In recent years, the neural-networks-based techniques are employed in the development of effective numerical schemes for solving the fundamental differential equations governing the laws in physical sciences. Subsequently, a minimization procedure is applied as e.g. the MERLIN code in Ref. [17], the simulated annealing method of Ref. [16], and others (see Reference list). The simulated annealing method, e.g. mimics the annealing process to solve an optimization problem. The temperature parameter T controls the search which typically starts from high values and is slowly “cooled” or lowered in every iteration. At each iteration a series of new points are generated. If the new point has a better function value it replaces the current point and the iteration counter is incremented. It is possible, however, to accept and move forward with a worse point, but the probability of doing so is directly dependent on the temperature T (this step, sometimes, helps identify a new search region in hope of finding a better minimum and protects the algorithm from being trapped in local minima).
During the last few decades, accurate solutions of specific differential equations (mostly non-linear), like the Dirac, non-linear Schroedinger, Klein-Gordon, etc., are required and the above computational methods have been successfully applied in conventional muonic atoms. The latter are normal atoms in which the electron cloud is partly or in total replaced by muons (it is also possible that electrons are replaced by other elementary particles of negative charge, as the τ–). The main observables of the muonic atoms in the X-ray (and γ-ray) spectroscopy are the energy and intensity of the emitted radiation during the de-excitation process of these atomic systems.
Theoretically, the relevant predictions require the knowledge of accurate muon wave-functions corresponding to some low-lying energy-levels. Existing codes need be refined, extended and improved in order to provide reliable theoretical predictions for the energy and intensity of X-rays and γ-rays emitted by various promising conventional muonic atoms as e.g. the 100Mo. The main shortcoming of the existing codes is related with the treatment of the relativistic effects and the absence of the consideration of the Breit-Darwin Hamiltonian. In our present project, we are going to devote special effort in order to take the Breit terms into account.
It is well known that, muons are important in the cutting-edge physics as they allow physicists to test with extremely high accuracy our best theories such as quantum electrodynamics (QED) or the Standard Model of the electroweak interactions (predicting various properties, matter, spectrum, etc.). In addition, the last decades, in various experiments leptonic exotic atoms have been produced in which the protons of the normal atoms have been replaced by leptons. For example, the Muonium (Mu) is like Hydrogen that has one electron in orbit around one proton, but Mu has a positively charged muon particle, μ+, in the place of the proton and an e– orbiting around μ+. Even though the study of the Mu leptonic atom is of great interest to the physics community, up to now it has not been spectroscopically/theoretically well investigated. This is a short-lived leptonic atom, so it is crucial for the relevant experiments to make quick observations with as much power as possible in order to obtain the best signal within the limited observation time.
It should be noted that, conventional spectroscopic methods require repeated observations across a range of frequencies to notice the particular key frequency (known as the resonance frequency). Recently, researchers find a new spectroscopic method that uses a well-understood physical effect, known as “Rabi oscillation spectroscopy”, which does not need to search for frequency signals in order to convey information about an atom. The new spectroscopy looks at the raw sensor or time-domain data over a shorter time duration and delivers information based on that. The latter method offers vast improvements in precision to “see” deeper into the world of exotic atoms containing unusual particles and isotopes, as well as other kinds of matter created at particle accelerators around the world. In the next few years, experiments (e.g. Mu-MASS at PSI) will probe new regions of the parameter space testing the existence of medium/short range (MeV and above) spin-dependent and spin independent “dark forces” between electrons and muons.
Furthermore, great improvements on the sensitivity of current precision spectroscopy, have been carried out for investigating the positronium system, Ps, too. For this leptonic atom, in order to reach high sensitivity needed for the spectroscopy bounds, a further improvement of roughly five orders of magnitude from the state-of-the-art precision is required. Based on the current technology, such an achievement would be a great challenge. Moreover, the low energy positive muons (up to 20 keV kinetic energies) coming out of the foils were investigated and the first observation of the negative muonium ion M– (produced by electron-capture in a beam foil experiment) was achieved. The results provide information on the energy distribution and the angular distribution of the emerging μ+ beam.
Regarding other similar leptonic systems, we mention that, in particle physics the exotic leptonic atom that represents a bound state of a muon and an anti-muon, i.e. the system (μ+,μ–) known as “true muonium”, is theoretically predicted. The existence of the true muonium is well-established theoretically within the Standard Model (see e.g. Ref. [8] and references therein of the Reference list). The properties of the true muonium are determined by quantum electrodynamics (QED) but in many models beyond the Standard Model (BSM) they may appear modified. The true muonium may has been produced in experiments involving collisions of electron and positron beams but it has not been observed experimentally yet.
It should be noted that, in Refs. [11, 23] (Reference list) the Breit terms of the Dirac Hamiltonian in exotic atoms has not been considered. In the present project, these terms are explicitly taken into account. Specifically, for the description of the (e+,e–), Ps system, the model presented in Ref. [5] (see Sect. 23, p. 114-118, and Sect. 42, p. 193-196) will operate as the starting point and as a guide in the initial modelling and development of the new algorithm in this project. The results of the code will be compared with those included in the above reference in order to estimate the portion of the new contributions. Furthermore, higher order terms are very complicated and extremely difficult to be considered within the framework of the present project. A special effort, however, will be devoted for the inclusion of specific low order terms in the case of hydrogenic type leptonic atoms.
The precise knowledge of the spectra and properties of exotic atoms, such as the Muonium (Mu), the Positronium (Ps), the true muonium (μ+,μ–), etc., will allow sensitive tests of quantum electrodynamics (QED) in bound states, complementary to other probes such as the anomalous magnetic moment of the muon (aμ) [coming out of the recent (g-2) measurement] and that of the electron (ae). In addition, exotic atoms can be used to probe new physics scenarios beyond the Standard Model theories (BSM). New, light mediators, responsible for a “fifth” force beyond the known four forces of the nature, give rise to shifts in the spectra, and can lead to novel decay modes not discussed up to now. Thus, exotic scalar, vector and spin-2 particles can be probed, if lighter than the corresponding inverse distance scale is involved.
Other new physics scenarios can be tested as well. For example, the Standard Model contains three light neutrinos that can propagate on the distance scale of (exotic) atoms and nuclei. Due to the fermionic nature of neutrinos, isolated neutrinos do not mediate a coherent force as their exchange leads to a change in the particle content or spin state of the constituents. The exchange of two, or more generally, an even number of, can be interpreted as a force, though, with a steeper distance dependence. Such exchanges are enhanced if neutrinos participate in exotic interactions beyond the Standard Model, specifically if they carry charged and neutral currents other than the left handed vector current of the Standard Model.
Existing measurements in exotic atoms, specifically Muonium, already provide meaningful constraints, which are expected to be improved with more precise calculations. Such calculations are planned to be performed within the context of the present project. Through the accurate muon wave functions derived as mentioned before we expect to extract more stringent constraints on the model parameters entering many particle models predicting properties of the Muonium.
In summary, the main innovations of this project are related with: (i) The modelling of predictions for the exotic atomic systems (like the purely leptonic atoms Mu and Ps) through accurate solution of the Dirac and Breit Darwin equations. Such atoms opened a new window on the research of low-energy (muon/lepton) physics and enable the assessment of various BSM models appeared during the last decade. (ii) The existing relevant codes mostly provide the ground state of exotic atomic (μ–nucleus) systems by solving the Dirac equations without taking into account explicitly the relativistic effects. The new codes will treat the Dirac equation with the Breit-Darwin potential (and possibly other terms) and will provide the excited states of: (a)the exotic leptonic atoms Mu and Ps which are two-body systems, (b) the exotic leptonic atom true-Muonium (μ+,μ–), etc., (c) the three-body system M- ion (μ+,e e–), (d) other exotic muonic systems.
