 4 shows the allowed region for AICs of NSs at  = 0.1  ⊙ .Usually, lower metallicity results in weaker wind mass loss of He stars and final carbon-oxygen cores with higher mass, so that the initial masses of He stars as a final fate of NS would be decreased overall, which is clearly shown in Figure 3.He stars formed at lowermetallicty environments also tend to be more compact and, hence, be more difficult to undergo MT, compared with equal-mass He stars at higher metallicty.In comparison to that at solar metallicity, Figure 4 reveals that the sub-solar metallicity AIC region for the same primary NS with a specific accretion rate moves toward a lower initial He star mass despite no significant change in the size of the region.
We now use an empirical formula from Osłowski et al. (2011), i.e., to consider possible accretion-induced magnetic field decay, where  NS,0 is the initial magnetic field which is listed in Section 3 for different super-Eddington accretion rates,  NS,min = 10 8 G is the minimal magnetic field of an NS,  is the accreted mass, and Δ d = 0.05  ⊙ given by Osłowski et al. (2011).By combining Equation ( 6) and Equation ( 8), we model the evolution of the NS magnetic field and accretion-induced with the accreted mass in MESA.
By setting three different initial critical accretion rates including 100  Edd , 300  Edd , and 500  Edd , our simulated results at  =  ⊙ are displayed in Figure 5.If we consider the possible accretioninduced magnetic field decay, Figure 5 demonstrates that NSs are unlikely to accrete enough materials to become mgBHs if the initial mass of primary NS is  NS 1,i ≲ 1.7  ⊙ since the magnetic field strength and accretion rate decay rapidly with the increase of the NS mass.AICs can only be achievable when the primaries of binary NS-He star systems are high-mass NSs with  NS 1,i ≳ 2.0  ⊙ and the accretion rates are ≳ 300  Edd .However, the AIC regions still reduce significantly relative to those without consideration of accretioninduced magnetic field decay.As shown in Figure 5, these AICs occur in very close-orbit binary systems with low-mass He stars which mainly undergo Case BB MT.Thus, if the magnetic field of NS could decay due to accretion, AICs could be hardly to happen via super-Eddington stable MT.
Accreting NSs during the Case BB/BC MT with super-Eddington accretion could form a population of mgBH-NS binaries.Because BHs formed after AICs of NSs could not continue to maintain super- Final remnant mass (the color bars) as a function of the initial mass of secondary He star  He 2,i , and orbital period  orb,i .Three different  NS 1,i , including 1.4  ⊙ (top panels), 1.7  ⊙ (middle panels), and 2.0  ⊙ (right panels) are considered.For each  NS 1,i , we set three different critical accretion rates, i.e., 100  Edd , 300  Edd , and 500  Edd from left to right panels.Black crosses, squares, triangles, diamonds, and circles represent the conditions of initial overflow, Case BA, Case BB, Case BC, and no MT, respectively.The points with pink interiors mark systems that can occur AIC of NS.Blue and green shading indicates where the He stars end up as WDs and BHs, respectively.We note that the color bars are different for each  NS 1,i .  .Left and right panels represent two different metallicities, i.e.,  ⊙ and 0.1  ⊙ , respectively.Squares, lower triangles, upper triangles, and circles are the outcomes of white dwarf, NS formed through electron-capture SN, NS formed through core-collapse SN, and BH.Since different initial primary NS masses and Eddington accretion limits have little impact on the mass of resulting compact objects,  NS 1, = 1.4  ⊙ and an accretion limit of 100  Edd are set here, while results by considering other different parameters limits are not shown.The maximum NS mass is set to  NS,max = 2.2  ⊙ .
Eddington accretions, the final masses of these mgBHs would be approach to or be a little higher than the Tolman-Oppenheimer-Vollkoff (TOV) TOV mass of NS.Furthermore, this formation channel could generate amount of massive BNS systems, in which the primary NSs could have a mass that is close to the TOV mass of NS, if AICs do not happen during Case BB/BC MT.In Figure 6, we show the post-MT separations of the evolved systems and then further estimate the insprial time of the massive BNS and mgBH-NS systems.The SN kicks of our simulations are ignored, since most of second SNe from the systems that undergo Case BB/BC MT would be ultra-stripped SNe (Tauris et al. 2015).Our simplified simulations suggest that massive BNS and mgBH-NS systems which can merge within the Hubble time require an initial orbital period of  orb,i ≲ 0.4 − 1 d.Tauris et al. (2017) and Hu et al. (2023) present that most of BNS and NSBH binaries would be survived after the second ultra-stripped SNe due to weak kicks (see discussions in Tauris et al. 2015) and can merge within the Hubble time if  orb,i ≲ 0.4 d which is consistent with our simplified simulations in Figure 6.Therefore, it is expected that these high-mass BNS and mgBH-NS binaries formed through super-Eddington stable MT are ideal GW sources in LIGO and LISA bands.

EM Signatures
It has long been proposed that disrupted BH-NS mergers can power gamma-ray bursts (GRB; e.g., Paczynski 1986Paczynski , 1991;;Eichler et al. 1989;Narayan et al. 1992;Zhang 2018;Gottlieb et al. 2023) and kilonova emissions (e.g., Li & Paczyński 1998;Metzger et al. 2010;Kyutoku et al. 2015;Kawaguchi et al. 2016;Kasen et al. 2017;Barbieri et al. 2019;Zhu et al. 2020Zhu et al. , 2022b;;Darbha et al. 2021;Gompertz et al. 2023).Whether an NS can be tidally disrupted by the primary BH and eject a certain amount of material to generate EM counterparts could be described by the total amount of remnant mass  total,fit outside the BH horizon as a nonlinear function of  BH , the dimensionless projected aligned spin  BH, ,  NS , and  NS1 , which has been given by an empirical model from Foucart et al. (2018), i.e.,  total,fit where is the normalized radius of the BH innermost stable circular orbit with 9), and  =  BH / NS is the mass ratio.The fitting parameters include  = 0.406,  = 0.139,  = 0.255, and  = 1.761.BH-NS mergers with system parameters falling into the parameter space where  total,fit = 0 correspond to plunging events, while those with  total,fit > 0 can allow tidal disruption to generate EM signals.
We adopt the AP4 model (Akmal & Pandharipande 1997) as the NS EoS, whose TOV mass is  TOV = 2.22  ⊙ nearly consistent with  NS,max we used in Section 2. AP4 is one of the most likely  EoSs allowed by the constraint on the observations of GW170817 (e.g., Abbott et al. 2018).We calculate  b NS as an empirical function of  NS given by Gao et al. (2020) NS , where  1 = 0.045 and  2 = 0.023, and  b NS and  NS in this function are in units of  ⊙ . NS ≈ 1.1056× ( b NS / NS −1) 0.8277 is estimated following the empirical formula constructed by Coughlin et al. (2017).In Figure 7, we display the tidal disruption region for BH-NS mergers dependent on  BH ,  BH, , and  NS by using Equation (9).
For the standard isolated binary formation channel of BH-NS mergers, BH component unusually forms firstly in a wide orbit and, hence, could have a negligible spin since the tides are too weak to  spin its progenitor up (see Figure 1 in Qin et al. 2018) 2 .As shown 2 Under extreme conditions, the NS in a BH-NS binary could be born first due to a reversal in the mass ratio of progenitor stars during the Case A MT stage (Pols 1994;Román-Garza et al. 2021;Broekgaarden et al. 2021;Mould et al. 2022;Adamcewicz et al. 2023).The BH progenitors can thus be tidally spun up after the common-envelope stage, finally remaining fast-spinning BHs (Hu et al. 2022).
in Figure 7, if the mass gap is intrinsically existent, disrupted events could only contribute a limited fraction of BH-NS population in the universe, which would require a mass space of  BH ≲ 6  ⊙ and  NS ≲ 1.2  ⊙ .For examples, GW200105 (GW200115) reported by the LVK Collaboration in O3 (Abbott et al. 2021b) is formed through a merger between a 8.9 +1.1 −1.3  ⊙ (5.9 +1.4 −2.1  ⊙ ) BH with a dimessionless projected aligned spin of −0.01 +0.10  −0.16 (−0.18 l o g 1 0 ( T i n s / M y r ) Figure 6.The left panel corresponds to the post-MT separation at the pre-SN stage and the right one refers to the insprial time of the massive BNS and mgBH-NS systems, respectively.The light gray region marks the parameter space in which the compact object binaries can merge within the Hubble time.Since different initial primary NS masses and Eddington accretion limits have little impact on post-ME separations and insprial times,  NS 1, = 1.4  ⊙ and an accretion limit of 100  Edd are set here, while results by considering other different parameters limits are not shown.and a 1.9 +0.2 −0.2  ⊙ (1.4 +0.6 −0.2  ⊙ ) NS, inferred from low-spin priors of the secondary.Mandel & Smith (2021) found that GW200115 could be more constrained, with  BH = 7.0 +0.4  −0.4  ⊙ and  NS = 1.25 +0.09 −0.09  ⊙ , by adopting a zero-spin prior preference.Thus, these two BH-NS mergers could possess BH components with mass above the mass gap, and their formation channels could align with the standard picture (e.g, Broekgaarden & Berger 2021;Chattopadhyay et al. 2022;Zhu et al. 2022a;Jiang et al. 2023).Apparently, Figure 7 suggested that these mergers could hardly make tidal disruption to generate EM counterparts, consistent with the conclusion from Zhu et al. (2021); Fragione (2021); Gompertz et al. (2022);D'Orazio et al. (2022); Biscoveanu et al. (2023).In contrast, mgBHs could be more likely to tidally disrupt more massive NSs. Figure 7 shows that mergers between zero-spin mgBHs and NSs can easily undergo tidal disruption as long as  NS ≲ 1.2 − 1.5  ⊙ .
Our detailed binary simulations shown in Section 3 suggest that mgBHs formed via super-Eddington accretion during the Case BB/BC MT could have a mass nearly equal to or slightly higher than  NS,max .Unless the mass of secondaries is ≳ 1.5  ⊙ , tidal disruption can always happen.As an example in Figure 7, we show all our MESA grid data capable of forming mgBH-NS mergers at a metallicity of  ⊙ in Figure 2.Although only four data points of mgBH-NS mergers are visible in Figure 7, we note that actually a total of 82 data points are included, as they are overlapping.One can find that the secondaries for those binary systems that experience Case BB/BC could mostly have a mass of ≲ 1.5  ⊙ , which are located in the parameter region that allows tidal disruption.Thus, it is expected that a large proportion of mgBH-NS mergers formed through super-Eddington stable MT would occur tidal disruption to generate EM signals.

Constraints on the NS TOV Mass and EoS
In Section 3, we suggest that binary systems undergoing super-Eddington stable MT can finally generate a population of GW NS binaries, comprising either an NS with a mass close to  TOV or a mgBH with a mass slightly higher than  TOV .Distinguishing between these high-mass BNS and mgBH-NS mergers using GW and EM observations can help constrain the NS TOV mass and EoS.
In the case of a mgBH-NS mergers, GW emissions would abruptly terminate upon the tidal disruption of the NS, without exciting quasinormal modes (Shibata et al. 2009).However, for a massive BNS involving a  TOV NS, the outcome of the merger is a prompt collapse accompanied with ringdown emission from the remnant BH.The difference of GW merger phase in the kilohertz range between these scenarios could be discerned with the advent of third-generation detectors (e.g., Shibata et al. 2009;Kyutoku et al. 2020).
Despite prompt collapse after merging, a typical-mass NS in an asymmetric BNS system with a companion of a  TOV NS can be tidally disrupted to power bright kilonova signal.These asymmetric BNS mergers can generate disk outflow and massive lanthanide-rich dynamical ejecta, whose kilonovae are similar to those from mergers between ≳ 5  ⊙ BH and NS (e.g., Sekiguchi et al. 2016;Bernuzzi et al. 2020).However, numerical relativity simulations revealed that the mass of dynamical ejecta from mgBH-NS mergers could be lower by an order of magnitude compared to that from corresponding BNS binaries, or even negligible (Foucart et al. 2019;Kyutoku et al. 2020;Hayashi et al. 2021).Thus, the kilonova emissions from low-mass mgBH-NS mergers could lack the contribution from lanthanide-rich dynamical ejecta, which can be used to distinguish mgBH-NS mergers from high-mass BNS mergers through follow-up observations.Thus, we expect that NS-He star binaries with super-Eddington stable MT could serve as a plausible formation channel to form a population of high-mass BNS and low-mass mgBH-NS GWs.Future GW and EM observations on a large number of these NS binaries could provide a precise mass boundary between NSs and BHs.

CONCLUSIONS
In this paper, we propose a new scenario for the formation of mgBH-NS binaries.A fraction of NSs with companions of He stars could have super-Eddington stable MT, as suggested by both observations and theory.We evolve NS-He star binaries by our detailed binary simulations to explore the parameter space that allows for the AICs of NSs.Since super-Eddington accretion can be maintained during the Case BB/BC MT stage, our simulated results reveal that AIC events tend to happen when the primaries NS have an initial mass ≳ 1.7  ⊙ with an accretion rate of ≳ 300  Edd .This formation scenario can thus generate a population of NS binaries consisting of a mgBH with a mass slightly higher than  TOV for AIC events or an NS with a mass close to  TOV if AICs do not happen, which can eventually merge within the Hubble time to become ideal GW sources.However, it is difficult to evaluate how common such a scenario is for forming mgBH-NS mergers based on our present knowledge.On the one hand, low-mass NSs are more common than high-mass NSs inferred by the Galactic pulsar observations (Lattimer 2012;Antoniadis et al. 2016;Alsing et al. 2018;Farr & Chatziioannou 2020) and population synthesis simulations (e.g., Giacobbo & Mapelli 2018;Vigna-Gómez et al. 2018;Broekgaarden et al. 2022).The lack of high-mass primary NSs in BNS systems makes it difficult for AICs to happen.On the other hand, GW190425 was measured to have a heavy primary NS with a mass of ∼ 1.61 − 2.52  ⊙ (Abbott et al. 2020a), accompanied by an unexpectedly high event rate density, if it had a BNS origin.Furthermore, the NSs observed in BNS and BH-NS mergers detected via GWs exhibit a uniform distribution in mass (Landry & Read 2021;Zhu et al. 2022a;Abbott et al. 2023a), contrary to the Galactic pulsar observations and population synthesis simulations.Systems between high-mass NSs and He stars may be widespread in the universe, suggesting that AICs via super-Eddington stable MT are not necessarily impossible.These mgBH-NS mergers can easily lead to tidal disruption, generating bright EM signals.Future GW and EM observations on the population of these NS binaries formed via super-Eddington stable MT could help us constrain the TOV mass and EoS of NSs.
Most recently, Ligo Scientific Collaboration et al. ( 2023) reported a GW compact binary merger candidate in the fourth observing run, S230529ay, which was observed solely by the LIGO Livingston Observatory.S230529ay could be a high-confidence GW candidate with a false alarm rate of 1 per 160.44 yr estimated by the online analysis.The probabilities of classifying this GW signal are 62% and 31% for BH-NS and BNS merger origins, respectively.Under the assumption that the source had an astrophysical origin, the probability that the mass of the primary compact object lied in the mass range of the lower mass gap is 98.5%, while the probability that the secondary compact object was an NS is > 99%.Based on present limited information on this GW, we could suspect that the primary of this GW could not be much higher than the TOV mass.Thus, S230529ay could be a GW candidate of mgBH-NS binary merger, and could possibly originate from super-Eddington stable MT formation channel.

Figure 1 .
Figure 1.MT rate of binary systems as a function of the star age.We set an initial NS mass of  NS 1,i = 1.4  ⊙ , an initial He star mass of  He 2,i = 2.5  ⊙ and  =  ⊙ .Seven different initial orbital periods of  orb,i = 0.02, 0.18, 0.34, 0.50, 0.65, 0.81, and0.97 d are considered.Three different super-Eddington accretion rates for a 1.4  ⊙ NS, including 100  Edd , 300  Edd , and 500  Edd , are marked with dashed lines.

Figure 5 .
Figure 5. Similar to Figure 2, but consider the accretion-induced magnetic field decay of NS.

Figure 7 .
Figure7.The source-frame mass parameter space for BH-NS merger systems to allow tidal disruption of the NS by the BH with consideration of the AP4 EoS model.The dashed lines represent mass ratio from  = 2 to 8. We mark several values of the primary BH spin along the orbital angular momentum from  BH, = −0.75 to 0.90 as solid lines.For a specific  BH, , the BH-NS mergers with component masses located at the bottom left parameter space (denoted by the direction of the arrows) can have  tot,fit > 0, indicating that these mergers can allow tidal disruption.The gray and white shadow regions represent that the mass of the primary compact objects falls below the NS maximum mass and into the mass gap, respectively.The binary systems that can form mgBH-NS mergers, as shown in Figure2, are marked as pink empty points.The 90% credible posterior distributions (colored solid lines) and the medians (colored stars) of GW200105 (orange), GW200115 (blue;Abbott et al. 2021b), and GW200115 (green) obtained by applying an astrophysically motivated priors(Mandel & Smith 2021) are displayed, while corresponding median values of  BH, for these two sources are marked as dashed-dotted lines.
Mass of second-born NS and BH (the color bars) as a function of the initial mass of secondary He star  He 2,i and initial orbital period  orb,i