---
abstract: 'A nonperturbative numerical evaluation of the one-photon electron self energy for the $3S$ and $4S$ states with charge numbers $Z=1$ to 5 is described. The numerical results are in agreement with known terms in the expansion of the self energy in powers of $Z\alpha$.'
author:
- 'Ulrich D. Jentschura'
- 'Peter J. Mohr'
title: Electron Self Energy for Higher Excited $S$ Levels
---

In this brief report, we consider the one-loop self-energy shift which is the dominant radiative correction to the energy of hydrogenic bound states. For high-precision spectroscopy, $S$ states are rather important because they can be excited from the ground state via Doppler-free two-photon spectroscopy. We calculate the self-energy numerically to high accuracy for $3S$ and $4S$ states (nuclear charge number $Z=1,\dots,5$). We follow the approach previously outlined for $1S$ (Ref. [@JeMoSo1999]) and $2S$ and $2P$ states (Ref. [@JeMoSo2001pra]).

The natural unit system with $\hbar = c = m_{\rm e} = 1$ and $e^2 = 4\pi\alpha$ is employed, as is customary in bound-state quantum electrodynamics. The (real part of the) energy shift $\Delta E_{\rm SE}$ due to the electron self-energy radiative correction is usually written as [@SaYe1990] $$\label{ESEasF}
\Delta E_{\rm SE} = \frac{\alpha}{\pi} \, 
\frac{(Z \alpha)^4 \, m_{\rm e}}{n^3} \, 
F(nl_j,Z\alpha) \,,$$ where $F$ is a dimensionless quantity. In writing the expression $F(nl_j,Z\alpha)$, we follow the usual spectroscopic notation for an atomic state with principal quantum number $n$, orbital angular momentum $l$ and total electron angular momentum $j$.

[l@r@[.]{}l@r@[.]{}l]{} $Z$ & &\

------------------------------------------------------------------------

$1$ & $10$ & $605~614~22(5)$ & $-31$ & $047~7(9)$\

------------------------------------------------------------------------

$2$ & $8$ & $817~615~14(5)$ & $-30$ & $512~6(2)$\

------------------------------------------------------------------------

$3$ & $7$ & $794~461~17(5)$ & $-30$ & $022~7(1)$\

------------------------------------------------------------------------

$4$ & $7$ & $083~612~42(5)$ & $-29$ & $564~53(6)$\

------------------------------------------------------------------------

$5$ & $6$ & $543~385~98(5)$ & $-29$ & $130~61(4)$\

The leading terms in the semi-analytic expansion of $F(n{S}_{1/2},Z\alpha)$ about $Z\alpha=0$ read $$\begin{aligned}
\label{defFLOnS}
\lefteqn{F(n{S}_{1/2},Z\alpha) = 
A_{41}(n{S}_{1/2}) \, \ln(Z \alpha)^{-2}} \nonumber\\[2ex]
& & \;\; + A_{40}(n{S}_{1/2}) +
(Z \alpha) \, A_{50}(n{S}_{1/2}) \nonumber\\[2ex]
& & \;\; + \, 
(Z \alpha)^2 \, 
\left[A_{62}(n{S}_{1/2}) \, \ln^2(Z \alpha)^{-2} \right. 
\nonumber\\[2ex]
& & \;\;\;\; \left. + A_{61}(n{S}_{1/2}) \,\ln(Z \alpha)^{-2} + 
G_{\rm SE}(n{S}_{1/2},Z\alpha) \right]\,.\end{aligned}$$ The $A$ coefficients have two indices, the first of which denotes the power of $Z\alpha$ \[including those powers implicitly contained in Eq. (\[ESEasF\])\], while the second index denotes the power of the logarithm $\ln(Z \alpha)^{-2}$.

We now list the analytic coefficients and the Bethe logarithms relevant to the atomic states under investigation [@Be1947; @Fe1948; @Fe1949; @FrWe1949; @KrLa1949; @Sc1949; @FuMiTo1949; @Ba1951; @KaKlSc1952; @BaBeFe1953; @FrYe1958; @FrYe1960; @La1960; @La1961a; @La1961b; @ErYe1965a; @ErYe1965b; @Er1971; @Sa1981; @Pa1993],

$$\begin{aligned}
\label{coeffs1S12}
A_{41}(n{S}_{1/2}) & = & \frac{4}{3}\,, \\[1ex]
A_{40}(n{S}_{1/2}) & = & 
  \frac{10}{9} - \frac{4}{3} \, \ln k_0(n{S})\,, 
\\[1ex]
A_{50}(n{S}_{1/2}) & = & 
4\pi\,\left[\frac{139}{128} - \frac{1}{2}\,\ln 2\right]\,,
\\[1ex]
A_{62}(n{S}_{1/2}) & = & -1\,.\end{aligned}$$

$A_{61}$-coefficients read

$$\begin{aligned}
A_{61}(1{S}_{1/2}) & = & \frac{21}{20} + \frac{28}{3} \, \ln 2 \,,\\[1ex]
A_{61}(2{S}_{1/2}) & = & \frac{67}{30} + \frac{16}{3} \, \ln 2 \,,\\[1ex]
A_{61}(3{S}_{1/2}) & = & \frac{6163}{1620} + \frac{28}{3} \, \ln 2 - 4\,\ln 3\,,
\\[1ex]
A_{61}(4{S}_{1/2}) & = & \frac{4}{3} \, \ln 2 + \frac{391}{80}\,.\end{aligned}$$

The Bethe logarithms $\ln k_0(n{S})$ are known [@KlMa1973; @BeBrSt1950; @Ha1956; @ScTi1959; @Li1968; @Hu1969], and we here present a re-evaluation,

$$\begin{aligned}
\label{BetheLog1S}
\ln k_0(1{S}) &=& 2.984~128~555~765~497~611(1)\,,\\[1ex]
\label{BetheLog2S}
\ln k_0(2{S}) &=& 2.811~769~893~120~563~520(1)\,,\\[1ex]
\label{BetheLog3S}
\ln k_0(3{S}) &=& 2.767~663~612~491~821~190(1)\,,\\[1ex]
\label{BetheLog4S}
\ln k_0(4{S}) &=& 2.749~811~840~454~057~422(1)\,.\end{aligned}$$

The results for $1S$, $2S$ and $3S$ are in agreement with values indicated for the “logarithmic sum” $\beta_1$ in [@GoDr2000 Tab. III]. Notice that the numerical difference of our above results for $\ln k_0$ in comparison to the results indicated for $\beta_1$ in [@GoDr2000] is entirely due to an additional contribution $\ln 2$ which is added to the Bethe logarithm of $S$ states according to the somewhat non-standard convention for $\beta_1$ used in [@GoDr2000].

[l@r@[.]{}l@r@[.]{}l]{} $Z$ & &\

------------------------------------------------------------------------

$1$ & $10$ & $629~388~4(2)$ & $-30$ & $912(4)$\

------------------------------------------------------------------------

$2$ & $8$ & $841~324~1(2)$ & $-30$ & $380~0(9)$\

------------------------------------------------------------------------

$3$ & $7$ & $818~078~5(2)$ & $-29$ & $892~4(4)$\

------------------------------------------------------------------------

$4$ & $7$ & $107~116~6(2)$ & $-29$ & $437~1(2)$\

------------------------------------------------------------------------

$5$ & $6$ & $566~758~8(2)$ & $-29$ & $006~0(2)$\

The evaluation of the coefficient $$A_{60}(nS_{1/2}) \equiv \lim_{Z\alpha \to 0} G_{\rm SE}(nS_{1/2},Z\alpha)$$ has been historically problematic [@ErYe1965a; @ErYe1965b; @Er1971; @Sa1981; @Pa1993], and it has therefore been a considerable challenge to reliably estimate the self-energy remainder function $G_{\rm SE}$, especially in the range of low nuclear charge number $Z$. Our calculation of the nonperturbative (in $Z\alpha$) electron self-energy for the $3{S}_{1/2}$ state (see Table \[tableF3S\]) has a numerical uncertainty of 2 Hz in atomic hydrogen. For the $4{S}_{1/2}$ state, the numerical uncertainty is 3$\times Z^4$ Hz (see Table \[tableF4S\]). The value of the fine-structure constant $\alpha$ employed in the calculation is $\alpha^{-1} = 137.036$; this is close to the 1998 and 2002 CODATA recommended values [@MoTa2000; @MoTaPriv2004]. The entries for the self-energy remainder function $G_{\rm SE}$ in Tables \[tableF3S\] and \[tableF4S\] are in agreement with those used in the latest adjustment of the fundamental physical constants [@MoTaPriv2004] (the $G_{\rm SE}$-values used in [@MoTaPriv2004] are based on an extrapolation of numerical data previously obtained [@MoKi1992] for higher nuclear charge numbers). Our all-order evaluation eliminates any uncertainty due to the unknown higher-order analytic terms that contribute to the bound electron self-energy of $3S$ and $4S$ states \[see Eq. (\[defFLOnS\])\]. This improves our knowledge of the spectrum of hydrogenlike atoms (e.g. atomic hydrogen, He$^+$).\
[**Acknowledgments.**]{} U. D. J. thanks the National Institute of Standards and Technology for kind hospitality during a number of extended research appointments. The authors acknowledge E.-O. LeBigot for help in obtaining numerical results for selected partial contributions to the electron self-energy, for the hydrogenic energy levels discussed in this work.

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