Split light composite supermultiplets

Nuclear Physics B249 (1985) 593-610 @ North-Holland Publishing Company

SPLIT

LIGHT

COMPOSITE

A. MASIERO CERN,

and G. VENEZIANO Geneva,

Received

SUPERMULTIPLETS

Switzerland

13 August

1984

The splitting induced by soft supersymmetry (SUSY) breaking inside the light composite supermultiplets of confining SUSY gauge theories is studied by effective lagrangian methods. Examples with and without unbroken chiral symmetries are considered. In the former case, for a suitable breaking, the lightest states are spin-f fermions. A prototype model for one leptonic family is discussed.

1. Introduction Supersymmetric (SUSY) composite models for quarks and leptons have been advertised in the past few years (for a recent review see [l]) for possessing several advantages over their non-SUSY competitors. It is becoming clear, however, that a SUSY preonic dynamics may exhibit one further, crucial property: theoretical predictivity. This comes from the recent observation [2-51 that the condensates characterizing the vacuum properties of SUSY gauge theories can be computed by convergent, reliable, small instanton calculations. These have confirmed and extended the results on SUSY gauge theory vacua previously obtained [6-81 by effective lagrangian methods. Of course, any realistic model of quarks and leptons must contain in one way or another a SUSY breaking mechanism allowing one to push the unseen scalar partners of the ordinary fermions to a sufficiently high mass. Since the road of spontaneous SUSY breaking [S] appears to be unavailable in the presently considered preon models [l], one is led to use instead some sort of soft SUSY breaking, as originates for instance from the supergravity Higgs effect [9]. Without committing ourselves to a particular origin of the soft SUSY breaking, we want to study here its interplay with the above-mentioned non-perturbative effects. The purpose of the exercise is that of gaining some general understanding of the low-lying spectrum of these theories as a function of the SUSY breaking parameters. This could eventually enable one to construct realistic and essentially calculable models for composite quarks and leptons. Unfortunately, the direct calculational method of refs. [2-51 cannot be easily extended to include soft SUSY breakings. For this reason, we shall rather employ the effective lagrangian techniques which have proven so useful in predicting the properties of SUSY vacua. 593

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We shall actually present three quite extreme examples. (i) Supersymmetric Yang-Mills (SYM) theory with a gaugino mass term (sect. 2). This theory does not possess any continuous chiral symmetry (even in the SUSY limit) and is therefore expected not to contain a light supermultiplet. Nevertheless, this case is interesting insofar as it allows us to see quite easily vacuum angle effects and splittings inside supermultiplets due to SUSY breaking. (ii) Supersymmetric QCD (SQCD) with two colours and one flavour (sub-sect. 3.2). This is the simplest case with a continuous chiral symmetry which is, however, spontaneously broken. A light Goldstone supermultiplet is thus expected, and splittings therein can be studied. Because of the chiral symmetry breaking, we expect the pseudoscalar member of the supermultiplet to be better protected than the scalar and the fermion, making life difficult for phenomenological applications. (iii) SQCD with three colours and three flavours (sub-sect. 3.3). This is a scaleddown version of the model of ref. [lo] in that it has only “leptons” and no “quarks”*. Yet the model turns out to be very instructive. It has a U( 1) axial R-symmetry which is unbroken in the SUSY limit and, consequently, some fermion masses enjoy a double protection [cf. ref. [lo]]. The splitting inside the light supermultiplets is now expected to favour the fermions and, indeed, for a wide range of SUSY breaking parameters, one is able to get rid of the bosonic partners of the leptons. The model allows for further gauging of an SU(2)L x U( 1) y subgroup of the flavour group with technicolour-type breaking down to U(l),,,,. and, finally, a non-vanishing tree-level photino mass is able to induce radiatively a mass for the charged (but not for the neutral) lepton. 2. Softly broken super-Yang-Mills

(SYM) theory

We shall start by illustrating the general considerations of sect. 1 in the case of softly-broken SYM theory which is defined by the lagrangian L =

Linv

+

Lbreak

(2.1)

*

Linv is the usual [I l] SYM lagrangian with gauge group SU(N)

and Lbreak is given

by L brcak= - m,:A w + h.c. ,

a=l,2, a=1 , a*-,N’-1.

(2.2)

Lbreak is the only possible gauge invariant SUSY breaking term, a gaugino mass. The phase of mh is arbitrary. By a U(l),

transformation

A, + Ah = exp (fi arg m,)A,, the phase of mA can be rotated away. Since U(l),

(2.3)

is anomalous, one generates,

* This simplified version of the model of ref. [IO] was conceived through discussions with R. Barbieri.

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however, a vacuum angle 6,: &=Nargm,.

(2.4)

Without loss of generality, we can work with a real m, > 0 and a vacuum angle 6, defined by

where W, is the gauge superfield and F denotes, as usual, the 19’ component. The low-energy physics of the supersymmetric theory (m,+ = 0) appears to be well described by the effective lagrangian [6] L :;sy=;(S*S);3+

(2.6)

where D denotes the t12ez component and the renormalization invariant composite superfield S is defined by*

group and gauge

(2.7)

Clearly, the effects of a non-zero vacuum angle & and of a non-zero mass m, can be accounted for by writing a new LeR: N - iB,

>> 1

+ h.c. + [m,S,,,f

h.c.1,

F

(2.8)

where m, now includes some g-dependent normalization factors that make it the physical (i.e., renormalization group invariant) SUSY breaking parameter. The scalar potential resulting from Lem is simply 2

N

V=~(T:T~)“~

where rA is the lowest component SUSY minima are located at (71*)=A3enp(i$)enp(

log%-& A

(2.9)

-mh(rrAfn:),

of S, i.e., (g2/32rr’)h~h”*“.

-$$K),

K=l,...,

For m, =O, the N

N.

This degeneracy corresponds to the spontaneous breaking of the non-anomalous i& subgroup of U( l)R down to a Z2(h + *,I). A small mass term breaks explicitly the ZZN symmetry and, indeed, it shifts the N local minima, leaving one of them

l

Note that our normalization

of S follows ref. [7] rather than ref. [6].

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as the true ground state. This is given by the value of K minimizing: -mn(7rA + 7~:)~ = -2m,fi3 cos EK mzn A

r2m:+

*“> *

(2.11)

At 8, = 0, the minimum with K = 0 clearly gives the ground state. From eq. (2.10), we see that it also gives a real (TV) and, hence, CP conservation. Indeed, K = 0 is the true vacuum in the range -r
x

cos

(e,/ N) ,

cos

= J% ,

(2.12)

ev=r.

The K = 9-l vacuum becomes the true ground state up to 0, = 3 r, where it crosses (and is replaced by) the K = +2 vacuum. This pattern continues up to BV= (2N - 1)~ where the K = 0 vacuum comes back again. Thus the whole pattern repeats itself with 0, periodicity 27rN, while the physics is periodic with period 27~ (it is the same at 8, = 0 in the K = 0 vacuum, at 8, = 27~ in the K = + 1 vacuum, etc.). The situation is analogous to the one known to hold in QCD [12] for N degenerate flavours, and is depicted in fig. 1 for the case N = 3. Notice that at 0” = r, one gets a complex (7~*), with (Im 7rA)=(iXy,A)=sin corresponding

to a violation

(

;

of CP. This violation

>

A3,

is “spontaneous”

for N odd

N=3

I -2mkh3 Fig. I. Vacuum energy as a function of f?” for SU(3), SYM theory. Solid lines denote the true ground state at each 0” while dashed lines represent the other two false vacua. The value of K for the true vacuum changes at 13= r(mod 2~).

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[Dashen’s phenomenon [13]], since it corresponds to having just changed the sign of mA (0,= NT is the same as 8,= r for N odd), and thus to a CP-conserving lagrangian. Again, the situation is similar to that of ordinary QCD [12] for an odd number of degenerate flavours. Let us now consider the expectation value of the auxiliary fields of S which are related to F’S iFk By SUSY, they were zero in the m, = 0 case. It is straightforward to see that they now become O(m,). More precisely, one finds Re (S,) = ;(v~v?)~” Im(S,)=

log

(dmjNt”:A A6N

(g(ri,n:)“l[log

A

>

(2)”

f+rA

(2.14)

+ 7.e) 9

-2&l)

= m,
(2.15)

Eq. (2.14) shows that Re SF = (g2/32r2)F2 gets a positive expectation value of O(m,A3) [the minimization of (2.11) yields a positive value for (r* + &]. This is consistent with the fact that for mh - A, the broken SYM theory should approach the ordinary YM theory (QCD without quarks) for which (F’) is believed to be positive [ 141 (condensate of magnetic monopoles). Turning to eq. (2.15), one obtains from it at small 0, Im (S,) = g’(FF)/(327?)

==, mJ3i3,/ Y

N* .

(2.16)

The result (2.16) again interpolates smoothly between the supersymmetric case m, = 0 and the ordinary YM case (m, = A ), where it is known [ 151 from the large-N solution to the U(1) problem that* -aF”,m’,, e,.=o

= O(A”) .

YM

Our last point about SYM concerns the mass splitting inside the massive WessZumino supermultiplet which, in the SUSY limit, contains the would-be Goldstone P of the anomalous U( l)R symmetry, a scalar S and a fermion F. A straightforward calculation yields, to first order in m,,, M;=

l

Note that, a - N-4’3

(r2N2A2,

M,a=crNA,

(2.18a)

Mg = a2N2A2+&m,A,

Ms=aNA+$,

ME= a2N2A2+~am,A,

M F =aN/i+:“h

with our normalizations, [see ref. [7]].

A3 and

mA are proportional

3N’

to N in the large-N

(2.18~) limit,

while

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We observe that STrM*=C(25+1)(-)‘ln/r,l’=O.

(2.19)

J

This result is a special case of a general super-trace mass formula which is derived in the appendix. 3. Softly broken SQCD

We are now coming to the physically more interesting case of softly broken SQCD. Such a theory is defined by the lagrangian L =

Linv

+

Lbrenk

,

(3-l)

where Linv is the usual SQCD lagrangian [ 1 l] containing, besides the gauge superlield W, the chiral matter super-fields @jh and &T((Y = 1,. . . , N; i, j, = 1,. . . , M) in the N + 15 representation of SU( N). Linv contains a SUSY mass term L!y

=

-;

m,i@b&~lF+h.c.

(3.2)

The possible gauge invariant soft breaking terms have been classified in ref. [16] for a general gauge theory. In the case of SQCD, they can be of one of the following types: (a) a gaugino mass term: L$Lnk = - m,AA + h.c. ;

(3.3a)

(b) scalar mass terms: (3.3b) where 4 and 4 are the scalar lowest components of @ and 6 respectively; (c) bilinear bosonic mass terms: (3.3c) Unlike the SYM theory, SQCD possesses, in the absence of mass terms, continuous chiral symmetries which, a priori, may or may not be spontaneously broken. This question has been studied, in the SUSY limit, both by the construction of effective lagrangians [7,8] and by direct (instanton) calculations [3,4], with consistent results. In any event, massless SQCD is expected to exhibit a rich structure of massless bound states, since spontaneous chiral symmetry breaking implies Nambu-Goldstone bosons (and their fermionic SUSY partners), whilst unbroken chiral symmetries imply massless anomaly matching fermions (“‘t hooftons”) and their bosonic SUSY partners.

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supennul~iplets

Following the discussion of sect. 1, we are interested in the splitting inside these supermultiplets in the presence of soft SUSY breakings. In particular with an eye on possible applications to composite models, we would like to understand the circumstances under which the fermionic bound states (quarks and leptons) remain considerably lighter than their bosonic partners (squarks and sleptons). For this purpose, we shall analyze two somewhat extreme situations in SQCD: (i) N=2,M=l, (ii) N=3, M=3. In the first case, the massless theory has just a chiral U(1) symmetry which is spontaneously broken by a @& condensate. As a consequence, we expect bosonic masses to be better protected than the fermionic ones when SUSY is broken. In the second case, on the contrary, some axial symmetries are expected to remain unbroken and this should lead to a more favourable situation for keeping the fermions lighter than their bosonic partners. Indeed, as we shall see, model (ii) can be seen as a simplified (i.e., purely leptonic) version of the model of ref. [lo].

3.1. SQCD

WITH

N=2

AND

An effective lagrangian in ref. [7]. It reads

M=l

for the SUSY limit of this theory has been constructed

L$“=~(S*S)‘d’+$(PT)Y’+[S(log($)-I)-mTIF+h.c., where besides S of eq. (2.7), we have introduced T=@cF=7r+ex+

(3.4) the composite superfield [7]

**a.

(3.5)

This lagrangian at m # 0 possesses two SUSY vacua with order parameters

(3.6) in agreement with explicit instanton calculations. We now add to L!$’ the soft breaking terms L ~~k=m,(~~+~~)-~'(~*~)"2-C22(~+$.*),' which have the same transformation terms of eqs. (3.3). l

For simplicity, however, we shall (i.e., of non-real mass parameters).

ignore,

(3.7)

properties as the three possible* soft breaking

for SQCD,

the possibility

of a non-vanishing

vacuum

angle

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The scalar potential originating

/ Composite

supermultiplets

from LcR = 'L$ + LzTk is

V=a(~~~:)Z'31L12+p(~*~)"2

I

z-m

I

2

-m,(~~++~)+~2(rr+rr*)+~2(~*~)“2, L=log[(~?T*)//P].

(3.8)

The boundedness of the potential requires (in the effective as well as in the underlying theory) a relation between m, p and fi, i.e., p~m~2+~2-2~~2~>0,

(3.9)

which we shall assume to be satisfied. Treating m, mA, P and fi as being all of the same order and much smaller than A, it is easy to find the minimum of the potential (3.8). Defining (3.10)

hHd=AV+4, (d/(4

= fi,

one finds

E(fi/p)'/6=-!g

[2~m-2m,-(m~+4~2m2-2/3mm,+3~~2+6/3~2)1'2]. (3.11)

Clearly, in the SUSY limit one gets fi + m and E + 0, thus recovering one of the SUSY vacua of eqs. (3.6). One also finds that, thanks to the constraint (3.9), this minimum always exists and that it is indeed the absolute minimum, with V,in=-2m*(7T*)<0.

(3.12)

At this point, we can compute the scalar, pseudoscalar and fermion mass matrices. A combination of S and T (predominantly S) gives a heavy [i.e., mass O(A)] split supermultiplet, whilst a different combination (predominantly T) gives a light split supermultiplet. Obviously, we are interested in this latter. Straightforward but lengthy calculations yield the following mass spectrum in the light supermultiplet:

(3.13)

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with & given by eq. (3.11). One immediately

supermulfiplefs

601

notices that (3.14)

The validity of such a simple supertrace mass formula is proven in the appendix. A simple numerical study of eqs. (3.13) shows that, in a range of the SUSY breaking parameters m,, p2 and k2, the lightest particle in the spectrum is the spin-4 fermion in spite of the fact that no unbroken chiral symmetry protects it. The most favourable situation for a light fermion appears to be the one in which M$= Mz and, from eq. (3.14), ME= ME-ap2. It can be shown, however, that irrespective of the values of the breaking parameters, it is not possible to get M$ less than -4 min (M& Mi). In other words, one cannot get rid of the scalar partners in this case. The opposite result will be achieved in the next model we are going to examine. 3.2. SQCD

WITH

N = M = 3: A MODEL

OF

COMPOSITE

LEPTONS?

The model we shall investigate now is the natural reduction of the one-family model of ref. [lo] after colour degrees of freedom (as well as hypercolour ones) have been eliminated. It can thus be seen as a prototype model for one family of leptons {say e, v,). The fundamental chiral preon superfields transform under the gauge group SU(3),,xSU(2),xU(l). as* @:=‘*‘=(3,2,

-f),

@p’,=(3,1,f),

6; = (3, 1, $, , &=(5,1,

-9x2.

(3.15)

As in ref. [lo], we shall first discuss the model in the absence of the SU(2)L X U( 1) y gauging, when it reduces itself to SQCD with N = M = 3. However, unlike what was done there, we shall introduce from the start soft SUSY breakings and determine (rather than guess) the resulting vacuum order parameters. We shall then compute the light spectrum before and after the SU(2)L x U( 1) y gauging. In the absence of the weak gauging, the effective lagrangian of this theory can be written as L,.*=

L%“,‘+ L$p,

(3.16)

where LrG can be taken from ref. [7] of the form inv L CR =-$(S*S)n+&(T$Tj)D+[S

logdet ($)-muqjIF+h.c*, (3.17)

* In order to make this model anomaly-free, two hypercolour singlet superfields (spectators) transforming as (I, 2, I) and (I, 1, -2) must be added. This also rids the model of a Witten SU(2) anomaly. We thank N. Seiberg for mentioning this last point to us.

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with s=rrA+e&+

**a,

7;is~;j5;=‘rrii+exij+.

...

(3.18)

The sum over repeated indices in (3.17) is understood. Before turning to LE;TCUk, two remarks are in order. The first one is that, for the sake of simplicity, we have taken the kinetic D-terms for S and T in (3.17) to be canonical up to a trivial resealing, rather than scale-invariant. The formulae would be somewhat more complicated with scale-invariant kinetic terms, such as those given in ref. [7], but the physics would remain the same. The second observation is that, in the case IV = M, the U( 1), symmetry is known [3] to be spontaneously broken by the condensates (det&L),

(det&P)=

This fact requires the introduction

0(/t’“)

.

(3.19)

of at least two more superfields*:

X = det,, @b ,

(3.20)

2 = detpj 6,P ,

in L$. When this is done, one recovers [ 171, in the chiral SUSY limit lztij= 0, enough massless fermions to saturate ‘t Hooft’s anomaly matching conditions and to agree with the counting of massless fermions given in ref. [lo]. On the other hand, the superfields X and X are neutral with respect to the SU(2)L x U( 1) ,, interactions. That is why we shall neglect them in the following discussion. The SUSY mass terms mVTjl, must be compatible with the SU(2),X u(l), gauging. This restricts the non-vanishing entries in mV to rn3, and rnJ3. For the moment, we shall take mu = 0. Similar restrictions hold for a SUSY breaking bilinear mass term &+i&j which we shall also neglect for the time being, together with a gaugino mass term. Some effects due to the switching-on of these mass parameters will be discujssed later on. In conclusion, in this first analysis we shall limit ourselves to soft breaking terms given by bosonic masses of the type** L break

=

-2

(3.21)

(p:f#$f$i+jif~:~i).

Using the symmetry properties of each term in (3.21), one can immediately they are reproduced, at the effective lagrangian level, by

(3.22)

d,/ao,

with the SU(2),. invariance forcing the constraint p, = p2. For simplicity, l l

A useful

discussion

* One can always Qi and &

with arrive

N. Seiberg

at such

diagonal

on this

point

hosonic

see that

we shall

is acknowledged. masses

through

an SU( M)

xSU(

M)

redefinition

of

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also take PI = b2, so that fi~,=j$2=fi~2=&‘p2.

(3.23)

The scalar potential resulting from Lcfl [eqs. (3.17), (3.22)] is

v=an

+&~12~ I I+pn27rgr:c l(7T-‘)ij12+

det rr 4 loq-

2

(3.24)

i,j

The stationarity

conditions

on V can be satisfied with i#j,

(~,\)=(nj)=o, (uA6L= -/ii*il(7Tii)12, where L= log (((rrll)(rr22)(~33))/A6).

Vi,

(3.25)

i= 1,2,3,

(3.26)

For small SUSY breaking, i.e., for &/AZ-e

1,

(3.27)

they yield (~,,)=(~22)-=(~)=(~L33/II)“3A’, (~33)

(3.28)

.

= Wc233)2’3~2

One can also argue that (3.25) and (3.28) provide the true minimum of V, at least in a large range of SUSY breaking parameters. We are now in a position to compute the various mass matrices. Let us start from the fermions xh and xP The off-diagonal fermions xii( i fj) are immediately seen to be massless, because of the vanishing of (71;\). The remaining mass matrix takes the simple form XII

x22

x33

MF =&+A3

,

(3.29)

which implies two massless states: x3”xII-x;2,

x8 = (‘dk,

I +x22)

--&‘&x33

,

(3.30)

and a Dirac particle of mass M .,,,,,,,.=~~A3(2(n)-2+(~,3)-2)“2.

(3.31)

Thus we get eight massless fermions (xv, i # j, x3 and x8). If we add to them the two fermions in X and X [eq. (3.20)], we end up with a total of ten massless fermions. This is precisely the number needed to saturate the ‘t Hooft conditions

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on the anomaly-free U(l), symmetry. In general, for N = M, this symmetry does not transform [7,8] the scalars 4 and 4 and is therefore unbroken by the condensates (3.25) and (3.26). One can actually show [IO] that the number of massless fermions is N”+ 1 in the absence of weak gauging. We now turn to the bosons. The off-diagonal sectors (i.e., i #j) give two-by-two mass matrices of the form nij

Tji

*

-PcLiiPj =

Mf.

?I?

(3.32)

PrZTi

where use has been made of the stationarity diagonal elements. Hence

condition

(3.26) in writing

the off-

(3.33) We see from (3.33) that, for generic values of p: and fi:, we have two massive eigenstates, whilst for pLi= pj (or 4; = pj) we have one massless and one massive eigenstate*. Because of the SU(2)L invariance, this latter possibility holds in the 1,2 sector (/.A,= pLz) for which we find the massless eigenstates: =I2 -?T&=n

(+I )

v21 -rry*=lT

(-) )

(3.34)

while the orthogonal combinations are massive, with M’=2Ppf,. The 1,3 and 2,3 sectors will be taken to have masses O(pp*). In the diagonal sector (i =j) it is more convenient to work with real (scalar) and imaginary (pseudoscalar) components. One then finds that the fields Im(m,,-

7r**) = 7r(O),

Im [(~lI+~22)(rr)-2rr33(~~3)1~n,

(3.35)

are massless, while all the remaining pseudoscalars and all the scalars take masses W/J’). It is easy to see that, indeed, four exact (massless) Goldstone bosons are expected with the soft breaking terms that have been chosen. One can also verify that STr M* = 2/3 C (pf+

b:) ,

(3.36)

ij

in agreement with the supertrace mass formula which is derived in the appendix. Before discussing the effects of gauging SU(2)L~U(1)y, we comment on those resulting from a non-vanishing gluino mass and/or SUSY mass mV = Si3ci3m. In the

l

Eq. (3.33) clearly shows that for certain ranges of the pLz and ,L2 parameters, one eigenvalue negative, signalling that the diagonal vacuum (3.28) has to be replaced by an off-diagonal

becomes one.

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presence of m,, and m, the scalar potential (3.24) is modified into

I I

det r ’ v-, V’= aA4 logn” +&i217rA7r~’ 1 +-pT+$)l~~12-

m,(r*

- m8,,iSj,~*

(3.37)

+ d).

As long as p:+-@f> m2+ mz, the values of (rtj.) given in (3.25) and (3.28) are left essentially unchanged, while 7rA develops an expectation value of the form

A-'(nd= hfi+mfi,

(3.38)

where f, and f2 are known functions of ratios of pf and bj. As a result of (3.38), the previously massless fermions pick up a mass of O(m, + m) from the F-terms in (3.17), in agreement with the fact that the R-symmetry [U(l),] protecting them nas been explicitly broken. We finally turn to the effects due to the SU(2)L x U( 1) y gauging, taking mh = m = 0 for simplicity. Considering the SU(2), x U( 1) y transformations of the composite superfields S and Ttii, it is straightforward to gauge SU(2)L xU( l)F One simply makes the replacements

(T$T’~)D+(T+)jiexp[(~~V2)ii’+(f(Yi+

yj))6ii~B]~~.

(3.39)

where V2 and B are the vector superfields containing the SU(2),- gauge bosons V, and the U( 1) y gauge boson BP respectively (they include the coupling constant factors g, and gv). One further adds kinetic terms for these gauge superfields and, possibly, SUSY breaking masses for their gauginos. The change in the scalar potential (3.24) is easily found to be

One can check that, as long as the minimum is flavour diagonal, the new term (3.40) does not affect the position of the minimum. The effects on the spectrum, however, are much less trivial. A technicolour-type Higgs mechanism occurs with the three goldstones rr(+), &-) and 7~“) of eqs. (3.34) an d ( 3.35) providing the longitudinal components to the massive W’ and Z”. These have the standard model expressions and masses M(W*)2

M(Z’)‘=

= pg2(42 _ pg2A2 lii3) 2’3 2A2 2 ( P > 9 M(W’)‘/cos’

&,

sin’ ew = 2

g:

gY+gz’

(3.41)

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There is also, of course, a massless photon, with the electric charge given by Q= TJL+fY. The spin-zero SUSY partners of v*, r” also get a mass’ term of 0(/3g*((~)‘/A*)) = 0(&f(W)*, M(Z’)*) on top of their previous masses O(pfi’). On the other hand, the pseudoscalar n8 remains massless at this stage. In the fermionic sector, xl2 = x(+), xrl =x’-’ and x3 combine with the fermionic partners of W* and Z”, \?r’ and go, to give three Dirac fermions whose tree-level masses (if m, = m =0) are the same as the W’ and Z” masses. When m, and/or m are non-vanishing, there is an extra x+x- and x0x0 mass term which is induced by (v,+) and the Dirac fermions become six Weyl particles with the mass splittings AM=O(m,, m). Of the remaining five composite massless fermions (at mh = m = 0), four represent a family of leptons, including an SU(2),xU( l),, singlet, the left-handed antineutrino. The identification would be (3.42) The fifth massless fermion is x8. Together with the fermions in X and X of eq. (3.20), we have a total of seven massless fermions. This is the correct number to match the U(l)“, anomaly, since the four massless weak gauginos now also contribute in the underlying theory, but three of them have acquired a mass from the dynamical Higgs phenomenon. The extra light states that we have obtained besides the wanted lepton family may turn out to be phenomenologically harmless, since they are either decoupled from the SU(2)L xU( l)y currents (X, 2) or coupled through them only to heavy systems. In any case, it seems to us premature to take the model seriously and try to defend it from phenomenological dangers. We plan to come back to these questions within a more complete scheme incorporating also colour (quarks) and, hopefully, families. Our last point concerns a mechanism for generating small masses for the composite leptons in the limit in which the U(l), symmetry is only broken at the weak gauging level (i.e., m, = mii = O)*. A photino mass is capable of generating a radiative mass to the charged lepton from the diagram of fig. 2. This diagram uses, besides a

Fig. 2. Diagram

l

For

m, this amounts

inducing

to a fine tuning

a radiative

since

mass

m4 induces,

to the charged

via

loops,

an m,

lepton.

of order

wni..

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photino mass, the SU(2),xU( l)y breaking (rrZ37rX2) propagator which can be obtained from eq. (3.32). The result of the (convergent)’ calculation is particularly simple for rntepf,@T: -e2 w33 mfl+=sm-,m:-mZ_

log 2 , ( )

m,=O >

(3.43)

where /!l, ml are the eigenvalues of the mass matrix MS of eq. (3.32) with i, j = 2,3. They are assumed to be both non-vanishing. Eq. (3.43) shows an interesting way of generating masses for fermions which enjoy the kind of double protection mechanism that was proposed in ref. [lo]. Indeed, the rn? factor signals the SUSY protection, whilst the iis factor is connected to the SU(2), xU(l). protection [SU(2), is unbroken for ~~~‘0, as can be seen from eq. (3.28)]. A similar effect would generate quark‘ masses through massive gluino exchange in a model incorporating the SU(3), gauging. In conclusion, our results can be viewed as an explicit dynamical realization of the mechanisms envisaged in ref. [IO] for generating a hierarchy of light mass scales. The crucial ingredients needed to achieve this goal have been the non-perturbative effects leading to the breakdown of the non-renormalization theorems and the inclusion of soft supersymmetry breaking terms in order to determine the vacuum expectation values. We hope that the results we have shown will provide useful hints as to which directions should be taken in the challenging search for a model of composite quarks and leptons. Note added

We have already checked that in the extension of the N = M = 3 case to the more realistic N = M = 6 case, where also quarks are present [cf. ref. [lo]], the anomalies which had forced us to introduce spectator fields are automatically absent.

Appkndix A SUPERTRACE

MASS

FORMULA

We shall derive here a general mass formula holding for any effective lagrangian of the type L=d(Si,

Sr)lD+S(Si)lF+h.C.+&g(~i,

46”) 3

(A.11

where Si are chiral superfields, c#Q(~“) is the 8 = 0 componeent of Si(ST) and cg denotes SUSY breaking terms which are characterized by a small parameter E.

608

A. Masiero,

Usi.ng the following

G. Veneziano

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formulae [ 181 for the spin-zero* and spin-4 mass matrices:

MZ,,,; = Ji’ v,+ = J;‘[(J;&+f”fz

a3d ad4 adI a+j

1 (A.3)

where

Jm =

a*d 84, a+: ’

(Jnm)j+=~Jnms J

(A-4) (sum over repeated indices is understood) one finds STr1M1*=C(-)*‘(2J+l)TrM: I = EJi’a@

a*g + Jii’fnfk T,.,,m,

(A.9

Qi

where

T,,,“,(J~~),i+-J~~(J”,)i+J;,‘(J,,),J;,l,.

L4.6)

In our case, J is diagonal, i.e., J,, = S,,,J,,,, and so one has T,,ii = (Jii3)((Jii)i12- Jii*(Jii)ii+ ,

while all the other entries are zero. In particular, took in the examples, one gets

if d(S, ST) = xi (SiS:)‘,

(A-7) as we

In any case, even if the additional term in eq. (AS) is present, it is of higher order in E, since, from the minimization conditions, one finds f=O(E).

(A.9

Notice that STr M* does not get contributions from SUSY breaking terms which are linear in the fields, such as gaugino mass terms, for instance. The final point concerns the validity of the supertrace formula for the light sector alone at O(E): (MF&=(M&++~gij+, l

(A.lO)

In the presence of SUSY breaking, there are also non-trivial mass matrices of M&,, and M$,3 type While relevant for the spectrum, such matrices do not contribute to the supertrace.

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where now we went over to the basis in which kinetic terms are canonical. Diagonalizing the SUSY mass matrices IMF12 and M& we find that to O(E) the shift of each eigenvalue is given by

WG)” 4MFIZ, = a”“+.

(A.1 1)

If we take the trace of this relation over the light sector, we get

In the case of SQCD with N = 2, M = 1 (subsect. 3.2), the soft breaking terms are all in the direction of the light eigenstates, so that one obtains STr M*l,,,= STr M211ight,explaining the validity of eq. (3.14). In the case of N = M = 3, part of the SUSY breaking lies in the direction of the component of 7rVwhich, together with rrA, acquires a heavy mass. In this case, the STr M211ightis slightly more complicated than the right-hand side of eq. (3.36). We wish to acknowledge many interesting discussions with R. Barbieri and M. Roncadelli on a number of alternative approaches to soft SUSY breaking which have finally led us to the method of this paper. References [I]

[2]

[3] [4] [S] [6] [7] [S]

[9]

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