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<jats:title>Abstract</jats:title> <jats:p>A production representation of partial-wave <jats:italic>S</jats:italic> matrix is utilized to construct low-energy elastic pion-nucleon scattering amplitudes from cuts and poles on complex Riemann sheets. Among them, the contribution of left-hand cuts is estimated using the <jats:inline-formula> <jats:tex-math><?CDATA ${\cal{O}}\left( {{p^3}} \right)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064110_M2.jpg" xlink:type="simple" /> </jats:inline-formula> results obtained in covariant baryon chiral perturbation theory within the extended-on-nass-shell scheme. By fitting to data on partial-wave phase shifts, it is indicated that the existences of hidden poles in <jats:italic>S</jats:italic> <jats:sub>11</jats:sub> and <jats:italic>P</jats:italic> <jats:sub>11</jats:sub> channels, as conjectured in our previous paper [Eur. Phys. J. C, 78(7): 543 (2018)], are firmly established. Specifically, the pole mass of the <jats:italic>S</jats:italic> <jats:sub>11</jats:sub> hidden resonance is determined to be (895±81)−(164±23)i MeV, whereas, the virtual pole in the <jats:italic>P</jats:italic> <jats:sub>11</jats:sub> channel locates at (966±18) MeV. It is found that analyses at the <jats:inline-formula> <jats:tex-math><?CDATA ${\cal{O}}\left( {{p^3}} \right)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064110_M3.jpg" xlink:type="simple" /> </jats:inline-formula> level improves significantly the fit quality, comparing with the previous <jats:inline-formula> <jats:tex-math><?CDATA ${\cal{O}}\left( {{p^2}} \right)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064110_M4.jpg" xlink:type="simple" /> </jats:inline-formula> one. Quantitative studies with cautious physical discussions are also conducted for the other <jats:italic>S</jats:italic>- and <jats:italic>P</jats:italic>-wave channels. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>The interaction of the pseudoscalar meson and the baryon octet is investigated by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation. In addition to the Weinberg-Tomozawa term, the contribution of the <jats:inline-formula> <jats:tex-math><?CDATA $s-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M1.jpg" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math><?CDATA $u-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M2.jpg" xlink:type="simple" /> </jats:inline-formula> channel potentials in the <jats:italic>S</jats:italic>-wave approximation are taken into account. In the sector of isospin <jats:inline-formula> <jats:tex-math><?CDATA $I=1/2$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M3.jpg" xlink:type="simple" /> </jats:inline-formula> and strangeness <jats:inline-formula> <jats:tex-math><?CDATA $S=0$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M4.jpg" xlink:type="simple" /> </jats:inline-formula>, a pole is detected in a reasonable region of the complex energy plane of <jats:inline-formula> <jats:tex-math><?CDATA $\sqrt{s}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M5.jpg" xlink:type="simple" /> </jats:inline-formula> in the center-of-mass frame by analyzing the behavior of the scattering amplitude, which is higher than the <jats:inline-formula> <jats:tex-math><?CDATA $\eta N$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M6.jpg" xlink:type="simple" /> </jats:inline-formula> threshold and lies on the third Riemann sheet. Thus, it can be regarded as a resonance state and might correspond to the <jats:inline-formula> <jats:tex-math><?CDATA $N(1535)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M7.jpg" xlink:type="simple" /> </jats:inline-formula> particle of the Particle Data Group (PDG) review. The coupling constants of this resonance state to the <jats:inline-formula> <jats:tex-math><?CDATA $\pi N$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M8.jpg" xlink:type="simple" /> </jats:inline-formula>, <jats:inline-formula> <jats:tex-math><?CDATA $\eta N$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M9.jpg" xlink:type="simple" /> </jats:inline-formula>, <jats:inline-formula> <jats:tex-math><?CDATA $K \Lambda$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M10.jpg" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math><?CDATA $K \Sigma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M11.jpg" xlink:type="simple" /> </jats:inline-formula> channels are calculated, and it is found that this resonance state couples strongly to the hidden strange channels. Apparently, the hidden strange channels play an important role in the generation of resonance states with strangeness zero. The interaction of the pseudoscalar meson and the baryon octet is repulsive in the sector of isospin <jats:inline-formula> <jats:tex-math><?CDATA $I=3/2$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M12.jpg" xlink:type="simple" /> </jats:inline-formula> and strangeness <jats:inline-formula> <jats:tex-math><?CDATA $S=0$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_6_064111_M13.jpg" xlink:type="simple" /> </jats:inline-formula>, so that no resonance state can be generated dynamically. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>We propose a novel mechanism for the production of gravitational waves in the early Universe that originates from the relaxation processes induced by the QCD phase transition. While the energy density of the quark-gluon mean-field is monotonously decaying in real time, its pressure undergoes a series of violent oscillations at the characteristic QCD time scales that generate a primordial multi-peaked gravitational waves signal in the radio frequencies’ domain. The signal is an echo of the QCD phase transition that is accessible by planned measurements at the FAST and SKA telescopes.</jats:p>

<jats:title>Abstract</jats:title> <jats:p>The diphoton invariant mass distribution from the interference between <jats:inline-formula> <jats:tex-math><?CDATA $gg\to H \to \gamma\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M2.jpg" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math><?CDATA $gg\to \gamma\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M3.jpg" xlink:type="simple" /> </jats:inline-formula> is almost antisymmetric around the Higgs mass <jats:inline-formula> <jats:tex-math><?CDATA $M_H$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M4.jpg" xlink:type="simple" /> </jats:inline-formula>. We propose a new observable <jats:inline-formula> <jats:tex-math><?CDATA $A_{\rm{int}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M5.jpg" xlink:type="simple" /> </jats:inline-formula>, the ratio of the sign-reversed integral around <jats:inline-formula> <jats:tex-math><?CDATA $M_H$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M6.jpg" xlink:type="simple" /> </jats:inline-formula> (e.g. <jats:inline-formula> <jats:tex-math><?CDATA $\int^{M_H}_{M_H-5~\rm{GeV}} -\int_{M_H}^{M_H+5~\rm{GeV}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M7.jpg" xlink:type="simple" /> </jats:inline-formula>) and the cross-section of the Higgs signal, to quantify this effect. We study <jats:inline-formula> <jats:tex-math><?CDATA $A_{\rm{int}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M8.jpg" xlink:type="simple" /> </jats:inline-formula> both in the Standard Model (SM) and new physics with various CP violating <jats:inline-formula> <jats:tex-math><?CDATA $H\gamma\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M9.jpg" xlink:type="simple" /> </jats:inline-formula> couplings. <jats:inline-formula> <jats:tex-math><?CDATA $A_{\rm{int}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M10.jpg" xlink:type="simple" /> </jats:inline-formula> in SM could reach a value of 10%, while for CP violating <jats:inline-formula> <jats:tex-math><?CDATA $H\gamma\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M11.jpg" xlink:type="simple" /> </jats:inline-formula> coupling <jats:inline-formula> <jats:tex-math><?CDATA $A_{\rm{int}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M12.jpg" xlink:type="simple" /> </jats:inline-formula> could range from 10% to −10%, which could probably be detected in the HL-LHC experiments. <jats:inline-formula> <jats:tex-math><?CDATA $A_{\rm{int}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M13.jpg" xlink:type="simple" /> </jats:inline-formula> with both CP violating <jats:inline-formula> <jats:tex-math><?CDATA $H\gamma\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M14.jpg" xlink:type="simple" /> </jats:inline-formula> and <jats:inline-formula> <jats:tex-math><?CDATA $Hgg$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073101_M15.jpg" xlink:type="simple" /> </jats:inline-formula> couplings is also studied, and its range of values is found to be slightly larger. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>The doubly weak transition <jats:inline-formula> <jats:tex-math><?CDATA $b\to dd{\bar s}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073102_M2.jpg" xlink:type="simple" /> </jats:inline-formula> is highly suppressed in the Standard Model, which makes it a potential channel for exploring new physics signals. We present a study of the exclusive two-body wrong sign weak decay <jats:inline-formula> <jats:tex-math><?CDATA $\smash{\overline B}^0\to K^+\pi^-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073102_M3.jpg" xlink:type="simple" /> </jats:inline-formula> , which belongs to this class, in the perturbative QCD framework. We perform a model independent analysis for various effective dimension-6 operators for which large effects are possible. We further analyze the considered process in the Randall-Sundrum model, including the custodially protected and the bulk-Higgs Randall-Sundrum models. Exploring the experimentally favored parameter space of these models leads to a large and significant enhancement of the decay rate compared to the Standard Model, which might be accessible in future experiments. We propose to search for the wrong sign decay <jats:inline-formula> <jats:tex-math><?CDATA $\smash{\overline B}^0\to K^+\pi^-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073102_M4.jpg" xlink:type="simple" /> </jats:inline-formula> via flavor-tagged time-dependent analyses, which can be performed at LHCb and Belle-II. </jats:p>

<jats:title>Abstract</jats:title> <jats:p> <jats:inline-formula> <jats:tex-math><?CDATA $\bar B_s^0\to (D^0,\bar D^0) \pi^+\pi^-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M261.jpg" xlink:type="simple" /> </jats:inline-formula> is induced by the <jats:inline-formula> <jats:tex-math><?CDATA $b\to c \bar us$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M262.jpg" xlink:type="simple" /> </jats:inline-formula>/ <jats:inline-formula> <jats:tex-math><?CDATA $b \to u\bar cs$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M263.jpg" xlink:type="simple" /> </jats:inline-formula> transitions, which can interfere if a CP-eigenstate <jats:inline-formula> <jats:tex-math><?CDATA $D_{\rm CP}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M264.jpg" xlink:type="simple" /> </jats:inline-formula> is formed. The interference contribution is sensitive to the CKM angle <jats:inline-formula> <jats:tex-math><?CDATA $\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M265.jpg" xlink:type="simple" /> </jats:inline-formula>. In this work, we study the <jats:italic>S</jats:italic>-wave <jats:inline-formula> <jats:tex-math><?CDATA $\pi^+\pi^-$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M266.jpg" xlink:type="simple" /> </jats:inline-formula> contributions to the process in the perturbative QCD factorization. In the factorization framework, we adopt two-meson light-cone distribution amplitudes, whose normalization is parametrized by the <jats:italic>S</jats:italic>-wave time-like two-pion form factor with resonance contributions from <jats:inline-formula> <jats:tex-math><?CDATA $f_0(500)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M267.jpg" xlink:type="simple" /> </jats:inline-formula>, <jats:inline-formula> <jats:tex-math><?CDATA $f_0(980),f_0(1500),f_0(1790)$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M268.jpg" xlink:type="simple" /> </jats:inline-formula>. We find that the branching ratio of <jats:inline-formula> <jats:tex-math><?CDATA $\bar B_s^0\to (D^0,\bar D^0) (\pi^+\pi^-)_S$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M269.jpg" xlink:type="simple" /> </jats:inline-formula> is of the order of <jats:inline-formula> <jats:tex-math><?CDATA $10^{-6}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M270.jpg" xlink:type="simple" /> </jats:inline-formula>, and that significant interference exists in <jats:inline-formula> <jats:tex-math><?CDATA $\bar B_s^0\to D_{\rm CP} (\pi^+\pi^-)_S$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M271.jpg" xlink:type="simple" /> </jats:inline-formula>. Future measurement could not only provide useful constraints on the CKM angle <jats:inline-formula> <jats:tex-math><?CDATA $\gamma$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_073103_M272.jpg" xlink:type="simple" /> </jats:inline-formula>, but would also be helpful for exploring the multi-body decay mechanism of heavy mesons. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>The chiral order-parameter <jats:italic>σ</jats:italic> field and its higher-order cumulants of fluctuations are calculated within the functional renormalization group approach by adopting the local potential approximation in this study. The influence of glue dynamics on fluctuations of the <jats:italic>σ</jats:italic> field is investigated, and we find that they are weakly affected. This is in sharp contrast to the baryon number fluctuations, which are sensitive to the glue dynamics and involve information on the color confinement. The implications of our calculated results are discussed from the viewpoint of the theoretical and experimental efforts in the search for the QCD critical end point. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>Spontaneous fission (SF) with a new formula based on a liquid drop model is proposed and used in the calculation of the SF half-lives of heavy and superheavy nuclei (<jats:italic>Z</jats:italic>= 90–120). The predicted half-lives are in agreement with the experimental SF half-lives. The half-lives of <jats:inline-formula> <jats:tex-math><?CDATA $ \alpha $?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074102_M1.jpg" xlink:type="simple" /> </jats:inline-formula> decay (AD) for the same nuclei are obtained by using the Wentzel-Kramers-Brillouin (WKB) method together with Bohr-Sommerfeld (BS) quantization condition considering the isospin-dependent effects for the cosh potential. The decay modes and branching ratios of superheavy nuclei (<jats:italic>Z</jats:italic>= 104-118) with experimental decay modes are obtained, and the modes are compared with the experimental ones and with the predictions found in the literature. Although some nuclei have predicted decay modes that are different from their experimental decay modes, decay modes same as the experimental ones are predicted for many nuclei. The SF and AD half-lives, branching ratios, and decay modes are obtained for superheavy nuclei (<jats:italic>Z</jats:italic>= 119–120) with unknown decay modes and compared with the predictions obtained in a previous study. The present results provide useful information for future experimental studies performed on both the AD and SF of superheavy nuclei. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>The cross sections for <jats:sup>59, 60</jats:sup>Ca, recently measured in the 345 <jats:italic>A</jats:italic> MeV <jats:inline-formula> <jats:tex-math><?CDATA $^{70}{\rm{Zn}}+^{9}{\rm{Be}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M3.jpg" xlink:type="simple" /> </jats:inline-formula> reaction, were estimated using the FRACS parametrization and an empirical formula, which are in good agreement. The FRACS parametrization and the empirical formula are combined to predict the cross sections for extreme calcium isotopes <jats:inline-formula> <jats:tex-math><?CDATA $^{66, 70}{\rm{Ca}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M5.jpg" xlink:type="simple" /> </jats:inline-formula> in the <jats:inline-formula> <jats:tex-math><?CDATA $^{70, 80}{\rm{Zn}}+^{9}{\rm{Be}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M6.jpg" xlink:type="simple" /> </jats:inline-formula> reactions at the incident energies of 60, 80, and 345 <jats:italic>A</jats:italic> MeV. The dependence of emperical formula parameters on the reaction system, as well as the incident energy, are discussed. The results indicate that <jats:inline-formula> <jats:tex-math><?CDATA $^{66, 70}{\rm{Ca}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M9.jpg" xlink:type="simple" /> </jats:inline-formula> can be discovered in reactions of 60, 80 <jats:italic>A</jats:italic> MeV <jats:inline-formula> <jats:tex-math><?CDATA $^{80}{\rm{Zn}}+^{9}{\rm{Be}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M11.jpg" xlink:type="simple" /> </jats:inline-formula>. The predicted binding energy for extreme neutron-rich isotopes by the spherical relativistic continuum Hartree-Bogoliubov theory was adopted in the calculation. Hence, the planned Beijing Isotope-Separation-On Line Neutron-Rich Beam Facility (BISOL), which is a third generation radioactive ion beam facility, could provide the opportunity to discover <jats:inline-formula> <jats:tex-math><?CDATA $^{66, 70}{\rm{Ca}}$?></jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cpc_43_7_074103_M13.jpg" xlink:type="simple" /> </jats:inline-formula> and neighboring neutron-drip line nuclei. </jats:p>

<jats:title>Abstract</jats:title> <jats:p>The radial basis function (RBF) approach is a powerful tool to improve nuclear mass predictions. By combining the RBF approach with the latest relativistic continuum Hartree-Bogoliubov (RCHB) model, the local systematic deviations between the RCHB mass predictions and the experimental data are eliminated, and the root-mean-square (rms) mass deviation is significantly reduced from 7.923 MeV to 0.386 MeV. However, systematic deviations between the RBF improved mass predictions and the experimental data remain for nuclei with four different odd-even parities, i.e. (even <jats:italic>Z</jats:italic>, even <jats:italic>N</jats:italic>), (even <jats:italic>Z</jats:italic>, odd <jats:italic>N</jats:italic>), (odd <jats:italic>Z</jats:italic>, even <jats:italic>N</jats:italic>), and (odd <jats:italic>Z</jats:italic>, odd <jats:italic>N</jats:italic>). They can be reduced by separately training RBF for the four groups of nuclei, and the resulting rms deviation decreases to 0.229 MeV. It is found that the RBF approach can describe the deformation effects neglected in the present RCHB mass calculations, and also improves the description of the shell effect and the pairing effect. </jats:p>