As mentioned above, previous relevant algorithms have been derived mostly for providing the ground state Dirac wave-functions of leptons (e–, μ–, τ–) bound in the electromagnetic field of Hydrogenic atoms (where an atomic nucleus of finite size with Z protons produces the attractive electrostatic potential). This was done because, e.g. in many muon-capture calculations, the initial atom is assumed to be in the ground state (g.s.) and its nucleus is represented by a finite charge density. In the majority of muon-nucleus processes a mean value for the g.s. wave-function of the muon was employed and not an accurate one.
In this project, one of our main aims is to construct new algorithms to become able to provide the following: (i) The bound states of exotic leptonic atoms like the Mu, Ps, M–, etc., which, though not precisely measured by experiments so far, they have predicted to exist within the context of beyond the standard model theories. (ii) The (e–, μ–, τ–) wave functions for any possible excited (bound) state of conventional muonic atoms (with atomic number Z and mass number A) in which an e– is replaced by a μ–.
The advanced algorithms, required for calculating (bound) energy-levels of conventional and exotic muonic atoms as described above, will respect the main advantages and benefits endowed in the neural network techniques. The programming language (initially) could be the C++, but the final codes will be available in Python language. Also, these software programs will have the advantage of providing the ground state energy as well as the energies of all possible (bound) excited states of few-body systems as the exotic atoms that consist of two leptons with opposite charges like: (a) the muonium (μ+,e–), (b) the true muonium (μ+,μ–), (c) the true positronium (e+,e–), etc. An attempt for including three-lepton system as the M–, etc. is going to be tested.
At this point we remind that, since the particles in these exotic systems are without structure (leptons), in the proposed code the attracting charged center will be taken as point-like (and not as in the DiracSolver where the nucleus is assumed with finite structure and not as point-like). Moreover, the code will have the possibility to take explicitly into account relativistic effects which means that, independent routines will be derived for both the Dirac and the Breit-Darwin equations (see Ref. [2] of the Reference list).
As input parameters, assuming use of natural units (c = ħ = 1, in the case of the two-lepton systems), in the new codes we are going to use: (i) The lepton’s masses (me, mμ), (ii) the electric charge e, which will be assumed as “free” parameter, so that the coupling parameter [number sqrt(1/137)] need not be taken as constant (in this case left- and right-margin limits will be used), and (iii) The quantum numbers determining the “excited” (resonant) states, i.e. the radial n quantum number and the angular momentum quantum number J so that the excited state will be represented as |n(ls)J>. The main output of the new code will be the energy-level (or equivalently the corresponding mass/resonance) of the bound particle but also other useful quantities needed.
Testing of the algorithm: The codes will be tested on the muonium Mu states (the antimuon-electron bound system), Among other systems, the algorithms will be used later to predict energy-levels of new bound systems (atomic systems involving only leptons). The electric charge of the new bound systems might be very different than the known electric charge e. Furthermore, because the Dirac equation must take explicitly into account relativistic effects, the electric charge might be much larger than the known value of e, e.g. the case when the relativistic effects appear to be of high significance.
Afterwards, the following concrete physical applications of current research interest may be implemented:
A. Exotic Atoms predicted by BSM of electroweak interactions theories: The anti-muon (μ+) is not a stable lepton, but it decays into a positron and two neutrinos (τμ = 2.2 μs). Hence, the muonium, Mu, is also unstable leptonic atom due to the weak decay of μ+. As an explanation, the difference between the rest masses of μ+, initial state, and the positron (e+), final state (assuming negligible the outgoing-neutrino masses) is mμ-me=105.658- 0.511=105.147 MeV. This decay would be forbidden only if the binding energy of Mu was -105.147 MeV or stronger but the binding energy of Mu is about equal to that of H-atom.
B. Conventional muonic atoms used as targets in current forefront experiments: The new code, may be easily applied (i) to obtain accurate muon wave functions for the low-lying energy-levels of the bound muon in muonic atoms, targets of current experiments (e.g. 100Mo). Previous calculations have either used a mean value of the μ– wave function or have neglected the small component for the muon (this is equivalent to solving the Schroedinger rather than the Dirac equation). Thus, e.g. the muon capture rates calculations of Ref. [13] (see Reference list) have ignored the bottom (small) component of the Dirac wave function. (ii) Furthermore, the accurate muon w-f of the low-lying excited states may be used for accurate predictions of high-energy astrophysical ν μ neutrinos according to the reaction (A,Z)+ νμ –> (A,Z+1)+ μ– (this reaction is a particle-conjugate reaction of the ordinary muon capture on nuclei).
