Two-Loop Bethe logarithms for higher excited $S$ levels

Processes mediated by two virtual low-energy photons contribute quite significantly to the energy of hydrogenic $S$ states. The corresponding level shift is of the order of ${(\alpha ∕\pi )}^2{\left(Z\alpha \right)}^6{m}_{\mathrm{e}}{c}^2$ and may be ascribed to a two-loop generalization of the Bethe logarithm. For $1S$ and $2S$ states, the correction has recently been evaluated by Pachucki and Jentschura [Phys. Rev. Lett. 91, 113005 (2003)]. Here, we generalize the approach to higher excited $S$ states, which in contrast to the $1S$ and $2S$ states can decay to $P$ states via the electric-dipole $\left(E1\right)$ channel. The more complex structure of the excited-state wave functions and the necessity to subtract $P$-state poles lead to additional calculational problems. In addition to the calculation of the excited-state two-loop energy shift, we investigate the ambiguity in the energy level definition due to squared decay rates.

Figure 1

Two-photon processes may be interpreted in terms of Feynman diagrams. The double line denotes the bound-electron propagator. In the figure, we display the crossed-loop (A), the rainbow (B), and the loop-after-loop (C) diagram.

Figure 2

(Color online) Hydrogenic $S$ levels have the same (radial) symmetry properties as the ground state. The wave function $\psi \left(r\right)$ of an $S$ level therefore depends only on the radial coordinate $r\equiv \sqrt{x^2+y^2+z^2}$. The function $f_{6\mathrm{S}}(x,y)\equiv {\int }_{-\infty }^{\infty }\mathrm{d}z{\mid {\psi }_{6\mathrm{S}}(x,y,z)\mid }^2$ is positive definite and constitutes effectively an integrated projection of the $6S$ electron probability density onto the $x$-$y$ plane. Indeed, we plot here the natural logarithm of this function, which is $\ln \left[f_{6\mathrm{S}}\right(x,y\left)\right]$, as a function of $x∊[-40a_{\text{Bohr}},40a_{\text{Bohr}}]$ and $y∊[-40a_{\text{Bohr}},40a_{\text{Bohr}}]$. Here, $a_{\text{Bohr}}$ denotes the Bohr radius $a_{\text{Bohr}}=\hslash ∕\left(\alpha mc\right)=\mathrm{0.529\hspace{0.17em}177\hspace{0.17em}2108}\left(18\right)\times {10}^{-10}\mathrm{m}$ 8.

Figure 3

(Color online) Plot of the large-${\omega }_1^{\prime }$ asymptotics of $g$ [see Eq. (29)] against numerical data obtained for $g_{3S}\left({\omega }_1^{\prime }\right)$ in the range ${\omega }_1^{\prime }∊[20,180]$. The numerical data are scaled by a factor of $3^3=27$. See also Table and Eq. (25a) where $g$ is defined. Dimensionless quantities are displayed in the figure [this statement relates to both the abscissa as well as the ordinate axis; see also Eq. (16a)].

Figure 4

(Color online) Large-${\omega }_1^{\prime }$ asymptotics of $g$ plotted against numerical data for the $6S$ state in the range ${\omega }_1^{\prime }∊[20,180]$. The numerical data are scaled by a factor of $6^3=216$. Explicit numerical sample values for $g_{6S}\left({\omega }_1^{\prime }\right)$ can also be found in Table . The apparent similarity of Figs. 3 and 4 is reflected in the scaled entries in this table. The difference between the numerical data (solid line) and the asymptotics (dashed line) is negative. The formula for the large-${\omega }_1^{\prime }$ asymptotics of $g$ is given in Eqs. (29, 30). The difference between the numerical data and the asymptotics gives rise to a negative contribution to the integral $J_2$ defined according to Eq. (33b) and to a negative value for the $B_{60}$ coefficient [see Eq. (52) below]. The scaled, primed quantities plotted along both the abscissa as well as the ordinate axis are dimensionless [see also Eq. (16a)].

Figure 5

Dependence of the two-loop Bethe logarithm $b_L\left(nS\right)$ on the principal quantum number $n$. The explicit numerical results [Eq. (34)] are displayed together with a three-parameter fit of the form $-57.4-13.7∕n-10.1∕n^2$ from which one may infer ${\lim }_{n\to \infty }{b}_L\left(nS\right)=-57\left(2\right)$. Quantities plotted along the abscissa and the ordinate axis are (of course) dimensionless.