Uehling correction in muonic atoms to all orders of $Z\alpha$

The Uehling correction to the energy levels is presented in terms of the hypergeometric functions ${{}_{2}F}_1$. This presentation allows to derive various asymptotics and approximations. Further applications of this method to other atomic characteristics are also considered.

Figure 1

Comparison of $K_{bc}\left({\kappa }_n\right)$ (lower panels) with its upper and lower estimations (middle panels) $C{Q}_{bc}$, with $C_{\text{up}}=\sqrt{2}$ and $C_{\text{lo}}=2/3$ [Eq. (8)], and the approximation ${\widehat{K}}_{bc}\left({\kappa }_n\right)$ [Eq. (11)] (upper panels) for the case of circular states $2p_{3/2}$, $3d_{5/2}$ and $4f_{7/2}$, for which energies are expressed via a single integral $K_{2,2\zeta }\left({\kappa }_n\right)$ [Eq. (12)].

Figure 2

Comparison of the convergence of the hypergeometric function in relations (9, 13, 15) for the $1s$ state: (a)—the exact function $Q_{bc}\left({\kappa }_1\right)$, (b)—the first five terms of expansion of ${{}_{2}F}_1$ in Eq. (9), (c)—the same for Eq. (13), (d)—the same for Eq. (15).

Figure 3

Comparison of the convergence of the hypergeometric function in relations (9, 13, 15) for the circular state with $n=10$: (a)—the exact function $Q_{bc}\left({\kappa }_{10}\right)$, (b)—the first five terms of expansion of ${{}_{2}F}_1$ in Eq. (9), (c)—the same for Eq. (13), (d)—the same for Eq. (15).

Figure 4

Relative errors for the hypergeometric function expansions in relations (13) (top) and (15) (bottom) as compared with the exact value for different number of nontrivial terms in the expansion (from bottom to up: 1,2,3,5 terms). The results are for the $1s$ state.

Figure 5

Relative errors for the hypergeometric function expansions in relations (13) (top) and (15) (bottom) as compared with the exact value for different number of nontrivial terms in the expansion (from bottom to up: 1,2,3,5 terms). The results are for the circular state with $n=10$.

Figure 6

Top: plots of $f\left(y\right)$ [Eq. (5)] (bold line), its partial sums with first the first three terms for series (21, 22) (thin lines) and the linear approximation (27) (dashes). The maximal and minimal values of $f\left(y\right)$ are denoted by the dotted lines. Bottom: the relative errors of the three-term partial sum (thin lines) and the linear approximation (dashes).

Figure 7

Top: the relative error of the approximative functions $K_{2,2n}\left({\kappa }_n\right)$, corresponding to function $\tilde{f}\left(y\right)=f_1\left(y\right)$, for $n=1$ (solid line) and $n=100$ (dots). Bottom: the same for function $\tilde{f}\left(y\right)=f_3\left(y\right)$.

Figure 8

Comparison of different approximations of Eq. (31) for $3d_{5/2}$ state: (a)—the relation (31), (b)—the first five terms of expansion for Eq. (9), (c)—the same for Eq. (13), (d)—the same for Eq. (15). Solid lines correspond to $Z=1$, dotted lines to $Z=92$.

Figure 9

Relative errors of different approximations of Eq. (31) for $3d_{5/2}$ state. (c1)—the first five terms of expansion for Eq. (13) and ${\tilde{\kappa }}_3=0.3$, (c2)—the same for ${\tilde{\kappa }}_3=0.9$, (d3)—the same for Eq. (15) and ${\tilde{\kappa }}_3=8$. The relative errors of the other approximations ($a$, $b$, $c$ at ${\tilde{\kappa }}_3=0.3,0.9,8$, which are not shown, are above 40%).

Figure 10

The relative error of $\delta {\Psi }_{1s}\left(0\right)/{\Psi }_{1s}\left(0\right)$ approximation as a function of ${\kappa }_1$ for $f=f_0$ (top) and $f=f_1$ (bottom) from Table .