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本文中对谱函数的计算分为两部分:MF部分$ {S_{\rm MF}}({\boldsymbol{p}},E) $和NN-SRC部分$ {S_{\rm corr}}({\boldsymbol{p}},E) $:
$$ S({\boldsymbol{p}},E) = {S_{{\text{MF}}}}({\boldsymbol{p}},E) + {S_{{\text{corr }}}}({\boldsymbol{p}},E), $$ (1) 谱函数归一化到核子数$ \int {} {{\text{d}}^3}p {\text{d}}E S({\boldsymbol{p}},E) $。本文采用轴向变形相对论平均场(relativistic mean field, RMF)模型计算MF部分。在低能$ E $和低动量p下,MF部分$ {S_{\rm MF}}({\boldsymbol{p}},E) $由单粒子性质主导:
$$ {S_{\rm MF}}({\boldsymbol{p}},E) = \sum\limits_i {{C_i}} \left( {{{\left| {{f_i}({\boldsymbol{p}})} \right|}^2} + {{\left| {{g_i}({\boldsymbol{p}})} \right|}^2}} \right){L_i}\left( {E - {E_i}} \right), $$ (2) 式中:$ {f_i}({\boldsymbol{p}}) $和$ {g_i}({\boldsymbol{p}}) $为单粒子态$ i $在动量空间中的二维狄拉克旋量;$ {C_i} $为单粒子态$ i $对应的占有数;核子的能量分布可以用洛伦兹函数$ {L_i} $表示[23]。
在高能和高动量区域,由于NN-SRC效应会将核子激发到费米面以上的状态,因此NN-SRC部分的谱函数$ {S_{\rm corr}}({\boldsymbol{p}},E) $起主导作用。$ {S_{\rm corr}}({\boldsymbol{p}},E) $由NN-SRC对和$ (A - 2) $剩余核子系统的基态构型决定,由下式计算得到[19]:
$$ {S_{{\text{corr}}}}({\boldsymbol{p}},E) = {n_{{\text{corr }}}}({\boldsymbol{p}})\frac{m}{{|{\boldsymbol{p}}|}} \sqrt {\frac{{{\alpha _{\rm P}}}}{\pi }} \times \big[ {\exp \left( { - {\alpha _{\text{P}}}{{\boldsymbol{p}}_{{{\min }^2}}}} \right) - \exp \left( { - {\alpha _{\text{P}}}{{\boldsymbol{p}}_{{{\max }^2}}}} \right)} \big], $$ (3) 其中:$ {{\boldsymbol{p}}_{\min }} $和$ {{\boldsymbol{p}}_{\max }} $分别是质心动量的上限值和下限值;$ {n_{{\text{corr }}}}({\boldsymbol{p}}) $为短程关联部分动量分布;$ m $为核子质量。 ${\alpha _{\rm{P}}} = {{\left[ {3(A - 1)} \right]} \mathord{\left/ {\vphantom {{\left[ {3(A - 1)} \right]} {\left[ {4\left\langle {p_{{\text{MF}}}^2} \right\rangle (A - 1)} \right]}}} \right. } {\left[ {4\left\langle {{\boldsymbol{p}}_{{\text{MF}}}^2} \right\rangle (A - 1)} \right]}}$,其中$ \left\langle {{\boldsymbol{p}}_{{\text{MF}}}^{}} \right\rangle $为平均场部分的平均核子动量[24]。
本文采用light-front dynamics (LFD)方法计算短程关联部分。在LFD方法中,$ {n_{\rm corr}}({\boldsymbol{p}}) $可以通过重新缩放氘核精确动量分布的NN-SRC部分来获得。动量分布可以表示为
$$ n_{{\text{corr }}}^\tau ({\boldsymbol{p}}) = {N_\tau }\tau {C_A}\big[ {{n_2}({\boldsymbol{p}}) + {n_5}({\boldsymbol{p}})} \big], $$ (4) 其中$ {n_2}({\boldsymbol{p}}) $和$ {n_5}({\boldsymbol{p}}) $两个分量反映氘核的短程关联部分,由LFD波函数给出[25]。比例因子$ {C_A} $代表了NN-SRC的强度。$ \tau $代表相应的质子数和中子数,$ {N_\tau } $为归一化因子。
在单举电子散射${\rm{(e,\, e')}}$过程中,入射电子与目标核相互作用,并有初始四动量$k \equiv ({E_{\rm{k}}},\;{\boldsymbol{k}})$和末态四动量$k' \equiv ({E_{\rm{k}}},\;{\boldsymbol{k}}')$。散射电子的四动量转移为$ q \equiv (\omega ,\;{\boldsymbol{q}}) $,其中$ \omega $为电子转移能量,q为电子的转移动量。忽略末态相互作用,双微分截面可表示为[26]
$$ \frac{{{{\text{d}}^2}\sigma }}{{{\text{d}}\varOmega {\text{d}}{E_{{{\mathbf{k}}^\prime }}}}} = \frac{{{\alpha ^2}}}{{{Q^4}}}\frac{{{E_{{{\mathbf{k}}^\prime }}}}}{{{E_{\mathbf{k}}}}}{L_{\mu \nu }}{W^{\mu \nu }}, $$ (5) 其中:$ \alpha $为精细结构常数;$ {Q^2} $为四动量传递的平方,${Q^2} = - {q^2} = {\omega ^2} - {{\boldsymbol{q}}^2}$。在式(5)中,轻子张量$ {L_{\mu \nu }} $包含了电子在散射过程中的所有信息,原子核的强子张量$ {W^{\mu \nu }} $包含了目标核的所有的核结构信息,因此原子核张量可以描述为
$$ {W}^{\mu \nu } = {\displaystyle \sum _{i}{\displaystyle \int {\text{d}}^{3}}p\text{d}ES({\boldsymbol{p}},E){w}_{i}^{\mu \nu }(q)\left(\frac{m}{{E}_{p}}\right)}\text{,} $$ (6) $ w_{}^{\mu \nu } $为核子的强子张量,反映了动量为p的束缚态核子的电磁结构。$ {E_p} $为散射前核子的初始能量。
结合式(5)和(6),${\rm{(e, e')}}$截面可以写成
$$ \begin{split} \frac{{{{\text{d}}^2}\sigma }}{{{\text{d}}\varOmega {\text{d}}{E_{{{\mathbf{k}}^\prime }}}}} =& \int {{{\text{d}}^3}} p {\text{d}}E\left[ {{S_{\text{p}}}({\boldsymbol{p}},E)\frac{{{{\text{d}}^2}{\sigma _{{\text{ep}}}}}}{{{\text{d}}\varOmega {\text{d}}{E_{{{\mathbf{k}}^\prime }}}}}} \right.\left. { + {S_{\text{n}}}({\boldsymbol{p}},E)\frac{{{{\text{d}}^2}{\sigma _{{\text{en}}}}}}{{{\text{d}}\varOmega {\text{d}}{E_{{{\mathbf{k}}^\prime }}}}}} \right]\\ & \delta \left( {\omega - E + m - {E_{\left| {{\boldsymbol{p}} + {\boldsymbol{q}}} \right|}}} \right), \end{split}$$ (7) 其中:基本截面$ {\sigma _{{\text{en}}}} $($ {\sigma _{{\text{ep}}}} $)描述电子与自由中子(质子)之间发生散射时的散射截面[19]。在准弹性电子散射过程和Δ共振电子散射中,$ {\sigma _{{\text{en}}}} $($ {\sigma _{{\text{ep}}}} $)的计算分别使用相应的形状因子,分别反映了核子和Δ(1232)粒子的电磁结构。$ {E_{\left| {p + q} \right|}} $为出射核子的能量。
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摘要: 核子-核子短程关联是核物理的关键问题之一,它会导致核子动量分布出现高动量尾。本工作首先基于轴向形变相对论平均场模型构建了平均场部分原子核的谱函数,并在此基础上引入了短程关联效应的修正。然后利用谱函数在平面波脉冲近似框架内计算了单举电子散射截面,包括准弹性散射部分和Δ共振散射部分。特别是在Δ共振散射区域,通过重新考虑了核子共振态Δ(1232)的电磁结构,有效地改进了理论计算,使理论散射截面与实验数据很好地吻合。本工作进一步将单举散射截面分为短程关联的贡献和平均场的贡献。研究发现,准弹性峰和Δ共振峰不仅反映了平均场结构,而且对短程关联信息敏感。最后,本工作提出了一种从实验截面中提取原子核短程关联强度的方法。Abstract: The nucleon-nucleon short-range correlation(NN-SRC) is one of the key issues in nuclear physics that cause high-momentum tails in the nucleon momentum distributions. In this paper, the nuclear spectral functions are constructed based on the axially deformed relativistic mean-field model, and the correction of the short-range correlation effect is introduced. Then, the inclusive scattering cross sections are calculated using the nuclear spectral function and the framework of the plane wave impulse approximation, including both the quasielastic and Δ resonance parts. In particular, in the Δ resonance region, the electromagnetic structure of the nucleon resonance state Δ(1232) is reconsidered, which effectively improves the theoretical calculations that can be in better agreement with experimental data. The paper further divides the inclusive scattering cross sections into the contributions of NN-SRC and mean-field. It is found that, the quasielastic peak and Δ resonance peak not only reflect the mean-field structure but also are sensitive to NN-SRC information. Finally, we propose a method for extracting the NN-SRC strength of nuclei from experimental cross-section data.
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图 1 根据形变的RMF模型和LFD方法计算的56Fe 原子核的计算结果(在线彩图)
(a) 平均场核子动量分布与包含关联的总动量分布;(b) 包含关联的原子核谱函数。图中核子动量分布的实验值取自文献[16]。
图 2 56Fe原子核在入射电子能量为Ek = 0.961 GeV,散射角θ = 37.5°时的散射截面(在线彩图)
(a) 准弹性散射截面、Δ共振散射截面与总截面;(b) MF部分的截面、NN-SRC部分的截面的贡献与两部分相加的总截面。实验数据来自参考文献[30]。
图 3 不同原子核在不同关联强度下的散射截面(在线彩图)
(a) 56Fe在入射电子能量为Ek = 0.62 GeV,散射角θ = 60°时的散射截面;(b) 12C在入射电子能量为Ek = 0.961 GeV,散射角θ = 37.5°时的散射截面。实验数据来自参考文献[34]。
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