Theory of highly charged subnuclear particles

ANNALS OF PHYSICS:

61, 315-328 (1970)

Theory

of Highly

Charged P. c. M.

Subnuclear

Particles

YOCK

Department of Physics, University of Auckland, Auckland, New Zealand Received June 22, 1970 “There are therefore Agents in Nature able to make the Particles of Bodies stick together by very strong Attractions. And it is the Business of experimental Philosophy to find them out. Now the smallest Particles of Matter may cohere by the strongest Attractions, and compose bigger Particles of weaker Virtue; and many of these may cohere and compose bigger Particles whose Virtue is still weaker, and so on for divers Successions, until the Progression end in the biggest Particles on which the Operations in Chymistry, and the Colours of natural Bodies depend, and which by cohering compose Bodies of a sensible Magnitude.” ISAAC

NEWTON,

Opticks(1704)

Recently the author has proposed a new quantum field theory of matter which predicts the existence of an as yet experimentally unconfirmed set of highly charged subnuclear particles. Previously only the mathematical foundation of the theory has been discussed. Here the more directly physical aspects of the theory are presented. In particular, the identification of nuclear particles as bound states of subnuclear particles is made. Conservation laws are also treated. The structure of the weak interactions implied by the new theory is derived. Possible experimental evidence of highly charged subnuclear particles is pointed out. Characteristic sizes of bound subnucleonic structures are discussed. Finally, comparison is made with conventional theory, and various distinguishing features are pointed out.

1. INTR~DUOTI~N Recently [l] the author has proposed a new quantum field theory of matter, which supposes that the commonly observed nuclear particles (proton, pion, etc.) are bound states of a set of six subnuclear fundamental particles. The purpose of this paper is to report some new results that have been obtained with this theory during the past two years. In particular we wish to present the more physical aspects of the theory. The present discussion should thus complement the discussion contained in Ref. [l], which is of a predominantly mathematical nature. 315

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YOCK

In the following sections we shall refer to the six subnuclear fundamental particles of the theory as “subnucleons.” This title, while perhaps somewhat clumsy, is clearly self-explanatory. For this reason it should suffice for present purposes. The basic properties of the six subnucleons are, of course, discussed in Ref. [l]. We list them here for convenience. (1) Spin and Statistics of Subnucleons. Each type of subnucleon is assumed to have spin = a/2, and to satisfy Fermi-Dirac statistics. (2) Masses of Subnucleons. The bare (i.e., mechanical) masses of subnucleons are required by the theory to vanish [l]. By virtue of the fact that subnucleons interact, their observed masses do not vanish. Their interactions produce self-energies. Within the framework of the present theory, the magnitudes of their observed masses are not calculable. In what follows they are determined empirically to be of order 102m,,. (3) Electric Charges of Subnucleons. For reasons stated in Ref. [l], their bare (i.e., actual) charges are assumed to be go, 2go , 3go, 4go, go + e, , and 4go + e. , respectively. Here e, denotes the bare charge of the positron. The magnitude of go is fixed by the assumption (or fact?) that the static polarizability of the vacuum is, like its nonstatic polarizability, finite. This implies, in that system of units in which e2/4&c PU l/137, that go2/4n2fic takes on some particular value which, although not yet exactly calculated, must be of order unity [l, 21. Thus subnucleonsare predicted to be highly chargedparticles.

The above refers to bare charges of subnucleons. These are not the so-called “observed charges.” Observed charges are those appropriate to Coulomb’s law for widely separated particles at rest in vacua. Because the vacuum is polarizable, bare and observed charges differ. Clearly (cf. Ward’s identity) the ratio of bare to observed charge is the same for all particles. Thus the observed charges of the subnucleons are g, 2g, 3g, 4g, g + e, and 4g + e, respectively, where e is the observed positron charge and g = ego/e0 . The ratio co/e = go/g is a measure of the static polarizability of the vacuum. It is calculable, although not easily so. As is implied by the above, it has not yet been experimentally measured. Beyond making our fundamental assumption that it is finite, we make no attempt here to calculate or estimate its precise magnitude. As will be seen, the present considerations do not depend crucially on its precise magnitude. It is noteworthy that, although subnucleons have large charges, they do not have large magnetic moments. This follows because they are highly massive. This important observation has been noted previously [I]. We shall turn to it again in the following.

317

THEORY OF SUBNUCLEONS

These properties describe the ingredients of the theory except for the weak interactions which are discussed in Section 3. Concepts such as baryon number, strangeness, and isospin will be seen to be contained implicitly in these properties. The dynamics of the theory are assumed to be governed by the familiar Dirac and Maxwell equations appropriate to a quantized system of charged spinor fields. These equations are discussed in Ref. [I]. We emphasize here that they are a mathematically consistent set of equations. They have finite solutions, and no problems with infinite renormalizations or infinite self-energies arise. Concerning the physical existence of subnucleons, we refer here to two earlier papers which discuss some experimental evidence for them [2, 31. In what follows we shall suppose that they do indeed exist. We shall also present further possible evidence of them. The plan of the paper is as follows. In Section 2 the identification of nuclear particles as bound states of subnuclear particles is made, and the conservation laws of strong interactions are derived. Section 3 contains the formulation of the theory of weak interactions in terms of subnucleons. In Section 4 new experimental evidence of subnucleons is presented. Section 5 deals with the masses of subnucleons, and Section 6 with the sizes of bound subnucleonic structures. Section 7 contains a general discussion.

2. PARTICLE

IDENTIFICATION

AND CONSERVATION

LAWS

In the present theory subnuclear binding, and indeed all strong interactions except for the G, interaction that is introduced in Section 3, are assumed to be mediated by the “strong electromagnetic” forces associated with the large charges carried by the subnucleons. Thus the theory is a unzjied theory of strong and electromagnetic forces. Associated with their large charges, subnucleons feel strong attraction for antisubnucleons and strong repulsion for subnucleons. Similarly, referring to units of g as units of (observed) “hadronic charge,” subnucleons feel strong attraction for systems of subnucleons and antisubnucleons having negative net hadronic charge, and vice versa. Thus the commonly observed nuclear particles are assumed to be hadronically neutral composite states of subnucleons. This assumption, together with a consideration of the isospins involved, enables the following particle identification to be made? 1 Actually there is a further assumption made each contain the same number of subnucleons, rationale for this assumption involves the weak satisfy the above assumption lead to complicated They are thus rejected.

here. This is that the baryons N, A, Z, .3, and Qand the same number of antisubnucleons. The interactions. Particle identifications which do not and inelegant schemes for the weak interactions.

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YOCK

l-r = 42

(isovector state)

N = 42 A = 431

(isoscalar state)

z=431

(isovector state)

s = 321 iI- = 211

(isoscalar state)

l-C,=432 I&j=3221 In this listing subnucleons with charges g, 2g, 3g, 4g, g + e, and 4g + e are denoted by 1, , 2, 3, 4, , 1, , and 4, , respectively. 1 and 4 denote, respectively, the members of the isodoublets (10 , 1,) and (4, , 4,). Antisubnucleons are distinguished by bars. Some features of this particle identification are immediately apparent. Thus, for each state, B = No. of antisubnucleons

- No. of subnucleons

Y = No. of [l’s + 4’s - i's - &I. Here B and Y denote baryon number and hypercharge, respectively. They are conserved because each type of subnucleon is separately conserved. For, we have thus far assumed that the subnucleons interact via their electrical charges only. It then follows that, for each type of subnucleon, “No. of subnucleons - No. of antisubnucleons” is conserved. By virtue of the above equations for B and Y it then follows that these quantities too are conserved. Isospin invariance follows in the present theory from the equality of the lo and 1, (and the 40 and 4,) charges in the limit e = 0. Thus, in the limit e = 0, the 1, and I, subnucleons (and likewise the 4, and 4, subnucleons) undergo identical interactions. The conservation laws of isospin and hypercharge are violated by the weak interactions. The nature of these violations is discussed in the following section. All hadronically neutral states not listed above are assumed to be resonances (like the p), or combinations of particles (ike the deuteron), or particles which are as yet undiscovered. Thus the isoscalar 44 ground state and the 53,22, and il ground states are all assumed to be multipion (or r’y, etc.) resonances. The isovector 2ii ground state is assumed to be a &-- resonance, and the 451 ground

THEORY

OF SUBNUCLEONS

319

state is assumed to be a KeKb resonance. The 44m and 33222 ground states are assumed to be KaN and (isoscalar) F8N resonances, respectively There are six remaining hadronically neutral states (excluding, that is, combinations of particles like the deuteron). They are 3111, 4112,41111, 43311, 441333, and 4443333 and they are all assumed to be resonances or undiscovered particles. Experimental searches have not been made for particles and resonances with the unusual quantum numbers that these last six states have. Thus their existence (and the same argument applies to the isovector 2i-l resonance) cannot be excluded. Experimental evidence of states with the quantum numbers of the other seven mentioned above does however presently exist. This will be discussed in detail in a future publication devoted especially to the task of classifying resonances (but see Ref. [4] and Section 3). The double kaon (i.e., K, , Kb) identification shown in the listing is required by the baryon (i.e., N, (1, Z, 9, a-) identification shown. Thus the kaon emitted in the (virtual) reaction N --f n + K is necessarily a 3221 state, which we have named Kb , whereas the kaon emitted in the reaction A + 8 + K is necessarily a 432 state, and this we have named K, . The kaon emitted in the reaction 9 + 52- + K is not a third kaon ; it is easily seen to be the 3221 or Kb state. Thus the theory predicts two states with the baryon number, hypercharge, and isospin of the kaon. Transitions between these two states are possible via the weak interactions (see Section 3) only. Clearly this implies that the present theory contains an extra conservation law additional to those usually ascribed to the strong interactions. For practical purposes this extra law may be regarded as the conservation of “purpose,” where we define “purpose” by the equation purpose = No. of [l’s - 4’s - i’s + 4’s]. Clearly purpose is independent of hypercharge, baryon number, and hadronic charge2. In strong interactions purpose is conserved. The purposes of the Ka and Kb states are - 1 and + 1, respectively. Thus conservation of purpose forbids the strong transition K, t) Kb , which transition is not forbidden by the conservation laws of baryon number, hypercharge, or isospin. As can easily be verified, the reaction

2 Note that, in our notation, hadronic charge = No. of [l’s - i’s] + 2 No. of [2’s - z’s] f 3 No. of [3’s - J’s] + 4 No. of [4’s - &I. It is, of course, zero for all normal particles. It is thus automatically conserved in experiments with normal particles. The name of the new conserved quantity predicted by the theory, viz., purpose, is chosen to contrast with strangeness.

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YOCK

necessarily takes the form according to the present theory. From the fact that the two kaons emitted in this process are observed [4] to have the same mass, we deduce that the 432 and 3221 ground states are degenerate (or nearly so): K, mass = Kb mass Because the physical masses of subnucleons are free parameters [ 11, this degeneracy is theoretically possible. (We note, however, that it would hardly be likely to happen by chance.) Such a degeneracy was originally postulated in 1965 to account for the unusual decay law exhibited by the kaon [5]. At that time the postulate appeared artificial; it was essentially made solely on the basis of the decay of the kaon and lacked independent support. The present theory appears to provide that support. The spectrum of baryons, the weak interactions as discussed in Section 3, and indeed the finiteness of vacuum polarization, all seem to require a kaon degeneracy. This completes the present discussion of the particle identification and of conservation laws in terms of the theory of subnucleons.

3. WEAK INTERACTIONS The so-called “weak interactions” are those interactions which are distinguished from other interactions by their ability to change particles of one identity into particles of another identity. This identity-changing phenomenon is of course the basic mechanism which allows particles which would otherwise be stable to decay. The structure of the weak interactions may be deduced from a knowledge of observed particle decays. We find that the weak interactions proceed via the following two loops, which carry single units of hadronic and electric charge, respectively:

Q+ t

G3

Pi, -

G3

G4

vv, -

G3

b40, t

THEORY

OF

321

SUBNUCLEONS

On the grounds of simplicity, we assume that the unrenormalized form of each interaction in these loops is the same. In muon decay, the form of the interaction is known to be a point or contact four-fermion interaction of the form (V - A), (V - A), . We thus assume that each of the interactions in the above loops is of this form. With this assumption strong, weak and electromagnetic interactions are all y5 invariant. The magnitudes of the coupling constants G, , G, , G, , and G4 are determined empirically. Thus the d1 = l/2 and dS # 2 rules require that G, = 0. In elegant contrast to this the approximate equality of the Z+ and Z- lifetimes requires that the coupling G, be strong. Determination of the precise value of G, involves the strong interactions, and is thus not attempted here. For a similar reason the value of G, is also not calculated here, except we note that G, would be of a similar magnitude (or perhaps somewhat larger) than G3 . This follows from the relative lifetimes observed in dS = 1 and LIS = 0 decays. The coupling constant G, is of course the muon decay coupling constant. This completes the description of the theory of weak interactions in terms of subnucleons. At least qualitatively, this scheme appears to account for all observed weak interactions. We list below some of the simpler decay modes as they appear in the present theory: ?r+ = 4+% ---e+ve G3 r+ = 4+4, strong , 1+1, ---;+v, Ga K, = 432 2

4343 strong t pions

Kb = 3221 G,

2izl

A = 4?;1 G,

425 431

Gl

9 = 3Z -

strong l pions strong

strong

Q- = 2101, G,

321,

23 = 4+x

4+22 strong -4,3 1,4,4+ 2

2- = 4,31+

G, strong

l

l

N

+

n

l f!l +n

strong t ST, /lKl

N

+

TT

4,224&

= n + 7T-

n = 4,22 A ” 4,ev,22 = p + e+ K G3 p-e+G+v, Three implications of the above scheme are noteworthy here. First, the inequalities G1 # Gz and G, # G4 forbid the existence of the long sought-after “intermediate vector bosons,” which have previously been postulated as the mediators of weak interactions. The second point concerns strangeness-changing semileptonic decays.

322

YOCK

Empirically, those decays which satisfy AS = -0Q are known to be very rare, and indeed their rates are consistent with being zero. On the other hand, those decays which satisfy AS = AQ are definitely observed, but at rates which are typically a thousand times slower than nonleptonic decays. These observations appear at first sight to invest the theory with a fatal flaw, for, as can easily be checked, the theory predicts strangeness-changing semileptonic decays, both AS = -AQ and AS = AQ, to be formally second-order weak. This would account for the nonobservation of AS = -0Q decays, but it poses a serious problem for the AS = AQ decays. Although these decays are rare, they are not as rare as would at first be expected for second-order transitions. Remarkably enough, however, this argument contains a loophole. Without exception, and this may easily be verified, the lowest-order nonvanishing AS = AQ and AS = -0Q Feynman diagrams differ topologically according to the present scheme. In the language [6] of quantum electrodynamics, the AS = AQ diagrams are “improper,” whereas the AS = -AQ diagrams are “proper.” The possibility thus clearly presents itself that the AS = AQ diagrams are dynamically enhanced (as would be the case, for instance, if subnucleonic wave functions are such that the improper line contained in AS = AQ decays approaches its mass shell). This possibility is being investigated. The third point concerns the G4 interaction which, we have noted, is strong. Clearly this interaction is not isospin or parity invariant. It thus predicts strong isospin and parity violations. These violations would be particularly evident in those processes which specifically involve the G, interaction. We briefly discuss here two such processes. The first is the decay into pions of the lightest 1,x resonance. This must necessarily involve the G, interaction. Hence it would not be expected to be parity invariant. In terms of conventional particle classifications, this would be interpreted as the existence of two charged multipion resonances having the same masses, spins, and widths, but opposite parities. For the reason given in Ref. [l], the common mass would be expected to be large. The ~~(1640) and p&660) resonances possibly have the above properties. We are thus led to suggest that they could be manifestations of one and the same resonance, viz., the lightest l+c resonance, and that the decay of this resonance is not parity invariant. We now consider evidence for strong isospin violations produced by the G4 interaction. In the previous section we noted that in the limit e = 0, the pairs I, and 1,) and 4, and 4, , each become isodoublets. Precisely because there are two such pairs, the symmetry of the theory at this stage is SU(2) @ SU(2) rather than just SU(2). A consequence of this is that, with the weak interactions turned off, the 2 and /I states become degenerate. It then follows that the 2-A mass difference of ~75 MeV/c2 must be due to the weak interactions. Such a large mass difference could not result if the weak interactions actually were all weak. In this way we arrive independently at the conclusion that some at least

THEORY OF SUBNUCLEONS

323

of the so-called “weak interactions” must be strong. We postulate, of course, that the G, interaction is the strong one. Indeed the nature of the G4 interaction is such that it could be expected to produce a Z-A mass difference without internally splitting the Z multiplet. This follows from the isotopic symmetries of the Z and d states. The actual magnitude of ~75 MeV/c2 observed for the Z-A mass difference provides a phenomenological measure of the effective strength of the isospin-violating G4 interaction. It is stronger than those interactions conventionally called electromagnetic, but not much stronger.3 Despite the qualitative nature of these considerations, we consider that the above scheme for weak interactions provides strong support for the theory as a whole. It would seem most unlikely that the simplicity and elegance of the numerics involved in both the fundamental loops (especially the first one) could have arisen “by chance” (as presumably it must have done if the theory is incorrect).

4. EVIDENCE OF SUBNUCLEONS Needless to say, the validity of the present theory must, in the end, be ultimately dependent on the existence of highly charged subnuclear particles. We have already referred in the Introduction to some possible evidence of such particles. We now present further possible evidence of them. Recently Fowler et al [7] have reported the observation of very highly ionizing relativistic particles at high altitudes. They furthermore gave arguments which suggest that these very highly ionizing particles are not heavy nuclei. We are thus led to suggest that they might be the subnucleons and antisubnucleons of thepresent theory. As a first test of this suggestion we note that the charges of the particles observed by Fowler et al. are consistent, within experimental uncertainties, with our estimate given previously in Ref. [2]. As a second test of this suggestion we may calculate an expected flux for the lightly ionizing particles of the variety recently reported by McCusker et al. [8] at the earth’s surface. According to an argument we have given previously [3], each relativistic antisubnucleon at high altitude produces a lightly ionizing particle at the earth’s surface. Thus the flux of the very highly ionizing particles ought to be twice (assuming an equal population of subnucleons and antisubnucleons) the flux of the lightly ionizing particles. The independent observations of Fowier et al. (at Texas in 1966) and McCusker et al. (at Sydney in 1968) are in agreement with this prediction of the theory, although we note that both the reported fluxes are small and subject to uncertainties. * Isospin violations of just this order of magnitude are required to account for the observed violations of the AZ = 4 rule.

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YOCK

5. MASSES OF SUBNUCLEONS For the reason stated in Ref. [l], the bare masses of subnucleons vanish. This implies that their observed masses are not calculable. One must, therefore, turn to experimental data. McCusker et al. quote a threshold of about 100,000 GeV for the production of the lightly ionizing particles. The argument of Section 4 requires that this must also be the threshold for free subnucleon-antisubnucleon creation. From this we deduce that the masses of the subnucleons must be of the order of 102mp . We note that neither theory nor experiment requires the masses of the different types of subnucleons to be identical. We note also that, according to the present theory, fundamental particles possess mass if and only if they are electrically charged.

6. SIZES OF SUBNUCLEONIC STRUCTURES The above estimate of the masses of subnucleons enables an estimate of the sizes of bound-state subnucleonic structures to be made. Consider, for example, the average separation of the 4 and the 3, which we assume comprise a pion. Without solving the Schwinger-Dyson equation appropriate to this system, we may anticipate that the calculated separation would be proportional to the subnucleon Compton wavelength, since this is the basic scale of length appearing in the equations. The dimensionless constant of proportionality multiplying the subnucleon Compton wavelength would be a (presumably nonpathological) function of go2/4n2fic only. Since this quantity is of order unity, it is reasonable to assume that the constant of proportionality is also of order unity. In this way we deduce that the characteristic size of bound-state subnucleonic structures is of order lo-l6 cm. This implies that direct effects associated with the large charges of subnucleons would be seen in experiments with normal (i.e., hadronically neutral) particles at only those energies for which the de Broglie wavelengths involved are less than, or of the order of 2 lo-l6 cm. The small size of subnucleonic structures compared to the separation of lo-l3 cm that is observed between nucleons in nuclei is evidently an indication of the fact that the mechanisms of subnuclear and nuclear binding are entirely different. In the case of subnuclear binding we have assumed that the fundamental mechanism is photon exchange, and that a typical bound state separation is lo-l6 cm. In the case of nuclear binding we of course assume that the dominant mechanism is pion exchange, which is well known to have a range M IO-l3 cm.

THEORY OF SUBNUCLEONS

325

There are then, according to the present theory, two phenomenologically distinct strong binding mechanisms. Associated with them are two distinct scales of length. There exists a further distinction between the two types of binding. Photon exchange is attractive between oppositely (hadronically) charged subnucleons only, whereas pion exchange is attractive between all nucleons. Subnucleonic binding is thus governed by the simple and systematic law of hadronic neutrality, whereas nucleons bind to form anything from deuterium to uranium nuclei. The situation is analogous to that which pertains in chemistry wherein the familiar mechanisms of atomic and molecular binding operate.

7. DISCUSSION The reader who has gotten this far will no doubt object that everything has been speculative (or “wishful” as it has been put by one critic), and that no quantitative calculations have been performed. In view of the fact that no existing (and widely known) theory can be considered both established and complete, we would suggest that new theories ought (being heedful of established laws) to be formulated, and that such theories are bound to be speculative at their inceptions. We furthermore offer no apologies for the present absence of quantitative calculations within the theory proposed here. The dynamical equations underlying the theory (i.e., the Schwinger-Dyson equations) are difficult to solve. For this reason it would seem worthwhile to consider qualitative estimates of various phenomena before proceeding to the difficult task of solving the equations exactly. Indeed we would hope that the publication of this paper might prompt some new ideas on the problem of extracting quantitative results from quantum field theories of strong interactions. Our viewpoint here is similar to that expressed by Sakurai [9] in connection with his gauge theory (1960) of strong interactions. Sakurai’s theory contained a basic problem, namely the problem of the masses of the so-called B quanta. The theory was, however, published with the expressed hope that its publication might prompt a solution to this problem. A solution has now indeed been obtained. In a similar fashion we hope that the present paper might generate fresh ideas in the problem of solving quantum field theories of strong interactions. It is perhaps worthwhile to summarize the present major successesof the theory. We have (in Ref. [l]) been able to put forward a reason as to why strongly interacting particles should exist, and we have accordingly formulated a theory of strong interactions. The theory is finite, simple, and consistent with established laws of physics (e.g., Coulomb’s law, Fermi-Dirac statistics). It yields the observed conservation laws, and appears to reproduce the observed particle spectrum. The weak interactions appear to take on a simple and elegant form in terms of the

326

YOCK

theory, and the theory appears to be able to account for the strange new particles recently reported by Fowler and McCusker from their studies of the cosmic radiation. But we would like to have some quantitative predictions. At this stage we offer just one quantitative statement of the theory. This is its capability [2] to account for the existence of particles which possess neither integral nor third-integral electric charge. This feature of the theory distinguishes it from previous theories. After the present scheme was formulated, three articles by J. Schwinger arrived in Auckland on a theory which is closely analogous in many aspects to the present theory [lo]. The following are the analogous and essential features of the two theories. Both are quantum field theories, contain two types of electromagnetic charge, one of which is large (g) and one of which is small (e), contain one eigenvalue equation for charge, and assume the existence of subnuclear particles. Both are unified theories; thus they assume that subnuclear binding is electromagnetic in origin. Both utilize the 2ii method for baryons and predict two types of weak interactions-those which involve fermion pairs of total charge g, and those which involve fermion pairs of total charge e. We note here, however, that whereas the theory of Schwinger encounters grave difficulties associated with its prediction of nuclear electric dipole moments [ll], the present theory encounters no such difficulties [l]. The particle classification which is proposed here embodies features that are completely foreign to the present conventional classification based on the Lie group SU(3). In particular, the total absence of any single overall Lie group underlying the present classification distinguishes it from the conventional classification. We consider that direct and very strong support for the present classification resides in the ease with which it appears to be able to accommodate the weak interactions4 Should, however, our classification turn out to be incorrect, then the underlying dynamical principle which is proposed here, viz., that of highly charged, massive subnuclear particles, is one which, it can be argued, merits consideration in its own right. To see this we can argue as follows. The abundance of varieties of nuclear particles suggests that they are composite states of a smaller set of subnuclear particles. The systematics (such as the dearth of mesons with positive parity) of nuclear particles suggests that the subnuclear binding mechanism follows a simple and systematic law. By a systematic binding law we mean, of course, a saturating binding law.5 Probably the only simple and saturating binding law 4 Including the absolute rates of nonleptonic decays. The W(3) theory fails completely to predict these correctly [12]. 6 It puzzles the author that this important property of the subnucleonic force has been largely ignored.

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THEORY OF SUBNUCLEONS

known to be operative is Coulomb’s law. 6 It thus seems not unreasonable to consider Coulomb’s law as the basic law of subnuclear binding. One must then necessarily consider subnuclear particles to be highly charged, since only in this way can the great strength of subnuclear binding be accounted for One would further assume that subnuclear particles possess spin = n/2, since this is the simplest way by which the spins of nuclear particles can be accounted for. It then follows that if subnuclear particles are highly charged and possess spin = fi/2, they must be very massive, since only in this way can the observed magnitudes of nuclear magnetic moments be accounted for [l]. By this purely empirical line of reasoning one is led, quite independently of the mathematical argument of Ref. [l], to consider the existence of highly charged, massive subnuclear particles. Further details along the above lines have been presented elsewhere [13]. ACKNOWLEDGMENT During the course of this work the author has benefited from remarks made by B. T. Feld, K. A. Johnson, R. C. Johnson, and R. F. Keam. REFERENCES 1. P. C. M. YOCK, Znt. J. Theor. Phys. 2 (1969), 247; Erratum 3 (1970), 83. 2. P. C. M. YOCK AND R. C. JOHNSON, Inr. J. Theor. Phys. 3 (1970), 80. 3. P. C. M. YOCK, Znt. J. Theor. Phys. 3 (1970), 82. 4. J. D. SORRELS, R. B. LEIGHTON, AND C. D. ANDERSON, Phys. Rev. 100 (1955), 1457; 0. I. DAHL, L. M. HARDY, R. I. HESS, J. KIRZ, AND D. H. MILLER, Phys. Rev. 163 (19671, 1377.

In Fig. 52 of Dahl et al. a KaKb resonance would manifest itself as a band of points at 135” to the horizontal axis. The data do seem to show such a band. 5. J. L. URETSKY, Phys. Left. 14 (1965), 154; H. EZAWA, Y. S. KIM, S. ONEDA, AND J. C. PATI, Phys. Rev. Lett. 14 (1965), 673; H. J. LIPKIN AND A. ABASHIAN, Phys. Lett. 14 (1965), 151. 6. F. J. DYSON, Phys. Rev. 75 (1949), 1736. 7. P. H. FOWLER, R. A. ADAMS, V. G. COWEN, AND J. M. KIDD, Proc. Roy. Sot. (London), Ser. A 301 (1967), 39 (we refer especially to the bottom paragraph on p. 44 of this article); L. CRODZINS, Comments Nucl. Particle Phys. 4 (1970), 11. 8. C. B. A. MCCUSKER AND I. CAIRNS, Phys. Rev. Left. 23 (1969), 658; I. CAIRNS, C. B. A. MCCUSKER, L. S. PEAK, AND R. L. S. WOOLCOTT, Phys. Rev. 186 (1969), 1394; I. CAIRNS AND AND L. PEAK, “Analysis of the Statistical Significance of the Five Sydney Quark Candidates,” University of Sydney, preprint, 1969; P. KIRALY AND A. W. WOLFENDALE, Phys. Lett. B 31 (1970), 410. 9. J. J. SAKURAI, Ann. Phys. (New York), 11 (1960), 1. The author has been greatly influenced

by this article. 10. J. SCHWINGER, Phys. Rev. 173 (1968),

1536;

Science

165 (1969),

757; 166 (1969),

690.

B The other binding laws that are known to be operative are Yukawa’s and Newton’s. These are simple enough, but they are not saturating.

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11. N. F. RAMSEY, P&w. Rev. 109 (1958), 225; M. Y. HAN AND L. C. BIEDENHARN, Phys. Rev. Lett. 24 (1970), 118; C. K. CHANG, “Magnetic Quarks-Bosons and Electric QuarksFermions in Dyons and Hadrons,” University of Houston, preprint, 1970. 12. L. WOLFENSTEIN, in “Proceedings of the 1967 Heidelberg International Conference on Elementary Particles” (H. Filthuth, Ed.), p. 289, North-Holland, Amsterdam, 1968. 13. P. C. M. YOCK, “The Proton: What Is It Made Of?,” to be published in H. G. Forder Festschrift, University of Auckland, 1970.