364 Progress of Theoretical Physics, Vol. 71, No.2, February 1984

Preon Composites and Their Interactions in a Unified Preon Model Masakuni IDA

Department of Physics, Kobe University, Kobe 657 (Received September 30, 1983) Hypercolor·singlet preon composites (or postons) and their interactions are studied in a unified preon model with the gauge group of 5U(7)(195U(7). The preonic structure of postons is combined with duality diagrams to examine features of a few condensations required by the model. Dual amplitudes for the scattering of two massless postons are also discussed. A switch· over phenomenon for two fermion trajectories crossing each other at s =0 and] = 1/ 2 is considered to understand a fa Regge how massless postons acquire masses.

§ 1.

Introduction

Among various attempts beyond the .standard electroweak theory there exist two streams of approaches very contrasting with each other. One is the grand unification theoreies (GUTs),l),2) in which unification of matter and that of its interactions are stressed. In this context supersymmetry may be included in the same category. The other is the composite models of quarks and leptons,3),*) in which the composite nature of the fundamental fermions is more emphasized than unification. Although GUTs have obtained physically important results, there remain unsatisfactory points such as the origin of generations and too many arbitrary parameters connected with Higgs fields. The composite models, on the other hand, seem to lack guiding principles. This is very serious because we cannot expect ample experimental information to check models. Unified preon models may be regarded as an attempt at unifying these two ideas. We characterize them by the following requirements. 5 ) i) All the primary interactions should be gauge ones. The original Lagrangian must contain fields of preons and gauge bosons but not of Higgs scalars. ii) The gauge interactions should be unified. This means that the gauge group G is either simple or quasi-simple, allowing a single gauge coupling constant. iii) The Lagrangian should be scale invariant. Possible representations for preon multiplets are restricted so as to forbid preon mass terms. iv) The theory should be anomalyjree. The subgroup of G responsible for preon confinement will be called hypercolor and· denoted by GhC. We also call the remaining degrees of freedom hyperfiavor. A quasisimple G has a form G= Ghc®G hf , where GhC and Ghf are isomorphic to each other and there must be a discrete symmetry (D) including the interchange of hypercolor and hyperfiavor. *)

For further references on composite models see, for instance, a review article, Ref. 4).

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The above four requirements are general enough to allow a variety of models. We add two more for further specification. v) GhC should be asymptotically free. vi) Preon multiplets should be irreducible under G if G is simple, and under G and D if G is quasi-simple. For quasi-simple G, it can be shown that the last requirement follows from the others.S ) It is not known if there exist preon models of simple G satisfying all the six requirements. Two types of models are possible for quasi-simple G. They are given by G =SU(7)®SU(7) with 1'L(21, 7) and 1'R(7, 21) (or %L(7*, 21*)) and G=S0(10)®SO(IO) with 1'L(10, 16*) and 1'R(16, 10) (or %L(16*, 10)).5),6) We do not consider the latter model because its generation structure for quarks and leptons looks unrealistic. In this work we thus restrict ourselves to the SU(7)®SU(7) model, which has non-sequential four generations. 7 ) But the basic idea presented in this work will be applicable to other preon models, too. Much attention will be paid to hypercolor-singlet preon composites or postons. Posthypercolor or postonic interactions among them are inherent in preon models,1) just as hadronic ones are in the quark model. We are necessarily led to concepts of Reggeized postons, duality and so on. A similar idea was already expressed six years ago by Tanikawa and Saito. B) We should note sharp contrast between hadron physics and 'poston physics'. The difference in characteristic energy scales is enormous. Bosonic postons are all superheavy, while there exist a relatively small number of massless fermions required by 't Hooft's anomaly conditions. 9 ) This makes it practically impossible to detect postonic interactions directly in the low-energy region. But they will be responsible for neutrino masses and proton decay as well as for quark-lepton masses. The author personally feels that further progress in preon models will be difficult without considering postonic interactions. The present article is organized as follows. In §2 an outline of the SU(7)®SU(7) model is given. The preonic structure of postons is studied in §3. Boson trajectories coupled to quarks and leptons are studied in §4. Natures of condensations required by our model are distinguished from a duality viewpoint. 10 ) We also discuss some properties of dual amplitudes in our concern. A switch-over phE!nomenon l l ) is considered in §5 between two fermion trajectories crossing each other at s = 0 and] = 1/ 2. This is done to understand in a Regge language how the massless fermions gain masses as a result of spontaneous breaking of the electroweak symmetry. § 2.

Outline of the SU(7)®SU(7) model

The SUhC(7)®SUhf (7) unified preon model has two kinds of gauge bosons A~c(48, 1) and A~A1, 48), while preon multiplets are given by 1'R(7, 21) and 1'L(21, 7). Its Lagrangian density is written as (2'1)

with the covariant derivative defined by

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(2·2) Here tR={(1+Y5)/2}t, lL={(1-Y5)/2}t, and t and Tdenotetherepresentationmatrices for 7 and 21, respectively. , Originally our Lagrangian has discrete symmetries PS and CS, where P stands for space reflection, C for charge conjugation and S for the interchange of hypercolor and hyperfiavor. They are assumed to be broken at a unification scale Au, where SUhf (7) breaks down spontaneously as

(2·3) The twenty gauge bosons transforming as (5, 2) EB(5*, 2) under SU(5)@SUg (2) obtain a mass of order g(Au )A u . The SU(5) subgroup further breaks down at a partial unification scale Apu, which is essentially equal to the GUT scale, as

(2·4) As is well known in the SU(5) GUT, twelve gauge bosons get a mass of order gsu(5)(A pu )A pu . There remain 16 massless hyperflavor gauge bosons, among which four are new ones characteristic of our model. SUhC(7) remains intact until it causes preon confinement at a lower scale AhC. If we take the theoretically most interesting possibility of Au=Mpl( ~1019GeV), then A hc is of the order of 1012GeV. Such a high value for AhC seems to be peculiar to. our model. Various kinds of postons are formed at AhC. As composite particles confined by hypercolor gauge interactions they will have strong postonic interactions, just as hadrons do. As is the case in QCD,does any spontaneous breaking of hyperfiavor symmetries occur at preon confinement? In this connection we should note an important difference between QCD and the present model. Our preons, transforming as cf;R(7, 21) and cf;L(21, 7), cannot get masses due to:hypercolor gauge invariance. Therefore, the mechanism(s) of spontaneous breaking of hyperfiavor symmetries should be different from that ofchiral flavor symmetry in QCD. We shall study the problem further in the next section. Useful information on massless composite fermions (MCF's) can be obtained from anomaly conditions of 't Hooft.9) We shall not repeat our previous discussion on a group 6 CAE for which anomaly equations are to be made. ) It is to be noted that a large symmetry obtained by switching off all the hyperfiavor interactions is not respected by strong postonic interactions. Due to algebraic properties of anomaly equations CAE could be larger than an exact symmetry at A hc . If we enlarge it too much, however, we shall lose solutions of physical interest. It should also contain a global symmetry in order . that anomaly equations are inhomogeneous. In our model such a global symmetry is provided by a U(1) charge defined by 6)

(2·5) where cf; stands for cf;L(21, 7) and X for XL(7*, 21*), the CP conjugate of cf;R(7, 21). X is anomaly-free with respect to hypercolor only, and it takes odd (even) integers for postonic fermions (bosons). We take CAE to be

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367 (2·6)

where SU(5) is regarded not as an exact symmetry but rather as a classification group. A solution found for the MCF's is given by7) 1. 2{(5*, 2t 3 EB(10, 2)1EB(1, 2)5h=1,

II. III.

{(5, 1t12 EB(5*, 1)12}1EB{(5,,)1)8EB(5*,lt8}_1, {4(1, 3)oEB10(1, 1)O}_1.

(2·7)

The next problem is whether all the states other than 2(5*EB10, 2)1 obtain large masses or not. The MCF II will become massive by spontaneous breaking of UQ(l), and the MCF III by that of Ux(1). In §3 we shall give a plausible reason to think that both AQ and Ax are close to A hc • The gauge boson VQ, transforming as (1, 1) under SU(5)Q!)SUg (2), acquires a massossible. We classify the cortiesponding states as Category II', separating them from Category II, for convenience. Examples of states belonging to Categories I, II and II' are shown for bosons in Fig. 1, and for fermions in Fig. 2. The values of X are also shown in parentheses. In particular, postons of BI have X=O, and those of FI have X=±1. N ext we study the hyperflavor nature of postons. We recall that preons behave

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368 (a)

(a)

BI (X=O)

X

FI (X

Xt

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