Vladimir Yershov- Dualism of spacetime as the origin of the fermion mass hierarchy

8/3/2019 Vladimir Yershov- Dualism of spacetime as the origin of the fermion mass hierarchy

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Apeiron , Vol. 12, No. 1, January 2005 1

Dualism of spacetime as the

origin of the fermion masshierarchy

Vladimir YershovUniversity College London,Mullard Space Science Laboratory,Holmbury St.Mary, Dorking RH5 6NT, UK

It is shown that hypothetical dualism of spacetime couldgive rise to the pattern observed in the properties of thefundamental fermions. By guessing at the basic symme-tries of space that underlie the properties of the simplestpossible (primitive) particle it is found that these symme-tries imply the existence of a series of structures equi-librium congurations of primitive particles whith prop-erties that cohere with those of the fundamental fermions,including their masses.

Keywords : Topology of space, fermion mass hierarchy

IntroductionDualism is known to be one of the general properties of matter.

Many successful ideas in physics were based on this property (thewell-known examples are the concepts of wave-particle duality and

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supersymmetry in quantum eld theories). It is conceivable that byunderstanding how spacetime dualism manifests itself on the sub-quark scale it would be possible to resolve some (if not all) of thepuzzles that plague modern physics.

One of such long-standing puzzles is the hierarchy of masses of the fundamental fermions [1]. The masses of quarks and leptons(Table 1) span a very wide range ( 105). They are not predicted

Table 1: Experimental masses of the fundamental fermions (in proton massunits). The data are taken from [2] for the charged leptons and from [3] foquarks.

First generation Second generation Third generatione 0.0005446170232(12) 0.1126095173(34) 1.8939(3)

e 3 10 9 2 10 4 2 10 2u 0.0047 c 1.6 t 189d 0.0074 s 0.16 b 5.2

from any application of rst principles of the Standard Model of particle physics, nor has any analysis of the observed data (see e.g.[4]) yielded an explanation as to why they should have strictly theobserved values instead of any others. Some theorists claim that thispattern could be randomly frozen in one of the early stages of theuniverse [5], much like the particular properties of the planets were

determined by the detailed history of the formation of the solar sys-tem. But most people believe that these masses can be explained bydifferent mechanisms of symmetry [6][7], hidden spatial dimensions[8][9], quantisation of energies of hypothetical exotic topologicalobjects [10], or some other geometrical properties of space [11][12].

However, the history of particle physics shows that, whenevera regular pattern was observed in the properties of matter (for in-



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stance, the eight-fold pattern of mesons and baryons or the period-ical table of chemical elements), this pattern could be explained byinvoking some underlying structures. According to this experience,the very existence of the pattern in the properties of the fundamentalfermions unambiguously points to their internal structures, as op-posed to the Standard Model where all the fundamental particles arepoint-like objects. In this paper we shall follow this lead and assumethat the fundamental fermions are bound states of smaller particles,which hereafter we shall call primitive (to distinguish them fromthe fundamental and elementary particles). The primitive parti-cles, by denition, should not decay into other particles, nor shouldthey be of more than one type. Their properties must be dened bythe very basic symmetries of spacetime, dualism being one of them.Then, based on these symmetries, one would expect to derive theproperties of all the known (supposedly composite) particles and,perhaps, to predict the existence of new, so far unknown, particles.

The UniverseLet us begin with the following conjectures (postulates) about

the general properties of matter and the universe:

1. Matter is structured, and the number of its structural levels isnite;

2. The simplest (and, at the same time, the most complex) struc-ture in the universe is the universe itself;

3. The universe is self-contained (by denition);

4. All objects in the universe spin (including the universe itself).



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The postulate (1) is as old as science itself. The idea of everythingconsisting of elementary indivisible atoms originates from Leucip-pus and Democritus. Since then, various structural levels of matterhave been found, from molecules and atoms to nucleons and quarks.Each time, when a lower level of matter has been revealed, it wasthought that this level is the ultimate and the simplest one. But pat-terns in properties of the elementary particles always indicatedthat there are underlying structures responsible for these properties.Fortunately, these patterns became simpler on lower structural lev-els, suggesting that matter is structured down to the simplest possiblelevel (entity).

The postulate (2), according to which the universe is consid-ered as the simplest possible structure, is not very new. In fact,this idea lies at the heart of modern cosmology. Theories of gen-eral relativity and of the Big Bang consider the universe to be asimple uniform object with curved space. Unfortunately, the char-acter of this curvature (shape of the universe) cannot be deduced

from Einsteins equations without additional observations. For sim-plicity, A.Einstein and A.Friedmann assumed the universe being athree-dimensional hyper-sphere, S 3, of positive, negative or zerocurvature. But if we take into account the denition of the uni-verse as a self-contained object (postulate 3), the spherical shape be-comes inappropriate because the sphere has two interfaces, inner

and outer (of course, in the case of S 3

these interfaces are three-dimensional hyper-surfaces). This might give rise to some doubtsabout the universes self-contained character. The topology avoidingthis problem is the well-known Klein bottle [13] usually visualisedwith the use of a two-dimensional surface (Fig.1). Similarly to S 3a three-dimensional Klein bottle, K 3, can be of positive, negativeor zero curvature. But K 3 has a unique hyper-surface, and its sym-



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Figure 1: Two-dimensional representation of the Klein-bottle.

metries are different from those of S

3. For simplicity, in Fig.2 wevisualise the hipothetical K 3-topology of the universe in the formof a one-dimensional scheme. An important feature of K 3 is theunication of its inner and outer hyper-surfaces. In the case of the universe, the unication might well occur on the sub-quark level,giving rise to the properties of elementary particles. In Fig.2 the uni-cation region is marked with the symbol (primitive particle). Of course, in order to form structures there should be more than oneprimitive particle in the universe. This makes a difference betweenthe topology of the universe and the conventional Klein bottle. Onecan call such an object a multi-connected hyper-Klein bottle. But itsmain features remain the same: dualism and unication of two op-posite manifestations of space, which result in identication of theglobal cosmological scale with the local scale of elementary parti-cles (supposedly, the distances comparable with the Planck-lengthscale).

Primitive particleLet us suppose that space is smooth and continuous, i.e., that



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.

R

r

{I

I

II

II

Scale of unication

Cosmologicalscale

Figure 2: One-dimensional scheme of the Klein-bottle topology of the universeThe inner (I) and outer (II) manifestations of space are unied throughthe region (which can be considered as a primitive particle). R is the radiusof curvature of the universe; r is the local curvature of the inverted space.

the local curvature of the medium cannot exceed some nite value(r = 0 ). Then, within the region space can be considered as beinglocally curved inside-out, which is also visualised in Fig.3 in theform of a two-dimensional surface.

Figure 3: Two-dimensional visualisation of the inversion region (theprimitive particle).

In this case, the primitive particles do not exist independentlyof the medium, but rather they are made of it. Then, the postulate(4) about the spinning universe gives us an insight into the possible



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origin of the particle masses. This postulate is not obvious, but itcomes from the common fact that so far non-rotating objects havenever been observed. The universe spinning with its angular veloc-ity (if considered from the embedding space) would result in thelinear velocity R of the medium in the vicinity of the primitiveparticle, where R is the universes (global) radius of curvature. It isseen in Fig.2 that the opposite signs in this expression correspondto two opposite manifestations of space, I and II. Thus, the spinninguniverse gives rise to an acceleration, ag , of the primitive particlebecause of the local curvature, 1/r , of space in the vicinity of (Fig.2). According to Newtons second law, this could be viewed asan action of a force, F g = mgag , proportional to this acceleration.The coefcient of proportionality between the acceleration and theforce can be regarded as inertial mass of the primitive particle. How-ever, for an observer in the coordinate frame of the primitive particlethis mass is perceived as gravitational ( mg ) because the primitiveparticle is at rest in this frame. Thus, the spinning universe implies

the accelerated motion of the primitive particle (together with theobserver) along its world line (time-axis). If now the primitive par-ticle is forced to move along spatial coordinates with an additionalacceleration a i , it resists this force in exactly the same way as it doeswhen accelerating along the time-axis. A force F i = m i a i , which isrequired in order to accelerate the primitive particle, is proportional

to a i with the coefcient of proportionality m i , and the observer willconclude that the primitive particle possesses an inertial mass m iBut, actually, in our model the particles inertial, m i , and gravita-tional, mg , masses are generated by the same mechanism of acceler-ation. Hence, they are essentially the same thing; m i mg , that is,mass is inertial.

Positive and negative signs of R do not affect the sign of thec 2005 C. Roy Keys Inc. - http://redshift.vif.com


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gravitational force F g = mgag because the second derivative ag = 2 (ict )

t 2 has always the same sign (the local curvature, 1/r , is theproperty of space and it does not depend on the direction of mo-tion). However, the rst derivative (ict )t (which is just another wayof expressing R) can be either positive or negative, and the cor-responding force, F e = q (ict )t , should be either repulsive or at-tractive (depending on the choice of the trial particle). It wouldbe natural here to identify F e with the electric force, and the co-efcient of proportionality, q , with the charge of the primitiveparticle. Although q is positive in this expression, one can equiv-alently consider that it is this quantity that changes its sign, but not (ict )

t . Within our consideration, mg and q are of the same mag-nitude. For simplicity, hereafter we shall regard these quantities asbeing normalised to unity and denote them as m and q . In fact,the above simple mass acquisition scheme should be modied be-cause, besides the curvature, one must account for torsion in thevicinity of the inversion region (corresponding to the Weyl tensor).

In the three-dimensional case, torsion has three degrees of freedomand the corresponding eld splits into three components (six, if bothmanifestations of space are taken into account). It is reasonable toidentify these three components with three polarities (colours) of thestrong interaction [14].

As we can see, dualism of space gives rise to a specic symmetry

of SU(3)/U(1)-type, which means that the primitive particles mustpossess both electric and colour charges ( , 3 c). We assumethat by rotational transformations (SU(3)-symmetry) the primitiveparticle can be translated into one or another colour-charge state.

The existence of the inversion region of space around the prim-itive particle means also that, together with distances measured



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from the particle centre, one can equivalently use the reciprocal dis-tances, = 1 , say, for the reciprocal manifestation of space (see,e.g., [3], p.p.239-249). Thus, any potential, which is proportional to1 , in the reciprocal manifestation of space will be proportional to

1

(or simply ) . For instance, the Coulomb-like potential, e 1

, inthe reciprocal manifestation of space should manifest itself strong-likely, e s , and vice versa. Note that due to the inversionof space the potentials cannot grow indenitely when 0 and,unlike the Coulomb potential, both e and s should decay to zeroat the origin. The equilibrium distance, , where | |= | | and| e |= | s | , establishes a characteristic scale (breakes the symme-try of scale invariance) on which the primitive particles form struc-tures (we assume the coupling constants to coincide, = s = 1 )A simple example of the split equilibrium eld, which decays at theorigin, is the eld F = s + e , where s = F (), e = F ()and F = s exp( 1). Here the signature s = 1 indicates thesign of the interaction (attraction or repulsion); the derivative of F is taken with respect to the radial coordinate . Far from the source,the second component of F mimics the Coulomb gauge, whereas therst component extends to innity, being almost constant (similarlyto the strong eld). Here we are not going into details of the spliteld; what does matter for our consideration is the anti-symmetriccharacter of two reciprocal elds due to the dual nature of space.

In order to formalise the use of the tripolar charges we shall in-troduce a set of auxiliary 3 3 singular matrices i with the follow-ing elements:

i jk = i

j ( 1)kj , (1)

where i j is the Kronecker delta-function; the -signs correspond tothe sign of the charge; and the index i stands for the colour ( i =



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1, 2, 3 or red, green and blue). The diverging components of theeld can be represented by reciprocal elements: jk = 1 jk . Then,we can dene the (unit) charges and masses of the primitive particlesby summation of these matrix elements:

q = u

u , q = u

u (2)

andm = | u

u | , m = | u

u | (3)

(u is the diagonal of a unit matrix; q and m diverge).Assuming that the strong (colour) and electric interactions are

opposite manifestations of the same geometrical feature of spaceand, taking into account the known pattern [14] of the coloure inter-action (two like-charged but unlike-coloured particles are attracted,otherwise they repel), we can write the signature, s ij , of the com-bined chromoelectric interaction between two primitive particles,say of the colours i and j , as:

s ij = u

i j u . (4)

This expression is illustrated by Fig.4: the left part of this Fig-ure shows the known pattern of attraction and repulsion betweentwo strongly interacting particles. The right part corresponds to thehypothetical chromaticism of the electromagnetic interaction, anti-symmetric (in our case) to the strong interaction. According to thispattern, under the electric force, two like-charged particles are at-tracted to each other if they are of the same colour charge and repelotherwise.



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Strong Electric

Figure 4: Pattern of the chromoelectric interaction between two primitiveparticles. Dashed and solid circles represent primitive particles with oppositcolour-charges.

Colour dipolesObviously, the simplest structures allowed by the tripolar eld

are the monopoles, dipoles and tripoles, unlike the conventional bipo-lar (electric) eld, which allows only the monopoles and dipoles.

For example, two like-charged particles with unlike colours shouldform an equilibrium conguration a charged dipole, g . The parti-cles in such a dipole will be mutually attracted because of the strongforce (increasing with distance). At the same time, the electric re-pulsive force, increasing when distance decreases, necessarily im-plies the existence of an equilibrium distance, , at which the at-

traction caused by the strong force will be cancelled. Similarly, aneutral dipole g0 of two unlike-charged primitive particles can alsobe formed.

Based on the analogy between the accelerated medium and theeld ow in the vicinity of the primitive particle we can dene themass of a system containing, say, N particles, as being proportionalto the number of these particles, wherever their eld ow rates are



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not cancelled. For this purpose, we shall consider the total eld owrate, vN , of such a system as a superposition of the individual volumeow rates of its N components. Then, the net mass can be calculated(to a rst-order approximation) as the number of particles, N , timesthe normalised to unity eld ow rate vN :

mN = | NvN | where vi =qi + vi 1

1+ | qi vi 1 |. (5)

Here vN is computed recursively as a superposition of the individualow rates, vi , where i = 2 , . . . , N ; and v1 = q1. The normalisationcondition (5) expresses the common fact that the superposition owrate of, say, two antiparallel ows ( ) with equal rate magnitudes| v |= | v |= v vanishes ( v = 0 ), whereas, in the case of parallelows ( ) it cannot exceed the magnitudes of the individual owrates ( v v).

When two oppositely charged particles combine (say red andantigreen), their oppositely directed velocities along the time-axis, (ict )

t , cancel each other (resulting in a neutral system). The corre-sponding, acceleration,

2 (ict )t 2 , also becomes almost zero, which is

implicit in (5). This formula implies the complete cancellation of masses in the systems with vanishing electric eld, but this is just anapproximation because the complete cancellation is possible onlywhen the centres of the interacting particles coincide, which is notexactly our case (the primitive particles are separated at least by theequilibrium distance ). However, for the sake of simplicity, weshall neglect the small residual masses of electrically neutral sys-tems.

In the matrix notation, the positively charged dipole, g+12 , formedof the red and green colour charges (indices 1 and 2) can be repre-



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sented as a sum of two matrices, 1 and 2:

g+12 = 1 + 2 =

1 +1 +1+1 1 +10 0 0

, (6)

with qg+ = +2 [ q ]. The mass and the reciprocal mass of the chargedcolour dipole will be mg+ 2 and mg+ = . If two componentsof the dipole are oppositely charged, say, g012 = 1 +

2(of what-

ever colour combination), then their electric elds cancel each other(qg0 = 0 ). The neutral colour dipole will be massless ( m g0 0), butstill m

g0 = due to the null-elements in the matrix g0 (the dipole

lacks, at least, one colour charge to make it colour-neutral). The in-nities in the reciprocal masses of the dipoles imply that neither g

nor g0 can exist in free states (because of their innite energies).In an ensemble of a large number of neutral dipoles g0, not only

electric but all the chromatic components of the eld can be can-celled (statistically). Then, any additional charged particle l be-longing to this ensemble, if coupled to a neutral dipole gik , will re-store the mass of the system:

m(i , k, l) = 1 , but still m( i ,

k, l) = . (7)

According to Stokes theorem, the charge of the new system willcoincide with the charge of the additional particle l .

Colour tripolesObviously, three complementary colour charges will tend to co-

here and form a Y -shaped structure with the distance of equilibrium, , between its components. Thus, by completing the set of colour-charges in the charged dipole (adding, for example, the blue-charged



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component to the system g+12 , one would obtain a colour-neutral (butelectrically charged) tripole:

Y = 1 + 2 + 3 = 1 +1 +1+1 1 +1

+1 +1 1

,

which is colour-neutral at innity but colour-polarised in the vicin-ity of its constituents. Both m and m of the particle Y are nite,mY = mY = 3 [m ], since all of the diverging components of itscombined chromoeld are mutually cancelled (converted into thebinding energy of the tripole). Locally, the red, green, and blue

colour-charges of the tripoles constituents will be distributed in aplane forming a ring-closed loop. Thus, a part of the strong eldof these three particles is closed in this plane, whereas another isextended (over the rings poles).Doublets of tripoles Two distant like-charged Y -particles will com-bine pole-to-pole with each other and form a charged doublet =

Y l Y l . Here we use the rotated symbol Yin order to indicate the factthat the second tripole is turned through 180 with respect to the rstone. The enhanced arm of Y marks one of the colours, say, red, in or-der to visualise the orientation of tripoles with respect to each otherin the structural diagrams. The charged doublet will have the chargeq = 6 [q ] and a mass m = m = 6 [m ]. Likewise, if twounlike-charged Y -particles are combined, they will form a neutral

structure = Y l Y l(with q = 0 , m = m = 0 ). One can show thatthe shape of the potential in the vicinity of the pair allows a certaindegree of freedom for the components of the doublet to rotate os-cillating within the position anlge 120 with respect to each other.The corresponding states of equilibrium,

and

, can be written as



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= Y l Yl , = Y l Y l, and = Y l Yl , = Y l Y l,which correspond to two possible directions of rotation, clockwise( ) and anticlockwise ( ).Triplets of tripoles The 23 -symmetry of the tripole Y allows up to

three tripoles to combine if all of them are like-charged. Necessarily,they will combine into a closed-loop (triplet), which can be in oneof its eight possible states, four of which correspond to Y :

e = Y l Y lYl , e = Y l Yl Y l, e = Y l Y l Yl , e = Y l Y l Yl ,

and four others correspond to Y . The upside-down position of thesymbol Yin the last two diagrams is used to indicate that the ver-tices of the tripole are directed outwards of the centre of the structure(see Fig.5b). Another possibility is the orientation of the vertices to-wards the centre (Fig.5a). These congurations can be viewed as

Figure 5: (a): Three like-charged Y -particles joined with their vertices directedtowards the centre of the structure and (b): outwards the centre.

two different phase states of the same structure with its componentsspinning around its ring-closed axis and, at the same time, translat-ing along this axis. The corresponding trajectories of colour-charges(currents) are shown in Fig.6. The structure e is charged, withqe = 9 [q ], and have a mass, m e = me = 9 [m ], correspond-ing to its nine constituents. The colour charges spinning around thering-closed axis of

ewill generate a toroidal (ring-closed) magnetic



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Figure 6: Trajectories of colour charges in the structure 3 Y = e. The chargesspin around the ring-closed axis of 3Y and, at the same time, synchronouslytranslate along this axis.

eld which, at the same time, will force these charges to move alongthe torus. This circular motion of charges will generate a secondary(poloidal) magnetic eld, contributing to the spin of these chargesaround the ring-axis, and so forth. The strength of the magneticeld will be covariant with the radius r e . The interplay of the vary-ing toroidal and poloidal magnetic elds, oscillating radius r e , andvarying velocities of the rotating charges converts this system into

a complicated harmonic oscillator with a series of eigenfrequencesand oscillatory modes.Hexaplets. The pairs of unlike-charged Y -particles can form chainswith the following colour patterns:

e = Y l Y l + Y l Yl + Yl Y l + Yl Y l + Y lYl + Y l Y l + . . . (8)

or

e = Y l Yl + Yl Y l + Y lY l + Y l Y l + Yl Y l + Y l Yl + . . . (9)

(depending on the chosen direction of rotation of the constituentswith respect to each other). The patterns repeat after each six con-secutive links. The sixth link is compatible (attractive) by the con-



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guration of its colour-charges with the rst link, which allows clo-sure of the chain 6 Y Y in a loop (hexaplet). The trajectories of thespinning colour charges in the structure are clockwise ( e ) or an-ticlockwise ( e ) helices. The hexaplet consists of twelve tripoles(n e = 36 primitive particles). It is electrically neutral and almostmassless, according to (5).

Combinations of the structures Y , 3 Y , and 6 Y Y

Unlike the structure Y , with its partially closed strong eld (inone plane), the strong elds of 3 Y (e) and 6 Y Y ( e ) are closed. Thus,these particles can be found in free states. They can combine witheach other, as well as with Y , due to residual chromaticism of theirpotentials. The conguration of colour charges in the particle 6 Y Ymatches (attracts) that of 3 Y if both of these particles have helicesof the same sign.

The repulsive and attractive forces between 6 Y Y and 3 Y are ofshort range: far from the particle the colour potentials mix togetherinto colourless electric potential. Separated from other particles, thestructure 6 Y Y behaves like a neutral particle. But, if two particles6Y Y approach one another, they will be either attracted or repelled.The sign of the interaction depends on the compatibility of the colourpatterns of both particles, that is, on the helicities of their currents.

The particle 6 Y Y , which has its red, green, and blue colour con-

stituents 120-symmetrically distributed along its closed loop, can

combine with the particles Y or 3 Y . The combined structure Y 1= +Ywill have mass and be charged, with its charge qY 1 = 3, corre-sponding to the charge of a single Y -particle, and a mass m Y 1 = 39(in units of m ), corresponding to n e + mY = 36 + 3 . The structure3Y 6Y Y = e e has a charge corresponding to its charged component,



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e ( 9), and a mass of 45 mass units ( m e e = me + n e = 9 + 36 ).The structure Y 1 cannot be free (similarly to Y ). It will cou-

ple further with other Y 1-particles (through an intermediate e ofthe opposite helicity) forming links Y 1 e Y1. A single link Y 1 e Y1, which can be identied here with the u-quark, will have thecharge derived from two Y -particles, qu = +6 [ q ], and a massof 78 mass units, mu = m(2Y 1) = 2 39 [m ]. The positivelycharged u-quark can combine with the negatively charged structuree e (with its 45-units mass), forming the d -quark of a 123-unitsmass, md = mu + me e = 78 + 45 = 123 [ m ], and with its chargeqd = qu + qe = +6 9 = 3 [q ] (see Fig.7). Expressing the

Charge:

Net charge: 3

9 + 3 + 3

Number of charges: 36 9

e e

3 36 (36) 36 3

u

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(mass 45) (mass 78)

Figure 7: Scheme of the d-quark. The symbol is used for the electron, thesymbols |and | stand for the tripoles ( Y -particles), and the symbols 0 repre-sent neutrinos. The symbols and denote the clockwise and anticlockwisehelicities of the colour currents in the particle components.

charges qu = +6 [ q ] and qd = 3 [q ] in units of qe (dividing thesevalues by nine) one would obtain the commonly known fractionalcharges of the u- and d-quarks ( + 23 and

13 ).



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The second and third generations of particlesIt is natural to suppose that the particles of higher (heavier) gen-

erations should be composed of simpler structures belonging to lower(lighter) generations. For example, the muon neutrino (a neutral

particle) can be formed of two unlike-charged structures Y 1 and Y 1intermediated by a neutral particle e :

= Y 6(YY ) 6(YY ) 6(YY ) Y = Y 1 e Y 1. (10)

The intermediate particle has the helicity sign opposite to that of the charged components of the structure, which creates a repulsive

potential maintaining the system in equilibrium. The structure of themuon (Fig.8) can be written as

= [6( YY ) + 3 Y Y ][6(YY ) 6(YY ) 6(YY ) Y ] = ee . (11)

Some of the binding forces between the components of the struc-ture can be weaker than others, depending on whether there exist an

extra electric repulsion between the components. Due to this factor,the second and third generations of particles can be considered asoscillating clusters, rather than rigid structures. In (11) the clusteredcomponents of the structure are enclosed in brackets. Computingthe mass-energies of such nonrigid systems with oscillating com-ponents is not a straighforward task. However, these masses are

computable in principle, which could be shown with the use of thefollowing empirical formula:

m =m1 + m2 + + mN

1/ m1 + 1 / m2 + + 1 / mN , (12)

where m i and m i are the masses and reciprocal masses of the com-ponents (all unit conversion coefcients are assumed to be set to



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unity). Alternatively, we shall use the following brief notation forthe above summation rule:

m = m1 + m2 + + mN .

The fermion masses computed with the use of (12) are summarisedin Table 2. As an example, let us compute the muons mass. Themasses of the muons components, according to its structure (Fig.8),are: m1 = m1 = 48 , m2 = m2 = 39 (in units of m ). And themuons mass is

m = m1 + m2 = 48 + 39 = 48 + 391/ 48 + 1/ 39= 1872 [m ]

Charge:

N1 =84

9 + 3

N2 =39

3

Number of chargess: 9 3 6 3

e e Y

36 36 3 36

Y 1

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(m 1 = 48 ) (m 2 = 39 )

Charge -6, Charge -3,

N 1 + N

2 = 123

Figure 8: Scheme of the muon.

For the -lepton (Fig.9), m1 = m1 = 201 , m2 = m2 = 156 , andits mass is m = 201 + 156 = 31356 [m ].

For the proton, the positively charged fermion consisting of twoup (N u = 2 ), one down (N d = 1 ) quarks and surrounded by a cloudof gluons

g0, the masses of its components are

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Charge:

q 1 = 9

3 + 3 9 + 3 3

q 2 =0

+ 3 3 + 3 3

N :39 (36) 39

45 39 (36) 39

39 (36) 39 (36) 39 (36) 39

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m 1 = 201m 2 = 156

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Figure 9: Scheme of the tau-lepton.

md = md = 123 . The total number of primitive charges comprisingthe protons structure is

N p = 2 mu + md = 2 78 + 123 = 279 ,

which would correspond to the number of gluons ( N g) interactingwith each of these charges (that is, N g = N p = 279 ). The masses

of these gluons, according to (7), are m g0 = 1 , mg0 = . Theresulting proton mass is

m p = N u mu + N dmd + N gmg = 16523 [m ]. (13)

With this value of m p one can convert me , m , m , and the massesof all other particles from units m into proton mass units, m p, thus

enabling these masses to be compared with the experimental data(Table 1). The computed fermion masses are listed in Table 2. Thevalue m p = 16523 [m ] also explains the well-known but not yet ex-plained proton-to-electron mass ratio, since m pm e =

16523 m 9 m 1836.

In this Table, the symbols Y 1, Y 2 and Y 3 denote the structurescontaining the triplets Y coupled to the ring-shaped neutral parti-cles

of increasing complexity. For example, the electron-neutrino,



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Table 2 and Fig.7-9 illustrate the family-to-family similaritiesbetween the particle structures. For example, in each family, thed-like quark appears as a combination of the u-like quark, witha charged lepton belonging to the lighter family. Thus, accord-ing to this scheme, the s-quark is composed of the c-quark and theelectron, s = c + e , then, its mass is m s = 165 + 165 + 9 =2751, derived from mc = 165 + 165 = 27225 and m e = 9 . Sim-ilarly, the mass of the b-quark, b = t + , is the combination of m t = 1767 + 1767 = 3122289 with m = 48 + 39 = 1872 , re-sulting in mb = 1767 + 1767 + 48 + 39 = 76061 .5. The overlinenotation here corresponds to the abbreviated form of (12). Eachcharged lepton is a combination of the neutrino from the same fam-ily with the neutrino and the charged lepton from the lighter family: = + ee , = + . Their masses computed accordingto the same scheme are also in a good agriment with experiment.

Conclusions

Based on the conjecture of spacetime dualism in the form of tworeciprocal manifestations of space unied through an inversion re-gion, our approach provides a reasonable explanation for the patternof the fermion masses. Being only a framework for building themodels of the composite fundamental fermions, our approach how-ever already yields the quantum numbers and masses of these par-

ticles that agree with the experimental values to an accuracy betterthen 0.5%. The masses are derived without using any experimentalinput parameters (in this sense our model is self-contained) The pro-posed framework can be used for exploring the details of the earlyuniverse evolution. Such things as ination, dark matter, dark en-ergy, and others can easily be explained within this framework.



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References

[1] Kobzarev, I. Yu.; Manin, Yu. I. Elementary particles ; 2nd Ed.,Phasis: Moscow, 2000; 208 p.

[2] Hagiwara, K., et al.(Particle Data Group), Phys. Rev. D 66(2002) 010001

[3] Greene, B.R., The elegant universe , Vintage, London (2000)448p.

[4] Pesteil, P., Numerical analysis of particle mass: the measure

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[10] Hoidn, P., Kusner, R.B., Stasiak, A., Quantization of energy and writle in self-repelling knots (2002) 16p.,arXiv:physics/0201018



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[11] Spaans, M., On the topological nature of fundamental interac-tions , arXiv:gr-qc/9901025 (2000) 18 p.

[12] Huber, S.J., Sha, Q., Fermion masses, mixing and proton de-cay in a Randall-Sundrum model , Phys.Lett, B 498 (2001) 256-

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[13] Kastrup, H.A., The contributions of Emmy Noether, FelixKlein and Sophus Lie to the modern concept of symmetriesin physical systems , in Symmetries in physics (1600-1980),Barcelona (1987), 113-163

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Vladimir Yershov- Dualism of spacetime as the origin of the fermion mass hierarchy

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Transcript of Vladimir Yershov- Dualism of spacetime as the origin of the fermion mass hierarchy