Testing fundamental interactions on the helium atom

TL;DR

This paper reviews high-precision theoretical calculations of helium atom energy levels, fine structure, and isotope shifts, comparing them with experiments to test quantum electrodynamics and determine fundamental constants.

Contribution

It provides the most accurate theoretical predictions for helium energy levels and fine structure, including next-order effects, and discusses discrepancies in nuclear charge radius differences.

Findings

Excellent agreement between theory and experiment for helium energy levels.

Fine-structure splitting predicted with 1.7 kHz accuracy, matching measurements.

Discrepancies in isotope shift-derived nuclear charge radii remain unexplained.

Abstract

We critically examine the current status of theoretical calculations of the energies, the fine structure, and the isotope shift of the lowest-lying states of helium, searching for unresolved discrepancies with experiments. Calculations are performed within the quantum electrodynamics expansion in powers of the fine structure constant $\alpha$ and the electron-to-nucleus mass ratio $m/M$. For energies, theoretical results are complete through orders $\alpha^6m$ and $\alpha^6m^2/M$, with the resulting accuracy ranging from $0.5$ to $2$~MHz for the $n=2$ states. The fine-structure splitting of the $2^3P$ state is predicted with a much better accuracy, 1.7~kHz, as a consequence of a calculation of the next-order $\alpha^7m$ effect. An excellent agreement of the theoretical predictions with the recent measurements of the fine structure provides one of the best tests of the bound-state QED in few-electron systems and determines the fine-structure constant $\alpha$ with an accuracy of 31~ppb. The isotope shift between $^3$He and $^4$He is treated theoretically with a sub-kHz accuracy, which allows for a high-precision determination of the differences of the nuclear charge radii $\delta r^2$. Several such determinations, however, yield results that are in a 4$\sigma$ disagreement with each other, what remains unexplained. Apart from this, we find no significant discrepancies between theory and experiment for the helium atom. In the future, a calculation of the next-order $\alpha^7m$ effect for energy levels will enable determinations of the nuclear charge radii from atomic transition frequencies with an accuracy better than 1\%. Combined with the complementary determinations from muonic atoms, this will provide a sensitive test of universality in electromagnetic interactions of leptons and contribute to the solution of the proton charge radius puzzle.

Figures

Tables

Table 1. Table 1: Breakdown of theoretical contributions to the ionization (centroid) energies of the lowest-lying states of 4 He, in MHz. The nuclear parameters used in the calculations are: M / m = 7 294.299 541 36 ​ ( 24 ) 𝑀 𝑚 7294.29954136 24 M/m=7\,294.299\,541\,36\,(24) , r = 1.6755 ​ ( 28 ) 𝑟 1.6755 28 r=1.6755\,(28)  fm. “NS” denotes the finite nuclear size contribution.

	${\left(m/M\right)}^0$	${\left(m/M\right)}^1$	${\left(m/M\right)}^2$	${\left(m/M\right)}^3$	Sum
$1^{1}S:$
${\alpha }^2$	$-\mathrm{5\hspace{0.17em}946\hspace{0.17em}220\hspace{0.17em}752.325}$	$\mathrm{958\hspace{0.17em}672.945}$	$-209.270$	$0.049$	$-\mathrm{5\hspace{0.17em}945\hspace{0.17em}262\hspace{0.17em}288.601}$
${\alpha }^4$	$\mathrm{16\hspace{0.17em}904.024}$	$-103.724$	$0.028$		$\mathrm{16\hspace{0.17em}800.327}$
${\alpha }^5$	$\mathrm{40\hspace{0.17em}506.158}$	$-10.345$			$\mathrm{40\hspace{0.17em}495.813}$
${\alpha }^6$	$861.360$	$-0.348$			$861.012$
${\alpha }^7$	$-71.(36.)$				$-71.(36.)$
NS	$29.7\left(1\right)$				$29.7\left(1\right)$
Total					$-\mathrm{5\hspace{0.17em}945\hspace{0.17em}204\hspace{0.17em}173}.(36.)$
$2^{1}S:$
${\alpha }^2$	$-\mathrm{960\hspace{0.17em}463\hspace{0.17em}083.665}$	$\mathrm{140\hspace{0.17em}245.887}$	$-37.131$	$0.010$	$-\mathrm{960\hspace{0.17em}322\hspace{0.17em}874.899}$
${\alpha }^4$	$-\mathrm{11\hspace{0.17em}971.453}$	$-3.344$	$0.003$		$-\mathrm{11\hspace{0.17em}974.795}$
${\alpha }^5$	$\mathrm{2\hspace{0.17em}755.761}$	$-0.627$			$\mathrm{2\hspace{0.17em}755.134}$
${\alpha }^6$	$58.288$	$-0.022$			$58.267$
${\alpha }^7$	$-3.7\left(1.9\right)$				$-3.7\left(1.9\right)$
NS	$2.007\left(7\right)$				$2.007\left(7\right)$
Total					$-\mathrm{960\hspace{0.17em}332\hspace{0.17em}038.0}\left(1.9\right)$
$2^{3}S:$
${\alpha }^2$	$-\mathrm{1\hspace{0.17em}152\hspace{0.17em}953\hspace{0.17em}922.421}$	$\mathrm{164\hspace{0.17em}775.354}$	$-30.620$	$0.006$	$-\mathrm{1\hspace{0.17em}152\hspace{0.17em}789\hspace{0.17em}177.680}$
${\alpha }^4$	$-\mathrm{57\hspace{0.17em}629.312}$	$4.284$	$-0.001$		$-\mathrm{57\hspace{0.17em}625.029}$
${\alpha }^5$	$\mathrm{3\hspace{0.17em}999.432}$	$-0.800$			$\mathrm{3\hspace{0.17em}998.632}$
${\alpha }^6$	$65.235$	$-0.030$			$65.205$
${\alpha }^7$	$-5.2\left(1.3\right)$				$-5.2\left(1.3\right)$
NS	$2.610\left(9\right)$				$2.610\left(9\right)$
Total					$-\mathrm{1\hspace{0.17em}152\hspace{0.17em}842\hspace{0.17em}741.4}\left(1.3\right)$
$2^{1}P:$
${\alpha }^2$	$-\mathrm{814\hspace{0.17em}848\hspace{0.17em}364.923}$	$\mathrm{153\hspace{0.17em}243.883}$	$-47.514$	$0.016$	$-\mathrm{814\hspace{0.17em}695\hspace{0.17em}168.538}$
${\alpha }^4$	$-\mathrm{14\hspace{0.17em}024.044}$	$-2.809$	$0.004$		$-\mathrm{14\hspace{0.17em}026.850}$
${\alpha }^5$	$38.769$	$0.470$			$39.240$
${\alpha }^6$	$8.818$	$-0.003$			$8.815$
${\alpha }^7$	$0.81\left(40\right)$				$0.81\left(40\right)$
NS	$0.064$				$0.064$
Total					$-\mathrm{814\hspace{0.17em}709\hspace{0.17em}146.46}\left(40\right)$
$2^{3}P:$
${\alpha }^2$	$-\mathrm{876\hspace{0.17em}178\hspace{0.17em}284.885}$	$\mathrm{61\hspace{0.17em}871.895}$	$-25.840$	$0.006$	$-\mathrm{876\hspace{0.17em}116\hspace{0.17em}438.823}$
${\alpha }^4$	$\mathrm{11\hspace{0.17em}436.878}$	$11.053$	$0.002$		$\mathrm{11\hspace{0.17em}447.932}$
${\alpha }^5$	$-\mathrm{1\hspace{0.17em}234.732}$	$-0.614$			$-\mathrm{1\hspace{0.17em}235.346}$
${\alpha }^6$	$-21.832$	$-0.001$			$-21.833$
${\alpha }^7$	$2.9\left(0.7\right)$				$2.9\left(0.7\right)$
NS	$-0.796\left(3\right)$				$-0.796\left(3\right)$
Total					$-\mathrm{876\hspace{0.17em}106\hspace{0.17em}246.0}\left(0.7\right)$

Table 2. Table 2: Comparison of the theoretical predictions for various transitions in 4 He with the experimental results, in MHz. IE denotes the ionization energy.

	Experiment/Theory/Difference	Ref.
$1^1{S}_0$ (IE)	$\mathrm{5\hspace{0.17em}945\hspace{0.17em}204\hspace{0.17em}212}.\left(6\right)$	kandula:11
	$\mathrm{5\hspace{0.17em}945\hspace{0.17em}204\hspace{0.17em}173}.\left(36\right)$
	$39.\left(36\right)$
$2^1{S}_0$ (IE)	$960332041.01\left(15\right)$	lichten:91
	$960332038.0\left(1.9\right)$
	$3.0\left(1.9\right)$
$1^1{S}_0$–$2^1{S}_0$	$\mathrm{4\hspace{0.17em}984\hspace{0.17em}872\hspace{0.17em}315}.\left(48\right)$	bergeson:98
	$\mathrm{4\hspace{0.17em}984\hspace{0.17em}872\hspace{0.17em}135}.\left(36\right)$
	$180.\left(60\right)$
$2^3{S}_1$–$3^3{D}_1$	$\mathrm{786\hspace{0.17em}823\hspace{0.17em}850.002}\left(56\right)$	Dorrer:97
	$\mathrm{786\hspace{0.17em}823\hspace{0.17em}848.4}{\left(1.3\right)}^a$
	$1.6\left(1.3\right)$
$2^1{S}_0$–$2^1{P}_1$	$\mathrm{145\hspace{0.17em}622\hspace{0.17em}892.886}\left(183\right)$	luo1
	$\mathrm{145\hspace{0.17em}622\hspace{0.17em}891.5}\left(2.3\right)$
	$1.4\left(2.3\right)$
$2^1{P}_1$–$3^1{D}_2$	$\mathrm{448\hspace{0.17em}791\hspace{0.17em}399.113}\left(268\right)$	luo2
	$\mathrm{448\hspace{0.17em}791\hspace{0.17em}397.4}{\left(0.4\right)}^b$
	$1.7\left(0.5\right)$
$2^3{P}_0$–$3^3{D}_1$	$\mathrm{510\hspace{0.17em}059\hspace{0.17em}755.352}\left(28\right)$	luo:16
	$\mathrm{510\hspace{0.17em}059\hspace{0.17em}754.0}{\left(0.7\right)}^a$
	$1.4{\left(0.7\right)}^a$
$2^{3}P$–$2^3{S}_1$	$\mathrm{276\hspace{0.17em}736\hspace{0.17em}495.649}{\left(2\right)}^c$	pastor:04
	$\mathrm{276\hspace{0.17em}736\hspace{0.17em}495.4}\left(2.0\right)$
	$0.2\left(2.0\right)$
$2^3{S}_1$–$2^1{P}_1$	$\mathrm{338\hspace{0.17em}133\hspace{0.17em}594.4}\left(5\right)$	notermans:14
	$\mathrm{338\hspace{0.17em}133\hspace{0.17em}594.9}\left(1.4\right)$
	$-0.5\left(2.2\right)$
$2^1{S}_0$–$2^3{S}_1$	$\mathrm{192\hspace{0.17em}510\hspace{0.17em}702.145\hspace{0.17em}6}\left(18\right)$	rooij:11
	$\mathrm{192\hspace{0.17em}510\hspace{0.17em}703.4}\left(0.8\right)$
	$-1.3\left(0.8\right)$

Table 3. Table 3: Breakdown of theoretical contributions to the energies of the 2 3 ​ P J superscript 2 3 subscript 𝑃 𝐽 2^{3}P_{J} fine-structure levels of 4 He, with respect to the centroid 2 3 ​ P superscript 2 3 𝑃 2^{3}P energy, in MHz. The value of α 𝛼 \alpha used in calculations is α − 1 = 137.035 999 139 ​ ( 31 ) superscript 𝛼 1 137.035999139 31 \alpha^{-1}=137.035\,999\,139\,(31) .

	${\left(m/M\right)}^0$	${\left(m/M\right)}^{1+}$	Sum
$2^3{P}_0:$
${\alpha }^4$	$\mathrm{27\hspace{0.17em}566.991\hspace{0.17em}8}$	$\mathrm{0.934\hspace{0.17em}0}$	$\mathrm{27\hspace{0.17em}567.925\hspace{0.17em}9}$
${\alpha }^5$	$\mathrm{36.102\hspace{0.17em}4}$	$-\mathrm{0.001\hspace{0.17em}6}$	$\mathrm{36.100\hspace{0.17em}8}$
${\alpha }^6$	$-\mathrm{5.043\hspace{0.17em}6}$	$-\mathrm{0.003\hspace{0.17em}8}$	$-\mathrm{5.047\hspace{0.17em}4}$
${\alpha }^7$	$\mathrm{0.077\hspace{0.17em}9}$		$\mathrm{0.077\hspace{0.17em}9}$
${\alpha }^8$	$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$		$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$
Total			$\mathrm{27\hspace{0.17em}599.057\hspace{0.17em}2}\left(8\right)$
$2^3{P}_1:$
${\alpha }^4$	$-\mathrm{1\hspace{0.17em}997.604\hspace{0.17em}0}$	$\mathrm{1.764\hspace{0.17em}2}$	$-\mathrm{1\hspace{0.17em}995.839\hspace{0.17em}8}$
${\alpha }^5$	$-\mathrm{18.605\hspace{0.17em}4}$	$\mathrm{0.002\hspace{0.17em}2}$	$-\mathrm{18.603\hspace{0.17em}2}$
${\alpha }^6$	$-\mathrm{3.436\hspace{0.17em}1}$	$\mathrm{0.006\hspace{0.17em}2}$	$-\mathrm{3.429\hspace{0.17em}9}$
${\alpha }^7$	$-\mathrm{0.022\hspace{0.17em}4}$		$-\mathrm{0.022\hspace{0.17em}4}$
${\alpha }^8$	$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$		$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$
Total			$-\mathrm{2\hspace{0.17em}017.895\hspace{0.17em}3}\left(8\right)$
$2^3{P}_2:$
${\alpha }^4$	$-\mathrm{4\hspace{0.17em}314.835\hspace{0.17em}9}$	$-\mathrm{1.245\hspace{0.17em}3}$	$-\mathrm{4\hspace{0.17em}316.081\hspace{0.17em}3}$
${\alpha }^5$	$\mathrm{3.942\hspace{0.17em}8}$	$-\mathrm{0.001\hspace{0.17em}0}$	$\mathrm{3.941\hspace{0.17em}8}$
${\alpha }^6$	$\mathrm{3.070\hspace{0.17em}4}$	$-\mathrm{0.003\hspace{0.17em}0}$	$\mathrm{3.067\hspace{0.17em}4}$
${\alpha }^7$	$-\mathrm{0.002\hspace{0.17em}1}$		$-\mathrm{0.002\hspace{0.17em}1}$
${\alpha }^8$	$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$		$\mathrm{0.000\hspace{0.17em}0}\left(8\right)$
Total			$-\mathrm{4\hspace{0.17em}309.074\hspace{0.17em}2}\left(8\right)$

Table 4. Table 4: Comparison of the theoretical predictions for the 2 3 ​ P superscript 2 3 𝑃 2^{3}P fine-structure intervals of 4 He with the experimental results, in kHz.

	$2^3{P}_0-2^3{P}_2$	$2^3{P}_1-2^3{P}_2$	$2^3{P}_0-2^3{P}_1$
Theory:
Pachucki and Yerokhin 2010 hefine	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}131.4}\left(1.7\right)$	$\mathrm{2\hspace{0.17em}291\hspace{0.17em}178.9}\left(1.7\right)$	$\mathrm{29\hspace{0.17em}616\hspace{0.17em}952.5}\left(1.7\right)$
Experiment:
Zheng et al. 2017 zheng:17	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}130.98}\left(13\right)$	$\mathrm{2\hspace{0.17em}291\hspace{0.17em}177.56}\left(19\right)$
Feng et al. 2015 feng:15		$\mathrm{2\hspace{0.17em}291\hspace{0.17em}177.69}\left(36\right)$
Smiciklas et al. 2010 smiciklas:10	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}131.25}\left(30\right)$
Smiciklas et al. 2010 smiciklas:10 reevaluated in hessels:15	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}131.25}\left(32\right)$
Borbely et al. 2009 borbely:09		$\mathrm{2\hspace{0.17em}291\hspace{0.17em}177.53}\left(35\right)$
Borbely et al. 2009 borbely:09 reevaluated in hessels:15		$\mathrm{2\hspace{0.17em}291\hspace{0.17em}177.55}\left(35\right)$
Zelevinsky et al. 2005 zelevinsky:05	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}126.8}\left(0.9\right)$	$\mathrm{2\hspace{0.17em}291\hspace{0.17em}175.6}\left(0.5\right)$	$\mathrm{29\hspace{0.17em}616\hspace{0.17em}951.7}\left(0.7\right)$
Zelevinsky et al. 2005 zelevinsky:05 reevaluated in hessels:15	$\mathrm{31\hspace{0.17em}908\hspace{0.17em}126.8}\left(3.0\right)$	$\mathrm{2\hspace{0.17em}291\hspace{0.17em}176.8}\left(1.1\right)$	$\mathrm{29\hspace{0.17em}616\hspace{0.17em}951.7}\left(3.0\right)$
Guisfredi et al. 2005 guisfredi:05			$\mathrm{29\hspace{0.17em}616\hspace{0.17em}952.7}\left(1.0\right)$
Guisfredi et al. 2005 guisfredi:05 reevaluated in hessels:15			$\mathrm{29\hspace{0.17em}616\hspace{0.17em}953}.\left(10.0\right)$
George et al. 2001 george:01			$\mathrm{29\hspace{0.17em}616\hspace{0.17em}950.9}\left(0.9\right)$
George et al. 2001 george:01 reevaluated in hessels:15			$\mathrm{29\hspace{0.17em}616\hspace{0.17em}950.8}\left(0.9\right)$
Castillega et al. 2000 castillega		$\mathrm{2\hspace{0.17em}291\hspace{0.17em}175.9}\left(1.0\right)$
Castillega et al. 2000 castillega reevaluated in hessels:15		$\mathrm{2\hspace{0.17em}291\hspace{0.17em}177.1}\left(1.0\right)$

Table 5. Table 5: Determinations of the nuclear charge difference of 3 He and 4 He, δ r 2 ≡ r 2 ( 3 He ) − r 2 ( 4 He ) \delta r^{2}\equiv r^{2}(^{3}\mbox{\rm He})-r^{2}(^{4}\mbox{\rm He}) from different measurements. Units are kHz if not stated otherwise. δ ​ E 𝛿 𝐸 \delta E is the part of the isotope shift induced by the finite nuclear size, represented as δ ​ E = C ​ δ ​ r 2 𝛿 𝐸 𝐶 𝛿 superscript 𝑟 2 \delta E=C\,\delta r^{2} , with C 𝐶 C being the coefficient calculated from theory.

Determination from Rooij et al. rooij:11 :
$E{(}^3\mathrm{He},2^1{S}^{F=1/2}-2^3{S}^{F=3/2})-E{(}^4\mathrm{He},2^{1}S-2^{3}S)$	$-\mathrm{5\hspace{0.17em}787\hspace{0.17em}719.2}\left(2.4\right)$	Expt. rooij:11
$\delta E_{\mathrm{hfs}}\left(2^3{S}^{3/2}\right)$	$-\mathrm{2\hspace{0.17em}246\hspace{0.17em}567.059}\left(5\right)$	Expt. schluesser:69 ; rosner:70
$-\delta E_{\mathrm{iso}}\left(2^{1}S-2^{3}S\right)$ (point nucleus)	$\mathrm{8\hspace{0.17em}034\hspace{0.17em}065.91}\left(19\right)$	Theory patkos1 ; patkos2
$\delta E$	$-220.4\left(2.4\right)$
$C$	$-214.66\left(2\right)\mathrm{kHz}/{\mathrm{fm}}^2$	heis
$\delta r^2$	$1.027\left(11\right){\mathrm{fm}}^2$	patkos2
Determination from Cancio Pastor et al. pastor:04 ; cancio:12:3he :
$E{(}^3\mathrm{He},2^{3}P-2S)$ (centroid)	$\mathrm{276\hspace{0.17em}702\hspace{0.17em}827\hspace{0.17em}204.8}\left(2.4\right)$	Expt. cancio:12:3he
$-E{(}^4\mathrm{He},2^{3}P-2S)$ (centroid)	$-\mathrm{276\hspace{0.17em}736\hspace{0.17em}495\hspace{0.17em}649.5}\left(2.1\right)$	Expt. pastor:04 ; smiciklas:10 a
$-\delta E_{\mathrm{iso}}\left(2^{3}P-2^{3}S\right)$ (point nucleus)	$\mathrm{33\hspace{0.17em}667\hspace{0.17em}149.3}\left(0.9\right)$	Theory patkos1 ; patkos2
$\delta E$	$-\mathrm{1\hspace{0.17em}295.4}\left(3.3\right)$
$C$	$-1212.2\left(1\right)\mathrm{kHz}/{\mathrm{fm}}^2$	heis
$\delta r^2$	$1.069\left(3\right)\text{fm}\text{<sup>2</sup>}$	patkos1
Determination from Shiner et al. shiner:95 :
$E{(}^3\mathrm{He},2^3{P}_0^{1/2}-2^3{S}_1^{3/2})-E{(}^4\mathrm{He},2^3{P}_2-2^3{S}_1)$	$\mathrm{810\hspace{0.17em}599.0}\left(3.0\right)$	Expt. shiner:95
$\delta E_{\mathrm{hfs}}\left(2^3{S}_1^{3/2}\right)$	$-\mathrm{2\hspace{0.17em}246\hspace{0.17em}567.059}\left(5\right)$	Expt. schluesser:69 ; rosner:70
$\delta E_{\mathrm{fs}}\left(2^3{P}_2\right)$	$-\mathrm{4\hspace{0.17em}309\hspace{0.17em}074.2}\left(1.7\right)$	Theory hefine
$-\delta E_{\mathrm{fs},\mathrm{hfs}}\left(2^3{P}_0^{1/2}\right)$	$-\mathrm{27\hspace{0.17em}923\hspace{0.17em}393.7}\left(1.7\right)$	Theory patkos1 ; patkos2
$-\delta E_{\mathrm{iso}}\left(2^{3}P-2^{3}S\right)$ (point nucleus)	$\mathrm{33\hspace{0.17em}667\hspace{0.17em}149.3}\left(0.9\right)$	Theory patkos1 ; patkos2
$\delta E$	$-1286.7\left(3.5\right)$
$C$	$-1212.2\left(1\right)\mathrm{kHz}/{\mathrm{fm}}^2$	heis
$\delta r^2$	$1.061\left(3\right){\mathrm{fm}}^2$	patkos1

Equations

$$r_p(\mathrm{H})$$

$$r_p(\mu \mathrm{H})$$

$$L$$

$$H$$

$$G(E)=\frac{1}{E-H_0-\Sigma (E)},$$

$$⟨\varphi |G(E)|\varphi ⟩$$

$$\sigma (E)=⟨\varphi |\Sigma (E)|\varphi ⟩+⟨\varphi |\Sigma (E)\frac{1}{(E-H_0{)}^{\prime }}\Sigma (E)|\varphi ⟩+\dots .$$

$$E=E_0+\sigma (E_0)+\sigma (E_0)\frac{\partial \sigma (E_0)}{\partial E_0}+\dots .$$

$$\displaystyle E\Bigl{(}\alpha,\frac{m}{M}\Bigr{)}$$

$$E^{(i)}\Bigl{(}\frac{m}{M}\Bigr{)}=E^{(i,0)}+\frac{m}{M}\,E^{(i,1)}+\Bigl{(}\frac{m}{M}\Bigr{)}^{2}\,E^{(i,2)}+\ldots\,.$$

$$H_0\equiv H=\frac{p_1^2}{2}+\frac{p_2^2}{2}-\frac{Z}{r_1}-\frac{Z}{r_2}+\frac{1}{r},$$

$$E^{(2,1)}$$

$$E^{(2,2)}$$

$$E^{(2,3)}$$

$${\delta }_{M}H=\frac{{\vec{P}}^2}{2}$$

$$E^{(4,0)}$$

$$E^{(4,1)}$$

$$E^{(4,2)}$$

$$H^{(4)}$$

$${\delta }_M{H}^{(4)}$$

$$H^{(5)}$$

$${\delta }_M{H}^{(5)}$$

$$\ln k_0$$

$$\displaystyle\biggl{\langle}P\left(\frac{1}{r^{3}}\right)\biggr{\rangle}$$

$$E^{(6)}=\langle H^{(6)}\rangle+\biggl{\langle}H^{(4)}\,\frac{1}{(E-H)^{\prime}}\,H^{(4)}\biggr{\rangle}.$$

$$E^{(7)}=\langle H^{(7)}\rangle+2\,\biggl{\langle}H^{(4)}\,\frac{1}{(E-H)^{\prime}}\,H^{(5)}\biggr{\rangle}\,.$$

$$\frac{\Delta r}{r}=\frac{1}{2}\frac{\delta E_{\mathrm{fs}}}{E_{\mathrm{fs}}}\approx \frac{1}{2}\frac{10}{3450}\approx 1.5\times 10^{-3}.$$

$$E(2^{2S+1}L)$$

Full text

I Introduction

Two-body bound systems such as hydrogen (H), hydrogenlike atoms ($\bar{\rm H}$, He+, $\mu$H), and pure leptonic atoms (positronium $e^{+}e^{-}$ and muonium $\mu^{+}e^{-}$) are commonly used for pursuing high-precision low-energy tests of the Standard Model. Comparisons of the measured transition frequencies with the theoretical calculations convey the extent to which the atomic energy level can be predicted by the Standard Model. If any discrepancy is found, it may be a signature of new physics or an indication that the values of physical constants are incorrect.

One of the best tests of fundamental physics in atomic systems is derived from the magnetic moment of the electron bound in the hydrogenlike carbon ion. The relative precision of the experiment of $3\times 10^{-11}$ sturm is matched by the complementary accuracy of ab initio theoretical calculations based on quantum electrodynamics (QED) pachucki:04:prl ; pachucki:05:gfact . Experiment and theory are in excellent agreement; their comparison is limited by the uncertainty of the electron mass, as taken from the best electron-trap measurement vandyck . In practice, one reverses the problem and determines nist the electron mass from the bound-electron $g$ factor, gaining an improvement in accuracy by two orders of magnitude. So, this test of fundamental physics is presently limited by the accuracy of the electron trap mass measurement vandyck .

Another prominent atomic test with a possible signature of new physics is based on the comparison of the Lamb shift of the muonic hydrogen $\mu$H pohl and the electronic hydrogen H (see Ref. nist for a review). The lepton universality in the Standard Model states that the coupling constants of the electron and muon are equal, so one must use the same physical laws to predict the energy levels in H and $\mu$H and the same physical constants. What came out in practice, however, was a surprise. The proton root-mean-square charge radius, treated as an unknown parameter and extracted from the comparison of theory and experiment for the Lamb shift, turned out to be significantly different for the electronic nist and muonic pohl spectra,

[TABLE]

This discrepancy became known in the literature as the proton radius puzzle. It may signal the existence of interactions that are not accounted for, a lack of universality in the lepton-hadron interaction, or incorrect values of physical constants. Attempts to resolve the proton radius puzzle are currently being made in several experiments, such as measurements of the $2S$–$4P$ transition energy in H hansch , the $1S$–$2S$ transition energy in He+ H1s2sK ; H1s2sU , transitions between circular Rydberg states in heavy-H-like ions tan:2011 , and the direct comparison of the cross sections of the $e$-$p$ versus $\mu$-$p$ elastic scattering mup_scattering . Preliminary results from the H($2S$–$4P$) experiment hansch suggest that the presently accepted value of the Rydberg constant $R_{\infty}$ might be incorrect, which would resolve the proton radius puzzle. Nevertheless, this suggestion needs to be checked by other experiments before definite conclusions can be drawn.

If the $\mu$H result for the proton radius is confirmed, a combination of the $\mu$H($2S$–$2P$) and H($1S$–$2S$) experiments will provide a much more accurate result for the $R_{\infty}$ constant than the previously accepted value. Its accuracy will be limited by uncertainties in the two- and three-loop electron self-energy in H and the proton polarizability in $\mu$H. The electron self-energy can be improved by extensive calculations based on QED theory, whereas an improvement of the polarizability is less likely, because a part of it (the so-called subtraction term) cannot be deduced from inelastic electron-proton scattering data. Nonetheless, the prospective improvement of the Rydberg constant will contribute to the extension of the atomic-physics tests of fundamental interactions.

The main goal of the present review is to demonstrate that accurate tests of fundamental physics can be obtained not only from the hydrogenic systems but also from the few-body atomic systems, such as He and He-like ions. In particular, the existing data of high-precision spectroscopy of helium can be used for setting constraints on spin-dependent forces between electrons ficek , for the determination of the nuclear charge radius and for the comparison with results obtained from muonic atoms and by the electron scattering. Such a determination would be of particular interest today, in the context of the proton radius puzzle and the ongoing experiment on muonic helium antognini:11 .

The accuracy achieved by the present-day theory is sufficient for accurate determinations of the differences of the nuclear radii from the isotope shift heis , but not the absolute values of radii. We demonstrate, however, that a calculation of the next-order $m\alpha^{7}$ QED correction to the energy levels will be sufficient for determination of the nuclear charge radii of helium isotopes on the level of 1% or better. Such a calculation is difficult but feasible, at least for the triplet states, and will provide a sensitive test of universality in electromagnetic interactions of leptons.

II Quantum Electrodynamics of Atomic Systems

We now summarize the theoretical method for calculations of bound-state energy levels of light atoms. It is based on the Nonrelativistic QED (NRQED) expansion, originally introduced by Caswell and Lepage caswell:86 . Although the method is applicable for an arbitrary light atom, we assume the simplest case of the two-electron atom in the formulas below.

The starting point of this approach is the NRQED Lagrangian $L$,

[TABLE]

where $\psi$ is the nonrelativistic fermion field and $H$ is the effective NRQED Hamiltonian, which is, in our implementation, derived from the Dirac equation by the Foldy-Wouthuysen (FW) transformation itzykson ,

[TABLE]

Here, $\vec{\pi}=\vec{p}-e\vec{A}$, $\vec{s}=\vec{\sigma}/2$ is the electron spin operator, and $e$ and $m$ are the electron charge and mass, respectively. The dots in the above equation denote the higher-order terms that include, in addition to the FW Hamiltonian, local counterterms originating from virtual electron momenta of the order of the electron mass. The counterterms are derived by matching the NRQED and the full-QED scattering amplitudes.

Once the NRQED Lagrangian is obtained, the Feynman path-integral approach is used to derive various corrections to the nonrelativistic multielectron propagator $G(t-t^{\prime})$, where $t$ and $t^{\prime}$ are the common time of the out and the in electrons, correspondingly. The Fourier transform of the propagator is written as

[TABLE]

where $H_{0}$ is the Schrödinger-Coulomb Hamiltonian for $N$-electrons. $H_{0}$ may also include the nucleus as a dynamic particle. The $\Sigma(E)$ operator incorporates corrections due to the photon exchange, the electron and photon self-energy, etc.

The energy of a bound state is obtained as a position of the pole of the matrix element of $G(E)$ between the nonrelativistic wave functions $\phi$ of the reference state,

[TABLE]

where

[TABLE]

The resulting bound-state energy $E$ (the position of the pole) is

[TABLE]

The basic assumption of the NRQED is that $E$ can be expanded in a power series of the fine-structure constant $\alpha$,

[TABLE]

where $m/M$ is the electron-to-nucleus mass ratio and the expansion coefficients $E^{(n)}$ may contain finite powers of $\ln\alpha$. The coefficients $E^{(i)}(m/M)$ are further expanded in powers of $m/M$,

[TABLE]

The expansion coefficients in Eqs. (6) and (7) can be expressed as expectation values of some effective Hamiltonians with the nonrelativistic wave function. The derivation of these effective Hamiltonians is the central problem of the NRQED method. While the leading-order expansion terms are simple, formulas becomes increasingly complicated for high powers of $\alpha$.

The first term of the NRQED expansion of the bound-state energy, $E^{(2,0)}\equiv E$, is the eigenvalue of the Schrödinger-Coulomb Hamiltonian in the infinite nuclear mass limit,

[TABLE]

where $r\equiv r_{12}$. The finite nuclear mass corrections to $E^{(2,0)}$ can be obtained perturbatively,

[TABLE]

where

[TABLE]

is the nuclear kinetic energy operator, with $\vec{P}=-\vec{p}_{1}-\vec{p}_{2}$ being the nuclear momentum.

The next term of the expansion, $E^{(4)}$, is the leading relativistic correction induced by the Breit Hamiltonian $H^{(4)}$ and the corresponding recoil addition $\delta_{M}H^{(4)}$,

[TABLE]

The Breit Hamiltonian (without spin-dependent terms that do not contribute to the centroid energies) is given by

[TABLE]

For a spinless nucleus, there is no nuclear Darwin correction and $\delta^{2}_{M}H^{(4)}=0$.

The leading QED correction $E^{(5)}$ is induced by the effective Hamiltonian $H^{(5)}$ araki:57 ; sucher:58 and the corresponding recoil addition $\delta_{M}H^{(5)}$,

[TABLE]

where $\ln k_{0}$ is the Bethe logarithm

[TABLE]

$P\left(1/r^{3}\right)$ denotes the Araki-Sucher term

[TABLE]

and $\delta_{M}\ln k_{0}$ is a correction to Bethe logarithm $\ln k_{0}$ induced by the nonrelativistic kinetic energy $\delta_{M}H$ in Eq. (12).

The next expansion term $E^{(6)}$ is the higher-order QED correction, whose general form is

[TABLE]

The complete derivation of $H^{(6)}$ was presented in the nonrecoil limit in Ref. hsinglet , whereas the recoil correction $\delta_{M}H^{(6)}$ was obtained recently in Refs. patkos1 ; patkos2 . The corresponding formulas are much too complicated to be presented here; we thus refer the reader to the original works. The main problem of the derivation of $E^{(6)}$ is that the first- and the second-order matrix elements in Eq. (21) are divergent; the divergences cancel only when both terms are considered together. In order to subtract the singularities algebraically, one derives $H^{(n)}$ in $d=3-2\,\epsilon$ dimensions, makes use of various commutator identities to eliminate divergences, and then takes the limit $\epsilon\to 0$.

The next term $E^{(7)}$ has the general form of

[TABLE]

So far $E^{(7)}$ has been calculated only for the fine structure of helium and heliumlike ions hefine . In the future it should be possible to extend this calculation to the energies of the triplet states of helium. The main problem would be the derivation of $H^{(7)}$ and the numerical calculation of relativistic corrections to the Bethe logarithm.

III Binding energies and transition frequencies

The numerical results of our calculations of the individual $\alpha$ and $m/M$-expansion contributions to the ionization energies of the lowest-lying states of the helium atom are listed in Table 1. These results mostly correspond to our calculations reported in Refs. hsinglet ; hsinglet2 ; helike , with two improvements: (i) we included the $\alpha^{6}\,m^{2}/M$ correction recently calculated in Refs. patkos1 ; patkos2 and (ii) we added the higher-order recoil corrections $E^{(2,3)}$ and $E^{(4,2)}$ computed according to Eqs. (II) and (II). The uncertainty of the total energies is exclusively defined by the $\alpha^{7}\,m$ contribution, whose complete form is unknown at present. The approximate results for this correction listed in Table 1 were obtained by rescaling the hydrogenic values as described in Ref. drake:88:cjp , and the uncertainty is assumed to be 50% for the singlet states and 25% for the triplet states (due to vanishing of all terms proportional $\delta(r)$). We note that in our previous compilation helike , the uncertainty was estimated as 50% for all states.

In Table 2 we compare our theoretical predictions for the ionization energy of the ground state and for transition energies of helium with the results of the most accurate measurements. Our results are in agreement with those of Drake drake:05:springer , but are significantly more accurate. Our theoretical predictions agree well with the experimental results for the $2^{1}S$–$2^{3}S$, $2^{3}S$–$2^{1}P_{1}$, and $2^{3}P$–$2^{3}S$ transitions and for the $1^{1}S$ ground-state ionization energy. For the $1^{1}S$–$2^{1}S$ transition, however, we find a deviation of $180(36)(48)$ MHz from the measured value. Bearing in mind the good agreement observed for the other transitions involving the $1^{1}S$ and $2^{1}S$ states, we believe that the problem is likely to be on the experimental side. Almost equal differences between experimental and theoretical values are observed for different transitions involving the $3D$ states namely, 1.6(1.3) MHz for the $2^{3}S$–$3^{3}D_{1}$ transition, 1.4(0.7) MHz for the $2^{3}P_{0}$–$3^{3}D_{1}$, and 1.7(0.5) MHz for the $2^{1}P_{1}$–$3^{1}D_{2}$ ones, which suggests a recalculation of the $3D$ energies (not calculated by us but taken from Ref. morton:06 ).

In all cases except for the $1^{1}S$–$2^{1}S$ transition theoretical predictions are less accurate than the experimental results, which indicates the importance of the yet unknown $\alpha^{7}\,m$ correction. As can be seen from Table 1, a calculation of this correction would improve the theoretical precision up to the level of about 10 kHz. Combining such theory with the available experimental result for the $2^{3}S$–$2^{3}P$ transition pastor:04 , one will be able to determine the nuclear charge radius with a sub-1% accuracy, which is comparable with the expected precision of the radius from the $\mu$He experiment antognini:11 . Specifically, the finite nuclear size contribution to the $2^{3}S$–$2^{3}P$ transition energy is $E_{\rm fs}=3\,450$ kHz$\times h$. Since $E_{\rm fs}$ is proportional to $r^{2}$, the assumed 10-kHz theoretical error corresponds to the following error of $r$

[TABLE]

A similar determination of the charge radii of nuclei of light elements should be possible from the $2^{3}S$–$2^{3}P$ transitions in He-like ions. Measurements at the required level of accuracy are planned for He-like boron and carbon noerters . On the theoretical side, the higher-order relativistic effects in He-like ions become much more important than in the helium atom, and thus the NRQED approach will need to be combined with the fully relativistic calculations based on the $1/Z$ expansion artemyev:95 , as it has already been demonstrated by Drake in Ref. drake:88:cjp . It is important that a determination of the charge radius of one stable isotope of a light element will give us access to the whole chain of radii of other isotopes, because the differences of the radii are nowadays very efficiently extracted from the isotope shift measurements noerters:09 .

IV Fine structure of $\bm{2^{3}P_{J}}$ state

The fine-structure transition frequencies between the $2^{3}P_{J}$ levels are presently the most accurately known transition frequencies in helium. On the theoretical side, these transitions are calculated rigorously within NRQED up to order $\alpha^{7}\,m$ drake:02:cjp ; pachucki:06:prl:he ; pachucki:09:hefs ; hefine ; pachucki:11 , with a resulting theoretical accuracy of about 1 kHz. A summary of the theoretical results for individual $\alpha$ and $m/M$ expansion fine-structure contributions is presented in Table 3. The small deviations from the values reported in our original calculation hefine are due to the updated value of $\alpha$.

The uncertainty of the theoretical predictions is fully defined by the unknown higher-order $\alpha^{8}\,m$ contribution. The uncertainties listed in the table are obtained by multiplying the corresponding values for the $\alpha^{6}\,m$ corrections by $(Z\alpha)^{2}$.

On the experimental side, there were numerous results for the fine-structure intervals of helium obtained during the last decades, some of them contradicting each other. Recently, it was pointed out hessels:15 that the effect of the quantum mechanical interference between neighboring resonances (even if such neighbors are separated by thousands of natural widths) can cause significant shifts of the line center. The reexamination of existing measurements presented in Ref. hessels:15 improved the overall agreement of the experimental results with the theoretical predictions, whereas the two latest measurements feng:15 ; zheng:17 are in excellent agreement with the theory. The comparison of the present theory with the experimental results for the fine-structure intervals of helium are presented in Table 4 and Fig. 1.

A combination of the experimental and theoretical results for the fine structure of helium can be used in order to determine the fine-structure constant $\alpha$ with an accuracy of 31 ppb hefine ; pachucki:11 , which is about two orders of magnitude less precise than the current best determination of $\alpha$ nist . Further improvement of theory by calculating the next-order $\alpha^{8}\,m$ correction appears to be too complicated to be accomplished in the near future. However, an identification of the $\alpha^{8}\,m$ contribution from experiments on light He-like ions and rescaling it to helium could provide an improvement of the theoretical precision and, therefore, the accuracy of the helium $\alpha$ determination.

V Isotope shift

The isotope shift is defined, for the spinless nuclei, as the difference of the transition frequencies of different isotopes of the same element. For the 4He and 3He isotopes, however, the comparison of the spectra is complicated by the presence of the nuclear spin in 3He and, as a consequence, by a large mixing of the fine and hyperfine sublevels. In order to separate out the effects of the nuclear spin in 3He, the isotope shift of the $2S$ and $2P$ levels is defined heis as the shift of the centroid energies, which are the average over all fine and hyperfine energy sublevels,

[TABLE]

where ${}^{2S+1}L_{J,F}$ denotes the state with electron angular momentum $L$, spin $S$, and total momentum $J$, whereas $F$ is the total momentum of the atom. The theory of the helium isotope shift was described in detail in our recent investigation heis , so it will not be repeated here.

A remarkable feature of the isotope shift is that the relative contribution of the finite nuclear size effect to it is much larger than that to the transition energies. In particular, for the $2^{3}S-2^{3}P$ transition energy, the finite nuclear size correction is only a $5\times 10^{-9}$ effect, while for the isotope shift it becomes as large as $4\times~{}10^{-5}$. Because of this, the $2^{3}S-2^{3}P$ transition is particularly suitable for determinations of the nuclear radii differences from the isotope shift.

The present theoretical accuracy for the isotope shift of the $2S$ and $2P$ states of helium is at a sub-kHz level heis ; patkos1 ; patkos2 , which enables precise determinations of the nuclear charge radius difference of 4He and 3He isotopes. Because the theory is supposed to be so very accurate, the precision of these determinations is orders of magnitude higher than the traditional determinations by means of the electron scattering sick , and it is limited only by the uncertainty of the frequency measurements.

Table VI reports the present status of the determinations of the 3He–4He nuclear charge radii difference $\delta r^{2}$ from the isotope shift of the $2^{3}S$–$2^{3}P$ and $2^{3}S$–$2^{1}S$ transitions, as obtained in different experiments. Surprisingly, these two transitions lead patkos1 ; patkos2 to contradicting results for $\delta r^{2}$; i.e. $1.069\,(3)$ fm2 and $1.061\,(3)$ fm2 from the $2^{3}S$–$2^{3}P$ transition versus $1.027\,(11)$ fm2 from the $2^{3}S$–$2^{1}S$ transition. Obviously, the nuclear charge radius has to be the same, provided that no new physics is involved. The numerically dominating part of the theoretical predictions for the isotope shift is verified by checking against independent calculations by G. Drake and co-workers morton:06 ; drake:10 (see the comparison in Tables 1 and 2 of Ref. heis ). The difference in the calculations is 2 kHz for the $2^{3}S$–$2^{1}S$ and 3 kHz for the $2^{3}S$–$2^{3}P$ calculations and cannot explain the 4$\sigma$ discrepancy between the results for $\delta r^{2}$. This unexplained discrepancy calls for the verification of the experimental results; first of all, the $2^{3}S$–$2^{1}S$ transition, for which only one measurement has been reported in the literature.

VI Summary

We have examined the current status of the theory and the experiment for the energies, the fine structure, and the isotope shift of the lowest-lying states of helium. The comparison of theoretical predictions and experimental results does not indicate significant discrepancies, apart from the one for the isotope shifts between 4He and 3He, which remains to be confirmed. With a further improvement of theory, i.e. with a calculation of the $\alpha^{7}\,m$ correction, it will be possible to determine the nuclear charge radii in He and He-like ions. Such a determination, combined with a complementary determination from muonic atoms, will provide a sensitive test of universality in electromagnetic interactions of leptons.

Acknowledgements.

This work was supported by the National Science Center (Poland) Grant No. 2012/04/A/ST2/00105. V.A.Y. acknowledges support by the Ministry of Education and Science of the Russian Federation Grant No. 3.5397.2017/BY.