The interaction between hyperons and the surface gravitational redshift of proto neutron star PSR J0740+6620

. An eﬀect of the contributions from σ ∗ and φ mesons on the surface gravitational redshift of PNS PSR J0740+6620 is studied with relativistic mean ﬁeld theory in consideration of a baryon octet. We ﬁnd that the energy density ε increases relative to the same pressure p as the contributions from σ ∗ and φ mesons are taken into account. Relative to the same central energy density, the radius R of the PNS increases while the mass M , the mass radius ratio M/R and the surface gravitational redshift z of the PNS decrease, taking into account the contributions from σ ∗ and φ mesons. Under the constraint of mass M =2.08 M (cid:12) , considering the contributions from σ ∗ and φ mesons, for the PNS PSR J0740+6620, the radius decreases by about -0.08%, the central energy density increases by about 0.89%, the central pressure increased by about 0.76%, the mass radius ratio increases by about 0.08%, and the surface gravitational redshift increases by about 0.17% from z =0.31663 to z =0.31718.


Introduction
Neutron stars (NSs) are extremely dense because of their large mass M and small radius R (Glendenning, 1997). The mass of an NS will provide constraints on its properties. For example, the mass of an NS would constrain the symmetry energy within it (Li et al., 2021), and perhaps the properties of an NS could also be explained by dark matter (Ding et al., 2022).
Whenever a new massive NS is found, it will inevitably provide some new restrictions on the properties of NS matter. In recent years, several massive NSs have been discovered in succession. In 2010, NS PSR J1614-2230 was observed (Demorest et al., 2010;Fonseca et al., 2016) and three years later another more massive NS PSR J0348+0432 was found (Antoniadis et al., 2013). In 2019, NS PSR J0740+6620 was observed, possibly the most massive NS to date, with a mass of M = 2.14 +0.10 −0.09 M (2020) (Cromartie et al., 2020), or M = 2.08 +0.07 −0.07 M (2021) (Fonseca et al., 2021). Why NS PSR J0740+6620 has such a large mass can be explained by scalar tensor theories of gravity by Degollado et al (Degollado et al., 2020). The mass of NS PSR J0740+6620 suggests that the equation of state (EoS) of NS matter must be hard enough, and it rules out some EoSs that are too soft (Zhou et al., 2019). Since NS PSR J0740+6620 is much more massive than an NS of mass 1.4 M , it is much denser than an NS of mass 1.4 M (Han et al., 2020).
The mass radius ratio M/R of an NS determines its surface gravitational redshift z. If we know the mass radius ratio M/R of an NS we can calculate its surface gravitational redshift z, and vice versa. NSs are relatively easy to observe. So if we know the surface gravitational redshift z of an NS and the mass of the NS, we can calculate its radius R. It can be seen that the surface gravitational redshift z of an NS is an important physical quantity to describe the properties of an NS (Glendenning, 1997).
An NS is formed from a proto neutron star (PNS) that emit energy through neutrino radiation. The PNS is formed in the center of a supernova explosion due to gravitational collapse. A PNS can have temperatures as high as T =30 MeV. The theoretical study of PNSs is helpful to understand the evolution of NSs (Burrows et al., 1986;Prakash et al., 1997).
The interaction between nucleons in PNS matter can be described in terms of σ, ω and ρ mesons. However, the interaction between the hyperons, which can be described in terms of mesons f 0 (975) (denoted as σ * ) and φ(1020) (denoted as φ), must also have some effect on the properties of the PNS matter (Schaffner et al., 1994;Mu et al., 2017).
In this paper, we study the effect of the contributions from σ * and φ mesons on the surface gravitational redshift of the PNS PSR J0740+6620 using the relativistic mean field (RMF) theory (Zhou, 2016).

RMF theory of finite temperature NS matter
The Lagrangian density of finite temperature NS matter (Glendenning, 1997;Schaffner et al., 1994) is The energy density and the pressure of mesons and baryons (Glendenning, 1987a,b) where n B (k) is the Fermi-Dirac distribution function of baryon The energy density and the pressure of leptons are respectively The mass and radius of the PNS are obtained by the TOV equation (Tolman, 1939;Oppenheimer et al., 1939) The PNS's surface gravitational redshift can be obtained by (Glendenning, 1997):

Parameters
In this work, we select eight sets of nucleon coupling parameters to calculate PNSs: DD-ME1 (Typel et al., 1999), GL85 (Glendenning, 1985), GL97 (Glendenning, 1997), TW99 (Typel et al., 1999), GM1 (Glendenning et al., 1991), FSUGold (Todd-Rutel et al., 2005), FSU2R (Laura et al., 2017) and FSU2H (Laura et al., 2017). In the early stage of formation, the temperature of the PNS can be as high as 30 MeV. We choose the PNS PSR J0740+6620's temperature as T =20 MeV (Burrows et al., 1986). Denoting hyperons Λ, Σ and Ξ) by h, we define gρ , which are in the range from 1/3 to 1 (Glendenning et al., 1991). The parameter x ρh is selected by quark SU(6) symmetry (Schaffner et al., 1996;Schaffner-Bielich et al., 2000). The calculation results show that the bigger x σh and x ωh are, the bigger PNS mass is (Zhao, 2019). Therefore, in order to obtain the largest possible mass of the PNS, we must choose the largest possible hyperon coupling parameters x σh and x ωh . We select x ωh =0.9 and x σh can be obtained by the following formula (Glendenning, 1997) where we choose hyperon-potentials as follows U  (Schaffner-Bielich et al., 2000;Weissenborn et al, 2012;Gal et al., 2016;Batty et al., 1997) and U (N ) Ξ = −14 MeV (Harada et al., 2010). We choose x σ * h and x φh as follows (Schaffner et al., 1994) x Without considering the contributions from σ * and φ mesons, we calculate the PNS by using the above eight groups of nucleon coupling parameters, and find that only the maximum masses (denoted as M max ) of the PNS obtained by DD-MEI, TW99 and GM1 are greater than that of the PNS PSR J0740+6620 (see Fig. 1). Among these three sets of parameters, the mass and  (Miller et al., 2021)). Therefore, we use GM1 to study the influence of the contributions from σ * and φ mesons on the surface gravitational redshift of the PNS PSR J0740+6620. We can see from Fig. 2(a) that the energy density ε of the PNS increases as the pressure p increases. We also see that the energy density ε increases relative to the same pressure p, taking into account the contributions from σ * and φ mesons. Under the constraint of mass M =2.08 M , considering the contributions from σ * and φ mesons, the central energy density of the PNS PSR J0740+6620 increases by about 0.89% from ε c =1.013×10 15 g.cm −3 to ε c =1.022×10 15 g.cm −3 (see Table 1); the central pressure of the PNS PSR J0740+6620 increased from p c =2.118×10 35 dyne.cm −2 to p c =2.134×10 35 dyne.cm −2 , increasing by about 0.76%.

The radius and the surface gravitational redshift of PNS PSR J0740+6620
The PNS's mass M and radius R as a function of the central energy density ε c are shown in Fig. 3. The left ordinate represents the mass of the PNS, and We see that the mass M of the PNS increases and the radius R decreases as the central energy density ε c increases. Relative to the same central energy density, the mass of the PNS decreases and the radius increases, taking into account the contributions from σ * and φ mesons. Under the constraint of mass M =2.08 M , considering the contributions from σ * and φ mesons the radius of the PNS PSR J0740+6620 decreases by about -0.08% from R=14.518 km to R=14.507 km (see Table 1). Figure 4 gives the mass radius ratio M/R and the surface gravitational redshift z of the PNS as a function of the central energy density ε c . The PNS's temperature is assumed to be 20 MeV. The red real curve does not consider the contributions from σ * and φ mesons, while the green virtual curve does. The curve ends at the central energy density ε c and the central pressure p c of the PNS PSR J0740+6620.
We see that the mass radius ratio M/R and the surface gravitational redshift z of the PNS all increase with the increase of the central energy density ε c . Relative to the same central energy density ε c , the mass radius ratio M/R and the surface gravitational redshift z of the PNS are reduced considering the contributions from σ * and φ mesons. As it can also be seen from Fig. 4 and Table 1, under the limit of mass M =2.08 M , the mass radius ratio of PNS PSR J0470+6620 increases from M/R=0.14327 M /km to M/R=0.14338 by about 0.08%, and the surface gravitational redshift of PNS PSR J0470+6620 increases by about 0.17% from z=0.31663 to z=0.31718, taking into account the contributions from σ * and φ mesons. Fig. 5, Fig. 6 and Fig. 7 respectively show the variation of the surface gravitational redshift z of the PNS with the radius R, the mass M and the mass radius ratio M/R. arate the curves that take into account the contributions from σ * and φ mesons and those that do not, we add 0.01 to the curve that takes into account the contributions from σ * and φ mesons, that is, z→z + 0.01.
Relative to the same radius R, the surface gravitational redshift z of the PNS decreases, taking into account the contributions from σ * and φ mesons (see Fig. 5). Similarly, given the contributions from σ * and φ mesons, the surface gravitational redshift z of the PNS increases relative to the same mass M (see Fig. 6). As it can also be seen from Fig. 7, considering the contributions from σ * and φ mesons, the surface gravitational redshift of the PNS does not change with respect to the same mass radius ratio M/R, which can be known by formula (9).
In addition, under the constraint of mass M =2.08 M , the radius R of PNS PSR J0740+6620 will decrease, and the mass-radius ratio M/R and surface gravitational redshift z of PNS PSR J0740+6620 will increase (see Fig. 5, Fig. 6, Fig. 7 and Table 1). Given the mass of the PNS PSR J0740+6620 by Fonseca et al. (Fonseca et al., 2021), and the radius by Miller et al. (Miller et al., 2021), we can calculate the mass radius ratio to be 0.1233 M /km<M/R<0.1762 M /km and the surface gravitational redshift of the PNS PSR J0740+6620 to be 0.2541<z<0.4440 (see the shaded boxes in Fig. 5-Fig. 7). It can be seen that our calculated results are in good agreement with the observed values.
The above results are valid for spherically symmetric static PNSs. For a rotating NS, the maximum mass increases by about 20%, and the NS becomes axisymmetric (Cook et al., 1994). The gravitational redshift of a rotating NS is a little more complicated than the previous NS as static and spherically symmetric, where we need to consider the polar redshift z p , the equatorial redshift in the backward direction z b , and the quatorial redshift in the forward direction z f (Tu et al., 2022).

Summary
We studied the effect of the contributions from σ * and φ mesons on the surface gravitational redshift of PNS PSR J0740+6620 with RMF theory in consideration of a baryon octet. Here, we use eight sets of nucleon coupling constants GL85, GL97, GM1, DD-MEI, TW99, FSUGold, FSU2R and FSU2H to calculate the PNS. The temperature of the PNS is chosen as T =20 MeV. We find that only three of them (DD-MEI, GM1 and TW99) can give the PNS PSR J0740+6620's mass. We use GM1 to describe the PNS PSR J0740+6620. The energy density ε of the PNS increases as the pressure p increases. The PNS's radius R decreases while the mass M , the mass-radius ratio M/R and the surface gravitational redshift of PNS increase with the increase of the central energy density ε c .
The energy density ε increases relative to the same pressure p, taking into  account the contributions from σ * and φ mesons. Relative to the same central energy density, the radius R of the PNS increases while the mass M , the massradius ratio M/R and the surface gravitational redshift z of the PNS decrease, taking into account the contributions from σ * and φ mesons. Under the constraint of mass M =2.08 M , considering the contributions from σ * and φ mesons, for the PNS PSR J0740+6620, the radius decreases by about -0.08% from R=14.518 km to R=14.507 km, the central energy density increases by about 0.89% from ε c =1.013×10 15 g.cm −3 to ε c =1.022×10 15 g.cm −3 , the central pressure increased by about 0.76% from p c =2.118×10 35 dyne.cm −2 to p c =2.134×10 35 dyne.cm −2 , the mass radius ratio increases from M/R=0.14327 M /km to M/R=0.14338 by about 0.08%, and the surface gravitational redshift increases by about 0.17% from z=0.31663 to z=0.31718.
Taking into account the contributions from σ * and φ mesons, the surface gravitational redshift z of the PNS decreases relative to the same radius R, and . The PNS's surface gravitational redshift z as a function of the mass radius ratio M/R. The PNS's temperature is assumed to be 20 MeV. The red real curve does not consider the contributions from σ * and φ mesons, while the green virtual curve does. The curves end at the mass radius ratio M/R of the PNS PSR J0740+6620. In order to separate the curves that take into account the contributions from σ * and φ mesons and those that do not, we add 0.01 to the curve that takes into account the contributions from σ * and φ mesons, that is, z→z + 0.01.
increases relative to the same mass M , but does not change with respect to the same mass-radius ratio M/R After accounting for the contributions of σ * and φ mesons, the radius and surface gravitational redshifts are slightly changed compared to the case without σ * and φ mesons. But the changes are too small to be inffered from astronomical observations.