Hyperfine structure of muonic mesomolecules tdμ, tpμ, dpμ in variational method

The hyperfine structure of energy levels of muonic molecules tdμ, tpμ and dpμ is calculated on the basis of stochastic variational method. The basis wave functions are taken in the Gaussian form. The matrix elements of the Hamiltonian are calculated analytically. Vacuum polarization, relativistic and nuclear structure corrections are taken into account to increase the accuracy. For numerical calculation, a computer code is written in the MATLAB system. Numerical values of energy levels of hyperfine structure in muonic molecules tdμ, tpμ and dpμ are obtained.


Introduction
The interest in hydrogen mesomolecular ions is primarily connected with muonic catalysis of nuclear fusion reactions [1]. Precise calculation of fine and hyperfine structure of muonic molecular ions with the inclusion of higher order QED corrections allows us to predict the rates of reactions of their formation and other parameters of the µCF cycle.
We use stochastic variational method [2][3][4] to calculate energy spectra of hydrogen mesomoleclular ions numerically. Variational method allows us to determine energies of threebody systems with very high accuracy. Random generation of variational parameters prevents possible convergence of the method to a local minimum and gives a much higher flexibility when applied to different three-body systems. In case of hydrogen mesomolecular ions the choice of variational method is also motivated by comparable masses of all three particles, which does not allow us to use adiabatic approximation effectively.
In this work not only do we calculate ground state energies and hyperfine structure of tdµ, tpµ and dpµ mesomolecular ions but also include important QED corrections to increase the accuracy. We calculate relativistic, vacuum polarization and nuclear structure corrections to the ground state energy as well as vacuum polarization and Zemach corrections to the hyperfine splitting. Inclusion of these corrections is crucial in achieving precise results. In all formulas throughout the paper we use muon-atomic units (e = m µ = = 1).

General formalism
To calculate ground state energy and hyperfine structure in muonic molecules tdµ, tpµ and dpµ we use stochastic variational method [4]. The trial wave function of muonic molecule in this approach has Gaussian form. The Gaussian-type basis function with non-zero angular momentum for nonidentical particles is the following: In the case of three nonidentical particles in S -state (L = 0, where L is total angular momentum of particles) basis functions take the form: where ρ and λ denote two Jacobi coordinates. Knowing the basis functions we can now perform analytical calculation of matrix elements of the Hamiltonian, which is an advantage of the Gaussian basis. The overlap matrix element has the form: where elements of matrix B are expressed in terms of matrix A as B i j = A i j + A i j . For the calculation of matrix elements of the Hamiltonian we use explicit expressions for potential and kinetic energy operators. In nonrelativistic approximation the Hamiltonian of the molecule without the account of hyperfine structure has the following form in Jacobi coordinates: where µ 1 = m 1 m 2 m 1 +m 2 , µ 2 = (m 1 +m 2 )m 3 m 1 +m 2 +m 3 , r 12 = r 1 − r 2 = ρ, r 13 = r 1 − r 3 = λ + m 2 m 12 ρ, r 23 = r 2 − r 3 = λ − m 1 m 12 ρ, e 1 , e 2 , e 3 are charges of the particles. Matrix elements of kinetic energy have the following analytical form: For potential energy matrix elements we present the following analytical expressions: < φ 0 |V|φ 0 >= e 1 e 2 I 12 + e 1 e 3 I 13 + e 2 e 3 I 23 .
where to get the expressions for F 13 1 and F 13 2 one should choose m 2 and the first sign in (10) and to obtain F 23 1 and F 23 2 m 1 and the second sign in (10) must be chosen. To further increase the accuracy of our calculation we take into account various QED corrections. Let us start with relativistic corrections. The explicit expression for relativistic corrections can be obtained from the Breit Hamiltonian for two interacting particles. In case of hydrogen mesomolecular ions Breit corrections have the form [5,6]: where p i are momentums of the particles, r i j = r i − r j are interparticle coordinates. Averaging over basis functions gives the following analytical expressions for matrix elements: Now let us consider another important QED correction -one-loop vacuum polarization correction. The potential in this case takes the form: where γ = m e m µ α , ρ(ξ) = ξ 2 − 1(2ξ 2 + 1)/ξ 4 . Analytical expression for matrix elements is the following: 2F 13,23 where G is the Meijer G-function.
It is important to also include nuclear structure correction, which in case of mesomolecular ions has the form: where indices 1, 2 denote nuclei of hydrogen isotopes, r 1e , r 2e are charge radii of the corresponding nuclei of hydrogen isotopes, index 3 denotes the muon. This correction can be obtained using the following expansion of electric form-factor: The calculation of the matrix elements of δ 3 (r i j ) will be presented explicitly in the next section. To calculate QED corrections numerically we use first order perturbation theory. General expression for an arbitrary potential: where V is an arbitrary potential, c i , c j are linear variational parameters, K is the number of basis functions. All numerical results for ground state energy and corrections after variational calculation are presented in Table 1 in lines 1 through 4. Results from paper [3] are presented in Table 1 in parentheses for comparison.

Hyperfine structure
Now let us calculate matrix elements of the hyperfine part of the Hamiltonian. The potential of hyperfine structure of L = 0 state, which is a part of the Breit Hamiltonian, can be written in the following simple form for three interacting particles with spins S 1 , S 2 , S 3 respectively [7][8][9]: where g 1 , g 2 , g 3 are gyromagnetic factors, indices 1,2 denote nuclei of hydrogen isotopes, index 3 denotes muon. Averaging procedure for such potential involves both averaging over radial and spin basis functions. Analytical integration of radial matrix elements of δ(r i j ) can be performed as follows: To perform averaging over spin functions Wigner -Eckart theorem [10] can be used with success. General formulas for spin averaging for any S 1 , S 2 and S 3 take the form: < S 12 , S |(S 2 S 3 )|S 12 , S >= (2S 12 + 1)(2S 12 + 1)(2S 2 + 1)(S 2 + 1)S 2 (2S 3 + 1)× × (S 3 + 1)S 3 (−1) 2S max 12 +S +S 1 +S 2 +S 3 +1 S 12 S 3 S S 3 S 12 1 In the case of S 1 = S 2 = S 3 = 1/2 the energy matrix has the form: After the matrix diagonalization we obtain the following eigenvalues: These eigenvalues are energies of hyperfine structure levels with respect to the total energy of ground state. In case of S 1 = S 2 = S 3 = 1/2, as it was already mentioned, we have 3 hyperfine levels. In the case of S 1 = 1, S 2 = S 3 = 1/2 the energy matrix is the following: After the matrix diagonalization the following eigenvalues are obtained: For S 1 = 1, S 2 = S 3 = 1/2 spin configuration we get 4 hyperfine levels. To calculate hyperfine structure of muonic molecular ions tdµ, tpµ, dpµ we use first order perturbation theory with a variational wave function obtained in variational calculation. For instance, to calculate < δ(r 12 ) > matrix element in first order perturbation theory one has to use the following expression: Other matrix elements are evaluated in a similar way. Thus we obtain numerical values of a, b and c coefficients and energies of hyperfine structure levels of tdµ, tpµ, dpµ mesomolecular ions.
To obtain a more precise result we include vacuum polarization and nuclear structure corrections to hyperfine structure. The potential of one loop vacuum polarization correction to the hyperfine structure has the form: Integration over the Jacobi coordinates can be performed analytically. Thus we obtain the following integral expressions for vacuum polarization correction to the hyperfine structure: