⇦Chinese Physics Letters, 2019, Vol. 36, No. 6, Article code 061301 Nucleonic $^{1}S_{0}$ Superfluidity Induced by a Soft Pion in Neutron Star Matter with Antikaon Condensations * Yan Xu (许妍)1,2, Qi-Jun Zhi (支启军)2, Yi-Bo Wang (王夷博)1, Xiu-Lin Huang (黄修林)1**, Wen-Bo Ding (丁文波)3, Zi Yu (喻孜)4, Cheng-Zhi Liu (刘承志)1** Affiliations 1Changchun Observatory, National Astronomical Observatories, Chinese Academy of Sciences, Changchun 130117 2Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guiyang 550025 3College of Mathematics and Physics, Bohai University, Jinzhou 121000 4College of Science, Nanjing Forestry University, Nanjing 210037 Received 18 January 2019, online 18 May 2019 *Supported by the Open Foundation of Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, the Youth Innovation Promotion Association of the Chinese Academy of Sciences under Grant No 2016056, the Development Project of Science and Technology of Jilin Province under Grant No 20180520077JH, and the National Natural Science Foundation of China under Grant Nos 11805022 and 11803057.
**Corresponding author. Email: huangxl@cho.ac.cn; lcz@cho.ac.cn Citation Text: Xu Y, Zhi Q J, Wang Y B, Huang X L and Ding W B et al 2019 Chin. Phys. Lett. 36 061301    Abstract The nucleonic $^{1}S_{0}$ superfluidity is investigated by solving the gap equation for the Reid soft-core potential as the nucleon–nucleon interaction in neutron star (NS) matter which is considered to be made up of n, p, e, $\mu$ and condensed antikaon matter. We mainly study the influence of the soft pion-induced potential on the nucleonic $^{1}S_{0}$ pairing gaps in the above NS matter. It is found that the intensities of the nucleonic $^{1}S_{0}$ pairing gaps including the soft pion-induced potential are smaller than those calculated in the case of not including the soft pion-induced potential. Furthermore, the nucleonic $^{1}S_{0}$ pairing gaps with the soft pion-induced potential fall into decline with the deepening of the optical potential of antikaons in the above NS matter, whereas they increase with the parameter $\eta$ for the fixed optical potential of antikaons. Due to the appearance of the soft pion-induced potential, the maximum values of nucleonic $^{1}S_{0}$ pairing gaps at parameter $\eta=0.20, 0.55$ are suppressed by 1.7%–6.8% with respect to the case without soft pion-induced potential in the above NS matter. DOI:10.1088/0256-307X/36/6/061301 PACS:13.75.Cs, 26.60.Dd, 26.60.Gj © 2019 Chinese Physics Society Article Text It is widely known that neutron stars (NSs) are ideal models for studies of dense matter physics, and the various superfluid phases play a significant role in our understanding of some phenomena of astrophysics in NSs.[1–4] Yet until now, a satisfactory description of superfluidity in NS matter has not yet been achieved despite the research carried out in the past six decades since the first application of the Bardeen–Cooper–Schrieffer (BCS) theory to nuclear systems in 1959.[5,6] It is largely due to the fact that superfluidity is an extremely subtle process when it is considered on an entirely microscopic level. In addition, the question of the screening of nuclear pairing, which has many uncertain factors such as non-direct observational data under extreme conditions, is a long-standing problem. Most theoretical calculations based on qualitative models and different model potentials of nuclear pairing cannot obtain accurate results for pairing gaps and estimate the quantitative influence of superfluidity on the properties of NS cooling.[7–9] Baryons in NSs can form several types of baryonic pairings on account of the attractive interactions between two baryons, which was firstly proposed by Migdal in 1959.[10] Specifically, neutronic superfluids may appear at relatively low temperatures by the generation of neutron–neutron (nn) Cooper pairs in NS matter. When the temperature of an NS is under the neutronic superfluid critical temperature within a couple years after its birth, neutrons probably form $^{1}S_{0}$ superfluids in the NS crust and $^{3}P_{2}$ superfluids in the NS core, respectively. Protons and possible hyperons are also likely to form $^{1}S_{0}$ superfluids with supranuclear density in the NS core.[11–17] It is generally believed that the presence of neutronic and protonic superfluids can suppress the neutrino energy losses of neutrino emission processes, thereby affecting the cooling rates of NSs remarkably. Kaplan and Nelson first proposed that K mesons could undergo Bose–Einstein condensation in dense matter in 1986.[18,19] Since then, studies have demonstrated that antikaon condensations, including both $K^{-}$ and $\bar{K}^{0}$ condensations, could occur in NS matter and change the bulk properties of NS matter.[20–23] The recent research of Xu et al. has also demonstrated that antikaon condensations lead to a strong suppression of the neutronic $^{1}S_{0}$ superfluid and an obvious enhancement of the protonic $^{1}S_{0}$ superfluid in NS matter, respectively.[24] Moreover, the condensation in a soft pionic mode can strongly modify the effective nucleon–nucleon (NN) interaction and is expected to appear in nuclear physics, hence the name 'induced interaction'.[25–33] It is well known that the soft pion-induced interaction holds for the Reid soft core (RSC) NN potential in NS matter. Up to now, we do not know how the nucleonic $^{1}S_{0}$ pairing gaps will change if the soft pion-induced interaction is included in NS matter with antikaon condensations. Our work mainly studies the nucleonic $^{1}S_{0}$ pairing in the vicinity of the $\pi^{0}$ condensation point, considers the pairing interaction induced by a soft pion, and discusses specific features of the solution of the gap equation with the different intensities of the optical potential of antikaons in NS interiors. The study is limited to the nucleonic $^{1}S_{0}$ pairings in the Landau state. An extension of the study on the other possible states of the Fermi surface will be performed in the future. We solve the BCS gap equation in the $^{1}S_{0}$ channel to calculate the nucleonic gap function ${\it\Delta}_{\rm N}({\boldsymbol p})$[5,6,8] $$ {\it\Delta}_{\rm N}({\boldsymbol p})=-\frac{1}{4\pi^{2}}\int{\boldsymbol p}^{'2}d{\boldsymbol p^{'}}\frac{V({\boldsymbol p},{\boldsymbol p^{'}}){\it\Delta}_{\rm N}({\boldsymbol p^{'}})}{\sqrt{\varepsilon^{2}({\boldsymbol p^{'}})+{\it\Delta}_{\rm N}^{2}({\boldsymbol p^{'}})}},~~ \tag {1} $$ where $\varepsilon({\boldsymbol p})=E_{\rm N}({\boldsymbol p})-E_{\rm N}({\boldsymbol p_{\rm F}})$, and $E_{\rm N}({\boldsymbol p})$ is the single-particle energy of the nucleon with momentum ${\boldsymbol p}$, $$ E_{\rm N}({\boldsymbol p})=\sqrt{p^{2}+m_{\rm N}^{*2}}+g_{\omega N}\omega+g_{\rho N}I_{3N}\rho ,~~ \tag {2} $$ with $m_{\rm N}^{*}=m_{\rm N}-g_{\sigma N}\sigma$ being the nucleonic effective mass.
Table 1. The set of parameters for the RSC potential.[34] In the two-neutron interaction, $a_{i}=a_{i}^{0}(1+a_{i}^{1}\alpha+a_{i}^{2}\alpha^{2}+a_{i}^{3}\alpha^{3})$, $b_{i}=b_{i}^{0}(1+b_{i}^{1}\alpha+b_{i}^{2}\alpha^{2}+b_{i}^{3}\alpha^{3})$. In the two-proton interaction, $\alpha$ should be substituted by $-\alpha$. The units of $\lambda_{i}$, $a_{i}^{0}$ and $b_{i}^{0}$ are fm, MeV and MeV$\cdot$fm$^{\frac{1}{2}}$, respectively. The remaining parameters of the RSC potential are dimensionless.
$\lambda_i$ 0.50 0.95 1.70 2.85 5.00
$a_{i}^{0}$ 2877.0 $-$797.1 29.87 $-$2.891 $-$0.02014
$a_{i}^{1}$ $-$0.07544 $-$0.3034 $-$1.02 $-$0.508 1.254
$a_{i}^{2}$ $-$0.59 0.3052 1.235 0.1453 0.4345
$a_{i}^{3}$ 0.4044 $-$0.1665 $-$0.7672 0.03999 $-$0.9284
$b_{i}^{0}$ $-$1305.0 365.7 $-$17.89 0.1370 0.0002224
$b_{i}^{1}$ $-$0.01357 $-$0.3856 $-$1.227 $-$7.498 94.17
$b_{i}^{2}$ $-$1.017 0.337 0.9234 $-$1.283 42.65
$b_{i}^{3}$ 0.733 $-$0.1108 $-$0.3478 4.386 $-$73.34
Table 2. The minimum values $V_{\rm nn}(r)$ for the nn interaction of the RSC potential plus soft pion-induced potential in coordinate space. We take nucleon density $n_{\rm N}=0.1$, 0.2, 0.3, 0.47 and 0.5$n_{0}$. The units of $V_{\rm nn}^{\min}(r)$ and $U_{\bar{K}}$ are MeV.
$V_{\rm nn}^{\min}(r)$ $0.1n_0$ $0.2n_0$ $0.3n_0$ $0.47n_0$ $0.5n_0$
$\eta=0.20$, $U_{\bar{K}}=-80$ $-$116.84320 $-$101.05681 $-$90.49664 $-$71.10552 $-$56.76825
$\eta=0.20$, $U_{\bar{K}}=-90$ $-$117.11489 $-$101.48751 $-$91.01336 $-$71.84778 $-$57.58524
$\eta=0.20$, $U_{\bar{K}}=-100$ $-$117.39032 $-$101.92515 $-$91.53926 $-$72.59868 $-$58.42316
$\eta=0.20$, $U_{\bar{K}}=-110$ $-$117.66953 $-$102.36985 $-$92.07458 $-$73.35754 $-$59.28247
$\eta=0.20$, $U_{\bar{K}}=-120$ $-$117.95826 $-$102.83076 $-$92.63051 $-$74.13895 $-$60.18155
$\eta=0.20$, $U_{\bar{K}}=-130$ $-$118.23957 $-$103.28090 $-$93.17446 $-$74.89633 $-$61.06782
$\eta=0.55$, $U_{\bar{K}}=-80$ $-$117.29626 $-$102.0542 $-$92.02746 $-$73.83032 $-$59.26385
$\eta=0.55$, $U_{\bar{K}}=-90$ $-$117.56084 $-$102.46611 $-$92.51499 $-$74.5391 $-$60.1361
$\eta=0.55$, $U_{\bar{K}}=-100$ $-$117.8291 $-$102.88487 $-$92.9306 $-$75.25409 $-$61.02478
$\eta=0.55$, $U_{\bar{K}}=-110$ $-$118.1012 $-$103.31061 $-$93.51721 $-$75.97467 $-$61.93025
$\eta=0.55$, $U_{\bar{K}}=-120$ $-$118.38261 $-$103.75211 $-$94.04273 $-$76.7147 $-$62.87151
$\eta=0.55$, $U_{\bar{K}}=-130$ $-$118.65688 $-$104.18356 $-$94.5573 $-$77.43021 $-$63.79359
Table 3. The minimum values $V_{\rm pp}(r)$ for the pp interaction of the RSC potential plus soft pion-induced potential in coordinate space. We take nucleon density $n_{\rm N}=0.5$, 1, 2.5, 3.5 and 5$n_{0}$. The units of $V_{\rm pp}^{min}(r)$ and $U_{\bar{K}}$ are MeV.
$V_{\rm pp}^{\min}(r)$ $0.5n_0$ $n_0$ $2.5n_0$ $3.5n_0$ $5n_0$
$\eta=0.20$, $U_{\bar{K}}=-80$ -207.36231 $-$154.50534 $-$93.91493 $-$74.24611 $-$60.37169
$\eta=0.20$, $U_{\bar{K}}=-90$ $-$208.50441 $-$155.66737 $-$95.03629 $-$75.55341 $-$61.37385
$\eta=0.20$, $U_{\bar{K}}=-100$ $-$209.66684 $-$156.89958 $-$96.18734 $-$76.86345 $-$62.39805
$\eta=0.20$, $U_{\bar{K}}=-110$ $-$210.85931 $-$158.15270 $-$97.36964 $-$78.17596 $-$63.44384
$\eta=0.20$, $U_{\bar{K}}=-120$ $-$212.16203 $-$159.45277 $-$98.60872 $-$79.50750 $-$64.52294
$\eta=0.20$, $U_{\bar{K}}=-130$ $-$213.44235 $-$160.72839 $-$99.83685 $-$80.80972 $-$65.59864
$\eta=0.55$, $U_{\bar{K}}=-80$ $-$208.80934 $-$156.53717 $-$96.63527 $-$77.00099 $-$62.96374
$\eta=0.55$, $U_{\bar{K}}=-90$ $-$209.93841 $-$157.71291 $-$97.74301 $-$78.31142 $-$64.01095
$\eta=0.55$, $U_{\bar{K}}=-100$ $-$211.08764 $-$158.93051 $-$98.87977 $-$79.61959 $-$65.07402
$\eta=0.55$, $U_{\bar{K}}=-110$ $-$212.27776 $-$160.16879 $-$100.04711 $-$80.92564 $-$66.15219
$\eta=0.55$, $U_{\bar{K}}=-120$ $-$213.5663 $-$161.45347 $-$101.27024 $-$82.28029 $-$67.25707
$\eta=0.55$, $U_{\bar{K}}=-130$ $-$214.83273 $-$162.71401 $-$102.48232 $-$83.59934 $-$68.35116
Table 4. The maximum values of nucleonic $^{1}S_{0}$ pairing gaps with and without the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter. The units of ${\it\Delta}_{\rm N}$, $U_{\bar{K}}$ and $T_{\rm CN}$ are MeV and $10^{9}$ K, respectively.
$U_{\bar{K}}$ ${\it\Delta}_{\rm N}^{\max}$ (no $V_{\pi}$) ${\it\Delta}_{\rm N}^{\max}$ ($\eta=0.20$) ${\it\Delta}_{\rm N}^{\max}$ ($\eta=0.55$) $T_{\rm CN}$ (no $V_{\pi}$) $T_{\rm CN}$ ($\eta=0.20$) $T_{\rm CN}$ ($\eta=0.55$)
$U_{\bar{K}}=-80$, n 0.29094 0.27722 0.28563 1.9202 1.8297 1.8852
$U_{\bar{K}}=-100$, n 0.28243 0.26838 0.27730 1.8640 1.7713 1.8302
$U_{\bar{K}}=-120$, n 0.27359 0.26015 0.26883 1.8057 1.7170 1.7743
$U_{\bar{K}}=-80$, p 0.35699 0.33272 0.34854 2.3561 2.1960 2.3004
$U_{\bar{K}}=-100$, p 0.34753 0.32393 0.33923 2.2937 2.1379 2.2389
$U_{\bar{K}}=-120$, p 0.33772 0.31489 0.32968 2.2290 2.0783 2.1759
The NN potential matrix element $V({\boldsymbol p},{\boldsymbol p^{'}})$ is written as $$\begin{align} V({\boldsymbol p},{\boldsymbol p^{'}})=\,&\langle {\boldsymbol p}|V({^1}{S}{_0})|{\boldsymbol p^{'}} \rangle\\ =\,&4\pi\int dr r^2 j_0(pr) V_{\rm NN}(r) j_0(p^{'}r),~~ \tag {3} \end{align} $$ where $V_{\rm NN}$ is the NN potential matrix element in coordinate space, and $j_0(pr)=\sin(pr)/(pr)$ is the zero-order spherical Bessel function. In this work, we use the RSC potential for the NN potential in the solutions[8,34–37] as an example to demonstrate the influence of the soft pion-induced potential on the nucleonic $^{1}S_{0}$ pairing gaps in npe$\mu K^{-}\bar{K}^{0}$ matter. The expression of $V_{\rm RSC}(r)$ is $$\begin{alignat}{1} \!\!\!\!\!\!V_{\rm RSC}(r;n_{\rm N},\alpha,\beta,\gamma)=\sum_{i=1}^5 c_i (n_{\rm N},\alpha,\beta,\gamma)e^{-(r/\lambda_i)^2}.~~ \tag {4} \end{alignat} $$ The RSC potential is described as the five-range Gaussian $c_{i}$, which depends on the two nucleon states $\beta$ and $\gamma$, total nucleon density $n_{\rm N}$ and asymmetry parameter $\alpha=(n_{\rm n}-n_{\rm p})/n_{\rm N}$. Here $c_{i}$ is expressed by $c_{i}(n_{\rm N},\alpha,\beta,\gamma)=a_{i}(\alpha,\beta,\gamma)+b_{i} (\alpha,\beta,\gamma)\sqrt{p_{\rm N}}$. The actual values of these parameters $\lambda_{i}$, $a_{i}$ and $b_{i}$ are listed in Table 1.[34] The nucleonic superfluid critical temperature $T_{\rm CN}$ is given by the corresponding pairing gap ${\it\Delta}_{\rm N}$ at zero temperature approximation,[38,39] $T_{\rm CN}\doteq 0.66\times 10^{10} {\it\Delta}_{\rm N}({\boldsymbol p})$.
cpl-36-6-061301-fig1.png
Fig. 1. The radial distribution derived for the nn interactions of the RSC plus soft pion-induced potentials in npe$\mu K^{-} \bar{K}^{0}$ matter. We take the optical potential of antikaons $U_{\bar{K}}=-80, -90, -100, -110, -120$ and $-130$ MeV. The black, red, blue, green and dark yellow lines represent the five cases of nucleon density $n_{\rm N}=0.1, 0.2, 0.3, 0.47$ and 0.5$n_{0}$, respectively.
The form of the soft pion-induced potential in coordinate space is as follows: $$\begin{align} V_{\pi}(r)=\,&\frac{C_{0}g_{\pi}}{\eta} \frac{p_{\rm F}^{2}}{4\pi r} \exp\Big(-\frac{\eta p_{\rm F}r}{2}\Big)\Big(\sin(p_{\rm F}r)\\ &+\frac{\eta}{2}\cos(p_{\rm F}r)\Big),~~ \tag {5} \end{align} $$ where $C_{0}=\pi^{2}/m^{*}_{\rm N}$ is the inverse unrenormalized density of states, $g_{\pi}=2.8$ is an effective coupling constant, and $\eta$ is a dimensionless measure of proximity of the system to the $\pi_{0}$ instability, $$ \eta=\sqrt{4.6(n_{\rm c}-n_{\rm N})/n_{\rm c}}.~~ \tag {6} $$ with the realistic critical density $n_{\rm c} \simeq 0.2$ fm$^{-3}$. One can see that the soft pion-induced potential is generally assumed to be dependent on its parameters only and we can reference Pankratov et al.[32] in 2015 for a detailed discussion. To estimate the influence of the soft pion-induced interaction on the neutronic and protonic $^{1}S_{0}$ pairing gaps in npe$\mu K^{-} \bar{K}^{0}$ matter, we extend our model by including the soft pion-induced potential and solving the gap equation with $V_{\rm NN}=V_{\rm RSC}+V_{\pi}$. Equation (2) is obtained in the relativistic mean field approximation using the parameter set GM1.[24,40–42] We will concentrate on the unwieldy calculations in the field in future work. The numerical results are discussed in the following.
cpl-36-6-061301-fig2.png
Fig. 2. The radial distribution derived for the pp interactions of the RSC plus soft pion-induced potentials in npe$\mu K^{-} \bar{K}^{0}$ matter. We take the optical potential of antikaons $U_{\bar{K}}=-80$, $-$90, $-$100, $-$110, $-$120 and $-$130 MeV. The black, red, blue, pink and green lines represent the five cases of nucleon density $n_{\rm N}=0.5$, 1, 2.5, 3.5 and 5$n_{0}$, respectively.
Once antikaon condensations appear in NS matter, the intensities of $K^-$ and $\bar{K}^{0}$ condensations are extremely sensitive to the optical potential of antikaons $U_{\bar{K}}$ at the saturation density of nuclear matter, which is limited to the range of $-$80–$-$150 MeV by the experimental studies. We choose $U_{\bar{K}}= -80$, $-$90, $-$100, $-$110, $-$120 and $-$130 MeV in the following numerical calculation. The possibility that the appearance of a pairing force induced by the exchange of a soft pionic mode will affect the nucleonic $^{1}S_{0}$ superfluids in NS matter cannot be ruled out all the time.[30–33] However, up until now we do not know how the nucleonic $^{1}S_{0}$ pairing gaps will change if the soft pion-induced interaction appears in npe$\mu K^{-} \bar{K}^{0}$ matter. The following theoretical results are obtained at parameter $\eta=0.20$ and 0.55 according to Eq. (6), to demonstrate the influence of the soft pion-induced interaction on the nucleonic $^{1}S_{0}$ pairing gaps in npe$\mu K^{-} \bar{K}^{0}$ matter. The properties of npe$\mu K^{-} \bar{K}^{0}$ matter are acquired by adopting the successful relativistic mean field parameter set GM1.[40,41] For the nn and pp interactions, the RSC potential plus the soft $\pi^{0}$ potential in coordinate space for the $^{1}S_{0}$ state at $\eta=0.20$ and 0.55 with the various optical potential depth of antikaons $U_{\bar{K}}$ are shown in Figs. 1 and 2. The two graphs give the nn and pp interactions for nucleon densities of $n_{\rm N}/n_{0}=0.1$, 0.2, 0.3, 0.47, 0.5 ($n_{0}=0.153$ fm$^{-3}$ is the saturation density of nuclear matter) and $n_{\rm N}/n_{0}=0.5$, 1, 2.5, 3.5, 5 in npe$\mu K^{-} \bar{K}^{0}$ matter, respectively. It is shown in Figs. 1 and 2 that the nn and pp interactions are repulsive at short distances. Yet, they turn attractive at middle distances which are enough to produce the nucleonic $^{1}S_{0}$ superfluids in npe$\mu K^{-} \bar{K}^{0}$ matter. As seen in Figs. 1 and 2, the nn and pp interactions are sensitively dependent on the nucleon density $n_{\rm N}$. The intensities of the nn and pp interactions all become weaker along with nucleon density $n_{\rm N}$ growth. To clearly see the influences of the optical potential of antikaons $U_{\bar{K}}$ and parameter $\eta$ on the nn and pp interactions in npe$\mu K^{-} \bar{K}^{0}$ matter more intuitively, the minimum values of the nn and pp interactions are listed in Tables 2 and 3. It can be seen from Figs. 1, 2 and Tables 2, 3 that the deeper the optical potential of antikaons $U_{\bar{K}}$ and the larger the parameter $\eta$, the stronger the nn and pp interactions in npe$\mu K^{-} \bar{K}^{0}$ matter. Meanwhile, the nn and pp interactions for the same optical potential of antikaons $U_{\bar{K}}$ obviously increase as the parameter $\eta$ increases in npe$\mu K^{-} \bar{K}^{0}$ matter.
cpl-36-6-061301-fig3.png
Fig. 3. The nucleonic $^{1}S_{0}$ pairing gaps ${\it\Delta}_{\rm N}({ p_{\rm F}})$ at the Fermi surface as a function of nucleon density $n_{\rm N}$ with the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter. The black, red, blue, dark cyan, pink and dark yellow lines are the cases of $U_{\bar{K}}=-80$, $-$90, $-$100, $-$110, $-$120 and $-$130 MeV, respectively. The green lines represent the nucleonic $^{1}S_{0}$ pairing gaps without the soft pion-induced potential in npe$\mu$ matter.
We take the parameter $\eta=0.20$ and 0.55 to show the changes of the nucleonic $^{1}S_{0}$ pairing gaps due to the presence of the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter, as shown in Fig. 3. In the figure, regardless of whether or not the pairing force induced by the exchange of a soft pionic mode has been included in NS matter, the trends of neutronic and protonic $^{1}S_{0}$ pairing gaps are essentially the same. The neutronic $^{1}S_{0}$ pairing gaps with the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter are clearly below the corresponding values without the soft pion-induced potential in npe$\mu$ matter, whereas the protonic $^{1}S_{0}$ pairing gaps with the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter are significantly higher than the corresponding values without the soft pion-induced potential in npe$\mu$ matter. Moreover, the neutronic and protonic $^{1}S_{0}$ pairing gaps with the soft pion-induced potential all fall into decline as the optical potential of $U_{\bar{K}}$ grows in npe$\mu K^{-} \bar{K}^{0}$ matter. In Fig. 4, the neutronic and protonic $^{1}S_{0}$ pairing gaps at parameter $\eta=0.20$ and 0.55 are compared with and without the soft pion-induced potential for the optical potential of antikaons $U_{\bar{K}}=-80$, $-$100, $-$120 MeV in npe$\mu K^{-} \bar{K}^{0}$ matter, respectively. The intensities of the nucleonic $^{1}S_{0}$ pairing gaps including the soft pion-induced potential are found to be significantly less than those calculated in the case without soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter. However, the differences between the nucleonic $^{1}S_{0}$ pairing gaps including and not including the soft pion-induced potential become smaller in npe$\mu K^{-} \bar{K}^{0}$ matter with the increasing parameter $\eta$. In addition, due to the appearance of the soft pion-induced potential, the maximum values of nucleonic $^{1}S_{0}$ pairing gaps are suppressed by 1.7%–6.8% with respect to the case without the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter. The maximum values of nucleonic $^{1}S_{0}$ pairing gaps and the corresponding nucleonic superfluid critical temperatures with and without the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter are given in Table 4. It can be seen from Fig. 4 and Table 4 that the nucleonic superfluid critical temperatures change with the intensities of the nucleonic $^{1}S_{0}$ pairing gaps, which will obviously impact on the NS cooling rate. We will carefully study the influence of the soft pion-induced potential on the properties of NS cooling in the future.
cpl-36-6-061301-fig4.png
Fig. 4. The nucleonic $^{1}S_{0}$ pairing gaps ${\it\Delta}_{\rm N}({p_{\rm F}})$ at the Fermi surface as a function of nucleon density $n_{\rm N}$ with and without the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter. We take the optical potential of antikaons $U_{\bar{K}}=-80$, $-$100, $-$120 MeV and the parameter $\eta=0.20$, 0.55, respectively. The black lines represent the nucleonic $^{1}S_{0}$ pairing gaps without the soft pion-induced potential in npe$\mu$ matter.
In summary, we have studied the effects of the soft pion-induced potential on the nucleonic $^{1}S_{0}$ superfluidity in npe$\mu K^{-} \bar{K}^{0}$ matter using the BCS theory. The results indicate that the intensities of the NN interactions become weaker with the increasing nucleon density $n_{\rm N}$ and stronger with the increasing optical potential of antikaons $U_{\bar{K}}$, respectively. According to the above calculations, the intensities of the NN interactions for the fixed optical potential of antikaons become stronger with the increasing parameter $\eta$ in npe$\mu K^{-} \bar{K}^{0}$ matter. It is also found that the appearance of the soft pion-induced potential makes the intensities of the nucleonic $^{1}S_{0}$ pairing gaps smaller than the corresponding values in npe$\mu K^{-} \bar{K}^{0}$ matter. The maxima of nucleonic $^{1}S_{0}$ pairing gaps including the soft pion-induced potential are suppressed by 1.7%–6.8% in comparison with the case without the soft pion-induced potential in npe$\mu K^{-} \bar{K}^{0}$ matter, whereas the distinctions between the nucleonic $^{1}S_{0}$ pairing gaps with and without the soft pion-induced potential become smaller along with the increase of the parameter $\eta$. Our model may seem oversimplified because it uses the lowest level of approximation in the BCS equation, and also neglects the possible appearance of other exotic matter (such as hyperons and quarks) in the NS core and the influence of inhomogeneity in the NS crust. However, it provides a relatively proper description of the effects of the soft pion-induced potential on the nucleonic $^{1}S_{0}$ pairing gaps in npe$\mu K^{-} \bar{K}^{0}$ matter. We will analyze more complicated models in future studies. References Study of measured pulsar masses and their possible conclusionsHyperon coupling dependence of hadron matter properties in relativistic mean field modelEffects of σ* and Φ mesons on the surface redshift of a neutron starEffects of Tensor Couplings on Nucleonic Direct URCA Processes in Neutron Star MatterPossible Superfluidity of a System of Strongly Interacting FermionsTheory of Superconductivity1S0 proton and neutron superfluidity in β-stable neutron star matterSinglet pairing gaps of neutrons and protons in hyperonic neutron starsEffects of tensor couplings of ω and ρ mesons on 1S0 nucleon superfluidity in neutron star matterSuperfluidity and the moments of inertia of nucleiNucleon Superfluidity in Kaon-Condensed Neutron StarsCooling of neutron stars and superfluidity in their cores3 PF 2 neutron superfluidity in neutron stars and three-body force effectPairing in nuclear systems: from neutron stars to finite nucleiInfluence of Hyperon on 1 S 0 Superfluidity of Nucleons in Neutron Star MatterCooling neutron star in the Cassiopeia A supernova remnant: evidence for superfluidity in the coreDirect Urca Processes Involving Proton 1 S 0 Superfluidity in Neutron Star CoolingStrange goings on in dense nucleonic matterStrange condensate realignment in relativistic heavy ion collisionsNegative kaons in dense baryonic matterIn‐Medium Kaon and Antikaon Production and Antikaon Condensation in Neutron Star MatterK̄ 0 Condensation in Hyperonic Neutron Star MatterTopological aspects in a two-component Bose condensed system in a neutron starThe influence of antikaon condensations on nucleon 1S0 superfluidity in neutron star matterPion fields in nuclear matterEquation of state for dense nucleon matterPion degrees of freedom in nuclear matterEquation of state of nucleon matter and neutron star structureInduced interaction in a Fermi gas with a BEC-BCS crossoverDifferent scenarios of topological phase transitions in homogeneous neutron matter S 0 1 pairing for neutrons in dense neutron matter induced by a soft pionPairing in high-density neutron matter including short- and long-range correlationsEffective Two-Nucleon Interaction in Asymmetric Nuclear MatterA density-dependent local effective interaction between nucleons in nuclear matterSuperfluidity of neutron matterQuasiparticle interactions in neutron matter for applications in neutron starsTemperature effects on hyperon superfluids in neutron starsBaryon superfluidity and neutrino emissivity of neutron starsThird family of superdense stars in the presence of antikaon condensatesEquation of state for neutron stars in SU(3) flavor symmetrySuperfluidity of Λ hyperons in neutron stars
[1] Zhang C M, Wang J, Zhao Y H, Yin H X, Song L M, Menezes D P, Wickramasinghe D T, Ferrario L and Chardonnet P 2011 Astron. Astrophys. 527 A83
[2] Mi A J, Zuo W and Li A 2007 Chin. Phys. B 16 3290
[3] Zhao X F 2011 Chin. Phys. B 20 039701
[4] Xu Y, Huang X L, Liu C Z, Bao T and Liu G Z 2016 Chin. Phys. Lett. 33 099701
[5] Cooper L N, Mills R L and Sessler A M 1959 Phys. Rev. 114 1377
[6] Bardeen J, Cooper L N and Schrieffer J R 1957 Phys. Rev. 108 1175
[7] Zuo W, Li Z H, Lu G C, Li J Q, Scheid W, Lombardo U, Schulze H J and Shen C W 2004 Phys. Lett. B 595 44
[8] Xu Y, Liu C Z, Fan C B, Han X W, Zhang X J, Zhu M F, Wang H Y and Liu G Z 2015 Res. Astron. Astrophys. 15 725
[9] Huang X L, Xu Y, Liu C Z and Liu G Z 2016 Int. J. Mod. Phys. D 25 1650059
[10] Migdal A B 1959 Nucl. Phys. 13 655
[11] Takatsuka T and Tamagaki R 1995 Prog. Theor. Phys. 94 457
[12] Yakovlev D G, Levenfish K P and Shibanov Y A 1999 Phys. -Usp. 42 737
[13] Cui C X, Zuo W and Schulze H J 2008 Chin. Phys. B 17 3289
[14] Dean D J and Hjorth-Jensen M 2003 Rev. Mod. Phys. 75 607
[15] Chen W, Lam Y Y, Wen D H and Liu L G 2006 Chin. Phys. Lett. 23 271
[16] Shternin P S, Yakovlev D G et al 2011 Mon. Not. R. Astron. Soc. 412 L108
[17] Xu Y, Yu Z, Zhang X J, Fun C B, Liu G Z, Zhao E G, Huang X L and Liu C Z 2018 Commun. Theor. Phys. 69 425
[18] Kaplan D B and Nelson A E 1986 Phys. Lett. B 175 57
[19] Nelson A E and Kaplan D B 1987 Phys. Lett. B 192 193
[20] Kolomeistev E E and Voskresensky D N 2003 Phys. Rev. C 68 015803
[21] Gu J, Guo H, Zhou R, Liu B, Li X G and Liu Y X 2005 Astrophys. J. 622 549
[22] Ding W B, Liu G Z, Zhu M F, Yu Z and Zhao E G 2008 Chin. Phys. Lett. 25 458
[23] Ren J R and Guo H 2009 Chin. Phys. B 18 3379
[24] Xu Y, Huang X L, Zhang X J, Yu Z, Fan C B, Ding W B and Liu C Z 2018 Mon. Not. R. Astron. Soc. 474 3576
[25] Migdal A B 1978 Rev. Mod. Phys. 50 107
[26]Ericson T and Weise W 1988 Pions and Nuclei (Oxford: Clarendon Press)
[27] Wiringa R B, Fiks V and Fabrocini A 1988 Phys. Rev. C 38 1010
[28] Migdal A B, Saperstein E E, Troitsky M A and Voskresensky D N 1990 Phys. Rep. 192 179
[29] Akmal A, Pandharipande V R and Ravenhall D G 1998 Phys. Rev. C 58 1804
[30] Yu Z Q, Huang K and Yin L 2009 Phys. Rev. A 79 053636
[31] Pankratov S S, Baldo M and Zverev M V 2012 Phys. Rev. C 86 045804
[32] Pankratov S S, Baldo M and Saperstein E E 2015 Phys. Rev. C 91 015802
[33] Ding D, Rios A, Dussan H, Dickhoff W H, Witte S J, Carbone A and Polls A 2016 Phys. Rev. C 94 025802
[34] Nishizaki S, Takatsuka T, Yahagi N and Hiura J 1991 Prog. Theor. Phys. 86 853
[35] Sprung D W L and Banerjee P K 1971 Nucl. Phys. A 168 273
[36] Amundsen L and Østgaard E 1985 Nucl. Phys. A 437 487
[37] Wambach J, Ainsworth T L and Pines D 1993 Nucl. Phys. A 555 128
[38] Takatsuka T and Tamagaki R 2003 Nucl. Phys. A 721 C1003
[39] Takatsuka T and Tamagaki R 2004 Nucl. Phys. A 738 387
[40] Banik S and Bandyopadhyay D 2001 Phys. Rev. C 64 055805
[41] Miyatsu T, Cheoun M K and Saito K 2013 Phys. Rev. C 88 015802
[42] Wang Y N and Shen H 2010 Phys. Rev. C 81 025801