Introduction

To understand the properties of nuclear many-body systems that start from realistic nucleon-nucleon (NN) interactions is still challenging. The repulsive core of realistic NN interactions causes a strong correlation for the many-body wave function and requires advanced many-body methods that go beyond mean field [1-6]. The Brueckner-Hartree-Fock (BHF) theory [7] is one of the representative nuclear many-body methods, which is characteristic for its capacity to soften the realistic NN interaction to an effective G matrix in nuclear medium. The BHF theory can be derived as a two-hole-line truncation to the general Bethe-Brueckner-Goldstone expansion theory [8, 1], where the ground-state properties of the nuclear many-body systems are calculated order by order according to the number of independent hole lines contained in the expansion diagrams [8]. Since 1960s, it has been found that the saturation points of symmetric nuclear matter (SNM) calculated using the BHF theory with different two-body interactions are located on a Coester line [9], which deviates systematically from empirical values. To address this issue within the nonrelativistic framework is to introduce three-body forces [10-15].

In 1980, a major modification of the saturation properties of SNM was obtained by including a relativistic description of the nucleon motion [16]. Through this pioneering work, significant efforts have been made to develop the relativistic Brueckner-Hartree-Fock (RBHF) theory [17-20]. Starting from the Bonn potential [21], the saturation points of SNM obtained using the RBHF theory have been shifted remarkably close towards the empirical values, without introducing explicit three-body forces. The success of the RBHF theory has been understood by the fact that, relativistic effects contribute a particular part of the three-body force [12, 14, 22] through virtual nucleon-antinucleon excitations in the intermediate states (known as Z diagrams) [23, 24].

An essential point of the RBHF theory is the use of the Dirac equation to describe the single-nucleon motion in the mean field, that is, the single-particle potential (SPP). An SPP operator is generally divided into scalar and vector components [25]. They can be obtained from an effective G matrix in a self-consistent manner. In principle, the calculations of SPPs and the G matrix should be performed in the full Dirac space, where both the positive-energy states (PESs) and negative-energy states (NESs) are included. However, owing to the complexities of these procedures, RBHF calculations are generally performed without NESs using approximate methods, such as the momentum-independent approximation (MIA) method [22, 26-30] and the projection method [20, 31-35], to perform the Hartree-Fock calculations.

The MIA method assumes that the SPPs are independent of the momentum and extracts the SPPs from single-particle potential energies calculated at two selected momenta. In the projection method, the elements of the G matrix are projected onto a complete set of Lorentz invariant amplitudes [19], from which the SPPs are calculated. Because the G matrix coupled to the NESs is not considered, the SPPs obtained using these two methods are ambiguous [36, 31, 37].

The Dirac mass $M_{\text{D}}^{*}$, a key quantity in relativistic nuclear physics, is defined using the scalar component of the nucleon self-energy within the Dirac equation. This quantity is crucial for describing medium effects in nuclear many-body systems, as highlighted in various studies [38-40]. An important aspect of the Dirac mass is its isospin splitting $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}$, which directly reflects the isovector properties of NN interactions [41, 42]. It is essential to clarify that the Dirac mass should not be confused with the non-relativistic effective mass, which characterizes the momentum and energy dependence of the Schrödinger equivalent potential [43, 44]. Theoretical predictions of the Dirac mass based on realistic NN interactions mainly rely on the RBHF theory. However, there is a controversy regarding the sign of this isospin splitting [33, 45]. As pointed in Ref. [46] in 1997, the MIA method leads to $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>0$, whereas the projection method indicates an opposite sign $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}<0$. This discrepancy demonstrates significant differences in the isovector properties of asymmetric nuclear matter (ANM) when employing the MIA method versus the projection method in RBHF calculations.

Recently, self-consistent RBHF calculations in the full Dirac space have been achieved [37, 47, 48], which avoid the ambiguities suffered from the RBHF calculations without NESs. In ANM, the full solution predicts the sign $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}<0$ [48] and clarifies the long-standing controversy between the MIA and projection methods on the isospin splitting of the Dirac mass. The RBHF theory has also been applied in the full Dirac space to study the nonrelativistic effective mass in nuclear matter [49], the properties of ²⁰⁸Pb using a liquid droplet model [50], the tensor-force effecs on nuclear matter [51], and the neutron star properties [52-54]. In this work, we aimed to thoroughly study the isospin splitting of the Dirac mass obtained using the RBHF calculations, particularly for the performance of the MIA method, in the full Dirac space viewpoint.

Theoretical framework

In the RBHF theory, the single-particle motion of a nucleon inside an infinite nuclear matter is described by the following Dirac equation: $\left\{\alpha \cdot p+\beta \left[M+{\mathcal{U}}_{\tau }\left(p\right)\right]\right\}{u}_{\tau }\left(p\mathrm{,}s\right)=E_{p\mathrm{,}\tau }{u}_{\tau }\left(p\mathrm{,}s\right)\mathrm{.}$ (1) where $\alpha$ and β are the Dirac matrices, M is the nucleon mass, $p$ and $E_{p\mathrm{,}\tau }$ are the momentum and single-particle energy, respectively, s denotes the spin, and τ=n,p denotes neutron n and proton p. The symbol uτ in Eq. (1) represents the positive-energy Dirac spinor, whereas the negative-energy Dirac spinor vτ is obtained using $v_{\tau }={\gamma }_5{u}_{\tau }$. The SPP operator ${\mathcal{U}}_{\tau }\left(p\right)$ can be decomposed into the scalar potential $U_{\text{S}\mathrm{,}\tau }\left(p\right)$, the timelike and spacelike components of the vector potential $U_{\mathrm{0,}\tau }\left(p\right)$ and $U_{\text{V}\mathrm{,}\tau }\left(p\right)$ [25], respectively. ${\mathcal{U}}_{\tau }\left(p\right)=U_{\text{S}\mathrm{,}\tau }\left(p\right)+{\gamma }^0{U}_{\mathrm{0,}\tau }\left(p\right)+\gamma \cdot \widehat{p}{U}_{\text{V}\mathrm{,}\tau }\left(p\right)\mathrm{.}$ (2) where $\widehat{p}=p/\left|p\right|=p/p$ is the unit vector parallel to the momentum $p$. By calculating the matrix elements of ${\mathcal{U}}_{\tau }\left(p\right)$ as expanded by the PESs and NESs, that is, ${\displaystyle {\sum }_{\tau }^{++}\left(p\right)}$, ${\displaystyle {\sum }_{\tau }^{-+}\left(p\right)}$, and ${\displaystyle {\sum }_{\tau }^{--}\left(p\right)}$, the momentum-dependent SPPs can be determined uniquely using the following equation [37, 52]: $U_{\text{S}\mathrm{,}\tau }\left(p\right)=\frac{{\displaystyle {\sum }_{\tau }^{++}\left(p\right)}-{\displaystyle {\sum }_{\tau }^{-}\left(p\right)}}{2}\mathrm{,}$ (3a) $U_{\mathrm{0,}\tau }\left(p\right)=\frac{E_{p\mathrm{,}\tau }^{*}}{M_{p\mathrm{,}\tau }^{*}}\frac{{\displaystyle {\sum }_{\tau }^{++}\left(p\right)}+{\displaystyle {\sum }_{\tau }^{--}\left(p\right)}}{2}-\frac{p_{\tau }^{*}}{M_{p\mathrm{,}\tau }^{*}}{\displaystyle {\sum }_{\tau }^{-+}\left(p\right)}\mathrm{,}$ (3b) $U_{\text{V}\mathrm{,}\tau }\left(p\right)=-\frac{p_{\tau }^{*}}{M_{p\mathrm{,}\tau }^{*}}\frac{{\displaystyle {\sum }_{\tau }^{++}\left(p\right)}+{\displaystyle {\sum }_{\tau }^{--}\left(p\right)}}{2}+\frac{E_{p\mathrm{,}\tau }^{*}}{M_{p\mathrm{,}\tau }^{*}}{\displaystyle {\sum }_{\tau }^{-+}\left(p\right)}\mathrm{.}$ (3c) where the plus and minus signs in the superscripts denote the PESs and NESs, respectively. The effective quantities in Eq. (3) are defined as follows: $p_{\tau }^{*}=p+\widehat{p}{U}_{V\mathrm{,}\tau }\left(p\right)$, $M_{p\mathrm{,}\tau }^{*}=M+U_{S\mathrm{,}\tau }\left(p\right)$, and $E_{p\mathrm{,}\tau }^{*}=E_{p\mathrm{,}\tau }-U_{\mathrm{0,}\tau }\left(p\right)$. The effective mass $M_{p\mathrm{,}\tau }^{*}$ is also known as the Dirac mass $M_{D\mathrm{,}\tau }^{*}$.

The quantities ${\displaystyle {\sum }_{\tau }^{++}\left(p\right)}$, ${\displaystyle {\sum }_{\tau }^{-+}\left(p\right)}$, and ${\displaystyle {\sum }_{\tau }^{--}\left(p\right)}$ in Eq. (3) are calculated by summing up the effective G matrix elements with all the nucleons inside the Fermi sea. $\begin{array}{c}{\displaystyle {\sum }_{\tau }^{++}\left(p\right)}={\displaystyle \sum _{s^{\prime }{\tau }^{\prime }}{\displaystyle {\int }_0^{k_F^{{\tau }^{\prime }}}}\frac{d^3{p}^{\prime }}{{(2\pi )}^3}}\frac{M_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}{E_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}\langle {\overline{u}}_{\tau }\left(p\mathrm{,1}/2\right){\overline{u}}_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\\ \left|{\overline{G}}^{++++}\left(W\right)\right|u_{\tau }\left(p\mathrm{,1}/2\right)u_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\rangle \mathrm{,}\end{array}$ (4a) $\begin{array}{c}{\displaystyle {\sum }_{\tau }^{-+}\left(p\right)}={\displaystyle \sum _{s^{\prime }{\tau }^{\prime }}{\displaystyle {\int }_0^{k_F^{{\tau }^{\prime }}}}}\frac{d^3{p}^{\prime }}{{\left(2\pi \right)}^3}\frac{M_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}{E_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}\langle {\overline{v}}_{\tau }\left(p\mathrm{,1}/2\right){\overline{u}}_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\\ \left|{\overline{G}}^{-+++}\left(W\right)\right|u_{\tau }\left(p\mathrm{,1}/2\right)u_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\rangle \mathrm{,}\end{array}$ (4b) $\begin{array}{c}{\displaystyle {\sum }_{\tau }^{++}\left(p\right)}={\displaystyle \sum _{s^{\prime }{\tau }^{\prime }}{\displaystyle {\int }_0^{k_F^{{\tau }^{\prime }}}}}\frac{d^3{p}^{\prime }}{{\left(2\pi \right)}^3}\frac{M_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}{E_{p\text{'}\mathrm{,}{\tau }^{\prime }}^{*}}\langle {\overline{v}}_{\tau }\left(p\mathrm{,1}/2\right){\overline{u}}_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\\ \left|{\overline{G}}^{-+-+}\left(W\right)\right|v_{\tau }\left(p\mathrm{,1}/2\right)u_{{\tau }^{\prime }}\left(p\text{'}\mathrm{,}{s}^{\prime }\right)\rangle \mathrm{.}\end{array}$ (4c) where $\overline{G}$ is the antisymmetrized G matrix, $k_{\text{F}}^{\tau }$ is the Fermi momentum, and W denotes the starting energy. The additional factors M^*/E^* in Eq. (4) are attributed to the normalization of the Dirac spinors, that is, $\overline{u}u=\mathrm{1,}\overline{v}v=-1$. The effective interaction G matrix is the solution of the in-medium Thompson equation [26], which describes the two-body scattering. $\begin{array}{l}{G}_{\tau {\tau }^{\prime }}\left(q\text{'}\mathrm{,}q|P\mathrm{,}W\right)=V_{\tau {\tau }^{\prime }}\left(q\text{'}\mathrm{,}q|P\right)\\ +{\displaystyle \int }\frac{d^{3}k}{{\left(2\pi \right)}^3}{V}_{\tau {\tau }^{\prime }}\left(q\text{'}\mathrm{,}k|P\right)\\ \frac{Q_{\tau {\tau }^{\prime }}\left(k\mathrm{,}P\right)}{W-E_{P+k\mathrm{,}\tau }-E_{P-k\mathrm{,}{\tau }^{\prime }}}{G}_{\tau {\tau }^{\prime }}\left(k\mathrm{,}q|P\mathrm{,}W\right)\mathrm{.}\end{array}$ (5) The labels of PESs and NESs are suppressed. In this study, the Bonn potential was chosen as the realistic NN interaction Vττ’ [21]. $P=\frac{1}{2}\left(k_1+k_2\right)$ and $k=\frac{1}{2}\left(k_1-k_2\right)$ are the center-of-mass and the relative momenta of the two interacting nucleons with momenta k₁ and k₂, respectively. The initial, intermediate, and final relative momenta of the two nucleons are $q\mathrm{,}k$, and $q\text{'}$, respectively. The NN scattering in the nuclear medium is restricted with the Pauli operator $Q_{\tau {\tau }^{\prime }}\left(k\mathrm{,}P\right)$.

Equations (1), (3), (4), and (5) constitute a coupled system that must be solved in a self-consistent manner. After the convergence of SPPs, the single-particle and bulk properties of nuclear matter can be calculated straightforwardly [32, 55, 29].

Results and discussion

In SNM, the Dirac masses for the nucleons calculated using the MIA and projection methods were both quantitatively close to the results obtained in the full Dirac space [48]. However, the situation changed dramatically in ANM. Figure 1 shows the isospin splitting of the Dirac mass $\left(M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}\right)/M$ calculated using the RBHF theory as a function of the asymmetry parameter α=(ρ_n-ρ_p)/ρ. The projection method finds that in ANM there is $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}<0$, whereas the MIA method resulted in an opposite sign $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>0$. The density shown in Fig. 1 is the empirical saturation density ρ=0.16 fm^-3, and the opposite sign persisted at higher densities. This contradiction, which is well known since 1997 [46], has been clarified recently in Ref. [48] by considering the PESs and NESs simultaneously. Comparing to the RBHF results calculated in the full Dirac space, as shown in Fig. 1, the projection method obtained a qualitatively consistent isospin dependence of the Dirac mass with overestimated amplitudes, whereas the MIA method resulted in an opposite sign.

Fig. 1Full-screen View

(Color online) The isospin splittings of Dirac masses $\left(M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}\right)/M$ as functions of the asymmetry parameter α obtained using the RBHF theory in the full Dirac space (red solid), the projection method (olive dashed), and the MIA method (blue dotted). The density was fixed at the normal nuclear saturation density of ρ=0.16 fm^-3. The NN interaction Bonn A was used. The results for pure neutron matter with α=1 were obtained using linear extrapolations

It is still unclear why the MIA method succeeded in SNM but failed in ANM. To reach the answer, we notice that this method has two essential procedures. The first procedure is known as the assumption procedure, which assumes that the scalar potential and the timelike component of the vector potential are momentum independent and the spacelike component of the vector potential is negligible, that is, $U_{\text{S}\mathrm{,}\tau }\left(p\right)\approx U_{\text{S}\mathrm{,}\tau }\mathrm{,}{U}_{\mathrm{0,}\tau }\left(p\right)\approx U_{\mathrm{0,}\tau }\mathrm{,}{U}_{\text{V}\mathrm{,}\tau }\left(p\right)\approx 0.$ (6) The second procedure is known as the extraction procedure, which extracts the two constants $U_{\text{S}\mathrm{,}\tau }$ and $U_{\mathrm{0,}\tau }$ from the single-particle potential energy $U_{\tau }\left(k\right)$ at two momenta $\left(k_1^{\tau }\mathrm{,}{k}_2^{\tau }\right)$. $U_{\tau }\left(k_1^{\tau }\right)=\frac{M+U_{\text{S}\mathrm{,}\tau }}{\sqrt{{\left(k_1^{\tau }\right)}^2+{\left(M+U_{\text{S}\mathrm{,}\tau }\right)}^2}}{U}_{\text{S}\mathrm{,}\tau }+U_{\mathrm{0,}\tau }\mathrm{,}$ (7a) $U_{\tau }\left(k_2^{\tau }\right)=\frac{M+U_{\text{S}\mathrm{,}\tau }}{\sqrt{{\left(k_2^{\tau }\right)}^2+{\left(M+U_{\text{S}\mathrm{,}\tau }\right)}^2}}{U}_{\text{S}\mathrm{,}\tau }+U_{\mathrm{0,}\tau }\mathrm{.}$ (7b) The quantity $U_{\tau }\left(k\right)$ in Eq. (7) is calculated as $\left(M_{\tau }^{*}/E_{k\mathrm{,}\tau }^{*}\right){\displaystyle {\sum }_{\tau }^{++}\left(k\right)}$.

The RBHF calculations in the full Dirac space provided an opportunity to analyze in detail the isospin splitting of the Dirac mass. In the following, we try to study further by testing separately the two procedures of the MIA method from the perspective in the full Dirac space.

First, we applied the assumption procedure (6) in the full Dirac space, that is, the quantities $U_{\text{S}\mathrm{,}\tau }\left(p\right)$ in Eq. (3a) and $U_{\mathrm{0,}\tau }\left(p\right)$ in Eq. (3b) were assumed to be momentum independent and were calculated at the Fermi momentum $k_{\text{F}}^{\tau }$, whereas $U_{\text{V}\mathrm{,}\tau }\left(p\right)$ in Eq. (3c) was set to zero. The newly obtained quantities $U_{\text{S}\mathrm{,}\tau }$ and $U_{\mathrm{0,}\tau }$ were used to update the Dirac spinors and G matrix for the next iterations. When SPPs converged, we calculated the binding energies per nucleon for SNM and pure neutron matter (PNM). Fig. 2 shows that the $E/A$ values for SNM and PNM were nearly similar to those obtained using the RBHF theory in the full Dirac space. This indicates that the assumption procedure is reasonable for describing the bulk properties of nuclear matter.

Fig. 2Full-screen View

(Color online) Binding energies per nucleon E/A of SNM and PNM as functions of the nucleon density ρ calculated in the full Dirac space using the assumption procedure. For comparison, the self-consistent results obtained using the RBHF theory in the full Dirac space are also shown

Figure 3(b) shows the Dirac masses of the neutron and proton obtained in the full Dirac space using the assumption procedure. The density was fixed at ρ=0.16 fm^-3. The relationship $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}<0$ was obtained, which was consistent with the result obtained using the RBHF theory in the full Dirac space, as shown in Fig. 3(a). This indicates that the assumption procedure is not sufficient to obtain an opposite sign when the isospin splitting of the Dirac mass in ANM is calculated.

Fig. 3Full-screen View

(Color online) Dirac masses for neutron (solid lines) and proton (dashed lines) as functions of the asymmetric parameter α at density ρ=0.16 fm^-3 calculated using the RBHF theory (a) in the full Dirac space, (b) using the assumption procedure, and (c) using the extraction procedure. Details can be found in the text

Second, we tested the influence of the extraction procedure (7) from the perspective in the full Dirac space. Starting from the converged $U_{\tau }\left(p\right)$ obtained using the RBHF theory in the full Dirac space, we extracted the two constants $U_{\text{S}\mathrm{,}\tau }$ and $U_{\mathrm{0,}\tau }$ using Eq. (7) with two selected momenta $\left(0.7k_{\text{F}}^{\tau }\mathrm{,}{k}_{\text{F}}^{\tau }\right)$ and subsequently calculated the Dirac mass. The resulting isospin splitting of the Dirac mass is shown in Fig. 3(c). It can be observed that $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>0$ was obtained for the entire region of the asymmetry parameter α, which is opposite to results obtained in the full Dirac space (Fig. 3(a)). This indicates that the extraction procedure (7) resulted in the opposite sign of the isospin splitting of the Dirac mass.

To further investigate how the opposite isospin dependence of the Dirac mass resulted from the extraction procedure, Eq. (7) was solved to obtain a formal expression of the Dirac mass. We start from the case of SNM and suppress the isospin indexes for moment. When $M\simeq 1000$ MeV, $U_{\text{S}}\simeq -400$ MeV, and $k_{\text{F}}=1.34{\text{fm}}^{-1}$, $k_1=0.7k_{\text{F}}$ and $k_2=k_{\text{F}}$ resulted in ${\left[k_1/\left(M+U_{\text{S}}\right)\right]}^2\simeq 0.1\ll 1$, ${\left[k_2/\left(M+U_{\text{S}}\right)\right]}^2\simeq 0.2\ll 1$. This allowed the square root in Eq. (7) to be expanded to first order, $U(k)=U_{\text{S}}-\frac{1}{2}\frac{U_{\text{S}}}{{\left(M+U_{\text{S}}\right)}^2}{k}^2+U_0\mathrm{,}$ (8) which indicates a quadratic form of the single-particle potential energy U(k). Using Eq. (8), the difference between U(k₂) and U(k₁) was obtained directly, as follows: $U\left(k_2\right)-U\left(k_1\right)=-\frac{1}{2}\frac{U_{\text{S}}}{{\left(M+U_{\text{S}}\right)}^2}\left(k_2^2-k_1^2\right)\mathrm{.}$ (9) For simplicity, we considered the limit that the two momenta k₁ and k₂ were extremely close, and thus were denoted as k. In this case, Eq. (9) can be written as a quadratic equation for $M_{\text{D}}^{*}/M$. $f(k){\left(M_{\text{D}}^{*}/M\right)}^2+M_{\text{D}}^{*}/M-1=\mathrm{0,}$ (10) where the dimensionless function f(k) is defined as $f(k)\equiv \underset{\begin{array}{c}{k}_2\to k\\ k_1\to k\end{array}}{{\displaystyle \lim }}2M\frac{U\left(k_2\right)-U\left(k_1\right)}{k_2^2-k_1^2}=M\frac{U^{\prime }\left(k\right)}{k}\mathrm{.}$ (11) The derivative U’(k) describes the momentum dependence of the single-particle potential energy U(k).

In Fig. 4(a), the single-particle potential energy Uτ(k), which was obtained in the full Dirac space for SNM and ANM with α=0.5 and ρ=0.16 fm^-3, is shown as a function of momentum k. It was found that Uτ(k) was a monotonic function of k. Therefore, the function f(k) was positive definite, and the solution of Eq. (10) is as follows: $M_{\text{D}}^{*}/M=\frac{\sqrt{1+4f(k)}-1}{2f(k)}\mathrm{.}$ (12) Figure 4(b) shows that the Dirac mass $M_{\text{D}}^{*}/M$ decreased with increasing f. Therefore, the sign of $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}$ was opposite to that of $f_{\text{n}}\left(k_{\text{n}}\right)-f_{\text{p}}\left(k_{\text{p}}\right)$, that is, $\{\begin{array}{c}{M}_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}<\mathrm{0,}\text{if}{f}_{\text{n}}\left(k_{\text{n}}\right)>f_{\text{p}}\left(k_{\text{p}}\right)\mathrm{,}\\ M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>\mathrm{0,}\text{if}{f}_{\text{n}}\left(k_{\text{n}}\right)<f_{\text{p}}\left(k_{\text{p}}\right)\mathrm{.}\end{array}$ (13) Figure 4(c) shows the function fτ(k) for nucleon τ obtained in the full Dirac space. It was observed that fτ(k) decreased with increasing k. Generally, there was f_n(k) < f_p(k) for $k<k_{\text{F}}^{\text{p}}$. This indicates that the momentum dependence of U_n(k) was weaker than that of U_p(k).

Fig. 4Full-screen View

(Color online) (a) Single-particle potential energy $U_{\tau }\left(k\right)$ obtained in the full Dirac space as a function of momentum k. The cases for SNM and ANM with α=0.5 and ρ=0.16 fm^-3 are shown. The momenta $0.7k_{\text{F}}^{\tau }$ and $k_{\text{F}}^{\tau }$ are highlighted with empty dots and solid squares, respectively. (b) Dirac mass $M_{\text{D}}^{*}/M$ as a function of dimensionless quantity f in Eq. (12). (c) Dimensionless function f (11) obtained in the full Dirac space as a function of momentum k. The notations for lines and symbols are the same as in panel (a)

In practice, one usually chooses $k_1^{\tau }=0.7k_{\text{F}}^{\tau }$ and $k_2^{\tau }=k_{\text{F}}^{\tau }$ [29, 48]. For ANM with α >0, the Fermi momentum for neutron $k_{\text{F}}^{\text{n}}$ was larger than that for proton, that is, $k_{\text{F}}^{\text{n}}>k_{\text{F}}^{\text{p}}$. The symbols in panels (a) and (c) in Fig. 4 indicate that this selection resulted in k_n >k_p and f_n(k_n) < f_p(k_p). Specifically, starting from the self-consistent Uτ(k) obtained in the full Dirac space, the choices $\left(0.7k_{\text{F}}^{\text{n}}\mathrm{,}{k}_{\text{F}}^{\text{n}}\right)$ for neutron and $\left(0.7k_{\text{F}}^{\text{p}}\mathrm{,}{k}_{\text{F}}^{\text{p}}\right)$ for proton resulted in $f_{\text{n}}\left(k_{\text{n}}\right)\approx 0.6$ and $f_{\text{p}}\left(k_{\text{p}}\right)\approx 0.8$. Based on the analysis in Eq. (13), the isospin splitting of the Dirac mass $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>0$ emerged. This explains the reason why the MIA method resulted in an opposite isospin splitting of the Dirac mass in ANM.

In Eq. (8), the extraction procedure forcibly assumes that the momentum dependence of the single-particle potential energy is in a quadratic form, where the strength $-\frac{1}{2}\frac{U_{\text{S}}}{{\left(M+U_{\text{S}}\right)}^2}$ is solely determined by the scalar potential U_S. In ANM, the momentum dependence of U_n(k) was generally weaker than that of U_p(k), as shown in Fig. 4(c). This resulted in U_S,n >U_S,p and the opposite sign of the isospin splitting of the Dirac mass $M_{\text{D,n}}^{*}-M_{\text{D,p}}^{*}>0$.

Based on the aforementioned analysis of the MIA method, the RBHF theory provides a robust understanding of the isospin splitting of the Dirac mass derived from realistic NN interactions. This framework effectively establishes ab initio predictions for the momentum dependence of nuclear mean fields and the isovector properties of in-medium NN interactions. Moreover, it potentially provides constraints on the relativistic energy density functional [56], enhances our understanding of pseudospin and spin-orbit splittings observed in exotic nuclei [57-59], and promotes the study of neutron-rich systems such as neutron stars [60, 40].

Summary

In summary, the relativistic Brueckner-Hartree-Fock (RBHF) theory plays an important role in deriving nuclear many-body properties from realistic nucleon-nucleon interactions. In comparison to the results obtained self-consistently in the full Dirac space, the momentum-independent approximation (MIA) method leads to opposite isospin splitting of the Dirac mass in asymmetric nuclear matter (ANM). The performance of this method was explored in detail in the full Dirac space viewpoint. The assumption procedure of the MIA method, which assumes that single-particle potentials are momentum independent, is not a sufficient condition that directly leads to the opposite sign of the isospin splitting of the Dirac mass, whereas the extraction procedure of the MIA method, which extracts single-particle potentials from single-particle potential energy, is found to be responsible for the opposite isospin splitting of the Dirac mass. A formal expression of the Dirac mass was obtained by solving approximately a set of equations involved in the extraction procedure. With the typical choice of momenta adopted in practical MIA calculations, the opposite isospin splitting of the Dirac mass was found. We conclude that the opposite isospin splitting of the Dirac mass emerges from the fact that the extraction procedure forcibly assumes the momentum dependence of the single-particle potential energy to be a quadratic form where the strength is solely determined by the constant scalar potential. This study substantially improves our understanding on the isospin splitting of the Dirac mass using the RBHF theory.