Search for composite models

Nuclear Physics B199 (1982) 119-167 © North-Holland Publishing Company

SEARCH FOR COMPOSITE MODELS* R. CASALBUONI

Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Florence, Italy and D~parternent de Physique Th~orique, Universit~ de Genbve, 1211 Genkve 4, Switzerland R. GATTO

D~partement de Physique Th~orique, Universit~ de Genkve, 1211 Gendve 4, Switzerland Received 3 July 1981 (Final version received 25 January 1982) Baryonic number conservation and 't Hooft anomaly equations select only a few candidate models for composite quarks and leptons. We carry out a systematic group-theoretical search and find that composite-component duality is the characteristic feature allowing for baryon conservation. Complementarity between the symmetric and the broken phases naturally fits within dual composite models and allows for the solution of the anomaly equations. The study suggests subcolor groups SU(9) or SU(7) with two color singlets and two color antitriplets of subcomponents, or with two singlets, one triplet and one antitriplet, or analogous situations with O(17) and O(15). Finally we discuss a dual-complementary pattern of different layers of compositeness using elementary Higgs at intermediate stages.

1. Introduction The present theory of elementary particles, based on SU(3)c ® S U ( 2 ) ® U(1), has achieved an impressive amount of success. It is plagued, however, by a theoretical disease related to its elementary scalars. The Higgs scalars are sources of various problems: naturalness, lack of predictivity, parameter instability. By naturalness we mean, following 't H o o f t [1], that the smallness of a physical parameter at s o m e energy scale must imply a larger symmetry when the parameter vanishes. The scalars of present electroweak theory do not satisfy such a requirement already at energies larger than the vacuum expectation value

v = (x/2G) 1/2 = 250 G e V .

(1)

The lack of predictivity for the parameters of the Higgs potential and of the Y u k a w a couplings is due to the fact that the gauge principle leaves them essentially arbitrary and unrelated. More fundamentally, one does not k n o w which scalars have to be introduced, and, especially in unified theories, the fact of introducing * Partially supported by the Swiss National Science Foundation. 119

120

R. Casalbuoni, R. Gatto / Search for composite models

a few particular ones to derive some satisfactory results sounds very much like an artefact. The parameter instability is macroscopically evident in unified theories where an almost unbelievable fine tuning of parameters is needed [2]. Such a fine tuning is essentially related to the almost thirteen orders of magnitude W-mass from the masses of leptoquarks, as they are predicted.

which separate the However, already

at the level of present electroweak theory, one notices that the Yukawa couplings require some small numbers of the order 10P5. In addition, for QCD + electroweak theory, a P-violating phase angle must be extremely small (-lO-*) in order not to violate the experimental bound on the electric dipole moment of the neutron. Various lines of work can be envisaged to improve on the Higgs situation. None has so far provided a solution, although they all are promising. The most suggestive approaches are supersymmetry, study of non-perturbative effects and dynamical symmetry breaking. Within schemes based on dynamical breaking, the symmetry breaking obtained by the introduction of scalars and their vacuum expectation values would only be a phenomenological description valid at relatively low energies [2, 31. Technicolor models, in particular, introduce new technifermions interacting through a new strong interaction, technicolor [2, 31. The scale of technicolor is in the TeV region, such as to be able to reproduce the phenomenological weak interaction scale, eq. (1) for the vacuum expectation value of the Higgs. These schemes have been extensively developed and are rich in interesting problems and implications [2-4]. In the original scheme of technicolor, quarks and leptons are elementary. New technifermions, which acquire a heavy dynamical mass, are added to the set of elementary fermions. The strong technicolor interactions provide for condensates of technifermions, allowing for a dynamical mechanism of symmetry breaking which gives masses to the electroweak gauge bosons. The problem of giving masses to the fermions is, however, more difficult in these schemes and requires the introduction of new massive gauge fields of an “extended technicolor” interaction. A more radical idea would be just to abandon the elementarity of quarks and leptons, or at least of quarks, and to think of them as composites of more elementary constituents. A new strong interaction would be needed, capable of binding together the elementary constituents. It would also give rise to condensates of these constituents and, thus possibly allow for dynamical symmetry breaking. Dynamical Higgs couplings would schematically arise through graphs such as quar.k

or

lepton

(2) quark

or

lepton

R. Casalbuoni, R. Gatto / Search for composite models

121

If such condensates have to account for the electroweak breaking, then, again, the scale for the new strong interaction must be in the range 1-100 TeV. The new gauge force, introduced for such composite models, could again be called technicolor, and from the point of view of dynamical symmetry breaking, indeed it plays a role similar to technicolor. However, we think it is better to keep the two schemes distinct and employ here a different name. We have called the new force subcolor [5] (metacolor [1] and hypercolor [6] are also used). In subcolor theories quarks and leptons are composite rather than elementary and there is no need to introduce "extended technicolor". The problems of subcolor are however, most probably, m o r e difficult than with technicolor, where more freedom is left in introducing elementary objects and elementary interactions. In general models of this type, which introduce a new strong interaction at a higher energy scale, the elementary fermions transform according to representations of the new gauge group (technicolor, subcolor, or something else). Some fermions may, however, in principle, transform trivially under such a group. In this case they would be elementary and non-confined by such an interaction and not coupled to its gauge bosons. They would be similar to ordinary quarks and leptons and perhaps identical to some of them. A theory containing such singlets would be of an " i n t e r m e d i a t e " type, in the sense that some singlets of the new gauge force would be composites and some of them elementary. It would, in this respect, be like Q C D where some of the color singlets are hadrons, which are composites, and the others are leptons which are elementary, according to the current picture of Q C D and electroweak theory. The dynamical problems posed by a composite description of quarks and leptons have in part been analyzed [7]. The essential new feature of such systems with respect to other compound systems we are familiar with (including hadrons as composite of quarks) is the fact that they would have to be small objects with C o m p t o n wavelength much larger than their size. Clearly no quasi-classical intuition would be of help to describe a similar situation. Comparison of subcolor to color would be quite misleading. In Q C D we think of the hadrons as composites and the typical Q C D scale is the p a r a m e t e r A which is derived, for instance, from deep inelastic scattering. The C o m p t o n wavelengths of the hadrons (kaons, protons, etc.) are not orders of magnitudes away from A -1. Similarly, the proton anomalous magnetic m o m e n t is compatible with a size of order A 1. On the other hand, a muon, for instance, has an anomalous magnetic m o m e n t entirely calculated within Q E D including hadronic vacuum polarization. The measurements agree with the 6th order Q E D calculations. If one accepts, as a satisfactory approximation, leptons and quarks as massless, as c o m p a r e d to the large scale of subcolor, then one has to look for some mechanism guaranteeing exact masslessness for the composite fermions or for a set of them in the subcolor theory. Such a mechanism could be exact chiral symmetry, although

122

R. Casalbuoni, R. Gatto / Search for composite models

we know that such symmetry is in fact spontaneously broken in the only example we have at hand, that is Q C D , as we observe it. We shall call Gsc the new confining gauge group which is responsible for the strong binding of the constituents. The subscript sc stands for subcolor. The global symmetry of the kinetic part of the lagrangian will be called the flavor group and denoted by GF. The subgroup of Gv which is not spontaneously broken is the chiral symmetry that would be able to prevent quarks and leptons from getting mass. In the particular case Gsc = SU(N), by taking n left-handed and n right-handed constituents, in order to cancel anomalies in Gsc we have Gv = U(n)L ~) U(n)R. The subgroup U(1)L--R is broken by strong Gsc anomalies, and we are left with GF/U(1)L-R. It m a y happen that GF/U(1)L-R is spontaneously broken, in which case the broken part gives rise to Goldstone bosons, whereas the unbroken part is realized through parity doublets. This can be very nicely illustrated within the o'-model [1]. 't H o o f t [1] has derived a necessary condition for the unbroken part of the flavor group to be a symmetry, preventing bound states from acquiring mass. To derive this condition one considers the triangular anomaly for three flavor currents belonging to the unbroken part of Gv. Then one finds that the contribution to the anomaly of the massless constituents must be equal to the contribution of the massless bound states. We will refer to this condition as to the " a n o m a l y equations". The anomaly equations are not terribly restrictive and they can be supplemented by the Appelquist-Carazzone theorem. In such a circumstance it has been shown [8] that the anomaly equations have no solutions for Gsc= SU(N). The subject has been widely discussed in different aspects and various models have been discussed [9]. However, the particular use that has been made in this context of the AppelquistCarazzone t h e o r e m has been criticized recently by Preskill and Weinberg [10]. In fact, 't H o o f t requires that, by giving a mass m to one of the constituents and letting m-~ oo, all the bound states containing such a particular constituent should also get a mass which goes to infinity. However, in order to m a k e this condition of some utility, one is forced to require that the pattern of spontaneous symmetry breaking does not change in the process. That is, one requires that the remaining chiral symmetry (after sending m-> oo) is not spontaneously broken, but, if this is the case, the remaining symmetry will generally be too large, and it will prevent the bound states containing the massive constituent from getting mass at all. It follows that, if the Appelquist-Carazzone theorem has to hold, then for m ~ oo the symmetry must undergo a spontaneous breaking in order to allow the previous bound states to acquire mass. But then we cannot extrapolate a statement valid in the "ultraviolet" region (m -~ oo) for getting information relative to the "infrared" region (m --, oo), due to the phase transition separating the two regions.

R. Casalbuoni, R. Gatto / Search for composite models

123

In order to reproduce 't Hooft results, Preskill and Weinberg have proposed a much stronger condition ("persistent mass" condition) which can be stated in the following abbreviated form: a composite state containing massive constituents should be massive. Unfortunately, this is not a theorem and it has been shown to be violated in a particular example [10]. Giving up this condition makes it necessary to find further constraints. An important constraint comes from proton decay [5]. In fact, it is easily seen that, if quarks and leptons are made up of the same elementary constituents, then proton decay occurs through a simple rearrangement of the subconstituent particles. These processes lead to a completely unacceptable life-time for the proton, if one insists on scales of order 1 T e V as our previous considerations seem to require. In fact, in order to respect the experimental limits, the subcolor scale Asc should be of the order 1015 GeV. But this would destroy the picture we gave at the beginning of this paper. Therefore, we are obliged to look for possible ways of suppressing p decay. We will see that this gives very strong restrictions on possible composite models at scales of order 1 TeV. The origin of the problem is that, in general, it is not possible to assign baryon number simultaneously to constituents and bound states without getting states with exotic baryon number. The solutions we shall find, after a systematic search, to the problem of massless composites and baryon conservation can be understood on the basis of two concepts which prove to be important in this domain. The first concept is that of composite-component duality which requires that composites and components transform in the same way under those factors of the flavor group which are non-trivial with respect to the SU(3) of color. The validity of composite-component duality directly implies the possibility of simultaneously assigning baryon number to both composites and components. What is more difficult, and indeed constitutes a great fraction of our effort here, is to show that under certain general hypotheses composite-component duality provides the only solution of the problem. The second concept is that of complementarity, which has already been proposed and largely discussed in connection with 't Hooft anomaly constraints, in particular by Dimopoulos, Raby and Susskind [9]. Complementarity enters in analysis, as will be discussed in detail in sect. 5, by allowing us, through the correspondence between the broken Higgs phase and the symmetric phase, to determine the residual exact flavor chiral symmetry responsible for the masslessness of the composite quarks and leptons. Composite-component duality and complementarity, taken together, allow for a particularly advantageous description in terms of a fictitious scalar composite field. This field, on one hand, allows for the realization of complementarity acting as an effective Higgs field through its vacuum expectation value. On the other hand, it allows for the dual correspondence between composites and components,

124

R. Casalbuoni, R. Gatto / Search for composite models

and its neutral character under baryon n u m b e r guarantees the possibility of simultaneous assignment of such a quantum number.

2. General outlines To provide a simple example, such as to illustrate the problem of proton decay in s u b c o m p o n e n t models, let us consider a very simple model. In the model the subconstituents are an (ordinary) color triplet ~, and one or m o r e (ordinary) color singlets, globally indicated ~:. Again for simplicity, let the subcolor group be SU(3). The possible fermionic bound states are (the indicated representations refer to ordinary color, i, j, k are color indices, and subcolor indices are antisymmetrized): ! , : eiikOi~j~k, _3: 4ti~:~,

!~: ~sc(,

(3)

3_*: eiik~Oi4tkl~.

(4)

Suppose one defines baryon numbers B(~0), B(~:) for the subconstituents. One has immediately B ( ! , ) - B(3*) = B(-3*) - B(-3) = B(3_) - B ( ! , ) = B ( O ) - B ( ~ ) ,

(5)

from which one gets B ( ! , ) + B(3_) - 2B (-3") = 0 ,

(6)

B ( ! , ) + B(3_*) - 2B(-3) = 0 .

(7)

For states with the usual baryon n u m b e r assignment (B = 0 for color singlets, B = for triplets and B = - ½ for antitriplets) these conditions cannot be satisfied. The impossibility of defining B at the level of the subconstituents implies that proton decay can go through simple rearrangements, such as

4"

(3")

3)

~

(11

3/

c~

t t

(1)

(9) (3")

R. Casalbuoni, R. Gatto / Search for composite models

125

These processes are originated by ~:- ~ rearrangement. One can estimate for them an effective coupling constant 2

2

gefr ~ g / A ~ c ,

(10)

where g is ~ 1 and A~c is the subcolor mass scale. The rough estimate for the p r o t o n lifetime would then be _4[ Asc'~4 1

implying the disastrous result % - 10 -4 sec for A , c - 10 s GeV. Clearly a conclusion of this type would completely destroy all programs of composite models and of dynamical symmetry breaking based on such models. A way out is then to look for a subset of composite states such that: (i) Ordinary quarks and leptons belong to the subset. (ii) It is possible to define the baryon numbers within such a particular subset without having to do with states of exotic baryon assignment (wild states). In subcolor models it is customary to call flavor degrees of freedom all the other internal variables except subcolor, such as ordinary color, SU(2)L quantum numbers, electric charge, etc. It is then natural to suppose that the above set of states can be identified with an irreducible representation of the flavor group. Therefore, the dynamics should be such as to keep these particular representations of the flavor group low in mass, whereas the representations containing wild states should remain high in mass. For consistency we have thus to require that the set of states for which baryon conservation holds must not contain exotic states of any kind. We have now a perfectly well defined mathematical problem which consists in looking for composite representations of the flavor group with assigned U(1) q u a n t u m numbers, such that these quantum numbers can be defined on the subcomponents. To be m o r e precise we consider the following class of models: the subcomponents are triplets, antitriplets and singlets of color all transforming according to the same subcolor representation: ~,

i = 1,2, 3 ,

a = 1, 2 . . . . , m ,

(12)

X i~ ,

i = 1, 2, 3 ,

cr = 1, 2 . . . . . p ,

(13)

a = 1, 2 . . . . . I.

(14)

~a,

If the subcolor group does not have anomaly-free representations, like S U ( N ) for N > 2, then we take n = 3(m + p ) + l left-handed and n right-handed Weyl spinors, belonging to the same representation of the subcolor group Gsc. In this case the flavor group is U ( n ) L ® U ( n ) R . If the representation of Gsc is anomaly free, the flavor group is U(n) and n = 3(m + p ) + l. We see that it is sufficient to study the problem for a group SU(n). One has to look for irreducible representations of

126

R. Casalbuoni, R. Gatto / Search for composite models

SU(n) with given quantum numbers assigned to its states, and such that they can be realized as additive quantum numbers (i.e. they can be assigned to the fundamental representation). Of course, this will in general be impossible because we will have too many equations to satisfy; however, there are particular cases that will be called e x c e p t i o n a l c a s e s , for which one can satisfy the above conditions. Mathematically, one can start by analysing all the completely antisymmetric representation of SU(n), because all the other representations can be obtained by applying a symmetrized Young product. This works very well also from a physical point of view, because the non-antisymmetric representations certainly will contain exotic states in color. However, these cases can also be discussed and we will come back to them later on. The technique that one uses is very simple. To illustrate it, let us again consider a model with only triplets and singlets. For an antisymmetric representation of rank k of SU(n) we can have the following particular states: !:

(~al~Otl~l)

" " "

(~1~1~0~,)=_

(~JOtbl~-]Otbl~l]Otbl)~ a l " " " ~ ab2 '

e~jkqtl~, ~Pj~1 a k~' ,

a l # ol2 ~ " " " ~ abl , ax ¢ a z # " • " # ab2 ,

etc.

(15)

3 b l + b2 = k '

where subcolor indices are suppressed and ]_1; bx, b 2 ) - (~04~O)b'~:b2 ,

(16)

-3: ~1(~0~0~4#2)... ( 4 ~ , ~ ° ~ , 4 , ~ , K ° , . . . ~ % , 3b~+b2+

(17)

l = k,

13_;b~,/,~)-- ~(~aa0)~ ~ , 3*:

~gal~al(Oa2O°:2l~°t2)

" ' "

(~°tbl~l~°tbl~ab')~al''"

3bl + b2 + 2 = k,

~ ab2 ,

(18)

]-3*; bl, b2)-= 0¢(4tO~0)b'{ b~ • One proceeds by looking for consistency relations for quantum numbers. For instance, by defining creation and annihilation operators, (~9, ~*) and (~:,~:*) one can write down the sequence !-3; bl, b2) ~*/ ]!; b~, b2+ 1)

~*'~

(19)

I-3"~bx, b z - 1).

Then, for a generic quantum number H which is constant on the -3's and on the 3_*'s, we again get the consistency relation H(_3*) - 2H(_3) +H(_I) = 0,

(20)

R. Casalbuoni, R. Gatto / Search for composite models

127

which is not satisfied for H = B, L. However, this equation holds only if the sequence is possible. In this particular case, the sequence can b r e a k down for b2 = 0 or b2 = I. Then, one examines these particular cases following analogous techniques. The result is that one can find, for this case, six exceptional cases for which it is possible to define the baryon number. Four of these cases are then ruled out; two because all the states transforming like _3 under SU(3)c have the same electric charge, and the two remaining ones are ruled out because they contain exotic states. In the case of only triplets and antitriplets we have four exceptional cases, with two ruled out from the exotics. Finally, in the general situation there are only two exceptional cases. Models with only color triplets of subcomponents can also be envisaged, but they only describe composite quarks (no leptons). The two exceptional cases that are always left correspond to the fundamental representation and to the complexconjugate of the fundamental representation of SU(n). One then proceeds by taking products of antisymmetric representations. It is easily seen that one can obtain exceptional cases only by taking products of exceptional antisymmetric representations. Again, unless one multiplies by singlets of SU(n), one gets cases in which there is charge degeneration or there are exotics in color. Tables 1-4 summarize the outcome of the study, which is detailed in the appendix. We are thus left essentially with two cases which can be described by the following Young tableaux:

_n =

(21)

This, of course, will imply a relation between the n u m b e r N of subcomponents giving a subcolor singlet and the n u m b e r of flavors. We get for the _n representation N = n + 1, and for the _n*, N = n - 1. The situations can be pictorially depicted in the two cases like a closed shell with one extra particle, n particles + 1 particle, or like a closed shell with a hole in it n particles + 1 h o l e . These models have thus a typical aspect, that we call composite-component duality. A closed shell of flavors plus one flavor is dual to that flavor and similarly for a closed shell of flavors minus a flavor. It is interesting that the c o m p o s i t e - c o m p o n e n t duality which has emerged, without any special assumption, from our investigation has a certain resemblance to the idea of c o m p l e m e n t a r y as proposed in the sense of Dimopoulos, R a b y and Susskind [9], and 't H o o f t [1]. The resemblance is apparently indirect, but still intriguing.

~0

#0 ~0

#0

#0 ~0 ¢0

0

0 0

0

0 0 0

0 0 0

0

0 0

0

p

U(3m) U(3m) U(3m)

U(3m)

U(3m) U(3m)

U(3m)

Flavor group

The quantum number H can be either B, L or (2.

m

l

TABLE

l

2 3rn-1 3b

b¢O,m-1

1 3m-2 rn#l 3b+2

b~O,m-1

3b+l

k

13*;rn I1;b)

I-3";0)

13"; b)

1)

I_3;0) 13;m-l)

13;b)

States

H(3*~)=2H(O ~') H(-3*~)=HT-H(g, ~)

H(_3*)=(3b+2)H(~)

H(3~,)=HT-2H(O")

H ( 3 . ) : H(O ~ )

H(3)=(3b+I)H(O)

H

Exceptional cases for models with m subcomponents transforming like color triplets

: 3 E ~H(4 ,~) color exotics H(~b ~) = H(~p) Va charge-degeneracy color exotics HT = 3 ]~H(4, ~) no triplets or antitriplets

H

charge-degeneracy

H(O~')=H(~b) Vo~

Comments

Fo

¢3

QO

U(3m + 1)

U(3m +2)

U(3m + 1)

U(3m + 1)

U(3m + I)

0

0

0

0

0

#O

#O

#O

f-0

#O

1

2

1

1

#O

U(3m + I)

#Cl

#O

0

m

1

Flavor group

TABLE

2

H

and m transforming

H(~,)=HT-H(C?

i?;m.I-1)

H($“)

H@:)

13*; m - 1, 0

= HT-

H(1) = (3bl+

I!_;b,+LO) 3mil-1

H@*) = (3b; + 2)H(#)

3b,+3

3)N(ti,)

H@*) = W, +2W(@)

\3*;bl, 0) iZ”; h, 1)

+ H&Y

H(3) = 1% + l)H(~) +H(c-1

13;bi, 1)

H(1) = 3b,H(@Fr)+H(() H(3) = C% + l)H(ti) H(L) = 3hff(J1)+H(5”) HZ*) = (3h - l)H(rL) + H(5’)f HE*)

H&l = H(&Y H(L) = WC’) H(3) = (3h + l)H(@)

(H = Q, B or L)

like color singlets

1) 13;bl, 0) 1%bl, 1) h*; b, - 132)

ik h,

13;0, 0) IL;(41) it?;bl, 0)

states

transforming

361+-Z

3b,+l h*#O

36,+1 b,fO

1

k

cases for models with 1 subcomponents

P

Exceptional

HT=~I:H(@~)+I:H($‘) CI a

H(ti? = H(G) Vff charge-degeneracy

H($“) = Wt(/) VLI color exotics

H(3) +W2*) = H(~I)+H(~A H($,*) = Wti) VCU charge-degeneracy

Va charge-degeneracy

H($,“) = H(ti)

Comments

like color triplets

n

g G-

3 h %. w 3

5 > 6 :

3

S F; .

3

E “jz.

a

Yj

2

l

TABLE 3

p

~0

~0

~0

~0

rn

~0

¢0

~0

~0

U(3rn +3p)

U(3m+3p)

U(3m +3p)

U(3m+3p)

Flavor group

3(re+p)-2

1

3(m+p)-i

2

k

~')

H(x ~')

H(3*) = (3m - 1)H(~b) + (3n - 1)H(x)

Va, charge-degeneracy

n(3*)

H(3)=(3m-2)H(~)+3nH(x) H(3)+H(3_*)=2H(1) H(1)=3mH(t~)+(3n-2)H(x ) H(0~)=H(~O) H(X ~) = H(X)

13"; 0, 0, 0)

HT= 3[~ H(O'~)+ ~ H(x~)]

charge-degeneracy

H(X ~) = HO¢)

H(3) + H(3_*) = 2H(_l) H(~b ") = H(~p)

Comments

13; 0, r n - 1, n) 11;2, m - l , n - 1 ) 13~; 0, m,n - 1)

=

H(3*) = H T - H ( O ~) H(3~) =H(~b ")

H(3*) = 2 H ( x ) H(3~) = HT-H(x

13'; 0, 0) 13; rn, n - 1)

I~*; rn - 1, n) I~; 0, 0, 0)

H(1)

=H(tO)+H(x)

H(3) = 2 H ( x )

H ( H = O, B or L)

I~; 1, 0, 0)

I-3; 0, 0)

states

Exceptional cases for models with m subcomponents transforming like color triplets and p transforming like color antitriplets

U.

m

#O

#O

1

#O

#O

#O

#O

P

Exceptional

U(3m + 3g + I)

U(3m+3p+l)

Flavor group

3(m+p)+l-1

1

k

cases for models with I subcomponents

o,o,

States

H

H(L)

II;O,m,n,I-1)

= HT-HE”)

H(3,*)=H,-H(ti”)

j?*;m-l,n,I)

Comments

and p like color antitriplets

HT=~H(~“)+~[FH(~^)+~H(X”)]

m like color triplets,

H(ZJ=H(@) NJ:) = H(x”) H(L) = H(5”) H&)=H,-H(x”)

like color singlets,

4

13; 0, 0) 13*; 0, 0, O,O) I!; o,o, 0,1) 13; m, n - 1)

transforming

TABLE

2 &. s

$

L? 2 s

8 F ,

E !% z % .? F

n

.R

132

R. Casalbuoni, R. Gatto / Search for composite models

According to complementarity, the confined fermions of the confining theory (which is here the subcolor theory) which are singlets under the gauge (subcolor) group correspond to elementary fermions in a picture ~ la Higgs of the theory with spontaneous breaking. The composite description, and an intuitive correspondence between Higgs vacuum expectation values and the condensates, illustrates the interpretation. Beyond this, an identification of massless composites through their complementarity to the possible massless fermions in the Higgs breaking mechanism satisfies the 't H o o f t anomaly criterion (anomaly from elementary fermions equal to anomaly from massless composites). In our case duality is strictly formulated for the global flavor group (rather than for gauged subcolor). Our closed shells are shells of closed flavor and they do not correspond to subcolor singlets, since the additional fermion, or hole, would then, with its subcolor, not allow for a singlet composite. Nevertheless, the similarity between the two concepts deserves further considerations, that we shall present in sect. 5. If we introduce a compact notation

~r=-(~O~,g~,6~),

r=l ..... n=3(m+p)+l,

(22)

then, the composite states for the two cases we have been left with are _n: qbs = e q'"r"-lX, 1 • • • X . . . . Xs,

n =N - 1,

(23)

n =N+I.

(24)

_rt*: (I)s = e r l " " r N + k ~ , r , " " " ~"rs "" " "~'. . . . .

A n y additive quantum n u m b e r assigned to the composites is given by the following expressions on the subcomponents: ]_ N - 1

n: H(Xs) = H(Os)-~ sY]=H(45s),

(25)

N+I

n * : H ' ( X , s ) = - H ( ~ J + ~-L- Y. H ( ~ ) , -

N

(26)

$=1

and one verifies that in both cases Tr

(composites)

H =

Tr

(subcomponents)

H.

(27)

For a flavor group U(n)L ® U(n)R we could have in principle representations of the kind

(_n, _n),

(_n*, _n*),

(_n, !)@(!, _n),

(_n, _n*)® (_n*, _n),

(_n*, 1)@ (_1, _n*).

(28)

The first three possibilities have to be rejected because B is a vector quantum number. These possibilities do not give rise to representations of the type _n or

R. Casalbuoni, R. Gatto / Search for composite models

133

n* with respect to the vector group U(n)L+R. This leaves us with only three possibilities:

(_n, _1)¢(!, v)=

(29) J

,30

(_n*, !)@(_1, _n*)=

(31)

For the first two cases N = n + 1, for the third N = n - 1. The first two cases are degenerate. The first two cases look like a generalization of the o--model. In fact, for Gsc = SU(3) we get N = 3 and n = 2. The doubling also occurs in the or-model, but the anomaly equations are satisfied with an index l = 1. There is not much sense in writing the Appelquist-Carrazone condition for these models, due to the rigid relation between the number of flavors and colors. One could hope that it would be possible to satisfy the anomaly equations. However, it is immediately seen that the anomaly equation for three SU(n)L currents, and that for two SU(n)L currents and a U(1) current, are incompatible. In the case of a color group like O(N), we have a flavor group U ( N + 1), that is U(2n) because N must be odd. In this case, the anomaly equation must be written only for three SU(2n) currents and they can be satisfied by an index l = N = 2n ± 1. We notice that there exists, in principle, a possible mechanism to generate families. In fact, in U(n)L ® U(n)R models, as long as one is interested in the baryon number, which is of vector type, one can classify the states with respect to the vector subgroup U(n)L+R. But the n or the n* representations of U(n)L+R can arise from various representations of U(n)L @ U(n)l~. The n can b¢ obtained from N representations of U(n)L ® U ( n ) a corresponding to Young tableaux of the type

134

R. Casalbuoni, R. Gatto / Search for composite models

whereas the _n* can be obtained from the ½(N + 1) representations of the type

In this section we have presented the general description of the problem, of the possible solutions, and we have illustrated some elements of the technical derivation. The full argument is developed in appendix A, since the rather long discussion of cases would distract from the simplicity of the main result. In the following we would like instead to concentrate on those specific models which possess attractive features as possible composite descriptions of the known quarks and leptons.

3. 't Hooft anomaly conditions for unitary subcolor groups For n subcolor multiplets of Dirac fermions, each transforming as the fundamental representation of subcolor, the corresponding flavor group is SU(n)L ® SU(n)R ® U(1). Let us start by introducing some general notations. We will denote an irreducible representation of SU(n)L ® SU(n)R by (R, S), where R and S are SU(n)-irreducible representations. Due to the left-right invariance, the bound states will occur in pairs of representations (R, S) G (S, R), and in general they will form a reducible representation E ni[(Ri, Si)~)(S,-, Ri)],

(34)

io

where n; is the multiplicity of the representation (R~, S~). The 't Hooft index, 1~, of the representation (R~, S~) is defined as the number of left-handed minus the number of right-handed fermions in the representation. It follows from the left-right symmetry that the index of (S i, Ri) is -li. 't Hooft [ 1 ] has derived necessary conditions to be satisfied in order that a subset of the representations (34) be massless. These conditions are that the anomalies for the flavor group currents, evaluated on the spin ½bound states, must be equal to the anomalies evaluated on the subconstituents fermions. Furthermore, 't Hooft requires that the Appelquist-Carrazone decoupling be satisfied. We are not going to require the validity of this condition. We cannot require it because of our relation between the number of flavors and the number of subcolors. In our case we have essentially two kinds of anomalies to evaluate: the anomaly due to three SU(n)L currents, and the anomaly due to two SU(n)L currents and one U(1) current. Let us consider the two cases separately.

(i) Three SU(n)L currents The anomaly A(R) of a given representation R of SU(n) is given by A(R)d,~bc = Tr (~) [{ha, hb}Ac],

(35)

R. Casalbuoni, R. Gatto / Search for composite models

135

where the A's are the generators of SU(n) in representation R. Then, for representations of the type (34), we get the following equation: acomposite~ = ~ l i [ A ( R i ) N ( S i ) - A(Si)N(R~)],

(36)

i

where N(R) is the dimension of the representation R. The anomaly for the subcomponents will be Acomp. . . . t~ = a(L)ns¢,

(37)

where L is the representation to which the left-handed subcomponents belong, and nsc is the number of subcolors. The ',t Hooft anomaly condition requires (36) to be equal to (37). (ii) T w o S U ( n ) L currents + one U(1) current The anomaly can be now expressed in terms of the quantity C(R)6ab = Tr (R) [AaAb],

(38)

and of the eigenvalues of the U(1) charge O. The quantity C(R) is called the Dynkin index of the representation and can be evaluated from the Casimir eigenvalues Cz(R). In fact, from (38) we get C(R) =

N(R) Cz(R), N(adj)

(39)

where N(adj) is the dimension of the adjoint representation. The normalization for Cz(R) is chosen in such a way that, on the fundamental representation of SU(n), C be equal to one. With these definitions we get

a~omposites = ~'~1 , [ C ( R , ) N ( S , ) - C ( S , ) N ( R ; ) ] O , ( R , ® S,),

(40)

i

because, from the vector nature of U(1), (2 can depend only on the tensor product Ri ® S~. Furthermore,

A'comp . . . . ts = C ( L ) O ( L ) n ~ .

(41)

Again the 't H o o f t condition is the equality of (40) and (41). As discussed in the previous section, it appears that certain strings of representations of S U ( n ) L ® S U ( n ) R contain the fundamental representation of SU(n)L+R or the complex conjugate representation, and therefore they could provide a mechanism for generating families. In the case of the fundamental representation the string is

t

n -k

(42)

k = 0 , . . . . n.

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R. Casalbuoni, R. Gatto / Search for composite models

In the next section, we will discuss a physical model of this type with n = 8, corresponding to a subcolor group SU(9). We are therefore interested in the anomaly conditions to be satisfied in this case, By using the general expression for A ( R ) given in ref. [11] and the values of the Dynkin indices given in ref. [12], and noticing that Q = 1 on the subcomponents and Q = n + 1 = 9 on the composites we get the equations 10- 63ll - 94812- 98013 + 525014 + 726615 + 221216 + 13217 + 18 = 9, (43) l0 + 65 ll + 80412 + 322013 + 497014 + 306615 + 70016 + 4417 + 18 = 1. In order to solve these equations for integers li, it is convenient to subtract them and to study the equation - 1 6 l l - 21912- 52513 + 3514 + 52515 + 18916 + 1117 = 1.

(44)

We have found all the solutions to this equation in the range - 3 ~

12 -1 0 -1

13

14

2 1 -1

-2 -1 2

ls 1 0 -2

16 2 3 1

17 -3 -7 0

10+ ls 101 -196 -288

(45)

In the case of the complex conjugate of the fundamental representation, the string is

(n-k-l{~,~}k)O(k{~,~}n-k-1),

(46)

k =0 ..... ½n-1. By working again with n = 8, we find the equations (this time we have on the composites Q = n - 1 = 7): - l o - 60ll - 36412 - 35013 = 7 , (47) 10+ 2011 + 8412 + 70•3 --- 1. By adding together the two equations we find - 4 0 l l - 28012- 28013 = 8 , which doesn't have any solution for integers li, i = 1, 2, 3. So much for the general discussion of anomalies. We shall see in sect. 5 that the anomaly problem has a natural solution in the context of the physical models proposed.

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4. Some specific models A t t r a c t i v e possibilities a r e o b t a i n e d for n = 8, b o t h for u n i t a r y o r o r t h o g o n a l s u b c o l o r g r o u p s , b e c a u s e o n e can r e p r o d u c e t h e o b s e r v e d s p e c t r u m of q u a r k s a n d l e p t o n s . F o r an u n i t a r y s u b c o l o r g r o u p we h a v e t h e two i n e q u i v a l e n t possibilities m =0,

p=2,

l=2,

re=l,

p=l,

l=2,

c o r r e s p o n d i n g to t w o a n t i t r i p l e t s a n d two singlets a n d to o n e triplet, o n e a n t i t r i p l e t a n d two singlets, r e s p e c t i v e l y . F o r an o r t h o g o n a l s u b c o l o r g r o u p we h a v e o n l y t h e possibility m =2,

p =2,

l=4,

i.e. two triplets, two a n t i t r i p l e t s a n d f o u r singlets. A l l t h e r e l e v a n t q u a n t u m n u m b e r s for t h e v a r i o u s cases a r e given in t a b l e s 5 - 7 . By c o n s t r u c t i o n , w e a r e g u a r a n t e e d t h a t t h e r e c a n n o t b e a n y B - v i o l a t i n g p r o c e s s a m o n g t h e states b e l o n g i n g to t h e e x c e p t i o n a l r e p r e s e n t a t i o n s . H o w e v e r , v i o l a t i o n of the c o n s e r v a t i o n rule can t a k e p l a c e t h r o u g h mixings with h i g h e r lying r e p r e s e n t a tions c o n t a i n i n g e x o t i c states. T h e s u p p r e s s i o n f a c t o r o n e gets f r o m s i m p l e d i m e n s i o n a l a r g u m e n t s is sufficient to e l i m i n a t e a n y p o s s i b l e c o n t r a d i c t i o n with e x p e r i m e n t s . F o r i n s t a n c e in t h e m o d e l c o n t a i n i n g o n e t r i p l e t a n d o n e a n t i t r i p l e t , TABLE

5

Models with two color singlets (~:1,~:2)and two color antitriplets (X1, X2) Subcolor group

Flavor representation

SU(9)

(8_,1) G (!, 8-)

SU(7)

(8-*, lJO(1, 8*)

Subconstit.

SU(3)c

O

B

L

4:1 c2 X1 X2

singlet singlet antitriplet antitriplet

0 1 2 -g ½

92 2_ 9 -~1 -~1

_7 -97

~1 ~2 X1 X2

singlet singlet antitriplet antitriplet

0 1 2_ -3 ½

27 2_ 7 1

-~

composite states SU(9) SU(7)

~1~ xl()(1 ~ lX 1)(X2~(2X2)~l~:2 a ~ x2(,1(1~ 1~(1)(,~2)(2,1(2)~1~:2 U~ )(lxl (X2X232)~l~2 d - - X2X2(j(IxIX1)~1~2

e ÷ ~ ~2(xlxlx1)(X2X2X2)~l~2 v ~ ¢2(XIXIXa)(X2X2X2)

e- ~ ~1()(1~(1X1)()(2X2)(2)

2 2

-~5 - - ~s 2 7 72_

138

R. Casalbuoni, R. Gatto / Search for composite models TABLE 6 M o d e l s with two color singlets (sel, ~:2), one triplet tp and one antitriplet X Subcolor group

Flavor representation

SU(9)

(-8, _1)E) (_1, 8)

SU(7)

( 8 ' , 1_)(~(1_, 8*)

Subconstit.

SU(3)c

Q

B

L

g:1 ~:2

singlet singlet

-~ 95-

4

0 0

1 - 1

¢t

triplet

X

antitriplet

g:l g:2

29 -~ 3

singlet singlet

¢J

triplet

x

antitriplet

-7 742~ 2

-2~

~

0

--31-

0

0 0

1 - 1

~

0

1

0

-~

c o m p o s i t e states SU(9) SU(7)

u ~ ~(~tb)(XXX)g:lg :2 a -- X( 6¢,¢ )( XXX )¢I ¢ 2

v - g:1(¢J4J¢J)(XXX)~:I~2 e + -- (2 ( ¢~6¢j)( XXX ) ¢ ~ 2

U ~ XX (I//~J///)~ 1~ 2

V -- ~ 1(~1//~//)( X,tv,~) e + ~ ~2(I/J(//I//)(X)(X)

a ~///(//(XXX)~I~ 2

which is a most dangerous one, because u and a are built with subconstituents belonging to the same subcolor representation, we have [we take Gsc = SU(7)] /x x

x \ x

q,

9

('

X/

e+

(48)

/X X u/

~ ~ ' ~ ! '

X \ I

iX~X i~--,# J

I

-8

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R. Casalbuoni, R. Gatto / Search for composite models

TABLE 7 Models with four singlets (~1, se2, ~:3,se4), two triplets (01, ~b2) and two antitriplets (X 1, X2) Subcolor group

0(17) 0(15)

Flavor representation

Subconstit.

SU(3)¢

Q

B

L

2n = 16 2n* = 16"

~:a ~:2 ~:3 ~74 01 02 X1 ~(2

singlet singlet singlet singlet triplet triplet antitriplet antitriplet

0 -1 0 +1 ~

0 0 0 0 ~

1 1 -1 -1 0

--31

31 1

0

or

-2 -g1

-3 -]

1

0 0

composite states O(17)

u~014~,

d~024~,

v~lq~,

e- ~ ~:24~

= (011//1 01)(1~21//21//2) (X 1X lX 1) (X2X2X2)(~l ~2~3~4 )

O(15)

U ~ xlxI[x2131011310213[~]

4

d ~ X2Xz[Xl]3[O']3[O2]3[gj]

4

/J ~ ~ l ~ 2 ~ 4 1 0 1 1 3 [ x l ] 3 [ X 2 1 3 e

~ ~1~2~3[I//113[)(113[X213

[1~/113~ (0101~1 )

etc.

fl ~ I//1 0 1 1 0 2 1 3 [ x l ] 3 [ ) ~ 2 1 3 [ ( ] 4 a ~

0202[@]3[xl]3[x213[~]"

17 ~ ,~:2~3~410113[x 11310213[x213 e + ~ ~l~3~4[t0113[xl]310213[x213

B y r e a r r a n g e m e n t of s u b c o m p o n e n t s , w e get t h e s t a t e ~1/1((~1~)~1~2 w h i c h is a 3_* w i t h B = ], L = 1, Q = ½, a n d t h e r e f o r e it is an e x o t i c s t a t e b e l o n g i n g t o t h e r e p r e s e n t a t i o n (216, 1)

--

8 " ® 216

8_

(49) •

.

8~®216

R. Casalbuoni, R. Gatto / Search for composite models

140

Then, in o r d e r to get the b a r y o n decay, we n e e d an e l e m e n t a r y B-violating process ~:~+ ~-'>X +X, which could be given, for instance, a c o n d e n s a t e of the type ~)?~lff having B = 1, L = 1, which therefore preserves B - L . T h e effective coupling constant of the process

(50) o/

x

is given by geff = g 4 h s S

,

(51)

giving rise to r p = g - 8 ( A s c / r n p ) ~ ° ( 1 / m p ) , which would give A , ~ = 1 0 0 - 1 0 0 0 T e V to be consistent with the present limit on the p r o t o n life-time. T h e case with two antitriplets is m u c h m o r e favorable, because quarks and antiquarks are m a d e up of subconstituents belonging to different subcolor representations. It follows that the m o s t favorable d e c a y in this case is B + L preserving. In fact we have the process (contributing to p --> u + 7r ÷) X2 \

/ X2

X2 X2 X'

X2 X2

u

X~

X1

X~

~2

('

.X 1

(52) X~ X1 2

x 2 x ~'-Y//~ (~.

~2

22

22 ,,

This process occurs t h r o u g h the mixing ~-2

1

1

1

2

2 ~--1 1

1

2

2~-"l~-'2

(~ 1(~ )(2 1(~? )(~ )0? )(~f )~ ~f ~? )? ~? ~

having B = - 1 , L = 1. H e r e the effective coupling constant is gefr = g giving a r p ~ g

28

(Asc/rnp)

40

14 A - 2 0

sc ,

(l/rap) ~ 1088 years for g ~- 1 and Asc = 1 TeV.

(53)

R. CasalbuonhR. Gatto/ Searchfor compositemodels

141

Let us now discuss the possible flavor gauge theories which are suggested by the various models. Let us start with the model with two antitriplets (the model of table 5). The maximal anomaly-free subgroup is SU(4)L+R ® SU(2)L ® SU(2)R with the subconstituents transforming as

(~., X.~)e (4, 2, 1), (54)

i , , (Xk)*) e (4", 1,2) . ((~:R)

The global symmetry group of the flavor interactions is then S w = SU(4)L+R ® SU(2)L @ SU(2)R ® U(1) @ U(1). One of these U(1) is violated by anomalies occuring in Gsc ® U(n)L @ U(n)R; the other one is generated by 3B + L and it is nothing but the total number of particles. Then, one can think about the possibility of maintaining such quantum number as exactly conserved. This would avoid the axion problem and would provide a mechanism for the survival of the cosmic baryon excess in spite of the possible existence of exotic particles in the model at an intermediate mass = T e V , along the line recently suggested by Weinberg [ 13]. Said roughly, the number of exotic particle providing for B-violating processes would be too small, but a conservation law creates a chemical potential such as to produce an excess of particles of a given type over their anti particles. The excess is proportional to the conserved quantum number

ni(p)-ni(p)=[exp(Ei(P)-lz(Bi+aLi))+ [

-Lexp[

l]

[E~(P)+tz(B~+aL~)~I]-~ ~-

)~-

-~ 2tz (Bi + aLi)/kT.

(55)

In the model we are considering B - L as one of the SU(4)L+R generators and the usual relation Q = TaL+ T3R +~(B - L )

(56)

holds. In particular, the condensate -2i

u

-2i

v

-2k

2

-2h

/a.

lh

(eijkeuv(XR~L)(XR~L)(XR Y,~R)(XR Y XR ))0

(57)

has B = 1, L = - 3 and it is an SU(2)L singlet. Therefore, it preserves 3B + L and breaks B - L . The gauge group would then break down to SU(3)¢ ® SU(2)L ® U(1). The bilinear condensates would provide for further symmetry breaking. In the model with one triplet and one antitriplet, the maximal anomaly-free flavor gauge theory containing SU(3)c is O(8)L @ O(8)R. However, the electric charge is not in the gauge group (we cannot add it because Q is not traceless on

142

R. Casalbuoni, R. Gatto / Search for composite models

left-handed subconstituents). It is given by ~1 R-½(B-L) O = 2N

(58)

where R is the total number of particles. Here B - L is a generator of the gauge group. The only possibility we are left with is O(8)L+R ® U(1) where U(1) is generated by Q - ½ ( B - L ) . Of course, SU(2)L is not part of the gauge group, and it would have to be dynamically generated. Here Sw is O(8)L+R @ U(1) ® U(1) but with one of the U(1) spoiled by subcolor anomalies. In fact our states belong to 8_1 ® 8_2 of 0(8). Therefore, there is no residual global symmetry. The condensate i i ,"T'k" ~--1 (8ijkXLXL (///R) ~R )0,

(59)

which is responsible for the proton decay, would preserve B - L . Therefore the gauge group would break down to SU(3)c @ U(1)o ® U(1)B-L. This would give rise to a massless gauge boson associated to B - L . However, if we want to break B - L we need a condensate which does not preserve the total number of subconstituents such as (~L~L)0

(60)

where ~7= ~(~2(t~4~4~)(£)797) ). In fact, it has L = - 2 , B = 0 and O = 0. In this case the gauge group is broken to SU(3)c ® U(1). Finally, about the models based on an orthogonal subcolor group we remark that they are especially attractive from the anomaly point of view (that is the 't Hooft problem [1]). H e r e one could envisage the gauging of O(10). No extra global symmetries would be left. Models with only color triplet subcomponents (m # 0, p = 0, l = 0) describe only composite quarks. They are possible with subcolor group SU(N) for N = 1 (mod 3) in which case the composite (quarks) transform as the fundamental representation of SU(3m) of flavor, or for N = 2 (mod 3) in which case the composites (antiquarks) transform like the complex conjugate of the fundamental representation of SU(3 m) of flavor. These models have no leptons in the composite spectrum. The two simplest cases, N = 7 and N = 5, are illustrated in table 8. The other class of models having only composite quarks is that with subconstituents transforming like triplets and antitriplets under SU(3)o. From table 3 we see that the only exceptional cases without charge degeneracy correspond again to the fundamental and its complex conjugate representation of the flavor group. The simplest model with one triplet and one antitriplet is illustrated in table 9 according to the two possible choices of the subcolor group, SU(5) or SU(7). Any of the models of tables 8 and 9 gives rise to corresponding models for orthogonal subcolor groups.

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R. Casalbuoni, R. Gatto / Search for composite models

TABLE 8 Leptonless models with two triplets (4/1, 4/2) Subcolor group

Flavor representation

Subconstit.

SU(3)c

SU(7)

(6, 1~(~)(1, 6)

~//1

~b2

triplet triplet

I//1

triplet

02

triplet

SU(5)

(6, 1 ) ( ~ ( 1 , 6 )

O 11 2-7 lO - 2-7 7 17

-~

B 1 ~-7 1 2-7 1 - 1-5 1

- 1-g

composite states SU(7)

U~1(010101)(0202~ 2) d~2(~1~101)(02~202 )

su(5)

a--~161(02~2~2)

S. Discussion O u r a p p r o a c h to t h e p r o b l e m of b a r y o n c o n s e r v a t i o n has b e e n to l o o k for a s u b s e t [ of p o s s i b l e b o u n d states such t h a t t h e b a r y o n i c n u m b e r B m a y b e a s s i g n e d to t h e e l e m e n t a r y f e r m i o n s , w i t h o u t having, within the subset, p a r t i c l e s with e x o t i c a s s i g n m e n t of B. T h e r e q u i r e m e n t t h a t t h e s u b s e t _f c o n t a i n s t h e o r d i n a r y q u a r k s TABLE 9 Leptonless models with a triplet and an antitriplet

Subcolor group

Flavor representation

SU(7) SU(5)

Subconstit.

SU(3)c

(6_,!)• (1, 6~

4t X

3 3*

(6, 1)~( !, 6)

0 X

3

composite states SU(7) SU(5)

u~(xxx)(~)

3*

Q

/3

2~ 2 -2~

gI 1 -g

14

-~

1

-~

144

R. Casalbuoni, R. Gatto / Search for composite models

and leptons implies that its states be light in comparison to the scale Asc. A state not belonging to the subset f will, by consistency, have to acquire a mass of the order of Asc. Otherwise, there would be mixings of exotic states with ordinary quarks and leptons. We have shown that the suppression factor to such mixings would indeed be consistent with experimental data for scales already of some TeV. We have divided the problem in two parts. We have first determined which representations of the maximal global symmetry group GF (flavor group) allow for a definition of the baryonic n u m b e r / 3 , In the second part of the problem, which will be developed mostly in this section, we have to see if the states found in this way satisfy the anomaly equations relative to GF or to some subgroup of GF. In our analysis we have considered constituents belonging to an anomaly-free representation R of Gsc and restricted ourselves to the following two possibilities: (i) R is irreducible, as for G~c = O(N); (ii) R is reducible but of the form R1 + R*, as for G~c = SU(N). Each subcolored constituent appears in n flavors, and we have assumed to have rn triplets, p antitriplets and l singlets In = 3(m + p ) + l] under SU(3)~ c GF. In case (i) GF = U(n), whereas in case (ii) GF = U(n)L × U(n)R. We have further restricted the possible representations by requiring: (1) absence of color exotics; (2) at least two quarks of different charge. The requirement (1) comes from the fact that chirality is supposed to keep m_every low, and then it would be difficult to find mechanisms to split color exotics from ordinary quarks and leptons. Furthermore, as we always take subcolor singlets, the eigenvalues of U(1) in case (i) and of U(1)L÷~ in case (ii) are fixed once for all to be equal to N. Taking into consideration that U(1), in case (i), and U(1)L-R in case (ii), are broken by the strong anomaly anyway, the result is better expressed in terms of the representations of C J F = G v / U ( 1 ) in case (i), and of GF = GF/U(1)'L X U(1)R in case (ii). Under our hypotheses we have found only two inequivalent solutions (see the second paper in ref. [5]): the possible representations of GF allowing for /3-conservation are the fundamental representation or its complex conjugate, times an arbitrary number of singlets of GF (which, however, are not singlets under OF). We see that constituents and bound states allowing for 13-conservation transform according to the same representation of (~F (or to complex conjugate representations). We have called this feature composite-component duality. This duality can be related to the complementarity principle which has recently been introduced for the study of composite models [9]. In fact, duality gives us a relation between constituents and bound states, whereas complementarity tells us about the relation between the confined and the Higgs phase. Furthermore, the constituents transforming like singlets under the unbroken part of the subcolor group are nothing but the elementary fermions in the Higgs phase.

R. Casalbuoni, R. Gatto / Search for composite models

145

For case (i), G~c = O(N), GF = U(n), the simplest representations satisfying "cornponent-composite" duality are:

_n=~}n=N-1,

(61)

A particular model is that of table 7. The anomaly equations must be evaluated relatively to GF/U(1), in this case. We have one equation for the vertex with the SU(n) currents, which is indeed satisfied if one takes N times the representation _n, or the representation _n*, of eq. (61). This means N = n + 1 "families" of states belonging to _n (or to _n*) of the SU(n) subgroup of GF. For case (ii), Gs~=SU(N), G F = U ( n ) L × U ( n ) R , the simplest possibilities are (62)

@ , []);

(63)

n*lO!n*(U}nltO(D1 We get, for this case, two anomaly equations: one for three SU(n)L currents; and one for two SU(n)L currents and one U(1)L+g current. These equations are mutually incompatible. However, if GF/U(1)L-R is broken down to GF/U(1)L-R ® U(1)L+R = SU(n)L ® SU(n)R, we are left with only one equation, which is again satisfied by a number of (_n,! ) + Q , _n) [or (n*,l)+(1,_n*)] [see eqs. (62)-(64)] equal to N. One can check that the "persistent mass" condition is not satisfied here. This, however, is not expected due to the rigid relation between N and n. For instance, in table 5 we have given the quantum numbers of a model with n = 8, and Gsc = SU(9) or SU(7), and constituents transforming like antitriplets (p = 2) and singlets (l = 2) under SU(3)color. This case reproduces a family of the standard model, plus an antineutrino. At this point one would expect 9 or 7 families. Similar considerations apply to the model in table 6. We have to justify the spontaneous breaking of U(1)L+R. TO do this let us analyze these models from the point of view of complementarity. We recall that complementarity is a statement about the relation between the symmetric and the broken phase of a gauge theory. It says that the two phases are not separated by a sharp phase boundary, but that

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one can indeed smoothly extrapolate from one phase to the other. This has been shown to happen in some lattice gauge theories [14]. In particular, the spectra of the two phases will be m a p p e d into each other. However, some state of the symmetric phase may b e c o m e unstable when we go from the symmetric to the broken phase. To be definite let us discuss the model of eq. (62). Let us first examine the model in the broken phase. The two-body maximal attractive channel (MAC) for SU(n) gauge theories is _ N + _ N * ~ ! . However, we would like to break U(1)L+R. This is possible only if m a n y - b o d y condensates form. The channel is uniquely determined by the requirem e n t of leaving SU(n)L ® SU(n)R unbroken. Because in the model (62) we have n = N - 1, the required channel is N+...+N+_N*. n~/V-1

By denoting the constituent fields by 0 ) , , r = 1 . . . . . n; A = 1 . . . . N, the condensate is Erl". . . . . eml"'mN 1A(~/~21 " " " ~/]~4Nl_ 1 )0 ~ ( ~ A ) 0 ~;~ 0 .

(65)

In order to justify why this m a n y - b o d y condensate should form, instead of the usual two-body condensate, we will try to estimate the potential for the channel (65) by a sort of average field approximation. We assume that each particle moves in the field created by a bound state belonging to the representation

of SU(N). The group theoretical factor of the potential is obtained by summing, over the (N - 1) particles, the difference of Casimirs:

C2(N- *)-C2( ( N2) *) -C2(N_ ) .

(66)

W e find VN-1---(N-2)

N 2-1 ~ .

(67)

This expression should be c o m p a r e d with the expression for the two-body M A C , which is V2 -

_ 2 N2 - 1 , N

(68)

R. Casalbuoni, R. Gatto / Search for composite models

147

and we see that for large enough N ( N > 3 ) , the ( N - 1 ) channel can be more attractive than the 2-body one. Therefore, we will assume that the condensate (65) can be formed. In this case, we can choose the breaking direction to be A = N and we get (~bA)o = 6Av.

(69)

The symmetry S U ( N ) s c ® S U ( n ) L ® S U ( n ) a ® U ( 1 ) L + R is broken down to SU(n)se ® SU(n)L @ SU(n)R ® U(1)u (n = N - 1), where U(1)u is generated by the number of ~0~v.We are left with a subcolor group SU(n)sc, whose bound states can be only meson states and a set of fermions O~v which are singlets under SU(n)sc. However, due to the global chiral symmetry S U ( n ) L ® SU(n)R, ~O~vis massless. That is, in the Higgs picture, the physical states are the massless fermions O~v and the boson states (singlets of SU(n)~e): mrl...r

n __

-

e

al..,a n

rI r ~bal ' " • ~b~,.,

(ai

=

1 .....

N-

1 = n).

(70)

The correspondences between the two phases and the fundamental fields are given in the following diagram:

¢,2

(~A)0///A = V~b[V

(~A~yr, _ (

(71)

complementarity symmetric phase

broken phase

In the symmetric phase, we need N times the s t a t e ~A~/~,~ in order to satisfy the 't H o o f t condition, whereas in the broken phase only one family is present. However, it is possible that the repetition o f ~ A ~ r occurs through some orbital excitation, and that all these excited states become unstable in the broken phase. Finally we notice that the condensate breaking U(1)L+Rin the symmetric phase is given by EAI...AN((D

A ....

~.)AN)o¢ O.

(72)

Notice that this equation gives a non-trivial constraint on the models, just because of the U(1) breaking. There are examples, for which this equation would imply a breaking of the electric charge (see the model in table 6). For the model of table 5, for instance, the implications are that B and L are broken, but the combination B - L is conserved. Finally, we would like to briefly summarize some other possible lines of development which follow from the preceding conclusions. The correspondence we have just discussed, between the confining and the broken phase, can be thought as due to the fact that n = N - 1 of the constituents, inside the bound state, glue together forming a condensate which gets a vacuum expecta-

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R. Casalbuoni, R. Gatto / Search ]:or composite models

tion value in the Higgs phase. We see that in order to discuss this p h e n o m e n o n it is very useful to introduce an effective Higgs field ~bA, together with the constituent fields qty. This suggests that in order to study possible patterns of complementarity, it can be useful to introduce effective Higgs fields, which are bound states of elementary fermions interacting through the gauge fields of Gsc. This way of looking at the problem can lead to interesting generalizations, like the possibility of having more than one energy scale. This can happen, for instance, if the effective fields are bound states of an interaction different from Gsc. Recently [15] we have looked at this problem by trying to build up a class of composite models, implementing complementarity by means of effective Higgs fields. Here the idea is to try to satisfy the anomaly equations automatically. To this end, we require that there be a one-to-one correspondence between constituents and light fermionic bound states (which is nothing but a strong version of the duality we introduced before). Then, by complementarity we must have a one-to-one correspondence between light bound states and fermions in the Higgs phase. It follows that one also has a one-to-one correspondence between constituents and fermions in the broken phase. But then this is equivalent to saying that the subcolor group Gsc is isomorphic to the "global classification group" G of the broken phase, which by complementarity can be identified with the flavor group of the symmetric phase. We can choose to implement this correspondence by using a linear local mapping between the fundamental representations of Gsc and G. For Gsc = SU(N), we introduce N z complex scalar fields ~b~, where A = 1 . . . . . N runs over the fundamental of Gsc, and i = 1 . . . . . N runs over the fundamental of G. (Here for convenience we have interchanged the roles of the indices i and A with respect to ref. [15].) The fields ~b~ are our effective Higgs fields. They play a role completely analogous to that of "vierbein" fields in gravity. This analogy suggests "orthogonality" conditions i

A

t

j

2 i

~)a(X) ~,~A(X)= V 6i,

(73)

where v is an energy scale. We also introduce constituent fields ~ belonging to the fundamental representation of G~c, and carrying additional indices u = 1 . . . . . m with corresponding global symmetry U(m)L × U(m)R. In the symmetric phase, we have bound states OU

~/ u ~ i t = ,eaOeA,

(74)

These states obviously satisfy the 't Hooft anomaly equations and they are candidates to be massless states. We see from (73), that, in the broken phase, it is consistent (but not necessarily implied by quantum theory) to assume that (¢b ~A)o = v 6 ~ .

(75)

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We thus get the following diagram:

(76) complementarity symmetric-phase

broken -phase

which should be compared with eq. (71). This diagram allows us to express all the content of complementarity by means of strong-duality; i.e., by means of the relation between 4~,~ and Q~'. At the level of constituents, apart from subcolor interactions one can have further interactions acting on the indices u. We suppose that also these interactions are of gauge type, mediated by fields W L and W R. These fields belong to a subgroup H of U(m)L X U(m)R and they can be both elementary, or effective composite gauge fields generated by bilinears in the fundamental fields [16]. Analogously, there may be a gauge interaction relative to a subgroup G0 of G = S U ( N ) , again due to elementary or composite gauge fields generated by bilinears in the Higgs fields. An interesting case is (~0 = S U ( N - 1) ® U(1), without kinetic term for the S U ( N 1) gauge fields. The breaking S U ( N ) ~ S U ( N - 1 ) × U(1) can then be described in a geometric way. In fact, by putting i

a

trA=--c~,a=

~)A ~((~A,O'A),

l .....

(77)

N-l,

with 1

Ora

~ ) ,N_I~!eAAr"AN-, e al...a "

1-- al t (PAl"

.

r~ a

1t

' - r ~4N~L1 ,

(78)

one can automatically satisfy eq. (73). In this way, one gets a non-linear realization of the direct product SU(N)sc x SU(N) which linearizes on the subgroup SU(N)sc x SU(N - i) × U(1). Needless to say, in these models baryon conservation is automatically ensured by strong-duality. We have applied these considerations to a particular model with Gsc = SU(4), introducing two subcolored spinors ~p~, u = i, 2, A = 1. . . . . 4. By choosing a gauge group G0 = SU(3)colorx U(1), the eight bound states (74) describe a quark-lepton family: qa------Oa =

(79) u

l u ~ QA

u

t

1

= I[IAOrA = - ~ E

ABCD

--a --a~b

~c

Eabc~IIA~B(DC~ D .

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R. Casalbuoni, R. Gatto / Search for composite models

TABLE 10 Quantum number assignments to the elementary fields

aI

O

B

L

½

}

}

1_

1_ 4

o~

-}

~i

_1

(g,)A

4

±

0 1

!

--12

4

0

0 1

Q = T3L+T3n-~N,, B=~EN,-gN¢], L= I[N~,+N4~]; (g,)na are the gauge fields of the confining subcolor group SU(4)sc.

The total group acting on Gsc singlets is

SU(3)sc ® SU(2)L @ SU(2)g @ U(1)F ® U(1)B, where U(1)B is generated byN~ = (number of 05) and U(1)F by N,. The quantum number assignments are given in table 10 where T3L and T3R are SU(2)L X SU(2)a generators. In the Higgs picture, the VEV (75) breaks SU(4),c x SU(4) -->SU(4)D, and furthermore it breaks N6. However, the combination N,~ + X is conserved. Here X is the subcolor generator acting on the fundamental of SU(4)~ as the diagonal matrix (-1, - 1 , - 1 , +3). The unbroken quantum numbers O', B', L' in the Higgs picture are obtained from the corresponding Q, B , L of the symmetric phase by the substitution N , -->N , + X. This is illustrated in table 1 1. By using a linear realization for the Higgs fields 05~ one would get one of the models proposed by Greenberg and Sucher [17]. TABLE 11 Complementary quantum number assignments in the confined and in the Higgs picture

Confined picture 1

at

Q 2

¢'A~A 2 a* ~,aa~ 1 t ¢'AO'A 2 + ~bAO"A at B b ~A (Dg)AfbB

~ --~

B a OrA(D~z)A~bB

_2

th,~*(D,~) ABO.B

0 --1 0

B 1

~ '-3 0 0 0 !

--3

2

31

L 0 0 1 1 0 1

-1

Higgs picture

~ ~ ~ t~ (g,)~ a

(go-)4

(g,~)4

Q' 2

~ _~ 0 --1 0 2

--3

2

B' 1

~ 3'0 0 0 !

--3

~

L' 0 0 1 1 0 1

-1

The quantum numbers Q', B', L' of the Higgs picture are obtained from Q, B, L (see table 10) by the substitution N,~ ~N,b +X, where X is the SU(4)sc generator acting as diag (-1, -1, -1, +3). The abc a t bt ct composite field O'A is O'A = (1/3!)eABcOe g)B ~bc ~bO.

R. Casalbuoni, R. Gatto / Search[or composite models

151

We have discussed these possible ideas essentially as a first step towards a m o r e complete theory. In particular one would try to understand if the Higgs fields &~ can be realized as composites of other elementary fermions. In particular one may have to introduce a new gauge interaction with a new scale. It may also be conceivable that only Gs~ would suffice. In any case, the general method of trying to implement each step by using effective fields and then try to see how to obtain them as composite may be useful in this problem.

6. Chirai models

In this p a p e r we have concentrated on two classes of models: (i) the subcomponents transform according to an irreducible real representation of G~c (as for Gsc = O(N)); (ii) the subcomponents transform according to the direct sum of an irreducible representation plus its complex conjugate (as for G,c = SU(N)). However, models not belonging to these classes have been discussed in the literature [9, 18]. The structure of such more general models starts from left-handed subcomponents Xl, 2;2 . . . . . X, transforming, respectively, according to irreducible representations R1, R2 . . . . . R , of Gsc. These representations have multiplicities N1 . . . . . Nn in such a way that the anomaly for any three Gsc currents vanishes. It follows that the maximal global symmetry which is free of G~c anomalies is SU(N1) @. • "® SU(N,) ® U(1)1 ® " ' • ® U(1),_> We can imagine S U ( 3 ) c o l o r t o be e m b e d d e d into a certain n u m b e r of SU(Ni) factors, let us say SU(N~) . . . . . SU(Nj), 1"~
.)®(.,M,.

.....

.)®...®(.,

....

.,D)

of SU(NI) ®" • • ®SU(Nj); that is, such bound states are duals of the s u b c o m p o n e n t states in the sense of the c o m p o s i t e - c o m p o n e n t duality explained in sect. 2. In such a situation all the derivation of the appendix could be immediately applied and baryon conservation would follow. According to our experience such a situation could be the only one guaranteeing baryon conservation in chiral models. The preceding discussion allows us to examine some of the proposed chiral models from the point of view of baryon conservation. In particular, we will examine a class of models [18] which makes use of the properties of the supergroup S U ( N + 4 / N ) [19] in order to satisfy the anomaly equations. Bars and Yankielowicz [18] discuss two models: model (1) and model (2), both with an S U ( K ) of subcolor.

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In model (1) the subcomponents P1,/'2, P3 are assigned to the SU(K) representations R1 ( ~] )*, R2 = ([-])*, R3 =[-7, with multiplicities N1 = 1, N2 = N + 4 - K, N3 = AT, corresponding to a flavor group S U ( N + 4 - K ) ® SU(N) ® U(1)I @ U(1)2. The composite fermions satisfying all anomaly constraints are the SU(K) singlets

fl

= P I P 2*P 2* = ( ~ ' , ' ) ,

fz=P*P2P*)=(f-]*,D*),

f3 = P1P3P3 = (', [~]

),

where the indicated representations are those of S U ( N + 4 - K ) @ SU(N). In model (2) the SU(K) representations of P~, P2 and P3 are instead R1 -- ( I ~ 7 ) * ,

R2 = ([-])*,

R3 = [ ] ,

with N1 = 1, N2 = N - 4 - K, N3 = N, corresponding to a flavor group S U ( N - 4 K ) @ S U ( N ) @ U(1)I @ U(1)2. The composite fermions, of model (2), are

fl=

P i P 2 P*2 * = ( r r T , . ) ,

= P

P2e*

=

f3=P1PsP, = ( ' , ~ ) . Various physical models can be obtained from these solutions of the anomaly constraints depending on the possibilities for embedding color and electroweak interactions. For instance, Bars and Yankielowicz [18] give examples in which SU(3)c is embedded in both flavor group factors S U ( N + 4 - K ) ® SU(N). For such examples it is indeed impossible to assign simultaneously baryon number to both composites and subcomponents. In another example of embedding, Bars [18] takes the model (2) and embeds SU(3)c into the single factor SU(N) of the flavor group. In this case compositecomponent duality holds between the composites fz and the subcomponents, and it is therefore possible to assign baryon number simultaneously to both composites and subcomponents.

7. Summary of the analysis and conclusion In this paper we have searched for subcomponent models of quarks and leptons, based on a gauge subcolor group, and which allow for conservation of the baryon number and for 't Hooft anomaly conditions. We have first assumed that the subcomponents transform according to an irreducible representation of the subcolor group [as for Gsc = O(N)] or to the direct sum of an irreducible representation and its complex conjugate [as for Gs¢ = SU(N)]. Such irreducible representations may or may not be the fundamental representations of Gs¢. We have first studied the constraints following from baryon conservation and found that in order that they be satisfied the composites must belong either to the fundamental representation of a restricted flavor group G~ or to its complex

R. Casalbuoni, R. Gatto / Search for composite models

15 3

conjugate (composite-component duality). The restricted flavor group GF is the maximal global symmetry group with all the U(1) factors taken out (see sect. 5). The second stage has been the verification of the 't I-Iooft anomaly conditions on such representations. The results are: (i) For G~c = O(N), for the Gsc anomaly-free part of the maximal global symmetry group (which in this case coincides with G_F=SU(n), n = N + I ) the anomaly equations are satisfied. (ii) For G,c = SU(N), the anomaly equations for the G,c anomaly-free part of the maximal global symmetry group (which in this case is given by SU(n)L@SU(n)R@U(1)v, n = N + 1) are not satisfied. For case (ii) the anomaly equations can be satisfied if the U(1)v factor of the flavor group is spontaneously broken. The condensate effecting such spontaneous breaking can have observable physical consequences• In tables 5 and 6 we have specified two models reproducing the correct physical family content. The unphysical character of the condensate, which would be required for the model in table 6 in order to b r e a k U(1), allows us to eliminate such a model. On the other hand, in the model of table 5 the condensate gives rise to a Goldstone boson which, however, has an extremely small unobservable effective coupling to (pne-u), violating B and L with selection rules A B = A L = + 2 . The conclusion is that two classes of models emerge from this analysis. The protoypes of these two classes are the models of tables 5 and 7. In any case, the 't H o o f t anomaly equations require a n u m b e r of families equal to the n u m b e r of subcolors. Finally, in sect. 6 we have enlarged our analysis to chiral models and again we have found that c o m p o s i t e - c o m p o n e n t duality provides a solution for baryon conservation.

Appendix Let us briefly recall the main assumptions. The left-handed subcomponents transform according to some irreducible representation of the subcolor gauge group Gsc. There must be no anomalies with respect to Gsc. This implies that whenever the representation of Gsc is not anomaly free, for each left-handed s u b c o m p o n e n t there is a corresponding right-handed one. A flavor group GF acts on the subcomponents and we assume that it contains ordinary color, SU(3)color. Our subcomponents are color triplets, antitriplets, and singlets, with the following notation: m color triplets 4'~i,A, p color antltrlplets X ~ , l color singlets sc.~,

Ot = 1, • . . , m ; tr = 1 . . . . . p ; a = 1. . . . . l ;

(A. 1)

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where i = 1, 2, 3 is a color index, A = 1, 2 . . . . . N is a subcolor index, and s o m e m, p, l can also be zero. These fields are all left-handed W e y l spinors. If the representation of Gsc is a n o m a l y free we have G F = U ( 3 m + 3 p + l ) ; otherwise we must add the corresponding right-handed spinors and GF = U ( 3 m + 3p + l)e ® U ( 3 m + 3p + I)R. O u r p r o b l e m is to find the possible irreducible representations of GF, on which a b a r y o n n u m b e r can be defined and which have no exotic states. W e first look for representations which allow for b a r y o n n u m b e r . W e start with Gv = U(n), since U ( n ) L ® U ( n ) R can then be r e c o v e r e d by tensor products. F u r t h e r m o r e , we will start by considering antisymmetric representations of o r d e r k of U(n), because n o n - a n t i s y m m e t r i c representations can be reconstructed by tensor products (and, as we will see they will generally contain color exotics). F o r technical reasons it is convenient to consider the following inequivalent possibilities: (i) m # 0, l = p = 0: only triplets; (ii) m, l ~ 0, p = 0: triplets+singlets; (iii) m, p ¢ 0, l = 0: triplets + antitriplets; (iv) m, l, p ~ 0: triplets + antitriplets + singlets. Let us start with case (i), m ~ 0, l = p = 0. W e can distinguish three cases: (1) k = 3b + 1. T h e possible color triplets are

(3)~,...~b = ~ ( ~ 1 ~ 0 ~ ' 4 , ° ' ) " " " (~0~b4,%~0~ ) ---13_, b ) ,

(A.2)

with a ¢ O/1 ~ " " " ~ O/b, and . . . . qJ . . . .~ ) = e ijk .qJi~iqJk. .....

(A.3)

T h e subcolor indices do not play any role in this kind of consideration, and f r o m now on they will be systematically omitted. N o p r o b l e m arises here f r o m possible b a r y o n n u m b e r n o n - c o n s e r v a t i o n because all the states in this representation have n o n - z e r o triality. If b ~ 0 a n d / 3 is any o n e of the indices (o/a. . . . . O / b ) , then O(3~,;,~1...t~ .... ~)- O(3t~.~. ...... ~) = 2 [ O ( ~ ~ ) - O(4t")] = 0, + 1 ,

(A.4)

where O is the electric charge o p e r a t o r and the last equality follows f r o m the r e q u i r e m e n t that all the triplets have electric charge + 2 or -½. If b < m - 1, we can take an index such that y ~ (o/1. . . . . O/b) and 3' # O/- Then, we get

O(3_~;~ .... ) - O(3_~;~ ~) = O(4, ~ ) - O(g,~) = 0, + 1.

(A.5)

W e can r o t a t e / 3 and y over all the possible indices different f r o m a. Therefore, we d e d u c e O ( O ~) = O ( ~ ) ,

Va.

(A.6)

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W e see t h a t all t h e triplets h a v e t h e s a m e electric c h a r g e , a n d t h e m o d e l is n o t p h y s i c a l l y satisfactory. If b = 0 w e get t h e f u n d a m e n t a l r e p r e s e n t a t i o n of S U ( 3 m ) w h i c h satisfies all o u r criteria. If b = m - 1 ~ 0, we g e t t h e r e p r e s e n t a t i o n 3 m - 2 a n d we n o t i c e t h a t c o l o r sextets a r e c o n t a i n e d in (O~14,~l)(4,~20°2)(g,%4,~3g,~3)...

(4,"m0"m~0om) ,

where (~ottc~

ijk~otto~

x --

(2) k = 3b + 2. T h e p o s s i b l e c o l o r a n t i t r i p l e t s a r e (3*L; ......... = (¢,~¢#)(0~1¢#10~9

• • • (0%¢,%¢,%)

--13_*; b ) .

(A.7)

U s i n g t h e s a m e a r g u m e n t s as b e f o r e a n d if b ~ 0, m - 1, w e get t h e e q u a t i o n s 0 ( 3 _ ~ ; o ~ . . . ~ . . . . ~) -

0(3~,~

Q ( 3*, ; ~ . .... ) - O ( 3

. . . . . . . . ~) =

O ( 0 ~ ) - O(~, ~) = 0, + 1,

.*. . . . . . . . ) = 2 [ O ( 0 * ) - O ( 0 ~ ) ] = 0 , + l ,

(A.8)

a n d again we g e t Q(~/,~) = Q(~p),

Va.

(A.9)

F o r b = 0 ~ m - 1, we g e t t h e r e p r e s e n t a t i o n k = 2, which c o n t a i n s c o l o r sextets. F o r b = m - 1, w e h a v e k = 3 m - 1 which is t h e c o m p l e x c o n j u g a t e r e p r e s e n t a t i o n of t h e f u n d a m e n t a l r e p r e s e n t a t i o n of S U ( 3 m ) , a n d satisfies all o u r criteria. (3) k = 3b + 3. A l l t h e states h a v e z e r o - t r i a l i t y , a n d we r e j e c t this case b e c a u s e t h e r e a r e no q u a r k states in it. In case (ii), m ,l ~ 0, p = 0, we t a k e a g a i n an a n t i s y m m e t r i c r e p r e s e n t a t i o n of o r d e r k. W e i n t r o d u c e t h e f o l l o w i n g n o t a t i o n for the c o l o r singlets, triplets a n d anti triplets which a r e p o s s i b l y c o n t a i n e d in the r e p r e s e n t a t i o n l ! ; b~, b~) = ( ~ 0 ~ ' g , ~ ° ' )

• • • (0~,~%a0

%1 ) ~ ° , • • • ~ %

(A. 10)

=-- (O~9)b~( b~ ,

k = 3bx+b2 ; 13; b l , bz> = ~ ° ( ~ , ° ~ J , ° ~ , ° ~ )

••

•

(~-~°tbl~tf°tbl~-]°tbl)~al" " " ~ ab2

=_ 0 ( 0 0 g , ) ~ l ~ b ~ ,

k = 3b1+b2+l [3"; bl, b~)=

;

(g,~g,~)(g~l~g,~l)

(A.11) • • (~%~0~,e,~1)~

(A.12)

=-- ( O 0 ) ( O 0 0 ) b l f b2 , a#al¢'''¢ab,;

" ~ . . - ~:%~

al¢'''¢abz.

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We can define creation and annihilation operators ( 0 ' , ~h, ~*, ~) operating on our states, and we notice that the operators 0 ' ~ and ~:'4' belong to the Lie algebra of GF = SU(3m + l). Therefore, by applying these operators to our states we remain inside the given representation of GF. Then our representation contains at least one triplet [eq. (A.11)] and we can apply to it the operators 0 ' ~ and so*O, getting the sequence 13; bl, b2) C% o

~

~°'

11; bl, b2+ 1)

(A.13)

[3-*; bl, b 2 - 1 )

For any quantum n u m b e r H which is constant on the triplets and on the antitriplets, we obtain the following equations from (A. 13): H ( 3 * ) - H ( 3 ) = H ( O " ) - H(~a') ,

H ( I ~ - H(3_) = H ( ~ as) - H ( O " ) .

(A.14)

These two equations hold for any value of r and s and therefore also for r = s, from which we get H(3-*) - 2H(3-) + H(_I) = 0 .

(A.15)

If H is a vector quantum n u m b e r we get (A.16)

H ( 1 ) = 3H(3-).

Then, the only consistent solution for baryon and lepton n u m b e r gives B(_I)= 1 and L(_I) = 0. It follows that either the singlets in the spectrum cannot be identified with leptons or we cannot define B or L. However, this conclusion is correct only and only if the sequence (A.13) does not b r e a k up. This can happen for b2 = 0, or b2 = l. In the case b2 = 0, we can take the sequence

[1; bl, 1) ~e*g, 13-*; b l - 1, 2)

13; bl, 0) ~

(A.17)

' 13; bl -- 1, 3) whereas for b2 = l we have

I_3;

l) -2r2,

3* ; b l , / - 1 ) I_

[1, b l + l , / - 2 ) (A.18)

,13;b1+1, I-3).

4,*~

- -

W e see that we are led back to states of the type considered in eq. (A.13), except for a certain n u m b e r of exceptional cases. In fact the sequence (A.17) breaks up

R. Casalbuoni, R. Gatto / Searchfor composite models

157

for b~ = 0, or l = 1, 2. F o r l = 3 the s e q u e n c e does not break, but we cannot apply the s e q u e n c e (A.13) to the final triplet. T h e r e f o r e , the s e q u e n c e (A.17) fails for b~ = 0 or l = 1, 2, 3. With analogous a r g u m e n t s we see that the s e q u e n c e (A.18) fails for b~ + 1 = m or l = 1, 2, 3 (notice that for the existence of 13; b~, l> we must have bl < m). L e t us e n u m e r a t e o n e by o n e the various cases in which the sequences (A.17) and (A.18) fail: (1) bl = b2 = 0, Vl; k = 3b~ + b2 + 1 = 1. This is the f u n d a m e n t a l r e p r e s e n t a t i o n of S U ( 3 m + l) and all the possible states are 15; 0, 0>,

[1; 0, 1>.

(A.19)

Clearly, this case can satisfy our criteria. (2) b ~ # 0 , b 2 = 0 , I = 1 ; k = 3 b ~ + l . [We assume b ~ 0 , because for b ~ = 0 we are back to case (1).] T h e possible relevant physical states are 1 3 ~ ; ~ . . . ~ ; b~, 0) = g ~ ( ~ l g ~ f f ~ ) . . .

(g~b"~O~b~), (A.20)

I!,~...~b~; bl, 1 ) = ~:(~k~b~'~a') - - " (~b'~/~b~t~%~). F o r an arbitrary index /3 ~ ( a b . . . , electric charge:

oeb,) we have the following relations for the

Q(3~:~r..a...~b,)- O(3~;-r..~ .... hi) -- 2[Q(tP ~ ) - O(ff~)] = 0, + 1 ,

(A.21)

Q(1,1...m..~bl) - Q ( I ~ , ........ bl ) = 3[Q(tp ~) - Q(tp~)] = 0, + 1 ,

(A.22)

f r o m which O(~)

= Q(gJB),

Va,/3.

(A.23)

T h e r e f o r e all the triplets and all the singlets have the s a m e electric charge, given by Q(t3; bl, 0 ) ) = (3b~ + 1 ) O ( O ) , (A.24) Q(II_; bx, 0)) = 3 b l Q ( O ) + Q ( 6 ) . (3) b~ ~ 0, b2 = 0, l -- 2; k = 3bl + 1. T h e possible relevant physical states are 13ot ; ~ l . . . O / b l ;

bl,

0 > = ~/]°t ( ~ / [ t ~ l ~ l ~ / ~ ° t l )

II~;~,...~,; b~, 1> = 6 ~ ( ~ b ~ 6 ~ ,

" " °

).

.

(~Ctbl~°tbl~°tbl),

"(~J°tbl~abl6°tbl),

.

(A.25)

°

In this case there are no p r o b l e m s , in principle, with the electric charge. In fact f r o m triplets and singlets we deduce, as in the previous case, that Q(tp ~) is i n d e p e n dent of a. Then, we get O(5) = (3ba + 1 ) Q 0 P ) , Q ( ! ~ ) -- 3b~ Q(O) + Q(6~), Q(3_*) -- (3bl - 1)Q0P) + Q ( ~ I ) + Q(sC2).

(A.26)

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R. Casalbuoni, R. Gatto / Search ]'or composite models

H o w e v e r , in this m o d e l we must have at least m = 2, and therefore color sextets are present f r o m (O~O~)(O~O~O(O~O~O~)

. . . (0~1&~10~1)

(A.27)

.

(4) b~ # 0, b2 = 0, l = 3; k = 3b~ + 1. T h e possible relevant physical states are

13; bl, 0},

[1; bl, 1},

13"; bl - 1, 2},

13; b~ - 1, 3). (A.28)

F r o m the last two states we get H(3*)

-

H(3_)

=

H ( 4 , ~)

-

H(6~),

(A.29)

which is valid for any a and any a. Then, we d e d u c e that H ( O " ) and H ( 6 ~) are i n d e p e n d e n t of a and a, respectively. F u r t h e r m o r e , f r o m (A.28) we get H(!) - H(3*) = H(&) -H(6),

(A.30) H(3*)-H(3)

= H(0)-H(6),

and therefore, H ( ! ) - 2 H ( 3 " ) + H(3_) = 0 ,

(A.31)

which again is not satisfied by n o n - e x o t i c states. (5) bl + 1 = rn, b2 = I; k = 3 m + l - 2. W e will assume here l > 2, because the case l = 1 will be considered next. T h e possible physical states are 13_; rn - 1, 1) = ,h~'(O'~=O'%h~=) " ' " (g,~ g,~ g,~ ) ~ 1 . . . 6 ' , 3* ; m - 1, 1 - 1) = ( ~ 1 0 ~ q ( 0 % 0 ~ = 0 , ~ ) . [_i

• • (///a

///c~l//a

)~1

. , . ~i

. . . ~l,

(A.32)

ILJ; m, t - e ) =

(6~l~,%0~9 . . . ( 4 , % 0 ~ 4 , ~ ) ~ ~ ' ' . £ ' "

g~"-6,

where the hat m e a n s that the c o r r e s p o n d i n g fields are missing. Then, we get H ( 3 ) - H ( 3 * ) = - H ( g , ~1) + H ( ~ i ) ,

Vffl, i, (A.33)

H(3*)-H(I_ij)

= -H(&"I)+H(~i),

(i # j ) .

H o w e v e r the first e q u a t i o n holds for any value of i and therefore (A.34)

H(3J - 2H(3") + H(l_if) = O. (6) Vbl, b2 = l, / = 1 ; k = 3bl + 2. T h e possible physical states are 13-'~;'~r.... 1; bl; 1) = tO'~(&~'lg,~l&"O • " ( 0 '~b14,% 1~0% 1)~,

I_~;~1 ..... l'bl;0)=(~0~6~)(J'%°%0~9 . . . (6

14, 1~ 1).

(A.35)

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L e t us again consider the electric charge o p e r a t o r ; we get f o r / 3 s ( a l . . . . .

abl)

Q(3_,;~l.-.t~ ..... 1 ) - Q(3t<~l-..~ .... bl) = 2 [ Q ( 0 ~ ) - Q ( 0 ~ ) ] = 0, + 1 , Q(3-,~;,~l.'.t3 ..... 1) - Q(3t~;,~r..=...,bl ) = [ Q ( 0 t~) - O(0'~)] = 0, 4-1, f r o m which 0 ( 0 ~) = Q ( 0 a ) , V a , / 3 . T h e r e f o r e , in this m o d e l we have O(3) = (3b1+ 1 ) O ( 0 ) + O ( ~ ) , (A.36) Q(3*) = (3b~ + 2 ) Q ( 0 ) • Clearly there are no p r o b l e m s with b a r y o n n u m b e r (no singlets present); h o w e v e r for b~ ~ 0 there are color sextets in the model: (0~0-)(0%0-1)(0-20%0-2)

• • . (0%~0-~10 %)~.

F o r bl = 0 there are exotics coming f r o m 0~0 ~ ~3"®6,

(a #/3).

W e could take m = 1, in which case the only states would be 0~ and 00. W e can add to these c o m b i n a t i o n s any n u m b e r of flavor singlets which are of the f o r m (000)~. T h e r e f o r e , the possible b o u n d states are not fermionic because they contain already an even n u m b e r of s u b c o m p o n e n t s . It follows that in this case there are always color exotics. (7) Vbl, b2 = l, l = 2; k = 3b~ + 3. T h e possible physical states are 130';0tl" .... 1; bl' 2)=

l/lOt (00~1~/)*0'1~/10~1) " " " ([llOtblOOtbl{llOlbl)~l~ 2 ,

[_~;~;-1. .... 1' b~, 1 ) = ( 0 ~ 0 " ) ( 0 " ~ 0 ~ R b ~ ) . . • (0

10

10

')~,

(A.37)

I1,,;,1. .... 1+1; b~, 1) = (0~10"10~*) • • • ( 0 % + ' 0 ~ " 1 + 1 0 % + ~ ) .

A s in case (5) we get H ( 3 ) - 2H(3_*) + H ( ! ) = 0 .

(A.38)

(8) Vbl, b2 -= l, l = 3; k = 3 b ~ + 4 . T h e possible physical states are [3; ba, 3) ,

3" ; ba, 2) , 1_

I1;

51 -[- 1, 1),

[3; bl + 1, 0)

which are the s a m e as in case (4). A g a i n we get the relation H(3)- 2H(3") + H(1) = 0.

(A.39)

It m a y h a p p e n that our r e p r e s e n t a t i o n contains antitriplets 13"; b~, b2); if this is the case we can apply the o p e r a t o r ~:*0 to go b a c k to the previous case; i.e. [3*; b~, b2) ~

[3; 51, b 2 + 1).

(m.40)

This will be always possible unless b2 = l. If b2 = l we consider the s e q u e n c e [3*;b~,/) ~

[1;b~+1, l-1)

~

13-;bx+l,/-2),

(A.41)

R. Casalbuoni, R. Gatto / Search for composite models

160

and we are back to the previous case again. H o w e v e r , the s e q u e n c e (A.41) breaks up for b~ + 1 = m, or l = 1. T h e r e f o r e , we have two m o r e cases to add to the previous eight. (9) b~ + 1 # m, b2 = 1, l = 1; k = 3b~ + 3. T h e possible physical states are: ]3";,,...~i; bl, 1) = ( 0 " 0 ~ ) ( 0 ' * ' 0 ~ ' 0 ' ~ ) ]!~, .... ~ + , ; b 1 + 1 '

. . . ( 0 % , 0 % 1 0 ' % )~,

0)~---(0al0al~/~°tl) ' " " ( O 0 ~ b l + l O O t b l + l O O t b l + l )

"

(A.42)

W e get for the electric charge o p e r a t o r O ( 3 o t ; a l " ' l ~ . . . . . 1) -

O(3-~;~x.-.~ .... ~,) :

0 ( 0 a) -

0 ( 0 ~) = 0,

+1

,

O(l~, .... ...,~,+x)- O(1,,..v...~,+, ) = 3[0(0 ~ ) - O(0V)] = O, + 1 ,

3'~ ( a l " • • a b e ) .

H o w e v e r , these f o r m u l a e hold for any a, fl, ~/therefore Q ( O " ) = Q ( O ) , i n d e p e n d e n t of a. W e see that 0 ( 3 * ) = (3bl + 2 ) 0 ( 0 ) + O ( ~ ) , (A.43) O(1) = (3bl + 3 ) 0 ( 0 ) . All q u a r k states (10) bl = m conjugate of the case (1). All the

would t h e r e f o r e have the s a m e electric charge. 1, b2 = / , Vl; k = 3m + l - 1. This r e p r e s e n t a t i o n is the c o m p l e x f u n d a m e n t a l of S U ( 3 m + l). T h e r e f o r e , this case is essentially like possible states are: 13"; m - 1, l ) ,

11; m, l - 1).

(A.44)

W e can call e x c e p t i o n a l the cases in which relations of the type (15) do not hold, that is those cases for which b a r y o n n u m b e r can be defined. W e can then s u m m a r i z e our results as follows: we have found six exceptional cases,which are (1), (2), (3), (6), (9), (10). In cases (2) and (9) all triplets (or antitriplets) have the s a m e electric charge. In cases (3) and (6) there are color exotics inside the representation. Finally, cases (1) and (10) c o r r e s p o n d to the f u n d a m e n t a l r e p r e s e n t a t i o n and to the c o m p l e x conjugate of the f u n d a m e n t a l r e p r e s e n t a t i o n of S U ( 3 m + l). T h e s e last two cases satisfy all o u r criteria. Their q u a n t u m n u m b e r structure will be discussed in the following. In case (iii), m, p # 0, l = 0 we introduce the following notations for color singlets, triplets and antitriplets which are possibly contained in the representation: I!, r, b , , b2> = ( ~ x ) ' ( 0 0 0 ) b l ( X X X )

b2 ,

t3; r, bl, b2) = O ( 0 X ) ' ( 0 0 0 ) b l ( x X X ) b2 ,

r = 0, 1, 2 ,

(A.45)

r = 0, 1,

(A.46) (A.47)

13_, bl, b2) = ( X X ) ( 0 0 0 )bI(xXX) b2 ,

13"; r, b,, b2) = X ( O X ) ' ( O o o ) b ' ( x X X ) b: ,

r = 0, 1,

(A.48)

R. Casalbuoni, R. Gatto / Search for composite models

161

l_3* ; bl, b z ) = (~b~b)(~b~P~P)bl(xXX) b2 ,

(A.49)

where 3

= ~., ,~PiX ,

(A.50)

1

and for the rest we use the same notations as in ( A . 1 0 ) - ( A . 1 2 ) . It is e n o u g h to consider the cases in which states of the type (A.46) or (A.47) are present in the representation, because the cases c o r r e s p o n d i n g to (A.48) or (A.49) can be o b t a i n e d by the exchange tP'~-~X. W h e n states of the type (A.46) are present, we take the s e q u e n c e

13-; r, bl,

/

b2)

(A.51) 3* ;r, bl, b2) 1_

St,oil_; 2, bl, b 2 - 1) -I-8r,all; O, b l + l , bE) A g a i n we derive H(3*)- 2H(3)-H(!)

= 0.

(A.52)

T h e s e q u e n c e breaks up for r = 0, bE = 0, or for r = 0, b2 = p. In the case r = 0, bE = 0 we take the sequence

13; o, b,, o~ ~

13"; o, b,, O~ ~

11; 2, bl - 1, 0)

(A.53)

whereas, for r = O, b2 = p, 13;0, bl, n) ~

11;2;bl, n-1)

13-*;0, b l + l ,

~,r > 13; 0, bl + 1, n - 1).

-1) (A.54)

In the case (A.53) we get the condition 2H(3-*) - H(3_) - H ( 1 ) = 0 ,

(A.55)

u n l e s s bx = 0. In the case (A.54) we go back to one of the original states but with

b2 ~ n, unless ba + 1 = m. T h e various cases are: (1) r = 0, b~ = 0, b2 = 0; k = 1. This is the f u n d a m e n t a l representation of S U ( 3 m + 3p), and all the possible states are 13; 0, 0, 0 ) , This case satisfies our criteria.

13-*; 0, 0, 0).

(A.56)

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162

(2) r = 0 , states are

b2=p; k = 3 ( m + n ) - 2 .

bl=m-1,

T h e possible relevant physical

13~,; 0, m - 1, p) = ~'(0~2~/,~2~=) • •. ( ~ - 0 ~ 4 F

[i. . . .

; 2, m - 1, p - 1)

=

~)

(~[1°tlxCrl)(O°tlxarl)(O°t2~J°t2~l] °t2) ...

(¢,~-¢,~¢j~)(x~x~2x~)

. . . (x~.x~ox~.),

(A.57) 13_*,; o , m , p -

l>=

x~'(6~'¢,~W,~I)

. . . (¢,~6""W,

...

. . . (x~.x~.x~.).

(x~x~x~)

°~)

F o r any possible q u a n t u m n u m b e r H we get H(3,,,) = - 2 H ( O ~ 0 + H T , H ( 1 ..... ) = - H ( 4 W 1) - H ( X ¢1) + H T , H(3_'1)

=

(A.58)

-2H(x¢O +HT,

where

HT=3

~ H(6~)+3 o-=1 ~ H(X¢).

(A.59)

o~=1

F o r H = B , (2, L, we require that on our representation H ( 3 ~ ) - H ( 3 o ) = O , +1, and H(I_~,~)-H(!~.~) = 0, +1, +2. Therefore, we get H(3_,,)-H(3~) = -2[H(g/*)-H(0a)]

= 0, 4-1, (A.60)

H(!~.~) - H ( ! e , ~ ) = - [ H ( 4 ~ ~) - H ( 0 ~ ) ] = 0, 4-1, + 2 . By c o m p a r i s o n we get H(O ~) = H(O) i n d e p e n d e n t f r o m a, and analogously H(X ~) = H(X) i n d e p e n d e n t f r o m o-. T h e result is H ( 3 ) = (3m - 2 ) H ( 0 ) + 3 n i l ( X ) ,

H(3*) = 3mH(O) + (3n - 2 ) H ( x ) ,

(A.61)

H(I_) = (3m - 1 ) ( H ( 0 ) + H ( x ) ) , with the constraint

H(3) +H(3*) = F o r the electric charge therefore 0 ( 3 ) + 0 ( 3 * ) = neutral and that all the that of the 3's. A g a i n we

2H(l~.

we must have 0 ( 3 ) + 0 ( 3 * ) = 0 , +1 and Q ( 1 ) = 0 , +1, 0 and ( 2 ( 1 ) = 0. It follows that all the singlets must be 3*'s have the same electric charge which is opposite to have charge degeneracy.

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163

If in the representation there are states of type (A.47), we take the sequence [3; b l ,

/

tO* ~ / "

I!, 1, bx, b2)

b2)

.vt4~

(A.62)

]-3*; b l - 1, b2+ 1>,

leading again to eq. (A.52). The sequence breaks up if bl = 0, or bl = m. In the case bl = 0 we take I1; 1, O, bz) ~

I-3; 0, b2> ~,, ~x

13"; o, b=>

, 13; 1, b : - 1>,

(A.63)

whereas for bl = m ]-3; m, b2>

, [3*;rn-l, b2+l)~

11;1, m - 1 , b2+1>. (A.64)

Using (A.63) we go back to the original case (A.62) with bl ~ 0, except if b2 = 0. In case (A.64) we get again the eq. (A.55) unless b2 = p - 1. Therefore, we get the other two possibilities: (3) bl = b2---0; k = 2. Notice that this representation of SU(3m + 3 p ) is the complex conjugate of the representation occurring in (2). It follows that the analysis is the same already done. In particular we get charge degeneracy in this case too. (4) b ~ = m , b 2 = p - 1 ; k = 3 m + 3 p - 1 . This is the complex conjugate of the fundamental representation of S U(3 m + 3 n); i.e. the complex conjugate of the case (1). All the states are 13_;m, n - 1>,

(A.65)

I-3"; m - 1, n ) .

This time we have found four exceptional cases. Finally, in case (iv), m, p, l ~ 0 we use the notations 1!; r, bl, b2, b3> = (x~)r(~oddt#)bl(xxx)b2~ b3 , 13; r, bl, b2, b3> = ~ ( X ~ O ) r ( ~ ) b ' ( x x x ) b 2 ~

b3 ,

(r = 0, 1, 2),

(A.66)

(r = 0, 1),

(A.67)

13; bl, b2, b3)-----XX(~ll¢~o)b~(xxx)bz~b3 , I-3"; r, bl, b2, b3> = X ( ~ x ) r ( ~ ) b ' ( X X X ) a 2 ~ 13"; b~, b2, b~> = O~(4,~O0)b~(XXX)~ ~ .

(A.68) b3 ,

(r = 0, 1),

(A.69) (A.70)

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If states of the type (A.67) are present, we take the sequence ~*x

13_;r, bl, b2, b3)

8r,013"; 1 ; bl, b 2 - 1 , b3 + 1) + 8,,d3"; bl, b2, b3+ 1)

x+4, •

b2, b3)

~~t~

(A.71)

&,o[3_; r + 1, bl - 1, b2, b3 + 1> + &.11_3;bl, b2, b3+1) from which H(3)-H(3*)

= H(~)-H(O)

,

H(3*)-H(3)

= H(,~)-H(~)

,

H(3*)-H(3)

= H(6)- H(X) ,

(A.72)

and, by comparison, H(3) = H(3*),

(A.73)

showing that it is impossible to define the baryon number or any other number constant on the 3's (and on the 3*'s). The sequence (A.71) breaks up for r = 0 , b 2 = 0 ; r = 0 , b l = 0 ; b 3 = / ; b 2 = p . Analogously, if the state ]_ 3* ; r, bl, b2, b3> is present in the representation, we can write a sequence which is obtained from (A.71) with the exchange 0 *-~X; it follows that this sequence will break up for r = 0, bl = 0; r = 0, b2 = 0; b3 = l; bl = m. We will call this sequence the conjugate of the sequence (A.71). It is convenient to distinguish the various possibilities in the following way: (1) r = 0 , b a # 0 , b 2 = 0 , b 3 # l ; (2) r = 0 , b l = 0 , b 2 ~ 0 , b 3 # l ; (3) r = O , b l = O , bE=O, b 3 # l ; (4) Vr, bl, bE, b3 = / . (5) r = 0, Vbl, b2 = p, b3 # l (r = 0 is implied by b2 = p).

The analysis case by case goes as follows: (1) r = 0, bl # 0, bE = 0, b3 ~ l. We take the sequence [3;O, bx, O, b3) ~

[3*;O, b1,0, b3> ~

]3_,l, b 1 - l , O , b 3 + l )

(A.74) and we go back to the sequence (A.71) but with r ¢ 0. (2) r = 0, bl = 0, b2 ~ 0, b3 ~ l. W e take the sequence 13;0,0, b2, b3> ~

[3_*;1,0, b E - l , b 3 + l > ,

and we go back to the conjugate of the sequence (A.71) with r ~ 0.

(A.75)

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165

(3) r = O, bl -- O, bE = O, b3 # I. We take the sequence

13; 0, 0, 0, b3) [3-*; O, O, b 3 - 1 )

13-*; O, O, O, b3)

13-; 0, 0, 0, b3 - 1), from which we again get eq. (A.73), except for the c a s e b3 = 0. This case is an exceptional one: (i) r = bl = b2 = b3 = 0; k = 1. This is the fundamental representation of SU(3m + 3p + l) and all the states are 13-; 0, 0, 0, 0),

[3-*; 0, 0, 0, 0),

[13 0, 0, 0, 1).

(A.76)

Continuing in our analysis we have: (4) Vr, bl, b2, b3 = 1. We take the sequence

8 r . O 1 3 - * ; b2, ~ b ll, - 1) "t"~r,113-*; O, b l + l , bE, l - 1 )

[3" ;~, 51, b2, l)

&.o[3-; bx, bE, l - 1) + &,113-, O, bl, b2 + 1, l - 1). This sequence always exists and we again get (A.73). (5) r = 0, Vbl, bz = p, b3 # l. We take the sequence 13-; O, bl, p, b3)

e*---~x13-*; 1, bl, p - 1, b3) (A.77)

, ]3-;1, bl, p - l ,

b3+l)

tO*X

and we go back to the sequence (A.71), except for b 3 + 1 = l, which, however, is case (4). If states of the type (A.68) are present, we consider the sequence 13; bl, bE, b3)

13"; 1; bl, b2, b3- 1)

[3-*; 0, bl, b2, b3+ 1),

and we go back to the sequence conjugate of (A.71), except for the case in which

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R. Casalbuoni, R. Gatto / Search for composite models

t h e original s t a t e has b l = m a n d b3 = l. I n this case we t a k e t h e s e q u e n c e 13_; m, b2, l> ~

13_*;m - 1 , b 2 + l , I ) .

T h e final state can b e r e c o n d u c e d to the s e q u e n c e (A.71) t h r o u g h t h e c o n j u g a t e of s e q u e n c e (A.78), e x d e p t for b 2 + 1 = n. This is a g a i n an e x c e p t i o n a l case: (ii) bl = m, b2 = n - 1, b3 = l; k = 3 m + 3n + l - 1. This is t h e c o m p l e x c o n j u g a t e of t h e f u n d a m e n t a l r e p r e s e n t a t i o n of S U ( 3 m + 3n + l) a n d all the states a r e 13; m, n - l ,

l),

13", m - l ,

n, l ) ,

11, 0, m, n, l - 1 ) .

This c o m p l e t e s o u r analysis for the case of a n t i s y m m e t r i c r e p r e s e n t a t i o n s of S U ( n ) . T h e s u m m a r y of t h e e x c e p t i o n a l cases is s h o w n in t a b l e s 1 - 4 . It is v e r y e a s y to r e a l i z e t h a t in o r d e r t h a t a p r o d u c t of a n t i s y m m e t r i c r e p r e s e n t a tions b e an e x c e p t i o n a l r e p r e s e n t a t i o n , all t h e a n t i s y m m e t r i c r e p r e s e n t a t i o n s m u s t b e e x c e p t i o n a l , e s s e n t i a l l y b e c a u s e if t h e r e is a n o r m a l r e p r e s e n t a t i o n o n e can always d e r i v e e q u a t i o n s of t h e t y p e (A.15) o r (A.73). F u r t h e r m o r e , in t h e p r o d u c t of a n t i s y m m e t r i c r e p r e s e n t a t i o n s w e will find always c o l o r exotics unless we m u l t i p l y an a n t i s y m m e t r i c r e p r e s e n t a t i o n b y a flavor singlet. W e see t h a t t h e o n l y cases w e a r e left with a r e t h o s e of t h e S U ( n ) f u n d a m e n t a l r e p r e s e n t a t i o n a n d of its c o m p l e x c o n j u g a t e , t i m e s an a r b i t r a r y n u m b e r of flavor singlets. C l e a r l y , t h e s a m e a r g u m e n t a p p l i e s to S U ( n ) L ® S U ( n ) R a n d t h e r e f o r e , t h e o n l y p o s s i b l e r e p r e s e n t a t i o n s will b e (n, 1), (1, n) a n d (_1, n*), (n*, 1).

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