The project will address specific challenges and define corresponding objectives as summarized below.
Challenge 1: Advanced Numerical Tools for Fundamental Differential Equations: In many modern physical problems and specifically those related to the structure and evolution of quantum systems, the evaluation of exact predictions for various physical observables is required. These predictions are based on wave functions describing the states (bound and excited states) of the quantum system in question that come out of accurate solutions of specific differential equations, like the Dirac, Breit-Darwin, Schroedinger, Klein-Gordon, etc. Because, analytic solutions of such equations, describing two- and many-body quantum systems, are not always possible, the application of advanced algorithms is required. Concrete examples of this category of physical systems (problems) are the hydrogenic-type lepton-nucleus or purely leptonic systems where the leptons (e–, μ–, τ–) are bound in Coulomb fields created by attractive charged centers (deduced by positive hadrons, as protons, or positive leptons, as anti-muons, μ+, and positrons, e+).
For high sensitivity descriptions of such systems, the use of advantageous numerical techniques is needed for obtaining the system’s wave functions. Then, simulated predictions of the lepton-nucleus (or lepton-lepton) atomic system come out by solving the Dirac (or Schroedinger) equation which may include relativistic as well as beyond the standard model (BSM) terms. Recently, there is intense interest in treating integration of partial/ordinary differential equations (PDEs/ODEs), like the above, through the use of neural network approaches or other schemes and application of a proper optimization process. In this context, one may optimize appropriate multi-parametric expressions representing the radial parts of the Dirac and Breit-Darwin wave functions, deduced form a system of coupled first order (ordinary) differential equations.
Objective 1: In this project, we aim to model and build advanced numerical codes (in Python language) to facilitate the solution of the fundamental differential equations Dirac and Breit-Darwin required for high accuracy description of exotic systems as the leptonic atoms: the muonium, the positronium, the true muonium, the muonium ion, etc.
Challenge 2: Precision spectroscopic data from studies of hydrogenic-type exotic atoms. In recent years, pure leptonic systems, such as the Muonium, (μ+,e–) or Mu, a hydrogen-like bound state of a positive muon, μ+, and an electron, e–, the positronium, (e+,e–) or Ps and others, which are hydrogenic-type exotic atoms composed of point-like particles, the leptons, have triggered the intense interest of researches from an experimental, phenomenological and theoretical physics point of view.
Experimentally, the recent and ongoing developments of low-energy muon beam lines (see below) opened a new era of precision Mu spectroscopy, e.g. at PSI, J-PARC, Fermilab, RCNP. The previous spectroscopic measurements on Mu were performed at statistically limited muon facilities. Recently, low-energy muon beams are available for current experiments which may improve the statistics and the precision by several orders of magnitude. Data from such precision experiments may be employed for comparison with predictions expected to come out by using the algorithms we will derive in the present project. Mu was first produced by the known beam-foil method and then, great effort has been devoted to perform Lamb shift measurements of this atom in the n=2 state.
The Mu and Ps leptonic atoms constitute ideal systems for precision spectroscopy to measure the muon mass, the muon magnetic moment, etc. A Mu beam with low-energy distribution can be formed when low-energy muons pass through thin foils or gas targets. It is worth mentioning that, when a μ+ beam passes through a thin foil, due to charge exchange of μ+ with the foil material, the outgoing species are μ+, Mu, and M–. The negative muonium ion (M–), which is the bound system of a positive muon and two electrons (three-body system), (μ+,e e–), has been recently produced and observed.
Objective 2: One of our main aims in the present project is to exploit appropriate high precision measurements, mostly related to the exotic leptonic systems Mu, Ps, and others, in order to check and assess the efficiency of the algorithms derived in Objective 1 and, subsequently, to test the QED and BSM theories.
Challenge 3: Description of the exotic Muonium (Mu) leptonic atom within BSM theories. During the last decades, our knowledge about the structure and composition of materials (based on the creation of new improved methods to observe, measure and compare with the previous values), offers a better understanding of the Universe’s matter. Thus, according to the physical theory of atomic systems, all the matter we see around us is made of the conventional atoms (systems of positive protons, neutral neutrons and negative electrons) that make up everything in our world.
With the current appreciably sensitive detectors and accelerators we discovered exotic forms of matter, including exotic atoms which are not made from the above three basic constituents. The modelling of exotic leptonic atoms, for example, requires knowledge of low-energy atomic physics and high-energy particle physics, a combination of disciplines within physics that favours deeper understanding of our material Universe too.
As such, the above mentioned leptonic atoms do not have finite-size effects, and they are largely free of other hadronic contributions, making them ideal for determining fundamental constants, testing bound-state quantum electrodynamics (QED), and searching for new physics beyond the Standard Model (BSM). Thus, the muonium atom is an ideal system for precision tests of the validity of Quantum Electrodynamics (or to extract fundamental constants like the muon mass, the muon magnetic moment, etc.), since the leptonic constituents of muonium behave as point-like particles.
Objective 3: Another main goal in this project is, through our new algorithms, to provide the possibility to test specific theoretical particle physics scenarios (models leading to exotic interactions that are searched for at several experiments (see Refs.[8, 10, 22] of the Reference list), as well as to test various symmetries in physics. Inspired by the recent advances mentioned above, after developing advantageous algorithms, we will be able to model the description within BSM theories of the Muonium spectrum and provide predictions for other properties of this exotic leptonic atom.
Challenge 4: Modelling the description of the exotic Positronium (Ps) leptonic atom within BSM theories. The Positronium, Ps (e+,e–) is a purely leptonic atom made of a positron and an electron in coupled orbit around their center of-mass. The Ps system is free of internal substructure that qualifies it as probe to investigate physics beyond the Standard Model (BSM). Recently, comprehensive studies of the sensitivity of Ps (and also Mu) precision spectroscopy to several new physics scenarios have been carried out. Thus, experimentalists consider the positronium Ps as promising target to probe new physics via precise spectroscopy. It has been estimated that, in order for the spectroscopy bounds to reach high sensitivity required for Ps, an improvement of roughly five orders of magnitude from the state-of-the-art precision is needed, which would be a challenge based on current technology.
It is worth mentioning that, Ps (e+,e–), like the muonium M(μ+,e–), is an important simple system for the study of fundamental interactions and atomic structure. Because of the absence of size effects, the structure of Ps can be computed very precisely by QED. Furthermore, experimental measurements related to the Ps structure provide rigorous tests of the QED theory and that is why Mu and Ps have been proposed as promising systems to test both QED and BSM physics. On the theoretical particle and atomic physics side, modern scenarios and several BSM models have been proposed for the open questions related to Ps and Mu systems. The corresponding experimental data/sensitivities, in connection with the predictions that will come out of our project’s calculations (based on our new codes), may offer significant help towards unravelling the relevant BSM scenarios.
Objective 4: Motivated by the above recent challenges and relying on the development of advantageous algorithms, we aim to model the description of the Ps (bound) spectrum and other properties within several BSM theories that will enable us to provide predictions for the main observables of this leptonic atom. Adhering to the results of Objective 3, we will provide, in addition, useful constraints corresponding to various exotic interaction mechanisms.
