Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

SUPERHIGGS EFFECT IN SU(5) SUSY GUTS ~

J. LEON, M. QUIROS Instituto de Estructura de la Materia, Serrano 119, Madrid-6, Spain

and

M. RAMON MEDRANO Departament de F{sica Te6rica, Facultad de Ffsicas, Universidad Complutense, Madrid-3, Spain

Received 16 November 1982

The coupling of the SU(5) SUSY GUT to N = 1 supergravity is considered. Super Higgs effect appears in a SU(5) singlet sector. The fine tuning which cancels the cosmological constant removes the degeneracy among degenerate vacua. Weinberg -Salam higgsinos acquire tree masses of the order of the gravitino mass. The scale of electro-weak breaking remains arbi- trary at the tree level.

A common problem to all supersymmetric Grand Unification Theories (SUSY GUTS) is to find a non supersymmetric stable vacuum. In particular, it is dif- ficult to apply the O 'Ra i fea r t a igh -Faye t - I l i opou los ' schemes [1 ] to realistic models. This, along with the possible influence of gravity effects in higher breaking hierarchies, led several authors [ 2 - 4 ] to consider the effects of local N = 1 supersymmetry in supersymmet- tic theories. These effects are characterized by a mass parameter t2 related to the scale of SUSY breaking A s through the relation/~ = A2/M, M being the Planck mass. The coupling of supersymmetric Yang-Mills theories to N = 1 supergravity has been recently stud- ied by Cremmer et al. [4]. These authors use a super- Higgs effect [5] to break local SUSY and cancel the cosmological constant. This breaking amounts to a shifting of scalar masses of the order of the gravitino mass/2. In such a scheme, the scale of SUSY breaking would be A S ~ 1010-11 GeV which gives/2 ~ 102 GeV suggesting a breakdown to SU~ X U v at the usual scale.

When trying to build realistic models along the above lines, it is necessary to take into account the hierarchy of scales present in nature: M >> AGU T >> A S

* Partly supported by CAICYT, under contract 3209.

>> A w. This makes the search for the true vacuum a very difficult task, and the question has been only part ly answered up to now. Thus, a SUSY standard model which breaks the electroweak group at a scale much lower than A s has been solved [6] . Also SUSY GUTS have been considered only in the case AGU T = M [7] . On the other hand, an iterative approach has been developed for the general case of a SUSY GUT [8]. It is based on a power series expansion in 1/M

around the global SUSY solution. Some of these works solve the problem only partially. The rest of them bypass the true parameter of SUSY breaking which is A S and not M. Moreover, even if Aw/A s ~ 1, we have AGuT/A S >> 1, and therefore the validity of a power series expansion around the SUSY solution is rather dubious.

In this work we shall adopt the point of view that the.superHiggs effect appears in a gauge singlet sector. As a consequence the Goldstino remains decoupled from the rest of the fields. The non singlet part of the theory satisfies local SUSY equations: F a = 0, D A = O. The case SU 5 offers an illustrative example for our method which does not use any power series expan- sion and it will be solved explicitly. The matter and Higgs field content is the usual one, and displays the different scales of gauge symmetry breaking.

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Volume 127B, number 1.2 PHYSICS LETTERS 21 July 1983

As it is well known [4], in the case of a Yang- Mills theory "minimally" coupled [2] to N = 1 super- gravity the action depends only on a real function G o f the scalar fields z which is given in terms of the superpotential f ( z ) by

G ( z , z * ) = - I z l 2 / M 2 - l o g ( l f ( z ) 1 2 / M 6 ) . (1)

In particular, the overall scalar potential is given by

V ( z , z * ) = L ~ 2 2 A D~4 + e - G ( G i G i - 3 / M 2 ) M 6 ' (2)

where Gi =- ~G/Oz i, G i = ~G/~z i and D A =~ - g G i T ~ ] z l . On the other hand, the Goldstino is a linear combina- tion of the chiral (X/) and gauge (X A ) fermion fields ap- pearing in the theory

77 = e - G/2Gixi + D A X A . (3)

By inspection of eq. (2), the conditions for a vanishing cosmological constant (V = 0) and a minimum (V/ = 0) are given by

GiG i = 3 /M 2 ,

D A D A i +M6e - G(Gi]G] - Gi /M2 ) = O . (4)

From now on we shall employ the following notation: z 1 =--z will indicate a gauge singlet which will be used for the super-Higgs phenomenon, while we shall indi- cate the rest of the scalar fields by z i g = 2 . . . . . n).

The case we are interested in is characterized by the decoupling of local SUSY breaking from the non singlet sectors, i.e.

G ; = 0 ( i=2 ..... n ) ,

DA = o , (s)

thus 77 = e-G/2G 1X. We notice that any solution of G i = O, for all chiral multiplets, trivially satisfies D A = 0 .

The equations for a null.cosmological constant and a minimum of the scalar potential can be written as:

V = O ~ G l g l = 3 / M 2 , (6)

V 1 =O=*G1 = M 2 G l l G 1 , (7)

V i = O = * G l i = O ( i = 2 ..... n ) . (8)

In particular, for real VEV's, eqs. (6), (7) lead to

G1 = ~v/3/M, G11 = 1/M 2 , (9)

while eqs. (8) are compatibility conditions to be held in any specific model. In terms of the superpotential, eq. (8) reads

f l i f - f l f i = O (i = 2, ..., n ) . (10)

Next, we shall study the application of the above scheme to the Sakai-Dimopoulos-Georgi [9] SU 5 SUSY GUT model, which has the usual chiral super- multiplets, E x l-I x ' ~xrn, y, ._ , Hx, X xy (m = 1,2, 3, family indices) along with a gauge singlet z for the superHiggs effect.

The superpotential which describes the model is a sum of two terms, f =fs + fns, given by

1 Z 2 + ½ a 3 Z 3 (11) fs = aoM2 + alMZ + ~a2

fns 1 = grn tr ~2 + ] t r ]~3 + H'NH + m 'H 'H

xy , + I '(1) ~xmXn Hy +P(2) e xXYxZUH ° + A (12) mn mn xyzyv m n •

Thus, while fns is nothing else than the Dimopoulos-Georgi superpotential, we have written for fs a general cubic superpotential. The parameters a O, a l , a2, a 3 will be tuned to have a vanishing cosmo- logical constant and satisfy conditions (9), (10).

In the class of models we are considering, i.e. no couplings of the type g i j z i z l z in the superpotential, we have f l i 0 and eq. (1 0) leads to f l f i = 0. However fi 4= 0 in general, since G i = 0 and f=/= 0, then the only solution ofeq. (10) isf l = 0. Thus, an f s linear inz is ruled out by this condition.

Working out in detail eqs. (9), (10) we obtain for the z-dependent part the following solution

z = V ~ ' M , a 2 = 2 a 0 ,

a 1 = Vt3(lp _ a0) , a3 = _ 3 - 1 / 2 M - 1 ( 1 p +a0) ,

(13)

where ta = fs(V~-) is a parameter which gives essentially the gravitino mass. Hence, A must be tuned to A = --fns((Zi~).

It is worth to point out that the solution given by (13) is a true minimum, in the complex z-plane, at z = x/c3 only if the further restriction

0 < v~f ' " (Vc3) M < 4p (14)

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holds. Eq. (14) is satisfied provided that the param- eter a 0 in (11) satisfies the condition, -lu < 2a 0 < -5/1.

Now, we shall study the equations G i = 0 q = 2,

..., n). Since G i = - F i l l , where F i = ~. + M - 2 f z T ,

they reduce to F i = 0 as in the case of local SUSY solu- tion (10). Let us point out that we have F i = f i + l i z* in our case, and therefore the expansion in powers of 1 /M does not make any sense.

We consider the breaking of SU 5 now. The equa- tion of minimum for ~ is given by:

1 t F z = ( m + / ~ ) G + G 2 - } t r G 2 + H H ' - g H H = 0 .

(15)

Eqs. (15) are equal, except for the substitution of m by m +/l, to the global SUSY ones that give rise to the well known degeneracy among vacuum states [10]. Later on we shall solve these ambiguities. For the sym- metry breaking pattern SU 5 --* SU 3 X SU 2 X U 1 we have:

( 0) 13 03 ~ + e , (16)

= w 0 - - ~ ' 1 2 ! 0 3

where we have included an SU 2 triplet with VEV e which takes into account the effect of higgses H and H' into the breaking of SU 5. Using eq. (16) we can write eq. (15)as:

(m + ~t)co - 16o2 - ~(e2 +a) = 0 , (17)

/all 5 + (6 - e)H~ = -P2(eXX)5 , (24)

where 6 = - ~6o + m' is the mass of the Higgs doublets. The physically interesting case corresponds to the

vanishing of the determinant of the system of eqs. (23), (24), i.e.

d e t ( 6-e / l 6-e/~ ) = 0 . (25)

In this case Pl (fiX)5 = P2(exx)5 = 0 for the system (23), (24) to be compatible, and it gives the solution H 5 = + H i arbitrary. For the system of eqs. (21), (22) we shall take the trivial solution H 4 = H i = Pl (fiX)4 = P2 (exx)4 = 0, i.e. a = - b -= ix .

The solutions of eqs. (23)- (25) should be written at an order of approximation enough to discuss the fine tuning which will allow the establishment of a hierarchy of breakings. First of all, let us notice that in the case of global SUSY (/a = 0) we have

6 = e ~ - x / l O m , r.o .~, 2 m - ~ x / m . (26)

• 2 • ' Taking now x ~ g - m20.e , m = 3m + O (m2w/m)),

we obtain a tree mass for higgsino doublets m 2 O ( m 2 / m ) ~ 10 -11 GeV. In the case of local SUSY

the situation is quite different due to the fact that higgsino doublets will acquire a mass of order #. In fact, we have from eqs. (23)-(25) :

6 = + / . t + c ,

(m + ~ ) e - 3 w e + b = 0 , (18)

where a and b are defined by HH' = al 2 + b o 3.

On the other hand, the equations of minimum for higgses H and H' are given by:

F H, = (G + m ' ) n +/IH'T + Pl (fiX) = 0 , (19)

F n = H'(Y~ + m') + uH + P2(eXX) = 0 . (20)

Inserting eq. (16) into eqs. (19), (20) we obtain dif- ferent equations for the components (1, 2) and (3, 1) ((5, 1)) of H(H'). In particular, we have for the dou- blets:

(6 + e)H 4 +/~H~ = - P l ( ~ X ) 4 , (21)

juH 4 + (6 + e)H~ = - F 2 ( e X X ) 4 , (22)

69 ~ 2(m + ~) - ~ x / ( m + I t ) , (27)

e ~ - x / l O ( m +l.t)

thus the higgsino doublet mass is m 2 = 3 ~ ~. Fixing again the value o f x to g - 2 m 2 w and taking the plus sign in eq. (27), we obtain:

m ' = 6 + 36o = 3m - 41a + O ( m 2 / ( m + U)). (28)

Letus notice that the scale x of electro-weak breaking is kept arbitrary at the tree level. The scale/a of local SUSY breaking is decoupled from x, but if we want the gauge hierarchy not to be spoiled by radiative cor- rections, we must assume/1 ~ m w [11 ].

The equations for Higgs triplets are:

(w + m ' )H 3 +/IH~ = - P l (6X)3 , (29)

(6 - e)H 5 + u H ; = Pl (~X)5, (23) pH 3 + (co + m')H~ = -P2(eXX)3 , (29)

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Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

admitting the trivial solution H 3 = H i = Pl (~//X)3 = I'2(eXX)3 = 0. On the other hand, for matter fields we obtain

F IXH' + / ~ = 0 ,

Pl , H ' + P2exH +/1X = 0 . (30)

The system (30) is homogeneous in matter fields, so that non trivial solutions are allowed only if

( " ~ l H ' ) det H' = 0 . (31)

P 1 /aI + r '2eH

As condition (31) does not hold, only trivial solu- tions ~ = X = 0 are allowed. This supports the choice of trivial solutions made in (21), (22), (29). This is a most nice separation from the case of global SUSY.

It is worth to point out here that the ambiguity among degenerate vacua is removed in local SUSY theories. In fact, by choosing for equations G i = 0 the solutions (z i) with ~ breaking SU 5 to SU 3 X SU 2 X U1, (16), and H, H' breaking SU 2 X U1, while keep- ing colour and electric charge unbroken, (25), we tune A to --fns((Zi)) in such a way that the cosmological constant vanishes. Thus, once fixed A the other vacua will get a non vanishing cosmological constant. Hence, the physical vacuum is stabilized by gravitation [12, 13] and can be chosen without ambiguities.

Let us complete this work with a brief description of the mass spectrum predicted by the model. As it is well known in this kind of schemes, scalar particles acquire a mass of the order of the gravitino mass

m3/2 = M e - G/2 =/ae3/2 . (32)

In fact, the splitting induced by (32) between scalar and fermion superpartners is one of the main motiva- tions to introduce local SUSY effects. Another fea- ture, due to the combined effect of local SUSY and the fine tuning, is that WS higgsinos acquire tree masses of the order of the gravitino mass. On the other hand; gauginos of unbroken gauge symmetries get radiative masses of the order of m3/2 coming from four-fermion non renormalizable interactions present in the local lagrangian.

Let us summarize the main results of this paper. We have solved, without recursion to any power series expansion, the equation of minimum for the overall scalar potential of a supersymmetric Yang-Mills the-

ory "minimally" coupled to N = 1 supergravity. Our starting point was to impose that the breaking of local SUSY kept decoupled from the gauge sector. The vac- uum invariant under SU~ X U~I m is selected by the condition of vanishing cosmological constant, i.e. fine tuned parameters. The condition of minimum at z = x/-3-was guaranteed by eq. (14). Furthermore, the other stationary points along z i, eqs. (16) - (31) , are obviously minima. In fact, the effective potential cor- responding to our model - the same as in ref. [14] fo rA = 3 and ref. [15] - is given by IFil 2, so that points satisfying F i = 0 are true minima.

Another result of this paper is that the higgsino acquires a tree mass of the order of the gravitino mass. Since there is not any higgsino Majorana mass term in the effective potential, it originates from the super- symmetric Higgs mass term IH'(I~ + m')HIF [16]. In this way, the higgsino mass is not expected to spoil the tree level hierarchy.

The scale x of electro-weak breaking remains ar- bitrary at the tree level. A way for fixing it should be the introduction of a light singlet Y, although this so- lution has been proved to spoil the gauge hierarchy [ 17]. Another possible safe way of fixing x should be by radiative corrections [ 18].

A possibility which has not been explored through- out this work is local SUSY breaking overlapping the gauge sector [19]. This amounts to a drastic modifica- tion of the scheme. In fact, being G i 4:0 we cannot infer DA = 0 so that we would get an additional split- ting of masses through the D-terms. In this case, small gravitino masses would be possible, i.e. we could be in the low mass, cosmologically acceptable, region m3/2 < 10 -6 GeV, [20], but the gauge group should be en- larged to allow D A 4:0 along directions orthogonal to SU 5 [21 ]. We should have at intermediate scales the gauge theory SU 5 X G 131 and local SUSY breaking would overlap with G.

A most striking consequence of supergravity is that the Dimopoulos-Georgi fine tuning does not keep the WS higgsinos massless but give them a tree mass of the order of the gravitino mass. On the other hand, it is well known that gauginos get radiative masses of order /a through a non renormalizable four-fermion interac- tion with the gravitino [3].

(8M2) -1 ~te~7~ o° I" ff, l ~z ~p3'u X~. (33)

This fact, which is welcome from the phenomenol-

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Volume 127B, number 1,2 PHYSICS LETTERS 21 July 1983

ogical po in t o f view, translates into some t roubles for

unders tanding chiral fe rmion masses. In fact the

lagrangian contains, along wi th (33), o ther non renor-

malizable interact ions like

(4M2)-I~LiXL]~ X1R . (34)

Therefore , once a tree mass mij has been generated,

the annihi lat ion o f two fermions through mi/would give a radiative cor rec t ion o f the same order o f mag-

ni tude. Self-consistent H a r t r e e - F o c k me thods [22]

would be necessary to unders tand fermion masses.

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