Mass without scalars

Nuclear Physics B155 (1979) 237-252 © North-Holland Publishing Company

MASS WITHOUT SCALARS Savas DIMOPOULOS Physics Dept., Columbia University, New York, N Y 10027, USA Leonard SUSSKIND Theoretical Institute, Physics Dept., Stanford University, Stanford, CA 943"05, USA Received 20 February 1979

We attempt to show that fundamental scalar fields can be eliminated from the theory of weak and electromagnetic interactions. We do this by constructing an explicit example in which the scalar field sectors are replaced by strongly interacting gauge systems. Unlike previous examples, our present work gives a natural explanation for fermion masses. The cost is a significant expansion of the size of the gauge group.

1. Introduction It has been argued that the presence o f scalar fields in unified weak-electromagnetic theories requires certain fundamental parameters to be adjusted to absurd precesion [1,2] .. Indeed it seems that to establish a hierarchy of mass scales, beginning at the Planck mass (10 Is GeV) and ending ar ordinary particle masses requires fundamental unrenormalized masses to be adjusted [1,2] to 30 decimal places! Perhaps in some future theory such adjustments will appear natural, but at present divine intervention is the only available explanation. F o r this reason we have sought an alternative mechanism for generating fermion and gauge boson masses in which no fundamental scalars are ever invoked. In a previous paper [2], one o f us showed how the generation o f intermediate vector boson (W e , Z) masses could arise without fundamental scalars or unnatural adjustments. However., this paper offered no explanation for the origin o f quark and lepton masses. It is the purpose o f this paper to fill that gap. We will present some examples which we argue are capable o f generating a set o f mass scales with realistic orders o f magnitude for both fermions and bosons. The examples in their present form are too simple to take completely seriously. We offer them as an "existence p r o o f " for a family o f theories which can produce reasonable scales without unnatural adjustments. We also feel that the examples offer valuable clues to the various mechanisms which m a y be needed. 237

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Before entering into a technical discussion of models, we must warn the reader that we will deal with complicated systems of interacting gauge sectors, some of which are strongly interacting. Thus we can only guess the patterns of spontaneous symmetry breaking which occur. Indeed our guesses are modelled on the usually assumed chiral behavior of QCD which is incompletdy understood [3 ]. The reader should keep in mind that by quark mass generation we mean "currentmasses" which are present in the absence of ordinary QCD strong interactions.

2. Technicolor In this section we will review the ideas of ref. [2]. There the standard SU(2) × U(1) theory of weak and electromagnetic forces in the absence of fundamental scalar fields was considered. The ordinary quarks and leptons are minimally coupled to the gauge fields We, W° , B. In this paper we ignore the usual strong interactions and color degrees of freedom which can easily be restored. To replace the scalar field sector, a new family of massless fermions called '~echniquarks" are introduced. The T-quark fields from electroweak doublets and N-tuplets under a new "Technicolor" group SU(N)Tc. They are also assumed to be color singlets. The T-quarks interact v/a an unbroken SU(N)Tc gauge interaction which we assume behaves in a manner which parallel QCD. The main difference is that the QTD running coupling becomes strong at a mass of order I TeV instead of 1 GeV. Thus the scale on which T-hadrons exist is 103 times heavier than ordinary hadrons. The T-color binding forces generate a spontaneous breakdown of chiral flavor SU(2) X SU(2) leading to the existence ofmassless T-pion Nambu-Goldstone bosons [3]. Now in ref. [2] it was shown how these massless T-pions replace the usual fundamental scalar fields in producing a mass matrix for the intermediate vector bosons. The graphs responsible for the shifts of mass and mixing of the boson fields are shown in fig. 1. The resulting mass matrix is identical to that in the Weinberg-Salam theory [4] except that the vacuum expectation value of the scalar field is replaced by F~r, the technicolor analog of the pion decay constant. Its value must be 250 GeV in order to give the Z, W± their expirical masses. This represents a rescaling of QCD by a factor of ~2000. The most interesting point about this mechanism is that it provides a simple explanation of the observed neutral current behavior. In general, the consistency of the Weinberg-Salam theory requires the scalar field sector to have SU(2) × U(1) symmetry only. The larger SU(2) × SU(2) symmetry of the T-quark sector insures a remaining unbroken T-isospin invariance after, spontaneous breakdown. The resulting equality of the F~r's for the neutral and charged T-pions gives the boson mass-matrix the same special form as in the usual theory [4]. Thus cos OwMz = M w ,

(2.1)

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W

~/

239

x

T-PION

W

B

r/

"

B

Fig. 1. Origin of W-S mass matrix.

where 0w is the usual weak angle. The reader may wonder about the origin of the enormous mass scale ~1 TeV. In particular, why is the ratio of scales o f QCD and QTD of order 103? To understand this, let us recall some facts from the renormalization group. For an asymptotically free theory, the mass scale at which the interaction becomes strong is given by [2,5] m = k exp

c

gg,

(2.2)

where k is the cutoff andgo the bare coupling. It is well-known [6] that in QCD if k is of order the Planck mass andgo ~ the electric charge then m ~ 1 GeV. Now suppose that a second sector exists with a somewhat different value of c/gg. The nature of eq. (1.2) is to magnify a small difference in c/g~o into an enormous difference in mass scales. Thus our expectation is that the strongly interacting sectors of a theory will have very different scales. In addition to generating gauge boson masses, the scalar fields in the WeinbergSalam theory are also responsible for the fermion mass matrix. Unfortunately, the T-sector as described in ref. [2] is not capable of replacing the scalar fields for this purpose. In order to see why this is so let us consider the simple case of a single doublet o f ordinary cluarks interacting with the above system through its SU(2) × U(1) charges. The system has an Abelian 3's symmetry * in the absence o f strong interactions. q -~ exp i07s q •

(2.3)

• We emphasize that by quark masses we mean current algebra masses which are present in the absence of ordinary strong interactions. Note also that electroweak instantons give minute masses to the NGB of the U(1) axial symmetry (2.3).

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This same symmetry is present in the Weinberg-Salam theory if the Yukawa couplings of the scalar and Fermi fields are absent. The symmetry in eq. (2.3) can either be realized algebraically or in the NambuGoldstone mode. In the first case the quarks remain exactly massless while in the second a massless *l-like boson would result. Either possibility is unacceptable. Actually the weakness o f the SU(2) X U(1) couplings make it very unlikely that the symmetry is spontaneously broken so that massless quarks are almost certainly the result. The purpose o f this paper is to argue that the above situation is not inevitable. We would like to construct a model with the following features. (i) Naturalness: no parameter needs to be adjusted to unreasonable accuracy. This presumably means the elimination o f scalars *. All mass scales should emerge from infrared instabilities of themassless theories [2]. (ii) Masses for the intermediate vector bosons o f order 100 GeV with the usual relationship between M z , Mw sin 0w. This means that the dynamical symmetry breaking mechanisms must have SU(2)L × SU(2)R flavor symmetry [2]. (iii) Dynamically induced quark and lepton masses. Furthermore it should occur naturally that these particles should be 1 0 - 2 1 0 - 3 times lighter than Z and W-+. (iv) No massless or almost massless Nambu-Goldstone bosons (NGB) should exist in the observable spectrum o f particles [5 ]

3. General hypotheses In this section we will formulate some general hypotheses about dynamical symmetry breaking. These hypotheses are suggested by analogy with the current theory of strong interactions: QCD. There, although also unproven, the assumptions about chiral symmetry breaking are supported b y empirical evidence [7]. For the sake of definiteness we consider an SU(N > 2) gauge theory with n left-handed fermion fields l / (i = 1, 2 ..... n) where a labels the fundamental representation of SU(N) and n right-handed fermions r~(j = k ..... n) **. In addition to the gauge symmetry the theory has an SU(n) X SU(n) "spectator" global chiral symmetry. (By spectator symmetry we mean that it commutes with the * In globally supersymmetric theories scalars are protected from receiving large masses. If supersymmetry is broken at a mass scale ~1 TeV then scalar mass scales ~100 GeV are not unnatural. If however, the breaking of supersymmetry occurs at a much higher scale, say ~ 1012 TeV, then absurdly unnatural adjustments are again needed to ensure scalar scales ~100 GeV. ** A non-trivial constraint in all of the models that we discuss is the absence of anomalies in the strongly interacting gauge sectors. For future reference note that the 5 and 10 representations of SU(5) have equal and opposite anomalies. This follows because, together with the singlet of SU(5), they constitute the 16-dimensional representation of the anomaly-free group SO(10). See ref. [10l.

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gauge interaction under consideration. At some later stage we may want to gauge some part of a spectator symmetry.) At the scale M when the coupling constant becomes large, non-perturbative phenomena occur. We shall assume the following hypotheses for the non-perturbative behavior. (i) Resistance to self-breaking: the non-Abelian gauge symmetry remains unbroken. The phenomenon of confinement occurs. The phsyical states are gauge singlets. (ii) Bilinear condensation: following Nambu and Jona Lasinio [3] we assume that the vacuum state rearranges by the formation of a condensate of pairs. The condensate is of the form

0.

(3.1)

(iii) The condensates forms in such a way as to give equal masses to n Dirac fermions. This means that.f/j is a unitary matrix. (iv) If the spectator symmetry or any part of it is gauged with a sufficiently weak coupling then the above pattern does not change. However those NGB's which correspond to gauged broken generators become the longitudinal components of massive spectator gauge bosons. The above hypotheses which are wellknown for QCD become more subtle when the strong group is SU(2). In this case the 2n chiral fermions can all be considered left-handed. This is because the antiparticle representation of SU(2) is unitarily equivalent to the particle representation. Accordingly the theory contains 2n left-handed fermions in the fundamental representation. Therefore the theory has an SU(2n) spectator symmetry instead of the SU(n)L × SU(n)R symmetry. Note that the U(1) symmetry is destroyed by the Adler-Bell-Jackiw anomaly. In analysing the SU(2) model we shall assume that the hypotheses (i)-(iv) generalize. Using a notation in which we have n left-handed and n right-handed fields, the condensate can be taken as (liff r i ) 4: O.

(3.2)

The fields r / may be replaced by their n left-handed antiparticle fields. Thus let i run from 1 to 2n and define (where c means charge conjugation) l ~ n = ea~(r~) i .

(3.3)

Thus the condensate in eq. (3.2) becomes i (ea¢ l~,(1) I~n+i ( 2 ) ) 4 : 0 ,

(3.4)

where the notations (1), (2) indicate upper and lower spin components of the lefthanded Weyl spinors. The expression in eq. (3.4) may be written in the compact form

(3.5)

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where the matrix B is defined by,

[o

~ - n --+

n=

+ - n ---~

(3.6)

-I The choice of condensate is ambiguous up to an SU(2n) transformation of the fields l~ under which n -~ UTrlU.

(3.7)

The subgroup of SU(2n) that leaves the copdensate invariant is the unitary symplectic group Sp(2n). This is to be contrasted with the situation when the strong group is SU(N > 2). In that case the spectator SU(n) X SU(n) breaks down to SU(n). In the SU(2) gauge theory an additional case is possible. Namely, the total number of chiral Weyl fields may be odd, equal to 2n + 1. (Anomalies forbid this for N > 2.) In this case the 2n + 1 left-handed Weyl fields cannot completely pair to form massive Dirac fields. The simplest assumption is that one field remains massless and the remaining 2n fields behave as above. That is they pair to form n massive Dirac fermions, thus breaking the spectator symmetry from SU(2n + 1) down to Sp(2n). One massless confined chiral field remains. Consider now the situation where a subgroup of the spectator group is gauged with a weaker coupling. According to hypothesis (iv) the symmetry breaking condensate persists. The orientation of the condensate in spectator space is not in general arbitrary. This is because the weaker interactions may define a direction in spectator space. The remaining freedom in the condensate orientation is the subgroup of the spectator group not explicitly broken by the weaker interactions. Determining the condensate is in general a difficult strong interaction problem [5,8]. Happily this problem does not arise from certain interesting special cases. For example if the weaker gauge group is the entire spectator group the orientation is arbitrary. Another example arises in the SU(N > 2) gauge theory if the weaker gauge group includes SU(n) L or SU(n)R. In both cases the (assuming hypothesis (iii) the weaker gauge group is sufficient to rotate the condensate to a standard form.

In general the condensate may break the @ectator symmetry down to a subgroup of the weaker gauge group. In this case the broken generators acquire a mass by eating the corresponding Nambu-Goldstone bosons. The subgroup of the weaker gauge group which is left invariant by the condensate will behave at lower energies like an unbroken gauge theory. The non-Abelian components may become strong and confining at a lower mass scale. Thus the process may repeat itself. In the following sections we shall apply these techniques to some primitive models of fermion and W-boson mass generation.

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4. A model In this section we will describe a model which we believe satisfies the four requirements listed in sect. 2. The model contains the following degrees of freedom. (a) The usual SU(2) × U(1) electroweak gauge bosons W~, B u. A is a weak isovector index. (b) A multiplet of Dirac fermions Qi. The index a is an electroweak doublet index. The index i labels another internal space called extended technicolor (ETC). In the present example ETC is an SU(3) space with i indexing the 3 representation

2

2

(4.1)

(qaJ

After spontaneous breakdown of ETC Q~ and Q~ will become techniquarks and qa an ordinary quark doublet. (c) ETC gauge bosons E~ w h e r e / i s an ETC adjoint index. (d) Gauge bosons L x and R y of two new strong SU(2) gauge groups SU(2)(L) and SU(2)(R). The generators are called left-spin and right-spin. The subscripts X, Y label L and R spin adjoint representations. (e) Chiral Weyl fermions £/x, ryi where £ and r are left- and right-handed. They are both ETC triplets and L or R spin doublets respectively. The usual minimal couplings between gauge bosons and fermions are assumed. This history has no triangle anomalies in the L, R and ETC sectors. Anomalies in the electroweak sector can be eliminated in one of two ways. The charges of the quark doublets can be chosen -2 +! or additional multiplets can be invoked. The coupling constants of the theory are gl, g2, gE, gL, gR corresponding to the electroweak SU(2) × U(1), ETC, L and R gauge groups. They can be defined as their running values at some mass scale. It is convenient to choose this scale to be larger than the largest symmetry breaking scale appearing in the models (>~100 TeV). The couplings gL, gR will be chosen equal and large enough so that they induce strong interactions at a scale ~ I 0 0 TeV *. The ETC coupling is somewhat smaller so that it becomes large at a few TeV. The entire structure is summarized in the diagram of fig. 2. The lines joining fermions to gauge groups indicate non-trivial transformation properties. The particular representation content of a fermion under a particular group is indicated by an integer labelling the bond. The next few sections explain how we believe this complicated system works.

* 100 TeV is a buzzword for a new mass scale much bigger than 1 TeV.

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21. _ ~ ~

su~(ETC)~ 3

2 "----~r

Q •

2

SU2 xU1 (E.W.) Fig. 2. A simple model of mass generation. 5. The strongest sector In this section we will consider the isolated system of £ fermions and L gauge bosom. (The considerations for the right sector are completely parallel.) The isolated L sector consists of L gauge bosons and 3 L spin doublets £%. We assume the couplingg L becomes large at about 100 TeV. At this scale a condensate forms. In particular (@~) = e x x , ( l i x ( 1 ) / ~ , ( 2 ) - / ~ ( 2 ) / ~ , ( 1 ) ) :/= O,

(5.1)

where 1 and 2 indicate the two spin components of the Weyl spinors. The condensate @~ transform under the spectator SU(3) group as a 3. Without loss of generality we may assume

<¢1,2> ~ o ,

<~,a> = <,[,3> = o .

(5.2)

That is, the 1 and 2 components pair to form a massive Dirac fermion and £3 remains massless. Here we must digress to discuss the fate of the massless field l 3. We shall assume that the strong L gauge group confines all non-singlet states. Thus the object £3 is not found in the physical space of states. Evidently, the surviving spectator symmetry is the SU(2) subgroup which leaves l 3 invariant. Five generators of the original SU(3) are broken. This results in the appearance of 5 NGB's. Note that the apparent global U(1) symmetry l i "~ e i° l ~

(5.3)

is broken by triangle anomalies. The right-hand sector R behaves in an analogous fashion. In particular ifg a = gL

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there exists an exact discrete symmetry L ~ R. The internal SU(3) directions of the condensates eL and CR are entirely arbitrary and will only become correlated when SU(3) is gauged. At that point we shall assume that the SU(3) dynamics aligns these directions.

6. ETC Now consider the SU(3) gauge interactions of the ETC group. The couplingg E is smaller than gL,R. Therefore it makes sense to treat ETC as a perturbation in the 100 TeV region. The invariant SU(2) subgroup of ETC plays the role of ordinary T-color. Renormalization effects will increase the strength of the SU(2)T c gauge coupling as the mass scale dimenishes. Thus at about 1 TeV the SU(2)TC becomes a strong interaction.

7. Quarks The quark multiplets

Ii

L'R1

qa

kUL,RJ

(7.1)

L,R .J

interact via their vector SU(3)ETC charges. When SU(3)ETC breaks down to SU(2)T c the T-quarks Q~,2 become strongly interacting. The quark q, being an SU(2)TC singlet communicates with the TC sector very weakly. This is because the 5 broken generators of SU(3) are extremely massive (~100 TeV). Thus it is appropriate to first study the isolated TC sector consisting of the three gauge boson E 1'2'3 and Q~,2. According to the discussion in sect. 3, the new spectator symmetry is SU(4), The four chiral components are

U=

,

U2 L

D=

,

ID2J L

a

~ ' ~ T. (7.2)

The generators of this SU(4) are 1, oi, ra, oi " ra where x changes U's to D's and a changes U to UC's and D to D O's. When the SU(2)TC forces become strong at a

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TeV, a condensate forms. Up to an SU(4) rotation the condensate has the form (UeU + DCD) 4= O .

(7.3)

The condensate can be rotated in two different ways: baryon conserving and baryon violating. The subgroup of SU(4) which conserves baryon number is the SU(2) × SU(2) X U(1) group generated by x and, xo3, and 03. The remaining baryon number violating generators are xol, xo2, O1, 02. We wish to show that the condensate (7.3) is an extremum of the energy when the electroweak interactions are turned on. Thus consider the eight-dimensional space spanned by the baryon number violating generators of SU(4). When there generators act they rotate the condensate changing the electroweak energy. This change in energy is easily shown to ~oe independent of the direction in the eightdimensional space of baryon violating condensates. All the remaining rotations of the condensate can be accomplished by using the electroweak SU(2) X U(1). Thus we conclude: (a) the condensate (7.3) lies on a trajectory of extrema; (b) the only ambiguity in the choice of the condensate is the usual SU(2) X U(O.

8. Pseudos In this section we will discuss the various Nambu-Goldstone bosons of the spectrum before the electroweak interactions are turned on. These correspond to the five generators broken by the condensate (7.3). Three of them have the quantum numbers of the pion and two of them have diquark quantum number UD, UCD c. When the electroweak interactions are turned on the diquark massless bosons acquire a finite mass of order ~ TeV ~ 100 GeV. Note the couplings of the broken ETC generators also give masses to the diquark bosons. This contribution to the mass is small. The three pion-like bosons are eaten by the weak vector bosons as in the standard technicolor picture, giving the usual pattern of masses to the W-+ and Z. The photon remains massless. Let us consider the 10 NGB's associated with the condensates ~L and t~R. Five of these bosons correspond to SU(3)(L+R) (generated by the sums of left and right spin). These give mass to the corresponding 5 ETC gauge bosons. The symmetries generated by the differences of L and R spin are explicitly broken. This has two effects of interest. First it gives mass to the remaining 5 NGB's. We expect this mass to be of order gE X 100 TeV. Secondly, the relative orientations of the condensates q~L and t~R become fixed. It is easy to see that parallel orientations in the ETC space are extrema of the energy.

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qL

QL

QR

247

qR

Fig. 3. The mass feed-down mechanism.

9. Quark masses In this section we describe a mechanism by which heavy techniquarks feed down masses to the light quarks. This mass feed-down mechanism is effected by the heavy (~100 TeV) generators that connect light quarks to techniquarks. To see how this happens consider the graph of fig. 3. The cross X indicates the chirality mixing caused by the condensate. The magnitude of this mixing is characterized by the 1 TeV mass scale. When loop momenta q2 much bigger than (I TeV) 2 flow through the internal fermion propagator of Q this soft chirality mixing goes away and the graph of fig. 3 vanishes. Thus the 1 TeV mass scale serves as an ultraviolet cutoff which renders the graph of fig. 3 finite. Since m E ~ 100 TeV > > mQ ~ 1 TeV, we can compute the corrections by approximating the ETC gauge boson exchange by a 4-Fermi type coupling with a loop cutoff of order 1 TeV ~ mQ. Thus we find

jYQ

mq

87r2M2o

d41 12 + m ~ mQ

g2

E 8ff2 m 2 ,

fg.:)

whereg~z/8n 2 is the coupling of the heavy ETC bosons and is expected to be somewhat smaller than 1. As stated in sect. 2, in general, the ratios of different strong interaction scales will be very large. For example, ifMQ ~ 1 TeV andME ~ 100 TeV we obtain mq ~ 100 MeV.

10. The problem of vertical splittings The previous model demonstrated the possibility of generating quark masses. This model however can not explain the large vertical splittings such as u-d, c-s, b-t and e-u,/a-t,. To be sure, the model would generate electroweak mass differences but these typically of order a. In fact this is a general problem. It is caused by the vertical SU(2) X SU(2) symmetry of the world without electroweak interactions. In order to consistently introduce electroweak interaction we only need a theory with

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global SU(2) X U(1) invariance. However, the global symmetry of the techniquarks (which are responsible for the W-+ and Z masses) has to be at least SU(2) X SU(2). This is necessitated by the successful neutral current phenomenology and in particular the relationMw =Mz cos 0 w [2].

11. A model of leptons In this section we construct a model with the following desirable features. (i) The light fermions exhibit vertical splittings. In particular the neutrino remains massless while the electon gains a mass. (ii) The heavy T-lepton world remains SU(2) X SU(2) symmetric thus guaranteeing a standard Z, W-+ mass matrix. The ETC group for this model will be SU(5). The breaking pattern needed is SU(5) --> SU(4). In subsect. 11.1 we shall show how this can be done by using the principles of sect. 3. 11.1. Breaking o f ETC

Consider a strong group SO(10)(L) or any other group (SO(4N + 2), N > 1) which is anomaly free and has complex representations [9]. Introduce chiral fermions £,, and r ei which belong to a complex representation of SO(10)(L) (index a). £a's are SU(5)ETC singlets while r/'s are SU(5)ETc quintets. Since ct labels a complex representation of SO(10)(L) the global spectator symmetry of the theory is SU(5) X U(1) and not SU(6). At the scale ~100 TeV when the SO(10)(L) forces become strong the following condensate forms: ( ~ ~a) 4= O.

(1 1.1)

This breaks the gauge SU(5)ETC symmetry down to SU(4)Tc. In order to avoid anomalies in the SU(5) sector we introduce a completely parallel strong SO(10)(R) sector with fermions re and £/a and a condensate (Va l~) 4= O.

(11.2)

When the condensates (11.1) and (11.2) align (L ,e, R symmetry) the desirable breaking pattern occurs. Another way to obtain the same breaking pattern is to replace the strong SO(10)(L) by an SU(5)(L) with chiral fermion content: £a, r~, r ~ , ~i t3. a and a3 label the 5- and 10-dimensional representations of SU(5)(L) and i labels the fundamental representation of SU(5)ETC- The condensates are: ( ~ r ~ 4= 0 and (F~t~l~/3) 4= 0. To avoid SU(5)ETC anomalies we introduce an SU(5)tR) sector with fermions: ra , £9, ~ , r/~.

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The condensates are (k-aft/) ve 0 and (-i-aflr~fl) 4=O. The desirable s y m m e t r y breaking occurs when all condensates align * The spectrum o f the strong L and R sector consists o f some massive ~ 1 0 0 TeV fermions together with massless and confined fermions and vector bosons. F o r the case when the strong groups L and R are SO(10) the model is illustrated in fig. 4.

11.2. Light spectrum o f the model The fermion content o f the model includes the following SU(5)ETC multiplets: "N 1

"0 NI2 N13 N I 4 NI5 ~

N2

0

N3

N23 N24 N25

,

O

N4 V

N34 N35 0

L,5

N4s 0

El 1 E2

,

R,IO

/~ /~

E3 [

,

L,5

e

R,5

The multiplets NiL, EiL and EiR are 5's under SU(5) while Ni/R is a 10. The notations are intended to suggest neutrino and electron. The left-handed multiplets form weak isospin doublets and carry weak hypercharge - 1 . The right-handed multiplets are isospin singlets. The multiplet E R has hypercharge - 2 while NRlO is neutral. The SU(4) content o f the fermions is 4 quartets: Ni, Nis, Ei, L, El, R , (i = 1 - - 4 ) , 1 sextet: Ni,], (i, ] = 1 - 4 ) , 3 sing,lets: v, e L, eR. * There are very many interesting patterns of symmetry breaking that can be realized dynamically by using the principles of sect. 3. Consider, for example, a strong SU(2) sector (Greek indices) and the chiral fermion doublets ~a transforming like the 10-dimensional representation of a weaker SO(10) gauge group. This model has no anomalies. Furthermore the condensate (e~flla(l) l~2) - (1 ~ 2)) ~ 0 breaks SO(10) down to SU(5). This is what happens in the first stage of breaking in the recently proposed Georgi-Nanapoulos unifoed model [ 10a]. Another academic exercise for the reader is to find strong sectors for the breaking of SU(5) down to SU(1) × SU(2) × U(1) which occurs in the Georgi-Glashow model [10b]. Notice that, in contrast to our examples in the text, the breaking SO(10) ~ SU(5) was aone through only one strongly interacting sector.

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250

S0~o(L)

10111

/

S010(R)

SUs(ETC)

EL

NL

ER

NR

SU2 xUI(E.W. )

Fig. 4. A model of leptons.

We assume that the non-Abelian SU(4) sector becomes a strong interaction at a scale ~1 TeV. The corresponding spectator symmetry of the four quartets and the sextet is SU(2) X SU(2) X U(1). The U(1) symmetry multiplies the quartet fields by a a phase and the sectet fields by another phase. The SU(2) X SU(2) chiral symmetry is the one needed to ensure the correct Z, W± mass matrix. Note that this full spectator symmetry is not exact. It is violated by the heavy broken ETC couplings. The only exact symmetry is SU(2) × U(1) electroweak. The violations are very small. Their magnitude is of order (MQ/ME)2 and they can be treated as small perturbations. By an SU(2) × U(1) electroweak rotation the condensate can always be brought in the form: (Ni5 ,RNiL ~-EiREiL) ~ 0 . This condensate yields the correct vector boson mass matrix. There are no pseudoGoldstone bosons left over. The lepton spectrum is as follows: the quartet N's and E's acquire constitituent masses of order 1 TeV. The sextet of Nifs remains massless and confined. The neutrino v L remains massless because there is no SU(4) singlet right-handed N particle that it can mix with. The electron acquires a mass by the mechanism of

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sect. 9 *. The model in general contains a dangerous anomaly in the hypercharge current arising from the strong ETC sector. The anomalies divergence is proportional to

2 au Yu ~ gEEuv E~u v { ~ Y(i) C2 (i) - ~ yq) C2 q ) } , L R where L, R means sum over the left (right) SU(5) multiplets of fermions, Y(i) is the hypercharge of a multiplet and C2 is the quadratic Casimir operator for the representation. With the standard Y assignments corresponding to an electrically uncharged neutrino, the anomaly vanishes.

12. Conclusions In this paper we have exhibited a class of models with dynamical symmetry breaking which induces both fermion and vector boson masses of reasonable magnitudes. This was accomplished without the appearance of true Nambu-Goldstone bosons or other unwanted light objects. Unlike the standard theory no unnatural adjustments need to be made. The models, as presently constructed are unrealistic on several accounts: the "XEROX" replication of e,/1, ~"... is not accounted for, color has not been incorporated. CP violation is absent. However, we feel that there is sufficient richness in this approach to include all the desirable features. The xerox duplication, for example, can be added onto the ETC sector ** We wish to thank Ken Lane for stimulating conversations. We are also indebted to J. Bjorken for helpful comments.

References [1 ] K. Wilson, private communication. [ 2] L. Susskind, Dynamics of spontaneous symmetry breaking in the Weinberg-Salam theory, SLAC-PUB-2142. [3] Y. Nambu and G. Jona Lasino, Phys. Rev. 122 (1961) 345. [4] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; * Had we not changed the strong L and R sectors of this model from SU2 to SO(10) (or SU(5) the breaking pattern would have been SU(5)ETC --~Sp(4)TC. Since Sp(4) has only real representations [9] it is easy to see that neutrino would become a Majorana fermion with a mass of order 1 eV. A Majorana neutrino would mediate double beta decay with an affective Fermi coupling of order G2mvAM. Here GF is the usual Fermi coupling, mv is the neutrino mass (~1 eV) and AM"the Q-value for the process. ** We thank Ken Lane for a conversation on this point.

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[9 ] [10]

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