Quark Mass, Quark Compositeness, and Solution to the Proton Spin Puzzle

Transcription

1 SCIENTIA PLENA VOL., NUM. 8 Quark Mass, Quark Copositeness, and Solution to the Proton Spin Puzzle Mário Everaldo de Souza Departaento de Física Universidade Federal de Sergipe 9-, São Cristóvão, SE, Brazil desouza@ufs.br (Recebido e 6 de fevereiro de 8; aceito e 7 de arço de 8 This work shows that the proton spin puzzle and the origin of quark ass are natural consequences of the copositeness of quarks which also generates a continuous syetry breaking. And is also quark copositeness what is behind the Kobaiashi-Maskawa atrix. Moreover, the paper shows that prequarks (prions have already been found by the first EMC experient in the 98s. Keywords: proton spin puzzle, origin of ass, prequarks; new SU(.. INTRODUCTION It has been proposed by de Souza - that Nature has six fundaental forces. One of the, called superstrong force, acts between any two quarks and between quarks constituents. This work will deal only with this new force. Actually, quark copositeness is an old idea although it had been proposed on different grounds -. A ajor distinction is that in this work leptons are supposed to be eleentary particles. This is actually consistent with the sallness of the electron ass which is already too sall for a particle with a very sall radius 5. In order to distinguish the odel proposed in this work fro other odels of the literature we nae these prequarks with a different nae. We call the prions, a word derived fro the Latin word prius which eans first. Let us develop soe preliinary ideas which will help us towards the understanding of the superstrong interaction. The work proposes that each quark is coposed of two prions that interact by eans of a new fundaental interaction called superstrong force. In order to reproduce the spectru of 6 quarks and their colors we need prions ( p, p, p, p in supercolor states (α,β,γ. Each color is fored by the two supercolors of two different prions that for a particular quark (Table.. Therefore, the syetry group associated with the supercolor field is SU(. As to charge, one has charge (+5/6e and any other one has charge (-/6e (Table.. Taking into account Tables. and. we construct the table of quark flavors (Table. which shows that the axiu nuber of quarks is six. There should exist siilar tables for the corresponding antiparticles of prions. Α β γ α blue green β blue red γ green red Table.. Generation of colors fro supercolors 8-

2 Mario E. de Souza., Scientia Plena, 8 (8 Superflavor p p p p charge Table.. Electric charges of prions p p p p p u c t p u d s p c d b p t s b Table.. Coposition of quark flavors. THE NEW HYPERCHARGE AND THE NEW SU( In order to find the new hypercharge let us recall the relation between electric charge and baryon nuber in quarks. Quark charges / and / are syetric about /6 and, since /-(-/==(/, we have Q = B ± ( because /6=(/(/=B/. Equation ( is in line with the forula Q = I + ( B + S + C + B * + T ( where I is the isospin coponent of the isospin I, B=/ is the baryon nuber and S,C,B*,T denote the quark nubers for the quarks s,c,b and t, respectively. C and T are equal to and S and B* are equal to. The above forula (Eq. can also be written as Y Q = I + ( which is the Gell-Mann--Nishijia relation where Y is the strong hypercharge. Since each prion has B=/6 Eq. is not valid. Instead of it we should have Q = B ± (

3 Mario E. de Souza., Scientia Plena, 8 (8 because electric charges are syetric about / since /=(/6=B. This iplies that for a syste of prions (a quark Q = B + ( P + P + P + P (5 where B is the total baryon nuber, and P =, P = P = P =. Fro this we note that we ay divide prions into two distinct categories and we should search for a new quantu nuber to caracterize such distinction. Therefore, we have and = + + ( + for u,c,t (6 6 6 = + + ( for d,s,b (7 6 6 Because quarks u and d have isospins equal to / and -/, respectively, we are forced to ake I = ±/ for prions p and p. We will see in detail below how I can be assigned to the. And we will be able then to find that the other prions also have I = ±/. Let us try to write a siple expression for the charge of prions like the one that is used for the nucleon. Following the footsteps of the strong hypercharge we can try to ake Y Q = I + (8 where Y is the new hypercharge (called superhypercharge and is given by Y = B + Σ (9 where Σ is a new quantu nuber (called supersiga to be found. Thus the forula becoes ( + Σ Q = I + B ( which is quite siilar to the Gell-Mann--Nishijia forula used for quarks Q = I + ( B + S. Fro now on instead of dealing with the new hypercharge Y = B + Σ we will deal directly with Σ. As was discussed above we should try = p and = p, p, p. Therefore, Σ + for Σ for and 5 = = for p ( for p, p, p (

4 Mario E. de Souza., Scientia Plena, 8 (8 As we will see shortly p, p, p can also have I = /. In this case we have = + + for p, p, p ( 6 6 This eans thus that Σ can assue the values,, and + and, thus, they can be considered as the projections of Σ =. Of course Σ = can also be the projection of Σ =. Putting all together in a table one has p p j ( j =,, I Σ Let us now find the values of Σ for quarks. The results are quite ipressive because they are directly linked to the Kobayashi-Maskawa atrix and to the quark doublets u c t,,. d s b As was seen above p can only have I = /, and p, p, p can have I = ±/. In the case of the u quark p has I = / and p has I = / because I = / for u, and then its value of Σ is + + ( =. For the d quark p has I = / and p has I = / since I for d is /, and its total Σ is thus + =. In the case of c and t quarks, since p has I = /, p and p should have I = / because the I of both quarks is equal to. In both cases the total Σ is equal to ++=+. Also s and b have opposite I s and Σ equal to either + = or to + ( =. Hence a syste of two prions (a quark has the four possible states Σ,Σ :, + for c,t, for d, for s,b, for u The choice, for d,, for u was ade considering that the u quark is the end product of the decays of quarks which eans that it should be singled out. Making a table with the results we obtain

5 Mario E. de Souza., Scientia Plena, 8 (8 5 I Σ c,t + u + / d / s,b and the quark doublets,,, +, +,,.,, which should be copared to u c t,,. d s b Putting the values of I and Σ in a diagra we obtain Fig. below Figure.. Diagra that shows how the decays of quarks are related to a new hypercharge and to isospin. It is directly related to the Kobayashi-Maskawa atrix. Coparing Figure with the Kobayashi-Maskawa atrix we note that the atrix eleents U cs and U tb (which are the largest ones and alost equal to satisfy the selection rule ΔΣ =, and the other large eleent U ud (which is also close to satisfies the selection rule ΔΣ =, ΔΣ = Σ =. The other large eleents U cd (=. and U us (=. obey, respectively, the selection rules ΔΣ =, ΔΣ =, and ΔΣ = +, ΔΣ =. The alost null eleents U ts and U cb can also be understood according to the above schee if we also take into account the three quark doublets. According to the schee flavor changing neutral

6 Mario E. de Souza., Scientia Plena, 8 (8 6 currents are forbidden when ΔΣ =. Another very iportant conclusion is that b and t quarks are heavier versions of the s and c quarks, respectively, and that is why U cs U tb. If we represent the values of Σ by arrows as we do with spin or isospin we have c p p ; t p p ;, + = = ( p p + p p d = ;, s = p p ; b = p p ;, and u = ( p p p p ;, and thus there is a SU( related to Σ and consequently a vectorial supercurrent Σ μ j = γ Σψ. ( ψ μ We clearly notice that since both t and c have p the ass difference between the is directly linked to how p and p interact with p because these two quarks have the sae Σ and the sae Σ. The sae reasoning happens between quarks s and b which have p in coon. As a given prion takes part in the coposition of heavy and light quarks prions probably have the sae ass which should be a light ass. More on this we will see in the next section. This new SU( is in coplete agreeent with weak isospin and, thus, the Weinberg-Sala odel (applied to quarks does not need any deep odification since the syety continues to be the sae. Concluding this section we note that what is behind the Kobaiashi-Maskawa atrix is the copositeness of quarks.. THE MASSES OF PRIMONS The agnetic oents of prions should be given by e 5 = ; μ = μ. μ = for p, p, p and hence μ μ 6 Considering that the spin content of quarks should be the sae we have 5 e μ = for p and 6 μ + μ = μ + μ d and since μ u / μ d = and using the above relations we obtain μ μ u (5

7 Mario E. de Souza., Scientia Plena, 8 ( =. (6 + Making = f and solving for the ratio / we arrive at 5 = (7 + f and as the ass of u ( p p and d ( p p are approxiately equal it is reasonable to suppose that and thus 5 f (8 + f which yields f and hence. Then it is reasonable to assue that prions have approxiately the sae ass.. SPIN AND SPACE ANISOTROPY IN THE NUCLEON What about spin? The spin of prions is a great puzzle in the sae way as the spin of a quark is since as is well known only half of the spins of quarks contribute to the total spin of the nucleon, as has been found by experients. Actually, the spin puzzle of the nucleon is directly linked to the spin of prions. Since they are eleentary ferions we expect the to be half spin ferions. And thus it is not an easy task to divise a way of aking two half spin ferions to copose another half spin ferion. We begin solving the puzzle in the following way. When we ake the addition of two different angular oenta we use the coutation relation [ S S ] (9, = which eans that their degrees of freedo are independent. But in the case of prions they are not independent since the two prions that copose a quark should always have their Z projections equal to ( + / + ( + / = + / or to ( / + ( / = / since the spin of each quak is /. This eans that when a prion (of the sae quark changes its spin state to ( / the other prion has also to change its spin state to ( /. Therefore, the spin of prions are not independent. Let us consider a u quark in the spin state + (/. Thus we have that S ϕ = + ϕ ( S ϕ = + ϕ ( S ϕ z ϕ = + (

8 Mario E. de Souza., Scientia Plena, 8 (8 8 where ϕ and ϕ are the spinors of p and ϕ S z ϕ = + ( p, respectively. Moreover we have that ( S + ϕ = + ϕ S ϕ = S. ( where ϕ is the u quark spinor. But we cannot say that ϕ = ϕ ϕ. This also eans that for the + two prions that copose a quark the raising and lowering spin operators Ŝ and Ŝ cannot be defined for each prion and therefore there is no way of finding Ŝ x and Ŝ y, separately. We can only obtain that (for (/ p, for exaple + ( ˆ ˆ ( ˆ ˆ S ϕ = S S ϕ = + ϕ = ϕ S x + y z. (5 6 6 ˆ ˆ And since x and y are equivalent directions we should have S x ϕ = S y ϕ = ϕ and. ˆ S ϕ = ϕ z. Thus Sˆ ˆ ˆ xϕ = S yϕ S z ϕ which eans that there is an intrinsic anisotropy of. space caused by the fact that there is a preferred direction which is exactly the direction of the spin of the nucleon. We can interpret this as a contiuous syetry breaking. It is a consequence of confineent and of having six half spin ferions foring three half spin ferions and these foring a nucleon which is also a half spin ferion. 5. SIZES AND MASSES OF QUARKS Having in ind what was developed in references and it is reasonable, therefore, to consider that the two prions that for a quark (inside a baryon are bound by the cobination of the strong and superstrong interactions. Therefore, they should generate an effective potential well. Each well has to have just one bound state which is the ass of each quark. The heavier a quark is the deeper should be the effective well between its two prions. Also, the well should be narrower because the heavier a quarks is the ore it should be bound. For a given quantu nuber, n, the energy of a well increases as it narrows. The effective potential of the top quark is extreely deep since it is uch ore assive than the other quarks. We are able, then, to understand the decays of quarks. The lowest level is, of course, the ground state of the u quark. The ground state of d is slightly above that of u and the ground state of s is above the ground state of the d. This should also happen for the other heavier quarks. These potentials and bound states are in line with the observed decay chain b c s u and with the decays d ueν e and b sγ. Let us now see what is behind quark asses. Several researchers have tried to relate their asses to soething ore fundaental. In order to do this let us consider that the asses are levels of effective wells and let us approxiate each well by an infinite potential well, that is, the ass of each quark (in units of energy, should be given by π E q = (6 8a

9 Mario E. de Souza., Scientia Plena, 8 (8 9 where is prion s ass and a is of the order of the average size of each quark. Since each ass corresponds to a single level in each well, and considering that prions have approxiately the sae ass, we obtain that each quark ass should be related to the average distance between each pair of prions, that is, to the width of each well. Therefore, we should have the approxiate relations: C C C C C. = ;.5 = ;.5 = ;5 = ; 5 = (7 ( R ( R ( R ( R ( R u s where C is a constant and R u, R s, R c, R b, R t are the widths of the wells. These relations can also be obtained, using Heisenberg s uncertainty principle. As is discussed in section of reference, R u.5f. We ay assue that R u R d. We arrive at the very iportant relations about quark sizes: one is: c ( Rs = ( Ru =.6( Ru.5F 5 (8 5 5 ( Rc = ( Rs = ( Ru =.( Ru.5F ( Rb = Rc = Ru = 6 Ru.5F ( = Rc = Ru = Ru.5F ( ( ( ( ( ( R ( ( ( t It is quite interesting that there are soe very fascinating relations. A very iportant d u c s c = = ( Ru ( Rd ( Rs ( Rc ( Rb ( R ( b ( Rc ( R t = = = = t = = ( b t and, thus, there is a factor of between the last two relations. Other quite iportant relations are: b ( Rs = = (5 c R t = = (6 which has the sae factor of. Therefore, the heavier a quark is the saller it is. The approxiation is copletely justified because we approxiated the finite well by an infinite well and also ade the top quark ass approxiately equal to 5GeV. The above results agree quite well with the work of Povh and Hüfner 6 that have found < r > u, d =.6 f and < r > s =. 6 f as effective radii of the constituent quarks. Also Povh 7 reports the following hadronic radii: < r h >=.7 f for proton, b t ( (

10 < r h >=.6 f for Σ, Mario E. de Souza., Scientia Plena, 8 (8 < r h >=.5 f for Ξ, < r h >=. f for π, and < r h >=.7 f for K. These radii clearly indicate that the s quark is saller than the u quark. In order to have very light prions we can consider that every pair of prions of a quark are bound by eans of a very strong spring. Since every potential well has just one level we have the ass of each quark equal to ω k qc = (7 μ p in which μ p is the reduced ass of the pair of prions and k is the effective constant of the spring between the. It is worth entioning that a quite siilar idea is used for explaining quark confineent and based on it a ter Kr is introduced in the effective potential. For the u quark, for exaple, we have c u. GeV. On the other hand if we consider a haronic potential we have qc ku Rq (8 where Rq is the size of the quark. For u we obtain k u J / Gev / f. Using this figure above we obtain μ p 8 kg which is about the proton ass. Therefore, in order to have light prions the effective well has to have a larger dependence with the distance between the two prions. Considering that the potential is syetrical about the equilibriu position we ay try to use the potential V ( x = au x (9 whose energy levels are given by ν /(+ ν π Γ + / ν ν ν /( ν E ν ( + + n = au n. ( μ Γ ν For ν = and n = we have / Γ + π / / E = νa u ( +. ( μ Γ fro which we obtain (aking E = qc a u μ p ~.5 ( c which can be very light depending on the value of a u. The above figure should be taken with caution because it is a result of nonrelativistic quantu echanics. Thus, if prions interact via a very strong potential such as V ( x = au x they can be very light ferions. We can then propose a ore general effective potential of the for q

11 V ( x x we can consider the potential energy Mario E. de Souza., Scientia Plena, 8 (8 = au x ku where the last ter is chose negative. Generalizing the coordinate x V ( φ = λ φ μφ ( where φ is a field related to the presence of the two prions (or other prions of the sae baryon and is connected to the scalar interaction between the, and μ and λ are real constants. Hence we can propose the Lagrangian ν L = ( νφ( φ + μ φ λ φ ( between the two prions of a quark. Since a quark only exists by eans of the cobination of the two prions we ay consider that its initial ass is very sall. The above Lagrangian is syetric in φ but let us recall that prions can interact by other eans, electroagnetically, for exaple. Therefore, we can ake the transforation φ φ + ηe where η e is a deviation caused by the electroagnetic field.and vacuu. The new potential energy up to second power in η e is V ( φ, ηe = V μ φηe μ ηe + λ φ ηe + λ φ ηe. (5 V φ, η has a iniu at ( e μ φ + λ φ ηe ( φ =. μ λ φ (6 As η e is sall let us ake μ φ λ φ = δ (a sall quantity. Then we can ake μ δ φ ± + ε and obtain ε and thus λ μ μ δ φ ± λ μ (7 and the syetry has disappeared. But it is not spontaneously broken, it is broken by the perturbation ε which can occur all the tie. Substituting the above value of φ into Eq 5 we obtain U ( η = V ( φ, η V μ η (8 and the approxiate Lagrangian is ν L = ( νη( η μ η ( which is a Klein-Gordon Lagrangian with ass = μ / c (5 which ay be an effective ass. This is in coplete agreeent with the ideas above discussed on prions for as we saw the two prions of a quark have to interact by eans of a scalar field because the z coponents of their spins are only ( / each. Taking a look at Table.5 we + observe that we need three scalar bosons, η, η and η. These are siilar to the Higgs bosons. The first and second particles are exchanged between the prions of the quarks p p ( u, p ( p c, and p p ( t, and the neutral boson is exchanged between the prions of the quarks p p, p p and p p. Therefore, three bosons produce the asses of ( d ( s ( b

12 Mario E. de Souza., Scientia Plena, 8 (8 quarks. This is also in line with the recent observations about the beginning of the Universe which have shown that the Universe expanded fro an initial ass 5. It is quite interesting that we should have a triplet of scalar bosons. And we notice iediately a very iportant trend: The charged bosons produce asses larger than those produced by the neutral boson, considering the quark generations u c t,, d s b Therefore, the origin of ass in quarks is also linked to the origin of charge. This is suarized below in Table.5. Quark Mass (GeV Charge Mass Generator u ( p p. c ( p p.5 t ( p p 7 d ( p p. s ( p p.5 b ( p p η +, η η +, η η +, η η η η Table.5. The asses of quarks and their generators. In general the quarks generated by charged bosons have larger asses and larger charges than those generated by the beutral boson. 6. THE COUPLING CONSTANTS OF MASS As was shown in section there is an intrinsic spin anisotropy in the quarks of the nucleon. If the spins of prions were isotropic we would have found < S z >= instead of and thus there is a lack of = for each prion. For each quark we 6 should have then a ΔS = ( z. We can consider that this as a continuous syetry 6 breaking that contributes to the foration of each quark ass. Taking into account the units of energy and angular oentu we can propose that the ass of each quark should be proportional to c ΔS z =, that is, c where l is of the order of the quark 8 8 l size. This equation was chosen linear in because in general energies related to spin are linear

13 Mario E. de Souza., Scientia Plena, 8 (8 in while energies related to angular oentu are in general proportional to or to. And as is seen above, in Table.5, the asses of quarks should also have a dependence on the charges of their prions. Thus the ass of a quark (in the nucleon should be given by c = K / c qq 8 K C (6 l l where K C is Coulob s constant, q and q are the prion charges, l is of the order of the quark size and K is a diensionless constant. Let us ake use of the above forula to calculate K for the u quark. In this case we have the following: c =. GeV, l. 6 F, q ( q 5 = C. We obtain K 5 6 u. Doing the sae for the other quarks we obtain the results shown below. Quark K c (GeV u 5. c 6.5 t 65 7 d 56. s 6.5 b 68.5 Table.6. The ass coupling constants of quarks. K is diensionless It is iportant to note that u, c, d and s have ass coupling constants of the sae order of agnitude and thus, t and b are separate.this is in line with what we found in section : t and b are heavier versions of c and s. 7. SOLUTION TO THE PROTON SPIN PUZZLE The latest results of CERN, DESY and SLAC have shown that quarks contribute to only % of proton s spin 8-. The experients have also revealed that gluons and the quark sea do not contribute uch to the proton spin. I will show below that the experients are revealing the copositeness of quarks in prions according to what was discussed in section and in reference. Reference proposed, based on the data of Hofstadter and Heran, that the nucleons are coposed of two layers of prions in which a prion fro the inner layer with another prion fro the outer layer for a particular quark. In this way we can for all three quarks which, in turn, copose each nucleon. What are DIS experients seeing? Any particle Physics text states the following: for low Q the virtual photon γ probes the nucleon as a whole, for ediu Q γ sees three assless quarks and for large Q γ probes the quark sea, and hadronic interactions show the constituent asses of quarks. For exaple, P. Renton s text Electroweak Interactions on p. x >. valence 6 says that...we can picture the nucleon as coposed of a core of hard ( / quarks with ( R. 5 f, surrounded by a large halo of sea quarks and gluons, the tails of which extend to infinity. Actually, this is quite a strange assertion because if the nucleon were coposed of only three particles they would be found anywhere and not only near its center.

14 Mario E. de Souza., Scientia Plena, 8 (8 Considering the results on the nucleon spin, the proposal of reference and the text of the above paragraph we clearly see that the valence quarks do not have ( R. 5 f because they are actually the very light prions of reference and that is why we only easure half of the predicted spin by QCD. Constituent quarks, that is, true quarks have / ( R. 5 f. And what about the other three prions? They are very light and get seared into the quark sea. Therefore, the solution to the proton spin puzzle is the following: DIS are easuring the net S z spin of the inner prions which is + = (7 but as it is well known by QCD only 6% of this goes to the nucleon spin and thus we obtain /.6 =. (8 This result clearly shows fro QCD is a great theory and works very well with prions because color states are coposed of supercolor states as is shown in reference. When we take into account the outer layer of prions we pick up another contribution and thus the total contribution is.6( in accord with QCD. Soe iportant results that coe out of this are: a the quarks seen by DIS are prions, actually; b quarks are not light and are not pointlike, that is, constituent quarks are just the true quarks; c bare (valence quarks are prions and are pointlike; c there is no issing spin when we take into account the two layers of prions; d LHC will find three quarks near the center of the nucleon (that is, three prions, actually and will find only half of the proton spin and, thus, will confir what is said above. 8. PRIMON: THE ULTIMATE UNIT OF HADRONS It is easy to see that prions cannot be coposed particles: atos are coposed of nucleons and electrons, nucleons are coposed of quarks and each quark is coposed of prions, and thus, a prion cannot be divided because then there would be a repetition in the nuber of units. That is, if a prion were a coposed particle it would be coposed of at least other particles but was already used above. This eans that the coposition of the units of atter is not ad infinitu. Thus prions are the ultiate units of hadrons and are as eleentary as the electron. As we see it is not an easy task to see a prion because they should be very light ferions that for a very heavy ferion which is a baryon, and experients ay be isleading. For this purpose the future experients should carefully copare the results of high, interediate and low energy inelastic lepton scattering. 9. THE CONFINED WORLD IS A STRANGE FRACTIONAL WORLD Taking into account what has been developed in the previous sections we arrive at the conclusion that the confined world, that is, the world inside baryons is a quite strange world where charge is fractional at the level of quarks and prions and spin is fractional at the level of prions. The sae holds for the isospins of prions which are ± /.

15 Mario E. de Souza., Scientia Plena, 8 (8 5 As we saw above prions only exist in pairs, foring quarks, and thus, a baryon is actually a very ordered set of six prions, arranged into three quarks. Therefore, free prions do not exist, and hence all this casts doubt on the existence of quark atter, that is, prionic atter. This is quite in line with the Universe having an initial baryonic ass constituted of squeezed nucleons. This is exactly what the four groups of RHIC have recently found at Brookhaven. As we see above the confined world is characterized by soe fractional quantities. And since this fractional character always existed this atter was never free and is, therefore, a noncreated atter.. DE SOUZA, M. E. The General Structure of Matter. Aracaju, Sergipe, Brazil, February.. DE SOUZA, M. E. The General Structure of Matter. Hep-ph/7.. DE SOUZA, M. E., The Ultiate Division of Matter, Scientia Plena, vol, no, 5.. DE SOUZA, M. E. Proceedings of the XII Brazilian National Meeting of the Physics of Particles and Fields. Caxabu, Minas Gerais, Brazil, Septeber 8-, DE SOUZA, M. E. th Interantional Conference on General Relativity and Gravitation, Huerta Grande, Cordoba, Argentina, June 8-July, DE SOUZA, M. E. Proceedings of the XIII Brazilian National Meeting of the Physics of Particles and Fields, Caxabu, Minas Gerais, Brazil, Septeber 6-, DE SOUZA, M. E. Proceedings of the XIV Brazilian National Meeting of the Physics of Particles and Fields, Caxabu, Minas Gerais, Brazil, Septeber 9-October, DE SOUZA, M. E. The Six Fundaental Forces of Nature, Universidade Federal de Sergipe, São Cristóvão, Sergipe, Brazil, February DE SOUZA, M. E. International Syposiu Physics Doesn't Stop: Recent Developents in Phenoenology, University of Wisconsin, Madison(Wisconsin, USA, April -, 99.. DE SOUZA, M. E. Proceedings of the XV Brazilian National Meeting of the Physics of Particles and Fields, Angra dos Reis, Rio de Janeiro, Brazil, October -8, 99.. EICHTEN, E..J., LANE, K. D., PESKIN, M.E. Phys. Rev. Lett. 5: 8, 98.. HAGIWARA, K., KOMAMIYA, S., ZEPPENFELD, D. Z. Phys. C9: 5, CABIBBO, N., MAIANI, L., SRIVASTAVA Y. Phys. Lett. 9: 59, 98.. FRITZSCH, H. Proceedings of the Twenty-Second Course of the International School of Subnuclear Physics, 98, Ed. ZICHICHI A., New York: Plenu Press, thooft, G. Recent Developents in Gauge Theories, Eds. thooft, G et al., New York: Plenu Press POVH, B., HÜFNER, J., Phys. Lett. B, 5, POVH, B., hep-ph/ ASHMAN et al. (European Muon Collaboration, Nucl. Phys. B, 8, ( ADEVA, B. et al. (Spin Muon Collaboration, Phys. Rev. D, 58, (998.. ABE, K. et al. (SLAC E Collaboration, Phys. Rev. D, 58, (998.. AIPAPETIAN, A. et al. (HERMES Collaboration, Phys. Rev. D, 75, 7 (7. HOFSTADTER, R., HERMAN, R., Phys. Rev. Lett., 6, 9, 96.. RENTON, P., Electroweak Interactions, Cabridge University Press, Cabridge, 99.. Brookhaven National Laboratory report no 5-8, /8/5.