NJL breaking of supersymmetric GUTs

N U CLEAR

P H VS I C S B

Nuclear Physics B409 (1993) 161—185 North-Holland

________________

NJL breaking of supersymmetric GUTs

*

E.J. Chun 1 and A. Lukas Physik Department, Technische Universitàt München, W-8046 Garching, Germany and Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, P.O. Box 40 12 12, W-8000 Munich, Germany Received 17 March 1993 (Revised 5 July 1993) Accepted for publication 7 July 1993

We analyze the breakdown of SUSY GUTs driven by Nambu—Jona-Lasinio condensates. Starting with the most general gauge invariant and pure kählerian lagrangian up to quartic order we solve the one-loop gap equation and determine the breaking direction. This is done for various classes of groups and spectra of fundamental particles which especially cover the most promising unifying groups SU(5), SO(1O) and E 6. Heavy masses for the fundamental as well as for the composite particles are calculated. The results are used to single out candidates which may lead to an acceptable low energy theory. In these models we discuss some phenomenological aspects and point out the difficulties in constructing phenomenological viable theories in our scenario.

1. Introduction The unification of gauge coupling constants in the minimal supersymmetric standard model (MSSM) [1] had directed great attention on supersymmetric theories. At present grand unified theories (GUTs) represent the only framework able to explain the meeting of coupling constants. Though GUTs are highly predictive in the gauge sector large arbitrariness enters the theory through the superpotential needed for spontaneous symmetry breakdown. Especially the explanation of fermion masses seems to be difficult without adding new structure like discrete symmetries or “ansätze” for the mass matrices to the theory. Dynamical symmetry breaking (DSB) can improve this situation and in principle

*

Supported by Deutsche Forschungsgemeinschaft. Supported by a KOSEF fellowship and the CEC science project no. SC1-CT91-0729.

0550-3213/93/$06.OO © 1993

—

Elsevier Science Publishers B.V. All rights reserved

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increase the predictivity of GUTs. The Nambu—Jona-Lasinio (NJL) mechanism [2] leads to scalar condensate formation through an effective four-fermion interaction which can trigger the symmetry breakdown. It was first shown by Bardeen, Hill and Lindner [3] that the standard model (SM) can be broken by this mechanism leading to a prediction for the top quark mass. However, a fine tuning caused by the quadratic divergence in the cutoff has to be made in order to obtain small SM masses. This can be avoided in the supersymmetric generalization of the NJL mechanism [4]. To form condensates supersymmetry (SUSY) has to be broken softly and the “four-fermion coupling” has to exceed a certain critical value. The quadratic dependence on the cutoff A is then replaced by that of the SUSY breaking scale ~i. Unfortunately, to reach the critical value the Fermi scale f has to be of the same order as L~i which seems to be an unpleasant feature for large cutoff values. This situation already appears in the dynamically broken MSSM [5,61 where a large cutoff is needed to produce acceptable small top masses. A solution to this problem in the framework of scaleless o--models was proposed by Ellwanger [7]. In these models the scale f is replaced by a dilaton field and therefore determined dynamically. Our motivation is to break the GUT symmetry using the NJL mechanism and to determine the phenomenological properties of the resulting highly predictive low energy theory. In this paper, however, we will concentrate on the first step in this scenario: breaking the GUT symmetry and analyzing basic structures of the low energy theory. This amounts to minimizing the self-consistent gap potential for various classes of groups similar to the analysis done by Li [8] for conventional GUTs with an explicit potential. The difficulties in building models with phenomenological acceptable low energy limit will already show up in this discussion. We start with the most general gauge invariant lagrangian up to quartic order which is pure kählerian. As an effective field theory it can be possibly induced by coset models or superstring compactifications below Planck scale. Apart from some connections to coset models we will not specify the origin. Likewise we do not address how the small Fermi scale can be produced but just put it in as a parameter. In this respect our scenario differs from a similar one discussed by Ellwanger [71in the context of an E8/SO(1O) x SU(3) x U(1) scaleless coset model. We adopt the idea to achieve a radiative low energy breaking triggered by condensate fields which was first proposed in ref. [12]. The plan of the paper is as follows. In sect. 2 the general structures and formulas are presented. Sect. 3 is mainly addressed to a discussion of adjoint condensates as used for the standard GUT breaking. We give a complete analysis for adjoint condensates formed out of the fundamental representations of SU(N) and SO(N) and some examples for the second rank tensors of SU(N). The formation of complex representations especially for the groups S0(1O) and E6 will be the subject of sect. 4. In sect. 5 we give the results for the heavy masses of the composites and discuss some phenomenological aspects.

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2. General framework Starting point of our analysis is the lagrangian

~=fd2O

~

+ (

2~A~AA~)~

fi

for for

(1)

çb~4’

4=~’’

where the gauging exponentials have been omitted. The two fields ~ and 4’ transform as representations r and r’ under the chosen gauge group G. Depending on the particle content and the mass structure of the theory they will be interpreted as new exotic fields or standard particles. In the latter case we will consider them as belonging to the heaviest family in the SM. group Soft breaking of supersymme2. The coefficients A split the try is parameterized scalar mass interaction term into byits the irreducible parts~i(labeled by r) corresponding to the decomposition of r X r’. We choose the normalization tr(A ~A) n. A cutoff A has to be introduced in order to define the theory. It refers to new physics at a higher energy scale which also represents the origin of the relative coupling strengths c~. E.g. in coset models related to the breakdown of certain larger groups G’ to G the lagrangian (1) can be identified with the leading terms of the Kähler potential on the coset space G ‘/G [9]. In such models the ratio of the couplings c~ is determined by group coefficients. In order to break our symmetry at the unification scale MGUT 10~GeV we have to demand A >> MGUT. A “natural” choice is A O(MPIanck). We see that the theory is specified by a small numbers of parameters, namely ~2, A, c~and the gauge coupling g. To study condensate formation one may introduce two auxiliary fields ~ and H~ in terms of which eq. (1) can be rewritten as [4] =

=

~‘=

fd2o d20[(44 + ~‘~‘)(1

+ [fd2O

H((H~

—

—

~2O2O2) + ~

-‘A~24)

+

h.c.J.

(2)

Eliminating H and H’ by their equations of motion leads back to the lagrangian (1) which implies the equivalence of the two formulations. The missing kinetic term for H can be developed at one-loop level. This can establish H and H’ as effectively independent degrees of freedom at energies below the cutoff A. Since

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only the fields H receive quantum corrections symmetry breaking is determined by their effective potential. We define the mass matrix M of the fundamental fields: 1 m(T)A

=

—

M

=

Crm(r)AA~).

(3)

In the following we will work in a one-loop approximation for chiral loop particles taking tree-level gauge effects (i.e. D-term effects) into account. The effective action can then be written as (4)

~ff”gap~”D,

where VD is the usual D-term VD

=

~g2D~D~, Da

=

r

The NJL self-consistency condition at one-loop level leads to the gap equation avgap

(r)B

r

87r2nf2

3m~* —

Xtr

AB*(~~)M (1M12+42)ln

A2

1M12

—

1MI2+42

In

A2

MI2

0.

=

For A>> M

I, ~

3Vgap

it can be approximated to ~2

—

Ca —~--—tr AB(t)*M~ln

A2 2

IMI

n

=0,

—

1

+~2

(5)

with a

=

~2/8~2f2.

Neglecting the group structure of eq. (5) it can be seen that a nontrivial solution presumes a small Fermi scale f ~ 0(4) [4].This expresses the fact that supersymmetry protects the theory against mass generation. A consequence is that possible higher dimensional operators are suppressed much weaker than in the non-super-

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symmetric case. E.g. Kähler potentials derived from coset models usually contain higher than quartic terms. Especially VEVs of the order MGUT naively inserted in the higher dimensional terms lead to a disastrous increasing. We do not see any satisfactory solution to this but we will just assume that either higher dimensional operators are absent or suppressed by some large scale 0(A). For the explicit solution of the gap equation we set a 1 so that —

—

=

f2=42/8~2.

(6)

The interaction strength is then determined by the dimensionless couplings 0(1). The rough behavior of the VEVs m is given by m2/A2

exp(

—

1/c2)

Cr

=

(7)

showing that even small changes in Cr (e.g. caused by group theoretical factors) can lead to large differences in the mass scale m. To achieve a breakdown scale safe below the cutoff c < has to be chosen. For a reasonable GUT scale the value of c should not be much lower. Integrating the LHS of eq. (5) leads to the potential ~-

1 “;ap=f2

A2

1m12ti

MI2 In

1

22

.

(8)

The minimum value can be obtained by combining eqs. (5) and (8):

~ap,min

=

—

~—tr I M

(9)

2•

It is well known that in the picture of renormalization group compositeness shows up as a Landau singularity in the Yukawa coupling. One may ask if three orders of magnitude below the cutoff (assuming A MPIanck and MOUT 10~ GeV) we are already in the nonperturbative region of this coupling. However, applying the naive bound aYUk ~ 1 and running down aYUk shows that we are safe below this value for typical examples. On the other hand we can be sure that we are dealing with a relatively large coupling aYUk and a gauge coupling of g 2/4~. 0(i). For that reason neglecting gauge loop effects in the gap equation is a much better approximation at the GUT scale than at the weak scale (assuming a comparable cutoff). For masses m O(MGUT) the two parts of the effective potential eq. (4)VDare 2M~UT)and obviously of very different order of magnitude: ~ O(4 O(M~UT). Therefore large VEVs can only be generated in directions with vanishing D-term (flat directions). This is required anyway in order not to break SUSY at high energy. Our strategy will be first to determine the flat directions and then solve eq. (5) restricted to them. =

=

=

=

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Some general remarks can be made about the condensate masses. High scalar masses of O(MGUT) will only be induced by the D-term contribution to the effective potential. The corresponding fermion masses arise from the coupling to the gauginos. Since our D-term does not break supersymmetry, we may expect that the heavy fields are those which become part of the massive gauge multiplets of broken generators. The remaining light scalars can receive masses of 0(42) from the gap potential. Masses of the same order for the fermionic partners are implied by the tree-level term in eq. (2).

3. Condensates in r X r In this section we consider the GUT breaking by adjoint condensates as is favored for the usual GUT breaking. To get adjoint condensates we choose a representation r and a complex conjugate representation r. The composite fields from r x i generally consist of a singlet, an adjoint and other fields if any. Tuning to a desirable condensate is now achieved by arranging the coefficients Cq where q runs for the singlet, the adjoint and the other condensates. First task is to pick up the flat directions. When cq is taken to be as large as to produce VEVs of O(MGUT), the absolute minimum of the effective potential is sitting on the flat direction as can be seen from. eq. (9). The obvious flat directions are those with zero weights. Taking these, the mass matrix becomes diagonal: M1~

=

~

(10)

Cqm(a)Ad~)i5ij.

A, q

Here d~ are the entries of the diagonal matrices A for a representation q. Note that a field in the adjoint representation can take the form of a diagonal matrix without loss of generality. Therefore, if only a singlet and an adjoint are present all flat directions are taken into account. If there appear more representations the directions with zero weights do not cover all flat directions. Including the missing directions makes the mass matrix off-diagonal. Since this complicates the gap equation much we restrict the discussion to the zero weight directions for simplicity. Since the mass matrix is now diagonal one can find explicit solutions for the gap equation. Let us now denote M,1 u,5,1 with u, Cqm~”~d~).Then the gap equation (5) reads =

=

*

u,+ ~Kjjuj(lnIujI2+1)=0,

*

Here we put A

=

K,3= ~

1 and neglect the SUSY breaking parameter ~i.

(11)

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167

where K~J9~ LA ~ The matrices denoted by K~”~ are the projectors for the subspaces corresponding to the representations q. They fulfill the following orthogonality and completeness properties: =

K~”~K~’~ ~5~P~(P),

~K(~) q

=

=

1.

(12)

To find the global minimum it is useful to notice the existence of a bound of the gap potential. Consider the case where only one cq is non-zero and define u z~exp[ +(1/C~+ 1)]. The variables z 1 being in the space defined by q should =

—

satisfy the constraint and the gap equations which read 2=0. ~

(13)

c~EKt9~z~ lnIz~l

Multiplying z

1 on both sides of the second equation results in L, I z1 I 2 in I I 2 0. Now one obtains the bound ~ap L, I z I 2 N. The value N is taken if and only if I z1 I 1 for all i. For the proof let us consider an (N 1)-dimensional surface defined by the equation La1 ln a1 0, where a1 denotes I I 2 On this surface the maximum value of La1 is given by N when a1 1 for all i. This can be shown by minimizing the function with Lagrange multiplier A: f(a1) La, + ALa, ln a1. Minimization shows that A 1 and a, 1 for all i. There can be possible minima on the boundary. However, the boundary is given by a, 0 for some i and is therefore described by a similar equation with lower N. Since our statement holds for N 2 an induction in N completes the proof. Now let us go to the case with several cq being non-zero. First we normalize the u, by taking the largest cq, say C1: u, z, exp[ ~(1/c? + 1)]. Then the gap equation becomes =

—

=

—

=

=

=

=

—

=

=

=

=

—

(14)

zi+~Kijzj(lnIzjI2_~)=0.

Multiplying the inverse K-matrix K~ tion leads to

I

2

z~I

ln I z,

2

=

Lq(1/c~)K~”~ and z to the above equa-

=

~

(I I) I -

I2,

(15)

where z~ K~z. Since Ci ~ cq we have La~ln a, ~ 0 with a, I z, 2~ The absolute maximum value of La, under this condition is again given by N and it is taken if and only if a, 1 for all i. Comparing with eq. (15) shows that the existence of such a solution implies the existence of a z with I z, I 1 for all i and z K~’~z. Obviously the converse also holds. If one, therefore, finds a solution to these two constraints it represents the absolute minimum of the gap potential. =

=

=

=

=

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NfL breaking SUSY GUTs

These minima correspond to condensate formation in the representation with the largest cq. If some cq take the same value the above argumentation holds for the sum of the corresponding representations. Clearly, for the singlet part such a solution always exists since its K-matrix is given by Kj”=l/d,

(16)

where d dim(r). This implies that for any representation r a dominating singlet coupling leads to a vacuum pointing into the singlet direction. Indeed, we find that it also exists for the other condensates formed out of the fundamental representations of SU(N) and SO(N). N of SU(N). Here we can have two condensates: singlet and adjoint. The adjoint projection matrix reads =

K~

=

—

K)~

=

—

1/d.

(17)

Depending on the couplings for the two condensates, denoted by ~ and Cadj, we find three different patterns of solutions. If c~1~ dominates, only a singlet solution =

is allowed. If

Cadj

exp[

—

~(1/c~~

+

1)]

for all i

(18)

is larger, the solution takes the form

u1=zjexp[—~(1/C~dj+1)I, with

~z,=0.

(19)

where the z, have modulus 1. Aside from a global phase, we have N 2 degrees of freedom which gives rise to degenerate vacua. Depending on the number of equal z, SU(N) can be broken down to lj~, (S)U(n~)X U(1Y where L~n~ N and the are constrained by ~ ~ Lothers n~.The number s is chosen to complete the 4, rank of the group to N 1. ForFinally, instance, SU(5) down to U(1) SU(2) X U(1)3 or SU(2)2 x U(1). in the casecan of ~be broken Cadj each phase z, can take an arbitrary value. Here, the breakdown of SU(5) to the SM can be possible in contrast to the previous case. N of SO(N). We restrict ourselves to even values of N The condensates of N x N’ consist of a singlet, an anti-symmetric, and a symmetric part. The projection matrices are given by —

=

—

=

~.

i)),

K~t1~5Ym=Diag((~ ~

K5Ym=Diag((~ *

~

~)).

(20)

For SO(N) with N odd the breaking patterns are similar to SO(N — 1) apart from an additional U(l) factor.

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breaking SUSY GUTs

If c51~is the largest, one finds a singlet solution as above. For dominating the solution takes the form

169

Cafltjsym

z= (e’~1,...,e’~N/2,_e’dl,..., _e’c’N/2),

(21)

where the values of ~ ‘s are arbitrary since the traceless condition is already fulfilled. The largest unbroken subgroup is SU(~N)x U(1) if all phases are the same. Depending now on the choice of i’s, SU(+N) can be broken further without any restriction. If c~,mexceeds the other couplings, the solutions are given by with

~

~e~’~=0.

(22)

p

They result in the breaking patterns SO(N) fl~,,SO(2n~,)where L~n~, with umax ~ Lothers np. For the SO(10) case and a dominating Cafltjsym the interesting patterns —~

S0(10)

—

=

JSU(5) xU(1)’ t.SU(3) xSU(2) xU(1) xU(1)

are allowed. Here we interpret 10 and 10’ as exotic fields to cause a breaking of SO(10). In the next section, we will consider the condensates 10 X 16 and 16 x 16 for a more realistic model with standard fermions and a broken U(1)’. Observe that for condensates in the symmetric representation 54, the conventionally allowed Pati—Salam path SO(10)

-~

S0(6) x SO(4) SU(4) x SU(2) x SU(2)

is excluded.

So far all our examples had a highly degenerate almost all of its T (r is vacuum the rankwith of the group). Only points leading to the unbroken subgroup U(1) for specific choices, larger unbroken groups could be obtained. This behavior is related to the existence of our “general” solution discussed above. As we will see for higher representations r it does not exist any longer in general and the vacuum structure changes. We restrict ourselves to the case where r is a second rank tensor representation of SU(N) in NXN=

[+(N—1)N]A+

[~(N+1)N] 5.

This may be especially of interest for the ‘°A representation in the standard SU(5) GUT. Moreover, we consider condensates in the singlet or adjoint part of r X r only. Couplings Cq of higher representations are set to zero. As a first step it is

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useful to work out the constraints on the vacuum which follow from this requirement. Here we denote the diagonal entries of the mass matrix by w~, i >~jand w,(f’), i >j for the symmetric and anti-symmetric part, respectively. The corresponding matrices d can be expressed by those of the fundamental representations like d~1J) dai + daj, i >~jand d~J) da, + d~1,i >1. This can be used to com‘~

pute the projection matrices K: 1

i>j, k>l,

K~J~~(kl)= ~

1

+K1k+K~~), i~j, k~ 1.

K8~(ki)= N+2j

(23) 5~)w~’~=

Here K is given eq. (17). Working out the constraints (Ks~~ + K’ w~5~ leads to thebyparametrization w~71=~(u,+u~),

i>j,

w~=~(u

1+u1),i>~j,

(24)

with arbitrary u,. The breaking pattern can be directly read off from the vector u. E.g. the singlet part vanishes for L u, 0. With this parametrization it can be shown that apart from trivial cases (3 E 3 X 3 of SU(3) and 6 E 4 x 4 of SU(4) SO(6)) our standard solution I w,1 I 1 for all i, j, does not exist for the adjoint. The gap equation and the gap potential are given by =

=

2(N+e)

~

N

(25)

and 2(N+e) N

~

2

(u)

,

(26)

I

with the rescaled quantities u, exp( -~-(1/C~d~1)) and c + 1 or 1 for the symmetric or anti-symmetric part, respectively. Both equations only depend on one combination of the couplings, namely =

~

—

1 2 C~1,,

—

1 2~ Cadj

=

—

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NJL breaking SUSYGUTs

TABLE

I

Vacuum patterns for adjoint condensates formed out of symmetric or anti-symmetric second rank SU(N) tensors N

Anti-symmetric (n 1, n2, n3 = n3)

(n1, n2, n3, a3,

fl4, p14)

3

(1,0,1)

(1,0,1,1,0,0)

4 6

see SO(6) (3,0, 1) (4, 0, 1)

7

(5, 0, 1)

8

(5,1,1) (4, 0, 2)

(2, 0, 1, 1, 0, 0) (1,0, 1, 1, 1, 1) (2, 0, 2, 2, 0, 0) (3, 0, 2, 2, 0, 0) (3,0,3,0,2,0)

5

*

Symmetric

*

The two patterns are degenerate.

Let us first look at the case ~ 0. The potential then consists of the first term in eq. (26). A numerical minimization for small values of N results in the following pattern for the absolute minimum: =

=

=

where p

=

3, 4, p~E ~

~, ~ ~,, ~,

for

n1

—p~

for

n2

~t~’p

for

n~

p~e’~ for

ii~

(27)

and ‘p~* 0, ~r. This causes a symmetry breaking

SU(N) —sSU(n1) XSU(n2)

X

flSU(n~) XSU(n~).

(28)

0A of SU(5) e.g. the induced The numbers nj,, are SU(3) given xinU(1)2. table We 1. For thethat ‘ breaking is SU(5) stress contrary to the case of fundamental representations we arrive at unique minima. As an example of what can happen for c~~ 0 we take again WA of SU(5). If f3 <0 (the singlet coupling dominates) we get a pure singlet solution with ~, exp( ~/3). In the opposite case the pattern of ~ and the unbroken group are unchanged with respect to the situation for c 1~ 0. However, there appears a small singlet admixture which vanishes for /3 (c~ 0). This is shown in fig. 1. To conclude this section we remark that in all our examples the fundamental fields receive high masses E1 u~4,. Therefore, e.g. the weak doublets in a 10 of SO(10) used to form condensates cannot serve as low energy Higgs fields. Composite fields as Higgs fields will be discussed in sect. 5. —*

—

—

=

—

=

—~ ~

—~

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/ NJL breaking SUSY GUTs

4. Condensates in r X r

—

the examples SO(10), E6

Unfortunately the flat directions for complex representations cannot be isolated in general. Moreover, since the mass matrix M will not be diagonal any longer, the consideration of all flat directions needs the calculation of logarithms of large matrices (e.g. 27 X 27 matrices in the E6 case). Therefore we restrict the discussion to the most important examples SO(10) and E6. Our method will be more pragmatic and guided by phenomenology: only the flat directions among the SM singlets will be taken into account. Unlike to the above section with r X r condensates, we have here different features of the flat directions and the solutions of the gap equation. The vacuum points into a D-flat direction if contributions from two or more irreducible representations are arranged to cancel each other. As a consequence we may expect some restrictions on the couplings c. Let us, as a toy example, consider the case_with condensates from 3 X 3 of SU(3). We introduce two couplings c1 and c6 for 3 and 6 respectively. Then, one can show that the flat direction is given by vacuum expectation values for the condensates 3 and 6 satisfying v611 ~fv~/ V~ [16]. The equality should also be compatible with the gap equation to produce the high scale. In general, this can be obtained in some restricted range of the parameter space for c1 and c6 only. Since the S0(10) model has to include fermions in the 16 representation, we consider 16 X 16 condensates. However, we will see that as the only condensates they possess no flat SM direction. The most economical way out is to add particles in the 10 which leads to condensates in the 16 x 10 and 10 X 10. The latter were =

~

~O~:

~81O

Fig. 1. Dependence of the singlet strength I u51,, 1/ uI on /3.

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173

already discussed in sect. 3. E6 is known as the unique GUT group which admits a spontaneous breakdown with mass giving fields only [13].For that reason we will in contrast to our other examples not discuss adjoint breaking of E6 but concentrate ourselves on condensates in 27 X 27. We begin our analysis with E6 and take r 27. Condensates can be formed in the representations —

=

27X27—~(~+351)s+351A. Since we consider one family only the two symmetric parts appear. Then the lagrangian possesses the structure

fd2o d2~(~727+ ~I(27x27)~2+

~‘=

~I(27x27)3~l~2).

(29)

3)R X For the explicit calculation we use the maximal subgroup SU(3)L X SU( SU(3)~of E 6. Notation and group properties are explained in the appendix. A lagrangian of the above type but with specific values for c~ and c~ is obtained in a coset model based on E7/E6 x U(1). General expressions for the Kähler potential of coset models involving E6 can be found in the literature [11]. The scaleless version produces no 351 part. For unbroken U(1) charge expansion up to quartic terms and rearranging the expression according to the representation contents gives eq. (29) with

c~,=4, c~=—1. Since c~is negative, no 351 condensates are formed. But with VEVs in the 27 only, no SM invariant flat directions occur (see below). Therefore a breakdown at high energy is impossible. In the form of eq. (2) the lagrangian looks like

~=

Jd20 d2~[(L~L~ + two perm.)

+

(~ci~ +

+

{fd2o{H:(H~

—

~

ab~~,aP +

+

(i~H~ + two perm.)

+

—

—

+

two

perm.)~

1EabCELI3LY))

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+~(~_

breaking SUSY GUTs

~

+~(~~_ ~(Q7-~Q~))

+two~erm.J +h.c.}.

(30)

~,

Leptons, left and right handed quarks are named by L, Q and respectively 351s D. The two omitted parts of the while the composites are called 27 H, lagrangian are obtained by the index permutation a a i a. We observe that the lagrangian (30) possesses an anomalous global U(1) symmetry with charges Qgi(27’ H, ‘1) (1, —2, —2). Obviously VEVs in H or 1 break this U(1), resulting in a large axion scale fa O(MGUT) which is excluded by cosmological considerations. One way out may be that higher dimensional operators with dimension less than 10 can break the U(1) symmetry. Even if they are strongly suppressed they can affect the upper cosmological bound on the axion scale [15]. The price will be that such an approximate U(1) will not solve the strong CP problem any longer. Seven SM invariant VEVs are contained in the two condensate representations: ‘~

—~

—‘

—~

=

=

m 1=H~],

m2=H~,

m3=cP~, m4=cP~ m m 5 ~a,b~3,3 m a,/3— 7 ~j~ab=2,2 a,~—3,3~ 3)R transformation which acts on the indices a 2, 3, the VEV m 3,3’

a,j3=3,3’ ~13a,b=2,3

6

=

=

=

=

Using an SU( 6 can be rotated to zero. We sum up the six remaining VEVs multiplied with the corresponding matrices A from the appendix, and arrive at 2 Diag(A, A, B, C, C, C, D, D, D), (31) =

MI

with

1m 2+Im 2 1I 0 2I

0

Imi_I

0

0 2

m 2m1 2 Ith

B=Diag(0, th~ C=Diag(0, 0,

D= 0 0

2)~

Imi+ 2+1m

2+ I

0

0

2),

5~

0 2

Im

2+I *

m1+m2+

*

m2~m1~

Imi+I 2

*

Im2_

2

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E.J. Chun, A. Lukas

175

NfL breaking SUSY GUTs

and the definitions m

12+= ~(CHml2+c~\/~

m34),

~

+

rn57

rn3,4),

(32)

c,1,m57.

=

The potential value at the minimum is given by

2) + 6(1 m 2)). m2 I 1~ 2 + Im2+ I Fortunately I M I 2 contains at most 2 x 2 blocks so that the computation of the logarithm is possible. After some algebra one obtains explicit gap equations for m 1, m.,. Since rn5 and m7 decouple they can be chosen to their trivial value zero or to 2/A~ exp( 1/c~ 1) (33) ~ap,min

. . .

=

_f2(I ~ 12 +

I

~

+

2

4(1 m1

~2 +

,

=

I

~

—

—

independently. The remaining VEVs are determined by m 12~

=

(3c~+ 2c~)l± 2(C~ c~)l. —

~ 3(c~ c~)l+

(3c~+ 2c~,)L

—

m12~

(34)

m12.

where 2 l~=ln42+ImI2+ImI A -

2~

These equations have to be combined with the flatness conditions which we extract from the D-term expressions given in the appendix: 2

2

+

3

3

2 ~

2

+

~2

—

~ (~‘,33 2 33

—

~ n~22 2

—

‘~~33

cD~cP~+H~H~=0.

(35)

As expected, we obtain four (real) conditions corresponding to the four broken E 6 generators which are SM singlets [16]. The first two equations belong to the broken central charges. They can be computed much easier using the charge values given in ref. [171.

E.f. Chun, A. Lukas /

176

NJL breaking SUSYGUTs

0.25 0.2

0.15 2 cc’s 0.1 0.05

0 0

0.025

0.05

0.075

0.1

0.125

0.15

0.175

2

Fig. 2. Relation between c~and

CH for E

4

6 breakdown.

The VEVs according to their symmetry properties can be divided into two groups: the VEVs with (3, 3) indices and the ones with (3, 2) indices. These groups also show up in the structure of the above set of equations. By direct calculation it can now be seen that gap equations and flatness conditions are non-compatible for non-zero VEVs in both groups. Therefore we end up with two possible patterns: —

—

(1)

‘I~, P~,H~# 0,

(2)

P~,

~,

H~# 0,

others zero, others zero.

The two cases are completely symmetric with respect to the remaining nontrivial equations: one of the first two flatness conditions has to be combined with one part of eq. (34) (the one with index 1 or 2) and with one choice in eqs. (33). Counting the degrees of freedom the system turns out to be overdetermined by one equation. Consequently a relation between c~and c~results which is shown in fig. 2. A breakdown only takes place if c~and c~are chosen to lie on this curve. We emphasize that this constitutes a fine tuning problem: already small deviations from the curve produce a D-term O(M~UT)which destroys the minimum at large VEVs. If, nevertheless, we accept the fine tuning, the breaking pattern reads 2)L X SU(2)R x SU(4)~. E6 X U(l)gi SU( One phase, say the one of H~,can be rotated away by gauge freedom. Then P~has to be real in *

—*

*

order to fulfill the gap equation.

E.J. Chun, A. Lukas /

NfL breaking SUSY GUTs

177

Some additional ZN symmetries arise from a mixing between U(1)~and internal U(1)’s. Their charges are generated by QN

=

‘~Qan+ aQgi,

with (N, n, a) (2, 0, 1), (3, 1, 1). Qan corresponds to the anomalous part of the global U(1) symmetries in the MSSM [18]. It is given by Qan ~(Q’+ Qt) where E 6 S0(10) x U~(1) SU(5) x U,(1) x U~(1).The detailed behavior of the VEVs is complicated, but for reasonable small values of c11 they follow eq. (7) quite closely. Large masses for all exotic particles in 27 except for the right handed neutrino (RHN) are guaranteed (see eq. (30)). The above analysis may be used to look for the breakdown of SO(10) as well. The relevant decompositions for SO(10) are =

=

—*

—

—~

120A’

16 X 16 (10 + 126)~+ 16x 10=T~+144. =

Without the terms forming adjoint condensates which were discussed in the last section, the lagrangian reads Sf=

fd20

d2~(T~ 16

2

(16 X 1:)1o12 +

+~.I(16x10)j~l+ ~I(16x10)

1(16 X 16)12612

).

1~l

(36)

The explicit expression can be obtained as a part of the E6 lagrangian (30). We denote the condensates by H 16, H 10, cl’i 144 and ‘J~ 126. Then the model possesses two independent global U(1) symmetries. They can be split into an anomalous and non-anomalous part: Qgian(16, 10, H, c~) (1, 1, —2, —2, —2, —2) and Qginonan(16, 10, H, I~,1, ~) (4, —5, 1, —8, 1, —8). The lagrangian (36) plus an additional I 10 x 10 2 term can be interpreted as part of the E8/SO(10) x SU(3) x U(1) coset model [191. In this model, however, the 10 x 10 coupling constant as well as CH and c~ are negative so that no consistent breakdown can occur. This situation changes in the scaleless version discussed by Ellwanger [7] where c~0~10/E~/c~ 1/2/1 and c~ 0 is obtained. Three SM invariants appear in the condensate representations which correspond to m2, m4 and m7 of the E6 example. The combinations m ± can be defined like in eq. (32). Independently the calculation can easily be performed using the maximal subgroup SU(5) x U(1). According to the decompositions ‘~

‘~

~,~,

=

=

=

=

10—~5(2)+5(—2), 16—*10(—1)+5(3)+1(—5),

(37)

178

E.f. Chun, A. Lukas

/

NfL breaking SUSY GUTs

we write

~ 1o=(~, The three SM invariants

1~1’

~‘).

H 0 and I~are then given by

rn2

H0

‘~

3

5

~

~a3

a=1

m7

—,

~

a=4

—~

Reading off the matrices A, 1

0010

0010 1 A~=7==- DO 5,

—

10

5,

00

1

—

00 5~

5~

A~ Diag(0,...,0, 10 =

1

o,...,o, 5

D=Diag(2, 2, 2, —3, —3), we arrive at

2=Diag(0,...,0, 2E~,m~),

IM16~16I

IM

2

=

16~10I

Diag(m~,rn~,m2, m~,m~,0, 0, 0, 0,0).

For rn 7 we get m~

1

1 (38)

and for rn ±the E6 result (34). The flatness condition is given by the second line in eq. (35). This can also be seen from the U(1) charges in eq. (37). The only difference to E6 is the appearance of a third independent coupling E.~,which determines rn7. Therefore the structure of solutions is similar: large VEVs are only

E.J. Chun, A. Lukas

/

179

NfL breaking SUSY GUTs

generated if the three couplings fulfill one constraint. For c~, E~this constraint is just represented by fig. 2. The breaking chain then reads =

SO(10) X U(1)gi,an

X

U(l)gi,non~an

4

SU(3)~ X

SU(2)L X U(1)~X

U(1)gi,rem,

where the remnant U(1) is given by Qgi,rem 3QR Qgi,an + Qgi,non-an and QR ~(Qr + Y). Moreover, discrete symmetries remain which we did not note explic=

—

=

itly. Observe that the additional gauged U(1)’ left over in the case of adjoint condensates only now is broken. Contrary to the E 6 example the RHN receives a heavy mass m7, and consequently no exotic fundamental field remains light. We finish this section with a remark about the fine tuning problem. Suppose in the E6 example an additional fermion—mirror-fermion pair 27’ and ~i’ is introduced. The condensates in 27’ x ~!‘ can generate heavy masses for these new fields. All complex condensate formation in 27 x 27, 27’ x 27’ and ~i’ x ~7’ with couplings (cH, c,~),(c~,c~) and (~,~) is described by eqs. (33) and (34). Choosing now (CH, c~) (~,F~,)* (c~,c~)produces a vacuum in a D-flat direction since the mirror contribution enters the D-term with the opposite sign. Moreover, all SM invariants can receive large VEVs allowing for a direct breaking to the SM ~. An analogous modification is possible for SO(10). In principle, of course, setting (CH, c~) (~,E~)is a fine tuning as serious as the former one. However, such a relation might be easier to realize in terms of an underlying theory than the complicated one represented in fig. 2. =

=

5. Masses of condensates and phenomenological discussion The general structure of condensate masses was already mentioned in the end of sect. 2. Let us now be more specific in order to discuss the possibility of radiative induced breakdown of the low energy theory triggered by condensates. Clearly the gauge singlets in r x i~receive no large mass. For the adjoint part it is easy to show that any supermultiplet which corresponds to a broken generator contains one (real) massive mode and a Goldstone boson. Looking at our examples we see that there exists only one way to achieve a breakdown of SU(5) to the SM: a “degeneracy” in the singlet and adjoint direction of 5 x 5 has to be implemented by setting c~, Cadj. Taking into account gauge corrections, however, we expect this degeneracy to be lifted in favor of the singlet direction. The smallest groups which can lead to SM(XU(1)’) by r X 1 condensates are therefore SU(6) and SO(10). But even if the problem of a highly degenerated vacuum in these cases will be solved by higher radiative corrections, no candidates for the low energy Higgses =

*

Here we assume that the breaking generated by condensates in 27’ x~’ leaves the SM invariant. This can e.g. be achieved for a dominating singlet coupling.

E.f.

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Chun, A. Lukas

/

NfL breaking SUSY GUTs

are present with adjoint condensates only. A discussion of condensates contained in complex representations is therefore in order. From the D-term expressions in the appendix we can read off the 57 massive condensate modes for the E6 model: m~’H~ + m3*~ 2rn5~~ + h.c., —

m~H~ +rn~~

—

y~ ~

m~H~ +rn~cJi~ ~ 3 V~~

+

m~H7+ ~

—

rn5~, a, a

—

~

b

=

1, 2, i

=

=

1,2,

1, 2, 3,

rn~J~/~

~

/3=1,2,i=1,2,3, m~u+rn 3b with

and

(ay, ca)

=

(u,v)=(,~~)

(11,22), (22, 11), (12, 21), (21, 12),

1

=

1,2,3.

One uncolored Goldstone mode is given by the imaginary part corresponding to the first expression. The others are obtained from the second and third expression by changing the sign of the last term ~. We do not quote the colored Goldstone modes which are a bit more complicated. The above results hold for the case rn1, rn3, m5 ~ 0. The other case is obtained by exchanging a 3 a 2. First we remark that the remaining SM singlets rn2, rn4, rn7 stay massless. They may induce a further breakdown to SM. If this happens at a scale lO~GeV a RHN mass desirable for the MSW solution of the solar neutrino problem is generated. Obviously both pairs of weak doublets which couple to the standard particles remain light. Clearly also in our modified E6 model with mirror-fermions light weak doublets are present. The SO(1O) result for 16 x 16 and 16 X 10 corresponds to the above E6 result for m2, m4, m7 ~ 0. A light pair of weak doublets is guaranteed. We see that the SO(1O) model with 16 and 10 fields possesses the desired features. Since the adjoint condensate in 10 x 10’ renders the 10 superheavy, all condensate formation has to take place at the GUT scale. The RHN mass will therefore be of O(MGUT). So far we did not discuss other condensate fields than weak doublets. We emphasize that the number of chiral condensate multiplets which disappear as massive or Goldstone states equals the number of broken generators. For realistic examples which need several condensate fields one may therefore expect many new particles at the SUSY breaking scale. Especially the color triplets belonging to =

*

~-‘

Their orthogonality to the massive modes is guaranteed by the flatness condition.

=

E.f. Chun, A. Lukas

/

NfL breaking SUSY GUTs

181

the weak doublets remain light. This is no surprise because we could not expect the dynamical mechanism to generate a doublet—triplet splitting which in ordinary GUTs is imposed “by hand”. Proton decay via triplet exchange needs a mixture of triplet and antitriplet. Though our global remnant symmetry forbids such a mixture for SO(1O) we expect it to be induced after elektroweak breakdown. Even if this occurs in higher loop order it will destabilize the proton at an unacceptable rate.

6. Conclusion

In this paper we analyzed the dynamical breakdown of supersymmetric GUTs triggered by NJL condensates. Motivated by the standard GUT breaking with adjoint representations we first looked at condensates in r x 1. For r being the fundamental representation of SU(N) or SO(N) we found general results: The vacuum is contained in the representation specified by the largest coupling. It turns out to be highly degenerated breaking to U(1Y~(r is the rank) for almost all of its points. Choosing special points in the vacuum can produce larger unbroken groups. For SU(5), however, only U(1)4, SU(2) x U(1)3 and SU(2)2 x U(1) are admitted whereas SO(10) can be broken to SM x U(1)’. The behavior of condensates of higher rank tensors were found to be very different. Adjoint condensates from second rank SU(N) tensors develop a unique vacuum and a small singlet admixture occurs for dominating adjoint coupling. The ‘°A of SU(5) e.g. generates a breakdown to SU(3) x U(1)2. A realistic dynamical breaking cannot be achieved using adjoint breaking only. First of all SU(5) cannot be broken to the SM in this way (using the small representations). Starting with larger groups, however, forces one to break the additional U(1) factors and to give mass to the exotic fermions. Moreover, with an adjoint only no candidates for the low energy Higgses are present. Therefore we investigated complex condensates for the examples SO(10) and E 6. Since we restricted the calculation to the SM invariant directions in a strict sence our results only give necessary but not sufficient conditions for the determined breakdown. To guarantee D-term flatness in these cases, VEVs in different representations have to cancel each other. We only found nontrivial solutions under this requirement if certain relations between the couplings hold. As a consequence the couplings have to be find tuned to obtain a breakdown at the GUT scale. Accepting this, E6 with the fermion field in 27 can be broken to SU(4)~x SU(2)R X SU(2)L. A breakdown of SO(10) to the SM may be achieved by taking an exotic field in 10 as well as the fermion field in 16. Typically our models possess additional global symmetries which after the breakdown lead to a large scale axion. Combinations of these symmetries with gauged U(1) factors can remain in the low energy theory. For SO(10) e.g. we find a remnant U(1). Alternatively we discussed a model with additional mirror fermions. Then setting

182

E.f. Chun, A. Lukas

/ NfL

breaking SUSY GUTs

coupling constants and mirror coupling constants equal guarantees D-term flatness. In this scenario E6 can be broken to the SM directly. In all cases we find light composite doublets which can trigger a further breaking. Large composite masses are generated through gauge terms only. Therefore the number of composite chiral fields which disappear from the low energy theory equals the number of broken generators. Consequently a large number of composite particles at the SUSY breaking scale can be expected in typical examples. Especially we find that the color triplets accompanying the weak doublets remain light. They lead to fast proton decay and are clearly unacceptable. Therefore, solving the doublet—triplet problem is no longer merely an aesthetical problem but necessary to obtain a realistic theory. We are grateful to Y. Achiman and H.P. Nilles for discussions and suggestions. A part of this work was performed when E.J.C. was visiting the Physics Department of the National Technical University in Athens. He thanks for its hospitality.

Appendix A. E6 group properties and notation In this appendix we summarize the main group properties of E6 used in sect. 4 in terms of the subgroup SU(3)L X SU(3)R x SU(3)~.The SU(3) factors are indexed by a, /3,.. . a, b,... and i, ~ respectively. Usually only one third of the results is written explicitly. The other two parts can be obtained by the permutation a a i a. The E6 representations decompose like

,

—~

—~

—~

27 —s (3,3, 1) + (3,1,3) + (1,3,3), 78

—~

(8, 1, 1,)

+

(1, 8, 1)

+

(1, 1, 8)

+

(3, 3, 3)

+

(3, 3, 3),

351~—~ (3, 3, 1) + (6, 6, 1) + (3, 3, 8) + two perm.

We denote 27

(u~’,u~,,u~’),

78

IQO ~ j3’

a6’

i j’

aai’

aai\)‘

35l~ (w:, w,aJ, w~j,two perm.). Using the E6 algebra given in ref. [14] its action on the 27 can be computed: Qaai(t~)

=

Qaaz(uf~)=

~~CabcU~~

E.f.

Chun, A. Lukas

/

NfL breaking SUSY GUTs

k)

183

~UC+~ISU~, =

Q~(u~) —~u~ =

—

Q~(u~) =

0.

(A.1)

The various components of u and w can be expressed as bilinears in v ~

a

(TI

Ua

a..... Wa —

*:

j

(~1)~ ~‘a

—

27 [10]

—

~EOPYEabcVi~V~),

+

~EYE~

~JL’k

Va~

\/~~V[’V~

6CV~Vy~),

~ ai... a Waj

L~Va

—

—

(A.2)

From these equations it is easy to read off the matrices A appearing in the lagrangian (1):

ii

A (a 27~,a)

I

—E”~”~E~~

7~I

0

0

0

°

~c5b

~k~y3~a jac

1

0

0 0a c 1~k~/36a

0

3

~

3a~I3+8a3/3)(6c8d+~c8d) ~YI

ji *

o

0

/I

o o’~

0

0

0

0

o ~13~a(615k

~

j

I

to A351 (al)IO ai

0

~af3y

A35i~a)=~

a/3’

I

/

abc

I

~

(

0

0)

0 —

I~1~J)

3ik

Observe that with these definitions w~and w~’Jare not normalized to one for a

=

b, a

=

/3 or i

=

j.

If we quote formulas with specific values for the indices we reintroduce the proper normalization.

E.f. Chun, A. Lukas / NfL breaking SUSY GUTs

184

Moreover, together with eq. (A. 1) they allow to compute the action of the group generators on the 351 components: Q~(w~’)

—8~w~ +

=

—o

Qa(WC~ 3\ k) — =

Q~(w~)

—

+

=

—

ki\_~

p~Wcd) — QaI

Q~ (we)

=

—

—

w~7+

+

Q~(W7~)=

Q~(w~) =

—

Q~(w~)

=

—

Qaai(~4’~) =

~

~

(A.3)

5~~abcWi— ~

=

—

Qaai(14’~L) =

—

—

+ ~w~)

~[Ea~(&~W~

~~abc(8iWk

Cabc(

—

—

~c~wf)

~6j~w~).

+

(/3

—

I/~&~EikIW~,

-

(A.4)

The action of the ~ is given analogously to that of the Qccai but with all coefficients replaced by their negatives. Finally all D-terms can be determined with the given formulas: D27()

=

D27(aai) =

=

—

—

i~u~’ + ~(i~u~

EabcUaUi

Ea$yU~Ui +

—

—

—

+

EjjkUaUa,

E.f. Chun, A. Lukas / NfL breaking SUSY GUTs D351()

=

—~w~°+i~w+ +

—

WCkYO W~

+ ~6~(~w~’— ~ D35l(aai)

=

~EabCWaWi

—

~

—

185

w~—~w~)

—

+ +

—

+

—

~dwk~C)

abc(W13 Wi~ +

aI3~a~c~’~’f+

~w~)

—

+

two perm.

(A.5)

Again the expression for D~j~ can be obtained as an analog of the one for D35laaj changing the coefficients to their negative values.

References [1] U. Amaldi, W. de Boer and H. Fürstenau, Phys. Lett. B260 (1991) 443; P. Langacker and MX. Lou, Phys. Rev. D44 (1992) 817 [2] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345 [3] WA. Bardeen, C.T. Hill and M. Lindner, Phys. Rev. D41 (1989) 1647 [4]W. Buchmüller and ST. Love, Nucl. Phys. B204 (1982) 213; W. Buchmüller and U. Ellwanger, NucI. Phys. B245 (1984) 237 [5] WA. Bardeen, T.E. Clark and ST. Love, Phys. Lefl. B237 (1990) 235 [6] M. Carena, T.E. Clark, C.E.M. Wagner, WA. Bardeen and K. Sasaki, NucI. Phys. B369 (1992) 33 [7] U. Ellwanger, Nucl. Phys. B356 (1991) 46 [8] L.F. Li, Phys. Rev. D9 (1974) 1723 [9] For a review see M. Bando, T. Kugo and K. Yamawaki, Phys. Rep. 164 (1988) 217 [10] H. Ruegg and T. Schücker, NucI. Phys. B161 (1979) 388 [11] Y. Achiman, S. Aoyama and J.W. van Holten, NucI. Phys. B258 (1985) 179; F. Delduc and G. Valent, NucI. Phys. B253 (1985) 494 [12] W. Buchmüller and U. Ellwanger, Phys. Lett. B156 (1985) 80 [13] R. Barbieri and D.V. Nanopoulos, Phys. Lett. B91 (1980) 369 [141 D. Delduc and G. Valent, Nucl. Phys. B253 (1984) 494 [15] R. Holman et al., Phys. Lett. B282 (1992) 132; M. Kamionkowski and J. March-Russell, Phys. Lett. B282 (1992) 137; SM. Barr and D. Seckel, Phys. Rev. D46 (1992) 539 [16] F. Buccella, J.P. Derendinger, CA. Savoy and S. Ferrara, Symmetry breaking in supersymmetric GUTs, presented at second Europhys. Study Conf. on The unfication of the fundamental interactions, Erice, Italy, October 1981 [17] R. Slansky, Phys. Rep. 79 (1981) 1 [18] L.E. Ibáñez and G.G. Ross, NucI. Phys. B368 (1992) 3 [19] K. Itoh, T. Kugo and H. Kunimoto, Prog. Theor. Phys. 75 (1986) 386