Alternative Implementation of Atomic Form Factors

A bound state of muon and anti-muon (true muonium or dimuonium), although predicted long ago [1–4], has never been observed experimentally. Many mechanisms for the production of dimuonium have been proposed in the literature. Dimuonium can be formed in fixed-target experiments [5–9], in electron-positron collisions [1, 10–12], in elementary particle decays [13–19], in a quark-gluon plasma [20, 21], in relativistic heavy ion collisions [21–23], in an astrophysical context [24], or in experiments with ultra-slow muon beams [25, 26]. As part of the first stage of the expensive and long-term super charm-tau factory project, Budker Institute of Nuclear Physics (Novosibirsk) is currently developing plans to build an inexpensive, low-energy μμ-tron machine [27]. Apart from purely accelerator studies, the μμ-tron will make it possible to produce and investigate dimuonium experimentally. Studying the interactions of dimuonium with ordinary atoms as it passes through the foil is an integral part of the planned experiments. Elementary atoms, such as dimuonium, when passing through the foil interact with ordinary atoms predominantly via the Coulomb potential [28–30]. Such an interaction is treated in terms of atomic form factors, and a comprehensive review of atomic form factor calculations can be found in [31]. General analytical formulas for calculating the form factor of a hydrogen-like atom were obtained in [32] by grouptheoretical methods. However, these formulas have a somewhat complicated form, requiring time-consuming calculations for each value of a transfer momentum [33]. A more convenient set of formulas was developed in [33, 34] and implemented as a FORTRAN program in [35]. Based on the mathematical results obtained in [36], in this article we present an alternative method for calculating the form factor, which in some sense complements the method presented in [33, 34]. As a byproduct of this reaearch, some trigonometric identities involving Chebyshev polynomials of the second kind were obtained in [37]. Throughout the paper, we use dimuonium atomic units, in which c = h̄ = 1, the unit of mass is 12mμ (reduced mass in the dimuonium atom), and the unit of length is the radius of the first Bohr orbit in dimuonium: aB = 2 me mμ a0 ≈ 512 fm, where a0 ≈ 5.29× 10 m is the usual Bohr radius. Although the motivation for the article was Novosibirsk dimuonium program, we emphasize that the results obtained are in fact of much broader interest, mainly in atomic physics, see Devangan’s review article [31].