On Hyperfine Structure of the Rotationally Excited 1s_μnℓ_e-States in Helium Muonic Atoms

1. Introduction

In this short communication we report our recent results and derive the formulas which can directly be used to determine the hyperfine structure of the rotationally excited $1s_{\mu }n{\ell }_e$-states in helium muonic atoms. As is well known the hyperfine structure of any one-electron atomic and/or quasi-atomic system can be analyzed by using the relativistic Dirac equation [1]. For one-electron atoms and ions this was done by E. Fermi in his works published in 1930 [2,3]. Here we want to generalize that method to more complicated three-body systems each of which contains one electron $e^{-}$ and two heavy particles: the `central’ helium nucleus He²⁺ and negatively charged muon ${\mu }^{-}$.

In our previous papers (see, e.g., [4,5]) we investigated the hyperfine structure of the three-body helium-muonic ³He $\mu e$ and ⁴He $\mu e$ atoms which are located in the $1s_{\mu }n{s}_e$-states. The general formula for the Hamiltonian of hyperfine structure in the case of bound $1s_{\mu }n{s}_e$-states written in atomic units, where $\hslash =1,m_e=1,e=1$ (see below), takes the form (see, e.g., [4,5]):

$$\begin{array}{ccc}\hfill {\widehat{H}}_{HFS}& =& -{\displaystyle \frac{8\pi {\alpha }^2}{3}}{g}_N{g}_{\mu }\left({\displaystyle \frac{m_e}{M_p}}\right)\left({\displaystyle \frac{m_e}{M_{\mu }}}\right){\left({\mu }_{B}c\right)}^2\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_{\mu })\hfill \\ & -& {\displaystyle \frac{8\pi {\alpha }^2}{3}}{g}_e{g}_{\mu }\left({\displaystyle \frac{m_e}{M_{\mu }}}\right){\left({\mu }_{B}c\right)}^2\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\hfill \\ & -& {\displaystyle \frac{8\pi {\alpha }^2}{3}}{g}_N{g}_e\left({\displaystyle \frac{m_e}{M_p}}\right){\left({\mu }_{B}c\right)}^2\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e),\hfill \end{array}$$

(1)

where $\alpha ={\displaystyle \frac{e^2}{\hslash c}}\approx {\displaystyle \frac{1}{137.04}}$ is the dimensionless fine-structure constant, c is the velocity of light in vacuum and ${\mu }_B={\displaystyle \frac{\hslash e^2}{2m_{e}c}}$ is the Bohr magneton, which is also often called the atomic magneton.

Almost everywhere in this study we apply the well known atomic units, where $\hslash =1,m_e=1,e=1$, where e is the absolute value of the electron’s electric charge, i.e., e is a positive value. Atomic units are very convenient for analytical and numerical investigations of various atomic and quasi-atomic systems. In particular, in atomic units one finds $c={\alpha }^{-1},{\mu }_B={\displaystyle \frac{\alpha }{2}}$ and ${\mu }_{B}c={\displaystyle \frac{1}{2}}$. This immediately leads us to the final formula

$$\begin{array}{ccc}\hfill {\widehat{H}}_{HFS}& =& -{\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_N{g}_{\mu }\left({\displaystyle \frac{m_e}{M_p}}\right)\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_{\mu })\hfill \\ & -& {\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_e{g}_{\mu }\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)-{\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_N{g}_e\left({\displaystyle \frac{m_e}{M_p}}\right)\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e),\hfill \end{array}$$

(2)

which is used in the direct numerical calculations of the hyperfine structure splittings in all bound $S(L=0)-$states of the helium-muonic atoms.

Note that the notation N in Eqs.(1) and (2) stands for the ³He and ⁴He nuclei, while symbols e and $\mu$ mean the electron and muon, respectively. The $g_e,g_{\mu }$ and $g_N$ factors are the gyromagnetic ratios of these particles, or $g-$factors, for short. Also in these equations the notation $M_p$ means the proton’s mass. The notation $\langle \delta \left({\mathbf{r}}_{ij}\right)\rangle$ traditionally denotes the expectation value of the $\left(ij\right)$ interparticle delta-function, while the symbols ${\mathbf{I}}_N,{\mathbf{s}}_{\mu }$ and ${\mathbf{s}}_e$ designate the spins of the helium nucleus, muon and electron, respectively. The absolute values of these three spins are always either integer, o r semi-integer. For the muonic helium-4 atom we have ${\mathbf{I}}_N=0$ and our formula, Equation(2), drastically simplifies, since it is written in the one-term form:

$$\begin{array}{c}\hfill H_{HFS}=-{\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_e{g}_{\mu }\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e),\end{array}$$

(3)

where all notations have the same meaning as in Equation(2).

In the both formulas, Eqs.(2) and (3), in this study we use the following values of the fundamental and physical constants [6]: $\alpha =7.2973525693·{10}^{-3},g_{e^{-}}=-2.00231930436256,g_{{\mu }^{-}}=-2.0023318418,{\mu }_{B}c={\displaystyle \frac{1}{2}}=0.5$ and $M_{\mu }$ = 206.7682830 $m_e$. Also, in these equations $M_p$ = 1836.15267343 $m_e$ is the proton mass, while $g_N$ is the $g-$factor of the helium-3 nucleus, which is determined by the formula: $g_N={\displaystyle \frac{{\mathcal{M}}_N}{I_N}}$ = - 4.255250615, where ${\mathcal{M}}_N$ = - 2.127625307 is the magnetic moments (in nuclear magnetons ${\mu }_N=\left({\displaystyle \frac{m_e}{M_p}}\right){\mu }_B$) of the helium-3 nucleus, while $I_N={\displaystyle \frac{1}{2}}$ is its spin. The expectation values of all two-particle delta-functions in Eqs.(2) and (3) must be expressed in atomic units. Note also that traditionally (in fact, since the end of 1940’s) the hyperfine structure splittings in atoms and molecules are expressed in $Hertz$ (or $Hz$), or in $MegaHertz$ ($MHz$), where 1 $MHz=1·{10}^{6}Hz$. To re-calculate the energies expressed in atomic units into $MHz$ we apply the conversion factor $Ry$ = 6.5796839205016$·{10}^{9}MHz{(a.u.)}^{-1}$ [6].

The hyperfine Hamiltonian, Eqs.(2) and/or (3), is the matrix with the dimension $(2I+1)(2s_{\mu }+1)(2s_e+1)$. To determine the hyperfine structure of helium muonic atoms we need to determine all eigenvalues (with the corresponding multiplicities) of this matrix. The differences between eigenvalues (or groups of eigenvalues) give us the corresponding hyperfine structure splittings, if they are expressed in $MHz$.

1.1. Results for Some Low-Lying $S(L=0)-$States

The formulas, Eqs.(2) and (3), presented in the first Section allow one to determine the hyperfine structure splittings of the bound $1s_{\mu }n{s}_e$-states in the helium-muonic ³He $\mu e$ and ⁴He $\mu e$ atoms. Currently, the hyperfine structure splittings $\Delta$ of the bound $1s_{\mu }n{s}_e$-states for $n=1,2,3$ are evaluated (in $MHz$) as:

$$\begin{array}{c}\hfill 4166.38685445\left(3\right)(\mathrm{for}n=1),520.78609121\left(5\right)(\mathrm{for}n=2),154.30584687\left(4\right)(\mathrm{for}n=3)\end{array}$$

for the helium-muonic ³He $\mu e$ atom and

$$\begin{array}{c}\hfill 4464.55409995\left(3\right)(\mathrm{for}n=1),558.05491883\left(5\right)(\mathrm{for}n=2),165.34833511\left(4\right)(\mathrm{for}n=3)\end{array}$$

for the helium-muonic ⁴He $\mu e$ atom. The overall numerical accuracy of our current variational computations for the bound $1s_{\mu }n{s}_e$-states in three-body helium-muonic atoms is very high and constantly increases. Briefly, we can say that as follows from numerical accuracy of these results the long standing problem of the hyperfine structure splittings for the bound $1s_{\mu }n{s}_e$-states in helium muonic atoms has been solved completely and accurately.

2. Hyperfine Structure of the Rotationally Excited States

The next step in research of the hyperfine structure of helium-muonic atoms is to consider and solve similar problem for the rotationally excited $1s_{\mu }n{\ell }_e$-states in these atomic systems. However, this is not an easy task, since currently we simply do not have any closed analytical formula(s) which allow one to determine the hyperfine structure splittings in the rotationally excited (bound) states of the helium-muonic atoms. Formally, such formulas have not been derived yet. The main goal of this Section (and this study) is to derive the closed analytical formulas which allow one to evaluate the hyperfine structure splittings for arbitrary rotationally excited $1s_{\mu }n{\ell }_e$-states in the helium muonic atoms. Currently, we know that many of these rotationally excited $1s_{\mu }n{\ell }_e$-states in the helium muonic atoms are stable (see, discussion and our results in [5]). It can be expected that the rotationally excited $1s_{\mu }2p_e$- and $1s_{\mu }3d_e$-states will be the first bound P- and D-states in helium muonic atoms for which it will be possible to study their hyperfine structure.

First, let us derive the correct formulas which can be used to determine the hyperfine structure in the rotationally excited $1s_{\mu }n{\ell }_e$ states in three-body helium muonic atoms. For this purpose let us consider the total vector potential $\overrightarrow{A}$ and magnetic field strength $\overrightarrow{H}$ which act on a single electron. The vector potential $\overrightarrow{A}$, which acts on a single electron $e^{-}$, is written in the form

$$\begin{array}{c}\hfill \overrightarrow{A}={\displaystyle \frac{({\overrightarrow{\mu }}_N\times {\overrightarrow{r}}_{31})}{r_{31}^3}}+{\displaystyle \frac{({\overrightarrow{\mu }}_{\mu }\times {\overrightarrow{r}}_{21})}{r_{21}^3}},\end{array}$$

(4)

where the particles 3 and 2 mean the atomic (helium) nucleus and negatively charged muon ${\mu }^{-}$, while the particle 1 always designates the electron $e^{-}$. The notations ${\overrightarrow{\mu }}_N$ and ${\overrightarrow{\mu }}_{\mu }$ stand for the magnetic moments of the nucleus and ${\mu }^{-}$ muon, respectively. This system of notation is used everywhere below in this study. As follows from these formulas the magnetic interaction between bound (atomic) electron and total vector potential equals to the scalar product of the vector potential $\overrightarrow{A}$ and electron’s momentum ${\overrightarrow{p}}_e$

$$\begin{array}{c}\hfill {\displaystyle \frac{e\hslash }{m_{e}c}}\overrightarrow{A}·{\overrightarrow{p}}_e={\displaystyle \frac{e\hslash }{m_{e}c}}\left[{\displaystyle \frac{1}{r_{31}^3}}({\overrightarrow{\mu }}_N·{\overrightarrow{\ell }}_{31})+{\displaystyle \frac{1}{r_{21}^3}}({\overrightarrow{\mu }}_{\mu }·{\overrightarrow{\ell }}_{21})\right].\end{array}$$

(5)

where $e=\mid e^{-}\mid$ is the absolute electric charge of the electron, i.e., it is a positive value. The mixed product included in the last equation is transformed in the following way: $({\overrightarrow{\mu }}_k\times {\overrightarrow{r}}_{k1})·{\overrightarrow{p}}_e={\overrightarrow{\mu }}_k({\overrightarrow{r}}_{k1}\times {\overrightarrow{p}}_e)={\overrightarrow{\mu }}_k·{\overrightarrow{\ell }}_{k1}$, where $k=(N,\mu )=(3,2)$ and ${\overrightarrow{\ell }}_{k1}$ is the electron’s angular moment in respect to the `central’ particle k. In our analysis below we shall follow [7,8,9].

The total magnetic field strength $\overrightarrow{H}$, which acts on the same electron, is

$$\begin{array}{c}\hfill \overrightarrow{H}={\displaystyle \frac{3{\overrightarrow{n}}_{31}({\overrightarrow{\mu }}_N·{\overrightarrow{n}}_{31})-{\overrightarrow{\mu }}_N}{r_{31}^3}}+{\displaystyle \frac{3{\overrightarrow{n}}_{21}({\overrightarrow{\mu }}_{\mu }·{\overrightarrow{n}}_{21})-{\overrightarrow{\mu }}_{\mu }}{r_{21}^3}},\end{array}$$

(6)

where the first term represents the nucleus magnetic component of the total magnetic field strength $\overrightarrow{H}$, while the second term describes similar contribution from the muon magnetic component. The corresponding part of the magnetic interaction between these two heavy particles and electron equals to the scalar product ${\overrightarrow{\mu }}_e·\overrightarrow{H}$, where ${\overrightarrow{\mu }}_e$ is the magnetic moment of the electron, while $\overrightarrow{H}$ is deined in Equation(6). Now, we can introduce the hyperfine Hamiltonian, or Hamiltonian of the hyperfine structure (see, e.g., [7]) for the electron-nucleus pair:

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(Ne\right)}=-{\displaystyle \frac{8\pi }{3}}{\overrightarrow{\mu }}_N·{\overrightarrow{\mu }}_e\delta \left({\overrightarrow{r}}_{31}\right)-{\displaystyle \frac{1}{r_{31}^3}}\left[3({\overrightarrow{n}}_{31}·{\overrightarrow{\mu }}_N)({\overrightarrow{n}}_{31}·{\overrightarrow{\mu }}_e)-({\overrightarrow{\mu }}_N·{\overrightarrow{\mu }}_e)+{\displaystyle \frac{e}{m_{e}c}}({\overrightarrow{\ell }}_{31}·{\overrightarrow{\mu }}_N)\right].\end{array}$$

(7)

Analogously, we can write the hyperfine Hamiltonian for the electron-muon pair

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(\mu e\right)}=-{\displaystyle \frac{8\pi }{3}}{\overrightarrow{\mu }}_{\mu }·{\overrightarrow{\mu }}_e\delta \left({\overrightarrow{r}}_{21}\right)-{\displaystyle \frac{1}{r_{21}^3}}\left[3({\overrightarrow{n}}_{21}·{\overrightarrow{\mu }}_{\mu })({\overrightarrow{n}}_{21}·{\overrightarrow{\mu }}_e)-({\overrightarrow{\mu }}_{\mu }·{\overrightarrow{\mu }}_e)+{\displaystyle \frac{e}{m_{e}c}}({\overrightarrow{\ell }}_{21}·{\overrightarrow{\mu }}_{\mu })\right]\end{array}$$

(8)

and for the muon-nucleus pair

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(N\mu \right)}=-{\displaystyle \frac{8\pi }{3}}{\overrightarrow{\mu }}_N·{\overrightarrow{\mu }}_{\mu }\delta \left({\overrightarrow{r}}_{32}\right)-{\displaystyle \frac{1}{r_{32}^3}}\left[3({\overrightarrow{n}}_{32}·{\overrightarrow{\mu }}_N)({\overrightarrow{n}}_{32}·{\overrightarrow{\mu }}_{\mu })-({\overrightarrow{\mu }}_N·{\overrightarrow{\mu }}_{\mu })+{\displaystyle \frac{e}{m_{\mu }c}}({\overrightarrow{\ell }}_{32}·{\overrightarrow{\mu }}_N)\right].\end{array}$$

(9)

The total hyperfine Hamiltonian is written as the sum of these three terms:

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}={\mathcal{H}}_{HFS}^{\left(Ne\right)}+{\mathcal{H}}_{HFS}^{\left(\mu e\right)}+{\mathcal{H}}_{HFS}^{\left(N\mu \right)}.\end{array}$$

(10)

However, in applications to the bound states in helium muonic atoms this Hamiltonian contains a number of terms which either equal zero identically, or provide only a very small (negligible) contribution. It is clear that similar terms must be excluded from the actual Hamiltonian of the hyperfine structure. In reality, this can be done by considering the internal structure of the three-body helium muonic atoms which is discussed in the Appendix.

First, let us consider the electron-nucleus part of our hyperfine Hamiltonian ${\mathcal{H}}_{HFS}^{\left(Ne\right)}$, Equation(7). In atomic units the first term of this Hamiltonian is

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{HFS}^{\left(Ne\right)}\left(1\right)=-{\displaystyle \frac{8\pi }{3}}{\overrightarrow{\mu }}_N·{\overrightarrow{\mu }}_e\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle & =& -{\displaystyle \frac{8\pi }{3}}{\mu }_B^2{\displaystyle \frac{g_e{g}_N}{M_p}}({\mathbf{I}}_N·{\mathbf{s}}_e)\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle \hfill \\ & =& -{\displaystyle \frac{2\pi {\alpha }^2}{3}}{\displaystyle \frac{g_e{g}_N}{M_p}}({\mathbf{I}}_N·{\mathbf{s}}_e)\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle ,\hfill \end{array}$$

(11)

where ${\mu }_B={\displaystyle \frac{e\hslash }{2m_{e}c}}={\displaystyle \frac{1}{2}}\alpha$ and $M_p={\displaystyle \frac{M_p}{m_e}}$. The second electron-nucleus term is written in a tensor form

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(Ne\right)}\left(2\right)=-{\displaystyle \frac{1}{4}}{\alpha }^2{\displaystyle \frac{g_e{g}_N}{M_p}}\left[3\langle {\displaystyle \frac{1}{r_{31}^3}}({\mathbf{I}}_N·{\mathbf{n}}_{31})({\mathbf{n}}_{31}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e)\right].\end{array}$$

(12)

The last (or third) term included in the electron-nucleus part of the hyperfine Hamiltonian is the pure orbital term

$$\begin{array}{ccc}\hfill {\mathcal{H}}_{HFS}^{\left(Ne\right)}\left(3\right)=-\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle \left({\displaystyle \frac{e\hslash }{m_{e}c}}\right){\overrightarrow{\mu }}_N·{\overrightarrow{\ell }}_{31}& =& -{\displaystyle \frac{1}{2}}{\alpha }^2{\mu }_B^2{\displaystyle \frac{g_N}{M_p}}\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{L}}_{31})\hfill \\ & =& -{\displaystyle \frac{1}{2}}{\alpha }^2{\displaystyle \frac{g_N}{M_p}}\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{L}}_{31}),\hfill \end{array}$$

(13)

where ${\displaystyle \frac{e\hslash }{m_{e}c}}=2{\mu }_B\alpha$ and ${\mathbf{L}}_{31}={\overrightarrow{\ell }}_{31}$ is the electron’s angular momentum in respect to the helium nucleus.

The electron-muon part of the hyperfine Hamiltonian ${\mathcal{H}}_{HFS}^{\left(\mu e\right)}$, Equation(8), is derived analogously. In particular, in this part one finds the explicit expression for the first term:

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(\mu e\right)}\left(1\right)=-{\displaystyle \frac{2\pi {\alpha }^2}{3}}{\displaystyle \frac{g_e{g}_{\mu }}{M_{\mu }}}({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle ,\end{array}$$

(14)

where $M_{\mu }={\displaystyle \frac{M_{\mu }}{m_e}}$. The second electron-muon term is also written in tensor form:

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(\mu e\right)}\left(2\right)=-{\displaystyle \frac{1}{4}}{\alpha }^2{\displaystyle \frac{g_e{g}_{\mu }}{M_{\mu }}}\left[3\langle {\displaystyle \frac{1}{r_{21}^3}}({\mathbf{s}}_{\mu }·{\mathbf{n}}_{21})({\mathbf{n}}_{21}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\right].\end{array}$$

(15)

The orbital term in the electron-muon part of the hyperfine Hamiltonian is

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(\mu e\right)}\left(3\right)=-\left({\displaystyle \frac{e\hslash }{m_{e}c}}\right){\mu }_B{\displaystyle \frac{g_{\mu }}{M_{\mu }}}\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{L}}_{21})=-{\displaystyle \frac{1}{2}}{\alpha }^2{\displaystyle \frac{g_{\mu }}{M_{\mu }}}\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{L}}_{21}),\end{array}$$

(16)

where ${\mu }_B={\displaystyle \frac{e\hslash }{2m_{e}c}}={\displaystyle \frac{1}{2}}\alpha$ in atomic units and ${\mathbf{L}}_{21}={\overrightarrow{\ell }}_{21}$ is the electron’s angular momentum in respect to the negatively charged muon ${\mu }^{-}$.

The muon-nucleus part of the hyperfine Hamiltonian ${\mathcal{H}}_{HFS}^{\left(N\mu \right)}$, Equation(9), in the helium-muonic atoms contains only one term which takes the familiar form:

$$\begin{array}{c}\hfill {\mathcal{H}}_{HFS}^{\left(N\mu \right)}\left(1\right)=-{\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_N{g}_{\mu }\left({\displaystyle \frac{m_e}{M_p}}\right)\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_{\mu }).\end{array}$$

(17)

The total hyperfine Hamiltonian of the helium-muonic atoms is the sum of three terms mentioned above, i.e., ${\mathcal{H}}_{HFS}^{\left(Ne\right)},{\mathcal{H}}_{HFS}^{\left(\mu e\right)}$ and ${\mathcal{H}}_{HFS}^{\left(N\mu \right)}$. However, some of the expectation values equal zero identically, i.e., they do not contribute into the final sum. In detail this problem is considered below (also see Appendix) .

3. Total Hyperfine Hamiltonian of the Helium-Muonic Atoms

The total hyperfine Hamiltonian, which is designated below as ${\widehat{H}}_{HFS}$, is the sum of all partial $HFS-$Hamiltonians ${\mathcal{H}}_{HFS}^{\left(Ne\right)},{\mathcal{H}}_{HFS}^{\left(\mu e\right)}$ and ${\mathcal{H}}_{HFS}^{\left(N\mu \right)}$ from Eqs.(11) - (17). This total hyperfine Hamiltonian is written in the form

$$\begin{array}{ccc}\hfill {\widehat{H}}_{HFS}& =& -{\displaystyle \frac{2\pi {\alpha }^2}{3}}\{{\displaystyle \frac{g_e{g}_N}{M_p}}\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e)+{\displaystyle \frac{g_e{g}_{\mu }}{M_{\mu }}}\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\hfill \\ & +& g_N{g}_{\mu }\left({\displaystyle \frac{m_e}{M_p}}\right)\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_{\mu })\}\hfill \\ & -& {\displaystyle \frac{1}{4}}{\alpha }^2\{{\displaystyle \frac{g_e{g}_N}{M_p}}\left[3\langle {\displaystyle \frac{1}{r_{31}^3}}({\mathbf{I}}_N·{\mathbf{n}}_{31})({\mathbf{n}}_{31}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e)\right]\hfill \\ & +& {\displaystyle \frac{g_e{g}_{\mu }}{M_{\mu }}}\left[3\langle {\displaystyle \frac{1}{r_{21}^3}}({\mathbf{s}}_{\mu }·{\mathbf{n}}_{21})({\mathbf{n}}_{21}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\right]\}\hfill \\ & -& {\displaystyle \frac{1}{2}}{\alpha }^2\left[{\displaystyle \frac{g_N}{M_p}}\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{L}}_{31})+{\displaystyle \frac{g_{\mu }}{M_{\mu }}}\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{L}}_{21})\right],\hfill \end{array}$$

(18)

where the first term essentially coincides with the Hamiltonian ${\widehat{H}}_{HFS}$ defined in Equation(2). The second term in this equation is a tensor which has zero contribution for all bound $S(L=0)-$states (or for the bound $1s_{\mu }n{s}_e-$states) in the helium-muonic atoms. The last (third) term is the spin-orbital term which describes interaction between the nuclear and/or muonic spins and electron which has non-zero angular momentum in respect to each of these particles. As expected the last term in Equation(18) does not include the $g_e-$factor. This term really contributes only for the rotationally excited $1s_{\mu }n{\ell }_e-$states, while for the $1s_{\mu }n{s}_e-$states in helium-muonic atoms its contribution equals zero identically.

As mentioned above in any bound state of the helium muonic atom the internal muonic orbital is always spherically symmetric, since it is the $1s_{\mu }-$orbital. For this orbital the $\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle$ expectation value is always a positive, non-zero value. Now, let us consider the case of rotationally excited electron’s states in helium-muonic atoms. In any of these states the both electron-nucleus and/or electron-muonic orbitals do not have spherical symmetry. Therefore, the both $\langle \delta \left({\mathbf{r}}_{Ne}\right)\rangle$ and $\langle \delta \left({\mathbf{r}}_{\mu e}\right)\rangle$ expectation values equal zero identically. For our Equation(18) this means that the first two terms equal zero, while all other terms are certainly not zero. Furthermore, in the last term from Equation(18) we always have ${\mathbf{L}}_{31}={\mathbf{L}}_{21}=\mathbf{L}$. This allows one to simplify Equation(18) even further. Indeed, by taking into account all arguments presented here we can write the explicit form of the total hyperfine Hamiltonian for the rotationally excited $1s_{\mu }n{\ell }_e-$states

$$\begin{array}{ccc}\hfill {\widehat{H}}_{HFS}& =& -{\displaystyle \frac{2\pi {\alpha }^2}{3}}{g}_N{g}_{\mu }\left({\displaystyle \frac{m_e}{M_p}}\right)\left({\displaystyle \frac{m_e}{M_{\mu }}}\right)\langle \delta \left({\mathbf{r}}_{N\mu }\right)\rangle ({\mathbf{I}}_N·{\mathbf{s}}_{\mu })\hfill \\ & -& {\displaystyle \frac{1}{4}}{\alpha }^2\{{\displaystyle \frac{g_e{g}_N}{M_p}}\left[3\langle {\displaystyle \frac{1}{r_{31}^3}}({\mathbf{I}}_N·{\mathbf{n}}_{31})({\mathbf{n}}_{31}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·{\mathbf{s}}_e)\right]\hfill \\ & +& {\displaystyle \frac{g_e{g}_{\mu }}{M_{\mu }}}\left[3\langle {\displaystyle \frac{1}{r_{21}^3}}({\mathbf{s}}_{\mu }·{\mathbf{n}}_{21})({\mathbf{n}}_{21}·{\mathbf{s}}_e)\rangle -\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·{\mathbf{s}}_e)\right]\}\hfill \\ & -& {\displaystyle \frac{1}{2}}{\alpha }^2\left[{\displaystyle \frac{g_N}{M_p}}\langle {\displaystyle \frac{1}{r_{31}^3}}\rangle ({\mathbf{I}}_N·\mathbf{L})+{\displaystyle \frac{g_{\mu }}{M_{\mu }}}\langle {\displaystyle \frac{1}{r_{21}^3}}\rangle ({\mathbf{s}}_{\mu }·\mathbf{L})\right],\hfill \end{array}$$

(19)

where $\mathbf{L}={\mathbf{L}}_{31}={\mathbf{L}}_{21}$. Note that the hyperfine Hamiltonian, Equation(19), is the matrix with the dimension $(2I+1)(2s_{\mu }+1)(2s_e+1)(2L+1)$ which includes (as coefficients) a few expectation values shown in Equation(19) inside of the $\langle \dots \rangle$ brackets. To determine the hyperfine structure of the helium muonic atoms we need to determine all eigenvalues (with the corresponding multiplicities) of this matrix. The differences between these eigenvalues give us the hyperfine structure splitting.

Note that our formulas, Eqs.(18) - Equation(19), can directly be used in numerical calculations of the hyperfine structure splittings for an arbitrary rotationally excited (bound) state in the both helium-muonic ³He $\mu e$ and ⁴He $\mu e$ atoms. To achieve this goal we need to know the highly accurate wave functions of the rotationally excited (bound) states in the helium muonic ³He $\mu e$ and ⁴He $\mu e$ atoms. In reality, these wave functions are used to determine all expectation values mentioned in Equation(18) and Equation(19). Possible generalizations of our formulas, Equation(18) and Equation(19), to the case of lithium muonic, beryllium muonic and other similar few-body systems with one additional negatively charged muon ${\mu }^{-}$ is straightforward and here we do not want to discuss it.

4. Approximate Analysis

As shown above in the helium-muonic atoms we always have ${\mathbf{L}}_{31}={\mathbf{L}}_{21}=\mathbf{L}$. Formally, this means that the original (quite complicated) problem is reduced to the one-center form. Therefore, we can apply some approximate methods developed earlier for atomic systems with one heavy center. The one-center approximation allows one to obtain some useful analytical expressions for the tensor terms included in the second term of Equation(18) and Equation(19) derived above. It is clear that these terms cannot be equal zero identically, if (and only if) the nuclear spin differs from zero. In respect to this, everywhere below in this Section we shall assume that ${\mathbf{I}}_N\ne 0$ and consider only such one-electron muonic atoms/ions.

In general, the best way to deal with the tensor terms is to average them over magnitude of the orbital (angular) momenta $\overrightarrow{\ell }$. The actual formula which is used in this case takes the form (see, e.g., [8,10]):

$$\begin{array}{c}\hfill \overline{{\mathbf{n}}_i{\mathbf{n}}_j}={\displaystyle \frac{1}{3}}{\delta }_{ij}-{\displaystyle \frac{1}{(2\ell -1)(2\ell +3)}}\left[{\widehat{\ell }}_i{\widehat{\ell }}_j+{\widehat{\ell }}_j{\widehat{\ell }}_i-{\displaystyle \frac{2}{3}}{\delta }_{ij}\ell (\ell +1)\right],\end{array}$$

(20)

where ${\widehat{\ell }}_i$ are the Cartesian components of the momentum vector $\overrightarrow{\ell }=({\ell }_x,{\ell }_y,{\ell }_z)$ and ${\mathbf{n}}_i$ are the Cartesian components of the unit vector $\mathbf{n}=(n_x,n_y,n_z)$, where for the helium muonic atoms we have to choose $\mathbf{n}={\displaystyle \frac{{\mathbf{r}}_{31}}{r_{31}}}$, where ${\mathbf{r}}_{31}$ is the vector directed from the atomic nucleus to the electron. Let $\mathbf{a}$ and $\mathbf{b}$ be two arbitrary spatial vectors, which are conserved during time-evolution of our two-body sub-system, then we can write Equation(20) in a slightly different form

$$\begin{array}{ccc}\hfill (\widehat{\mathbf{a}}·\overline{\mathbf{n}\left)\right(\mathbf{n}}·\widehat{\mathbf{b}})& =& {\displaystyle \frac{1}{3}}(\widehat{\mathbf{a}}·\widehat{\mathbf{b}})-{\displaystyle \frac{1}{(2\ell -1)(2\ell +3)}}\left[{\widehat{a}}_i{\widehat{\ell }}_i{\widehat{\ell }}_j{\widehat{b}}_j+{\widehat{a}}_i{\widehat{\ell }}_j{\widehat{\ell }}_i{\widehat{b}}_j-{\displaystyle \frac{2}{3}}\ell (\ell +1)(\widehat{\mathbf{a}}·\widehat{\mathbf{b}})\right]\hfill \\ & =& {\displaystyle \frac{1}{3}}(\widehat{\mathbf{a}}·\widehat{\mathbf{b}})-{\displaystyle \frac{2}{(2\ell -1)(2\ell +3)}}\left[(\widehat{\mathbf{a}}·\widehat{\ell })(\widehat{\ell }·\widehat{\mathbf{b}})-{\displaystyle \frac{1}{3}}\ell (\ell +1)(\widehat{\mathbf{a}}·\widehat{\mathbf{b}})\right],\hfill \end{array}$$

(21)

where any of these two vectors $\mathbf{a}$ and $\mathbf{b}$ cannot be zero-vector and they cannot be orthogonal to each other. In our case in Equation(21) we have to choose $\mathbf{n}={\mathbf{n}}_{31},\mathbf{a}={\mathbf{I}}_N$ and $\mathbf{b}={\mathbf{s}}_e$:

$$\begin{array}{c}\hfill ({\widehat{\mathbf{I}}}_N·\overline{{\mathbf{n}}_{31}\left)\right({\mathbf{n}}_{31}}·{\widehat{\mathbf{s}}}_e)={\displaystyle \frac{1}{3}}({\widehat{\mathbf{I}}}_N·{\widehat{\mathbf{s}}}_e)-{\displaystyle \frac{2}{(2\ell -1)(2\ell +3)}}\left[({\widehat{\mathbf{I}}}_N·\widehat{\ell })(\widehat{\ell }·{\widehat{\mathbf{s}}}_e)-{\displaystyle \frac{1}{3}}\ell (\ell +1)({\widehat{\mathbf{I}}}_N·{\widehat{\mathbf{s}}}_e)\right].\end{array}$$

(22)

To obtain the final expression for this term one needs to multiply this term by the factor ${\displaystyle \frac{3}{r_{31}^3}}$ and calculate the corresponding expectation value, i.e., the $\langle {\displaystyle \frac{3}{r_{31}^3}}\rangle$ expectation value.

Now, consider the second tensor term which contains the $(\widehat{\mathbf{a}}·{\mathbf{n}}_{21})({\mathbf{n}}_{21}·\widehat{\mathbf{b}})$ scalar product. This term cannot be determined directly as it was done in Equation(22), since the two unit vectors ${\mathbf{n}}_{31}$ and ${\mathbf{n}}_{21}$ differ from each other and we cannot simply replace the unit ${\mathbf{n}}_{31}$ vector by the different ${\mathbf{n}}_{21}$ vector during our averaging of any expression over magnitude of the orbital (angular) momenta $\overrightarrow{\ell }$. However, we can write the following formula

$$\begin{array}{ccc}\hfill (\widehat{\mathbf{a}}·{\mathbf{n}}_{21})({\mathbf{n}}_{21}·\widehat{\mathbf{b}})& =& (\widehat{\mathbf{a}}·{\mathbf{n}}_{31})({\mathbf{n}}_{31}·{\widehat{\mathbf{n}}}_{21})({\mathbf{n}}_{21}·{\widehat{\mathbf{n}}}_{31})({\mathbf{n}}_{31}·\widehat{\mathbf{b}})\hfill \\ & =& (\widehat{\mathbf{a}}·{\mathbf{n}}_{31})({\mathbf{n}}_{31}·\widehat{\mathbf{b}}){\left({\displaystyle \frac{r_{31}^2+r_{21}^2-r_{32}^2}{4r_{31}^2{r}_{21}^2}}\right)}^2,\hfill \end{array}$$

(23)

where we used the well known three-body vector identity ${\mathbf{r}}_{31}+{\mathbf{r}}_{12}+{\mathbf{r}}_{23}=0$. The right-side of this equation explicitly depends upon the unit vector ${\mathbf{n}}_{31}$ and some true scalar only. Therefore, the last equation is reduced to the form

$$\begin{array}{c}\hfill (\widehat{\mathbf{a}}·\overline{{\mathbf{n}}_{21}\left)\right({\mathbf{n}}_{21}}·\widehat{\mathbf{b}})=(\widehat{\mathbf{a}}·\overline{{\mathbf{n}}_{31}\left)\right({\mathbf{n}}_{31}}·\widehat{\mathbf{b}}){\left({\displaystyle \frac{r_{31}^2+r_{21}^2-r_{32}^2}{4r_{31}^2{r}_{21}^2}}\right)}^2,\end{array}$$

(24)

where in our present case $\mathbf{a}={\mathbf{s}}_{\mu }$ and $\mathbf{b}={\mathbf{s}}_e$. Finally, one finds

$$\begin{array}{ccc}\hfill ({\widehat{\mathbf{s}}}_{\mu }·\overline{{\mathbf{n}}_{21}\left)\right({\mathbf{n}}_{21}}·{\widehat{\mathbf{s}}}_e)& =& ({\widehat{\mathbf{s}}}_{\mu }·{\widehat{\mathbf{s}}}_e)-{\displaystyle \frac{1}{2(2\ell -1)(2\ell +3)}}\left[3({\widehat{\mathbf{s}}}_{\mu }·\widehat{\ell })(\widehat{\ell }·{\widehat{\mathbf{s}}}_e)-\ell (\ell +1)({\widehat{\mathbf{s}}}_{\mu }·{\widehat{\mathbf{s}}}_e)\right]\hfill \\ & & \times \langle {\left({\displaystyle \frac{r_{31}^2+r_{21}^2-r_{32}^2}{r_{31}^2{r}_{21}^2}}\right)}^2{\displaystyle \frac{1}{r_{21}^3}}\rangle .\hfill \end{array}$$

(25)

This formula can be used to determine the $({\widehat{\mathbf{s}}}_{\mu }·\overline{{\mathbf{n}}_{21}\left)\right({\mathbf{n}}_{21}}·{\widehat{\mathbf{s}}}_e)$ expectation value in our approximate approach.

5. Conclusion

We developed the original approach which allows one to determine the hyperfine structure splittings for the rotationally excited $1s_{\mu }n{\ell }_e$-states in the three-body helium muonic atoms. Rigorously, his problem has never been considered in earlier studies. Our approach is based on generalization of the method originally proposed by E. Fermi in 1930 [2,3] for one-electron atomic systems. It is shown that the final formula for the hyperfine Hamiltonian is represented as a sum of three scalar terms, two tensor terms and two spin-orbital terms. Diagonalization of the corresponding ${\widehat{H}}_{HFS}$ matrix, which has the dimension $(2I_N+1)(2s_{\mu }+1)(2s_e+1)(2L+1)$, allows one to determine all possible hyperfine structure splittings and investigate the actual hyperfine structure in the three-body helium muonic ³He $\mu e$ and ⁴He $\mu e$ atoms. In reality, we need to determine all eigenvalues of the ${\widehat{H}}_{HFS}$ matrix. Differences between these eigenvalues (expressed in $MHz$) coincide with the corresponding hyperfine structure splittings. All formulas obtained in this study can be used in direct numerical calculations of the hyperfine structure splittings of the helium muonic atoms. Note also that generalizations of our formulas Equation(18) and Equation(19) to the cases of lithium-muonic, beryllium-muonic ions and other similar few-body systems with one negatively charged muon ${\mu }^{-}$ is relatively easy, but it requires a different system of notation.