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根据RBHF理论,介质中的核子在一个平均场中运动,该平均场由该核子与其他所有核子的相互作用提供。核子的正能态旋量
$ u_{\tau}^{}({\boldsymbol{p}},s) $ 满足如下的Dirac方程:$$ \left[ {\boldsymbol{\alpha}}{\boldsymbol{\cdot}}{\boldsymbol{p}}+\beta \left(M+{\cal{U}}_{\tau}^{}\right) \right] u_{\tau}^{}({\boldsymbol{p}},s) = E_{{\boldsymbol{p}},\;\tau}^{}u_{\tau}^{}({\boldsymbol{p}},s), $$ (1) 其中:
$\boldsymbol\alpha$ 和$ \beta $ 是Dirac矩阵;$ M $ 是核子的静止质量;$ s,\tau( = {\rm{n}},{\rm{p}}) $ 分别表示自旋和同位旋。根据对称性,Dirac方程中描述平均场的单粒子势算符$ {\cal{U}}_{\tau}^{} $ 可以做如下分解:$$ {\cal{U}}_{\tau}^{}({\boldsymbol{p}}) = U_{S,\;\tau}^{}(p)+ \gamma^0U_{0,\;\tau}^{}(p) + {{\boldsymbol{\gamma}}}{\boldsymbol{\cdot}}{\boldsymbol{\hat{p}}}U_{V,\;\tau}^{}(p), $$ (2) 上式中
$U_{S,\;\tau}^{}$ ,$U_{0,\;\tau}^{}$ ,和$U_{V,\;\tau}^{}$ 分别表示标量势、矢量势的时间分量和空间分量;$ \hat{{\boldsymbol{p}}} $ 是动量$ {\boldsymbol{p}} $ 方向的单位矢量。利用单粒子势的不同分量可以定义有效质量
$M_{{\boldsymbol{p}},\;\tau}^{*}$ 、有效动量$ {\boldsymbol{p}}_{\tau}^{*} $ 和有效能量$E_{{\boldsymbol{p}},\;\tau}^{*}$ ,$$ M_{{\boldsymbol{p}},\;\tau}^{*} = M+U_{S,\;\tau}^{}(p), \tag{3a} $$ $${\boldsymbol{p}}_{\tau}^{*} = {\boldsymbol{p}}+\hat{{\boldsymbol{p}}}U_{V,\,\tau}^{}(p), \tag{3b} $$ $$ E_{{\boldsymbol{p}},\;\tau}^{*} = E_{{\boldsymbol{p}},\;\tau}^{}-U_{0,\;\tau}^{}(p) , \tag{3c} $$ 进而将Dirac方程(1)改写成类自由的形式:
$$ \left( {\boldsymbol{\alpha}}{\boldsymbol{\cdot}}{\boldsymbol{p}}_{\tau}^{*}+\beta M_{{\boldsymbol{p}},\;\tau}^{*} \right) u_{\tau}^{}({\boldsymbol{p}},s) = E_{{\boldsymbol{p}},\;\tau}^{*} u_{\tau}^{}({\boldsymbol{p}},s)。 $$ (4) 于是可以解析求解得到核子的正能态和负能态:
$$ u_{\tau}^{}({\boldsymbol{p}},s) = \ \sqrt{\frac{E_{{\boldsymbol{p}},\;\tau}^*+M_{{\boldsymbol{p}},\;\tau}^*}{2M_{{\boldsymbol{p}},\;\tau}^*}} \left[ \begin{array}{c}1 \\ \dfrac{{\boldsymbol{\sigma}}{\boldsymbol{\cdot}}{\boldsymbol{p}}_{\tau}^{*}}{E_{{\boldsymbol{p}},\;\tau}^*+M_{{\boldsymbol{p}},\;\tau}^*} \end{array} \right]\chi_s^{}\chi_{\tau}^{}, \tag{5a} $$ $$ v_{\tau}^{}({\boldsymbol{p}},s) = \ \gamma^5u_{\tau}^{}({\boldsymbol{p}},s), \tag{5b} $$ 其中:
$ \chi_s^{} $ 和$ \chi_{\tau}^{} $ 分别表示核子$ \tau $ 的自旋和同位旋波函数。根据定义,单粒子势算符
$ {\cal{U}}_{\tau}^{} $ 在完备的Dirac空间中有如下矩阵元:$$ \begin{split} \varSigma_{\tau}^{++}(p) =& \bar{u}_{\tau}^{}({\boldsymbol{p}},1/2) {\cal{U}}_{\tau}^{}({\boldsymbol{p}}) u_{\tau}^{}({\boldsymbol{p}},1/2) \\ =&U_{S,\;\tau}^{}(p) + \dfrac{E_{{\boldsymbol{p}},\;\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{0,\;\tau}^{}(p) + \dfrac{p_{\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{V,\;\tau}^{}(p), \end{split} \tag{6a}$$ $$ \begin{split} \varSigma_{\tau}^{-+}(p) =& \bar{v}_{\tau}^{}({\boldsymbol{p}},1/2) {\cal{U}}_{\tau}^{}({\boldsymbol{p}}) u_{\tau}^{}({\boldsymbol{p}},1/2) \\ =&\dfrac{p_{\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{0,\;\tau}^{}(p) + \dfrac{E_{{\boldsymbol{p}},\;\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{V,\;\tau}^{}(p), \qquad\quad\;\;\; \end{split} \tag{6b}$$ $$\begin{split} \varSigma_{\tau}^-(p) =& \bar{v}_{\tau}^{}({\boldsymbol{p}},1/2) {\cal{U}}_{\tau}^{}({\boldsymbol{p}}) v_{\tau}^{}({\boldsymbol{p}},1/2) \\ =& - U_{S,\;\tau}^{}(p) + \dfrac{E_{{\boldsymbol{p}},\;\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{0,\;\tau}^{}(p) + \dfrac{p_{\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} U_{V,\;\tau}^{}(p)。 \end{split}\tag{6c} $$ 利用这些矩阵元可以计算得到单粒子势的不同分量:
$$ U_{S,\;\tau}^{}(p) = \ \frac{\varSigma_{\tau}^{++}(p)-\varSigma_{\tau}^-(p)}{2},\qquad\qquad\qquad\qquad\quad\;\; \tag{7a} $$ $$ U_{0,\;\tau}^{}(p) = \ \frac{E_{{\boldsymbol{p}},\;\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}}\frac{\varSigma_{\tau}^{++}(p)+\varSigma_{\tau}^-(p)}{2} - \frac{p_{\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}}\varSigma_{\tau}^{-+}(p),\quad\;\; \tag{7b} $$ $$ U_{V,\;\tau}^{}(p) = \ -\frac{p_{\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}}\frac{\varSigma_{\tau}^{++}(p)+\varSigma_{\tau}^-(p)}{2} + \frac{E_{{\boldsymbol{p}},\;\tau}^{*}}{M_{{\boldsymbol{p}},\;\tau}^{*}} \varSigma_{\tau}^{-+}(p)。 \tag{7c} $$ 另一方面,单粒子势算符的矩阵元可以通过对介质中的有效核力进行积分得到。在RBHF理论中,这一有效核力称为G矩阵,它对应现实核力
$ V $ 的无穷阶梯形图求和(图1)。这一求和过程等价于求解描述介质中两核子散射的Thompson方程[27]$$ \begin{split} G_{\tau\tau'}^{}({\boldsymbol{q}}', {\boldsymbol{q}}|{\boldsymbol{P}},W) =& V_{\tau\tau'}^{}({\boldsymbol{q}}',{\boldsymbol{q}}|{\boldsymbol{P}}) + \displaystyle\int \dfrac{{\rm{d}}^3k}{(2\pi)^3} V_{\tau\tau'}^{}({\boldsymbol{q}}',{\boldsymbol{k}}|{\boldsymbol{P}}) \times \\ &\dfrac{Q_{\tau\tau'}^{}({\boldsymbol{k}},{\boldsymbol{P}})}{W-E_{{\boldsymbol{P}}+{\boldsymbol{k}},\tau}^{}-E_{{\boldsymbol{P}}-{\boldsymbol{k}},\tau'}^{} + i\epsilon} G_{\tau\tau'}^{}({\boldsymbol{k}},{\boldsymbol{q}}|{\boldsymbol{P}},W), \end{split} $$ (8) 其中
$\tau\tau' = {\rm{nn}}$ ,${\rm{pp}}$ 或者${\rm{np}}$ 。简洁起见,式(8)中略写了正负能态的指标。两核子散射的质心动量用$ {\boldsymbol{P}} $ 表示,散射初态、中间态和末态的相对动量分别用${\boldsymbol{q}}, {\boldsymbol{k}}$ 和$ {\boldsymbol{q}}' $ 表示。泡利算符$ Q({\boldsymbol{k}},{\boldsymbol{P}}) $ 要求中间态必须是费米面之上未被占据的状态,$ W $ 是初始能量。利用有效核力G矩阵可以容易地得到单粒子势算符的矩阵元:
$$ \begin{split} \varSigma_{\tau}^{++}(p) =& \sum\limits_{s'\tau'}^{} \int\nolimits_0^{k_F^{\tau'}} \dfrac{{\rm{d}}^3p'}{(2\pi)^3} \dfrac{M_{{\boldsymbol{p}}',\tau'}^*}{E_{{\boldsymbol{p}}',\tau'}^*} \big\langle \bar{u}_{\tau}^{}({\boldsymbol{p}},1/2) \bar{u}_{\tau'}^{}({\boldsymbol{p}}',s')\big| \times\\ & \bar{G}^{++++}(W)\big| u_{\tau}^{}({\boldsymbol{p}},1/2)u_{\tau'}^{}({\boldsymbol{p}}',s') \big\rangle, \end{split} \tag{9a}$$ $$ \begin{split} \varSigma_{\tau}^{-+}(p) = & \sum\limits_{s'\tau'}^{} \int\nolimits_0^{k_F^{\tau'}} \frac{{\rm{d}}^3p'}{(2\pi)^3} \dfrac{M_{{\boldsymbol{p}}',\tau'}^*}{E_{{\boldsymbol{p}}',\tau'}^*} \big\langle \bar{v}_{\tau}^{}({\boldsymbol{p}},1/2) \bar{u}_{\tau'}^{}({\boldsymbol{p}}',s')\big|\times \\ & \quad \bar{G}^{-+++}(W)\big| u_{\tau}^{}({\boldsymbol{p}},1/2)u_{\tau'}^{}({\boldsymbol{p}}',s') \big\rangle, \end{split} \tag{9b} $$ $$ \begin{split} \varSigma_{\tau}^-(p) =& \sum\limits_{s'\tau'}^{} \int\nolimits_0^{k_F^{\tau'}} \frac{{\rm{d}}^3p'}{(2\pi)^3} \dfrac{M_{{\boldsymbol{p}}',\tau'}^*}{E_{{\boldsymbol{p}}',\tau'}^*} \big\langle \bar{v}_{\tau}^{}({\boldsymbol{p}},1/2) \bar{u}_{\tau'}^{}({\boldsymbol{p}}',s')\big|\times \\ & \bar{G}^{-+-+}(W)\big| v_{\tau}^{}({\boldsymbol{p}},1/2)u_{\tau'}^{}({\boldsymbol{p}}',s') \big\rangle 。 \end{split} \tag{9c}$$ 进而提取单粒子势的不同分量。其中
$ k_{\rm F}^{\tau} $ 表示核子$ \tau $ 的费米动量。对于给定的总密度$\rho = \rho_{\rm{n}}^{}+\rho_{\rm{p}}^{}$ 和不对称度$\alpha = (\rho_{\rm{n}}^{}-\rho_{\rm{p}}^{})/\rho$ ,费米动量满足$k_{\rm{F}}^\tau = \left(3\pi^2\rho/2\right)^{1/3}(1+\tau_3^{}\alpha)^{1/3}$ ,其中对于中子/质子,$ \tau_3^{} = \pm1 $ 。在式(9)中,$ \bar{G} $ 表示反对称化的$ G $ 矩阵,上标中的$ \pm $ 号分别表示正能态和负能态。初始能量W等于散射初态两粒子的单粒子能量之和[37]。单粒子势的提取依赖于G矩阵,而G矩阵的计算又依赖于单粒子势,因此方程(1)和(7)~(9)需要联立迭代求解。求解过程严格地在完备的Dirac空间中执行,不需要引入动量无关近似或者投影方法,能够唯一地确定单粒子势的不同分量,因而避免了计算结果对不同近似方法的依赖性。自洽迭代收敛后,考虑两空穴线Goldstone图对多体系统基态能量的贡献(图2), 可以得到核物质系统的每核子结合能:
$$ \begin{split} E/A =& \dfrac{1}{\rho} \displaystyle\sum\limits_{s,\tau}^{} \displaystyle\int\nolimits_0^{k_F^\tau} \dfrac{{\rm{d}}^3p}{(2\pi)^3} \dfrac{M_{{\boldsymbol{p}},\tau}^{*}}{E_{{\boldsymbol{p}},\tau}^{*}} \big\langle \bar{u}_{\tau}^{}({\boldsymbol{p}},s)\big| {{\boldsymbol{\mathbm{γ}}}}{\boldsymbol{\cdot}}{\boldsymbol{p}} + M |u_{\tau}^{}({\boldsymbol{p}},s) \big\rangle -\\ &M + \dfrac{1}{2\rho} \displaystyle\sum\limits_{s,s',\tau,\tau'}^{} \displaystyle\int\nolimits_0^{k_F^\tau} \dfrac{{\rm{d}}^3p}{(2\pi)^3} \displaystyle\int\nolimits_0^{k_F^{\tau'}} \dfrac{{\rm{d}}^3p'}{(2\pi)^3} \dfrac{M_{{\boldsymbol{p}},\tau}^{*}}{E_{{\boldsymbol{p}},\tau}^{*}}\frac{M_{{\boldsymbol{p}}',\tau'}^*}{E_{{\boldsymbol{p}}',\tau'}^*}\times \\ & \langle \bar{u}_{\tau}^{}({\boldsymbol{p}},s) \bar{u}_{\tau'}^{}({\boldsymbol{p}}',s') \big|\bar{G}^{++++}(W)| u_{\tau}^{}({\boldsymbol{p}},s) u_{\tau'}^{}({\boldsymbol{p}}',s') \big\rangle。 \end{split} $$ (10) 关于完备Dirac空间中现实核力矩阵元的计算、初始能量的选取和Thompson方程的数值求解等细节,可以参考文献[37]。
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摘要: 相对论Brueckner-Hartree-Fock(RBHF)理论是相对论框架下重要的第一性原理方法,仅包含两体力即可以满意描述核物质的饱和性质。在完备的Dirac空间中自洽求解核物质RBHF方程,唯一确定了单粒子势的标量和矢量分量,避免了已有工作中由于忽略负能态所导致的不确定性,解决了40多年来RBHF计算中不能唯一确定单粒子势的问题。本工作简要回顾RBHF理论的发展历史,阐述包含负能态做RBHF计算的必要性,介绍利用完备Dirac空间的RBHF理论,研究核物质性质和中子星物质性质的最新进展,包括有效质量、纯中子物质的每核子结合能、对称核物质和纯中子物质的压强、中子星物质的粒子分数和状态方程、中子星的质量-半径关系和潮汐形变等。最后展望了完备Dirac空间的RBHF理论在确定密度泛函理论参数、微观描述核子-原子核弹性散射和研究中子星内部强子-夸克相变等方面的可能应用。
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关键词:
- 相对论Brueckner-Hartree-Fock 理论 /
- 完备的Dirac空间 /
- 纯中子物质 /
- 中子星状态方程
Abstract: Relativistic Brueckner-Hartree-Fock(RBHF) theory is one of the most important ab initio methods in the relativistic framework, where the saturation properties of nuclear matter could be described satisfactorily with only considering two-body forces. By achieving the self-consistent solution of the RBHF equations for nuclear matter in the full Dirac space, the scalar and vector components of the single-particle potential have been determined uniquely, the uncertainties caused by the neglect of negative-energy states(NESs) have been avoided, and the long-standing problem over 40 years of not being able to uniquely determine the single-particle potential has been solved. The history of the RBHF theory is briefly reviewed, and the necessity of considering NESs is illustrated. The latest results of nuclear matter and neutron star matter by the RBHF theory in the full Dirac space are discussed, including the effective mass, the binding energy per particle of pure neutron matter, the pressure of symmetric nuclear matter and pure neutron matter, the particle fractions as well as the equation of state for neutron star matter, and the mass-radius relation as well as the tidal deformability of a neutron star. Possible applications of the RBHF theory in the full Dirac space are also discussed, including the calibration of the parameters in density functional theory, the microscopic description of nucleon-nucleus elastic scattering, and the research on the hadron-quark transition inside neutron stars. -
图 3 完备Dirac空间的RBHF理论得到的核物质结合能和单粒子势(Full Dirac space),以及与动量无关近似(MIA)和投影方法(Proj.)的对比[37]
图 4 完备Dirac空间的RBHF理论计算得到的Dirac质量随同位旋不对称度的变化(a),以及与投影方法(b)和动量无关近似方法(c)所得结果的比较[38]
图 5 完备Dirac空间的RBHF理论计算得到的非相对论有效质量随同位旋不对称度的变化(a),以及与投影方法(b)和动量无关近似方法(c)所得结果的比较。图中还给出了基于AV18和微观三体力(TBF),非相对论BHF的计算结果(d)[43]
图 6 完备Dirac空间的RBHF理论计算得到的纯中子物质的每核子结合能随密度的变化(红色实线),以及与动量无关近似(灰色阴影)和投影方法(绿色虚线)的比较[41]
图中还给出了非相对论第一性原理方法的计算结果,包括考虑(蓝色菱形)和不考虑(蓝色虚线)三体力的BHF理论、非相对论变分方法(APR,黑色圆点)、辅助场蒙特卡罗方法(AFDMC)和多体微扰理论(MBPT)。
图 7 完备Dirac空间的RBHF理论计算得到的对称核物质(a)与纯中子物质(b)的压强
$ P $ 随密度$ \rho $ 的变化(红色实线),以及与投影方法(绿色点虚线)、动量无关近似方法(灰色阴影)、非相对论变分计算(APR)和基于重离子碰撞流数据所提取的约束(HIC)[47]的对比“Asy_soft”和“Asy_stiff”分别表示在输运模型中采取较软和较硬的对称能。
图 8 完备Dirac空间的RBHF理论计算得到的中子星物质的粒子分数
$Y_i^{} (i = {\rm{n}},\;{\rm{p}},\;{\rm{e}},\;\mu)$ 随重子数密度的变化[39]图 10 完备Dirac空间的RBHF理论计算得到的中子星质量-半径关系(红色实线)、与投影方法(绿色虚线)、动量无关近似方法(灰色点线)的对比
实验数据取自大质量中子星的天文观测(水平阴影带)和NICER团队关于脉冲星PSR J0030+0451质量和半径的联合约束(蓝色阴影带)[57]。
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