# Impact of the nuclear charge distribution on the $g$-factors and ground
state energies of bound muons

Atakan Çavuşoğlu  Max Planck Institute for Nuclear Physics, Saupfercheckweg 1,
69117 Heidelberg, Germany  Department of Physics, Boğaziçi University,
Istanbul, Turkey  Department of Electrical and Electronics Engineering,
Boğaziçi University, Istanbul, Turkey  Bastian Sikora  Max Planck Institute
for Nuclear Physics, Saupfercheckweg 1, 69117 Heidelberg, Germany

(November 28, 2023)

###### Abstract

The finite nuclear size corrections to the ground state energies and
$g$-factors in muonic atoms are investigated for several elements. Both
approximative and exact solutions of the one-particle Dirac equation with both
the homogeneous sphere nucleus model and the Fermi distribution nucleus model
are presented, and the leading nuclear deformation effects on the $g$-factors
and ground state energies are also evaluated. The electronic, muonic, and
hadronic electric-loop vacuum polarization corrections are calculated for
point-like, spherical, and Fermi nucleus models. The obtained results show a
heavy dependence on the chosen nuclear model, and highlight the importance of
constructing precise theoretical models for the nucleus for accurate QED
predictions of the observables of muonic atoms.

††preprint: APS/123-QED

##  I Introduction

In comparison to the electron, the muon’s $\sim$207 times heavier mass renders
bound muonic systems an ideal testing field for investigating the nuclear
effects on bound particles. Qualitatively, the orbital radii of bound
particles are inversely proportional to their mass [1], which, in the case of
the bound muon, causes the muonic orbit radii to be comparable to the nuclear
radii, which significantly amplifies the nuclear effects on the properties of
the bound muon in comparison to the electron, especially for muons bound to
heavier nuclei.

Nuclear effects on the $g$-factors of bound electrons have been a major
theoretical [2, 3, 4, 5, 6, 7, 8, 9, 10] and experimental [11, 12, 13, 14, 15,
16, 17] research interest in the past years, and significant progress has been
made in determining the nuclear contributions to the energy spectra of muonic
atoms [18, 19, 20, 21, 22, 23, 24, 25], which has been a major tool in
determining the properties of various nuclei [26, 27, 28, 29, 30, 31].
Nevertheless a thorough investigation of the nuclear corrections to the bound
muon $g$-factor still remains untouched, with the exception of the very light
muonic 4He+ ion [32].

In the present work, we investigate both the direct nuclear effects on the
ground state energies and $g$-factors in muonic atoms, as well as the nuclear
corrections to the one-loop vacuum polarization contributions. We consider
heomogeneous sphere, two-parameter Fermi, and deformed Fermi charge
distribution models for the nucleus, and we calculate the first-order
electronic, muonic, and hadronic electric-loop vacuum-polarization
corrections. Throughout this work we use muonic natural units with
$m_{\mu}=c=\hbar=1$ and the charge units $\alpha=e^{2}/4\pi$.

##  II Dirac Equation for Central Potentials

The stationary states of a bound fermion can be obtained from the solutions of
the time independent Dirac equation, which is the relativistic wave equation
for spin-1/2 particles:

$\left[{\bf\it\alpha}\dotproduct\mathbf{p}+M\beta+V(\mathbf{r})\right]\psi(\mathbf{r})=E\psi(\mathbf{r}),$
(1)

where ${\bf\it\alpha}$ and $\beta$ are Dirac matrices and $M$ is the mass of
the bound particle. The eigenvalues $E$ of this equation are the bound state
energies, and the eigenfunctions $\psi(\mathbf{r})$ are bound state
wavefunctions.

In the case of a spherically symmetric potential, i.e. $V(\mathbf{r})=V(r)$,
the wavefunctions can be separated into radial and angular components [1]

$\psi(\mathbf{r})=\begin{pmatrix}\frac{i}{r}G(r)\Omega_{\kappa
m_{j}}(\theta,\phi)\\\ -\frac{1}{r}F(r)\Omega_{-\kappa
m_{j}}(\theta,\phi)\end{pmatrix},$ (2)

where $\kappa$ is the relativistic angular momentum quantum number with
$\kappa=-l-1$ for $j=l+1/2$ and $\kappa=l$ for $j=l-1/2$, and $m_{j}$ is the
z-component of the total angular momentum $\mathbf{J}=\mathbf{L}+\mathbf{S}$.
In the case of the $1s$ (ground) state, $\kappa=-1$ and $m_{j}=1/2$.
$\Omega_{\kappa m_{j}}$ are the spherical spinors. $G(r)$ and $F(r)$ are the
solutions of the radial part of the Dirac equation (1):

$\begin{split}\derivative{G(r)}{r}+\frac{\kappa}{r}G(r)-(M-V(r))F(r)&=EF(r),\\\
-\derivative{F(r)}{r}+\frac{\kappa}{r}F(r)+(M+V(r))G(r)&=EG(r).\end{split}$
(3)

In the case of the point-like nucleus, the potential is the Coulomb potential

$V_{\text{C}}(r)=-\frac{Z\alpha}{r},$ (4)

and the eigenfunctions and eigenvalues of the radial Dirac equation (3) are
known analytically (see [1]). The energy eigenvalues express the fine
structure of the energy spectra of atoms:

$E_{\text{C}}=\\\
M\left[1+\frac{(Z\alpha)^{2}}{\left[n-j-\frac{1}{2}+\left[\left(j+\frac{1}{2}\right)^{2}-(Z\alpha)^{2}\right]^{1/2}\right]^{2}}\right]^{-1/2}$
(5)

###  II.1 $g$-Factor of a Bound Dirac Particle

The so-called $g$-factor characterizes the coupling strength of a particle to
an external magnetic field. For a bound Dirac particle in a state
$\ket{\psi(\mathbf{r})}$, to first order in perturbation theory, it is defined
by the relation

$\frac{e\mathbf{B}\dotproduct\expectationvalue{\mathbf{J}}}{2M}g=e\bra{\psi(\mathbf{r})}{\bf\it\alpha}\dotproduct\mathbf{A}\ket{\psi(\mathbf{r})},$
(6)

where $\mathbf{A}$ is the vector potential corresponding to the constant
magnetic field $\mathbf{B}$, i.e.
$\mathbf{A}=1/2(\mathbf{B}\crossproduct\mathbf{r})$. For a spherically
symmetric potential and a particle in a stationary state, using the properties
of the spherical spinors, it can be shown that (see [33])

$g=\frac{2M\kappa}{j(j+1)}\int\differential{r}rG(r)F(r),$ (7)

which, by using the radial Dirac equations (3) and the mass derivative of the
Hamiltonian $\partialderivative*{H}{M}=\beta$, can be expressed as [34]

$g=-\frac{\kappa}{2j(j+1)}\left[1-2\kappa\partialderivative{E}{M}\right],$ (8)

for any central potential that is independent of the mass of the particle. For
a particle in the ground state, $\kappa=-1$ and $j=1/2$, so the formula
simplifies to

$g=\frac{2}{3}\left(1+2\partialderivative{E}{M}\right).$ (9)

If we express any central potential as a deviation from the Coulomb potential

$V(r)=V_{C}(r)+\Delta V(r),$ (10)

the ground state energy for this potential can also be expressed as a
deviation from the Coulomb ground state energy

$E=E_{C}+\Delta E,$ (11)

with $E_{C}=M\sqrt{1-(Z\alpha)^{2}}$. As a result, the deviations of the
$g$-factors from the Coulomb $g$-factor can also simply be expressed as

$g_{C}+\Delta
g=\frac{2}{3}\left(1+2\sqrt{1-(Z\alpha)^{2}}\right)+\frac{4}{3}\partialderivative{\Delta
E}{M},$ (12)

and

$\Delta g=\frac{4}{3}\partialderivative{\Delta E}{M}.$ (13)

##  III Finite Nuclear Size Corrections

One of the simplest nuclear models that take the finite size of the nucleus is
the homogeneously charged sphere, with the charge distribution

$\rho_{\text{Sphere}}(r)=\begin{cases}\frac{3Ze}{4\pi r_{0}^{3}}&\text{if
}r\leq r_{0}\\\ 0&\text{if }r>r_{0}\end{cases}$ (14)

where $r_{0}$ is the radius of the nucleus, related to the root-mean-squared
radius of the nucleus by [2]

$r_{0}=\sqrt{\frac{5}{3}}r_{\text{rms}}.$ (15)

For the homogeneously charged sphere, the potential outside the nucleus is the
same as the Coulomb potential (4), but the potential inside the nucleus is
different: [20]

$V(r)=\begin{cases}-\frac{Z\alpha}{2r_{0}}\left(3-\frac{r^{2}}{r_{0}^{2}}\right)&\text{if
}r\leq r_{0}\\\ -\frac{Z\alpha}{r}&\text{if }r>r_{0}\end{cases}$ (16)

The radial Dirac equation (3) can be solved for the region inside the nucleus
($r\leq r_{0}$) in power series, and since the potential (16) is a polynomial
in $r$, a simple recursion relation can be obtained for the coefficients of
the power series. The wavefunction inside the nucleus is [20]

$\begin{pmatrix}G_{r<r_{0}}(r)\\\
F_{r<r_{0}}(r)\end{pmatrix}=N_{1}r^{\absolutevalue{\kappa}}\sum_{i=0}^{\infty}\left[a_{i}\pm(-1)^{i+1}\frac{\kappa}{\absolutevalue{\kappa}}a_{i}\right]r^{i},$
(17)

with the recursion relation

$a_{i}=\frac{a_{i-1}\left[E+\frac{3Z\alpha}{2r_{0}}-M(-1)^{i}\frac{\kappa}{\absolutevalue{\kappa}}\right]-\frac{Z\alpha}{2r_{0}^{3}}a_{i-3}}{\kappa+(-1)^{i+1}\frac{\kappa}{\absolutevalue{\kappa}}\left(i+\absolutevalue{\kappa}\right)}.$
(18)

Since the potential outside the nucleus is the same as the Coulomb potential,
the solution of the Dirac equation in this region follows similarly to the
solution for the point nucleus (see [1]), with the difference being that the
solutions singular at $r=0$ are also allowed. For a given E, the radial Dirac
equation (3) has two linearly independent solutions, but because of the
normalizability of the wavefunctions, only the specific linear combinations
that are regular at infinity are allowed. This condition reduces the number of
possible solutions to one, and the wavefunction outside the nucleus can be
written as [20]

$\begin{pmatrix}G_{r>r_{0}}(r)\\\
F_{r>r_{0}}(r)\end{pmatrix}=\frac{N_{2}}{\kappa+\frac{MZ\alpha}{\lambda}}\left(2\lambda
r\right)^{-\frac{1}{2}}\sqrt{M\pm E}\\\
\times\left[\left(\kappa+\frac{MZ\alpha}{\lambda}\right)W_{q,\gamma}(2\lambda
r)\pm W_{q+1,\gamma}(2\lambda r)\right],$ (19)

where $\lambda=\sqrt{M^{2}-E^{2}}$, $q=Z\alpha E/\lambda-1/2$,
$\gamma=\sqrt{\kappa^{2}-(Z\alpha)^{2}}$, and $W_{a,b}(x)$ are the Whittaker
functions of the second kind [35].

The quantization of the energy is obtained from the continuity of the
wavefunction at the edge of the nucleus,

$\begin{pmatrix}G_{r<r_{0}}(r_{0})\\\
F_{r<r_{0}}(r_{0})\end{pmatrix}=\begin{pmatrix}G_{r>r_{0}}(r_{0})\\\
F_{r>r_{0}}(r_{0}),\end{pmatrix}$ (20)

or, alternatively,

$\frac{G_{r<r_{0}}(r_{0})}{F_{r<r_{0}}(r_{0})}=\frac{G_{r>r_{0}}(r_{0})}{F_{r>r_{0}}(r_{0})}$
(21)

which can be solved numerically to obtain the energy eigenvalues. After the
eigenvalues are found, the normalization constants $N_{1}$ and $N_{2}$ in
equations (17) and (19) can also be calculated numerically from the
normalization condition of the wavefunction and the continuity condition at
$r=r_{0}$.

After the ground state energy is found, it is also straightforward to find the
finite size correction to the $g$-factor. By taking the mass derivative of the
quantization condition in equation (21) and using equation (13), we find

$\partialderivative{E_{Sph}}{M}=-\frac{\partialderivative{M}(\frac{G_{r<r_{0}}(r_{0})}{F_{r<r_{0}}(r_{0})}-\frac{G_{r>r_{0}}(r_{0})}{F_{r>r_{0}}(r_{0})})}{\partialderivative{E}(\frac{G_{r<r_{0}}(r_{0})}{F_{r<r_{0}}(r_{0})}-\frac{G_{r>r_{0}}(r_{0})}{F_{r>r_{0}}(r_{0})})}$
(22)

and

$\Delta
g_{\text{Sph}}=-\frac{4}{3}\frac{\partialderivative{M}(\frac{G_{r<r_{0}}(r_{0})}{F_{r<r_{0}}(r_{0})}-\frac{G_{r>r_{0}}(r_{0})}{F_{r>r_{0}}(r_{0})})}{\partialderivative{E}(\frac{G_{r<r_{0}}(r_{0})}{F_{r<r_{0}}(r_{0})}-\frac{G_{r>r_{0}}(r_{0})}{F_{r>r_{0}}(r_{0})})}-\frac{4}{3}\sqrt{1-(Z\alpha)}.$
(23)

Having analytical expressions for the wavefunctions in the case of the
homogeneous sphere model allows us to take the mass and energy derivatives
analytically, and the corrections to the $g$-factors can be calculated
exactly, without having to normalize the wavefunctions or take integrals of
the form (7).

###  III.1 Deviations from the Sphere Model

Although the sphere model for the nucleus allows simple calculations of the
finite nuclear size effects on the energy levels and $g$-factors, it is
important to investigate the dependence of finite size corrections on the
specific nuclear model used, especially for muonic atoms, such as more
realistic charge distributions like the Fermi distribution or deformed Fermi
distribution. Furthermore, quantum electrodynamic (QED) effects that can be
expressed as an effective potential, such as one-loop vacuum polarization
corrections, need to be investigated for extended nuclei rather than point-
nuclei. As we will show in this paper, in the case of muonic atoms, the
corrections due to these effects can be larger than the uncertainties to the
finite size corrections due to the rms radius uncertainties, and therefore
require a detailed consideration and understanding.

A method for estimating corrections for a wide range of nuclear charge
distributions, by finding an effective radius and using the sphere model with
that radius, was proposed in [36], and this method was applied to electronic
atoms for calculating the finite nuclear size effects and nuclear deformation
effects [34, 37, 38, 39]. But since the formulas derived in [36] are based on
the approximation that the nuclear radius is much smaller than the Bohr radius
of the bound particle, they are not applicable to muonic atoms, and other
approximation methods need to be used.

Here, we present a discussion of different methods for handling effects that
can be represented as a correction to the sphere model potential, in which
case the potential becomes

$V(r)=V_{\text{Sph}}(r)+\delta V(r)$ (24)

In the following discussion, uppercase $\Delta$ will be used to denote the
leading finite size correction to the point-nucleus case, and lowercase
$\delta$ will be used to denote the corrections to the extended nucleus
quantities.

####  III.1.1 First Order Perturbation Theory

If $\delta V(r)$ is small in comparison to $V_{\text{Sph}}(r)$, it can be
regarded as a perturbation and first order perturbation theory can be applied,
as

$\begin{split}\delta
E_{\text{pert}}&=\bra{\psi_{\text{Sph}}(\mathbf{r})}\delta
V(r)\ket{\psi_{\text{Sph}}(\mathbf{r})}\\\
&=\int\differential{r}\left(G_{\text{Sph}}(r)^{2}+F_{\text{Sph}}(r)^{2}\right)\delta
V(r).\end{split}$ (25)

The perturbation integral can be evaluated numerically.

For calculating the corrections to the $g$-factor, equation (13) can be used.
If $\delta V(r)$ is independent of the mass of the bound particle, the
derivative with respect to mass only affects the wavefunctions:

$\partialderivative{\delta
E_{\text{pert}}}{M}=2\frac{\bra{\psi_{\text{Sph}}}\delta
V(r)\partialderivative{M}\ket{\psi_{\text{Sph}}}}{\bra{\psi_{\text{Sph}}}\ket{\psi_{\text{Sph}}}}\\\
-2\frac{\bra{\psi_{\text{Sph}}}\delta
V(r)\ket{\psi_{\text{Sph}}}\bra{\psi_{\text{Sph}}}\partialderivative{M}\ket{\psi_{\text{Sph}}}}{\left[\bra{\psi_{\text{Sph}}}\ket{\psi_{\text{Sph}}}\right]^{2}}$
(26)

This expression is valid for normalized, as well as for unnormalized
wavefunctions, which avoids the calculation of the derivative of the
normalization constant. In the derivatives of the wavefunctions, the
derivatives of $E$ needs to be taken into account as well, which can be
calculated from equation (22). Afterwards, the correction to the $g$-factor is
simply

$\delta g_{\text{pert}}=\frac{4}{3}\partialderivative{\delta
E_{\text{pert}}}{M}.$ (27)

####  III.1.2 Variational Approximation

Another approximation method is the variational method that is based on the
fact that the expectation value of the Hamiltonian in any state other than the
ground state is always greater than the ground state energy [40]. This fact is
exploited by choosing a trial wavefunction that is dependent on one or more
parameters, and minimizing the expectation value of the Hamiltonian in that
state with respect to the free parameters. In our case, we can choose the
trial wavefunction as the ground state of the sphere nucleus Hamiltonian, with
the radius of the nucleus left as a free parameter with respect to which the
expectation value of the true Hamiltonian is to be minimized.

Specifically, the Hamiltonian for the sphere nucleus potential, with variable
radius $R$, is

$H_{\text{Sph},R}={\bf\it\alpha}\dotproduct\mathbf{p}+M\beta+V_{\text{Sph},R}(r),$
(28)

and the ground state energy depends on $R$. The exact Hamiltonian of the
system can be written as

$H=H_{\text{Sph},R}+H^{\prime}_{R},$ (29)

with

$H^{\prime}_{R}=V(r)-V_{\text{Sph},R}(r).$ (30)

We are looking to minimize $\expectationvalue{H}$, which is

$\begin{split}\expectationvalue{H}&=\bra{\psi_{\text{Sph},R}}\left(H_{\text{Sph},R}+H^{\prime}_{R}\right)\ket{\psi_{\text{Sph},R}}\\\
&=E_{\text{Sph},R}+\int\differential{r}\left(G_{\text{Sph},R}(r)^{2}+F_{\text{Sph},R}(r)^{2}\right)H^{\prime}_{R}.\end{split}$
(31)

For finding the minimum of this expression, any numerical opimization method,
such as the golden-section search [41] can be used.

If the value of $R$ for which $\expectationvalue{H}$ reaches its minimum is
denoted as the effective radius $R_{\text{eff}}$, then the correction to the
ground state energy is given approximately as

$\delta E_{\text{var}}=E_{\text{Sph},R_{\text{eff}}}-E_{\text{Sph},r_{0}}\\\
+\int\differential{r}\left(G_{\text{Sph},R_{\text{eff}}}(r)^{2}+F_{\text{Sph},R_{\text{eff}}}(r)^{2}\right)H^{\prime}_{R_{\text{eff}}},$
(32)

and the $g$-factor correction is

$\delta g_{\text{var}}=\frac{4}{3}\partialderivative{\delta
E_{\text{var}}}{M}.$ (33)

Again, the derivative of $E$ can be calculated from equation (22), but $r_{0}$
needs to be replaced with the effective radius $R_{\text{eff}}$.

####  III.1.3 Numerically Solving the Dirac Equation

The ground state wavefunctions and the corrections to the ground state
energies can also be obtained from the numerical solution of the radial Dirac
equation (3). The wavefunctions are necessary for calculating the corrections
to the $g$-factor using equation (7). For the numerical results presented in
this paper, we used the finite difference method to solve the radial Dirac
equation (3), and computed the $g$-factor corrections from

$\delta
g_{\text{num}}=-\frac{8}{3}M\int\differential{r}rG(r)F(r)-g_{\text{Sph}}.$
(34)

##  IV Alternative Charge Distribution Models

###  IV.1 Two-Parameter Fermi Distribution

Table 1: Comparison of the corrections to the ground state energies for sphere and Fermi models, in units of the muon rest energy. $\Delta E_{\text{Sph}}$ is the leading order finite size correction (difference between the homogeneous sphere and point nucleus values), and $\delta E_{\text{Fermi}}$ are the Fermi distribution nucleus corrections to the sphere nucleus values. Values for the nuclear radii are taken from Ref. [42]. The parameter $a$ in Eq. (35) is taken to be $2.3\text{fm}/4\log{3}$ for all elements. | Z |  $r_{\text{rms}}$ (fm) | $\Delta E_{\text{Sph}}$ | $\delta E_{\text{Fermi}}^{\text{pert}}$ | $\delta E_{\text{Fermi}}^{\text{var}}$ | $\delta E_{\text{Fermi}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{}{\text{C}}$ | 6 | 2.4702(22) | $3.8967(66)\times 10^{-6}$ | $-2.3079(8)\times 10^{-8}$ | $-2.3290(8)\times 10^{-8}$ | $-2.3727(7)\times 10^{-8}$
$\prescript{16}{}{\text{O}}$ | 8 | 2.6991(52) | $1.4057(50)\times 10^{-5}$ | $-9.4133(63)\times 10^{-8}$ | $-9.4931(60)\times 10^{-8}$ | $-9.6493(57)\times 10^{-8}$
$\prescript{20}{}{\text{Ne}}$ | 10 | 3.0055(21) | $4.0175(50)\times 10^{-5}$ | $-2.7638(6)\times 10^{-7}$ | $-2.7854(5)\times 10^{-7}$ | $-2.8240(5)\times 10^{-7}$
$\prescript{28}{}{\text{Si}}$ | 14 | 3.1224(24) | $1.5229(20)\times 10^{-4}$ | $-1.2805(1)\times 10^{-6}$ | $-1.2920(1)\times 10^{-6}$ | $-1.3090(1)\times 10^{-6}$
$\prescript{38}{}{\text{Ar}}$ | 18 | 3.4028(19) | $4.4039(38)\times 10^{-4}$ | $-3.8294(1)\times 10^{-6}$ | $-3.8660(1)\times 10^{-6}$ | $-3.9098(2)\times 10^{-6}$
$\prescript{40}{}{\text{Ca}}$ | 20 | 3.4776(19) | $6.6509(55)\times 10^{-4}$ | $-5.9452(4)\times 10^{-6}$ | $-6.0055(5)\times 10^{-6}$ | $-6.0708(6)\times 10^{-6}$
$\prescript{66}{}{\text{Zn}}$ | 30 | 3.9491(14) | $3.2385(14)\times 10^{-3}$ | $-2.8141(5)\times 10^{-5}$ | $-2.8487(6)\times 10^{-5}$ | $-2.8730(6)\times 10^{-5}$
$\prescript{86}{}{\text{Kr}}$ | 36 | 4.1835(21) | $6.3388(35)\times 10^{-3}$ | $-5.2290(19)\times 10^{-5}$ | $-5.2994(20)\times 10^{-5}$ | $-5.3395(21)\times 10^{-5}$
$\prescript{90}{}{\text{Zr}}$ | 40 | 4.2694(10) | $9.1096(22)\times 10^{-3}$ | $-7.3846(15)\times 10^{-5}$ | $-7.4906(15)\times 10^{-5}$ | $-7.5446(16)\times 10^{-5}$
$\prescript{120}{}{\text{Sn}}$ | 50 | 4.6519(21) | $1.9954(8)\times 10^{-2}$ | $-1.3942(7)\times 10^{-4}$ | $-1.4157(7)\times 10^{-4}$ | $-1.4241(7)\times 10^{-4}$
$\prescript{136}{}{\text{Xe}}$ | 54 | 4.7964(47) | $2.5930(21)\times 10^{-2}$ | $-1.6995(20)\times 10^{-4}$ | $-1.7262(21)\times 10^{-4}$ | $-1.7357(21)\times 10^{-4}$
$\prescript{142}{}{\text{Nd}}$ | 60 | 4.9123(25) | $3.6374(14)\times 10^{-2}$ | $-2.2449(15)\times 10^{-4}$ | $-2.2817(15)\times 10^{-4}$ | $-2.2935(16)\times 10^{-4}$
$\prescript{176}{}{\text{Yb}}$ | 70 | 5.3215(62) | $6.0941(44)\times 10^{-2}$ | $-3.0765(53)\times 10^{-4}$ | $-3.1265(55)\times 10^{-4}$ | $-3.1397(55)\times 10^{-4}$
$\prescript{185}{}{\text{Re}}$ | 75 | 5.3596(172) | $7.5168(139)\times 10^{-2}$ | $-3.6513(181)\times 10^{-4}$ | $-3.7125(187)\times 10^{-4}$ | $-3.7277(189)\times 10^{-4}$
$\prescript{208}{}{\text{Pb}}$ | 82 | 5.5012(13) | $9.9579(12)\times 10^{-2}$ | $-4.4041(17)\times 10^{-4}$ | $-4.4789(17)\times 10^{-4}$ | $-4.4958(18)\times 10^{-4}$
$\prescript{209}{}{\text{Bi}}$ | 83 | 5.5211(26) | $1.0346(2)\times 10^{-1}$ | $-4.5137(35)\times 10^{-4}$ | $-4.5904(36)\times 10^{-4}$ | $-4.6076(36)\times 10^{-4}$
$\prescript{212}{}{\text{Rn}}$ | 86 | 5.5915(176) | $1.1588(18)\times 10^{-1}$ | $-4.8286(254)\times 10^{-4}$ | $-4.9108(262)\times 10^{-4}$ | $-4.9284(264)\times 10^{-4}$
$\prescript{238}{}{\text{U}}$ | 92 | 5.8571(33) | $1.4530(4)\times 10^{-1}$ | $-5.2566(51)\times 10^{-4}$ | $-5.3428(53)\times 10^{-4}$ | $-5.3598(53)\times 10^{-4}$
Table 2: Comparison of the corrections to the $g$-factors for sphere and Fermi models. $\Delta g_{\text{Sph}}$ is the leading order finite size correction (difference between the homogeneous sphere and point nucleus values), and $\delta g_{\text{Fermi}}$ are the Fermi distribution nucleus corrections to the sphere nucleus values. Values for the nuclear radii are taken from Ref. [42]. The parameter $a$ in Eq. (35) is taken to be $2.3\text{fm}/4\log{3}$ for all elements. | Z |  $r_{\text{rms}}$ (fm) | $\Delta g_{\text{Sph}}$ | $\delta g_{\text{Fermi}}^{\text{pert}}$ | $\delta g_{\text{Fermi}}^{\text{var}}$ | $\delta g_{\text{Fermi}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{}{\text{C}}$ | 6 | 2.4702(22) | $1.5029(25)\times 10^{-5}$ | $-1.1643(3)\times 10^{-7}$ | $-1.1755(3)\times 10^{-7}$ | $-1.1977(3)\times 10^{-7}$
$\prescript{16}{}{\text{O}}$ | 8 | 2.6991(52) | $5.3243(183)\times 10^{-5}$ | $-4.6250(24)\times 10^{-7}$ | $-4.6676(23)\times 10^{-7}$ | $-4.7447(21)\times 10^{-7}$
$\prescript{20}{}{\text{Ne}}$ | 10 | 3.0055(21) | $1.4873(18)\times 10^{-4}$ | $-1.3140(2)\times 10^{-6}$ | $-1.3256(2)\times 10^{-6}$ | $-1.3440(1)\times 10^{-6}$
$\prescript{28}{}{\text{Si}}$ | 14 | 3.1224(24) | $5.4320(65)\times 10^{-4}$ | $-5.7628(1)\times 10^{-6}$ | $-5.8237(2)\times 10^{-6}$ | $-5.8996(3)\times 10^{-6}$
$\prescript{38}{}{\text{Ar}}$ | 18 | 3.4028(19) | $1.5004(12)\times 10^{-3}$ | $-1.6118(2)\times 10^{-5}$ | $-1.6308(2)\times 10^{-5}$ | $-1.6490(3)\times 10^{-5}$
$\prescript{40}{}{\text{Ca}}$ | 20 | 3.4776(19) | $2.2191(16)\times 10^{-3}$ | $-2.4263(5)\times 10^{-5}$ | $-2.4569(5)\times 10^{-5}$ | $-2.4832(5)\times 10^{-5}$
$\prescript{66}{}{\text{Zn}}$ | 30 | 3.9491(14) | $9.6827(35)\times 10^{-3}$ | $-9.7629(29)\times 10^{-5}$ | $-9.9198(30)\times 10^{-5}$ | $-10.0011(31)\times 10^{-5}$
$\prescript{86}{}{\text{Kr}}$ | 36 | 4.1835(21) | $1.7838(8)\times 10^{-2}$ | $-1.6561(9)\times 10^{-4}$ | $-1.6857(9)\times 10^{-4}$ | $-1.6977(9)\times 10^{-4}$
$\prescript{90}{}{\text{Zr}}$ | 40 | 4.2694(10) | $2.4763(5)\times 10^{-2}$ | $-2.2189(6)\times 10^{-4}$ | $-2.2614(7)\times 10^{-4}$ | $-2.2766(7)\times 10^{-4}$
$\prescript{120}{}{\text{Sn}}$ | 50 | 4.6519(21) | $4.9641(14)\times 10^{-2}$ | $-3.6545(25)\times 10^{-4}$ | $-3.7303(26)\times 10^{-4}$ | $-3.7503(26)\times 10^{-4}$
$\prescript{136}{}{\text{Xe}}$ | 54 | 4.7964(47) | $6.2469(35)\times 10^{-2}$ | $-4.2363(66)\times 10^{-4}$ | $-4.3259(69)\times 10^{-4}$ | $-4.3471(69)\times 10^{-4}$
$\prescript{142}{}{\text{Nd}}$ | 60 | 4.9123(25) | $8.4164(21)\times 10^{-2}$ | $-5.2508(46)\times 10^{-4}$ | $-5.3669(47)\times 10^{-4}$ | $-5.3909(48)\times 10^{-4}$
$\prescript{176}{}{\text{Yb}}$ | 70 | 5.3215(62) | $1.3139(6)\times 10^{-1}$ | $-6.4182(141)\times 10^{-4}$ | $-6.5585(147)\times 10^{-4}$ | $-6.5813(148)\times 10^{-4}$
$\prescript{185}{}{\text{Re}}$ | 75 | 5.3596(172) | $1.5801(18)\times 10^{-1}$ | $-7.3148(459)\times 10^{-4}$ | $-7.4795(477)\times 10^{-4}$ | $-7.5041(481)\times 10^{-4}$
$\prescript{208}{}{\text{Pb}}$ | 82 | 5.5012(13) | $2.0174(1)\times 10^{-1}$ | $-8.3094(40)\times 10^{-4}$ | $-8.4983(41)\times 10^{-4}$ | $-8.5231(42)\times 10^{-4}$
$\prescript{209}{}{\text{Bi}}$ | 83 | 5.5211(26) | $2.0856(3)\times 10^{-1}$ | $-8.4463(81)\times 10^{-4}$ | $-8.6385(84)\times 10^{-4}$ | $-8.6632(85)\times 10^{-4}$
$\prescript{212}{}{\text{Rn}}$ | 86 | 5.5915(176) | $2.3004(21)\times 10^{-1}$ | $-8.8124(578)\times 10^{-4}$ | $-9.0125(601)\times 10^{-4}$ | $-9.0369(604)\times 10^{-4}$
$\prescript{238}{}{\text{U}}$ | 92 | 5.8571(33) | $2.7897(4)\times 10^{-1}$ | $-9.0619(110)\times 10^{-4}$ | $-9.2593(114)\times 10^{-4}$ | $-9.2804(114)\times 10^{-4}$

A more realistic charge distribution model for the nucleus is the Fermi
distribution,

$\rho_{\text{Fermi}}({\bf\it r})=\frac{ZeN}{1+\exp(\frac{r-c}{a})},$ (35)

which depends on two adjustable parameters $a$ and $c$. Here, $N$ is a
normalization constant that is chosen such that the normalization of the
charge distribution is satisfied:

$\int\differential[3]{r}\rho(\mathbf{r})=Ze.$ (36)

The parameter $a$ is related to the skin thickness $t$ of the nucleus, which
is the distance over which the charge density falls from 90% to 10%, as
$t=4a\log{3}$. For most nuclei, the skin thickness is approximately equal to
2.3fm [2], which was the value used for all of our calculations. With $a$
known, $c$ can be determined by requiring that the rms radius of the charge
distribution (35) matches the rms radii of elements found in the literature,
such as in [42]. The rms radius of a charge distribution is defined as

$r_{\text{rms}}=\sqrt{\expectationvalue{r^{2}}}=\left(\frac{\int\differential[3]{r}r^{2}\rho(\mathbf{r})}{\int\differential[3]{r}\rho(\mathbf{r})}\right)^{1/2},$
(37)

which, in the case of a spherically symmetric charge distribution, can be
written as

$r_{\text{rms}}=\sqrt{\expectationvalue{r^{2}}}=\left(\frac{\int_{0}^{\infty}\differential{r}r^{4}\rho(r)}{\int_{0}^{\infty}\differential{r}r^{2}\rho(r)}\right)^{1/2}.$
(38)

In the case of the Fermi distribution (35), the integrals can be evaluated
analytically, and either approximative analytical formulas for $c$ and $N$,
such as those given in [2], can be used, or, for more accurate results, $c$
and $N$ can be found by solving equations (38) and (36) numerically, which was
the approach used for the results presented here.

The potential for the Fermi charge distribution (35) can also be calculated
analytically, and is

$V_{\text{Fermi}}(r)=-4\pi a^{2}N\alpha
Z\left[\text{Li}_{2}\left(-e^{(c-r)/a}\right)\right.\\\
\left.+\frac{2a}{r}\left[\text{Li}_{3}\left(-e^{(c-r)/a}\right)-\text{Li}_{3}\left(-e^{c/a}\right)\right]\right],$
(39)

where $\text{Li}_{n}(x)$ is the polylogarithm function [35], which frequently
appears in integrals involving the Fermi distribution.

An analytical solution of the Dirac equation with the Fermi distribution
potential (39) is not known. Therefore, either approximation methods need to
be used, or the Dirac equation needs to be solved numerically.

If the Fermi distribution potential is taken to be a perturbed sphere
distribution potential, then $\delta V(r)$ in equation (24) becomes

$\delta V(r)=V_{\text{Fermi}}(r)-V_{\text{Sph},r_{0}},$ (40)

and if the variational approximation is to be used, $H^{\prime}_{R}$ in
equation (30) becomes

$H^{\prime}_{R}=V_{\text{Fermi}}(r)-V_{\text{Sph},R}.$ (41)

The results for both approximation methods, and for the numerical solution of
the Dirac equation using finite difference method, is presented in Table 1 for
the ground state energies, and in Table 2 for the $g$-factors.

###  IV.2 Deformed Fermi Distribution and Nuclear Deformation Effect

Table 3: Some isotopes and their nuclear deformation parameters. Values for the nuclear radii are taken from Ref. [42], and nuclear deformation parameters are taken from [43], unless indicated otherwise. Only the elements with non-vanishing deformation parameters and even number of nucleons from Tables 1 and 2 are considered. | Z |  $r_{\text{rms}}$ (fm) | $\beta_{2}$ | $\beta_{4}$
---|---|---|---|---
$\prescript{12}{}{\text{C}}$ | 6 | 2.4702(22) | 0.44111Ref. [38] | 0.00111Ref. [38]
$\prescript{16}{}{\text{O}}$ | 8 | 2.6991(52) | 0.021 | -0.108
$\prescript{20}{}{\text{Ne}}$ | 10 | 3.0055(21) | 0.335 | 0.428
$\prescript{28}{}{\text{Si}}$ | 14 | 3.1224(24) | -0.478 | 0.250
$\prescript{66}{}{\text{Zn}}$ | 30 | 3.9491(14) | -0.215 | 0.005
$\prescript{86}{}{\text{Kr}}$ | 36 | 4.1835(21) | 0.082 | 0.012
$\prescript{176}{}{\text{Yb}}$ | 70 | 5.3215(62) | 0.278 | -0.071
$\prescript{208}{}{\text{Pb}}$ | 82 | 5.5012(13) | 0.061 | 0.000
$\prescript{212}{}{\text{Rn}}$ | 86 | 5.5915(176) | 0.000 | 0.008
$\prescript{238}{}{\text{U}}$ | 92 | 5.8571(33) | 0.280222Ref. [39] | 0.070222Ref. [39]
Table 4: Comparison of the corrections to the ground state energies for the deformed Fermi model, in units of the muon rest energy. $\delta E_{\text{DefFermi}}$ are the differences between the deformed Fermi distribution nucleus results and the homogeneous sphere nucleus results, whereas $\delta E_{\text{ND}}$ are the differences between the deformed and undeformed Fermi distribution nucleus results. Nuclear radii and deformation parameters are listed in Table 3. The parameter $a$ in Eq. (42) is taken to be $2.3\text{fm}/4\log{3}$ for all elements. | $-\delta E_{\text{DefFermi}}^{\text{pert}}$ | $-\delta E_{\text{DefFermi}}^{\text{var}}$ | $-\delta E_{\text{DefFermi}}^{\text{num}}$ | $-\delta E_{\text{ND}}^{\text{pert}}$ | $-\delta E_{\text{ND}}^{\text{var}}$ | $-\delta E_{\text{ND}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $2.354(1)\times 10^{-8}$ | $2.377(1)\times 10^{-8}$ | $2.423(1)\times 10^{-8}$ | $4.663(38)\times 10^{-10}$ | $4.756(39)\times 10^{-10}$ | $4.983(40)\times 10^{-10}$
$\prescript{16}{8}{\text{O}}$ | $9.435(7)\times 10^{-8}$ | $9.515(6)\times 10^{-8}$ | $9.672(6)\times 10^{-8}$ | $2.157(29)\times 10^{-10}$ | $2.196(30)\times 10^{-10}$ | $2.285(30)\times 10^{-10}$
$\prescript{20}{10}{\text{Ne}}$ | $3.271(3)\times 10^{-7}$ | $3.301(3)\times 10^{-7}$ | $3.357(3)\times 10^{-7}$ | $5.069(20)\times 10^{-8}$ | $5.157(20)\times 10^{-8}$ | $5.326(20)\times 10^{-8}$
$\prescript{28}{14}{\text{Si}}$ | $1.4117(6)\times 10^{-6}$ | $1.4258(6)\times 10^{-6}$ | $1.4472(6)\times 10^{-6}$ | $1.311(5)\times 10^{-7}$ | $1.337(5)\times 10^{-7}$ | $1.382(5)\times 10^{-7}$
$\prescript{66}{30}{\text{Zn}}$ | $2.9421(4)\times 10^{-5}$ | $2.9801(4)\times 10^{-5}$ | $3.0071(4)\times 10^{-5}$ | $1.281(1)\times 10^{-6}$ | $1.314(1)\times 10^{-6}$ | $1.341(1)\times 10^{-6}$
$\prescript{86}{36}{\text{Kr}}$ | $5.285(2)\times 10^{-5}$ | $5.357(2)\times 10^{-5}$ | $5.398(2)\times 10^{-5}$ | $5.584(7)\times 10^{-7}$ | $5.740(7)\times 10^{-7}$ | $5.835(7)\times 10^{-7}$
$\prescript{90}{40}{\text{Zr}}$ | $7.399(1)\times 10^{-5}$ | $7.506(2)\times 10^{-5}$ | $7.560(2)\times 10^{-5}$ | $1.4721(8)\times 10^{-7}$ | $1.5157(8)\times 10^{-7}$ | $1.5400(8)\times 10^{-7}$
$\prescript{176}{70}{\text{Yb}}$ | $3.767(4)\times 10^{-4}$ | $3.843(4)\times 10^{-4}$ | $3.864(5)\times 10^{-4}$ | $6.905(10)\times 10^{-5}$ | $7.164(10)\times 10^{-5}$ | $7.240(10)\times 10^{-5}$
$\prescript{208}{82}{\text{Pb}}$ | $4.459(2)\times 10^{-4}$ | $4.536(2)\times 10^{-4}$ | $4.554(2)\times 10^{-4}$ | $5.535(1)\times 10^{-6}$ | $5.729(1)\times 10^{-6}$ | $5.776(1)\times 10^{-6}$
$\prescript{212}{86}{\text{Rn}}$ | $4.830(25)\times 10^{-4}$ | $4.912(26)\times 10^{-4}$ | $4.930(26)\times 10^{-4}$ | $1.059(3)\times 10^{-7}$ | $1.096(3)\times 10^{-7}$ | $1.104(3)\times 10^{-7}$
$\prescript{238}{92}{\text{U}}$ | $7.380(4)\times 10^{-4}$ | $7.551(4)\times 10^{-4}$ | $7.585(4)\times 10^{-4}$ | $2.124(1)\times 10^{-4}$ | $2.209(1)\times 10^{-4}$ | $2.225(1)\times 10^{-4}$
Table 5: Comparison of the corrections to the $g$-factors for the deformed Fermi model. $\delta g_{\text{DefFermi}}$ are the differences between the deformed Fermi distribution nucleus results and the homogeneous sphere nucleus results, whereas $\delta g_{\text{ND}}$ are the differences between the deformed and undeformed Fermi distribution nucleus results. Nuclear radii and deformation parameters are listed in Table 3. The parameter $a$ in Eq. (42) is taken to be $2.3\text{fm}/4\log{3}$ for all elements. | -$\delta g_{\text{DefFermi}}^{\text{pert}}$ | -$\delta g_{\text{DefFermi}}^{\text{var}}$ | -$\delta g_{\text{DefFermi}}^{\text{num}}$ | -$\delta g_{\text{ND}}^{\text{pert}}$ | -$\delta g_{\text{ND}}^{\text{var}}$ | -$\delta g_{\text{ND}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $1.1878(5)\times 10^{-7}$ | $1.1995(5)\times 10^{-7}$ | $1.2229(5)\times 10^{-7}$ | $2.355(19)\times 10^{-9}$ | $2.405(19)\times 10^{-9}$ | $2.520(20)\times 10^{-9}$
$\prescript{16}{8}{\text{O}}$ | $4.636(3)\times 10^{-7}$ | $4.678(2)\times 10^{-7}$ | $4.756(2)\times 10^{-7}$ | $1.062(14)\times 10^{-9}$ | $1.082(14)\times 10^{-9}$ | $1.126(15)\times 10^{-9}$
$\prescript{20}{10}{\text{Ne}}$ | $1.555(1)\times 10^{-6}$ | $1.571(1)\times 10^{-6}$ | $1.598(1)\times 10^{-6}$ | $2.411(9)\times 10^{-7}$ | $2.458(9)\times 10^{-7}$ | $2.539(9)\times 10^{-7}$
$\prescript{28}{14}{\text{Si}}$ | $6.355(2)\times 10^{-6}$ | $6.429(2)\times 10^{-6}$ | $6.525(2)\times 10^{-6}$ | $5.919(21)\times 10^{-7}$ | $6.057(22)\times 10^{-7}$ | $6.259(23)\times 10^{-7}$
$\prescript{66}{30}{\text{Zn}}$ | $1.0210(2)\times 10^{-4}$ | $1.0382(3)\times 10^{-4}$ | $1.0472(3)\times 10^{-4}$ | $4.469(4)\times 10^{-6}$ | $4.622(4)\times 10^{-6}$ | $4.713(4)\times 10^{-6}$
$\prescript{86}{36}{\text{Kr}}$ | $1.6739(9)\times 10^{-4}$ | $1.7041(9)\times 10^{-4}$ | $1.7164(9)\times 10^{-4}$ | $1.777(2)\times 10^{-6}$ | $1.842(2)\times 10^{-6}$ | $1.871(2)\times 10^{-6}$
$\prescript{90}{40}{\text{Zr}}$ | $2.2233(6)\times 10^{-4}$ | $2.2660(6)\times 10^{-4}$ | $2.2813(7)\times 10^{-4}$ | $4.449(2)\times 10^{-7}$ | $4.625(2)\times 10^{-7}$ | $4.695(2)\times 10^{-7}$
$\prescript{176}{70}{\text{Yb}}$ | $7.875(13)\times 10^{-4}$ | $8.089(13)\times 10^{-4}$ | $8.125(13)\times 10^{-4}$ | $1.457(1)\times 10^{-4}$ | $1.531(1)\times 10^{-4}$ | $1.544(1)\times 10^{-4}$
$\prescript{208}{82}{\text{Pb}}$ | $8.415(4)\times 10^{-4}$ | $8.609(4)\times 10^{-4}$ | $8.634(4)\times 10^{-4}$ | $1.0535(2)\times 10^{-5}$ | $1.1032(2)\times 10^{-5}$ | $1.1102(1)\times 10^{-5}$
$\prescript{212}{86}{\text{Rn}}$ | $8.814(58)\times 10^{-4}$ | $9.015(60)\times 10^{-4}$ | $9.039(60)\times 10^{-4}$ | $1.952(3)\times 10^{-7}$ | $2.045(5)\times 10^{-7}$ | $2.056(3)\times 10^{-7}$
$\prescript{238}{92}{\text{U}}$ | $1.272(1)\times 10^{-3}$ | $1.312(1)\times 10^{-3}$ | $1.316(1)\times 10^{-3}$ | $3.6606(8)\times 10^{-4}$ | $3.8561(7)\times 10^{-4}$ | $3.8767(7)\times 10^{-4}$

Both the homogeneous sphere model, and the Fermi distribution model assume a
spherically symmetric charge distribution for the nucleus, but the
deformations of the nucleus can break its spherical symmetry. Therefore, it is
more realistic to introduce a deformed Fermi distribution, in which the
parameter $c$ has angular dependency. If the nucleus is still assumed to
preserve its axial symmetry, then the dipole and the octupole deformations
vanish, and in a first approximation, it is sufficient to consider quadrupole
and hexadecapole deformations only [37, 38, 39]. This is achieved by replacing
$c$ by $c_{0}(1+\beta_{2}Y_{20}+\beta_{4}Y_{40})$, and writing the charge
distribution as

$\rho_{\text{DefFermi}}(\mathbf{r})=\frac{ZeN}{1+\exp(\frac{r-c_{0}(1+\beta_{2}Y_{20}+\beta_{4}Y_{40})}{a})},$
(42)

where $Y_{20}$ and $Y_{40}$ are spherical harmonics, and $\beta_{2}$ and
$\beta_{4}$ are quadrupole and octupole deformation parameters. The
deformation parameters for some isotopes are listed in Table 3.

The potential of the deformed Fermi distribution simplifies if the nucleus is
spinless and is assumed to be in its rotational ground state, which means the
nucleus has vanishing total angular momentum, which can only be satisfied if
the nucleus has an even number of protons and neutrons. In the rotational
ground state of the nucleus, the nucleus is found in all orientations with
equal probability. If the nuclear and bound particle degrees of freedom are
assumed to be independent, which is the approximation we consider in this work
(this approximation is less applicable to muonic atoms as the excitation
energies of the bound muon is similar in magnitude to the nuclear excitation
energies, and the corrections due to the excitations of the nucleus are
collectively denoted as the nuclear polarization effect, see the discussion in
the last section), then the potential affecting the bound particle is the
expectation value of the potential due to the nuclear charge distribution in
the considered nuclear state [37]. In the case of the nuclear ground state,
the expectation value averages out the angular dependence of the potential,
and only the spherically symmetric component survives, in which case the
potential for any charge distribution becomes

$V(r)=\frac{e}{4\pi}\int\differential[3]{r^{\prime}}\frac{\rho(r^{\prime})}{r_{>}},$
(43)

where $r_{>}=\max(r,r^{\prime})$.

In the case of the deformed Fermi distribution (42), the potential can be
written as

$V_{\text{DefFermi}}(r)=-2\pi a^{2}N\alpha
Z\int_{0}^{\pi}\differential{\theta}\sin{\theta}\left[\text{Li}_{2}\left(-e^{(c(\theta)-r)/a}\right)+\frac{2a}{r}\left[\text{Li}_{3}\left(-e^{(c(\theta)-r)/a}\right)-\text{Li}_{3}\left(-e^{c(\theta)/a}\right)\right]\right],$
(44)

where $c(\theta)=c_{0}(1+\beta_{2}Y_{20}(\theta)+\beta_{4}Y_{40}(\theta))$.
$c_{0}$ and $N$ can be found numerically using the formulas (37) and (36),
with $a=\frac{2.3\text{ fm}}{4\log{3}}$.

As in the Fermi distribution potential, the deformed Fermi distribution can
also be considered a perturbation on top of the sphere nucleus potential with
the same nuclear rms radius and charge, and perturbative and variational
approximations can be applied, and the Dirac equation can also be solved
numerically. The results for the ground state energies are presented in Table
4, and the results for the $g$-factor corrections in Table 5. Nuclear
deformation parameters are listed in Table 3 and are usually taken from Ref.
[43] unless stated otherwise.

It is important to notice that the corrections due to the difference between
Fermi and Sphere charge distributions are larger than the uncertainties in the
leading order finite nuclear size corrections, and for some elements large
nuclear deformation parameters, even the nuclear deformation corrections, by
which we refer to as the difference between the finite size corrections for
deformed Fermi and un-deformed Fermi distributions, are larger than the
uncertainties due to the rms radii, which is due to the nuclear effects being
amplified in muonic atoms.

Figure 1: Feynman diagrams depicting the one-loop QED corrections to the
$g$-factors: (a) electric-loop vacuum polarization correction, (b) magnetic-
loop vacuum polarization correction, and (c, d) self-energy corrections.
Double lines represent bound-particle wave functions and propagators, the wave
line ending with a triangle represents the constant external magnetic field.
For the corrections to the energy levels, only the analogues of (a) and (c)
appear, with the magnetic field line removed.

##  V Nuclear Size Corrections to the QED Effects

So far, the corrections to the g-factors and energy levels have only been
considered in the framework of relativistic quantum mechanics, but quantum
electrodynamic (QED) effects also introduce corrections to the leading order
finite size correction. At the one-loop level, these effects are the self-
energy correction and the vacuum-polarization corrections. In the case of the
$g$-factor, vacuum polarization manifests itself as electric-loop vacuum
polarization and magnetic-loop vacuum polarization, where the magnetic-loop
introduces corrections to the interaction with the external magnetic field,
and the electric-loop introduces corrections to the interaction with the
nuclear potential, and therefore, there is no analog of the magnetic loop in
the Feynman diagrams for the QED corrections to the energy levels. The Feynman
diagrams for the one-loop QED corrections to the $g$-factors are depicted in
Figure 1.

Electric-loop vacuum polarization describes the QED corrections to the
electric field of the nucleus, and therefore it can be modelled as a
correction to the nuclear potential. Since the bound-particle propagator is
significantly more complicated than the free-particle propagator, the common
approach in electric-loop vacuum polarization calculations is to
perturbatively expand the loop in interactions with the nucleus, as depicted
in terms of Feynman diagrams in Figure 2. To first order in the interaction
with the nucleus, this correction potential is called the Uehling potential
[44]. The Uehling potential changes depending on the particle that is pair-
created, and we investigate the electronic, muonic, and hadronic vacuum
polarization effects, as well as the dependence on the nuclear charge
distribution. In the case of the electronic vacuum polarization effect in
bound electrons, the leading Uehling contribution is found dominate over the
higher-order electric loop corrections, the so-called Wichmann-Kroll term [2].
We therefore focus on the Uehling correction in this work.

Both the magnetic loop vacuum polarization correction and the self-energy
correction require using the bound-muon propagator for extended nuclei for an
accurate description [4]. In this work, we limit ourselves to the electric-
loop vacuum polarization corrections only, and leave the determination of
finite nuclear size effects on the magnetic-loop vacuum polarization and self-
energy corrections to the $g$-factors of bound muons for a future work.

###  V.1 Electronic and Muonic Vacuum Polarization

Figure 2: Perturbative decomposition of the electric-loop vacuum polarization diagram into Uehling and Wichmann-Kroll diagrams. The first term on the right hand side is the Uehling diagram, and the remaining terms are collectively denoted as the Wichmann-Kroll terms. A double line represents the bound-particle propagator, and a single line represents the free particle propagator. Table 6: Electronic vacuum polarization corrections to the ground state energies in units of the muon rest energy. The results for both sphere-Uehling corrections ($\delta E_{e\text{VP,Sph}}$), and the Fermi-Uehling corrections ($\delta E_{e\text{VP,Fermi}}$) are presented and compared to the results in the case of a point-like nucleus ($\delta E_{e\text{VP,PN}}$). The nuclear radii and uncertainties are taken from Ref. [42], same as in Table 1. | $\delta E_{e\text{VP,PN}}^{\text{num}}$ | $\delta E_{e\text{VP,Sph}}^{\text{pert}}$ | $\delta E_{e\text{VP,Sph}}^{\text{var}}$ | $\delta E_{e\text{VP,Sph}}^{\text{num}}$ | $\delta E_{e\text{VP,Fermi}}^{\text{pert}}$ | $\delta E_{e\text{VP,Fermi}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $-3.874\times 10^{-6}$ | $-3.8040(1)\times 10^{-6}$ | $-3.8075(1)\times 10^{-6}$ | $-3.8138(1)\times 10^{-6}$ | $-3.8045(1)\times 10^{-6}$ | $-3.8144(1)\times 10^{-6}$
$\prescript{16}{8}{\text{O}}$ | $-8.132\times 10^{-6}$ | $-7.8757(8)\times 10^{-6}$ | $-7.8852(8)\times 10^{-6}$ | $-7.8974(8)\times 10^{-6}$ | $-7.8779(7)\times 10^{-6}$ | $-7.8997(8)\times 10^{-6}$
$\prescript{20}{10}{\text{Ne}}$ | $-1.430\times 10^{-5}$ | $-1.3556(1)\times 10^{-5}$ | $-1.3575(1)\times 10^{-5}$ | $-1.3594(1)\times 10^{-5}$ | $-1.3562(1)\times 10^{-5}$ | $-1.3601(1)\times 10^{-5}$
$\prescript{28}{14}{\text{Si}}$ | $-3.298\times 10^{-5}$ | $-3.0040(3)\times 10^{-5}$ | $-3.0092(3)\times 10^{-5}$ | $-3.0128(3)\times 10^{-5}$ | $-3.0071(3)\times 10^{-5}$ | $-3.0160(4)\times 10^{-5}$
$\prescript{38}{18}{\text{Ar}}$ | $-6.102\times 10^{-5}$ | $-5.2362(7)\times 10^{-5}$ | $-5.2461(7)\times 10^{-5}$ | $-5.2513(7)\times 10^{-5}$ | $-5.2454(6)\times 10^{-5}$ | $-5.2605(7)\times 10^{-5}$
$\prescript{40}{20}{\text{Ca}}$ | $-7.883\times 10^{-5}$ | $-6.5658(9)\times 10^{-5}$ | $-6.5784(9)\times 10^{-5}$ | $-6.5844(9)\times 10^{-5}$ | $-6.5799(9)\times 10^{-5}$ | $-6.599(1)\times 10^{-5}$
$\prescript{66}{30}{\text{Zn}}$ | $-2.103\times 10^{-4}$ | $-1.4558(2)\times 10^{-4}$ | $-1.4586(2)\times 10^{-4}$ | $-1.4595(2)\times 10^{-4}$ | $-1.4620(2)\times 10^{-4}$ | $-1.4657(3)\times 10^{-4}$
$\prescript{86}{36}{\text{Kr}}$ | $-3.273\times 10^{-4}$ | $-2.0060(5)\times 10^{-4}$ | $-2.0096(5)\times 10^{-4}$ | $-2.0106(5)\times 10^{-4}$ | $-2.0170(5)\times 10^{-4}$ | $-2.0217(6)\times 10^{-4}$
$\prescript{90}{40}{\text{Zr}}$ | $-4.233\times 10^{-4}$ | $-2.4058(3)\times 10^{-4}$ | $-2.4100(3)\times 10^{-4}$ | $-2.4111(3)\times 10^{-4}$ | $-2.4210(3)\times 10^{-4}$ | $-2.4263(4)\times 10^{-4}$
$\prescript{120}{50}{\text{Sn}}$ | $-7.355\times 10^{-4}$ | $-3.356(1)\times 10^{-4}$ | $-3.361(1)\times 10^{-4}$ | $-3.362(1)\times 10^{-4}$ | $-3.382(1)\times 10^{-4}$ | $-3.388(1)\times 10^{-4}$
$\prescript{136}{54}{\text{Xe}}$ | $-8.934\times 10^{-4}$ | $-3.727(3)\times 10^{-4}$ | $-3.733(3)\times 10^{-4}$ | $-3.734(3)\times 10^{-4}$ | $-3.758(3)\times 10^{-4}$ | $-3.765(3)\times 10^{-4}$
$\prescript{142}{60}{\text{Nd}}$ | $-1.172\times 10^{-3}$ | $-4.334(2)\times 10^{-4}$ | $-4.340(2)\times 10^{-4}$ | $-4.341(2)\times 10^{-4}$ | $-4.373(2)\times 10^{-4}$ | $-4.381(2)\times 10^{-4}$
$\prescript{176}{70}{\text{Yb}}$ | $-1.768\times 10^{-3}$ | $-5.137(5)\times 10^{-4}$ | $-5.144(5)\times 10^{-4}$ | $-5.145(5)\times 10^{-4}$ | $-5.188(5)\times 10^{-4}$ | $-5.195(5)\times 10^{-4}$
$\prescript{185}{75}{\text{Re}}$ | $-2.144\times 10^{-3}$ | $-5.655(16)\times 10^{-4}$ | $-5.661(16)\times 10^{-4}$ | $-5.662(16)\times 10^{-4}$ | $-5.714(17)\times 10^{-4}$ | $-5.722(17)\times 10^{-4}$
$\prescript{208}{82}{\text{Pb}}$ | $-2.782\times 10^{-3}$ | $-6.284(1)\times 10^{-4}$ | $-6.291(1)\times 10^{-4}$ | $-6.292(1)\times 10^{-4}$ | $-6.353(1)\times 10^{-4}$ | $-6.361(1)\times 10^{-4}$
$\prescript{209}{83}{\text{Bi}}$ | $-2.886\times 10^{-3}$ | $-6.373(3)\times 10^{-4}$ | $-6.380(3)\times 10^{-4}$ | $-6.380(3)\times 10^{-4}$ | $-6.443(3)\times 10^{-4}$ | $-6.451(3)\times 10^{-4}$
$\prescript{212}{86}{\text{Rn}}$ | $-3.219\times 10^{-3}$ | $-6.622(20)\times 10^{-4}$ | $-6.629(20)\times 10^{-4}$ | $-6.630(20)\times 10^{-4}$ | $-6.696(20)\times 10^{-4}$ | $-6.704(20)\times 10^{-4}$
$\prescript{238}{92}{\text{U}}$ | $-4.003\times 10^{-3}$ | $-6.956(4)\times 10^{-4}$ | $-6.963(4)\times 10^{-4}$ | $-6.964(4)\times 10^{-4}$ | $-7.034(4)\times 10^{-4}$ | $-7.041(4)\times 10^{-4}$
Table 7: Electronic vacuum polarization corrections to the $g$-factors of particles in the ground state. The results for both sphere-Uehling corrections ($\delta g_{e\text{VP,Sph}}$), and the Fermi-Uehling corrections ($\delta g_{e\text{VP,Fermi}}$) are presented and compared to the results in the case of a point-like nucleus ($\delta g_{e\text{VP,PN}}$). The nuclear radii and uncertainties are taken from Ref. [42], same as in Table 2. | $\delta g_{e\text{VP,PN}}^{\text{num}}$ | $\delta g_{e\text{VP,Sph}}^{\text{pert}}$ | $\delta g_{e\text{VP,Sph}}^{\text{var}}$ | $\delta g_{e\text{VP,Sph}}^{\text{num}}$ | $\delta g_{e\text{VP,Fermi}}^{\text{pert}}$ | $\delta g_{e\text{VP,Fermi}}^{\text{num}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $-8.288\times 10^{-6}$ | $-8.0072(4)\times 10^{-6}$ | $-8.0185(4)\times 10^{-6}$ | $-8.0314(4)\times 10^{-6}$ | $-8.0100(4)\times 10^{-6}$ | $-8.0343(4)\times 10^{-6}$
$\prescript{16}{8}{\text{O}}$ | $-1.673\times 10^{-5}$ | $-1.5708(3)\times 10^{-5}$ | $-1.5736(3)\times 10^{-5}$ | $-1.5757(3)\times 10^{-5}$ | $-1.5720(3)\times 10^{-5}$ | $-1.5769(3)\times 10^{-5}$
$\prescript{20}{10}{\text{Ne}}$ | $-2.861\times 10^{-5}$ | $-2.5710(3)\times 10^{-5}$ | $-2.5762(3)\times 10^{-5}$ | $-2.5790(3)\times 10^{-5}$ | $-2.5742(3)\times 10^{-5}$ | $-2.5823(3)\times 10^{-5}$
$\prescript{28}{14}{\text{Si}}$ | $-6.353\times 10^{-5}$ | $-5.256(1)\times 10^{-5}$ | $-5.267(1)\times 10^{-5}$ | $-5.271(1)\times 10^{-5}$ | $-5.270(1)\times 10^{-5}$ | $-5.286(1)\times 10^{-5}$
$\prescript{38}{18}{\text{Ar}}$ | $-1.146\times 10^{-4}$ | $-8.406(2)\times 10^{-5}$ | $-8.425(2)\times 10^{-5}$ | $-8.429(2)\times 10^{-5}$ | $-8.444(2)\times 10^{-5}$ | $-8.468(2)\times 10^{-5}$
$\prescript{40}{20}{\text{Ca}}$ | $-1.466\times 10^{-4}$ | $-1.0130(3)\times 10^{-4}$ | $-1.0152(3)\times 10^{-4}$ | $-1.0156(3)\times 10^{-4}$ | $-1.0187(3)\times 10^{-4}$ | $-1.0214(3)\times 10^{-4}$
$\prescript{66}{30}{\text{Zn}}$ | $-3.775\times 10^{-4}$ | $-1.8264(5)\times 10^{-4}$ | $-1.8294(5)\times 10^{-4}$ | $-1.8295(5)\times 10^{-4}$ | $-1.8468(5)\times 10^{-4}$ | $-1.8501(6)\times 10^{-4}$
$\prescript{86}{36}{\text{Kr}}$ | $-5.791\times 10^{-4}$ | $-2.236(1)\times 10^{-4}$ | $-2.239(1)\times 10^{-4}$ | $-2.239(1)\times 10^{-4}$ | $-2.269(1)\times 10^{-4}$ | $-2.271(1)\times 10^{-4}$
$\prescript{90}{40}{\text{Zr}}$ | $-7.429\times 10^{-4}$ | $-2.5024(6)\times 10^{-4}$ | $-2.5051(6)\times 10^{-4}$ | $-2.5049(6)\times 10^{-4}$ | $-2.5443(7)\times 10^{-4}$ | $-2.5470(6)\times 10^{-4}$
$\prescript{120}{50}{\text{Sn}}$ | $-1.270\times 10^{-3}$ | $-2.909(2)\times 10^{-4}$ | $-2.911(2)\times 10^{-4}$ | $-2.910(2)\times 10^{-4}$ | $-2.971(2)\times 10^{-4}$ | $-2.972(2)\times 10^{-4}$
$\prescript{136}{54}{\text{Xe}}$ | $-1.534\times 10^{-3}$ | $-3.018(4)\times 10^{-4}$ | $-3.020(4)\times 10^{-4}$ | $-3.019(4)\times 10^{-4}$ | $-3.087(4)\times 10^{-4}$ | $-3.088(4)\times 10^{-4}$
$\prescript{142}{60}{\text{Nd}}$ | $-1.996\times 10^{-3}$ | $-3.218(2)\times 10^{-4}$ | $-3.219(2)\times 10^{-4}$ | $-3.218(2)\times 10^{-4}$ | $-3.299(2)\times 10^{-4}$ | $-3.299(2)\times 10^{-4}$
$\prescript{176}{70}{\text{Yb}}$ | $-2.979\times 10^{-3}$ | $-3.255(6)\times 10^{-4}$ | $-3.255(6)\times 10^{-4}$ | $-3.254(6)\times 10^{-4}$ | $-3.345(6)\times 10^{-4}$ | $-3.344(6)\times 10^{-4}$
$\prescript{185}{75}{\text{Re}}$ | $-3.593\times 10^{-3}$ | $-3.388(16)\times 10^{-4}$ | $-3.387(16)\times 10^{-4}$ | $-3.386(16)\times 10^{-4}$ | $-3.487(17)\times 10^{-4}$ | $-3.485(17)\times 10^{-4}$
$\prescript{208}{82}{\text{Pb}}$ | $-4.629\times 10^{-3}$ | $-3.461(1)\times 10^{-4}$ | $-3.460(1)\times 10^{-4}$ | $-3.459(1)\times 10^{-4}$ | $-3.569(1)\times 10^{-4}$ | $-3.567(1)\times 10^{-4}$
$\prescript{209}{83}{\text{Bi}}$ | $-4.797\times 10^{-3}$ | $-3.470(3)\times 10^{-4}$ | $-3.468(3)\times 10^{-4}$ | $-3.467(3)\times 10^{-4}$ | $-3.578(3)\times 10^{-4}$ | $-3.576(3)\times 10^{-4}$
$\prescript{212}{86}{\text{Rn}}$ | $-5.336\times 10^{-3}$ | $-3.481(17)\times 10^{-4}$ | $-3.479(17)\times 10^{-4}$ | $-3.478(17)\times 10^{-4}$ | $-3.592(18)\times 10^{-4}$ | $-3.589(18)\times 10^{-4}$
$\prescript{238}{92}{\text{U}}$ | $-6.596\times 10^{-3}$ | $-3.370(3)\times 10^{-4}$ | $-3.368(3)\times 10^{-4}$ | $-3.367(3)\times 10^{-4}$ | $-3.479(3)\times 10^{-4}$ | $-3.476(3)\times 10^{-4}$
Table 8: Muonic vacuum polarization corrections to the ground state energies and $g$-factors. The energies are in units of muon rest mass. The results for both sphere-Uehling corrections and the Fermi-Uehling corrections are presented and compared to the results in the case of a point-like nucleus. Results obtained from the perturbative and variational approximations agree with the numerical results for up to 4 significant digits, therefore, only the numerical results are presented. | $\delta E_{\mu\text{VP,PN}}$ | $\delta E_{\mu\text{VP,Sph}}$ | $\delta E_{\mu\text{VP,Fermi}}$ | $\delta g_{\mu\text{VP,PN}}$ | $\delta g_{\mu\text{VP,Sph}}$ | $\delta g_{\mu\text{VP,Fermi}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $-2.177\times 10^{-9}$ | $-1.9142(3)\times 10^{-9}$ | $-1.9206(3)\times 10^{-9}$ | $-8.540\times 10^{-9}$ | $-7.179(1)\times 10^{-9}$ | $-7.212(2)\times 10^{-9}$
$\prescript{16}{8}{\text{O}}$ | $-6.801\times 10^{-9}$ | $-5.584(3)\times 10^{-9}$ | $-5.608(3)\times 10^{-9}$ | $-2.654\times 10^{-8}$ | $-2.029(1)\times 10^{-8}$ | $-2.042(1)\times 10^{-8}$
$\prescript{20}{10}{\text{Ne}}$ | $-1.643\times 10^{-8}$ | $-1.2328(3)\times 10^{-8}$ | $-1.2391(3)\times 10^{-8}$ | $-6.376\times 10^{-8}$ | $-4.305(1)\times 10^{-8}$ | $-4.338(1)\times 10^{-8}$
$\prescript{28}{14}{\text{Si}}$ | $-6.205\times 10^{-8}$ | $-4.025(2)\times 10^{-8}$ | $-4.056(2)\times 10^{-8}$ | $-2.379\times 10^{-7}$ | $-1.3158(7)\times 10^{-7}$ | $-1.3311(8)\times 10^{-7}$
$\prescript{38}{18}{\text{Ar}}$ | $-1.674\times 10^{-7}$ | $-8.969(3)\times 10^{-8}$ | $-9.059(4)\times 10^{-8}$ | $-6.338\times 10^{-7}$ | $-2.702(1)\times 10^{-7}$ | $-2.744(1)\times 10^{-7}$
$\prescript{40}{20}{\text{Ca}}$ | $-2.539\times 10^{-7}$ | $-1.2443(5)\times 10^{-7}$ | $-1.2583(5)\times 10^{-7}$ | $-9.554\times 10^{-7}$ | $-3.610(2)\times 10^{-7}$ | $-3.673(2)\times 10^{-7}$
$\prescript{66}{30}{\text{Zn}}$ | $-1.274\times 10^{-6}$ | $-3.759(2)\times 10^{-7}$ | $-3.823(2)\times 10^{-7}$ | $-4.642\times 10^{-6}$ | $-8.908(5)\times 10^{-7}$ | $-9.157(5)\times 10^{-7}$
$\prescript{86}{36}{\text{Kr}}$ | $-2.657\times 10^{-6}$ | $-5.754(4)\times 10^{-7}$ | $-5.870(4)\times 10^{-7}$ | $-9.491\times 10^{-6}$ | $-1.214(1)\times 10^{-6}$ | $-1.255(1)\times 10^{-6}$
$\prescript{90}{40}{\text{Zr}}$ | $-4.084\times 10^{-6}$ | $-7.344(3)\times 10^{-7}$ | $-7.508(3)\times 10^{-7}$ | $-1.439\times 10^{-5}$ | $-1.4490(7)\times 10^{-6}$ | $-1.5038(7)\times 10^{-6}$
$\prescript{120}{50}{\text{Sn}}$ | $-1.035\times 10^{-5}$ | $-1.0985(8)\times 10^{-6}$ | $-1.1269(9)\times 10^{-6}$ | $-3.523\times 10^{-5}$ | $-1.811(2)\times 10^{-6}$ | $-1.894(2)\times 10^{-6}$
$\prescript{136}{54}{\text{Xe}}$ | $-1.440\times 10^{-5}$ | $-1.238(2)\times 10^{-6}$ | $-1.272(2)\times 10^{-6}$ | $-4.828\times 10^{-5}$ | $-1.908(4)\times 10^{-6}$ | $-2.001(5)\times 10^{-6}$
$\prescript{142}{60}{\text{Nd}}$ | $-2.285\times 10^{-5}$ | $-1.488(1)\times 10^{-6}$ | $-1.532(2)\times 10^{-6}$ | $-7.493\times 10^{-5}$ | $-2.105(3)\times 10^{-6}$ | $-2.217(3)\times 10^{-6}$
$\prescript{176}{70}{\text{Yb}}$ | $-4.634\times 10^{-5}$ | $-1.739(4)\times 10^{-6}$ | $-1.793(4)\times 10^{-6}$ | $-1.460\times 10^{-4}$ | $-2.096(7)\times 10^{-6}$ | $-2.217(7)\times 10^{-6}$
$\prescript{185}{75}{\text{Re}}$ | $-6.467\times 10^{-5}$ | $-1.957(13)\times 10^{-6}$ | $-2.020(14)\times 10^{-6}$ | $-1.994\times 10^{-4}$ | $-2.229(20)\times 10^{-6}$ | $-2.365(21)\times 10^{-6}$
$\prescript{208}{82}{\text{Pb}}$ | $-1.017\times 10^{-4}$ | $-2.186(1)\times 10^{-6}$ | $-2.260(1)\times 10^{-6}$ | $-3.039\times 10^{-4}$ | $-2.284(2)\times 10^{-6}$ | $-2.432(2)\times 10^{-6}$
$\prescript{209}{83}{\text{Bi}}$ | $-1.084\times 10^{-4}$ | $-2.217(2)\times 10^{-6}$ | $-2.292(2)\times 10^{-6}$ | $-3.224\times 10^{-4}$ | $-2.289(3)\times 10^{-6}$ | $-2.439(3)\times 10^{-6}$
$\prescript{212}{86}{\text{Rn}}$ | $-1.313\times 10^{-4}$ | $-2.299(16)\times 10^{-6}$ | $-2.378(17)\times 10^{-6}$ | $-3.847\times 10^{-4}$ | $-2.288(21)\times 10^{-6}$ | $-2.441(23)\times 10^{-6}$
$\prescript{238}{92}{\text{U}}$ | $-1.924\times 10^{-4}$ | $-2.333(3)\times 10^{-6}$ | $-2.412(3)\times 10^{-6}$ | $-5.465\times 10^{-4}$ | $-2.130(4)\times 10^{-6}$ | $-2.275(4)\times 10^{-6}$

Since electrons and muons are particles with identical properties except for
their mass, electronic and muonic Uehling potentials are given by the same
expression. The Uehling potential for the case of leptonic vacuum polarization
and extended nucleus can be expressed as [45]

$V_{\mathrm{Ueh}}=\frac{2}{4\pi^{2}}\int_{2m_{l}}^{\infty}\mathrm{d}q\,\rho_{\mathrm{int}}(r,q)\imaginary\Pi(q^{2}).$
(45)

The imaginary part of the leptonic vacuum polarization function is

$\imaginary\Pi(q^{2})=\frac{\alpha}{3}\sqrt{1-\frac{4m_{l}^{2}}{q^{2}}}\left(1+\frac{2m_{l}^{2}}{q^{2}}\right).$
(46)

where $m_{l}$ is the mass of the virtual lepton that is pair-created, being
either $m_{e}$ or $m_{\mu}$. The integral over the nuclear charge distribution

$\rho_{\mathrm{int}}(r,q)=-\frac{2\pi
e}{q^{2}r}\int_{0}^{\infty}\mathrm{d}r^{\prime}r^{\prime}\rho(r^{\prime})\left(e^{-q|r-r^{\prime}|}-e^{-q(r+r^{\prime})}\right)$
(47)

was performed analytically for the sphere distribution, and numerically for
the Fermi distribution.

Depending on whether the homogeneous sphere model, or the Fermi distribution
model is used, the total potential becomes either
$V(r)=V_{\text{Sph}}(r)+V_{\text{Ueh-Sph},m}(r)$, or
$V(r)=V_{\text{Fermi}}(r)+V_{\text{Ueh-Fermi},m}(r)$. In the case of the
homogeneous sphere model, perturbative and variational approximations can be
applied, or the Dirac equation can be solved numerically. In the case of the
Fermi distribution model, the Fermi-Uehling potential can be considered as a
perturbation on top of the Fermi nucleus potential (as in Eq. (24), but with
$V_{\text{Sph}}$ replaced by $V_{\text{Fermi}}$), and perturbation theory
using the numerical wavefunctions can be applied, or the Dirac equation can be
solved numerically for the total potential. For the perturbation theory
calculation of the Fermi-Uehling contribution to the $g$-factor, Eq. (26)
needs to be used, in which the mass derivatives of the wavefunctions appear.
If the Fermi-distribution nucleus wavefunctions are found numerically, their
mass derivatives can be obtained by solving the mass derivative of the Dirac
equation

$\left(H-E\right)\partialderivative{M}\ket{\psi(\mathbf{r})}=\left(\partialderivative{E}{M}-\beta\right)\ket{\psi(\mathbf{r})},$
(48)

or in terms of the radial wavefunctions,

$\begin{pmatrix}M+V(r)-E&-\derivative{r}+\frac{\kappa}{r}\\\
\derivative{r}+\frac{\kappa}{r}&-M+V(r)-E\end{pmatrix}\begin{pmatrix}G^{\prime}(r)\\\
F^{\prime}(r)\end{pmatrix}=\\\ \begin{pmatrix}(E^{\prime}-1)G(r)\\\
(E^{\prime}+1)F(r)\end{pmatrix}$ (49)

and requiring that
$\bra{\psi(\mathbf{r})}\partialderivative{M}\ket{\psi(\mathbf{r})}=0$.

Electronic vacuum polarization results for the ground state energies are
presented in Table 6, and for the $g$-factors in Table 7. Muonic vacuum
polarization results are presented in Table 8.

Table 9: Hadronic vacuum polarization corrections to the ground state energies and $g$-factors. The energies are in units of muon rest mass. The results for both sphere-Uehling corrections and the Fermi-Uehling corrections are presented and compared to the results in the case of a point-like nucleus. Results obtained from the perturbative and variational approximations agree with the numerical results for up to 4 significant digits, therefore, only the numerical results are presented. | $\delta E_{h\text{VP,PN}}$ | $\delta E_{h\text{VP,Sph}}$ | $\delta E_{h\text{VP,Fermi}}$ | $\delta g_{h\text{VP,PN}}$ | $\delta g_{h\text{VP,Sph}}$ | $\delta g_{h\text{VP,Fermi}}$
---|---|---|---|---|---|---
$\prescript{12}{6}{\text{C}}$ | $-1.492\times 10^{-9}$ | $-1.2838(2)\times 10^{-9}$ | $-1.2885(4)\times 10^{-9}$ | $-5.904\times 10^{-9}$ | $-4.829(2)\times 10^{-9}$ | $-4.854(1)\times 10^{-9}$
$\prescript{16}{8}{\text{O}}$ | $-4.710\times 10^{-9}$ | $-3.752(2)\times 10^{-9}$ | $-3.770(2)\times 10^{-9}$ | $-1.860\times 10^{-8}$ | $-1.368(1)\times 10^{-8}$ | $-1.378(1)\times 10^{-8}$
$\prescript{20}{10}{\text{Ne}}$ | $-1.150\times 10^{-8}$ | $-8.293(2)\times 10^{-9}$ | $-8.339(2)\times 10^{-9}$ | $-4.528\times 10^{-8}$ | $-2.908(1)\times 10^{-8}$ | $-2.932(1)\times 10^{-8}$
$\prescript{28}{14}{\text{Si}}$ | $-4.434\times 10^{-8}$ | $-2.717(1)\times 10^{-8}$ | $-2.738(3)\times 10^{-8}$ | $-1.735\times 10^{-7}$ | $-8.931(5)\times 10^{-8}$ | $-9.037(10)\times 10^{-8}$
$\prescript{38}{18}{\text{Ar}}$ | $-1.222\times 10^{-7}$ | $-6.068(2)\times 10^{-8}$ | $-6.131(3)\times 10^{-8}$ | $-4.748\times 10^{-7}$ | $-1.839(1)\times 10^{-7}$ | $-1.869(1)\times 10^{-7}$
$\prescript{40}{20}{\text{Ca}}$ | $-1.875\times 10^{-7}$ | $-8.428(4)\times 10^{-8}$ | $-8.527(5)\times 10^{-8}$ | $-7.253\times 10^{-7}$ | $-2.461(1)\times 10^{-7}$ | $-2.507(2)\times 10^{-7}$
$\prescript{66}{30}{\text{Zn}}$ | $-9.955\times 10^{-7}$ | $-2.554(1)\times 10^{-7}$ | $-2.600(1)\times 10^{-7}$ | $-3.771\times 10^{-6}$ | $-6.102(3)\times 10^{-7}$ | $-6.279(4)\times 10^{-7}$
$\prescript{86}{36}{\text{Kr}}$ | $-2.152\times 10^{-6}$ | $-3.914(3)\times 10^{-7}$ | $-3.996(3)\times 10^{-7}$ | $-8.038\times 10^{-6}$ | $-8.333(8)\times 10^{-7}$ | $-8.625(8)\times 10^{-7}$
$\prescript{90}{40}{\text{Zr}}$ | $-3.390\times 10^{-6}$ | $-5.000(2)\times 10^{-7}$ | $-5.116(2)\times 10^{-7}$ | $-1.254\times 10^{-5}$ | $-9.955(5)\times 10^{-7}$ | $-1.0346(5)\times 10^{-6}$
$\prescript{120}{50}{\text{Sn}}$ | $-9.162\times 10^{-6}$ | $-7.480(6)\times 10^{-7}$ | $-7.681(6)\times 10^{-7}$ | $-3.296\times 10^{-5}$ | $-1.245(1)\times 10^{-6}$ | $-1.304(1)\times 10^{-6}$
$\prescript{136}{54}{\text{Xe}}$ | $-1.308\times 10^{-5}$ | $-8.431(15)\times 10^{-7}$ | $-8.667(16)\times 10^{-7}$ | $-4.652\times 10^{-5}$ | $-1.311(3)\times 10^{-6}$ | $-1.377(3)\times 10^{-6}$
$\prescript{142}{60}{\text{Nd}}$ | $-2.165\times 10^{-5}$ | $-1.014(1)\times 10^{-6}$ | $-1.044(1)\times 10^{-6}$ | $-7.548\times 10^{-5}$ | $-1.447(2)\times 10^{-6}$ | $-1.526(2)\times 10^{-6}$
$\prescript{176}{70}{\text{Yb}}$ | $-4.722\times 10^{-5}$ | $-1.183(3)\times 10^{-6}$ | $-1.220(3)\times 10^{-6}$ | $-1.589\times 10^{-4}$ | $-1.437(5)\times 10^{-6}$ | $-1.523(5)\times 10^{-6}$
$\prescript{185}{75}{\text{Re}}$ | $-6.849\times 10^{-5}$ | $-1.331(9)\times 10^{-6}$ | $-1.376(10)\times 10^{-6}$ | $-2.259\times 10^{-4}$ | $-1.529(14)\times 10^{-6}$ | $-1.625(15)\times 10^{-6}$
$\prescript{208}{82}{\text{Pb}}$ | $-1.141\times 10^{-4}$ | $-1.4865(8)\times 10^{-6}$ | $-1.5381(8)\times 10^{-6}$ | $-3.650\times 10^{-4}$ | $-1.565(1)\times 10^{-6}$ | $-1.671(1)\times 10^{-6}$
$\prescript{209}{83}{\text{Bi}}$ | $-1.226\times 10^{-4}$ | $-1.508(2)\times 10^{-6}$ | $-1.560(2)\times 10^{-6}$ | $-3.906\times 10^{-4}$ | $-1.568(2)\times 10^{-6}$ | $-1.675(2)\times 10^{-6}$
$\prescript{212}{86}{\text{Rn}}$ | $-1.524\times 10^{-4}$ | $-1.563(11)\times 10^{-6}$ | $-1.618(12)\times 10^{-6}$ | $-4.785\times 10^{-4}$ | $-1.567(14)\times 10^{-6}$ | $-1.676(16)\times 10^{-6}$
$\prescript{238}{92}{\text{U}}$ | $-2.359\times 10^{-4}$ | $-1.584(2)\times 10^{-6}$ | $-1.639(2)\times 10^{-6}$ | $-7.182\times 10^{-4}$ | $-1.455(2)\times 10^{-6}$ | $-1.558(3)\times 10^{-6}$

###  V.2 Hadronic Vacuum Polarization

Unlike leptons, hadrons are composite particles that interact and are bound
together via the strong interaction, which is decribed by the theory of
quantum chromodynamics (QCD). In QCD, unlike in QED, the perturbative
calculation methods fail due to the relatively strong coupling of the quark
and gluon fields [46], which renders the calculation of the interaction of
virtual hadronic particles a difficult task, and the hadronic vacuum
polarization function appearing in the formula for the Uehling potential is
difficult to calculate from theory alone. Instead, a common method in the
calculations of the hadronic vacuum polarization is to construct a
polarization function from experimental data [47]. Using this polarization
function, it is possible to calculate the Uehling potential, as was done in
[48, 49].

The Uehling potential for an arbitrary polarization function $\Pi$ and a
spherically symmetric charge distribution can be expressed as [50]

$V_{\text{Ueh}}(r)=-\frac{e}{2\pi^{2}}\int_{0}^{\infty}\differential{q}j_{0}(qr)\tilde{\rho}(q)\real[\Pi^{R}(-q^{2})],$
(50)

where $j_{0}(x)$ is the spherical Bessel function of the first kind,
$\tilde{\rho}(q)$ is the Fourier transform of the charge distribution
$\rho(r)$, and $\Pi^{R}(-q^{2})$ is the regular part of the polarization
function.

In the case of the hadronic vacuum polarization, the polarization function is
written as [51]

$\real[\Pi^{R}(q^{2})]=A_{i}+B_{i}\log(1+C_{i}\absolutevalue{q^{2}}),$ (51)

where $A_{i}$, $B_{i}$, and $C_{i}$ have different values depending on the
range of $q$, as given in Table 10.

Table 10: Piecewise parametrization of the hadronic polarization function in Eq. (51), as given in [51, 48] with the mass of the Z boson $m_{Z}=91.1876\text{ GeV}$. Adapted from Ref. [48]. $q$ (GeV) | $A_{i}$ | $B_{i}$ | $C_{i}\,(\text{GeV}^{-2})$
---|---|---|---
0.0$-$0.7 | 0.0 | 0.0023092 | 3.9925370
0.7$-$2.0 | 0.0 | 0.0022333 | 4.2191779
2.0$-$4.0 | 0.0 | 0.0024402 | 3.2496684
4.0$-$10.0 | 0.0 | 0.0027340 | 2.0995092
10.0$-$$m_{Z}$ | 0.0010485 | 0.0029431 | 1.0
$m_{Z}$$-$$10^{4}$ | 0.0012234 | 0.0029237 | 1.0
$10^{4}$$-$$10^{5}$ | 0.0016894 | 0.0028984 | 1.0

In our investigation, we obtained results both by numerical integration of the
expression in Eq. (50) using the piecewise defined polarization function in
Eq. (51) and Table 10, and by using the analytical formula for the hadronic
Uehling potential for homogeneous sphere charge distribution nucleus, derived
in [48] with the approximation that the low energy region of $q$ in Eq. (51)
can be extended to infinity, and the values $A=0$, $B=0.0023092$, and
$C=3.9925370\text{ GeV}^{-2}$ can be used for the entire range of $q$. This
approximation works better for the extended nucleus Uehling potential than the
point-like nucleus Uehling potential, as the Fourier transform
$\tilde{\rho}(q)$ of the nuclear charge distribution appearing in Eq. (50) is

$\tilde{\rho}(q)=3Ze\frac{j_{1}(qr_{0})}{qr_{0}},$ (52)

which is a function approaching $0$ as $q\rightarrow\infty$ for the extended
nucleus, but becomes $\tilde{\rho}(q)=Ze$ for the point-like nucleus
($r_{0}\rightarrow 0$), meaning that the contributions of the large $q$
regions in Eq. (50) is greater for the point-like nucleus than the extended
nucleus. We found that the results obtained from both methods agree in the
first three decimal digits for the extended nucleus case, and the difference
between both methods is smaller than the rms radius uncertainties in the
hadronic vacuum polarization corrections. In Table 9, we present the results
that were obtained numerically with the piecewise defined polarization
function in Eq. (51) and Table 10.

##  VI Discussion of Results

Figure 3: Comparison of the magnitudes of various contributions to the ground
state energies of bound muons for several atoms. The plotted vacuum
polarization corrections are for the spherical nucleus case. The leading
uncertainty is the uncertainty in the leading finite size corrections due to
the rms radius uncertainties. Figure 4: Comparison of the magnitudes of
various contributions to the $g$-factors of bound muons for several atoms. The
plotted vacuum polarization corrections are for the spherical nucleus case.
The leading uncertainty is the uncertainty in the leading finite size
corrections due to the rms radius uncertainties.

An overview of the corrections to the bound muon ground state energies and
$g$-factors that were discussed so far are depicted in Figures 3 and 4,
together with the uncertainty in the leading finite size corrections due to
the known rms radius uncertainties.

In comparison to electronic atoms, nuclear effects in muonic atoms are
amplified due to the larger mass of the muon, which manifests itself clearly
in the corrections to the ground state energies and $g$-factors. The nuclear
charge distribution model dependence is amplified, and is larger than the
leading uncertainties for all elements, and for nuclei with large octupole and
quadrupole deformations, even the nuclear deformation corrections can be
larger than the uncertainties due to the rms radii, which is importantly the
case for uranium. This is different from the case of the bound-electron
$g$-factor in light ions, where the nuclear model uncertainty is much smaller
than the uncertainty due to the rms radii [11, 38, 4].

Unlike the nuclear model dependence, the finite size of the nucleus causes the
vacuum polarization corrections to be suppressed in comparison to the vacuum
polarization corrections in the case of a point-like nucleus, and the
suppression becomes more pronounced for heavier elements. This can be
conceptually understood as follows: for a finite-size nucleus model, the
potential is the same as the Coulomb potential outside the nucleus, but is
smaller in magnitude inside the nucleus. Therefore, virtual particles that are
created inside the nucleus is under the effect of a weaker potential, which
results in a weaker vacuum polarization and shielding of the nuclear charge.
This suppression of the electric-loop vacuum polarization corrections become
more evident for the $g$-factors, and for heavier elements, difference of
leading finite size corrections for different nuclear models due to the
different nuclear charge distribution models become even greater than the
electronic vacuum polarization corrections, as can be seen in Figure 4.

In the case of the free muon $g$-factor, electronic vacuum polarization is
found to be strongly enhanced compared to electronic vacuum polarization in
the free-electron $g$-factor [52], since vacuum polarization effects depend
only on the mass ratio of the reference lepton and the lepton in the vacuum
polarization loop. Similarly, we find electronic vacuum polarization in the
bound-muon $g$-factor to be enhanced compared to the bound-electron $g$-factor
(see e.g. results in Ref. [2].) However, for heavy muonic atoms, for $Z>86$,
we find the suppression of the vacuum polarization effect due to the finite
nuclear size to be stronger than the expected enhancement due to the lepton
mass ratio. Therefore, surprisingly, the absolute impact of electronic vacuum
polarization is smaller in the heaviest muonic atoms than in the corresponding
electronic H-like ions.

The muonic vacuum polarization contribution to the $g$-factor in muonic atoms
is found to be smaller than electronic vacuum polarization in electronic ions,
due to the stronger suppression due to the finite nuclear size. (In point-
nucleus calculations, electronic vacuum polarization in electronic ions and
muonic vacuum polarization in muonic ions are identical.)

The suppression in the muonic and hadronic vacuum polarization corrections are
even larger than the suppression to the electronic vacuum polarization
corrections: the point-like nucleus approximation overestimates the muonic and
hadronic vacuum polarization corrections to the $g$-factors by more than two
orders of magnitude for heavy elements. In the point-like nucleus
approximation, the hadronic vacuum polarization corrections also appear to be
larger than the muonic vacuum polarization corrections for heavy elements
(compare Tables 8 and 9), which is due to the hadronic Uehling potantial for
point-like nucleus being larger in magnitude than the muonic Uehling potential
in the small $r$ region (see e.g. Figure 1 in Ref [48]), but our results
confirm that for extended nuclei, muonic vacuum polarization corrections are
strictly larger than the hadronic vacuum polarization corrections for all
elements (see Figures 3 and 4).

Following the results we obtained, it becomes clear that the dependece on
nuclear charge distribution model significantly affects the properties of
bound muons, and it is necessary to construct precise theoretical models for
the nucleus in order to be able to have accurate theoretical predictions for
the properties of the bound muon. A significant contributor that was not
considered in this work is the nuclear polarization effect, which has been
studied for electronic atoms [53, 54] and for the energy levels of bound muons
[55, 24], but remains undetermined for the $g$-factors.

Among the QED effects, the self-energy corrections and the magnetic-loop
vacuum polarization corrections to the $g$-factors were not considered in this
work. Both of these effects have been studied heavily in the point-nucleus
case (such as in [2, 56, 57]), but it becomes significantly more difficult to
account for the finite size of the nucleus in those effects. Nuclear model
dependence of the self-energy correction to the energy levels of muonic atoms
has recently been studied in [25], but the finite-nuclear-size self-energy
corrections to the $g$-factors have still not been worked out.

###### Acknowledgements.

A.Ç. would like to thank the Theoretical Quantum Dynamics and Quantum
Electrodynamics division of the Max Planck Institute for Nuclear Physics for
their hospitality during his stay and internship in Heidelberg, where this
work was completed. Supported by the Deutsche Forschungsge- meinschaft (DFG,
German Research Foundation) Project-ID 273811115 - SFB 1225.

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