Mass spectrum of elementary particles

. In this study, we discuss different methods for quantifying the mass of elementary particles. The Barut's model approach gives us a more comprehensive understanding of elementary particles and their formation through magnetic interaction. Through this model, we can understand the mechanism of the formation of hundreds of unstable elementary particles whose components are interconnected through positive energy and thus quickly disintegrate into their basic components .


Introduction
The problem of studying the mass spectrum of elementary particles is considered one of the unsolved problems in physics, as it involves many methods and unresolved results.One of the first to mention this issue was Nambu [1], who correlated the masses of elementary particles (known at the time) with the fine structure constant.After that, Barut succeeded in developing a formula that includes leptons, based on Nambu's idea, where he was able to obtain the mass of the taun and predict the mass of the fourth lepton [2]: ) ,  = 0,1,2,3, … … ..
Later, a Japanese physicist named Yoshio Koide was able to find a relationship between the three previous leptons, and it is called the Koide formula [3]: (√  + √   + √  ) 2 . ( And despite its accuracy, there is still no theoretical basis from which the Koide formula can be deduced.There is another approach in the study based on group theory SU(n) (Gell-Mann) and also a geometric approach (Polokhov-Vladimirov).Later , Varlamov introduced a different formula based on two parameters [4]: ). ( The mass (state energy) is determined by the " ,  -cyclic representation" of the Lorentz group, and the electron mass   plays the role of the "mass quantum".Cyclic representations and the number s=|-| are determined by the value of the particle spin.We presented the most important models known so far to study the mass spectrum of elementary particles, but we will choose Barut's approach (the dynamic approach), as from our point of view, this approach is an intuitive approach and close to reality to describe the spectrum of elementary particles, including hadrons and mesons.Barut's model is based on the following assumption [5]: "If the proton , electron, and neutrino (and their antiparticles) are indestructible, they must also be indestructible inside the hadrons", And these particles bind together by ordinary magnetic forces and make all the other particles.The mass of most particles is much greater than the mass of their components, and therefore, in order for Barut's assumption to be correct, the particles must be connected with positive energy.

Materials and methods
The dynamics of magnetic interactions (Barut's model): At small distances, magnetic forces become very strong and dominant, when treated nonperturbative which in turn provide a deep enough well to give rise to high-mass narrow resonances.To clarify the previous idea, we will study two particles ( 1 ,  2 ), each of which has a charge ( 1 ,  2 ) and the spin of each of them (s 1 , s 2 ), The Hamiltonian of the previous system is given by the following formula [6]: Where: electromagnetic field created by one particle at the location of another particle, Moving to the center of mass system, we get the following: Where: S 12 (r 1 ⃗⃗⃗ − r 2 ⃗⃗⃗ ) = .Formula ( 5) gives the effective interaction potential for the radial function in the form: The coefficients a, b, c, d take the form: We can represent the effective interaction potential between an electron and a proton, for example, in the following figure: We made the replacement X=Rr .To solve the previous equation, the behavior of the wave function must be discussed: We rewrite equation (7) as: ) When r→0, the fourth term, dominates the rest of the terms, equation ( 8) becomes: This equation has the following solution: When ( → ∞); the wave function ( →  ∞ ), and we can write the following equation: The solution to the previous function is given in the following form: Hence, the final solution is given in the following form [7,8]: Where () = { ∏ ( −   () ),  > 0; =1 1,  = 0. Let's consider the situation  = 0, Putting equation (13) into equation (8), and taking into account formula (6), we find the following relations between potential parameters: ∈= − 2 ,  = −2,  = ( − 1) − 2,  = 2( − 1),  =  2 .The above parameters achieve the following relationship: The corresponding energy levels are given by the following formula: The previous energy levels(when  = 0), is composed of several other levels that depend on (), correspond to the following wave function: Where  0 , is normalization constant.For  = 1, () =  −  1 (1) , we find the next level of energy: 3

Results and discussions
The energy levels(∈ 0 ± , ∈ 1 ± , … … ., ∈  ± ) are the interaction energy between the two imposed particles, each of them consists of other sublevels that depend on (, ), and for the bonding energy between two particles to form a third one, we have two options: the first option is for the bonding energy to be negative, the former bonding occurs at the expense of mass (as in nuclear forces); in other words, the mass of the resulting particle must be less than the mass of its constituent particles: So far, there is no limit on the number , but we can get this limit by noting that  ≥ 0, that is, the following inequality is fulfilled: In other words, the bonding energy (in absolute terms) cannot exceed the total mass of the interacting particles.Equality is achieved in the case when ( →   ).Nature tends to be in the lowest energy state possible.Therefore, stable bound states must have negative bonding energy [9], as an example, deuterium is stable against breaking up into a proton and a neutron [10], since   −   −  = 1876 − 938.27 − 939.57 = −1.72.
Same for the hydrogen atom and stable chemical compounds.For elementary particles, we notice two things: 1-The mass of any particle is much greater than the mass of its components.2-Elementary particles (except the proton, electron, and neutrino) quickly disintegrate, meaning that they do not form a stable physical system.The previous two observations assure us that the bonding between elementary particles occurs through positive energy.The total energy  =∈  ± =  + (), is conserved.Since kinetic energy is not negative, motion is possible in cases that satisfy the relationship  = ∈  ± − () ≥ 0.
If  0 <∈  ± <   ≤ 0, The physical system is stable, see figure (2).This state corresponds to the nucleus of an atom, such as the nucleus of deuterium, and compatibility of atoms such as the hydrogen atom and most chemical compounds If 0 ≤  0 <∈  ± <   , the physical system is not stable; see figure (3).This state corresponds to elementary particles like the muon, neutron, and so on.The bonding energy is positive, which means that the mass of the formed particle is greater than the sum of the masses of the particles forming it.In this case: The case (20) is for unstable particles, the total mass is in general greater than the sum of the constituents, which means positive bonding energy.Final formulas for the mass spectrum takes the following expression:

Conclusion
The nuclei of atoms, atoms, and other compounds bond through negative energy and form stable physical systems that do not disintegrate except with an external influence.On the other hand, elementary particles bond through positive energy to form an unstable physical system that disintegrates into its elementary components (which do not disintegrate), such as the proton, electron, and neutrino.