MRSU 2020 IOP Publishing Journal of Physics: Conference Series 1560 (2020) 012005 doi:10.1088/1742-6596/1560/1/012005

Influence of the magnetic interactions on the mass spectrum of elementary particles

N V Samsonenko 1, F Ndahayo 2 and M A Alibin 1 1 RUDN University (Peoples’ Friendship University of Russia), Moscow, Russia 2 University of Rwanda, Kigali, Rwanda

E-mail: [email protected]; [email protected]; [email protected] Abstract. Different approaches to the problem of mass quantization are discussed. The Barut ideas of crucial influence of magnetic forces for explaining the properties of the strong interaction are considered in details. It is shown that this approach gives the possibility to understand the enormous number of elementary particles (about 400) as the excited states of stable fundamental particles ( , , ) , bounded by magnetic interactions.

𝑒𝑒 𝑝𝑝 𝑣𝑣 1. Introduction The description of the mass spectrum of the observed elementary particles is included in the Ginzburg list of 30 most important unsolved problems in theoretical physics [1]. There are numerous approaches to its solution: group methods based on SU (N) - symmetries (Gell-Mann); dynamic (Barut); relational (Vladimirov); geometric (Bolokhov, Vladimirov) and many others. Interesting formulas for the masses of leptons and hadrons are obtained. One of the people who “laid the foundation” was Nambu [2], whose idea was to connect the masses of all elementary particles known at that time with the fine structure constant. Barut was also a supporter of that idea, and in 1979 obtained a formula in the form of an empirical dependence related to lepton masses [3]: = 1 + (1) 3 𝑛𝑛 4 where is the electron mass, is the𝑛𝑛 fine structure𝑒𝑒 constant.𝑘𝑘=0 This formula is in a good agreement𝑚𝑚 with𝑚𝑚 the� observed2𝛼𝛼 ∑ masses𝑘𝑘 � of leptons. For example, for n = 0 𝑒𝑒 . we get the𝑚𝑚 electron mass 𝛼𝛼. = 0.510999 ( = 0.510999 ); for n = 1 - the muon . mass . = 105.549 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 ( = 105.658 )𝑒𝑒𝑒𝑒𝑒𝑒; for n = 2 the mass of the tauon . = . 𝑒𝑒 𝑒𝑒 1786.155𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 =𝑚𝑚1776.822𝑒𝑒𝑒𝑒𝑒𝑒 𝑀𝑀𝑀𝑀𝑀𝑀 𝑚𝑚 𝑀𝑀𝑀𝑀𝑀𝑀 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝜇𝜇 ( 𝜇𝜇 ). With a value of n = 3, the 4th lepton with a mass𝜏𝜏 of 10293𝑚𝑚.711 is 𝑒𝑒𝑒𝑒𝑒𝑒predicted,𝑀𝑀𝑀𝑀𝑀𝑀 which𝑚𝑚 has not yet been𝑀𝑀𝑀𝑀 observed.𝑀𝑀 𝑚𝑚 A little 𝑀𝑀later,𝑀𝑀𝑀𝑀 a𝑚𝑚 Japanese𝜏𝜏 physicist𝑀𝑀 Yoshio𝑀𝑀𝑀𝑀 Koide discovered the following relationship between the masses of𝑀𝑀 leptons𝑀𝑀𝑀𝑀 [4]: + + = + + (2) 2 2 𝑒𝑒 𝜇𝜇 𝜏𝜏 𝑒𝑒 𝜇𝜇 𝜏𝜏 Expression (2) is true with a very high𝑚𝑚 accuracy.𝑚𝑚 𝑚𝑚 Based3 � �on𝑚𝑚 experimental�𝑚𝑚 � data𝑚𝑚 � (2016), the ratio of the left side of (2) to the right side (on the right side without taking into account the coefficient) is obtained and is equal 0.6666605 ± 0.0000068 (in theory that ratio is 0.666666(26)). Despite of 3 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 MRSU 2020 IOP Publishing Journal of Physics: Conference Series 1560 (2020) 012005 doi:10.1088/1742-6596/1560/1/012005

this, no reasonable theoretical explanation of formula (2) has yet been obtained. The predicted mass of the -lepton from the Koide formula, turns out to be . = 1786.968884 ±0.000065, and . the experimental value is = 1776.822. 𝑡𝑡ℎ𝑒𝑒𝑒𝑒𝑒𝑒 𝜏𝜏 Varlamov𝜏𝜏 [5] also presented𝑒𝑒𝑒𝑒𝑒𝑒 his own formula for the particles𝑚𝑚 masses. If we take into account the principle of equivalence between𝑚𝑚𝜏𝜏 mass and energy, it can be argued that the mass formula

= + + (3) 1 1 𝑒𝑒 defining the mass (energy) of the state (cyclic𝑚𝑚 Lorenz𝑚𝑚 representation�𝑙𝑙 2� �𝑖𝑖 2 �( , )), is, in a sense, similar to the well-known relation = , where the electron mass plays the role of “quantum of mass” . Up to present it is difficult to give preference to any of the existing𝑙𝑙 app𝑖𝑖 roaches. In our opinion, Barut approach is more promising,𝐸𝐸 ℎ𝜈𝜈 as it allows one to naturally extend it to describe the spectrum𝑚𝑚 of𝑒𝑒 the hadron sector, which is much richer in the number of observed states.

2.Materials and methods From the very beginning we shall indicate some not well known facts about the magnetic interaction.

2.1. Unusual (little known) properties of magnetic forces 2.1.1 Attraction is possible at different orientation of magnetic moments.

= 2 < 0 (4) 𝜇𝜇1𝜇𝜇2 𝑖𝑖𝑖𝑖𝑖𝑖 3 𝑊𝑊 − 𝑟𝑟

Figure 1. Coaxial parallel orientation of magnetic moments

= < 0 (5) 𝜇𝜇1𝜇𝜇2 𝑖𝑖𝑖𝑖𝑖𝑖 3 𝑊𝑊 − 𝑟𝑟

Figure 2. Misaligned antiparallel orientation of magnetic moments

2.1.2. Different dependence upon a distance.

~ ± ± + (6) 𝑏𝑏 𝑐𝑐 𝑑𝑑 𝑖𝑖𝑖𝑖𝑖𝑖 2 3 4 2.1.3. Emergence of a repulsive core independently𝑊𝑊 of 𝑟𝑟 orientation𝑟𝑟 𝑟𝑟 .

[ × ] ~ +𝜇𝜇⃗ 2 (7) 𝜇𝜇� 𝑟𝑟̅ The presence of terms with different signs in the𝑖𝑖𝑖𝑖𝑖𝑖 interaction6 potential makes it possible to obtain for different particles with different masses and charges𝑊𝑊 a large𝑟𝑟 number of potential wells in which the bound states of particle systems can exist and may be observed experimentally as resonances.

2.2. Barut mass formula

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To illustrate the effectiveness of the Barut method, let us deduce the above mass Barut formula (1) for leptons [3]. For a particle of mass with a charge moving in the field of a magnetic dipole we have: 𝑚𝑚 = 𝑒𝑒 𝜇𝜇 2 (8) 𝑚𝑚𝑣𝑣 𝑒𝑒𝑒𝑒𝑒𝑒 In the nonrelativistic case, the Bohr-Sommerfeld quantization3 rule can be applied: = 𝑟𝑟 , 𝑟𝑟 = 0,1,2, … (9) From (9) we find = , then we substitute this expression into (8) and we obtain: 𝑚𝑚𝑚𝑚𝑚𝑚 𝑛𝑛ħ 𝑛𝑛 𝑛𝑛ħ 𝑟𝑟 𝑚𝑚𝑚𝑚 = 2 (10) ħ For kinetic energy, we have the expression: 2 𝑣𝑣𝑛𝑛 𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛

= = = 2 4 (11) 𝑚𝑚𝑣𝑣𝑛𝑛 ħ 4 4 On the other hand, adding the rest mass of the electron2 2 to the Nambu formula for the muon [2]: 𝐸𝐸𝑛𝑛 2 2𝑒𝑒 𝑚𝑚𝜇𝜇 𝑛𝑛 𝜆𝜆𝜆𝜆

= (12) 3 we get 𝜇𝜇 𝑒𝑒 𝑚𝑚 2𝛼𝛼 𝑚𝑚 = 1 + , = 1 (13) 3 Using (11) and (13) we get the Barut𝜇𝜇 formula𝑒𝑒 [3]: 𝑚𝑚 𝑚𝑚 � 2𝛼𝛼� 𝑛𝑛 = 1 + (14) 3 𝑛𝑛 4 Here = 0 for electron; = 1 for𝑛𝑛 muon;𝑒𝑒 = 2 for𝑘𝑘= 0tauon; = 3 for the 4-th lepton. 𝑚𝑚 𝑚𝑚 � 2𝛼𝛼 ∑ 𝑘𝑘 � 2.3. The 𝑛𝑛main ideas of Barut 𝑛𝑛 𝑛𝑛 𝑛𝑛 As it is well known, the quarks show no existence in nature (nobody observed them up to now). Observed elementary particles (more than several hundreds) according to Barut can be described as bounded states of a small number of really stable particles , , . The features of strong interactions, such as: − 1) Short interaction range; 𝑝𝑝 𝑒𝑒 𝜈𝜈 2) Saturation; 3) Charge independence (isotopic invariance); 4) Strong spin dependence; 5) Pairing; 6) Pauli principle; 7) Experimentally observed quark potential;

( ) = + + (15) can be explained strictly by electromagnetic forces only.𝑎𝑎 𝑉𝑉 𝑟𝑟 𝑟𝑟 𝑏𝑏𝑏𝑏 𝑐𝑐 2.4. Dirac equation with electromagnetic interaction From the Dirac Equation

+ m = 0 (16) describing a free particle, one can obtain an𝜇𝜇 extended Dirac equation for a charged particle 𝜇𝜇 interacting with an external electromagnetic �field𝛾𝛾 𝜕𝜕 in two� steps.𝛹𝛹 1) The extension of derivatives results in the appearance of additional terms

𝜕𝜕𝜇𝜇 → 𝜕𝜕𝜇𝜇 − 𝑖𝑖𝑖𝑖𝑖𝑖𝜇𝜇

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( ) + ( ) (17) in the Hamiltonian operator, where is the normal magnetic dipole moment of the charged particle. 𝜇𝜇𝑛𝑛 𝜌𝜌3 𝜎𝜎� ∙ 𝐻𝐻� 𝜇𝜇𝑛𝑛 𝜌𝜌1 𝜎𝜎� ∙ 𝐸𝐸� 2) For a neutral particle with an abnormal𝜇𝜇𝑛𝑛 (anomalous) magnetic dipole moment (such as a neutron), one has to add on the right hand side of the Dirac Equation , the Pauli coupling term

0 = ( ) + ( ) (18) 𝜇𝜇𝜈𝜈 𝑎𝑎 𝜇𝜇𝜈𝜈 𝑎𝑎 3 𝑎𝑎 1 where is the abnormal→ magnetic𝜇𝜇 𝐹𝐹 𝜎𝜎 dipole𝜇𝜇 moment𝜌𝜌 𝜎𝜎� ∙. 𝐻𝐻� 𝜇𝜇 𝜌𝜌 𝜎𝜎� ∙ 𝐸𝐸� The above two steps, when applied to the electron moving around a proton, yield a radial equation with an effective𝜇𝜇𝑎𝑎 potential of the following form:

( ) = ± + ± + (19) 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑 2 3 4 The coefficients , , , are obtained automatically𝑉𝑉 𝑟𝑟 𝑟𝑟 and𝑟𝑟 they𝑟𝑟 are fixed𝑟𝑟 in the model and their explicit form will be given below. 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑑𝑑

Figure 3. Effective interaction potential of the electron with the proton in the Barut model

Figure 3 shows two potential wells that are obtained for the “electron-proton” system, for given values of the parameters , , , which are automatically fixed in the model as a result of the above procedure (two steps) to turn on the interaction. The right potential well𝑎𝑎 has𝑏𝑏 𝑐𝑐 a 𝑑𝑑minimum at 10 cm. In this well the terms + play the −8 𝑎𝑎 𝑏𝑏 main role. We call this domain electric. Other terms in (19) related to magnetism give small2 corrections. It is in this well that a familiar bound𝑟𝑟 ≈ state the hydrogen atom arises. In− the𝑟𝑟 left𝑟𝑟 potential well with a minimum at 10 10 cm, the short-range magnetic forces play the main role, and the electric forces give small− 13corrections.−14 Here the formation of𝑟𝑟 heavy≈ particles− is possible due to magnetism, which in the experiment will look like resonances. In the case of a coupled system of two heavy particles (for example, a neutron- proton), the left well is interpreted in the Barut model as a well that reproduces all the properties of strong interaction with small electromagnetic corrections from the right well.

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2.5. Effective potential for two charged particles with normal and anomalous magnetic moments Let us consider two interacting charged particles with normal and anomalous magnetic moments. The full Hamiltonian of the system in this most general case is as follows [6]:

= + + + ( ) | | (20) 1 𝑒𝑒1 2 1 𝑒𝑒2 2 𝑒𝑒1𝑒𝑒2

2𝑚𝑚1 1 𝑐𝑐 ⃗2 2𝑚𝑚2 2 𝑐𝑐 ⃗1 𝑟𝑟⃗1−𝑟𝑟⃗2 12 1 2 ×( 𝐻𝐻 ) �𝑝𝑝⃗ −×( 𝐴𝐴 �) �𝑝𝑝⃗ − 𝐴𝐴 � 𝑆𝑆 𝑟𝑟⃗ − 𝑟𝑟⃗ = , = Here | | | | are vector potentials of the electromagnetic field created 𝑀𝑀��⃗1 𝑟𝑟⃗2−𝑟𝑟⃗1 𝑀𝑀��⃗2 𝑟𝑟⃗1−𝑟𝑟⃗2 3 3 by one particle𝐴𝐴⃗1 𝑟𝑟at⃗1− 𝑟𝑟the⃗2 point𝐴𝐴⃗2 of location𝑟𝑟⃗2−𝑟𝑟⃗1 of the other particle; = (1 + ) is the intrinsic magnetic moment of a charged particle with spin 1/2, proportional to𝑒𝑒 ħthe Bohr magneton; is a ��⃗ parameter that determines the magnitude of the intrinsic anomalous𝑀𝑀 magnetic2𝑚𝑚𝑚𝑚 moment𝑎𝑎 𝜎𝜎⃗ of the particle; is the particle spin operator; , ( = 1,2) are charges and masses of particles; the last𝑎𝑎 term describes the spin-spin interaction of the intrinsic magnetic moments of the particles. It is usually 𝜎𝜎written⃗ as: 𝑒𝑒𝑖𝑖 𝑚𝑚𝑖𝑖 𝑖𝑖

( ) = 3 , = | | (21) 1 𝑟𝑟⃗ 3 𝑆𝑆12 𝑟𝑟⃗1 − 𝑟𝑟⃗2 𝑟𝑟 ��𝑀𝑀��⃗1𝑀𝑀��⃗2� − �𝑀𝑀��⃗1𝑟𝑟⃗0��𝑀𝑀��⃗2𝑟𝑟⃗0�� 𝑟𝑟⃗0 𝑟𝑟⃗ After the transition to the center of mass system, the Hamiltonian takes the form:

1 × × = + + + 2 + 2 22 2 2 2 2 𝑝𝑝⃗ 𝑒𝑒1𝑒𝑒2 𝑒𝑒1𝑀𝑀��⃗2 𝑒𝑒2𝑀𝑀��⃗1 𝑒𝑒1 𝑀𝑀��⃗2 𝑟𝑟⃗ 𝑒𝑒2 𝑀𝑀��⃗1 𝑟𝑟⃗ 3 �⃗ 2 3 2 3 𝐻𝐻 − �𝐿𝐿 � 1+ 2 �� 3 1 � � 1 � � (22) 𝜇𝜇 𝑟𝑟 𝑟𝑟 𝑚𝑚 𝑐𝑐 1 𝑚𝑚2 𝑐𝑐 𝑚𝑚 2𝑐𝑐 𝑟𝑟 𝑚𝑚 𝑐𝑐 𝑟𝑟 𝑀𝑀 𝑀𝑀 2 3 𝑟𝑟 �𝑆𝑆⃗ − �𝑆𝑆⃗𝑟𝑟⃗0� � Here = is the reduced mass; = ; = ; = + ; = + ; 𝑚𝑚1+𝑚𝑚2 𝑚𝑚1𝑝𝑝⃗1−𝑚𝑚2𝑝𝑝⃗2 = 𝜇𝜇 𝑚𝑚 1is𝑚𝑚 2the radius of the mass 𝑟𝑟⃗center𝑟𝑟⃗2 ;− 𝑟𝑟⃗1= 𝑝𝑝⃗× ; 𝑀𝑀= ( 𝑀𝑀+ 𝑚𝑚) 1is the𝑚𝑚 2operator𝑃𝑃�⃗ 𝑝𝑝⃗1 of 𝑝𝑝the⃗2 1 1 2 2 total 𝑚𝑚spin𝑟𝑟⃗ +of𝑚𝑚 a𝑟𝑟⃗ system of two particles. 1 �⃗ �⃗ ⃗ 2 1 𝑅𝑅 Formula𝑀𝑀 (22) gives the effective potential of interaction𝐿𝐿 𝑟𝑟⃗ 𝑝𝑝for⃗ 𝑆𝑆the radial2 𝜎𝜎� function𝜎𝜎� in the form:

( ) = + + + (23) 𝑏𝑏1 𝑏𝑏2 𝑏𝑏3 𝑏𝑏4 2 3 4 𝑉𝑉 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 𝑟𝑟 In this expression the centrifugal potential ~ appeared as a result of variables separation of the 1 = , , 2 Laplacian inside the term 2 2 . 𝑟𝑟 𝑝𝑝⃗ ħ ∆𝑟𝑟 𝜑𝜑 𝜃𝜃 The coefficients , , , are of the form [6]: 2𝜇𝜇 − 2𝜇𝜇

1 2 3 4 𝑏𝑏 𝑏𝑏 𝑏𝑏 𝑏𝑏 =

𝑏𝑏1 = 𝑒𝑒1𝑒𝑒(2 + 1) 2 2 ( )( ) = ( + ) + 𝑏𝑏 ħ 𝑙𝑙 𝑙𝑙 + 3 2 1 2 1 2 1 2 2 1 2 𝑒𝑒 𝑒𝑒 ħ 𝑒𝑒 𝑒𝑒 ħ 𝑒𝑒 𝑒𝑒 ħ 1+𝑎𝑎 1+𝑎𝑎 2 2 2 2 3 2𝑚𝑚1𝑚𝑚2𝑐𝑐 �⃗ 2 2 1 1 𝑚𝑚1𝑚𝑚2𝑐𝑐 �⃗ ⃗ 4𝑚𝑚1𝑚𝑚2𝑐𝑐 ⃗ ⃗ 0 𝑏𝑏 �𝐿𝐿 𝑎𝑎 𝜎𝜎� 𝑎𝑎 𝜎𝜎� � ( �𝐿𝐿)∙ 𝑆𝑆 �( ) �𝑆𝑆 − �𝑆𝑆 ∙ 𝑟𝑟⃗ � � = + 2 2 2 (24) 𝑒𝑒1𝑒𝑒2ħ 1+𝑎𝑎1 1+𝑎𝑎2 4 𝑏𝑏4 4𝑚𝑚1𝑚𝑚2𝑐𝑐 � 𝑚𝑚1 𝑚𝑚2 �

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In general case, both the relativistic and nonrelativistic descriptions of two interacting fermions do not allow to completely solve the problem analytically. The main advantage of a relativistic description is a wide range of allowable energies. However, as a result, the final equations are complex and analytically solvable only for a small number of fields, often with a very specific choice of parameters. The scope of nonrelativistic equations is naturally limited by the range of permissible energies. The advantages of a nonrelativistic analysis are the relative simplicity of the equations and the possibility of easy comparison with known results, as well as the ability to use fewer “adjustable” parameters.

3. Conclusion We presented in a brief form the main ideas of Barut and gave a general view of the potential of interaction of two charged particles with normal and anomalous magnetic moments. Using the general expression (20) for the Hamiltonian, a number of problems were solved earlier. In the case of the “ep” system, it was predicted the possible existence of small Barut-Vigier atoms with a size of 10 cm [7]. The use of expression (20) for the bound “neutron-proton” system (deuterium nucleus)− 11turned out to be very effective [8], allowing us to describe the basic properties of the 𝑟𝑟deuteron.≈ For example, the absence of a singlet state in a deuteron on the experiment is easily proved. In this case, the spins of the particles are antiparallel (which means that the magnetic moments are parallel due to the fact that = +2.7 , = 1.9 ) and there will be no potential well due 𝑒𝑒ħ 𝑒𝑒ħ to the repulsion of the magnetic moments. There is hope using the general formula (20) to obtain a 𝜇𝜇𝑝𝑝 2𝑀𝑀𝑝𝑝𝑐𝑐 𝜇𝜇𝑛𝑛 − 2𝑀𝑀𝑛𝑛𝑐𝑐 more accurate mass spectrum of light and heavy particles.

Reference [1] Ginzburg V L 1999 What problems of physics and astrophysics seem now to be especially important and interesting (thirty years later, already on the verge of XXI century)? Physics- Uspekhi 169 pp 419–41 [2] Nambu Y 1952 An empirical mass spectrum of elementary particles Prog. Theor. Phys. 7 pp 595–96 [3] Barut A O 1979 Lepton mass formula Phys. Rev. Lett. 42 p 1251 [4] Koide Y 1983 New view of quark and lepton mass hierarchy Physical Review D 28 pp 252 – 54 [5] Varlamov V V 2017 Mass quantization and lorentz group Mathematical Structures and Modeling 2(42) pp 11–28 [6] Tahti D V 2000 PhD Thesis “The consequences of the accurate accounting of the magnetic moments in the two-body problem for interacting particles” Peoples’ Friendship University of Russia (RUDN) Moscow pp 1-112 [7] Samsonenko N V, Tahti D V and Ndahayo F 1996 On the Barut-Vigier model of the hydrogen atom Physics Letters A 220 pp 297-301 [8] Samsonenko N V, Tahti D V and Ndahayo F 1998 On the consequences of the accurate accounting of the “spin-spin” and “spin-orbit” interactions in deuteron Proceedings of the International conference on Physics of atomic nuclei Saint-Petersburg RAN p 45

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