An attempt at realistic supercompositeness

Nuclear Physxcs B261 (1985) 633-650 © North-Holland Pubhshmg Company

AN ATTEMPT AT REALISTIC SUPERCOMPOSITENESS A MASIERO1 CERN, Geneva, Switzerland

R PETTORINO Dtpartlmento dt Ftswa, Unwerstth dz Napoh and INFN, Sezwne dt Napoh, Naples, Italy

M RONCADELLI Dlparnmento & Fistea Nucleare e Teonca and INFN, Unwerslth dz Pavia, Pavta, Italy

G VENEZIANO CERN, Geneva, Swttzerland

Recewed 17 June 1985

Composite models of quarks, leptons and Haggs bosons based on softly broken supercolour are both predlctave and capable of giving some of the desired mass hlerarchaes In the SQCD examples we have explored, however, masswe neutnnos (m~ - me) and hght spin-~ leptoquarks (mlq _<1 GeV) can only be avoided at the price of introducing further elementary particles (spectators)

1. Introduction It is possible that we have discovered i n the usual q u a r k - l e p t o n fanulles the u l u m a t e c o n s t i t u e n t s of matter a n d m the gauge bosons of S U ( 3 ) c × SU(2)L X U ( 1 ) y the basis of their interactions. It looks m o r e hkely, however, that, as has b e e n the case before, new structures will emerge w h e n scales smaller t h a n those yet tested, say 10-16 cm or less, are explored If thts should t u r n out to be so, one would be left with the puzzle of explaining the smallness of the dimensionless q u a n U t y m q , £ . rq,£< 10 -6, together with the fact that the p r e s e n t l y " o b s e r v e d " n o n - g a u g e e l e m e n t a r y particles have spin ~. t It has b e e n a n 1 Permanent address Istatuto da Flslca, Umvers~thdl Padova, and INFN, Sezlonedl Padova, Italy 633

634

A Maslero et a l /

Supercompostteness

excmng, though frustrating, adventure to look for confining gauge theories that predict such a spectrum In a recent paper [1], two of us have argued that confining supersymmetnc gauge theories (supercolour or SC), endowed with soft SUSY-breakmg terms, can be promising candidates for such preomc theories. A few examples supporting this contention were given. In this p a p e r we report on a (non-exhaustwe) search for a phenomenologlcally consistent model of tlus type for one family of composite quarks, leptons and Hlggs particles. Unfortunately, the best model we are able to come up with Is still unrealistic, it is ruled out (unless ad hoc spectators are added) by its prediction of some new hght fermlons We wish nonetheless to present at for the following reasons: (a) It shows the predictwe power of supersymmetnc composite models [2]. (b) It exhibits the multiple protecnon mechanism for quark and lepton masses suggested in ref [3]. (c) It can remove from the hght spectrum the supersymmetnc partners of quarks and leptons (d) It exhlbxts dynamical SU(2)L × U ( 1 ) r breaking with a technlcolour [4] scale G~ t/2 which is related (but not identical) to the supercolour composlteness scale A SC"

(e) It is able to provide small, calculable masses for quarks and leptons without m v o k m g new ("extended technlcolour" [4]) interactions. (f) It has automatic strong CP conservation at the price of an "invisible" axlon

[5] The phenomenologlcal problems of the model reside in the fact that its fermlomc content is ncher than that of the usual model Furthermore, the mass protection mechanism, being too democratic, tends to give the neutrino a mass O(me) and some leptoquark fermaons ( Q = + ~, B = + I, L = g l objects) with masses O(1 GeV) In order to avoid ttus problem one can only revoke "spectators", 1.e, elementary SC slnglet fields that are able to form large Dlrac masses with the leptoquarks and restore an almost massless neutnno These modifications of the model look rather contrived, especially for the spectator leptoquarks Although we have not been able to avoid these hght states m a natural way, we do not thank that there is a general no-go theorem in th~s respect We hope that, by a more judicious choice of the gauge group a n d / o r of the preon quantum numbers, this problem can be circumvented, possibly within a scheme incorporating famlhes. This brings us to the second disappointing feature of our construction' farmhes are not predicted, although there are straightforward ways of incorporating them by adding a fanuly label to some of the preons The situation with respect to honzontal symmetries may turn out to resemble that of flavour m Q C D where confinement and hadron composlteness do not lead to much " e c o n o m y " . flavour has to be added at the consntuent level.

A Mastero et al / Super~ompostteness

635

In the following sections, we shall present our specific model with this outhne (A) Definmon of the model, through specifications of the gauge group and the preomc quantum numbers (sect 2) (B) Discussion of the model at three successive levels of approximation (1) level I, 1.e., in the absence of supersymmetry breaking and of "weak" gauging (sect. 3); (11) level II, 1.e., in the presence of SUSY breaking but still w~thout weak gauging (sect 4), (in) level III, i.e., after weak gauging and R-symmetry breaking (sect 5) (C) Finally, we show how to free the model of its main phenomenologlcal problems by adding spectators (sect 6) and give some conclusions (sect 7)

2. Definition of the model

The model we consider is of the type originally proposed in refs [6]. It 1s essentially the specific model of ref. [3] supplemented with a specific set of soft (mass) terms through which SUSY breaking is introduced [1] The preon binding supercolour group is taken to be SU(6)s c to which a standard " w e a k " gauge group O w = SU(3)c X SU(2)L X U ( 1 ) y

(2 1)

Is added. All gauge particles are elementary both m SU(6)s c and in G w Supersymmetry will be broken at tree level in G w (through glulno, photlno, etc mass terms), but only radlatlvely in the SU(6)sc gauge sector. The preon quantum numbers are given in table 1 and correspond to an anomaly-free theory with asymptotically free (and hopefully confining) supercolour. The model is not completely specified until one gives the explicit form of the superpotentlal and of the soft SUSY-breaklng terms For pedagogical reasons, we postpone the definitions and discussion of these terms until we study the theory at level II. For the time being we will just recall that, m the absence of mass (or any other superpotentlal) and of weak gauging, tins model has a global symmetry G ~ = O F × U(1)A= SU(6)L X SU(6)R X U ( 1 ) v X U ( 1 ) x X U(1)A ,

(2 2)

in which the U(1)~ factor Is SC-anomalous, and the non-anomalous U(1)x is an appropriate combination of axial U(1)'s that does not commute with SUSY (R-symmetry) We recall (see, for example, ref [7]) that, in this particular case, the

636

A Masteroet al / Supercomposlteness TABLE 1

The SU(6)sc, SU(3)c, SU(2)t"U(1)y and O(1)e m quantum numbers of the preomc superfields (4, ~) and of the composite superfields(4~) Transform er

SU(6)sc

4~ 23 44

SU(3)c

SU(2)I_

U(1)r

Q=

1

+ 5Y

T3

6 6 6

3 1 1

1 1 2

13 1 0

6 2×6 6

3 1 i

1 1 1

13 -1 1

1

3

2

I

1

2

-I

o

~ ~ 4.1 2 3 ~

I

3

i

-~

-](

4al 2 3'~ff 4 ~4~5 4~2 e+ 4 ~~ 424g lq - 4,",41,2 3 17:t= 4~23~,~ Oct = 4~ 2 3 ~ 2 3

1 1 1 1 I 1

3 i 1 3 3 8

1 1 1 1 1 1

456

~['23 4,~5 ~' (d)

(:)

a¢ 695a ~1 2

= ~

= 4 : ,o-o q~,~

d=

v ~- ~

-

__

3

4

~

l

3

-'2 12 3

1 --3

i

23 0 2 ~ ~ 0 4

/

~3 0 +1 2

2 0

1 family (with va)

B=t3, L= - 1 B= 3~' L = I B=L=O

R-charge is zero for all the scalar fields and equal a n d opposite for the gaugino a n d the m a t t e r fermlons [our c o n v e n t i o n will have q x ( q ' ) = - q x ( X ) = 1]

3. Level I: the supersymmetric m a s s l e s s limit W e shall first analyze our model m the h n u t m whach supersymmetry ts exact a n d weak gauge couplings are turned off The idea is that such a limit should already give us an idea of the fmal spectrum at least all the hght degrees of freedom should be p r e s e n t i n that crude approximation, while states of mass O(A so) can be d r o p p e d from further discussion Of course, we can only argue m d u s way if G w m t e r a c t m n s are small at distances O ( A sc) -1 a n d If the S U S Y - b r e a k m g mass parameters are chosen to be small with respect to A sc A t this level, our model reduces to the m u c h - s t u d i e d case of S Q C D with a n Identical n u m b e r of (super)colours a n d flavours ( N - Nf = 6 m our case) Various a r g u m e n t s [7,8] a n d m s t a n t o n - t y p e calculauons [9-11] have led to an essentially

A Mastero et a l /

Supercompostteness

637

complete understanding of the non-perturbatlve vacua of this theory and of ~ts low-energy exotatlons. We clmm that they are both given consistently by an effectwe lagranglan generahzmg slightly the one of ref. [7] and contalmng the following N z + 3 chiral superfields. T , j = * , , , , ~7 ,

S--

,,J=l

g

.N,

W ,2

X = det,, • ....

A"= det,, ~7

(3.1)

Invanance requirements plus fulflment of (anomalous and non-anomalous) Ward ldentiues restrict Leff to be of the form: (3 2)

Leff = L D + L F

where L D a r e a set of invarlant loneUc terms and the superpotentlal fixed to have the form

L F IS

umquely

L F = S [ l o g d e t T / A Z N + f ( Z ) ] [ o 2 - Y ' ~ m , T , Io2+h.c ,

(3.3)

l

with f an arbitrary funcUon of the chiral superfield Z - X- X / d e t T In order to be general, we have added the most general superpotentlal, which in this case consists of supersymmetnc masses m, for our superfields ~,, ~,. The case m, 4:0 is the simplest one to describe. In this case, Komshi-type arguments [8] imply

(T,j) = 8,jm~-l(s),

,, j = 1

U

( X ) = (.~) = 0

(3 4)

Indeed, mlnmuzlng the potential that originates from (3 3), one finds the condmons

S X f ' / d e t T = S X f ' / d e t T = 0, =

(3.5)

which give (3.4) if f ' :# 0 and, in any case, are solved by (3 4). One also finds logdet T / A zN + f ( 0 ) = 0,

(3 6)

638

A Mastero et a l / Supercompostteness

wbach fixes det (T,j) = A2Ne-f{°)

(3.7)

an agreement wath anstanton calculations [10,11]. The massless situation as more complicated. In ttus case, both our Laf and anomaly arguments [8] require ( S ) = 0 Besides, one just gets the further condltaon log det T / A 2u + f ( Z ) = O,

(3 8)

which does not fix ( d e t T ) and ( Z ) Thas appears to contradict the results of ref [11] if taken at face value. However, both the method of constrained lnstantons [9] and a recent extension of the ITEP approach [12] show that, in a Hlggs phase with (q~)~ ' ) > > A s o there are stall SUSY vacua with arbitrarily large values for (det T ) . The calculataons of ref. [11] would then be justified only for the confimng vacua at (q,,,~), (~j')<< Asc and would be bhnd to the exastence of Hlggs-type vacua (the opposite being true for the calculataons of refs [9]) The amusing thing here as that our Laf smoothly interpolates between the two pactures, since at possesses a flat direction an det T, XX space gaven by (3.8). To illustrate this poant, take as an example

LF = S l o g

f(Z)=log(1-Z),

det AT2- N X~'] -M,T,, o2 + h c .

(3 9)

In this case, the perturbatlve Haggs vacua (q~,,.,) = (~7) = v3,~,,

v >> A,

(3.10)

are recovered as (det T )

>> A 2N ,

XJ(/det T ~ 1

(3 11)

One can also verify that the SC slnglets T,j, X, X contain, an the sense of complementarity*, the massless superfields of the Higgs picture [including the Goldstone of the SC anomalous U(1)~, symmetry] In the confining picture we are adopting, S and one combination of det T and X.g pick up a mass O(Asc) an agreement wath R-invaraance [recall that q x ( S o ) = -qx(To, (XA')0) = - 1 ] , while the orthogonal combanatlon and the remaining N 2 superfaelds remain massless The resulting N 2 + 1 massless superfields are precisely what is needed [3] in order to satisfy all the appropriate 't Hooft consistency conditions [14]. The actual vacuum stabihty group H depends, of course, on which * For the use of complementaritym this context,see ref [13]

A Mastero et a l / Supercomposlteness

639

one of the many degenerate vacua (3.8) we choose If, for instance, we take

= c ,jv, 2,

(x),(2) ,o,

(3 12)

we break U(1)v and SU(6)R )< SU(6)L down to a subgroup of SU(6)v, whach 1s SU(6)v itself if V, = V and U(1) 5 if each V, is different. It can be seen, however, that in each case, the manifold of the vacua is the same, thas being a consequence of the fact [15] that such a mamfold represents the complexlflcatlon of G F and is lnvarlant under a suitable complex extension of H. Thas is why the same set of degrees of freedom describes simultaneously all the inequivalent vacua which are present in the absence of mass and SUSY breaking. 4. Level II: adding SUSY breaking

We are now ready to add soft SUSY-breakmg terms. We shall do thas via charal lnvarlant bosonlc mass terms with the hope that fermlon masses will keep their charallty based protection It is at this point that the effect of the supercurrent anomaly comes into play Whale the tree-level vacua all go to the zero value of q~ and for non-vanlshang (real) bosomc masses, the lnstanton-lnduced term (3 3) makes such a trivial vacuum disappear Rather, the true ground state onglnates from playing the mass term against the effect of the anomaly In order to study how thangs go quantitatively, we have to represent the additional bosonic mass terms of the underlying theory

- E (.2.,**, +

(4 1)

In ref. [1], such terms were represented In Leef by -2

,

(4 2)

~Leff = -- Etl~/ri.zjq.r~j '

where we have denoted by ~r,j the lowest component of T,j Indeed, 8L and 6Lef f have the same transformation properties under G F A more convincing way to take the perturbation (4.1) into account would be to introduce further (vector) superfields into Leff. Following ref. [16], these are R,j - O* eVOj,

/~,j -- ~* ef'~j,

(4 3)

where V,/7" are the supercolour gauge superfields in the N, N (6, 6) representations The terms (4.1) can then be directly represented as 8Leff = - Y'. (/t,R,, +/22R,, 2

)o

# o

(4 4)

640

A Mastero et al / Supercomposlteness

Following the methods of ref. [16], we have studied the consequences of adding (4.4) to our Lcff and we have checked that the results thus obtained agree with those derived from the simpler term (4 2), if R and R carry only particles of mass O(A sc), as expected on general grounds. In the following, we shall thus use the approach of ref. [1] based on the SUSY-breaklng term (4.2). Since we want SUSY to be essentially unbroken at the A sc scale, we shall take all the ~'s much smaller than A sc Also, lnvanance under G w forces the conditions

(4 5)

115 = ~ 6 ,

while in general P'5 ~ ~6 Since the "diagonal" masses ~,, will play an important r61e in the following, we shall also adopt the simphfied notation ~,,=~,

t=1,2,3,

~z = ~j,

j = 4,5,6

(4 6)

Adding the extra term (4.2) to the previous supersymmetnc lagrangaan yields 1

L . . = ~-a--i( S ' S )

1

~ +

1

-2 7/'*~ ~ - ~ T r ( r * r ) o + [S logdet T/A I2 ] 02 + h c . - ~-7 Z.,t~,,j ,j ,j , 1,J

(4.7) where we have taken canomcal kinetic terms (up to a rescallng) and have omitted for slmphclty the X, ~" superfields Neither one of these slmphficatlons affects the main results of our subsequent discussion The effective potential following from (4.7) is "-:2 ~r*Ir Vcff= aA4llogdet ~/A1212 + flA2El~rxl2l(~r 1)u[ 2 + A -2V"~ 2.,t% ,~ u ' l,J

(4 8)

I,y

where ~rx = So= o. It is then easy to see that, for g2 << A z, Veff leads to a candidate diagonal vacuum where

(,~) = 0,

(%)

- a,,,~,,

(~4~5~6 tl/6.2

( ~ 3 ~ 5 ~ 6 1 1/6

l--Z-4 ] a c, --3-- --

P" #4P'6

1/6

2

(4 9)

A Mastero et a l /

Supercompostteness

641

TABLE2 Global and local U(1) quantum numbers of the preomc superfields U(1) symmetnes and hght bosons boson fields #,23 +4 +56

~,23 ~4 ~5 ~6 Status Lght bosons

B

L

U(1) r

U(1) x

U(1)A

U(1)A

1 1

~ -~

~ -~

-~ 1

0 0

0 -2

1 -3

1

--~

~

0

0

1

0

U(1) v

-1 -1 -1 -1 broken spon X, X

-~ ~ ~ ~ unbroken -

-~ ~ -~ -~ unbroken -

~ 0 -1 0 -1 0 1 0 gauged, unbroken broken eaten -

0 -2 1 1 broken spon Im ~

1 -3 0 0 colouran broken axaon

The hne denoted by "status" describes the fate of each U(1) symmetry,of the theory Note that B and L are exact symmetnes, the same occurs for U(1)x as long as we do not introduce some exphclt breaking, such as a mass for glulnos

I n a r a n g e of values for the /t 1, ~1 parameters, tills can be shown to be the true g r o u n d state (see also ref. [1]) The v a c u u m (4 9) possesses some interesting features (a) It respects the R - s y m m e t r y ( ( ~ r x ) = 0) This Is of course i m p o r t a n t for ferrmonlc m a s s protection (b) It preserves o r d i n a r y colour as a consequence of the vector-like n a t u r e of the p r e o n s u n d e r SU(3)c(c) It b r e a k s SU(2)L × U ( 1 ) y (through (~rs) , (~r6)) as a consequence of the chlral n a t u r e of the p r e o n s u n d e r the electroweak group Since the F e r r m scale ( - 100 GeV) is related directly to ~r5 a n d ~r6 a n d we wish it to b e (muchg) smaller than A s o we have to play o n the mass parameters ~ a n d m a k e ~rs,6 << A 2. Ttus is possible due to the original huge v a c u u m degeneracy of the massless theory Small masses can reduce a widely asymmetric v a c u u m , as Is evident f r o m eqs. (4.9) W e are d e a r l y led to the choice* -2

g2 _ g~ < ( << )/zs,6 '

(4 10)

w t u c h yields 71"1 = '/1"2= ~3)"/7"4 > ( >> ) A ~ c ,

7r5, %

< ( << ) A 2 s c .

(411)

* Inslsttng on a huge ratxo ~5,6/~ could turnout to be unnatural due to the couphng of thfferent masses through hagher orders in asc

A Mastero et al / Supereompostteness

642

Proceeding as in ref. [1], we study now the light spectrum Fermwns They all remain massless except for the Dlrac fermlon, which had already become massive at level I. This result ~s hardly surprising, since the R-symmetry has not been broken, 37 massless fernuons are still necessary in order to fulfil the U(1)3x anomaly constraint Bosons. Only the Goldstone bosons remain massless, while the others pick up masses O(~) For/~ 4:#4 :g/z5 = 1£6,/~ ~ ~4 :# ~5' ~6' the expected Goldstone bosons are. (i) A colour octet from SU(3)L X SU(3)R D SU(3)v-- SU(3)c (n) Three "techmplons" from SU(2) L X U(1)y D U(1)Q. (uI) Two more pseudoscalars associated with the spontaneous breaking of the two non-anomalous axial symmetnes U(1)A, U(1) x of table 2

TABLE 3 Order of magmtude of the ferrmomc and bosomc masses at the levels II and III of the theory Fermaons

Level II mass

leptons quarks span- ½ leptoquarks octet hxggsmos Xx, X0

0 0 0

X'O, X'o'

0

XF,X v glulno "~q,£

0 -

a~ m ~ (~4/~5 6 ) a~mg(g/P'5 6) a~ m~ ( ~ 4 / ~ 2 + ~2) a c ( CA/ CF ) mglog(f~/m~) eaten by W, Z

0 0 Asc

Asc

acing, acmg(p,/~5 6) 2 a~mgf(~,/fi 9) mg (input) mw z + max(acm~,(~ 2 - ~2)//~2 6row,z)

Bosons

Level II (mass) 2

sleptons squarks scalar leptoquarks

~5,6 ~2 6 (~ + ~4) 2

scalar octet haggses ,try, ¢r0

Level III mass

Level III (mass) 2

,i/,2 6 (~ + ~4) 2

pseudoscalar

scalar

pseudoscalar

~2 ~t5, 6-2

0 0

~2 fi,2 6 + O ( m 2 )

(a~ mglog(~/mg))2 eaten by W,Z

a~c

X~c

a~c

A~c

-2 '/~5,6 -2

0,0

~2, ~2 6

o

~,: 2 2 ~,~3(;~

< A4QcD/A2so 0

X+X

X-Y(

~,:

W,Z

-

-

See sects 4 and 5 of the text for detatls

g2 A sc t

o

+~) m w/COS 0w

A Maswro et al / Supercompostteness

643

gc

?'-S--.,* o;l

"

gc Fig

i

Example of radiative corrections anducing the superglulno mass mx once the R-symmetry is broken by the glulno mass mg

(iv) A c o m b i n a t i o n of I m X, I m .~" for the breaking of U(1) v The second column of table 3 summarizes the mass spectrum at level II

5. Level III: "weak" gauging and R-symmetry breaking W e finally gauge the subgroup of G F G w = S V ( 3 ) c x SU(2)L X U ( 1 ) r , and, at the same time, break the R-symmetry by introducing explicit masses for the g a u g m o s of G w, m p a m c u l a r for the glumo ~. Here we encounter the problem that the hyperglulno mass m x is no longer protected b y the R-symmetry (unlike that of the fermlomc preons) and is generated b y radiative corrections such as that of fig 1. Setting m x = 0 or rn x << acre ~ a m o u n t s thus to an unnatural fine tuning I n conclusion, weak gauging and glumo mass b n n g m the following modifications of Left: (1) A direct (super)glumo mass term providing a correction [1] 8Leff = - m x l r x + h.c ,

m x - acm~, <<

(5 1)

(2) W e a k gauge interactions for our composite fields Neglecting Wess-Zumlno type terms [17] (which should be irrelevant for our present purposes) we follow ref [1] a n d m a k e the replacements (S*S)D--) ( S * S ) D ,

(T,*T,s) D -+ T * , j [ e V l , y T k , ,

(5 2)

where V includes all the gauge fields of G w and is, in general, in a redumble representation of it The modification (5 2) b n n g s in a d d m o n a l c o n t n b u t l o n s to the scalar potential m the form of (non-negative) D-terms. We have to see whether these m o d i f y the previously-found vacuum (4 9) The colour D-term vamshes in the v a c u u m (4.9) and hence tends to stabilize it. The SU(2)L x U ( 1 ) r D-terms would

644

A Mastero et a l /

Super~ompostteness

tend to favour a vacuum in which

(~55) "~" (~66)"

(5 3)

F o r ~5 = ~6, eq. (5.3) is obviously saUsfied by the v a c u u m (4 9). W e shall see, however, that in order to dlstmgmsh charge 2 from charge ~ quarks, it is i m p o r t a n t to allow ~5 ~ ~6 In thts situation, there is a competition between the SU(2)L D-term"

g2( ~2-

q'g,]2 66'2A£1.SC,4

(5.4)

2 -2 2 -2 2 ASC (~5q'/'55 + ~6"ff66) .

(5 5)

and the mass term

The ratio of the two terms is of the order of

VD 2 -2"'2 Vmass-gwAsc['rr55+~2)-2-

('rr2--~)2 m~v{ 1"£-5 -* -- 1~6-2 2 ~ 2 T 2 < 1 ~5,6~55 (,//.55 ..[._7/.66) -- T

(5 6)

where we have a n t l o p a t e d the expression for the W, Z mass Since ~5,6 are the large masses, we can assume Vo/Vmass << 1, In wbach case the level II vacuum is not appreciably modified T h e extra term (5.1) induces the n o n - v a m s h m g VEV for ~rx mx ( S ' / lot- 11)2) (~'x) = A 2 ~ . , \ , ,j

-1

m?~

-~-T mln(~- ) 2

=

(5 7)

,

as can be seen from the extremum condition Consequently, the S log det T term generates the fermlomc mass term

(';'rkk)

rnln(';'rkk) 2 X,jXj,

L ~ s = (°rx) k.,.jI-I (det 'n') XuXj, = mx (u.)(.,'rjj)

A2

'

(5 8)

where Xu 1s the 0-component of Tu One also gets extra masses for the spin-zero b o u n d states, but these are small compared w~th the level II masses, since m x << The true G o l d s t o n e bosons are not affected by these terms The other effects of weak gauging are of two types: the usual Hlggs m e c h a m s m and the radlauve contributions to fermlomc and b o s o m c masses T h r o u g h the Hlggs mechanism, the W and Z bosons and their SUSY partners, "qq and Z, become massive. The longltudmal components of W and Z are provided by the would-be G o l d s t o n e bosons in the ~r~b sector (a, b = 5, 6) which play the role of techmplons. W e obtain

M ~ / = l ~ g 2 5 ~2 7./ /.2 +

eff62)/A2=cos20wM2

'

g? sm20w g12+g

(5 9)

645

A Maswro et al / Supercompostteness

m~

X

gc

n

gt

n

X

Fig 2 Radmtave correctmns ~xang nse to fermmmc masses These dmgrams gave the contributions m eqs (5 12)

A n a l o g o u s l y , the " ~ / a n d Z get a m a s s b y eating up the ferrmomc p a r t n e r s of the t e c h n l p i o n s . I n the limit m x = 0 a n d ~r5 a n d ~r6 the two elgenvalues of the "~ m a s s m a t r i x coincide. I n the case of a mass splitting m a i n l y due to m x ~ 0, we get

Mqvl ~ = M w +_acre ~ ,

(5.10)

w h e r e a s in the situation where rr5 ~ ~r6 IS the m a j o r source of flus sphtting, we o b t a i n

~

_

-2

]16

M~V, 2 = M w + ~2 + ~ 6 M w •

(5.11)

H e n c e , in a n y case, one of two m a s s elgenvectors is h g h t e r t h a n the W b o s o n with r e l e v a n t p h e n o m e n o l o g l c a l consequences. A n a l o g o u s results a p p l y to the Z case. F i n a l l y , o n e gets [1] fermtonlc masses from r a d i a t i v e corrections such as that of fig. 2*. K e e p i n g SU(2)L x U ( 1 ) r gaugino masses of O ( m g ) or less, their effect will b e n e g h g l b l e c o m p a r e d to that o f m~ or mx. I n the a p p r o x i m a t i o n m g < / ~ , , one finds

(rad)= O,

m v

m(rad)

Otc

--(rad)---- O (3~ mgfi4/~6 ) ,

,,,e±

_ _

u,d = - - CFmglt/l'ts,6,

'"lq'~(rad)= __ CFm ~log _

_

/J,4 -- ~I/,

m(orad)= --~-CAmglog(#/mg),

(5 12)

w h e r e C F = ( N 2 - 1 ) / 2 N , CA = N are the SC CasImlrs The i d e n t i f i c a t i o n o f the o r d i n a r y f e r m l o n s with a p p r o p r i a t e c o m p o s i t e s is, of course, d i c t a t e d b y the q u a n • Presumably these dmgrams have to be dressed with form factors [18] Since they already converge, however, the final outcome walldiffer at most by a factor O(1) from the answer given below

646

A Maswro et al / Supercompostteness

tum numbers and is shown m table 1. Comblmng tree level [eq. (5.8)] and radiative [eq (5.12)] masses, we get the final result [m x = O(acm~) ] 0te

]~4]£6

rne±=--CFrn~a-G-~= 2 + O( a m , ) , etc t~4~5 m~=--CFrn~a ~ ,

~

[ ~+ ~ ]

~

[ ~g6 ~] ] L

mu=--~-Cvmg[a-~2j

-~5 '

m d = --Cvm~la~-~= 2 + - -

mlq

=

,aa(rad) "~lq ,

moct

=

'

__(rad) moct

(see 5 12)

(5 13)

where a is a number O(1) related to the ratio mx/otcm ~ Notice that m , = rn d unless ~ 5 ~ ~ 6- Concerning the bosons, we stall had, at level II, a massless octet coming from the breaking of SU(3)R × SU(3)L to S U ( 3 ) o Since tins symmetry is exphcitly broken by colour interactions, we expect these bosons to acquire masses O(ac). Indeed, the loop &agram of fig 3, after inserting the octet fernuon mass obtained beforehand [eq (5.13)] gives

m(%ct)=m(xo,:t )

(5.14)

In other words, the two bosons of each member of the octet get masses O(/~) and O(acm ?CAlog ~/mg). In the diagonal sector (%,, X, X), the only remaining hght bosons consist of (a) The true Goldstone bosons of U(1)v and U(1)A (see table 2). (b) The Goldstone of the U(1)A, symmetry which, because of a color anomaly, behaves precisely as an amon. Since the scale of breaking of U(1)A, IS not the same

rn~

ffoct Fig 3

gc

Xoct

Xoct gg

Hoot

Ra&atavemass of the tree-levelmasslessoctet winchcomesfrom the breakingof SU(3)L X SU(3)R to SU(3)c

A Masteroet al / Supercomposlteness

647

as that of SU(2)L × U(1)y breaking, the axlon is of the "invisible" [5] type with mass and couphngs depressed with respect to the ordinary axaon by a factor O(~rs,6/~r, ~r4) = ~/~5,6 << 1. Furthermore its bare couphng to 2"y is depressed by the presence of hght composite fermtons which essentmlly saturate the relevant nonabelian anomahes* As far as the true (massless) Goldstone boson Im(~r~') (table 3) Is concerned, one may worry about its couplings to the weak gauge bosons. One finds, however, that the only relevant couphng is through the Z 0 which connects Im ~r~' to Re ~r~' and to other massive states. Also the other true Goldstone boson I m ( X - .~') is found to decouple from all our hght states, in particular from quarks and leptons Turning to the fermions, we find that the partner of 7r~', Le., X~ of table 3, can present a vertex ~'~'/~x~Z~ ff ~r5 :~ ~r6. This could simulate some extra neutrinos contributing to the Z 0 total w~dth. On the other hand, the results (5.13) for the fermmns exhibit two conspicuous problems" 0) The neutnno gets a Dxrac mass of O(me) from the tree-level mechanism due to rn x 4:0 [eqs. (5.8)]. (u) The spin-1 leptoquarks (which have charge -+ 2) 3 receive [eqs (5.13)] masses O(acmg) which could be dangerously small. In order to see this pomt, let us try to keep the leptoquarks heavier than 10 GeV It follows that mg > 102 GeV, but also, from eq. (5 1), that ~, ~4 > 102 GeV. We also want mq/mlq ----P'/P'5 ~ 10-3 ~ P'5 > 102 TeV,

(5.15)

and, on the other hand, we have from (5.9) and (4 9),

- Mw ( A2sc ) l/2..~ (3OOGeV)(~5,6/~t)2/3

A sc-

g2 ~ 7/'5,6

30 TeV < ~5,6-

(5.16)

We are thus led to a sxtuatlon where ~5,6 >~" Asc, which 1s the opposite of our initial assumption (SUSY breaking small compared to A sc ). Suppose in fact that we require ~5,6/Asc _< 10-1;

(5 17)

g2 < (30 GeV)3/g5

(5.18)

we then get from (4 9)

* Thas questaon has been studied meanwlule m some detaal by E Guadagmm, K Komsba and M Mmtchev (private commumcataon)

A Maswroet a l / Supercompostteness

648

Since we want /t < / t s , we must have 55 > 30 GeV (1 e , Asc > 300 GeV = GF1/2), but the higher we take ~5, the smaller ~ [eq (5 18)] and m l q [eqs. (5.13)] wall be We are squeezed into a compromise, e.g., Asc > 1 TeV = 55.6 >-- 100 GeV ~ ~ < 16 GeV,

(5 19)

which gives (for mg =/~, ~4) mlq ~

1.5 GeV,

me, m q = 250 MeV

(5.20)

The quark and lepton masses could be fit for the second generation, but the leptoquark mass is obviously too small

6. An unappealing remedy: spectators We now present the only remedy we have been able to find for the problems (1) and (n) mentioned above. This consists of adding spectators, i e., SC-slnglet elementary fields, to our list of preons Let us look at the neutrino sector in some detail. The slnglet N which is needed is neutral with respect to the entire gauge group. It is easy to see that, if we want the R symmetry, as well as B and L, not to be broken by the interactions of N with @ and ~, we can give charges - 2, 0 and + 1 (under R, B and L ) to N and add the gauge mvarlant interaction

8L= f daON,N + f d 2OgN@~@sN+ 4~a h.c

(6 1)

to the fundamental lagranglan. Writing this new term in the effectwe theory as

8Laf = ( N*N)D + gNT45NI02 + h.c

(6 2)

we find easily that the new vacuum is consistent with the old vev's, with the addition of ( N ) = 0 For the masses nothing new happens in the bosonlc sector, while in the ferrmonlc sector one gets a 3 × 3 neutrino mass matrix of the form lp

o

o

0(too)

N

0

0

gu A

/,c

O(me) gNA

(6.3)

0

Thus the combination Vphys = /-' - - ( O ( m e)/gN A)N remains strictly massless (as long as L number is conserved) We have just rediscovered in our context one of the known mechanisms [19] for ehliunatmg the right-hand neutrino and its dangerous Dlrac mass with the ordinary neutrino

A Maswro et al / Supercompostteness

649

The situation with leptoquarks is sirmlar, except that now the spectators carry colour and hypercharge. We need the set of spectators [under SU(6)s c × G w ] M , = (1,3,1, 4),

37/, = (1,3,1, - 4 )

(6.4)

whach evidently do not change (gauge) anomaly cancellations Again one can use symmetries to restrict the possible interactions of M,, IVl, with ~, ~, and obtain large Dlrac masses between the composite leptoquarks and the spectators (which have been chosen ad hoc for that purpose). Of course, this way of proceeding is far from satisfactory A possible alternative to the M,, 1VI, spectators is to start from a gauge group G w m which SU(3)c is part of a Patl-Salam SU(4) If we could thank of a mechanism that spontaneously breaks SU(4) PS to SU(3)c, the usual supersymmetrlc Hlggs mechanxsm would be able to eat up spin-zero and spin-1 leptoquarks and given them masses O(A PS). Unfortunately, with the order parameters at our disposal, we cannot find a way to achieve such a breaking.

7. Conclusions In spite of several encouraging features, hke the predtctton of light composite fernuons, the models we have been able to construct are ruled out by lack of sufficient haerarchy m the fermlonlc mass spectrum. It seems to us that, in order to avoid the above problems, one must start with SUSY gauge theories in wtuch the preon-quantum numbers lead to a radically different global symmetry structure One posslbdlty would be to enlarge the flavour group (and possibly G w itself) by adding a family label to some of the preons whale keeping the size of the SC group unaltered It is known [7, 20] that in this case the vev's of the relevant order parameters are controlled by the masses in such a way that they are typically much smaller than A sc Tins could avoid the unwanted constraint ~ 5 . 6 > A s c whach we encountered m order to get heavy enough leptoquarks. More radical alternatives, hke that of a set of preons which are charal under SC, should also be explored. Unfortunately, we are short of hants as to whach precise direction to follow m the search for modifications At the same time, the techmques for analyzing a gwen model in sufficient detail are still primitive and reqmre m a n y long and tedious calculations We hope, however, to have convinced the reader that supercomposlteness is not a game m handwavlng and parameter adjustment. Its predictwlty is what d o o m e d our attempt, but it can be invaluable m providing a unique, workable scheme, if ~t exists. The stakes are well worth the effort, we beheve R.P. would hke to acknowledge the hospltahty of the Theoretical Physics Division at C E R N d u n n g part of thas work. M.R. would like to thank H. Frltzsch for

650

A Mastero et al / Supercompostteness

h o s p ~ t a h t y at t h e Unlvers~ty o f M u m c h , w h e r e p a r t o f the p r e s e n t w o r k was d o n e , a n d t h e " F o u n d a z l o n e A n g e l o D e l l a Raccla" for p a r n a l s u p p o r t .

References [1] A Maslero and G Venezlano, Nucl Phys B249 (1985)593 [2] R D Peccel, Max-Planck-Instltute preprlnt MPI-PAE/PTh 35/84 (1984), W Buchmuller, CERN prepnnt TH 4004/84 (1984), G Venezlano, Proc 1st Capri Symp (1983), to be pubhshed [3] R Barblen, A Maslero and G Venezlano, Phys Lett 128B (1983) 179 [4] E Farha and L Susslond, Phys Reports 74 (1981) 277, H Georgt and D B Kaplan, Phys Lett 145B (1984) 216 [5] J E Klm, Phys Rev Lett 43 (1979)109, M Dine, W Flschler and M Sredmckl, Phys Lett 104B (1981) 99, M B Wise, H Georg~ and S L Glashow, Phys Rev Lett 47 (1981) 402 [6] W Buchmuller, S T Love, R D Peccel and T Yanaglda, Phys Lett 115B (1982) 233, W Buchmuller, R D Peccel and T Yanaglda, Phys Lett 124B (1983) 67 [7] T R Taylor, G Venezlano and S Yanklelowlcz, Nucl Phys B218 (1983) 493, G VenezlanO, Phys Lett 124B (1983) 357 [8] K Komsh~, Phys Lett 135B (1984)439 [9] I Affleck, M Dine and N Selberg, Phys Rev Lett 51 (1983) 1026, Nucl Phys B241 (1984) 493 [10] V A Novtkov et al, Nucl Phys B229 (1983) 381,394, 407, E Cohen and C Gomez, Phys Rev Lett 52 (1984) 237 [11] G C Ross1 and G Venezlano, Phys Lett 138B (1984) 195, D Amatl, G C Rossx and G VenezlanO, Nucl Phys B249 (1985) 1, D Amatl, Y Meunce, G C Ross1 and G Venezaano, CERN preprlnt, Th 4201 (1985) [12] V A Novlkov, M A Shafman, A I Valnshtem and V I Zakharov, Nucl Phys B 260 (1985) 157 [13] T R Taylor, Phys Lett 125B (1983)185 [14] G 't Hooft, Proc 1979 Carg~se Summer School, ed G 't Hooft et al (Plenum Press, New York and London, 1980) [15] B A Ovrut and J Wess, Phys Rev D25 (1982) 409, C K Lee and H S Sharatchandra, Max-Planck-Instltute preprmt MPI-PAE/PTh 54/83 (1983), W Lerche, Nucl Phys B238 (1984) 582, T Kugo, I Ojlma and T Yanaglda, Phys Lett 135B (1984)402 [16] E Guadagmm and K Komsha, Umverslty of Plsa prepnnt IFUP-TH-1/84 (1984) [17] J Wess and B Zummo, Phys Lett 37B (1971) 95, E Wttten, Nucl Phys B223 (1983) 422, 433 [18] W Lerche, D Lust and H Steger, Max-Planck-Instatute preprlnt MPI-PAE/PTh 91/84 (1984) [19] H Georga and D V Nanopoulos, Nucl Phys B159 (1979) 16, D Wyler and L Wolfenstem, Nucl Phys B218 (1983) 205 [20] M G Schmldt, Phys Lett 141B (1984)236, J M G6rard and H P Nllles, Phys Lett 129B (1983) 243