Nuclear structure effects to atomic Lamb shift and hyperfine splitting

Ji Chen

doi:10.7498/aps.73.20241063

Abstract

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1. 引　言
2. 兰姆位移与超精细结构中的双光子交换效应
2.1 兰姆位移中的双光子交换效应
2.2 超精细结构中的双光子交换效应
3. 核结构第一性原理计算
4. TPE理论研究现状
5. 结　论

References

Abstract

The development of precision atomic spectroscopy experiments and theoretical advancements plays a crucial role in measuring fundamental physical constants and testing quantum electrodynamics (QED) theories. It also provides a significant platform for studying the internal structure of atomic nuclei and developing high-precision nuclear structure theories. Nuclear structure effects such as charge distribution, magnetic moment distribution, and nuclear polarizability have been accurately determined in many atomic spectroscopy experiments, significantly enhancing the precision of nuclear structure detection.This paper systematically reviews the theoretical research and developments on the corrections of two-photon exchange (TPE) effects on the Lamb shift and hyperfine structure (HFS) in light ordinary and muonic atoms. Advanced nuclear force models and ab initio methods are employed to analyze the TPE nuclear structure corrections to the Lamb shift in a series of light muonic atoms. The paper compares the calculation of TPE effects from various nuclear models and evaluates the model dependencies and theoretical uncertainties of TPE effect predictions.Furthermore, the paper discusses the significant impact of TPE theory on explaining the discrepancies between experimental measurements and QED theoretical predictions in atomic hyperfine structures, resolving the accuracy difficulties in traditional theories. Detailed analyses of TPE effects on HFS in electronic and muonic deuterium using pionless effective field theory show good agreement with experimental measurements, validating the accuracy of theoretical predictions.The theoretical studies of TPE effects in light atoms are instrumental for determining nuclear charge radii and Zemach radii from spectroscopy measurements. These results not only enhance the understanding of nuclear structure and nuclear interactions but also offer crucial theoretical guidance for future experiments, thereby advancing the understanding of the proton radius puzzle and related studies.

Keywords:

PACS:

21.45.-v	(Few-body systems)
21.60.De	(Ab initio methods)
25.20.-x	(Photonuclear reactions)
36.10.Ee	(Muonium, muonic atoms and molecules)

Authors and contacts

Ji Chen^1,2

1.
Key Laboratory of Quark and Lepton Physics, Institute of Particle Physics, Central China Normal University, Wuhan 430079, China
2.
Southern Center for Nuclear-Science Theory, Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, China

Corresponding author: Ji Chen, jichen@ccnu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12175083, 12335002, 11805078).

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1. 引　言

原子精密光谱实验与理论的发展在测量基本物理常数和检验量子电动力学 (quantum electro dynamics, QED) 理论中发挥关键作用, 同时也为研究原子核内部结构与发展高精度核结构理论提供了重要的观测平台. 近十年来, 随着原子光谱测量精度的不断提高, 核电荷分布、磁矩分布、核极化度等核结构效应在许多原子精密光谱实验中得到了精确测定, 从而显著提升了核结构探测的精度.

传统测量质子电荷半径的方法主要依赖于普通氢原子光谱和电子-质子散射数据^[1]. 2010年至2013年间, 瑞士保罗·谢勒研究所 (PSI) 的CREMA (charge radius experiment with muonic atoms) 合作组通过测量缪氢原子 (µH) 兰姆位移, 首次获得质子电荷半径的新值0.84087(39) fm^[2,3], 其精确度比原有实验提高约1个数量级, 但数值比CODATA 2010推荐值小4%, 相差7个标准差. 这一发现被称为“质子半径难题”, 对传统实验与QED理论的精确性提出了挑战, 并推动了测量核电荷和磁矩分布的新实验研究. CODATA 2014更新后, 这一分歧缩小至5.6个标准差^[4].

质子半径难题引发了广泛的研究兴趣, 因为它不仅影响对质子基本性质的理解, 还指向新物理的存在可能. 为解决质子半径难题, 研究者们提出并实施多种新的实验方案, 包括精确测量低动量转移下的电子-质子散射^[5,6]、以及研究缪子-质子散射^[7]. 另一种方法是通过测量不同核电荷或质量数的缪原子兰姆位移, 研究其他轻核的均方根电荷半径 (r_E). 通过系统比较电子-核与缪子-核系统中提取的 r_E, 可以检验核电荷半径的测量差异是否在不同质子数 (Z)、中子数 (N) 或核质量数 (A) 的轻子-核系统中仍然存在甚至有所增加.

CREMA合作组已开展一系列轻质量缪原子兰姆位移实验, 取得氘核半径难题的新发现^[8]. 同时^{3, 4}He核电荷半径也在相关缪原子兰姆位移的光谱测量中得到确定^[9,10]. 在兰姆位移测量中, 核电荷半径的准确性不仅依赖于实验精度, 还取决于QED和核结构修正的计算精度. 电子氢原子兰姆位移的QED贡献主要来源于束缚态电子的自能修正. 与之不同的是, 轻质量缪原子兰姆位移的QED修正主要由光子真空极化 (Uehling效应) 产生, 并且其2S和2P态的能级顺序与电子原子相反. 由于缪子的质量较重, 它的原子轨道半径远小于电子, 因此缪原子兰姆位移的核结构修正比在电子原子中显著增强.

缪原子兰姆位移与核电荷半径之间的关系可表示为

$\begin{aligned} E_\text{LS} = E_\text{QED} + \mathcal{A}_\text{OPE} r_{{\mathrm{E}}}^2 + E_\text{TPE}, \end{aligned}$

(1)

其中QED贡献 $E_\text{QED}$ 由光子真空极化、束缚态缪子自能修正和相对论反冲修正组成, 其领头阶Uehling效应的量级为 $\alpha\left(Z\alpha\right)^2$ ^[11]. 方程中的其他两项来自核结构修正. 与 $r_{{\mathrm{E}}}^2$ 成正比的项由缪子和核之间的单光子交换主导, 而 $E_{\rm TPE}$ 源自双光子交换贡献, 可以分为弹性与非弹性部分.

精密激光光谱技术的应用, 使得能够通过测量原子跃迁, 深入了解核结构并检验束缚态QED的准确性. 通过对轻质量缪原子兰姆位移测量, 获得高精度的核电荷半径. 同时, 高精度的超精细结构 (hyperfine splitting, HFS)测量为研究原子核磁矩结构提供重要信息, 相关测量已经或将要在轻质量电子原子 (如 ^1,2H, ³He, ^6,7Li)^[12–17]以及相应的缪原子中开展^{[3,8,10,18–22]}. HFS主要由核磁矩与轻子磁矩短程相互作用主导^[23–25], 因此它是研究原子核磁结构的理想探针.

理论上, 高精度HFS理论预测受限于由双光子交换 (two-photon exchange, TPE)过程所主导的核结构效应修正. 同理, 超精细劈裂的主要贡献可由如下公式表示:

$\begin{aligned} E_{\rm HFS} = E_{\mathrm{F}} + E_\text{QED} + E_{\rm TPE}, \end{aligned}$

(2)

其中 $E_\text{QED}$ 与 $E_{\rm TPE}$ 分别对应HFS的QED与TPE修正. 与兰姆位移相似, TPE修正可进一步分解为弹性与非弹性贡献. 弹性TPE贡献由Zemach半径表征, 来源于核电荷和磁密度的卷积, 而非弹性TPE修正则来自核极化.

由轻子-原子核TPE过程所产生的核结构效应对原子能谱的修正至关重要. 这不仅体现在缪原子兰姆位移的测量上, 也体现在电子原子与缪原子HFS的测量中. 通过对不同轻核及其缪原子的兰姆位移和HFS的精密测量, 可以深入分析核结构效应, 为提高核理论的预测精度和理解核力机制提供重要的高精度实验平台. 同时, 结合量子多体计算与先进核子-核子相互作用理论的原子核第一性原理计算的发展, 也为更加精确地预测原子能谱中的TPE修正提供了理论工具.

本文接下来的内容将分为几个部分进行详细阐述. 第2部分阐述轻质量原子兰姆位移与HFS中的TPE过程的理论基础. 第3部分介绍原子核第一性原理计算在研究TPE效应中的应用. 第4部分介绍核结构TPE理论研究的最新进展. 第5部分, 总结本文的主要内容, 并展望未来在提升实验精度和理论计算准确性方面的潜在研究方向及其对核物理和新物理探索的意义.

2. 兰姆位移与超精细结构中的双光子交换效应

轻子-原子核系统中的TPE效应对原子能谱在 $\alpha^5$ 量级产生贡献, 如所示. 考虑到相对论效应, 其对应的费曼图包括箱图、交叉图以及双光子顶点图 (海鸥图). TPE对原子能谱的贡献可以分为两部分: 一部分依赖于单个核子的基态与激发态内部结构, 称为单核子TPE效应 $E_{1{\mathrm{ N}}}$ ; 另一部分依赖于原子核的整体结构与反应, 称为核结构TPE效应 $E_{\rm nucl}$ , 并可进一步分解为弹性与核极化贡献项 $E_{\rm nucl}=E_{\rm el}+E_{\rm pol}$ . 本文主要讨论核结构TPE效应. 由此, 原子能谱中TPE总贡献可表示为

$图 1 轻子-原子核系统中的双光子交换费曼图, 从左至右依次对应箱图、交叉图与海鸥图. 波浪线、细直线、粗直线与椭圆分别对应光子、轻子、核基态与核激发态\r\nFig. 1. Two-photon exchange diagrams in lepton-nucleus systems. The diagrams from left to right are respectively the box, cross and seagull diagrams. Wiggled, thin-straight, thick-straight lines and ellipse represent respectively the photon, lepton, nuclear ground state and nuclear excited states.$

图 1 轻子-原子核系统中的双光子交换费曼图, 从左至右依次对应箱图、交叉图与海鸥图. 波浪线、细直线、粗直线与椭圆分别对应光子、轻子、核基态与核激发态

Fig. 1. Two-photon exchange diagrams in lepton-nucleus systems. The diagrams from left to right are respectively the box, cross and seagull diagrams. Wiggled, thin-straight, thick-straight lines and ellipse represent respectively the photon, lepton, nuclear ground state and nuclear excited states.

下载: 全尺寸图片幻灯片

$\begin{aligned} E_{\rm TPE} = E_{\rm 1N}+ E_{\rm el} + E_{\rm pol}. \end{aligned}$

(3)

利用费曼规则, 与图1对应的算符在洛伦兹规范下表示为^[26]

$\begin{split} \mathcal{H}_{2\gamma} =\;& {\mathrm{i}} Z^2 \left(4 \pi\alpha\right)^2 \phi_n^2 (0) \int \frac{{\mathrm{d}}^4 q}{\left(2\pi\right)^4} \\ &\times \frac{\eta_{\mu\nu}\left(q\right) T^{\mu \nu}\left(q,-q\right) }{\left(q^2+{\mathrm{i}}\epsilon\right)^2\left(q^2-2m_{{\mathrm{l}}} q_0+{\mathrm{i}}\epsilon \right)}, \end{split}$

(4)

其中 η 和 T 分别代表轻子和原子核的张量, $\phi_n\left(0\right)= \sqrt{\alpha^3 m_{{\mathrm{r}}}^3 / n^3\pi}$ 是原子nS态在原点处的波函数. $m_{\mathrm{l}}$ 与 $m_{\mathrm{r}}=m_{\mathrm{l}} M/\left(m_{\mathrm{l}}+M\right)$ 分别为轻子的静止质量与其在原子系统中的约化质量. T对应原子核的前向虚康普顿散射振幅^[27]:

$\begin{split} &T_{\mu\nu}\left(q,-q\right)=\text{seagull} \\ &+ \sum_{N} \Bigg\{ \frac{ \left\langle N_0\left| J_\mu \left(0\right)\right| N \boldsymbol{q} \right\rangle \left\langle N \boldsymbol{q} \left| J_\nu\left(0\right) \right| N_0 \right\rangle}{E_0-E_N +q_0 +{\mathrm{i}}\epsilon} \\ &+ \frac{ \left\langle N_0\left| J_\nu \left(0\right)\right| N - \boldsymbol{q} \right\rangle \left\langle N - \boldsymbol{q} \left| J_\mu \left(0\right) \right| N_0 \right\rangle}{E_0-E_N -q_0 +{\mathrm{i}}\epsilon} \Bigg \}, \end{split}$

(5)

其中J为原子核电磁流算符, $|N_0 {\rangle}$ 与 $|N \boldsymbol{q} {\rangle}$ 分别代表核基态与TPE过程中的核中间态; $E_0$ 与 $E_N$ 为相应的原子质心坐标系下本征能量. 在计算TPE的弹性贡献时, 中间态保持原子核基态, 本征能 $E_N= \boldsymbol{q}^2/ (2 M)$ 对应TPE中间过程的核反冲动能, M为原子核质量. 在计算核极化效应时, 中间态包括除基态外的所有满足电磁激发选择定则的核激发态和连续谱态. $\mathcal{H}_{2\gamma}$ 算符产生对原子光谱的核结构修正, 分别体现在对原子兰姆位移谱和超精细结构谱能级的改变. 以下就这两种情况分别展开讨论.

2.1 兰姆位移中的双光子交换效应

TPE对兰姆位移的修正可利用算符 $\mathcal{H}_{2\gamma}$ 的微扰计算获得. 引入轻子自旋z分量平均后, 相应的轻子张量可约化为^[27]

$\eta_{\mu \nu}(q) = (k\cdot q) g_{\mu \nu} + (k-q)_\mu k_\nu + k_\mu (k-q)_\nu,$

(6)

其中 $k=\left(m_{\mathrm{l}}, 0\right)$ 为近似静止的轨道轻子四动量. 将其表达式代入(4)式进行进一步计算, 可获得TPE效应对兰姆位移的能量修正^[27]:

$\begin{split} \;& E_{\rm TPE} = Z^2 \left(4\pi \alpha\right)^2 \phi_n^2\left(0\right)\, {\rm Im} \bigg\{ \int \frac{{\mathrm{d}}^4 q}{(2\pi)^4} \\ &~\times \frac{2m_{\mathrm{l}}}{(q^2 + {\mathrm{i}}\epsilon )^2-4m_{\mathrm{l}}^2 q_0^2} \left[ \frac{1}{ \boldsymbol{q}^2} T_{\mathrm{L}} + \frac{q_0^2}{ (q^2 + {\mathrm{i}}\epsilon )^2} T_{\mathrm{T}} \right] \bigg\},\end{split}$

(7)

其中 $T_{\mathrm{L}}=T_{00}$ 与 $T_{\mathrm{T}}=\left(\delta_{ij}-q_i q_j/ \boldsymbol{q}^2\right)T_{ij}$ 对应原子核向前虚康普顿振幅的纵向与横向极化部分. (7)式中的非弹性贡献项来自核的电磁极化, 并体现为核极化响应函数的求和规则计算^[27,28]:

$\begin{split} E_{{\rm pol}} =\;&-8 Z^2\alpha^{2} \phi_n\left(0\right)^{2}\int_{0}^{\infty} {\mathrm{d}} \left| \boldsymbol{q}\right| \int_{\omega_{{\mathrm{th}}}}^{\infty} {\mathrm{d}} \omega \\ &\times[ K_{{\mathrm{L}}}\left(\omega, \left| \boldsymbol{q}\right|\right) S_{{\mathrm{L}}}\left(\omega, \left| \boldsymbol{q}\right|\right)\\ &+ K_{{\mathrm{T}}}\left(\omega, \left| \boldsymbol{q}\right|\right) S_{{\mathrm{T}}}\left(\omega, \left| \boldsymbol{q}\right|\right)\\ &+ K_{{\mathrm{S}}}\left(\omega, \left| \boldsymbol{q}\right|\right) S_{{\mathrm{T}}}\left(\omega, 0\right) ]. \end{split}$

(8)

变量 $\left(\omega, \left| \boldsymbol{q}\right|\right)$ 是交换光子携带的四动量. $S_{{\mathrm{L}}}$ 和 $S_{{\mathrm{T}}}$ 分别是核纵向和横向极化响应函数:

$S_{{\mathrm{L,T}}} = \sum_{N\neq N_0} \delta (\omega-E_N + E_0) | a \langle N|\mathcal{O}_{{\mathrm{L,T}}} |N_0 \rangle|^2,$

(9)

其中电磁激发算符 $\mathcal{O}_{\mathrm{L}} = \rho (\boldsymbol{q})$ 和 $\mathcal{O}_{\mathrm{T}} = \boldsymbol{q} \times \boldsymbol{J}(\boldsymbol{q})/|\boldsymbol{q}|$ 分别与核电荷密度与电磁流密度相关. ()式中的积分权重函数 $K_{\mathrm{L }}$ , $K_{\mathrm{T}}$ 与 $K_{\mathrm{S}}$ 分别为^[27]

$\begin{split} K_{{\mathrm{L}}}(\omega, | \boldsymbol{q}| ) =\;& \frac{1}{2 E_{{\mathrm{q}}}} \bigg[ \frac{1}{(E_{{\mathrm{q}}} - m_{\mathrm{l}} ) (\omega + E_{{\mathrm{q}}} - m_{\mathrm{l}})} \\ &- \frac{1}{(E_{{\mathrm{q}}} + m_{\mathrm{l}}) (\omega + E_{{\mathrm{q}}} + m_{\mathrm{l}})} \bigg], \end{split}$

(10)

$\begin{aligned} K_{{\mathrm{T}}}\left(\omega, \left| \boldsymbol{q}\right|\right) =& -\frac{\omega+2\left| \boldsymbol{q}\right|}{4m_{\mathrm{l}} \left| \boldsymbol{q}\right|\left(\omega+\left| \boldsymbol{q}\right|\right)^2}+ \frac{ \boldsymbol{q}^2}{4m_{\mathrm{l}}^2}K_{{\mathrm{L}}}\left(\omega, \left| \boldsymbol{q}\right|\right), \end{aligned}$

(11)

$\begin{aligned} K_{{\mathrm{S}}}\left(\omega, \left| \boldsymbol{q}\right|\right) =& \frac{1}{4m_{\mathrm{l}} \omega }\left(\frac{1}{\left| \boldsymbol{q}\right|} -\frac{1}{E_{\mathrm{q}}}\right), \end{aligned}$

(12)

其中 $E_{\mathrm{q}} = \sqrt{m_{\mathrm{l}}^2 + \boldsymbol{q}^2}$ 为轻子相对论能量. ()式中引入双光子顶点海鸥项, 确保规范不变性并在动量积分中减除 $\left| \boldsymbol{q}\right|=0$ 处的发散. 在库仑规范中, 海鸥项仅贡献于横向极化^[27].

(7)式中的弹性贡献部分由原子核电磁分布产生, 其主要贡献项可表示为

$\begin{split} E_{\rm el} =\;& -8 Z^2\alpha^{2} \phi_n\left(0\right)^{2}\int_{0}^{\infty} {\mathrm{d}} \left| \boldsymbol{q}\right| \bigg[ K_{{\mathrm{L}}}\left(\frac{ \boldsymbol{q}^2}{2M}, \left| \boldsymbol{q}\right|\right) \\ &\times F_{\mathrm{E}}^2\left( \boldsymbol{q}^2\right)-\frac{2 m_{\mathrm{r}} }{ \boldsymbol{q}^4}+ \frac{2 m_{\mathrm{r}} r_{\mathrm{E}}^2}{3 \boldsymbol{q}^2} \bigg],\\[-1pt] \end{split}$

(13)

其中后两项减除积分在 $| \boldsymbol{q}|=0$ 处的发散, 同时避免单光子交换中非微扰部分的重复计算. $F_{\mathrm{E}}$ 为原子核的电荷形状因子. 在 $M\rightarrow \infty$ 近似下, 该贡献与Zemach moment $\left\langle r_{{\mathrm{E}}}^{3}\right\rangle_{\left(2\right)}$ 成正比^[26]:

$\begin{split} &E_{\rm el} \approx -m_{{\mathrm{r}}}^4\frac{\alpha^5}{24}\left\langle r_{{\mathrm{E}}}^{3}\right\rangle_{\left(2\right)}\\ =\;&-m_{{\mathrm{r}}}^4\alpha^5\frac{2}{\pi}\int_{0}^{\infty}\frac{{\mathrm{d}}\left| \boldsymbol{q}\right|}{ \boldsymbol{q}^4}\left[F_{\text{E}}^{2}\left( \boldsymbol{q}^2\right)-1+\frac{r_{{\mathrm{E}}}^2 \boldsymbol{q}^2}{3}\right]. \end{split}$

(14)

2.2 超精细结构中的双光子交换效应

由于超精细结构 (HFS) 涉及轻子自旋与核自旋的耦合, () 式中轻子张量可约化为仅与轻子自旋相关的部分 $\tilde{\eta}^{\mu \nu} = {\mathrm{i}} q_0 \epsilon^{0 \mu \nu i}\sigma_i^{\left({\mathrm{l}}\right)} + {\mathrm{i}} \epsilon^{\mu \nu i j} \sigma^{\left({\mathrm{l}}\right)}_iq_j$ ^[29]. 中海鸥图对应的核张量为 $B_{\mu \nu}$ , 其中电荷-电流干涉项 $B_{0 m}$ 提供领头阶在 $1/M^2$ 量级的相对论修正, 而电流-电流项 $B_{ij}$ 由于交叉对称性相互抵消^[26,30].

由此可得到TPE对HFS修正的非弹性部分贡献, 并表示为核响应函数的求和规则^[29]:

$\begin{split} \;& E_{\rm pol} = \frac{ 6\alpha \mu_{\mathrm{l}} E_{\mathrm{F}}}{\pi \mu_I} \int_0^\infty {\mathrm{d}} \left| \boldsymbol{q}\right| \int_{\omega_{\rm th}}^\infty {\mathrm{d}}\omega \big[ h^{(0)} (\omega, | \boldsymbol{q} |) \\ &~~~~\times S^{\left(0\right)}(\omega,| \boldsymbol{q}|)-h^{(1)} (\omega, | \boldsymbol{q}|) S^{(1)} (\omega,| \boldsymbol{q}|)\big],\\[-1pt] \end{split}$

(15)

其中核响应函数 $S^{\left(0\right)}$ ( $S^{\left(1\right)}$ )由电荷(电流)算符与磁流算符之间的干涉项产生:

$\begin{split} \;& S^{(0)}(\omega,| \boldsymbol{q}|) \\ =\;& {\rm Im}\bigg\{ \int \frac{{\mathrm{d}} \hat{q}}{4\pi \boldsymbol{q}^2} \sum_{N\neq N_0} \left\langle N_0 I I | \rho(- \boldsymbol{q}) | N \right\rangle \langle N |\\ &\times \left[ \boldsymbol{q} \times \boldsymbol{J}_{\mathrm{m}}( \boldsymbol{q})\right]_3 | N_0 I I \rangle \delta(\omega-E_N+E_0)\bigg\},\\[-1pt] \end{split}$

(16)

$\begin{split} &S^{(1)}(\omega,| \boldsymbol{q}|) \\ =\;& {\rm Im}\bigg\{ \int \frac{{\mathrm{d}} \hat{q}}{4\pi} \epsilon^{3jk} \sum_{N \neq N_0} {\langle} N_0 I I | \boldsymbol{J}_{{\mathrm{c}},j}(- \boldsymbol{q}) | N {\rangle} \\ &\times{\langle} N | \boldsymbol{J}_{{\mathrm{m}},k}( \boldsymbol{q}) | N_0 I I {\rangle} \delta(\omega-E_N+E_0) \bigg\}. \end{split}$

(17)

式中 ${\mathrm{ d}} \hat{q}$ 定义为动量 $\boldsymbol{q}$ 所对应立体角的积分变量, $J_{\mathrm{c}}$ 与 $J_{\mathrm{m}}$ 分别为电流与磁流密度算符. $|N_0 II {\rangle}$ 定义自旋为I且z分量投影为I的原子核基态. ()式中能量与动量积分权重函数 $h^{(0, 1)}$ 的表达式为^[29]

$\begin{aligned} h^{(0)} (\omega,| \boldsymbol{q}|) = \left(2+ \frac{\omega}{E_{\mathrm{q}}}\right)\frac{E_{\mathrm{q}}^2+m_{{\mathrm{l}}}^2+E_{\mathrm{q}} \omega}{\left(E_{\mathrm{q}}+\omega\right)^2-m_{\mathrm{l}}^2 } - \frac{2 | \boldsymbol{q} |+\omega}{| \boldsymbol{q}| + \omega}, \end{aligned}$

(18)

$h^{\left(1\right)} (\omega, | \boldsymbol{q}| ) = \frac{1}{E_{\mathrm{q}}} \frac{E_{\mathrm{q}}^2+m_{{\mathrm{l}}}^2+E_q \omega}{\left(E_{\mathrm{q}}+\omega\right)^2-m_{\mathrm{l}}^2 } - \frac{1}{\left| \boldsymbol{q}\right|+\omega}.$

(19)

当考虑弹性贡献时, 原子核在中间态获得动量 $\boldsymbol{q}$ 及反冲动能 $\boldsymbol{q}^2/(2 M)$ . 因此, TPE效应对HFS的弹性修正为^[29]

$\begin{split} & E_{\rm el} = \frac{ 2 \alpha E_{\mathrm{F}}}{ \pi m_{\mathrm{l}} } \int_0^\infty {\mathrm{d}} \left| \boldsymbol{q}\right| \bigg\{ \bigg[h^{\left(0\right)}\left(\frac{ \boldsymbol{q}^2}{2M}, \left| \boldsymbol{q}\right|\right)-\frac{ \boldsymbol{q}^2}{2M} \\ & \times h^{(1)}\left(\frac{ \boldsymbol{q}^2}{2M}, | \boldsymbol{q} |\right) \bigg] F_{{\mathrm{E}}} (\boldsymbol{q}^2) F_{{\mathrm{M}}} (\boldsymbol{q}^2) - \frac{4 m_l m_{\mathrm{r}}}{ \boldsymbol{q}^2} \bigg\},\\ \end{split}$

(20)

其中 $F_{{\mathrm{M}}}$ 为原子核的磁形状因子. 当取 $M \gg m_{\mathrm{l}}$ 时, $E_{\rm el}{(0)}$ 可近似为与Zemach半径成正比 $E_{\text{Zem}} = -2\alpha m_{\mathrm{r}} r_{\mathrm{Z}}$ ^[31,32]. (20)式中最后一项抵消了动量积分的红外发散, 并避免与单光子交换的非微扰计算重复考虑^[33].

3. 核结构第一性原理计算

在计算TPE核结构效应对原子光谱的修正时, 需要计算原子核的电磁形状因子以及核响应函数的能量与动量积分, 可使用核结构第一性原理方法进行计算. 核电磁形状因子是原子核基态的可观测量, 因此需要精确求解核基态波函数. 如今, 量子多体理论与算力的发展为通过多体哈密顿量求解轻核( $A\leqslant 4$ )基态性质提供了便利^[34,35]. 相比之下, TPE效应对原子能谱的非弹性贡献可通过计算一系列核响应函数 $S_{\hat{O}}\left(\omega\right)$ 的广义求和规则 (generalized sum rule, GSR) I获得:

$I =\int_{\omega_{\rm th}}^\infty {\mathrm{d}}\omega g (\omega)S_{\hat{O}} (\omega).$

(21)

在计算核响应函数时, 必须计算原子核完整的激发谱, 其中包括众多连续态. 原子核第一性原理计算方法通常基于具有局域性且平方可积的完备基展开, 可严格计算束缚态问题, 但在求解散射态时则存在收敛困难. 洛仑兹积分变换(Lorentz integral transform, LIT)方法可绕开对连续谱的直接计算, 从而间接求解核响应函数^[36,37]. 然而, 在计算TPE中的求和规则时, LIT仍存在计算效率与精度的挑战.

对于²H的质子-中子两体问题, 一方面可通过谐振子基展开获得²H的基态信息与GSR. 随着谐振子基空间的扩大, 数值计算结果逐步收敛, 实现高精度计算. 另一方面, 也可通过Lippmann-Schwinger (LS)方程严格求解²H基态(束缚态)与激发态(散射态), 从而实现对TPE的严格求解.

对于三体和四体问题, 运用有效相互作用超球谐基(effective interaction hyperspherical harmonics, EIHH)方法可高效计算原子核多体束缚态问题. 但在计算GSR时需要庞大的基空间, 直接对角化的方法通常很难实现数值收敛以达到高精度. 因此, 通过发展“Lanczos求和规则”技术^[38], 可实现对GSR的高精度计算, 并同时避免LIT直接求解响应函数时的困难. 当()式中能量权重函数 $g(\omega)$ 连续可导时, Lanczos方法将完整的哈密顿量空间投影至与电磁激发算符 $\hat{O}$ 相关的Krylov子空间, 从而缩小基空间有效维度, 并通过对子空间对角化获得GSR的高精度计算结果. Lanczos方法的引入显著加速了计算的收敛过程, 能够在较短时间内获得核多体系统中GSR的高精度计算结果.

通过结合谐振子基函数、LS方程、EIHH方法, 能够准确求解原子核两体、三体和四体问题, 结合Lanczos求和规则方法, 可以高效地计算响应函数的广义求和规则, 从而准确、高效地计算TPE核极化效应对一系列原子光谱的影响. 这些数值方法为深入理解核结构和核反应机制以及理解原子能谱中的TPE效应提供了有力的帮助.

4. TPE理论研究现状

结合手征核力与量子多体方法的第一性原理计算, TPE核结构效应对兰姆位移的修正已经在一系列缪原子中得到系统研究. 在µ²H中, $E_{\rm TPE}$ 在手征有效场理论 (χEFT) 核力下的计算结果^[39,40]与唯象核力模型 (Argonne V18, AV18)^[41,42]、零程核力模型^[43,44]以及无π介子有效场理论^[45–47] 的计算结果高度一致, 并且与通过对 ${\text{e-D}}$ 散射数据进行色散关系分析所提取出的 $E_{\rm TPE}$ 预测结果^[48]相符. 在 $\text{µ}^2 \rm{H}$ 中, 通过手征有效场理论的幂次规则 (power counting)估计手征核力在第三阶展开下对 $E_{\rm TPE}$ 理论预测的误差为0.6%, 相比之下, 由于缺乏高精度的散射数据, 基于色散关系的TPE计算误差为20%. 结合EIHH量子多体方法与Lanczos求和规则算法, µ³H, µ^{3, 4}He⁺中TPE效应对兰姆位移的修正也得到了计算^[49–51]. 通过比较AV18+UIX唯象核力模型^[52,53]与χEFT^[54,55]手征核力对 $E_{\rm TPE}$ 的计算结果, 可以评估核结构理论的模型依赖性对 $E_{\rm TPE}$ 的影响, 从而估算理论误差. 列出了上述缪原子中 $E_{\rm TPE}$ 的理论预测结果和误差, 并分解为核结构弹性贡献、核极化贡献、单核子弹性TPE贡献与单核子极化贡献四部分. 通过比较分析, 核结构理论模型计算TPE效应的误差随质量数A的增大而增加. 研究不仅验证了不同核力模型在计算 $E_{\rm TPE}$ 时的一致性, 还深入分析了这些模型的理论误差来源. 这为实验数据的精确分析和核半径的准确提取提供了重要的理论支持.

表 1 不同μ原子中

$\delta_{\rm TPE}$

的计算结果和理论误差(单位meV). 结果被分解为弹性部分和核极化部分, 以及单核子部分. 数据来源于文献[51]

Table 1. Theoretical prediction and uncertainty of

$\delta_{\rm TPE}$

in various muonic atoms (in unit of meV). The results are decomposed into elastic, polarizability, and single-nucleon parts. Data collected from Ref. [51].

	$\delta^{N}_{\rm el}$	$\delta^{N}_{\rm pol}$	$\delta^{A}_{\rm el}$	$\delta^{A}_{\rm pol}$	$\delta_{\rm TPE}$
$\text{µ}^2{\rm H}$	–0.030(02)	–0.020(10)	–0.423(04)	–1.245(13)	–1.718(17)
$\text{µ}^3{\rm H}$	–0.033(02)	–0.031(17)	–0.227(06)	–0.480(11)	–0.771(22)
$\text{µ}^3 {\rm He}^{+}$	–0.52(03)	–0.25(13)	–10.49(23)	–4.23(18)	–15.49(33)
$\text{µ}^4 {\rm He}^{+}$	–0.54(03)	–0.34(20)	–6.14(31)	–2.35(13)	–9.37(44)

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CREMA合作组在2016年测量了 $\text{µ}^2 \rm{H}$ 的兰姆位移, 从中提取出氘核的电荷半径为 $r_{\mathrm{d}} = 2.12562(78)$ fm^[8]. 其系统误差主要源于 $E_{\rm TPE}$ 理论预测的不确定性. 该结果比CODATA标准值小6.0σ^[4], 与 ${\mathrm{e}}^2 {\rm H}$ 精密谱测量结果相差3.5σ^[56]. 并且 $\text{µ}^ {\rm H} -\text{µ}^2 {\rm H}$ 兰姆位移测量核电荷半径同位素位移的结果与相应的电子原子能谱测量值相差2.6σ^[57]. CREMA合作组又于2021年与2023年分别测量了 $\text{µ}^4{\rm He}^+$ 与 $\text{µ}^3{\rm He}^+$ 的兰姆位移, 并确定了 $^4{\rm He}$ 与 $^3{\rm He}$ 的核电荷半径分别为 $r_\text{α}=1.67824(83)$ fm与 $r_{\mathrm{h}}= 1.97007(94)$ fm^[9,10]. 实验同时确定了 ${}^{3, 4} {\rm He}$ 电荷半径的同位素位移, 这一结果为电子原子精密谱测量该同位素位移时存在的争议^[58–63]提供了新的参考数据.

传统理论在对HFS中TPE效应的预测精度上存在困难. 利用核力零程近似计算²H 1S HFS中的TPE效应^[64,65]与 $\nu_{\rm exp}(^2\text{H})-\nu_\text{QED}(^2\text{H})= 45.2\text{ kHz}$ 在5%范围内一致^[13,25]. 该理论方法又被文献[66]扩展到估算µ²H 2S态HFS中的TPE效应^[67,68]. 然而, 核力零程近似在对氘核长程渐近行为的描述上存在33%的偏差, 并且理论中能量积分截断的选取具有任意性, 因而理论存在很大的不确定度, 与实验的符合存在偶然.

另一种传统理论方法则基于Low-term公式, 利用准完备性近似合并TPE中的弹性与非弹性贡献, 使TPE在该近似下仅依赖原子核基态的波函数. 然而, 当核激发的动量尺度与轻子质量相当时, 该近似改变了积分的低动量依赖. Low-term公式预测的²H 1S态的TPE效应为46 kHz^[26,69]. 然而, 由于计算中忽略了单核子TPE贡献的反冲与极化效应, 与 $\nu_{\rm exp}-\nu_\text{QED}$ 的符合具有偶然性. 考虑单核子贡献修正后的TPE预测结果应该为 $E_{\rm TPE, mod}(^2\text{H})= 64\text{ kHz}$ , 与 $\nu_{\rm exp}-\nu_\text{QED}$ 相差43%. Kalinowski等^[70]通过引入高阶极化修正扩展了Low-term公式, 并用于计算µ²H 2S态中的TPE效应, 得到的计算结果为 $E_{\rm TPE}({\text{μ}}^2\text{H})= 0.0383(86)\text{ meV}$ , 仅占 $\nu_{\rm exp}({\text{μ}}^2\text{H})- \nu_\text{QED}({\text{μ}}^2\text{H})=0.0966(73)\text{ meV}$ 的40%^[68]. 由于这一计算引入了原子核激发能的幂次展开, 可能过分强调了极化修正中的高能贡献, 并导致与Low-term贡献的显著抵消, 造成与实验结果的差异.

利用无π介子有效场理论核力, TPE核结构效应对 $\text{e}^2{\rm H}$ 与 ${\text{μ}}^2{\rm H}$ 中HFS的修正, 分解为核结构弹性贡献与核极化贡献. 单核子TPE效应的贡献可通过将质子与中子的TPE效应^[71–74]经由原子波函数的比例关系转化为对 $\text{e}^2{\rm H}$ 与 ${\text{μ}}^2{\rm H}$ 中的贡献. 表2总结了²H和µ²H中的弹性、核极化和单核子TPE效应及其理论误差, 并与测量值和其他理论预测进行了比较. 理论误差主要来源于无π介子有效场理论在次次领头阶截断的高阶修正误差^[29]、核子电磁形状因子参数误差^[75–78]、色散关系分析提取单核子TPE效应的理论误差、以及高阶三光子交换效应修正误差 $\left(\varDelta_{3 \gamma}\right)$ ^[70]. 这项最新研究中, 计算的²H 1S HFS中的TPE贡献为41.7(4.4) kHz. 这和HFS的实验测量值与QED理论预测值之间的差异 $\nu_{\rm exp}-\nu_\text{QED}$ 在 $1\sigma$ 内符合. 预测的µ²H 2S HFS中的TPE贡献为0.117(13) meV, 超出实验-QED差异17%, 但在 $1.3\sigma$ 范围内符合.

表 2 单质子、单中子、核弹性和核极化TPE效应对²H与µ²H中HFS的修正. 数据来源于文献[29]

Table 2. The single-proton, single-neutron, nuclear elastic, and nuclear-polarizability TPE contributions to HFS in ²H and µ²H. Data from Ref. [29].

	²H (1S)/kHz	µ²H (1S)/meV	µ²H (2S)/meV
$E_{\rm el}$	–42.1(2.1)	–0.984(46)	–0.123(6)
$E_{\rm pol}$	109.8(4.5)	2.86(12)	0.358(14)
$E_{\rm nucl}=E_{\rm el}+E_{\rm pol}$	67.7(4.2)	1.878(88)	0.235(11)
$E_{\rm 1 p}$ ^[71]	–35.54(8)	–1.018(2)	–0.1272(2)
$E_{\rm 1 n}$ ^[72]	9.6(1.0)	0.079(32)	0.010(4)
$\varDelta_{3\gamma}$	±0.49	±0.052	±0.0065
$E_{\rm TPE}$	41.7(4.4)	0.94(11)	0.117(13)
Ref. [64,65]	43
Ref. [26,69] mod	64.5
Ref. [70]		0.304(68)	0.0383(86)
$\nu_{\rm exp}-\nu_\text{QED}$ ^[25,68]	45.2		0.0966(73)
注: “mod”对原文献修正核子反冲与极化效应; TPE效应在µ²H的1S和2S态中相差8倍.

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5. 结　论

本文系统回顾并深入探讨了TPE效应对轻质量电子原子及缪原子兰姆位移和HFS修正的理论框架及研究进展. 探讨了利用核子-核子间相互作用模型, 并结合LS方程、EIHH量子多体方法和Lanczos求和规则算法, 对TPE效应在一系列原子光谱中的贡献的第一性原理研究. 通过对不同核力模型的比较分析和误差来源的系统评估, 有效地分析核理论误差对TPE效应的影响, 从而提供高精度且可靠的理论预测结果.

研究表明, 深入理解TPE效应对于提高原子光谱实验对核电荷半径及Zemach半径等核结构信息的测量精度至关重要. 为了进一步提升TPE效应的理论精度, 未来的研究需要发展更精确的核力理论, 提升量子多体计算的效率和精度, 并扩展核理论计算对TPE效应研究的应用范围. 通过与高精度原子光谱实验的进一步结合, 这些努力将有助于深化对核结构和核子-核子相互作用的理解, 检验高阶QED理论, 为解决质子半径难题及相关核半径测量问题提供新的见解.

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Figure 1. Two-photon exchange diagrams in lepton-nucleus systems. The diagrams from left to right are respectively the box, cross and seagull diagrams. Wiggled, thin-straight, thick-straight lines and ellipse represent respectively the photon, lepton, nuclear ground state and nuclear excited states.

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不同μ原子中 $\delta_{\rm TPE}$ 的计算结果和理论误差(单位meV). 结果被分解为弹性部分和核极化部分, 以及单核子部分. 数据来源于文献[51]

Theoretical prediction and uncertainty of $\delta_{\rm TPE}$ in various muonic atoms (in unit of meV). The results are decomposed into elastic, polarizability, and single-nucleon parts. Data collected from Ref. [51].

	$\delta^{N}_{\rm el}$	$\delta^{N}_{\rm pol}$	$\delta^{A}_{\rm el}$	$\delta^{A}_{\rm pol}$	$\delta_{\rm TPE}$
$\text{µ}^2{\rm H}$	–0.030(02)	–0.020(10)	–0.423(04)	–1.245(13)	–1.718(17)
$\text{µ}^3{\rm H}$	–0.033(02)	–0.031(17)	–0.227(06)	–0.480(11)	–0.771(22)
$\text{µ}^3 {\rm He}^{+}$	–0.52(03)	–0.25(13)	–10.49(23)	–4.23(18)	–15.49(33)
$\text{µ}^4 {\rm He}^{+}$	–0.54(03)	–0.34(20)	–6.14(31)	–2.35(13)	–9.37(44)

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表 2 单质子、单中子、核弹性和核极化TPE效应对²H与µ²H中HFS的修正. 数据来源于文献[29]

Table 2. The single-proton, single-neutron, nuclear elastic, and nuclear-polarizability TPE contributions to HFS in ²H and µ²H. Data from Ref. [29].

	²H (1S)/kHz	µ²H (1S)/meV	µ²H (2S)/meV
$E_{\rm el}$	–42.1(2.1)	–0.984(46)	–0.123(6)
$E_{\rm pol}$	109.8(4.5)	2.86(12)	0.358(14)
$E_{\rm nucl}=E_{\rm el}+E_{\rm pol}$	67.7(4.2)	1.878(88)	0.235(11)
$E_{\rm 1 p}$ ^[71]	–35.54(8)	–1.018(2)	–0.1272(2)
$E_{\rm 1 n}$ ^[72]	9.6(1.0)	0.079(32)	0.010(4)
$\varDelta_{3\gamma}$	±0.49	±0.052	±0.0065
$E_{\rm TPE}$	41.7(4.4)	0.94(11)	0.117(13)
Ref. [64,65]	43
Ref. [26,69] mod	64.5
Ref. [70]		0.304(68)	0.0383(86)
$\nu_{\rm exp}-\nu_\text{QED}$ ^[25,68]	45.2		0.0966(73)
注: “mod”对原文献修正核子反冲与极化效应; TPE效应在µ²H的1S和2S态中相差8倍.

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