Patterns of local supersymmetry breaking in SUSY GUTS

Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

PATTERNS OF LOCAL SUPERSYMMETRY BREAKING IN SUSY GUTS ~

J. LEON, M. QUIR6S Instituto de Estruclura de la Materia, Serrano 119, Madrid-6, Spain

and

M. RAMON MEDRANO Departamento de Ft'sica Te6rica, Facultad de Ciencias F[sicas, Universidad Cornplutense, Madrid-3, Spain

Received 23 December 1982 Revised manuscript received 24 May 1983

The super-Higgs effect is analyzed in the framework of SUSY GUTS. We consider the cases where: (i) local SUSY is broken through a gauge singlet field, and (it) local SUSY and the grand unification group are simultaneously broken when a non singlet field takes a non zero VEV. The second case is illustrated with the SUs SUSY GUT where the adjoint plays the role of the super-Higgs fields.

Nowadays it is widely accepted that global supersymmetry (SUSY) [1 ] can solve the gauge hierarchy problem [2] along with the strong CP problem [3]. How- ever one does not observe supersymmetry at present energies. Unfortunately SUSY breaking seems a hope- less unsurmountable barrier in a global SUSY scheme. It is not only difficult to get rid of colour and/or lep- ton number violating vacua [4], but, even worse, the present cosmological constant cannot be made to vanish. Furthermore we should include gravitational ef- fects as the scale of SUSY breaking is of the order 1010 GeV [5]. Nevertheless SUSY Yang-Mil ls theories coupled to N = 1 supergravity (SUGRA) [6] provide us with a soft SUSY breaking and we are also allowed to cancel the cosmological constant by means of an additional fine-tuning [7].

The coupling of chiral multiplets (S) to N = 1 SUGRA and Yang Mills fields (V) is given by the ac- tion [7]:

f d 4 x d40 E(q~(Se 2gV, S)

+ R e R I[2f(S)+fAB(S)WaAeabwB]} , (1)

" Partly supported by Comisi6n Asesora de Investigaci6n Cientlfica y T6cnica, under contract 3209.

where 4~(g, S) is an arbitrary vector mul t ip le t , f (S) the chiral superpotential,fdB(S ) is a chiral function trans- forming as the symmetric product of the adjoint representation of a gauge group ~ , E the superspace de- terminant, R the chiral superfield of scalar curvature and Wa "4 the gauge covariant chiral superfield which contains the field strength tensor Fffv. After substitu- tion of the auxiliary fields and Weyl rescaling, one ob- tains from (1) the scalar potential

I 2 - 1 A B V=eG[GiGI (GS) - I 3] +~g R e f A B D D (2)

where

G(s, s*) = 3 In [ -3 /~ ( s , s*)] + In If(s) I 2, (3)

D A =Gi T A I s / , (4)

where G i = 3G/3s i, G i = 3G/3s i Is i = (s i )*] and (s i, X i) are the components of the chiral superfield S i. Notice we work i n M units, whereM = Mp 1 / V ' ~ = 2.4 X 1018 GeV.

The relations 3 In (-3/q~) = sisi andfAB = 6AB define the "minimal" coupling to supergravity [7]. We will consider only this framework along the paper. Thus the potential (2) can be written as

0 .031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 61


i . 2 n 2 V = eG(GiGi - 3) + ~ "-"A , (5)

whereG i = f i f 1 + si" As is well known when supersymmetry breaks down,

we have [8] (0 ]J u It/) = c'y u. Where Ju is the vector- spinor supersymmetric current and r/the goldstone fer- mion (goldstino). In the case of local SUSY, Ju contains the gravitino fields and then we can identify the goldstino field through the coupling fu"),or/in the la- grangian. The degrees of freedom of the goldstino are "eaten" by the gravitino that acquires a mass (super- Higgs effect). From the quadratic terms in the lagran- gian, we write the following expression for the goldstino [6,7].

rl = eG/2Gix i + 1D A X A , (6)

where X A are the gauginos and functions G i and D A are evaluated at the VEVs of the scalar fields s i. Eq. (6) tells us that the super-Higgs effect, i.e. G i --/= 0 o1

D A 4= O, for a Yang Mills theory coupled to SUGRA is, in general, associated with a spontaneous breaking of a gauge symmetry. The unique way to avoid this situation is to impose G i :~ 0 just for a gauge singlet s i. In general, the goldstino will be a combination of different chiral fermions X i which belong to some representation R of the gauge group ~ , gauginos X A and possible additional gauge singlet fermions X. This combination will be defined by the ground state coming from the potential [9] (5). In general, D A = 0 at the minimum and thus the goldstino does not couple to gauginos. On the other hand, for any sensible physical theory we will have G i = 0 at least for matter fields S i. For instance the Yukawa couplings of ordinary quarks and leptons (XL, XR) to the Higgs H contribute to the superpotential by an amount 8 f = FSRHS L where S R and S L are squarks or sleptons. Thus we get GSL = ( r S R I t ) f 1 + S~ = 0 since (S R ) = (S L) = 0.

We choose as the ground state of a spontaneously broken locally SUSY theory the minimum of the potential which has a cosmological constant A equal to zero. (The stability of this ground state, against decay into other minima of negative energy, has been proved by Weinberg [10] in the case of locally SUSY vacua). Here it is convenient to introduce some additional no- tation. We use latin indices for the mul t ip le t sy i with G i = O, and greek indices for the multiplets z a with Gc, 4: O. For simplicity we restrict ourselves to the case D A = O.

The condition for a vanishing cosmological constant is

Ge~G ~ = 3 , (7)

and the equations for a mininmm (Vc~ = 0) in the sector {z ~} read

G~ + G ~ G ~ = 0 . (8)

On the other hand, for the sector {yi} which pre- serves local SUSY i.e.

G i = 0 ~ f i = -£v i (9)

we have the minimum condition ( V i = O)

G~i Ge' = 0 , (10)

where

God = ( f fa i f a f i ) f 2

Since local SUSY breaking produces a shift on the scalar masses of the order of the gravitino mass (If[

~) [11 ] , and the mass of the W - S Higgs doublets must be of the order mw, the physically interesting case which will be studied throughout this letter is

In this letter we shall study two different cases. First, (z}reduces to a gauge singlet; in this case we also analyze the possibility of mixing between(z} and (y} in the superpotential. Second, (z} is a non- trivial representation of the gauge group; here several physically interesting examples are analyzed in detail.

Let us consider now the simplest example, i.e. when the sector (zc~) is reduced to a gauge singlet z. Now eq. (7), (8), (10) are given by

G z = V ~ , G z z = - l , G z i = O ( l l a , b , c )

for real VEVs. Eq. (1 lc) can be written as

f f z i = g f i (12)

whose solution depends on the particular superpotential. For a superpotential where z is not coupled to the y i ' s

f ( z , y ) =g(z ) + h ( y ) , (13)

we have f z i = 0 and eq. (12) becomes

gz = 0 (14)

62

Volume 129. number 1,2 PHYSICS LETTI';RS 15 September 1983

and we get from ( 11 a), ( 1 I b)

z : x / 7 , g : ~ = - - / . (15)

Notice that, due to tile presence of a local SUSY sector ( y i } , we have eq. (14).

A linear potent ial cannot satisfy this condi t ion and we need, at least, a bil inear term in z. A general analytic solution to this problem can be wri t ten as follows

g ( z ) = a + b ( z Zo)+C(Z z 0 ) 2 + ( z Zo)3P(z) , (16)

where z 0 = x,/3is tile m i n i m u m with A = O;a, b and c are real constants and P(z) is any funct ion o f z regular at z 0. Eqs. (14) and (15) translate into

I g(z)=u--(/,> ~,u(z v% 2+(z ~)3t"(z1, (17)

where/1 = If(x/3, (Y))I. We redefine h(v) in eq. (13) namely h O: ) (h) -+ hO:). Then, we have (h) = 0 [:ronl n o w o i l ,

Diagonalization of tile mass matrix provides tile condi t ion tor z = x / ~ to be a true m i n i n m m in tile complex z plane and we have

0 < ? ( , / 3 ) < :UI3 , /33 . ( J 8)

On the other band, the masses of the singlet fields are given hy nz/~ = 6X,/33 P,u, m~ = 4p 2 - - m 2 , which satisfy tile Cremmer el al. mass sun>rule [6] .

In addit ion to the mininunn at z = x /~ the re wi l l be, m general, other stationary points which can be classified a s :

(i) supersymme lric points which are solutions of G z = 0 and (ii) non-supersymmetr ic points which are solu-

tions of Gzz 2 + GrG z = 0. Tile existence and prop- erties of these points depend on the choice of tile function P(z). We lake as an illustrative example tile simplest case where P(z) is a constant , for instance P(z) = ,u/3X/'3. Apart from tile m i n i m u m at z = W:'3, there are, in this case, a non supersymmetr ic m i n i m u m [A = 5.25 (gM) 2] at z = 0 and a max i mu m at z ~ 0 .37as

is shown in fig. 1. Other min ima for non real VEVs should be investigated in each part icular application.

For tile supersymmetr ic sector {.vi}, eq. (9) becomes./'} = la.V i after local SUSY breaking. Tiffs equa- tion coincides with the global SUSY fornlula in lhe lilnit ,u = 0. This has impor tanl consequences for gauge breaking, as we already pointed out in re(. [9] f o r a SU 5 local SUSY GUT that breaks to SU~' X U] "m.

We can check that G i = 0 is a true n ml i mu m by

It

_3

i

:111

I:ig. 1. Potential corresponding to eq. (17), with f ' ( z ) = l a / 3 , ~ .

studying the mass matr ix for this sector which is given

by

<>,>) v'](<z>, <y>) j m2Cy,.y *) = V//((z)' (Y)) viJC(z), ( y ) ) / . (19)

Using eqs. (9) and (1 1) in eq. (19) we get

o + o ' % / \ 2G~!

which can be wri t ten as

m 2 ( 3 , j,* )

2G I + GG + G

2Gi/

~)Ji + Gik Gkj ) ' (20)

G +,9

i )- (:,t)

Finally, as tile eigenvalues of the square of an hermit- Jan matrix are senti-positive definite, the solut ion G i = 0 corresponds to a tree min inmm.

We consider now a more complicated case. Namely when there is a mixing between the sector ( y i } and z, For simplicity, we will s tudy the case where the sector ( v i} does not conta in gauge invariant fields. The super. potential is then

.f(Z, .I 'i ) = g'(Z ) + h ( V ) + zlnijyi3 '/. (22)

From eqs. (11), (12) we deduce:

&+yv,,o,:/'(Vq z), &~=.tl(x/.~-z) 2 I].

( 2 3 )

63


m y = - ½ ( x / ~ z ) f y . (23 cont'd)

Now eq. (14) is no longer required. Therefore eq. (23) can be solved for the simple linear case g(z ) = az + b. Eqs. (23) become

z = x / ~ - 1 ,

a = f ( l +_~yTy) , b = ( 2 x / r S ) f _ h ( y )

1 my = - ~ f y . (24)

A detailed analysis shows that the superpotential eq. (17), along with the choice given by eq. (24), pre- sents a unique minimum at z = x /~ - 1.

Notice that for the relevant physical case, f = t*, the adimensional matrix m should have an eigenvalue - ½ f = t~/2M ~ 10 -17 , which is different from zero. The presence in the superpotential of adimensional parameters with such small values seems rather unnatu- ral, although technically possible.

For the other solution o feq . (23) ,z = v r 3 + 1, the conditions of zero cosmological constant at this point give a = _f(1 + ½yVy), m y l = ~fv, b = f ( 2 + X/~-). How- ever, z = x/~ + I is a maximum. Also, there are two non supersymmetric minima at z = 1.227 and z = 2.227 and one supersymmetric minimum at z = 3.4415; all of them with negative cosmological constants. Therefore, this case is rather uninteresting from the physical point of view.

We could think about the possibility of breaking local SUSY through a singlet z and with a constant superpotential g(z ) = A [12]. However, this case leads to f = h i = 0, i.e. neither local nor global SUSY are broken. Furthermore, matrix m, that appears in the mixing z m i j y i y / , must be singular [det(m) = 0] and y t is the eigenvector associated with the zero eigenvalues (my = 0).

In the previous cases, local SUSY has been broken in a gauge singlet sector and the mininmm equations concerning gauge non-singlet fields were equal to the local SUSY ones, G i -- 0. Thence the goldstino was a gauge singlet.

Next we want to consider the more general case where the goldstino transforms in a non trivial way under the gauge group or, in other words, when the local SUSY breaking sector {z a } is not a gauge singlet. This situation could be particularly interesting to de- scribe the physics associated with a group that con-

tains a grand unification group ~1 (SU5, SO10, E6 ...) [13] and a horizontal group ~ 2 ( S U ~ SU~ .. . . ) and breaks down spontaneously to ~1 X ~2 at tire Planck nlass.

We are going to consider an SU 5 gauge theory. The best candidate for the super Higgs field seems to be the adjoint representation E whose VEV breaks the unification group as well. In this case, the {z ~ } sector is the E field and the superpotential is written as an arbitrary function [ 14].

cxa

g ( E ) = ~ X n t r ( E n ) . (25) n = 0

A remarkable point now is that we can treat the problem as in the previous case if we particularize (25) to breakings which are along one direction of the Car- tan subalgebra E = coH(t r H 2 = 1), i.e. playing the role of a singlet. For each direction H we will get a different g(co) from eq. (25). First, we will choose a g (E ) such that there will be a minimum at co o with vanishing cosmological constant and which breaks SU 5 to SU 3 X SU 2 X U 1 . Then, we will analyze the other stationary points along this direction. Finally, the be- haviour of the potential along the direction which breaks SU 5 to SU 4 X U 1 will complete our study.

In our SU 5 model the sector {z '~ } = {E} and the {yi} sector contains the higgses 5(H), 5 (H)and the matter fields (~, X), i.e. O ,i} = (H, fi, ~, X}. The mixing between both sectors is given by f i r= H2H. On the other hand, at the grand unification scale (yi) = 0 and therefore q'wy ) = ( f y ) = 0. Then the minimum condition V i = 0, eq. (12), is automatically satisfied for arbitrary values of (leo)-

Now we consider the case where SU 5 breaks to SU 3 X SU 2 X U 1 and write the adjoint as

E = c o H , H = ( ~ ) l / 2 ( "13 ) . (26) - ( 3 / 2 ) 1 2

The superpotentia[ is given by eq. (25) and z reads CO n o w ,

Eq. (25) becomes

g(co) = ~ an(CO - COO) n (27) n

and eqs. (11) fix the values of the coefficients a l , a 2.

a 1 = a o x , a2=aO(x 2 1)/2. (28)

64


where x = k / J - coO' Condi t ion tr E = 0 translates into

n a n ( - c o 0 ) n - I = 0 , (29) n

which implies the existence o f a supersymmetr ic mini-

m u m at co = 0. As supersymmetr ic min ima have A ~< 0,

then we shall impose A = 0, i.e. V(co = 0) = 0 in order

to achieve the max imum stabili ty for the broken phase

(coO) at zero temperature . This implies a further condi-

t ion g(co = 0) = 0, i.e.

an( COO)" =0. (30) 11

Up to now the potent ia l has two stat ionary points

at co = 0 and coO. Let us impose the condi t ions on these points to be true minima in the complex plane.

The condi t ions fw = f = 0 at co = 0 [eqs. (29), (30)] guarantee that this is a true supersymmetr ic min imum.

Moreover, we need an additional condi t ion on coO sin>

ilar to eq. (18), i.e.

0 < Goowc o < 4/~/3-3, (31)

which leads to

a3/a O=~(x 3 3x)+(2/3x/3)a, 0 < a < l . (32)

Notice that a 0 is fixed by the gravitino mass through

the relation m3/2 = exp(co2/2)a0 . Summing up, we have three equat ions (28) (30) and one constraint eq.

(32) to fix the coeff icients ofg(co) . So we need, at

least, a four th order polynomia l for the superpotent ia l .

In this minimal case, only a restr icted range o f values

of coO satisfies eq. (32). For a higher order polynomia l

one can succeed in having an arbitrary value for co o

after a suitable choice o f parameters.

We concent ra te now on a particular example, name-

ly the minimal case where g(co) is a four th order poly-

nomial . Using the above results we have

g ( E ) = b 2 t r (E 2) + b 3 t r (E 3) + b 4 t r (E 4) , (33)

where

b 2 = a 2 - 3 a 3 6 o 0 + 6a46o 2 ,

= 30 (34) b 3 x / / ~ ( a 3 4a4co0) , b 4 = ~-a 4 .

As we have already ment ioned not all the values o f

SUsx SUSY SU# S~ A=O A=O

Z / N

Fig. 2. Potential corresponding to eq. (33) for the breaking SUs--+SU3 x SU 2 x U I.

coO satisfy eq. (32); a numerical analysis shows that

the allowed range is 1.48 < co O < 2.89. In fig. 2, we

illustrate the particular case co o = x/'-3. The supersym-

metr ic min immn, i.e. solution of Gto = 0, is unique

and at co = 0. Non supersymmetr ic s ta t ionary points,

i.e. solutions o f Gww - 2 + G 2 = 0, are a min imum at

co = x /~ and a m a x i m u m at co ,-,-, 1.0798. It can be

proved that there are no other s tat ionary points along

the real axis.

The next step is to look at the potent ia l , eqs. (33),

(34), when H = ( 1 / x / - ~ ) d i ag ( l , 1, 1, 1, 4), i.e. along

the SU 4 × U l direct ion. In this case, we have three su-

persymmetr ic minima: (i) co = 0 with A = 0 (ii) co I =

-0 .303 , A 1 = -6 .1 X 10 - 3 (m3/2M)2 and (iii) °°2 =

7.03, A 2 = 1.5 X 10 24 (m3/2M)2 The height o f the barrier be tween co = 0 and col is of the order o f a

0.048 (m3/2M)2 and the barrier be tween col and

co2 is b ~ 6.4 X 1024 (m3/2M) 2. The si tuation is de-

picted in fig. 3. There we see that the last min imum is

separated, by an enormous barrier, f rom the region

near the origin. Therefore it is inaccessible not only

by quan tmn tunnelling but also by thermal f luctat ions for temperatures below T c _ (b) 1/4 ~ 1016 GeV.

To summarize, we have studied the super-Higgs ell

feet in a GUT coupled t o N = 1 supergravity. Scalar

fields are classified in two sectors: (a) st, perpartners

of the goldst ino (zC~}, i.e. Gc~ 4: 0, and (b) fields ~ i }

65


~2 SU4: U, ~ SUSY

A z

b

a S%,SUST

Su4, U,, SuSv A I

(o /b ) ~ ~" ( I 0 -26 )

{Ai /AZ}~ ~(10 -26"~ I

Z I M

Fig. 3. Sketch of the potential corresponding to eq. (33) for the SU4 X U1 breaking.

which do not break local SUSY, i.e. G i = 0, with arbitrary VEVs. For the sake o f def ini t ion we have restrict-

ed ourselves to the SU 5 gauge theory which is spon-

taneously broken to SU 3 X SU 2 X U 1. In this case,

the VEVs (z '~) nmst be SU 3 X SU 2 X U 1 smglets bul

not necessarily SU 5 invariant, Then a ques l ion arises

112] : is it possible to break SU 5 and local SUSY wilh the same sel (z~)? We have given an affirmative answer to this quest ion and worked out in detail the case where

{z c~} = Z, the adjoint representat ion o f S U 5 . The possibil-

ity of obtaining (X) ,~Mp [ 15] appears as an interesting by-producl of our analysis. This would happen lor superpotent ia ls o f degree higher than four.

Lately, renewed inlmest was shown [16 18] in

the case where local SUSY is broken by a hidden sec-

tor, i.e. {z ~ } interacts with the rest o f the fields {3 ,c~ )

only through supergravity, f ( z , y ) = g(z) + h (y). We have considered the example in which (z a } reduces

to a gauge singlet. There the potent ials were very flat

(fig. I ), a fact which might be useful for inflat ionary

scenarios. The generalization o f lhe above example to

arbitrary {z c~ } is s t raighlforward. The Higgs X, in lhe adjoint representat ion, belongs 1o the sector ( y i } and

we have. l~ 4- 0 and.[,_:z = 0 by def ini t ion of lhe hidden sector. Now condi t ions (7), (8) and (10) for the

A = 0 ground state read as

z~GC~ = 3 , j'~GC~ = 0 , ( J ~ +.18c~) ( f a + )'kt3) = 0 .

In particular, the lhs o f the first equat ion is the

pa rame te rA in t roduced in ref. [18] in order to write

the low energy effective potent ia l , and we are led to

the conclusion A = 3.

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Patterns of local supersymmetry breaking in SUSY GUTS