INIS-mf—9691
THEORETIC PAPERS THE BLINDERN THEORETIC RESEARCH TEAM P.B. 1029. BLINDERN, OSLO 3, NORWAY Theoretic Papers 1983 Nr 2
THE MASSES OF ELEMENTARY PARTICLES. APPLICATIONS OP SEMI-CLASSICAL ENERGY LEVELS IN MAGNETIC MONOPOL SYSTEMS by Nils Aall Barricelli Institute of Matbenatics University of Oslo, Blindern, Norway THE MASSES OF ELEMENTARY PARTICLES. APPLICATIONS OF SEMI-CLASSICAL ENERGY LEVELS IN MAGNETIC MONOPOL SYSTEMS.
by
Nils Aall Barricelli Department of Mathematics, University of Oslo, Blindern, Norway
Abstract
The semi-classical method of calculating the energy levels in a system of two magnetic monopoles, presented in a preceding paper (Barricelli, 1982) is applied in the interpretation of the masses and angular momenta of various elementary particles. The interpretation involves new proposals concerning the organization of magnetic monopoles (configurations) in various elementary particles. It is worth noticing that no other theory known today has been capable of calculating the masses of so many elementary particles by using so few free parameters (meaning parameters neither expressed in terms of universal constants nor calculated by the theory) and with a precision comparable to that obtained in the model presented here. - 1 -
1. Introduction
In a preceding paper (Barricelli, 1982) the problem of calculating the energy levels by semi-classical (Bohr and Sommerfeld) methods in a system of two magnetic monopoles was examined. It was found that if semi-classical theory is applied rigorously the magnetic monopole can not be considered a point
charge, and must be assigned a finite (classical) radius rQ related to its magnetostatic mass m and its magnetic charge g by the formula
(The ro~radius defined this way is one half of what is commonly called "classical radius". We will call it "semi-classical radius" because it presents some advantages in semi-classical theory). When the magnetic monopoles are ascribed this kind of a finite radius, the energy levels in a system of two magnetic monopoles can easily be calculated by semi-classical theory in spite of their large magnetic charges which according to Dirac are expressed by integer numbers in terms of an elementary charge g (Dirac monopole) about —p— times larger than the charge of the electron e:
(2) q _ 137.036 c _ \/137.036fic where c=Speed og light, f^-j^' h being Plank's constant. If we decided to ignore the finite size of the magnetic monopoles and tried to calculate the energy levels assuming a coulombian field all the way to the lowest distances, we would find that no less than 17 energy levels would give for example circular orbits with a radius lower than rQ. The potential and bindings energies at these low levels would have negative values far greater than the magnetostatic energies (or masses) of the two magnetic monopoles orbiting each other. The system would have negativ energy if we use non relativistic formulas, and imaginary or complex energies if we use relativistic formulas. The result would be completly absurd. - 2 -
This is the reason of earlier failures to calculate such energy levels by semi-classical theory. If we want to obtain meaningful results, as shown in the preceding paper (Barricelli, 1982) we have to take into account
that inside the semi-classical radius rQ the magnetic field does not grow to infinity when the distance r between the two charges approaches 0, but on the contrary the force becomes smaller as a growing portion of the magnetic charge remains out- side and does not contribute to the field. The attraction force goes to 0 and the potential energy for two equal magnetic charges of opposite sign and equal mass (or magnetostatic energy) m goes to -2mc2 when r-*0. This makes it impossible to find an energy level with negative total energy if the potential field is consistent with semi-classical theory because the minimum poten- tial energy -2mc2 is barely sufficient to make up for the sum 2mc2 of the magnetostatic energies of the two magnetic charges. The absurdities which have frustrated earlier attempts to calculate energy levels by semi-classical theory are due to a mistaken application of the theory. In wave mechanics it is customary to consider both electrons and magnetic monopoles as point charges. As long as that approach is used there is perhaps no much possibility of finding meaningful
energy levels inside the semi-classical radius rQ. However, if the same potential fields which give meaningful energy levels by a semi-classical approach were used in a wave mechanical(for example a Dirac equation) approach, we see no reason why meaning- ful energy levels could not be obtained also by the wave mechan- ical approach. The argument that a potential with a minimum value -2mc2 can not give negative energies in a system of two magnetic monopoles with masses equal to m is valid for wave mechanics as well as for semi-classical mechanics. A machine program is being developed for the purpose of solving by wave mechanical methods the same problems which have been solved by semi-classical methods. That can give us the possibility to verify whether the wave mechanical methods will give results substantially different from those obtained by semi-classical methods. The purpose of this paper is to show how the possibility of calculating the energy levels in a system of magnetic monopoles can be helpful in the development of a magnetic quark theory for - 3 -
the interpretation of the properties of elementary particles. Particularly the masses of elementary particles are a character- istic which has been difficult to calculate earlier in the absence of a method for identifying meaningful energy levels. In order to carry out this program we will, however, have to introduce a set of new assumptions both concerning the nature of the magnetic potential fields inside the semi-classical radius and concerning the way in which quarks and other magnetic monopoles (gluons) are supposedly put together in the various elementary particles. One may agree or disagree on the particular assumptions we are going to introduce and perhaps others will be able to do a better job. But the investigation we are going to present seem to give evidence that the calculation of energy levels made possible by a rigorous application of semi-classical theory is the way to obtain information concerning the possibility of interpreting the masses and other properties of elementary particles by a magnetic monopole theory. While in the earlier paper (Barricelli, 1982) no new assumption was introduced; only a rectification of a common error and a self- consistent application of semi-classical theory was involved, in the present paper we will have to introduce a series of new assumptions in order to achieve an interpretation of elementary particles properties.
2. The quest for a potential distribution
The first thing to do in order to identify a suitable dis- tribution was to write machine programs capable of calculating energy levels and other orbital characteristics of two magnetic monopoles for any given asymptotic coulombian potential distribu- tion fulfilling the requirements imposed by semi-classical theory. Two programs, a relativistic and a non relativistic one, whose characteristics are described in the preceding paper (Barricelli, 1982, table 1A,1B and 1C) , solve this problem for circular orbits. Of course the relativistic one is the one which is normally used. The other one has the purpose of verifying the validity of the relativistic one for energy levels involving low velocities compared with the velocity of light. Table 1A
Masses of binary systems of magnetic monopoles with respective rest masses M »M9n and the respective magnetic charges S-iig» (s being the Dirao monopole, and — the nonopolar unit of mass). 0 Energy level n-1 Energy level n«2
4M 9 4M 9 «20 «o Q «o "o 0 "o S "2g «10 «1 2 -g -3g -g "2g -3g «o e 0. 08307 1. 07933 4.O8467 0. 19111 1. 187 23 4. 20033 2 1. 07933 0. 1. 18723 0. 1. 12998 4MO « 05671 I.O5279 13884
9M0 3g 4. 08467 1. 05279 0.04416 4. 20033 1. 12998 0. IO952
Table IB
Maximum reciprocal distance (rQ»l) reached by the two monopoles in their linear oscillations.
Energy level n«1 Energy level n-2
M 4M 9Mo M 9M «20 o O 0 4«0 O "2g -3g «1 g2 -g -g -2g -3g M g 0. 36882 0.29541 0.26802 0. 48906 0.38309 0.34473 0 4Mo 2g 0. 29541 0.22168 0.19500 0. 3830^ 0.284a> 0.2489 i
9M 0. 26802 O.195OO 34473 tr. 24893 o 3g 0.16717 Or. 21268 Table 1C
Maximum velocities v./c , vg/c of the two monopoles (c 'beiRg the speed Of light).
Energy level n«l Energy level n=2
4Mo 9 M 4M «20 «0 «o o Q 9Mo
0 4 -3g «10 *1 g2 -a« -g -2g -3g M g 0.27955 0. 33946 0. 36895 O.4O042 0.49080 0. 52941 0 T 0.27955 0. O8966 0. 64407 O.4O842 0.13946 0. 06917 4M. 2g v-i/° 0.08986 0. 11848 .0. 13418 0.13943 0.18392 0. 20833 0.33946 0. 11848 0. O6OO7 O.4908O 0.18392 0. 09425
9M0 3g O.O44O7 0. 06 007 0. 06991 0.06917 O.O9425 0. IO98I 0.36895 0. 13418 0. 06991 0.52941 0.20833 0. IO98I * - 4 -
It was immediately clear that in order to interpret the low spins of many elementary particles, circular orbits could not be used. Orbits with low angular momentum, often with angular momentum equal to 0, were needed. The next two programs, a relativistic and a non relativistic one, solve the same problem for linear oscillation orbits, which are orbits of angular momentum equal to 0. In such orbits the two particles move on a straight line through their common barycenter. Both particles move simultaneously through their centre of gravity in opposite directions. They reach simultaneously their respective maximum distances from the centre of gravity and are pulled back by their reciprocal attraction to repeat in reverse the same movements (fig. 1). In these programs the Sommerfeld conditions are used for quantization. The theory for the relativistic one is presented in the appendix. Also this program will calculate the total mass M of the system (see table 1A which is calculated by using a potential to be described in the appendix, identified by formula (T) and fig. 2). It will also calculate the maximum distance between the two monopoles (table 1B) and their maximum velocities v. and v_ which are reached when they move through the barycenter (table 1C). Because of the different potential distributions used no far reaching conclusions should be drawn from a comparison between these tabels and the tables 1A,1B,1C of the preceding paper (Barricelli, 1982). The masses in table 1A are measured by using the magneto- static rest mass MQ defined by the following formula as a unit of mass:
(3)
where rQ is the semi-classical radius of the electron given by formula
(4)
M@ being the mass of the electron and e its charge. This choice of the semi-classical radius is the most obvious one, in view of the fact that the proton and other elementary particles also have a radius and a crossection comparable to those ascribed to the •lectron in semi-classical theory. Fig. 1 Linear oscillation movements in s binary system in vhich the rtut-aass of one particle is 4 times greater than tho rest-mass of the othar one (H2O-4M1O). - 5 -
Thus our first hypothesis is that all magnetic monopoles have a common semi-classical radius, which is equal to the semi- classical radius of the electron. This is part of a more general hypothesis holding that the
potential energy in-a system of two magnetic monopoles g1 and
g2 depends only on the distance r and their magnetic charges, being proportional to the product 9-i'32 °*" t*ie two charges. The magnetic monopoles we are going to consider may also have electric charges besides magnetic ones, and the electric charges may also have an influence on the potential energy U, which is supposedly proportional to 9i9o + e1e2 in a svstem °^ two magnetic monopoles with the respective magnetic and electric charges 9i»e-i (first monopole) and 9j/e2 (second monopole) . Otherwise the potential energy U is supposed to depend only on the distance r between the two particles' barycenters:
(5) U = (g1g2+ele2)V(r)
The requirements which must be fulfilled by the distance- dependent function V(r) are: 1 . Since U must approach a coulombian potential when r goes to infinity, it follows
rV(r) -» 1 for r -> æ
r 2. Moreover in the case g.,=g, g2=-g, e^=e= ) the potential U 2 2 2 must be -2Mnc for r=0, meaning -g V(0)=-2Mnc = - 2— (two u 0 rQ coinciding opposite magnetic charges cancel out, see Barricelli, 1982, section 3), it follows: 1 2M c (6) V(0) =-L = —Qn — ro g*
However, in order to find a specific formula for the potential U and the distance-dependent function V(r) we will have to test different potentials by applying them to the calculation of the masses of different elementary particles and compare the calcu- lated masses with the observed ones. This requires that some tentative proposals concerning the magnetic monopoles involved in the respective elementary particles and the way in which they are arranged must be made and each proposal must be tested together with the various potential formulas. In the next section we will outline some of the procedures used in this inquiry and a few of the results. We have no machine program for orbits intermediate between linear oscillation and circular ones. The energy levels for such orbits will have to' be calculated by interpolation between circular orbits and linear oscillation orbits' energies, by taking the orbital angular momentum into account (see Barricelli and Kolset, 1982).
3. The baryon interpretation problem. Once the programs were available the next step in our quest for a potential distribution was to select a few elementary par- ticles suitable for the purpose and try to make some tentative assumptions concerning the magnetic monopoles involved and the way in which they are kept together in each particle. Let us first consider the baryons. Our assumptions concerning the magnetic monopoles involved are inspired by the quark models. In all quark models, chromodynamic as well as magnetic ones, the baryons are assumed to be formed by three fermions of spin ^ > designated as "quarks" and one or several particles capable of "gluing" the quarks together, which are called "gluons". In one of the simplest magnetic quark models, which is the one we are going to use, the three quarks are assumed to be magnetic mono- poles with equal magnetic charges and there is a single gluon* assumed to be a boson of spin 0 with a three times larger magnetic charge of opposite sign. Different magnetic charge assignement and different arrangements of energy levels (see next section) were tested, some of them with more than 40 different potential distributions. The only acceptable results obtained so far are based on the assumption that the magnetic charge of each quark is equal to a Dirac monopole g, whereas the gluon, hereafter called "Baric" and designated by the symbol B3, has a three times larger opposite magnetic charge -3g. Another magnetic monopole, desig- nated L1, a boson of spin =0, magnetic charge equal, but opposite to that of the quarks and electrically uncharged is also involved. But this monopole, which is called "light boson", could hardly be considered a gluon, since it does not have the function of
* Our use of this term is different from common practice. Our B3 is not being exchanged between particles; it is gluing together other monopoles by its magnetic field. - 7 -
gluing other monopoles together. Four types of qurks have been
used: the u-guark designated as Ux , the d-quark designated as
Dj, the strange quark designated as S1 and the charmed quark designated as C . These particles are listed in table 2 which presents some of their properties. The reader may notice that the electric charges ascribed to the various quarks in table 2 are expressed by whole numbers in terms of e instead of the fractional numbers commonly used. This has been made possible by ascribing an electric charge -e to the Baric thus allowing the addition of an electric charge e/^ to each one of the three quarks in every baryon without changing its total electric charge. This convention has no effect on the electric charges of baryons and mesons and little if any on the calculated masses. But it has the advantage of allowing an inter- pretation of leptons and a calculation of their masses without running into the problem of leptons with fractional charges. A major problem which arose at the very beginning of our inquiry was the selection of a distribution of the quarks in dif- ferent energy levels around the Baric and the use of our machine program in this kind of energy level calculations. We opted for a distribution analogous to that of the three electrons in a Litium atom, namely two quarks with opposite spin in an internal orbit about the Baric, and a third quark in an external orbit.* Baryons with higher energy could be interpreted by rising one quark from the internal to the external orbit or by some other increase of energy level. This, however, would seem to imply that different quarks will have to be treated in the same way as identical quarks as far as Pauli's exclusion principle is concerned. However, since atoms with three or four different kinds of electrons are not a common object of experience, we can not say that the exclusion principle which actually applies, hereafter designated as "the extended exclusion principle", should not be interpreted this way as far
* An other alternative might have been the assumption that all of the three quarks would be moving in a common orbit. Such a possi- bility is supposedly not excluded by Pauli's exclusion principle if the three quarks are not all identical, but is excluded if they are. Since two of the A-baryons have three identical quarks and are energetically not substantially different from the other A-baryons, this seems to rule out this solution (unless we want to introduce colours also in magnetic quark theory where chromo- dynamic interactions are not needed, being replaced by magnetic interactions). Table 2 Description of magnetic monopoles.
Name Symbol Mass Electric Magnetic Strangeness Charm Spin Configuration charge charge E«l brief notation e-1 g-1
Baric B3 9.000213 -1 -3 0 0 0 B Light boson L1 1.000000 0 -1 0 0 0 L u-quark 1.000213 1 1 0 0 1/2 U Ul d-quark 1.000000 0 1 0 0 1/2 (BUL)O s-quark (compact) 1.079326 0 1 -1 0 1/2 ((BU)OL)1 Sl s-quark (.split) 1.068 0 1 -1 0 1/2 — Tl c-quark (normal) 1.572278 1 1 0 1 1/2 ((BS)2L)3 c-quark (l-version) 1,562069 1 1 0 1 1/2 ((BT)2L)3 Zl
The respective antiparticles B , L ,U,D,S,T,C,I have opposit magnetic and electric charges, and opposit strangeness ani charm. Lower indexes identify positive magnetic charges; upper indexes identify negative ones. _ Q i>
as quarks are concerned. An other problem was created by the very nature of our machine programs, which are designed to deal with systems of two magnetic monopoles, not with systems of three or four monopoles. The very nature of the systems we are dealing with suggested a possible solution to this problem. The tentative solution adopted was to assume that the two quarks moving in a common (for example internal) orbit could be kept together and move together (keeping a reciprocal distance r=0) as a single particle by the very dynamical properties of the sytem. This looks dynamically possible for example in a linear oscillation orbit system in which the two
quarks, say Ux and D1, could keep together moving as a single partner (UJDJ^ of magnetic charge 2g to the Baric B3 of opposite magnetic charge -3g. Of course it is possible and very likely that the two quarks would not constantly keep a distance r=0, but their distance would oscillate about this value. Our solution was therefore expected to yield only an approximate result. A similar solution would not give an acceptable precision in the case of two electrons moving in a common orbit about an atomic nucleus because of the high energy required in order to bring the two electrons close to one another, compared with the other energies involved. But in a system of monopoles with recip- rocal distances far shorter than the classical radius, the repul- sion force between two quarks becomes weaker (approaching zero) the closer they are to each other. As a result the energy required in order to bring their distance close to 0 is not large compared with the other energies involved. We tried this solution in order to calculate the energy of 3 a system, hereafter designated as (B U1D1)n consisting of a Baric B3, and two "positionally associated" quarks (UjDi) at the energy level n, designated (L-n) level. Once this energy is calculated, the total energy of a baryon 3 ((B U1D1)nUj)1 containing an extra Uj in an external orbit of energy level 1 - designated (L-1) level - can be calculated by the not unusual procedure of treating the internal system 3 (B U1D1)n and the external quark U1 as two separate particles moving about each other in an orbit of energy level 1. The next question was to identify the energy level n of 3 the (B U1D1)n system. If the two positionally associated quarks - 9 -
(U..D..) could truely have been treated as a single magnetic mono- pole with a charge 2g, twice that of a single quark, and a mass
4MQ, the solution would obviously have been n=1 for baryons of lowest mass (Proton and Neutron). That solution was actually tried to begin with; but the results were not encouraging. The masses of baryons became much too low compared with those of other elementary particles (mesons and leptons). They were also much lower than the values any regular treatment of the problem of a Baric with two quarks at the lowest energy level could be expected to give. It was therefore decided that if our tentative approach were to be used at all, the energy level should be con- sidered a free parameter, allowed to take only integer values, to be determined together with the potential distribution. Some of the more than 40 asymptotic coulombian distributions we tested had free parameters of their own. To begin with also the masses of the quarks were treated as free parameters before they were identified as Dirac monopoles or systems of several monopoles whose masses can be calculated (see next section). By using what- ever free parameters were available we attempted to fit the cal- culated masses of a series of elementary particles to the observed ones, after selecting for each elementary particle an arrangement of magnetic monopoles consistent with the one described above, which was ascribed to the Proton. Some of the main results are presented in the following sections. We shall start by describing the properties of the quarks and magnetic monopoles which have been used in our interpretation of elementary particles.
4. Monopoles and quarks. The set of quarks and magnetic monopoles which was eventually adopted is presented in table 2. The baric and the quarks are not the only magnetic monopoles involved in the theory. Other magnetic monopoles can be con- structed by so-called "zero-level" or (L-0) associations between charges of different sign. Also associations at an energy level higher than zero, namely (L-1),(L-2) and (L-3) associations are involved in the Sx and Cx quarks (see table 2). In these cases the result will not be truely monopoles, but may still be magnetically charged objects. - 10 -
An (L-0) association is a monopole whose electric and magnetic charges are the sums of the respective charges in the associated monopoles. Its spin is either J or 0 depending on whether the association includes an odd or an even number of particles of spin J (fermions). .Particles of spin greater than | can not be members of an (L-0) association. For example the (L-0) association of a baric B3 with the 3 quark Ut will be a monopole designated by the symbol (B U],)0f with spin J, magnetic charge -3g+g=-2g and electric charge -e+e=0. Schwinger (1969) introduced monopoles with similar mag- netic charges and spin properties, when he assumed that a quark could absorb a magnetically triply charged boson of spin zero. But in our semi-classical interpretation we consider (L-0) asso- ciations as the result of a binding at the lowest possible energy level, namely level zero. This energy level is characterized by resting associated monopoles at the lowest energy position, namely the systems barycenter. More (L-0) associations will be introduced later on. We may, however, give notice that (L-0) associations of a monopole and 1 x 3 its anti-particle, such as (U U1)0, (D Di) 0 or (B B3)0 will be con- sidered as annihilations. This kind of association does not give a true particle. Moreover, will the (L-0) association of a mono- pole such as for example B3, with an other (L-0) association- 1 product, such as (B3U )0, which includes its anti-particle, be considered as equivalent to the result obtained by removing 3 (annihilating) the two monopoles (B and B3) from the result:
(B'BjUMO = U1
An other monopole, a boson of spin equal to zero, magnetic charge -g and no electric charge, which is called "light boson" and is designated by the symbol L1 (see table 2) was originally intro- duced as a means to interpret the properties of the strange quark Sj (see below). The three monopoles B^UIJL1 are the primary monopoles we will use in order to construct all the other particles by (L-0) or higher energy associations. The masses of magnetically and electrically charged monopoles are calculated as follows: - 11 -
Our unit of mass M,, to be designated as monopolar mass unit, is
defined by the preceding formula (3). The mass Me of the elec- tron is then according to (4) and (3):
e (7) M = 2 •= 0.000213M = 0.511 M.E.V. e 2rQc ° { if the mass M of a magnetic monopkle with a magnetic charge ig and an electric charge je is originated exclusively by its mag- netostatic and electrostatic energy, it can be calculated by the formula:
2 2 (8) M = i MQ + j Me
which in monopolar units becomes:
2 2 (9) M = (i + j «0.000213)MQ
This is assumed to be the case for the primary monopoles B3,Ui, L1 and their (L-0) associations, whose masses, calculated this way, are given in table 2. Also the mass of a positional associa- tion such as (UJDJ^) is calculated by the same rule. In the brief notations used in table 2 in order to define the various particles the indexes are omitted. For example (BU)0 3 stands for (B Ux)0 or its anti-particle (BjUMO; { (BU) 0L) 1 stands 3 for (B3UMOLM1 or its anti-particle ((B UX) 0L1) 1, etc. Except for the ambiguity between a particle and its anti-particle, which is unimportant for the calculation of masses, there are no other ambiguities created by the use of brief notations. How the masses of associations at an energy level higher than 0, namely (L-1) associations, (L-2) associations etc. were calculated will now be explained. Two of the quarks listed in table 2, namely the s-quark and the c-quark, are ascribed masses which are larger than those cal- culated by formula (9). It was soon discovered that in order to interpret the masses of "strange" and "charmed" particles it would be necessary to ascribe these two quarks greater masses than those required by their magnetostatic and electrostatic energies, and approaching those indicated in table 2. This led to an interesting discovery. As soon as the poten- tial distributions we were testing started showing satisfactory - 12 -
performance it was found that the mass one would have to ascribe to the s-quark would, in many cases, lie close to the mass of a binary system formed by an (L-1) (energy level 1) association of two monopoles with the respective magnetic charges -g and 2g. The reader can veri.fy that the mass 1.07933 M ascribed to the Si quark in table 2 is identical to the mass of an (L-1) system of two monopoles with the charges -g and 2g which is listed in table 1A. This led to the hypothesis that the s-quark is not a single monopole, but an (L-1) association of a fermion and a boson with the mentioned magnetic charges, as for example the associa- tion ((BaUMOLMi or ((BU) 0L) 1 which is indicated in table 2 as a definition of the s-quark (compact). An other alternative to be considered is the {L-1) association 1 1 of the u-antiquark U and the monopole (B3L )0, which would ascribe to the s-quark a configuration ((BL)0U)1 instead of ((BU)0L)1. Perhaps the two alternatives should be considered two different states of the same system (involving the same three monopoles B,L and U) and oscillations between the two states can not be ruled out, since they have the same energy. We have a machine program which can calculate the mass of such an association when its definition is indicated in an input card by its "configuration" ((BU)0L)1. This interpretation of the s-quark was what led to the introduction of the light boson La, which later on also proved useful in the interpretation of the c-quark and other particles. This way was the identity between the mass 1.07932M of the s-quark and the mass of a binary (-g,2g) system at the enrgy level 1 - which is indicated in table 1A - explained. The d-quark (see table 2) is interpreted as an (L-0) associa- tion (BUL)O involving the same monopoles as the s-quark. Later on also the mass of the c-quark (compact) was explained by assuming that it is an (L-3) association ((BS)2L)3 between (BS)2 and L.
The masses ascribed to the S, and Cx quarks are found to be adequate for the interpretation of several strange and charmed particles of lowest energy and lowest spin, such as the spin i baryons Z(1190),A(1115) and Ac(2260) (see table 3, section 5).
However, these Sx and C1 masses seem too high for the inter- pretation of particles with higher energy and/or higher spin, such as the strange, spin 3/2 decuplett baryons Z(1385),E(1530), - 13 -
(2(1672) (see section 7) and many other mesons and baryons. These particles have masses too low to be interpreted correctly by the
compact S1 or Cx quarks. The lowered mass exhibited by several strange hadrons can be interpreted by assuming that some of the s-quarks involved may behave in these cases as if they were particles with a lower mass than that ascribed to the normal (or compact) s-quark. The dif- ference between the two is not the same for every hadron showing this phenomenon, and seem to be slightly higher for baryons than for mesons. Nevertheless it is possible in most cases to obtain predictions of hadron masses with errors not substantially greater than 1% by substituting for the normal s-quark of mass 1.0793259 M
a so called "split" s-quark* which is ascribed a mass 1.068 MQ.
This split s-quark is designated by the symbol Tt (see table 2), and its mass and other properties are specified on top
of table 7, where the symbol Tx is replaced by its brief nota- tion T. Once the T properties are defined, the machine is capable of calculating the masses of strange hadrons with assigned configurations in which T is substituted for S. The masses of several baryons and mesons involving the split s-quark are cal- culated this way in tables 7 and 8. Splitting of the c-quark might also have to be considered as a possible way to interpret the masses of several charmed hadrons. We have limited our study to the use of a c-quark substitute 1 1 I1-{(B3T )2L )3 designated as "I-version" (see table 2), which is 1 1 obtained from the normal c-quark C1=((B3S )2L )3 by substituting the split s-quark T for its normal version S (see top of table 7). T and I are used instead of S and C respectively in tables 7 and 8 whenever required in order to obtain a better fit between calculated and observed masses. Each monopole or particle described in table 2 has a corre- 1 sponding anti-particle. The respective anti-particles B, ,LirU r D^S^C1, etc. have opposite magnetic and electric charges, and
* No satisfactory explanation of the phenomenon is found yet. In a few hadrons we have obtained a mass-reduction in the proper range by substituting for the S, quark its two components 1 1 (B3U )0 and L added separately (Barricelli, 1980). But there is no evidence that splitting is the correct explanation in all cases or even in some cases. - 14 -
opposite strangeness and charm. Lower indexes identify positive magnetic charges; upper indexes identify negative ones.
5. Baryon configurations. At this point we may present a first tentative interpreta- tion of a few baryons. As mentioned above, (section 3) some of the baryons with lowest energy, and lowest mass - such as the Proton P(938), the Neutron N(939) and the lambda A(1115) par- ticles (the numbers between brackets identify the respective mass in M.E.V.) - are assumed to consist of an internal system, 3 say (B U1D1)n, associated at the energy level 1 with an external quark. In order to interprete the electric charges and strangeness
of the three particles the external quark would have to be Ux
for the Proton, Dx for the Neutron and S-^ for the A (1115) baryon. These three baryons may conveniently be designated by the respective symbolic representations (configurations): ((B'UjDjInUjKUB^j.DiJnDiM and ( (B^DJnSi) 1. Likewise, the expression (B'UjD^n will be the configuration of the internal system formed by the association at the energy level n - (L-n) 3 association - of the Baric B and the two quarks Ux and Dx. By looking at the configuration the reader can identify the mag- netic monopoles included in an elementary particle, and the way in which they are organized in one or several systems with their respective energy levels. Baryons with larger masses, such as £°(1190) or A0(1232), might tentatively be interpreted by rising one of the quarks from the internal to the external orbit, for example by putting 3 E°(1190)=((B DJ)1UlSl)n, and/or by increasing an energy level, 0 3 for example A (1232) = ( (B DX) il^Dj) (n + 1) . If these tentative suggestions had nothing to do with phys- ical reality, one might expect: 1. That it would be extremely difficult to find a potential dis- tribution giving theoretic calculated masses related to those observed in elementary particles, by following an approach like this. 2. In order to obtain any similarity at all one would have to make a free use of all parameters available. For example the magnetic charges and masses ascribed to the Baric B3 and the - 15 -
quarks U1,D.,S1 would have to be used as free parameters and could hardly be expected to be related to the Dirac monopole like those presented in table 2. Exactly the opposite is actually found. Many of the potential distributions one may select more or less at random fulfilling the conditions required for asymptotic coulombian distributions and ordinary smoothness requirements give results comparable to those obtained by using the exponential coulombian distribution (see Barriceili, 1982, formula (6)):
(10) U(r) = ^(1-Exp(-ro/r))
This distribution was selected very early as an example of asymptotic coulombian distribution only because of its simplicity, at a time when the possibility of calculating the masses of ele- mentary particles by semi-classical theory was pure speculation. Nevertheless if we use this distribution (designated as Exp. Distr. in table 3) in order to calculate by our linear oscillation machine program the masses of the elementary particles whose con- figurations are tentatively proposed above, table 3 shows what we find, for example by assuming n=4, and using the masses and charges given in table 2 for the baric and quarks.
Table 3 Calculated and observed masses of selected baryons.
Name Configuration Spin Observed Calculated masses E-l mass Expon. distr. U_ distr. M.E.V. M.E.V, M.-l M..É.V. M0«l (L-4) BUD stm. (BUD)4 0 1.3736 1.3144 (L-l) BD stm. (BD)1 1/2 4.0971 4.O848 Proton (938) ((BUD)4U)1 1/2 938 1115 0.4649 939 0.3914 Neutron (939) ((BUD)4D)1 1/2 939 1115 0.4647 938 0.3911 Lambda (1115) ((BUD)4S)l 1/2 1115 1331 0.5549 1124 O.4685 Lambda (2260) ((BUD)4C)l 1/2 2260 2702 1.1264 2287 0.9532 Sigma (1190) ((BD)IUS)4 1/2 1190 1399 0.5830 1180 0.4920 Delta (1232) ((BD)IDD)5 3/2 1232 1480 0.6170 1245 O.5188
It Masses are calculated by the linear oscillation program, For particles involving non-linear oscillation orbits (Delta baryon of spin 3/2) a better fit can be obtained by the interpolation program (see section 6). - 16 -
3 In table 3, instead of the explicit configurations (B U1D1)4, 3 (B DX) 1, ( (B^JDJ) 4U )1, etc. we have used their respective brief notations (BUD)4,(BD)1,((BUD)4U)1, etc. A brief notation such as
((BUD)4U)1, can not discriminate between a particle ( (B^JDJ) 4UZ) 1 1 I 1 and its anti-particle ((B3U D )4U )1. But otherwise it leads to no other ambiguities. Using the configuration, punced for example on an input card, our machine programs are able to calcu- late the theoretic mass of an elementary particle, when the po- tential distribution is identified. The calculated masses for the Exp. Distr. given in table 3 are obviously out of range, and either the potential distribution or something else will have to be changed. But notice that they are nearly proportional to the observed masses. Many other dis- tributions were found which yielded theoretically predicted masses nearly proportional to the observed ones (see Barricelli, 1978, Table 4), although the proportionality factor could be drastically different for different distributions. This surprisingly good result obtained by randomly selected potential distributions suggested that the approach tentatively proposed above is sound. The next step would have to be a fitting procedure designed to find a best possible potential distribution by testing various possibilities. The fitting process led eventually to a potential distribu- tion (designated as U Distribution) which is defined in the appendix, formula (T). The theoretical masses calculated by using the U Distribution for the same baryons described above are also g given in table 3. The reader may notice that the fit obtained with the U Distribution is better than in any previous theory. Par- ticularly the substantial difference between the masses of the A (1115) and the 1(1190) baryons earlier considered a mystery (see Feynman, 1973) is beautifully interpreted. The masses of many other elementary particles, including leptons, mesons and baryons will be interpreted in the following sections by the U Distribution. An other singular property of the U Distribution is the value of the only free parameter S involved in this dis- tribution. With an error lower than 0.5%, the best fitting value we have found for this parameter is:
(11) S = 5.85 M 1V137.O - 17 -
Notice that -1^137.031 6 = -J= is the value of the Dirac monopole g * vlic — measured in terms of the natural unit VTic . This is the reason why the distribution is called U . We have no explanation for this coincidence and we do not know whether it is accidental. The formula (T) which defines the U distribution does not suggest any obvious physical interpretation for this coincidence, or the function of S as a measure of g.
6. The extended exclusion principle. Several questions remain to be answered particularly with respect to the arrangement of the various quarks in a baryon, and the spins of the various baryons. The interpretation we will suggest as an answer to the various questions is an extention of Pauli's exclusion principle. The first question one may want to ask is why (BUD)4 is the only internal association of a Baric and two quarks used in the interpretation of the three baryons P{938),N(939) and A(1115). Could not (BUU)4 or (BDD)4 be used? The usual interpretation of Pauli's exclusion principle would allow two identical fermions in a common orbit, provided they have opposite spins. However, Pauli did never tell us whether they would be allowed to be positionally associated in the lowest (L-4) orbit about a Baric. There are no positional associations in atomic theory. The question can be decided by considering that if (BUU)4 or (BDD)4 were allowed, we would find baryons of the type ((BUU)4U)1 with two positive elec- tric charges or of the type ((BDD)4D)1 with one negative electric charge, in both cases with nearly the same mass as the proton or neutron. The external quark can namely be of any one of the types U,D,S or C, since it has no companion sharing its orbit. But no baryons corresponding to the descriptions ((BUU)4U)1 or ((BDD)4D)1 have been found. This can be explained if we assume that the extended exclusion principle to be applied for quarks in posi- tional association not only requires opposite spins but excludes identical quarks in the lowest (L-4) energy level. The other peculiarities of the octett and decuplett with lowest masses can be interpreted if we adopt an "extended exclu- sion principle" described by the following rules. 1. Positionally associated quarks must have opposite spins. - 18 -
2. The lowest energy level permitted for two positionally associ- ated quarks* is (L-4) (energy level 4 5. 3. At the (L-4) level - but not necessarily at higher levels, (L-5),(L-6) etc. - the two positionally associated quarks must be different ones.1 4. In the internal (L-4) orbit - but not in an external (L-4) orbit - the S- and C. quarks are not allowed. It is obvious how these rules can be used in the interpreta- tion of the spin 1/2 octett. Since the two quarks in the (L-4) orbit can not be identic and S- and C. are not permitted in the internal orbit, the only baryons which are possible in this group are those whose configurations are presented in table 4A. Moreover, since the two quarks in the (L-4) orbit have opposite spins and all orbits are supposed to be linear oscillation orbits which have no angular momentum, the baryons have angular momentum 1/2 which is the spin of the third quark isolated in (L-1) orbit. The calculated masses of these baryons are compared with observed ones in the machine output presented in Table 7 (oscilla- tion orbit program). The program works as follows: For every particle whose mass one wishes to calculate an input card must be entered in the program punched with the name of the particle, its configuration defining its composition and energy levels and, if desired, one may include its spin which may be printed in the last column of the output table as in table 7. The configuration will be printed in the output table's next-last column and the particle's name in the third-last. At the top of the input cards one must include a few cards which define the monopoles and quarks one wishes to use among them the three pri- mary monopoles B,U,L. These definitions are printed at the top of table 7, which is an example of the kind of listings one ob- tains in the output or reply from the machine. The other columns from left to right contain the electric and magnetic charges and the masses of the two monopoles or associations which are part of the system, their maximum distance and maximum velocities, and the mass of the system both in monopolar units (M) and in
* This exclusion rule is supposed to apply for two fermions (quarks) but not for two bosons or for a boson and a fermion positlonally associated to each other. Table 4 Baryon configurations A. Octett of spin l/2 Strangeness Name and Configurations mass -2 5(1321) ((BS)1DT)4 ((BS)1UT)4 -1 ((BD)1D3)4 ((BD)1US)4 ((BU)1US)4 -1 A(1115) ((BUD)4S)l N,P(938) ((B0D)4D)l ((BUD)4U)1 Electric charge •* •-1 0 +1 Charmed Lambda baryon of spin l/2, A (2260) ((BUD)4C)l C B. Decuplett of spin 3/2 Strange- Name and Configurations ness mass -3 fl(l672) ((BT)1TT)5 -2 5(1530) ((BD)1TT)5 ((BU)1TT)5 ••1 (1385) ((BD)1DT)5 ((BD)1UT)5 ((BU)lUT)5 (1232) ((BD)1DD)5 ((BD)lUD>5 ((Ba)lUD)5 ((BU)lUU)5 Electric charge •* -1 +2
C. Octett of spin 5/2 Strangeness Name and Configurations mass -2 E (2030) ((BT)2DT)6 ((BT)2UT)6
-1 2 (1915) ((BD)2DS)6 ((BD)2US)6 ((BU)2US)6 -1 A (1830) ((BUD)4S)4 M (1670) ((BUD)4D)4 ((BUD)4U)4 Electric charge +1 - 19 -
millions electron volts (MEV), together with the energy level of the system listed under the designation N. The MEV-masses can be directly compared with the observed masses of the particles, which are indicated between brackets after the names of the respective particles. This way, one can verify the ability of the theory and/or the proposed configurations to predict the masses of the various particles. One of the strange quarks in the XI(1321) baryons is iden-
tified as a split Tx quark instead of the compact Si version. This is not unusual for strange baryons of higher mass in which split quarks are more frequent than in baryons of lower mass. The use of a compact quark instead of the split one would yield a calculated mass substantially higher than observed. If we apply the above exclusion rules to the spin 3/2 decup- lett baryons we find the configurations listed in table 4B. In this case the two positionally associated quarks are located in an (L-5) orbit. In this orbit there is no prohibition for the two quarks to be identical ones. But they must still have opposite spins. That rules out the possibility of interpreting the spin 3/2 of these baryons by ascribing parallel spins to the three quarks. The missing angular momentum must be obtained by ascribing to one of the orbits an angular momentum equal to 1 instead of 0. That would probably be the (L-5) orbit if we can apply the rule valid in atomic theory that the energy level 1 orbits (L-1) have angular momentum equal to zero. The mass of an (L-5) system of angular momentum 1 can be calculated by interpolation between the mass for the case of a linear oscillation (L-5) orbit (angular momentum 0) and the case of a circular (L-5) orbit (angular momentum 5). Such interpolated masses can be calculated by a recently developed machine program which uses the spin of the particle in order to calculate the in- terpolation (Barricelli and Kolset, 1982). The interpolated masses calculated for the decuplett baryons of table 4B are presented in table 8. All the strange quarks involved in the decuplett baryons are split (Tx quarks) instead of compact (Sx quarks). Among the interpolated baryon masses calculated in table 8 are included those belonging to the octett of spin 5/2 presented in table 4C. Table ft Meson oonfigurations trari- Name and1 Configurati ons Name and Configurati one sne t33 mass Nonett of spin 0 mass Nonett of spin 1 0 n'(958) ((BL)0UT)4 ? * (1020) (TT)3 0 n (549) (ST)l « (783) (ST)2 tlfc ±1 K*(494) ((BT)l(BU)O)l K (886) ((BT)l(BU)l)2 ±1 K°(498) ((BT)l(BD)O)l ((BU)1TL)1 K|O(892) ((BT)l(BD)l)2 0 n*(140) ((BU)O(BD)O)I M77O) (((BD)1B)1U)2 0 no(l35) ((BU)O(BU)O)1 P °(77O) (((BU)1B)1U)2 Charmed triplett of spin 0 Charmed triplett of spin 1 0 D°(1863) ((BC)l(BU)l)l D'°(2006) ((BI)1(BU)1)2 0 D*(l868) ((BC)1(BD)1)1 D'*(2OO9) ((BI)1(BD)1)2 ± ±1 P (2O4O) \ { a\j ) JL\ DO J1 J1 P'*(214O) ((BI)1(BT)1)2 Charm-anticharm of spin 0 Charia-anticharm of spin 1 0 n (2970)? (cc)i ? ¥(3095) (CC)2
Table 6 Lepton configurations
N ame and Configurations Electric Strangeness Charm mass charge T (1807) (B(BL)0C)3 ±1 0 *1 S° (B(BL)0S)2 0 ±1 0 V±(l06) (B(BL)0U)l or (B(BU)OL)l ±1 0 0 (B(BL)OD)l or (BULL)l 0 0 0 %(°) (BULL)O 0 0 0 e(0.51l) (UL)O r- l 0 0
ve(0) (DL)O 0 0 0 - 20 -
7. Mesons. The magnetic quark theory interpretation of mesons is not identic to the conventional one. The mesons are still supposed to contain only two fermions, but they do not have to be quarks or at least not only one quark and one anti-quark. If we try to calculate the mass of a H°(135) meson assuming 1 1 a configuration (l^U )! or (D1D )1 in order to comply with conven- tipnal theory, we find a much too high mass of 0.08307 M (see table 1A) which according to formula (3) is equal to 199.28 M.E.V. If we want to obtain a mass consistent with observation we may replace the two quarks with two heavier fermions or (L-0) associ- ations with a twice as large magnetic charge and ascribe to the n° meson the configuration ((B3U1)0 (B3Ui)0)1. With this configu- ration the mass becomes according to table 1A (charges 2g and -2g) and formula (3):
0.05671 MQ = 136.04 MEV
showing a better fit with the observed mass of 135 M.E.V. A similar interpretation can be given to the II+ (139) meson by ascribing to it the configuration ((BsD*)0(B3Ui)0)1 yielding the mass
0.05692 MQ = 136.55 MEV
The difference between the calculated masses is due to the electric charge of the JI+ meson, which causes an increment equal
to the mass of the electron Me=0.000213 MQ=0.511 MEV according to formula (7). Why the observed mass difference between the two mesons is greater than the calculated one is still an unanswered question (see next section). One of the two fermions in the n+ configuration is the 1 1 monopole (B3D )0. We notice that if D is replaced by its con- figuration (B'UiLiJO given in table 2, this monopole becomes 3 (B3B U1L1) 0=(U1L1) , where (l^Li) designates the positional associ-
ation of the two monopoles U.x and Li. These two monopoles have equal magnetic charges of the same sign. If a dissociation of 1 + (B3D )0 into (ULI^) is possible in the n system, this would be a case in which two positionally associated monopoles of magnetic charge 1 are (L-1) associated with a monopole (B3Ui)0 of twice as large magnetic charge. In the proton and neutron we found that I the lowest energy level for an associateion of two quarks with a - 21 -
Baric B3 was (L-4) not (L-1). This rule may, however, apply for two fermions but not necessarily for one fermion positionally
associated with a boson such as (U1L1) (see section 6, rule 2). The configurations of the most common mesons are presented in table 5, and their masses are calculated in table 7 (oscillation orbit interpretation, mostly low spin) or in table 8 (interpolated masses, for higher spin). The interpolated masses can be calcu- lated by the machine when the configuration and spin are indicated in the input. For spin =0 the calculated masses are not different from those calculated by the oscillation orbit program. But for spin =1 the two programs yield different masses. Since there is no clear-cut rule prohibiting parallel spins in a fermion-anti- fermion system (as there is in a fermion-fermion case) this can lead to ambiguities. The K1 mesons present an ambiguity of this kind. For example K'°(892) is ascribed in table 7 a configura- tion ((BT)1(BD)1)2 where the two fermions (BT)1 and (BD)1 are ascribed parallel spins and a linear oscillation orbit. In table 8 it is ascribed a configuration ((BS)1(BD)1)2, anti-parallel spins and an orbit of angular momentum 1. The meson interpretations presented in table 5 are clearly different from the conventional ones. However, the obvious symmet- ries and relationships displayed by the configurations presented can not be accidental. Neither can the good fit obtained by the calculated masses be accidental. The only configuration which does not fit the pattern is the one ascribed to the n'(958) meson, namely ((BL)0UT)4. According to our interpretation this particle does not belong to the group. Another particle whose interpretation is uncertain is the nc(2970). The observed mass of 2970 M.E.V. we have ascribed to it is the one estimated by using measurements from the Crystal Ball detector at SLAC. The earlier accepted mass value was 2830 M.E.V. Because of this large uncertainty the discrepancy between theore- tic (2896 M.E.V., table 7) and observed mass for this particle is not surprising. If the theory had not been sound, one might hardly have expected to find plausible fitting configurations for the various mesons; and interpretation difficulties like the one we have found for nc would have arised for almost every particle no matter how well their masses and other properties are known. TABEL 7 Fl,E2,Gl,G2,eLECmc ANO NACNETIC CWAftCES, «I,P2- MASSES, R,V1,V2 PAXIKU* OISTANCE AND VELOCITIES OF ODJECTS IN (EXTERNAL! ORBIT
PARTICLE DEFINED: B HASSS 9.00021301 EL.CHARSE: -1 RAtN.CttAftCE: -3 SPIN: 0 PARTICLE DEFINED: L NASSs 1.00000000 EL. CflARCE: 0 MGN.CHIRCE: -1 SPIN: 0 PARTICLE DEFINED: U MSSS 1.00021301 EL. CHARCE: 1 MCN.C'ARCEs 1 SPINS 1/2 PARTICLE DEFINED: T MASS! 1.06900000 EL-ClfARf E: 0 »A6». CfARCE: 1 SPIWr 1/2
El E2 ci 62 111 (12 R «1 V2 M H M NAPE CONFIGURATION SPIN
0 0 -l a 1.00000000 4.0000*001 0.00000000 O.OOOOOOGO 0.00000000 1.00000001 0 2399.916 D t(nU)OL)O 1/2 0 0 -i 2 1.00000000 4.0*000001 •29531403 .33936446 .08982903 1.07932591 1 2389.320 S 1/2 0 1 -i 2 t.00000000 4.27224383 .44*33314 .389619 72 .16843280 1.57227903 3 3771.920 C (CBS)2L>3 1/2 0 1 -i 2 1.00000000 4.26194400 .44838048 .58958027 .16881144 1.56206891 3 3747.428 I CCHT)2L»3 t/2
3ART0NS
1 0 l -1 1.00021301 1.31436132 .35987412 .28547849 .22107036 .30135060 1 938.856 P<938) <
UIV9
0 0 2 -2 4.00000001 4.00000001 .22164150 .11804138 .11844138 .05671083 1 136.050 PI0U35) (CBU)OCBU)O>1 0 1 0 2 -2 4.00021302 4.00000001 .22164046 .11843570 .11844192 .056922*6 1t 136.559 PIM140» ({ou)o
LEPTQNS
0 t -1 i 1.00000000 1.000001)01 0., 00000000 O.ODOOOaOQ o.osooooeo .00000001 0 .000 NUE(O) (OL50 1/2 0 i -I i 1.00000000 1.00021301 0.00000000 0.0000000* 0.30000000 .00021301 0 .511 EC0.511» (UL)O 1/2 0 fi a -2 4.00000000 4.0OO0O0O1 0.00000000 c.onoooooo 0.00000000 .00000001 0 .000 NUPU(O) (BULLIO t/2 2 1 3 -3 9. 000*3203 9.00021301 .16714174 .069884C7 .06988901 .04433446 1 106.359 CY0N(106> CPtflDODH 1/2 1 1 3 -3 ".00021302 ^.00021301 .16114344 .06?38t31 •06°88131 .04412213 1 105.450 PYO (B(tiLVOD)l 1/2 1 - I 3 -3 ''.07953892 J.00021301 .16701771 •069332C9 .06994019 .12332123 1 295. 8<5O SO (B(BL)OS)1 1/2 1 [ 3 -3 9.07953«92 -: 00021301 .2I247'20 . 10 • 'JO 829 ,109»56«5 .19«46200 2 452. 123 SO (B(nL)0S)2 1/2 2 L 3 -3 '•37291T05 .00021301 •24413n41 .13473792 .14314109 .7540200" 3 1*08.906 TAUC1807) (B(OL)CC )3 1/2 TABLE 8. IftTEIPftETATtOH «F SO*E ««BROWS OF S*I«f GREATER T««K 1/2
El E2 «1 «2 V2 «A»E CONFIGURATION ORE SPIN flAMONS «ECUPLETT OF SPIN 3/2
0-1 2-2 4.00000002 4.09*84345 .28692972 .24680216 .24197921 .4*043614 5 1192.374 CTR 0-1 2-8 4.0000tO»2 4.0*484245 .40181463 .31804333 .31212630 .51878061 5 1244.563 OSC INTERPOLATE» RASS .51111172 1226.165 01TAU232) < OCTET! OF SPIfl 5/2 PESONS 0 0-11 1.06800000 1.06*00900 .40136721 .37321745 .37821749 .41551745 3 996.433 CIR 0 0-11 1.06800000 1.06(100000 .57825323 .4733406* .4753406* .42781938 3 1026.346 OSC INTERPOLATED RASS .42371874 1016.508 PHK1020) (TT)3 0 0-11 1.06400000 1.07932591 •33968*53 .30635975 .30344240 .31*30203 2 763.612 CIR 0 0-11 1.06800000 1. 079325" 1 • 48187968 .39071144 .38722903 .33087871 2 793.783 OSC INTERPOLATED «ASS .32439037 778.697 0i»C743> (STI2 0-12 -2. 4.01462164 4.16069666 • 20 43*>07S .13850520 .13602002 .36607616 2 474.224 CIR 0-1 2-2 4.08462164 4.16069666 .28263395 .14110508 ,177B9°46 .3R144281 2 915.087 OSC INTERPOLATED «ASS .37375984 896.656 (tns)iC«u»i)2 1-1 2-2 4.08*84245 4.16069666 .20438732 .1384998* .13602194 .36587177 2 877.732 CIR 1-1 2-2 4.08414249 4.16069666 .28262968 .14109813 .17790193 .38123783 2 914.596 OSC 1HTERPOLATE0 «ASS .37353460 896.164 K0*(892) <(nS)l 8. Leptons. In conventional quark theory the leptons are treated as pri- mary particles at the same level as the quarks, and no attempt is made to interprete their properties in terms of quarks and/or gluons. The magnetic quark theory we have developed is capable of interpreting some of the main properties of leptons just as well as it can interprete the corresponding properties for baryons and mesons. The main difference between leptons and the other parti- cles, from the point of view of magnetic quark theory is that the constituents of a lepton include only one fermion, whereas two fermions are included among the constituents of a meson, and three among the constituents of a baryon. The proposed configurations of various leptons are listed in table 6 together with other properties. The masses are listed in the machine output presented in table 7. The distinction between the v neutrino and v neutrino is not clear since both include the same magnetic monopoles. The respective configurations we have indicated in table 6 are an attempt to interprete their role in decay processes. But that is a question which will be discussed in the next paper (Barricelli, 1983). Besides the well-known leptons observed so far, two neutral still undiscovered leptons M° and S° predicted by the theory with the respective predicted masses 106 and 452 M.E.V. are listed in table 6 and 7. p° differs from y+ by the substitution of D for U; and S° by the substitution of S for U and by an in- crease of the energy level from 1 to 2. All of the 4 leptons \i°, \i +, S° and T are systems including B and a positional associ- ation of the monopole (BL)0 with magnetic charge 2g and a fermion with a single magnetic charge g, namely D,U,S or C. Together the two positionally associated objects have a magnetic charge 3g opposite to that of B. Notice that the light boson L is present in all the lepton configurations presented in table 6. The same applies to many systems containing no more than one fermion, namely D=(BUL)0, S=((BU)0L)1, C=((BS)2L)3 and (BL)0. On the other hand a glance at the tables 4A,B,C and 5 shows that L has never been found as a free monopole in systems containing more than one fermion. Only as a member of an internal system or a positional association with - 23 - no more than one fermion is L found in these elementary parti- cles. Another peculiarity about this monopole is that the observed masses of particles including (L-0) systems or positional associ- ations involving I, are usually higher than their theoretically predicted masses, and the difference (mostly greater than 1 M.E.V.) is the larger, the greater the number of such (L-0) systems in- volved. For example the observed mass of the IT* meson with con- figuration ((BU)O(BD)O) 1 or ((BU)OULM (see preceding section) is 139.5 M.E.V., substantially greater than its theoretically calcu- lated mass of 136.5 M.E.V. and also substantially greater than the observed mass, 135.0 M.E.V., of the 11° = ((BU) 0 (BU) 0) 1 meson. Also the mass of the neutron with the configuration ((BUD)4D)1 involv- ing two D-quarks, which are (L-0) systems, (BUL)O, containing L, has an observed mass 939.6 M.E.V., greater than the predicted one 938.3 M.E.V. and greater than the observed mass 938.3 M.E.V. of the Proton, whose configuration ((BUD)4U)1 involves only one D- quark. Likewise the n'(958) meson with the tentative configuration ((BL)0UT)4 has an observed mass 957.6 M.E.V., greater than the predicted one 955.7 M.E.V. This rule does not apply if the internal system containing L is not an (L-0) system - is for example an (L-1) system like Sx or and (L-3) system like Ci. The reason for this peculiar behaviour of the monopole L is unknown. But if this effect is taken into account it might be possible to improve the theoretic calculation of masses. The subject of this paper has been restricted to an inter- pretation of the masses and angular momenta of elementary parti- cles, by identifying their configurations in terms of the basic monopoles U,B,L. Important evidence supporting the theory, which has not been discussed, is the conservation of the three basic monopoles during decay processes. This conservation law and the way in which it can be used in order to interprete various features, including the rapid (strong) and slow (weak) decay processes will be presented in the next paper (Barricelli, 1983), in which various questions raised by this paper will also be discussed. Appendix We shall nov present the relativistic semi-classical theory for linear oscillation in a two-body system. 4 Tbf binding» energy between tht two particle» will be a constant E defined by: 2 (A) E - U(r) + (M, + M2)c - (il10 + c* where u(r) is the potential energy, II and M are the masses of the two particles, while M10 and M2Q are their rest masses related to M and H2 by the formulae M W20 fl l0 ( ) H,- -__ Jr__^.. ? V v / "2" 2" T1 and T? being their respectire absolute velocities relative to the centre of grarity. The laws of movement (impuls » 0, barycenter at rest) are expressed by: (C) M1T1 - M2v2 and (D) Miri - M2r2 (B) r, + r2 - r r being the distance between the two partial©» (centres), r. and r_ tbeir respective distances from oentre of gravity. Somaerfeld's condition for quantitatlon can now be expressed byi (P) i MiVrl+ yrM2V2dr2" nh or according to (C) and (E) nh II If we oall r the maximum distance periodically aohiered by the two partiolerT% s (maximum r-value), this formula booomest (H) 4/ M v dr - nh The maximum distanoe r is characterized by the condition v. a r m Qt which according to formulas (A) and (B) givesi (I) B - U(rx) which is a way of defining r in terns of the bindings energy E. In equation (c) we may now replace M. and U by their values obtained from formula (-B) and then solve the equation with respeot to v? or v1: 2 2 2 Vl (L) T2 2, 2^ , 2, 2. 2 . 2 ' 1 2 , 2 2. 2. 2 .2 Tl/O (1~YC )M2O/M1O V° + (l"V° )M10/M20 which aocording to (c) and (B) gives: 2 , y—z > 2 M20/Ml0 ' V 20\/ 2 2 Ml(/ M20 c 7 c - v1 |) " 2 If we put U(r ) - U(r) M(rx,r) g rx formula (A) becomes according to (i) i (0) M(rx,r) » M1 + M2 rx, In this formula we may replace M? by (M) and then eliminate v.. by the following formula derives from (B) (P) r,- cl/1 - M^/4 nr Bj solving the result with respect to M. we obtain: < 4 g V1 2H(rx,i This way M1 is expressed as a function of r and r only. According to formula (F) also the product «".T. can therefore be expressed as a function of r^ and r only, which will be designated as P(r ,r): (B) P(rx,r) . « and (H) becomes /p* (S) 4 / P(r_,r)dr - nh 4/ P(V 'o Using these formulae our machine program can calculate M(r ,r) for any given r and r values by formula (N) then by formula (Q) • it obtains H. and by formula (P) it obtains v1«and P(r .r) is then given by formula (R) . An r value fulfilling formula (s)' can then be obtained by successive approximations. Once r is determined, formula (i) gives the bindings energy E ,and the mass M of the two-body system is then given by. M - M, • M2 + 5_ This way all of the parameters! maximum distance r , velocities v1 and v^f'mase of the system H are obtained. The asymptotic coulombian potential U (r) we are going to use in order to calculate the masses of elementary particles is given by the following formula fulfilling the requirements (see Barrioelli 1982) , provided the substitution indicated in the footnote below is applied: This formula gives tho potential U»(r) only for r substituting the infinite series —!^-( 1 + £R2 + ( JR2) 2 + ( \R2) 3 + . . . ) S.R for the expression 1/SR2(1-^R2) in formula IV 1/C1+R2) + g., g2 are the magnetic charges, e^, e2 the electric charges of two interacting magnetic monopoles, R= ~ , r being the ro distance between their centers and rQ their standard radius. S is a free parameter whose best value (giving the best predic- tions for the masses of elementary particles including the elec- tron) is found to be S=5,853« ^f°5T. This value of S is susDiciouslv close to a Dirac monopola expressed in units of /lie, namely -=å=: = 4/i37* and might not be an accident, even though we do not know what this coincidence means. The above potential u (r) and the force field generated by it are shown in fig." 2. Potential field U -U (r) defined in appendix , formula (T), and its derivative P identifying the force field generated by it, for tiro Dirac monopoles of opposit magnetic charge. REFERENCES Barricelli, N.A. (1980) The application of Bohr and Sommerfeld methods in the theory of magnetic quark models. Preprint series No. 11 (Appl.Math.), University of Oslo. Barricelli, N.A. (1981) The masses of elementary particles cal- culated by a magnetic monopole model. Theoretic implications. Preprint series No. 4 (Appl.Math.), University of Oslo. Barricelli, N.A. (1982) The calculation of energy levels in a system of two magnetic monopoles in semi-classical theory. Theoretic papers No. 3, 1982, P.B. 1029, Blindern, Oslo 3, Norway. Barricelli, N.A. (1983) Conservation of basic monopoles in decay processes (in preparation). Dirac, P.M. (1931) Proc.Roy.Soc.A., 133, 60. Dirac, P.M. (1948) Phys.Rev., 74, 817. Feynman, R.P. (1973) Protonens struktur. Kosmos. Utgiven av svenska fysikersamfundet. Band 50. Schwinger, I. (1969) Science, 165, 757.