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Spontaneously broken supersymmetric models free of fine-tuned parameters and ΔB, ΔL troubles (I). General survey
Nuclear Physics B222 (1983) 104-124 © North-Holland Publishing Company
SPONTANEOUSLY BROKEN SUPERSYMMETRIC MODELS FREE OF FINE-TUNED PARAMETERS A N D ~B, ~L TROUBLES* (I). General survey J. LEON and M. QUIROS Instituto de Estructura de la Materia, CSlC Serrano 119, Madrid 6, Spain
M. RAMON MEDRANO Departamento de Ffsica Te6rica, Facultad de F(sicas, Universidad Complutense, Madrid 3, Spain
Received 2 August 1982 (Revised 21 March 1983) We propose a class of supersymmetric models based on canonical U(n) gauge groups with the following properties: (i) absence of anomalies; (ii) no quadratic renormalizations for the D term; (iii)absence of fine-tuned parameters; (iv)spontaneous SUSY breaking h la Fayet-Iliopoulos; (v) free of unobserved effective AB ~ 0 and AL ~ 0 interactions.
1. Introduction The main underlying problem in gauge theories is the generation of masses for elementary particles. In the last years, the traditional f r a m e w o r k ~ la Higgs has been questioned and other schemes with dynamical generation of masses have been favoured. The latter scheme in its two versions of technicolor, and extended technicolor, was able to predict masses for gauge bosons and fermions. Unfortunately, problems with F C N C darkened the validity of this scheme. Recently, a renewed interest for supersymmetric gauge theories has emerged. The idea is that theorems which provide criteria on non-renormalization for supersymmetric theories allow the existence of mass hierarchies, and the rather unnatural procedure of fine tuning is not needed. H o w e v e r the real world does not seem to be supersymmetric. H e n c e the first trouble of these theories: supersymmetry has to be broken in such a way that it is not observable at low energies, while the non-renormalization theorems should keep playing their role. This has led us to consider spontaneous SUSY breaking. Two kinds of spontaneous breakings have been proposed: (i) the O'Raifeartaigh method [1], where the true ground state cannot annihilate the F terms of the lagrangian, and (ii) the Fayet-Iliopoulos method [2], where the presence of a l~l(1) * Part•ysupp•rtedbytheC•misi6nAses•radeInvestigaci6nCienti•cayT6cnica•underc•ntract32•9.
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J. Le6n et al. / Spontaneously broken supersymmetric models
105
gauge group prevents the cancellation of the D terms. Nevertheless, it is very difficult to find a supersymmetric model with a non-supersymmetric vacuum being the most stable one, Hence, explicit soft breakings, that do not destroy the theorems of non-renormalization, have been proposed [3]. Unfortunately, these breakings show problems connected with the masses of the fermion partners of the Higgs scalars. There is another additional question which is common to all SUSY GUTs proposed up to now, namely [4]: the mass parameter, in the Higgs superpotentiaJ, has to be fine-tuned in order to keep massless the Higgs doublets needed for the breaking of SU(2)Lx U(1) Y. Some progress along this line can be achieved by the introduction of a singlet [5] decoupled from the rest of the fields of the theory, i.e. the v.e.v, of the singlet produces the same results as a mass parameter, or by considering further Higgs supermultiplets [6]. In this work we propose a class of SUSY GUTs related to the U(n) gauge groups with a canonical assignment of lJ(1) charges. This will allow us to understand the U(n)'s as grand unification groups at energies of the order of the Planck mass. Canonical U(n) theories share the following properties: (i) absence of anomalies; (ii) absence of quadratic renormalizations for the D term related to 1~(1); (iii) no fine tuned parameters; (iv) spontaneous SUSY breaking ~ la Fayet-Iliopoulos at a convenient scale; (v) no effective interactions with dB ~ 0 and AL ~ 0 and dimension lower than 6. By a fine-tuning problem we understand here an unnatural adjustment of the parameters in the superpotential in order to keep Weinberg-Salam higgses massless. In other words, once a particular model is given the fine-tuning problem does not exist if the WS Higgs doublets are massless without any need of fine adjustments of the parameters in the corresponding superpotential. The choice of a model amounts to the writing of a particular superpotential, eventually setting certain couplings allowed by gauge symmetry equal to zero, which is technically possible by non-renormalization theorems. We present the scheme step by step. We show in sect. 2 that U(5) (and not SU(5)) is the gauge group able to overcome the fine-tuning problem. In sect. 3, we consider a model invariant under SU(5 + n) which is similar to the one studied in sect. 2. We prove that the fine tuning problem remains unsolved in this case. In sect. 4 we propose canonical U(n) groups that satisfy properties 1, 2 and 3 already mentioned. The SO(4n + 2) groups are considered in sect. 5. They verify properties 1, 2 and 3 as well, but are unable to accommodate a large va Majorana mass and property 4 simultaneously. In sect. 6, we demonstrate that canonical U(n) theories can satisfy properties 4 and 5. We analyze several questions on mass splittings and the presence of heavy multiplets in the theory. Finally, we finish with our conclusion in sect. 7.
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2. Supersymmetric U($) gauge theory without fine-tuned parameters There are three fundamental problems in the minimal supersymmetric SU(5) model of Sakai [7] and Dimopoulos and Georgi [4, 8]. (a) An unnatural fine tuning is necessary to adjust to zero the masses of the Higgs doublet [4]. (b) There is not an invariance that prevents a violation of the baryon number [9, 10] at d < 6. (c) Supersymmetry has to be broken softly. Hence it is not possible to give a mass to the fermionic partners of the Higgs supermultiplet [3]. Problems (b) and (c) have been already considered by Weinberg [9] in the supersymmetric SU(3)× SU(2)× U(1) model. He comes to the conclusion of the necessity of introducing an additional 0(1) that, while breaking the supersymmetry la Fayet-Iliopoulos [2], prevents the presence, in the superpotential, of terms that violate the baryon number. Solutions to problem (a) have been considered by several authors recently [5]. They start from the fact that the SU(5) SUSY model needs the inclusion of two higgses H and/-I, in the 5 and 5 representation, to break SU(2)L×U(1), and also an adjoint ,~ to break SU(5). Furthermore, as 5 x 5 = 1 + 24, the introduction of the term/-I,~H in the superpotential is unavoidable. This term would provide masses for/-t and H of the order of the grand unification mass. In order to obtain massless Higgs doublets, avoiding at the same time the introduction of mass parameters that require fine tuning, Ibafiez et al. [5] proposed the introduction of a new singlet that allows the introduction of a term/-7~H in the superpotential. This singlet is not coupled to the rest of the fields, and therefore it can acquire an arbitrary v.e.v. without breaking supersymmetry and gauge symmetry. Obviously, a suitable choice of this v.e.v, seems to solve the problem. Nevertheless, the introduction of the singlet • is somewhat artificial. In short, a fine tuning of a mass parameter is transferred to a fine adjustment of an otherwise arbitrary v.e.v. Another solution to the fine-tuning problem has been advocated by Masiero et al. [6] in the SU(5) SUSY GUT. They use a Higgs supermultiplet in the 75 representation to break SU(5) and need the further introduction of higgses 50 + 50. However, apart from the fact that their solution is highly uneconomical from the point of view of the number of involved fields, they need to impose the existence of an R symmetry or an extra 1~I(1) gauge symmetry to avoid the appearance of disastrous dimension-five operators. We would like to present here what, we believe, is the most natural solution to the problem. Namely, a replacement of the adjoint of SU(5) by the adjoint of U(5)(,~). Now ~b(=tr ~) is associated with the new generator in the algebra and corresponds to the direction 0(1). In this way, 0 does not remain uncoupled from the rest of the fields in the superpotential and its v.e.v, is determined by the supersymmetry conditions.
3. Ledn et al. / Spontaneously broken supersymmetric models
107
We propose now a model invariant under SU(5) x 1~1(1)that involves H , / 4 and ,~: V = A1/-[x~H y + A 2 ~ x y + A 3 , ~ Y ~ z .
(2.1)
This is not the most general superpotential, but WS doublets will remain massless in this particular model without any need of fine tunings. Therefore the model does not need any unnatural adjustment of parameters in the sense we stated in sect. 1. In eq. (2.1) x
1
x
X~ = try + 76 rob,
(2.2)
~r~ being the adjoint of SU(5). The superpotential can be written in terms of the above fields as -x y x y V =AI(Hxo'yH + g1 H-- x H x ¢)+A2(o'yo'x +½~b2)+
x y ~ + 73¢ 0 , x (2.3)
The conditions for a supersymmetric vacuum to exist are given by V,,~ = V~ = Vu = Vg = 0.
(2.4)
For the breaking of SU(5) into SU(2)× S U ( 3 ) x U ( 1 ) [11] we take the v.e.v. cr = w diag (1, 1, 1, - 2 , - 2 ) and obtain two sets of solutions: (i)
H =/-I = 0
¢ '
(ii)
H=/-I=0,
2A2 A3
& ~ - - - - 4 -A2 3
A 3 '
4 /~2 15 h3 0.) ~ -
(2.5) 4 -A2 15
A 3 '
Now the mass term for the higgses is m3/~aH3 + rn2I~EHE,
(2.6)
where the mass of the triplet i s m 3 = (~b +to))tl and the mass for the doublet is m2 =
Using (2.5) we see that solutions (i) cancel m2 independently of the values A2 and ,~3, and m3 = ~eoh~ where w is the grand unification scale. On the other hand, solutions (ii) cancel m3, also independently of h2 and h3, being rn2=-~a~Aa. Obviously, the former solution is the one we are interested in and it solves the problem satisfactorily. Therefore there is no fine tuning involved in our model since the supersymmetric solution should be naturally preferred by nature. Also, we would like to stress two virtues of our model: the economy in the number of involved fields and the simplicity of the superpotential. As we have just seen, the natural solution to the problem of the massless Higgs doublet led us to the replacement of an adjoint of SU(5) by an adjoint of U(5). This poses new aspects we will be treating in the following sections. In the first place, as the charge 1~(1) of the adjoint is zero, the gauge symmetry will break to
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Jr. Ledn et al. / Spontaneously broken supersymmetric models
SU(2)Lx SU(3)CxU(1) Y x0(1). A suitable choice of the charges 1:1(1)* should forbid possible baryon number violations as it was pointed out by Weinberg [9]. Also the presence of an additional 1~(1) could be associated with a breaking of the supersymmetry /~ la Fayet-Iliopoulos. Apart from these low-energy constraints, there are additional general restrictions due to the requirement of a U(5) anomaly free theory. Furthermore, being SU(5)xl~I(1) a non-semi-simple group, it seems logical to ask oneself whether it is possible to find a unification group at higher energies. On the other hand, let us stress that although SU(5) x l:l(1) is not semisimple, i.e. it has two different coupling constants gsu~5~ and go~l), strong and electro-weak interactions remain unified in SU(5) at the scale mx. The new hypercharge I7" obviously commutes with all SU(5) generators and does not affect all the good properties (electric charge, Weinberg angle,...) of the minimal DimopoulosGeorgi-Sakai SUSY SU(5) model. A separate study of all the above questions will be carried out in the following sections.
3. Discarding SU(n) SUSY GUTS Supersymmetric gauge models could also be considered as effective GUTs of a supergravity theory, and this is certainly a very attractive feature. This possibility has generated many works [12], but its analysis is not the subject of this paper. Nevertheless, we would like to emphasize here that N = 8 extended supergravity contains an effective SU(8) gauge theory at energies higher than Mp [13]. Furthermore, the possible breaking of supersymmetric theory was speculated recently [14]. The above ideas seem to suggest that the SU(n) groups would be possible unification groups. In this section, we will study the consequences of such statement. SU(n) groups (n > 5) have been somehow traditional models for grand unification. The idea is to reach the existence of three families, but not as a mere repetition of the structure 5+ 10. King [15] tabulated asymptotically free and anomaly free representations able to contain at least one family of light fermions without exotic states. Leaving aside asymptotic freedom that leads to different results for the supersymmetric case and is anyhow of rather dubious necessity, we find an ample variety of solutions in the form:
R =~.Ck[n,k],
(3.1)
k
where [n, k] is the k-fold antisymmetric tensor: [n, k] = 4~ac''~k. Apart from these complex representations, we need a gauge vector supermultiplet and a chiral Higgs supermultiplet, both in the adjoint of SU(n). The D terms of these representations
* T h e 1:1(1) factor was first introduced by Fayet in order to obtain a correct low-energy spectrum, otherwise scalar quarks or leptons would be lighter than their spinor partners.
J. Ledn et al. / Spontaneously broken supersymmetn'c models
109
can be written as -k6r,...,,,,To,
D,~[,~r,...,.,,]
,.~c~
= k6r'"'r"T~,r~,-,,~,.~...,.,,
,
,
D ~ [ e # ba ] = T~~c,[~b, - t#]~',
(3.2)
while the superpotential will depend on n. Now, we could write not only terms like ~ba~babSU(5) x G--> SU(3) x SU(2) x U(1) x G
(3.3)
For the first breaking we will need the Higgs supermultiplets ~b~1.... -, ~ba,. . . . , ~ba, that take non-zero v.e.v.'s for the components al . . . . . a , = 6 . . . . . 5 + n . Their ratios will be determined by the supersymmetry condition D~ = 0 [11, 16]. The introduction of an additional group G reflects the possibility that a certain linear combination of generators of the Cartan subalgebra, which are not contained in SU(5), annihilates each one of the v.e.v.'s. Therefore G would be of the form (U(1))", and a straightforward analysis shows that for our particular v.e.v.'s m = 0. As usual, the second breaking comes through a non-zero v.e.v, of an adjoint , ~ of SU(5) which is contained in the adjoint , ~ of SU(5 + n). Our notation goes as follows r, s, t = 1 . . . . . 5 + n ; x, y, z = 1 . . . . . 5; a, b, c = 6 . . . . . 5 + n . We consider a SU(5 + n) superpotential similar to the one we wrote in eq. (2.1). However, we will show that, unlike U(5), WS doublets will not be "natural" massless in the SU(5 + n) model. Our superpotential is V
= A 1/r'~VH +
A2 tr X 2 + A3 tr Z 3 ,
(3.4)
.~ contains, among others, (1, n2--1)(trb~,), (24, 1)(try) and (1, 1)(~b) with respect
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J. Le6n et al. / Spontaneously broken supersymmetric models
tO SU(5)x SU(n). We have the relations: X~ = o'~ + 1 8 ~ b , n
(3.5a)
2yx =cryx - ~l ~ox ~ .t
(3.5b)
In terms of representations of SU(5)x SU(n), the superpotential (3.4) can be written in the form:
+A2 tr bO'a + 2 ~ , ~ r
a
b
c
+ o'ytrx + a
b
x
:"1-A3tO'bO'cOv a + 3trb.~,x,,~,a + r /1
1\
a x
1
+ a
x
y
x
y
z
3~O'y.,~a + Orytrztrx
,, b
1
~ y'~
1
1
a
The condition D~ = 0 allows us to diagonalize simultaneously the real and imaginary parts of ,~. We take the following set of v.e.v.'s: ,~, ~ = Z ~ = n '
,
trb=X
(li, 0
0
= ISl, = O,
--p/(n--p)l,,_p
o',=to 0
),
0
(3.7)
-312
and, using the conditions of the supersymmetric vacuum V ~ = V~; = 0, we obtain for x and to the relations: 4../h2 3 \ to
=
n-p
Notice that o-~ in (3.7) has been chosen in a way that will break SU(5) to SU(3) x SU(2)xU(1)(to # 0). The masses of the colored triplet and the SU(2) doublet contained in H are given by m3 = (--51-¢ +to)}[1 ,
m2 = (--51-¢- 3to)A1.
Using (3.8), together with the supersymmetric vacuum condition V, = 0, the results of the table 1 are obtained. It is easy to see that no couple of values (n, p) exists such that the mass of the Higgs doublet vanishes. We then deduce that the
111
J. Ledn et al. / Spontaneously broken supersymmetric models TABLE 1
Vacuum expectation values and doublet Higgs masses for the breaking SU(5 + n) ~ SU(5) ~ SU(3) x SU(2) x U(1) x 0
~
~o
m2
n n+l n 3A - n-1 A 2n-5p n-2p+l A 3- n --5 p n-2p-1
n+5 n+l 2 n +5 ~A n-1 ~a n+__.___~_5 n-2p+ l 2 ° n+5 _---3A n ---2p- 1
(n+3) n+l AAa 3-n-1 -,~,~1 n + 3 - p n-2p+l AAx- p+3 n-2p-1
-¼A(n -5)
~A(n +5)
-¼AAI(n+7)
-½hA
~A(n +5)
-½aa l(n +6)
2/~ - -
0 -a (n-p)(n+5) n ( n - 2 p + l) (n -p)(n+5) n(n-2p-1) ±A (n-p)(n+5)* 2n ±½A(n +5) **
2~
--/~ 1 -
The parameter A = 2A2/3A3. *The ± sign is valid for p =[(n ±1), n odd. **Is valid for p = ½n, n even. The rest holds for the other values 0 < p < n. s y m m e t r y b r e a k i n g p a t t e r n (3.3) does not solve the p r o p o s e d p r o b l e m without fine tuning. A m o r e general situation that the one outlined in (3.3) could be o b t a i n e d passing through SU(5+m)×G instead of S U ( 5 ) x G . In this case, the second b r e a k i n g would give a colour SU(3 + m) (an example could be the SU(4) of P a t i - S a l a m [18]). T h e r e f o r e , we will analyze the s y m m e t r y b r e a k i n g pattern given by SU(5+n)-->SU(5+rn)xG-->SU(3+m)xSU(2)xU(1)xG,
(3.9)
with n > m. W e use the index notation: r,s,t=l .... ,5+n, a,b,c =6+m .....
x,y,z=l
.....
5+m,
5+n.
A s before, the first s y m m e t r y b r e a k i n g needs Higgs supermultiplets $ a c a " - m , $al . . . . . $ . . . . that take n o n - z e r o v.e.v.'s, for the c o m p o n e n t s a l . . . . . a , _ , , = 6 + m . . . . . 5 + n , with suitable p r o p o r t i o n s [16]. T h e g r o u p G would be again (U(1)) s, and we have s = 0 with the p r o p o s e d v.e.v.'s. T h e second b r e a k i n g of (3.9) would p r o c e e d t h r o u g h the n o n - z e r o v.e.v, of an adjoint 2 ~ of S U ( 5 + m ) that is c o n t a i n e d in the adjoint , ~ of S U ( 5 + n ) . T h e relevant superpotential is given by (3.4). ,~ contains, w h e n d e c o m p o s e d with respect to S U ( 5 + m ) x S U ( n - m ) , (1, ( n - m ) 2 - 1 ) ( o ' ~ ) , ( ( 5 + m ) 2 - 1 , 1) (o"s) and (1, 1) ($). T h e y are related a m o n g themselves through: 1
" ~:~ = orb+ n-m
" 864',
1
-8#. ~ = o'y~ ' - -5+m
(3.10)
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Y.Le6n et al. / Spontaneously broken supersymmetric models
In terms of SU(5 + m ) x S U ( n - m) representation the superpotential (3.4) can be written as V =/~1
(O.ba "4-
1
8~
n-m
)fib
1
+ Hx ( o ' , -x ___5+m 8~qb)HY
]
[ as / 1 1 ~ 2"] +A2 ort~ra+2~'a~"x + c r y - : +kn----~+5-V~m)~b J a
b c
a
b
x
a
+As o'w~o-a + 3cr~,~:.~ + + 3
4,((~__~__1 m
1 ) 5
1
m
x
y
1 .yay~+
1
n -m
+ or,~r~rx
~ b
OgbO'a
(3.1"1)
1 _ 5 _ +_m
. ,)
OryKFx
3
+ ( ( n - m ) 2 (5+m)2) ~b ]" The supersymmetric vacuum conditions V~g = V ~ = D e = 0 diagonalize X~ and obtain the following set of v.e.v.'s: xa
o(1
Crb=X 0
= .y, ~ ,~ =
o
H r =fir=O,
)
0
--p/(n--m--p)l .... p
o'y =to
-
permit us to
o)
(3.12)
to#O
I(3 +m)12
with to and x given by the relations 1 [4 A2 (n +5)_44~] ' to = (n + l ) ( n +5) L3 A3 2(A2+n3A m ~b)x+ 3A3 n --m--2Px2 = 0. -
n -m
(3.13)
-p
The masses of the coloured triplet and the SU(2) doublet contained in H are given by ma=(to-qb/(5+m))A1 and m 2 = ( - ~ ( 3 + m ) t o - d ~ / ( 5 + m ) ) a l . Using (3.13), together with the supersymmetric vacuum condition V~ = 0, we obtain the results of table 2. Notice again that is impossible to get m2 = 0 for any set of values (n, re, p). Summing up, the SU(n) SUSY G U T models considered above require a fine tuning or an arbitrary introduction of singlets in order to keep the Higgs doublet massless. It is amusing to verify that in the limit n ~ 0o, the rest of the parameters fixed, there are solutions with m2 = 0 in tables 1 and 2. This suggests that the problem proposed in this section might have a solution in the framework of a non-semi-simple group such as U(n). We will solve this question in sect. 4.
J. Le6n et al. / Spontaneously broken supersymmetric models
>(
I
~
4-
~.
t"~ l h-
÷
÷
v
.z
>(
113
v
7
÷
"t
~+
÷
÷
.=. t.q~'
r
1.:.
"0 ¢. t~
I
4-
I +1 II cO
¢q I t'q 44-
~
:> ..<
÷
ra
4-
t"4
~ea q'l
•z . . ~
I
[...,
114
3. Ledn et al. /Spontaneously broken supersymmetric models
4. Canonical U(n) SUSY GUTS In this section we will study a supersymmetric theory based on U(n) groups. In particular we will analyze the symmetry breaking pattern given by U(5 + n) ~ U(5) × G ~ SU(3) × SU(2) x U(1) × 1](1) x G ,
(4.1)
where we do not exclude, as in sect. 3, the existence of a G factor whose presence should be investigated in each particular case. The U(1) group could be the one, frequently proposed [9, 10, 19], that is able to depress the A B ~ 0 processes in a way required by experiment, and be the origin of a spontaneous supersymmetry breaking in a framework free of the problems exhibited by the theorem of Fayet et al. (see appendix of ref. [8]). As the U(n) groups are not semi-simple we move on to the old problem of obtaining an anomaly free theory. Moreover, the D term presents two additional problems: (i) it has to remain free of quadratic renormalizations [11] and (ii) supersymmetric vacuum conditions have to be enlarged with equation Do¢~)= 0. We will show further on that we have positive answers to all these problems, however we would like to start by showing that this framework solves the fine-tuning problem. Following a similar treatment to that of sect. 3, the superpotential relevant to our problem is given by (3.4). The adjoint of U(5 + n) contains, among others, the adjoint of U(5)(.~) and the adjoint of U(n)(,~). Performing now the decomposition: =o'y-5Oy~,
Z b =trb +
8
(4.2)
,
n
where d~ ~ d~ indicates that we are in U(5 + n) instead of SU(5 + n). The potential (3.4) decomposed with respect to U(5)× U(n) is written as
n [
a
b
c
a
b
"JI-A3[O'bO'cO'a . t _ 3 O r ~ , , ~ x , ~ '
3 * n
a
x
x a
3
J
a x~,~ y + O'r xO'~ yO"x z + 32~"
a b 3 x a +o'b~ra)--~(.V~.Vx +crycrx)
(4.3)
3". Ledn et al. / Spontaneously broken supersymmetric models
115
TABLE 3 Vacuum expectation values, doublet and triplet Higgs masses for the breaking U(5 + n) ~ U(5) ~ SU(3) x SU(2) x U(1) × l~l(1) t~
tO
2A
~A -~A
3A
m2
-AAI
0
0
-AAI
mp
m n_p
0 -AA1
0 -hA1
-AA1
0
0
-AA1
.
X
0 -nA
0 0 n-p -A--
-Ap
m3
n
n-p A- -
-A(n -p)
n
The parameter A = 2A2/3A3.
choosing those v.e.v.'s that lead us to SU(3)c x S U ( 2 ) L × U ( 1 ) Y, i.e. o'y=.,
-312
0
[-p/(n-p)]l._p
o(1 o'b=x
0)
0
'
0
) '
(4.4)
and imposing supersymmetric vacuum conditions V, = Vg = 0, we obtain the results of table 3. Notice that, in the case (U(n)), the sets of v.e.v.'s (~b,to) and (~, x) are decoupled. For the decomposition associated with (4.1) the Higgs field H is given by H r = H x ~ ) H a = ( H ~2)( ~ H ~3))~ ( H
¢p}~)H("-P)),
(4.5)
following a similar notation to the one used before./-It will be written analogously. The physically acceptable solution is the one for which only the H ~2) doublet is massless. Then, the solution to the proposed problem can be read from table 3. It will be given by the v.e.v.'s 4, = 3 A , = -nA,
m3 = -AA1,
to = -52-A,
m2=0,
x = 0,
mp = m,_ o = -hAl.
(4.6)
Next, we will concentrate on the other problem of the supersymmetric vacuum: the cancellation of D terms. The first breaking in (4.1) needs Higgs fields ~brc''r", ~b~1). . . . . ~b~") with non-zero v.e.v.'s ~b~i) =l~65+i.,, ~ 6 .... =tz. This assures D,, = 0, a being an index of the Lie algebra of SU(n) [16]. On the other hand, the D term for the higgses, associated
J. Le6n et al. / Spontaneouslybrokensupersymmetricmodels
116
to the group l~l(1), is given by --
I=1
i
1
with an obvious, but not unique, solution: I7"1. . . . .
I7",= ___1 17". n
(4.8)
Then, we propose a canonical assignment for the charges 0(1)(17") as a simple non-trivial solution for the problem of the cancellation of D terms. A canonical assignment means that the generator of 0(1) acts in a similar way to the rest of the generators of U(5 + n), i.e. 8,d~ r'''''m = T,~",~d~ r~'2"'''m + " • • + T , , " , ~ , ~ ' ' ' ' ' ' m - ' ' ; ~ ,
(4.9)
The U(1) generator is 7~ = I, therefore
g¢,c",. = m&rc"',,
t" = m.
(4.10)
As the complex conjugate representation ¢,, ..... transforms with - T t, we have g&,,...,, = -m&,, ......
I7"= - m .
(4.11)
Analogously, for a representation with m upper and m' lower indices the hypercharge 0 ( 1 ) is I7"= m - m'. Now, we have the necessary ingredients to build an anomaly-free U(n) theory. Let us assume, in general, that R of eq. (3.1) is a set of Nk supermultiplets & , c , k (k = 1 , . . . , n), i.e. Ck = Nk, where negative values of Nk will indicate the conjugate representation ~,,...,k. The anomalies of a U(n) theory can be classified into three classes: (0(1)) 3, U(1)(SU(n))2 and (SU(n))3. Their cancellation will be assured, respectively, by the equations:
~lk3N~(n)=o,
i klkNk=O
k
k
k=l
~ AkNk=O, '
(4.12)
k=l
where
(n-2k)(n-3)!
Ak=(n_k_l)!(k_l)!,
lk =
(n--2) k-1
(4.13)
"
The solution to the system of equations (4.12) is given by k=4
N2 =
k=4
(k - 3)
_ 2 Nk,
2/n-1\ N3 = -
k=4 k - 3
Nk.
(4.14)
3".Ledn et al. / Spontaneously broken supersymmetric models
117
Surprisingly, there are admissible solutions (Nk integer) for any group U(n). Before analyzing in detail the obtained representations, we will study the problem arisen from the/9 term. The problem for a supersymmetric gauge theory with a l~l(1) factor is well known: the /~ term of Fayet-Iliopoulos is generated perturbatively through quadratic divergences i.e./~ ocM~. This is a disaster for supersymmetric GUTs. Witten showed [11] that if a 1~1(1) factor gets unified in a semi-simple group G, then the term/~ is not generated perturbatively. More recently, Fischler et al. [20], obtained a less restrictive condition: It is sufficient to demand a cancellation of the l~l(1) hypercharge, tr I7"= 0, to protect the theory from a perturbative generation of the/~ term. With the canonical assignment (4.10) (4.11) tr I7" can be written as: tr f" =
k=l
k
k
Nk,
(4.15)
where N1, N2 and N3 are fixed by (4.14). We rewrite (4.15) in the form: tr f ' - - ~ d k N k . k~4 Remarkably enough, an explicit calculation shows dk=O,
(4.16)
k = 4 . . . . . n.
then tr I7"= 0. Now we can state the following theorem. Theorem. "Any supersymmetric gauge theory based on a gauge group U(n), with canonical assignment of charges and free of anomalies, is protected from quadratic renormalizations". The origin of this beautiful result can be traced back to the fact that eq. (4.15) is a linear combination of eqs. (4.14). Thus, we have generalized the results of Kaul and Majumdar [21] to the general case of arbitrary canonical U(n) groups. Still, two questions do remain: (i) supersymmetry breaking (ii) depression of AB ~ 0 processes. We postpone their discussion to sect. 6. Finally, we would like to discuss the relation between the supersymmetric GUTs of this section and the effective gauge theories contained in N supergravities [13]. Discarding N = 8 supergravity (with an effective SU(8) theory), N supergravities (N = 5, 6, 7), with effective U(N) gauge theories, .do remain as possible candidates for a superunification. Among these, N = 7 supergravity is the candidate that presents a greater degree of convergence in its perturbative expansion. Therefore, a supersymmetric gauge theory based on a U(7), and unified to N = 7 supergravity at the Planck mass, seems to be the most attractive candidate for the superunification. Work along this direction is now in progress by the authors. 5. Naturalness in SO(4n + 2) SUSY GUTS The well known properties of this series of groups have made them serious candidates for grand unification [22]. In this section we show that SO(4n + 2) groups
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are free from fine-tuning problems. However, if we want a very massive rightneutrino we will lose the U(1) invariance at low energies. Although our reasoning and results are completely general, we will centre our treatment on SO(10) for the sake of clearness. In SO(10), the content of our theory will be: (i) a gauge vector supermultiplet in the adjoint (45), (ii) matter supermultiplets in the spinor representation (16), (iii) Higgs supermultiplets: several in the 10 representation (vectors that break SU(2)L× U(1)Y), one in the 45 (antisymmetric second rank tensor) and another in the 54 (traceless, symmetric second rank tensor). They form a minimal set of higgses whose SU(5) decomposition is 10a = {H x,/-Iax},
a = 1. . . . . h ,
45 = {A Ex'yl,AE~.yl, .,~zxy, 4,},
(5.1)
54 = {S ~x'y~, S~x,y~, .,~l"y}, where H~ = 5 ,
I7Iax=5,
a=l ..... h,
A tx'yl = 10,
/~[x,y]= 1--0,
,,~2xy = 24,
S ~x'y~= 15,
go,.y~ = 1--5,
.2"lXy= 24.
4,=1 (5.2)
Notice that fine tuning demands the presence of more than one 10, due to the fact that the natural masslessness of the doublet is achieved through the coupling of the 10's with the antisymmetric 45. The more general superpotential compatible with supersymmetry and gauge symmetry is V = A 1(54) 2 + A 2(45) 2 + A 3(10 × 10)s × 5 4 + A4(10 × 1 0 ) a
× 45 + A5(54) 3 + A654 × (45 × 45)s
(5.3)
and the SU(5) decomposition of these terms is (54) 2 = 2 tr ($S+~V21), (45) 2 = 2 tr (AA + ~ + ~ 4 , 2 ) , (10 × 10)s × 54 = dab (HaSHb + 2I'7I~.Y,IHb + HaSHb ) , (10 × 10)A × 45 = fab (H, ug.Hb + I4aAI~b + I'tb (~,Z + kd~)Ha), (54) 3 = 2 tr (3Sg,~1 + ~ ] ) , 54 × (45 × 45)s = 2 tr (AASa +~1-~z~2 + 20-~z-~1),
where lab (dab) is an antisymmetric (symmetric) arbitary matrix.
(5.4)
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119
The supersymmetric vacuum conditions give us: A = A = S = S = H a =/-I= = 0 , V~b= ~A2¢~"1"2A6 tr (X2X~) = O, Vx, = 4A 1,~1+ 6A s (XxX1 - ~ tr X~) + 2A64~$2 q.. ~6(,~2,~ 2 -- 1
tr $2) = 0,
V,~1 = 4 A 2 5 2 + 2 A 6 ( $ 2 5 1 -
51-tr $2X1) + 52-X6~b$1= 0 .
(5.5)
We are interested in the v.e.v.'s that break SU(5) to SU(3)c x SU(2)L × U(1)v:
.~l = 0)1( ~ __23_O2),
$2 = ~o2(10 _~O2}.
(5.6)
The solutions to eqs. (5.5) are given in table 4. Notice that A2 and A6 should be different from zero; nevertheless A1, A3 and A5 could be zero. Obviously, the only condition for •2 # 0 will be A1 or A5 different from zero. The mass term is rn~j-7~Hb, where the mass matrix rn~b can be expressed as mab = A 3dabS1 + A4fab ( $ 2 + l~b ) .
(5.7)
Let us consider the c a s e / ~ 3 = 0. Then, by using the relations between ~b and to2 given in table 4, we obtain for the doublet (triplet) mass m2(m3) the following values: ~b = -5to2, --'~~¢M2,
m3 = 0,
m2 = -5to2,
(5.8a)
m 3 = 5¢-D2,
m2 = O.
(5.8b)
Choosing, hence, solution (5.8b), the fine-tuning problem is solved. Another possibility, that will lead us to natural results as well, could be accomplished with the introduction of two higgses 10 and 120 through the couplings: 8V = A4(120 x 10 x 45) +~ ~ (120 x 120L × 54+A~(10 x 10L x 54
(5.9)
The SU(5) content of 120 is given by t~ [ijk] ~- {D[,,yz],
D[,,yz],Dzxy + 8 ~xD y - 8 ~ yD -
,l)yz+Sy1~-8715y},
TABLE 4 Vacuum expectation values for the breaking of SO(10).
4,2 3 A6
-5.2
2 15 \A 2 A6/\A6] A2
A6]\A6]
(5.10)
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J. Le6n et al. / Spontaneously broken supersymmetric models
where Dt~yzl = 1 - 0 ;
/~txy~l= 10 ;
D~Z+87DY-6~DX=45+5,
and Dy~ -x +8~/9z x - - 8 ~xD-y = 4 5 + 5 . The relevant mass terms in the superpotential, i.e. /~4 terms in eq. (5.9), are 1 0 x 5 x 1 0 + 1 0 x 5 x 1 0 + ( 4 5 + 5 ) x 5 x 1 0 + ( 4 5 + 5 ) x 5 x 1-0 + 2(45 + 5) × 5 x (2~2+~d,) + 2(45 + 5) x 5 x (2~2+~b).
(5.11)
Again, as the non-zero v.e.v.'s are in the adjoint Z and in the singlet ~b, the appearance of a structure (Z +½~b) assures us the repetition of results of the type (5.8) as requested. We are analyzing a symmetry breaking pattern that breaks SO(10) to SU(2)L× SU(3)c× U(1)Y × 0(1) directly, without an intermediate step through SU(5). This framework presents a disadvantage: The right-neutrino cannot acquire a large Majorana mass. In order to circumvent this problem Harvey et al. [23] proposed the introduction of a Higgs in the 126 with a non-zero v.e.v, for the SU(5) singlet. This procedure gives a Majorana mass to the right-neutrino, of the order of the grand unification scale, while the left-neutrino remains massless. Unfortunately, the singlet in the 126 is not invariant under 0(1). Thus, we lose the possibility of breaking supersymmetry, h la Fayet-Iliopoulos, at lower energies, along with the depression of AB ~ 0 terms. The symmetry breaking pattern is, in this case, SO(10)-, SU(5) ~ SU(3) × SU(2) × U(1),
(5.12)
where the first breaking is due to the non-zero v.e.v.'s of the 126 and 16. Both v.e.v.'s conspire to keep supersymmetry unbroken.
6. Spontaneous SUSY breaking and ~ B ~ 0 suppression Having in mind the results of sects. 3, 4 and 5 we would like to present here some general results on spontaneous supersymmetry breaking and on suppression of processes with AB ~ 0 for supersymmetric canonical U(n) theories. In the canonical U(n) theories, we can introduce a "small" spontaneous symmetry breaking of supersymmetry. This possibility comes from the absence of perturbative quadratic divergences as we showed in previous sections. On the other hand, as the supersymmetric vacuum is the most stable one, we should prevent the appearance of representations whose v.e.v.'s protect supersymmetry from breaking. We present here a study on the different patterns of gauge and supersymmetry breaking for a supersymmetric U(n) gauge theory. Supersymmetry breaking comes h la Fayet-Iliopoulos, i.e. from the non-cancellation
J. Ledn et al. / Spontaneously broken supersymmetric models
121
of the L) term. In general, we write the D terms as /9 =• IT"(~l~b)+~,
D~ = Y <61T I >
(6.1)
for a theory with higgses (~b) of charge I7".Let us consider the case where supersymmetry breaking appears through D~ = 0 a n d / ~ # 0. The SU(n) gauge symmetry breaking with D,, = 0 has been classified in ref. [16] as follows. (i) Gauge invariance is not broken: tb = 0. (ii) U(n) gauge invariance breaks spontaneously to U(m) (m < n ) , through the higgses ~btrc''~-J, ~ b(1)r , i = l . . . . . m - n . (iii) U(n) gauge invariance breaks spontaneously through higgses in real representations. For example U(n) goes to U ( m ) x U ( n - m) (m ~ ½n) through a Higgs in the adjoint of U(n). (iv) U(2n) gauge invariance breaks spontaneously to Sp(2n) through a Higgs (v) U ( 2 n + 1) gauge invariance breaks spontaneously to Sp(2n) through higgses ¢~ [rl'r2] and
Notice that we are not including, in the U(n) theory, neither fields with an exotic colour content (i.e. symmetric tensors and so on) nor real reducible representations. The higgses conjugate to the above ones would lead to the same gauge symmetry breaking and then we do not consider them as separate ones. In cases (i), (ii) and (iii), condition D~ = 0 implies ~ I7"~<= 0 [16]. Therefore, /9 = ~ and supersymmetry is not restored by gauge symmetry breaking. However, supersymmetry is protected (/9 = 0) by ~=-214,t~"r2112 in case (iv). Then, the condition that allows supersymmetry breaking is given by N 2 -- N 2 n - 2 = 0 ,
(6.2)
where AT/is the number of fields 4~n-,,. Notice that Ni < 0 indicates conjugate fields. In case (v), supersymmetry is broken (LJ # 0) iff: N1 • N2 ~<0.
N2,-1 • N2, ~<0.
(6.3)
We would like now to present the symmetry breaking pattern that emerges from the above results. At the mass Planck scale, the effective gauge theory coming from supergravity would be U(n). At lower energies, we would have a series of successive steps described by the chain: SO(n) supergravity ~ U(n) x SUSY
A,
~ U(m) x SUSY
Am
SU(m - 2) x SU(2) x U(1) x l~l(1) x SUSY a~.~~ -~ SU(3) c x SU(2) x U(1) x I~l(1) x S U S Y -~ SU(3)c × SU(2) x U(1) x I~l(1)
) S U ( 3 ) C x U(1) .... Aw
(6.4)
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I. Le6n et al. / Spontaneously broken supersymmetric models
with n > m 1>5. The relevant cases would be m = 5 (Georgi-Glashow type) and m = 6 where an SU(4) colour group is obtained (Pati-Salam type). The first breaking, at scale An, is produced through a set of higgses ~b'l'~-m, 4~ ~) (i = 1 . . . . . n - m) with v.e.v.'s that guarantee D~ = 0. T h e n / 5 -- (, that points out the appearance of a spontaneous SUSY breaking which should not be observable up to energies of the order of (1/2((6.4)). With a canonical assignment of U(1) charges in U(n), the proposed v.e.v.'s lead to a canonical assignment of charges for the U(1) contained in U(m). Therefore, the 1~1(1) that appears later on corresponds to the canonical U(m). The second breaking, at a scale Am, happens through v.e.v.'s in the adjoint of U(n). The criteria, that we studied in sect. 4, assures the natural masslessness of the Higgs doublets. Now, by passing Pati-Salam breaking, about which we have nothing new to say, we could choose an energy scale (~/2 suitable for solving the mass hierarchy problem and making sfermions massive enough. Our guess is ~>A2w. Then, either SUSY and electroweak breakings occur simultaneously, or standard breaking of SU(2) L x l~I(1) x U(1) .... appears immediately after SUSY breaking. Regarding mass splittings and the existence of superheavy fermion families, there are certain constraints which are present in the models ~tla Fayet-Iliopoulos. Firstly, we consider the mass splitting formula for a supermultiplet [24]:
Y~(-1)2'(2J + 1) tr rn~ = 2g2gtr I7".
(6.5)
J
Eq. (6.5) shows that the order of magnitude of m a2 - r n ~ is determined by the breaking parameter ~ Moreover, either we have a light multiplet (mr2 <>m ~f), depending on the sign of I7". On the other hand, our model is free from the anomaly U(1) × SU(P) x SU(P) i.e. ~Ro C2(R) tr IT"R= 0. This takes us, using eq. (6.5), to the condition: Y. C2(R)(m~ R - m~R) = 0,
(6.6)
Rp
from which we can easily see the joint presence of light and heavy multiplets. Building up realistic models where SUSY is spontaneously broken is a difficult task. This difficulty has been stressed particularly by Buceella et al. [16] in the context of the Dimopoulos-Georgi-Sakai model. The problem is that matter fields prefer to break colour and/or letpon number, in order to preserve a supersymmetric vacuum state, unless the couplings and fields present in the superpotential were able to prevent the above mechanism. However, there is not any no-go theorem that forbids a SUSY breaking ~ la Fayet-Iliopoulos and realistic standard SUSY models have been drawn by Barbieri et al. [25] and Hall and Hinchliffe [24]. Another possible mechanism of SUSY breaking is through N = 1 supergravity effects in SUSY Yang-Mills theories [26] although, in this case, R-invariance shQuld be imposed as deduced by Barbieri et al. [27].
J. Le6n et al. / Spontaneously broken supersymmetric models
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With the previous assignment of charges I7" (i.e. light multiplets have IT"'s with the same sign), there are not any terms (dimension <6) that violate the baryon number. This problem has been studied by Weinberg and we refer the reader to ref. [9] for further details.
7. Conclusions
The relevance of SUSY GUTs, as a description of the real world, has been darkened, up to now, by two annoying facts: (i) the need for an "unnatural" fine tuning of the mass parameters of the theory, and a more serious point, (ii) the difficulty to generate a SUSY breaking with a suitable mass hierarchy. In this work, we proved that both facts are indeed closely related and found a simultaneous solution for both of them. We began undertaking the problem of fine tuning, whose solution led us in a direct and natural way to models of the U(n) type. Specific difficulties connected with D terms, along with the anomalies present in U(n) theories, guided us to a canonical assignment for l~l(1) charges, recovering for these the character of grand unification at high energy scales. The absence of quadratic renormalizations for the D term permitted us to break supersymmetry spontaneously h la Fayet-Iliopoulos at an energy scale ~1/2. The value of ~ can be chosen conveniently in order to obtain realistic mass splittings for the supermultipiers. SO(4n +2) groups solved the fine-tuning problem. However, as we showed explicitly in our example SO(10), a large Majorana mass for UR is incompatible with l:l(1) invariance. Therefore we lose the possibility of spontaneous supersymmetry breaking. U(1) invariance is able to forbid the appearance of effective interactions of low dimension (d < 6) with A B ~ 0 and A L ~ O. Our treatment is connected with the one proposed by Weinberg [9] for a supersymmetric standard model. Summing up, the moral that follows from this work should be: canonical U(n) groups are the most suitable ones in order to build up a supersymmetric gauge theory, free of unnatural tunings, with a realistic spontaneous SUSY breaking and superunified at Mp with n supergravities. Now, an open question would be to find, within the proposed framework of this paper, a realistic model able to reproduce low-energy physics. Some work along this line is now in progress.
References
[1] [2] [3] [4]
L. O'Raifeartaigh,Nucl. Phys. B96 (1975) 331 P. Fayet and J. Iliopoulos,Phys. Lett. 51B (1974) 461 L. GirardeUoand M.T. Grisaru, Nuci. Phys. B194 (1982) 65 H. Georgi, in Grand unifiedtheories and related topics, p. 109 ed. M. Konuma and T. Maskawa (World Scientific,1981) p. 109
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[5] L. Iba'nez and G. Ross, Phys. Lett. 110B (1982) 215; Phys. Lett. 105B (1981) 439; D.V. Nanopoulos and K. Tamvakis, Phys. Lett. 113B (1982) 151; K. Tamvakis and D. Wyler, CERN preprint TH-3240 (1982) [6] A. Masiero, D.V. Nanopoulos, K. Tamvakis and T. Yanagida, Phys. Lett. 115B (1982) 380 [7] N. Sakai, Z. Physik C l l (1981) 153 [8] S. Dimopoulos and M. Georgi, Nucl. Phys. B193 (1981) 150 [9] S. Weinberg, Phys. Rev. D26 (1982) 287 [10] G. Costa, F. Feruglio and F. Zwirner, Nuovo Cim. 70A (1982) 201 [11] E. Witten, Nucl. Phys. B185 (1981) 513 [12] J. Ellis, Phenomenology of unified gauge theories, preprint CERN TH-3174 and LAPP TH-48 (1981) [13] E. Cremmer and B. Julia, Nucl. Phys. B159 (1079) 141 [14] J. Ellis, Grand unification in extended supergravity, preprint CERN TH-3206 (198t) [15] R.C. King, Nucl. Phys. B185 (1981) 133 [16] F. Buccella, J.P. Derendinger, C.A. Savoy and S. Ferrara, Symmetry breaking in supersymmetric GUTs, preprint CERN TH-3212 (1981); Phys. Lett. llSB (1982) 375 [17] J. Le6n, M. Quir6s and M. Ram6n Medrano, Phys. Lett. l18B (1982) 365 [18] J. Pati and A. Salam, Phys. Rev. D8 (1973) 1240 [19] P. Fayet, Supersymmetry, particle physics and gravitation, in Unification of the fundamental interactions, Erice (1980); preprint CERN TH-2864 (1980). [20] W. Fischler, M.P. Nilles, J. Polchinski, S. Raby and L. Susskind, Phys. Rev. Lett. 47 (1981) 757 [21] R.K. Kaul and P. Majumdar, Nucl. Phys. B199 (1982) 36 [22] P. Langacker, Phys. Reports 72 (1981) 185; R. Slansky, Phys. Reports 79 (1981) 1 [23] J.A. Harvey, P. Ramond and D.B. Reiss, Phys. Lett. 92B (1980) 309 [24] S. Ferrara, L. Girardello and F. Palumbo, Phys. Rev. D20 (1979) 403; L. Hall and I. Hinchliffe, Phys. Lett. l12B (1982) 351 [25] R. Barbieri, S. Ferrara and D.V. Nanopoulos, Z. Phys. C13 (1982) 267; Phys. Lett. I16B (1982) 16 [26] E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Phys. Lett. l16B (1982) 231; Nucl. Phys. B212 (1983) 413 [27] R. Barbieri, S. Ferrara, D.V. Nanopoulos and K.S. Stelle, Phys. Lett. l13B (1982) 65
SPONTANEOUSLY BROKEN SUPERSYMMETRIC MODELS FREE OF FINE-TUNED PARAMETERS A N D ~B, ~L TROUBLES* (I). General survey J. LEON and M. QUIROS Instituto de Estructura de la Materia, CSlC Serrano 119, Madrid 6, Spain
M. RAMON MEDRANO Departamento de Ffsica Te6rica, Facultad de F(sicas, Universidad Complutense, Madrid 3, Spain
Received 2 August 1982 (Revised 21 March 1983) We propose a class of supersymmetric models based on canonical U(n) gauge groups with the following properties: (i) absence of anomalies; (ii) no quadratic renormalizations for the D term; (iii)absence of fine-tuned parameters; (iv)spontaneous SUSY breaking h la Fayet-Iliopoulos; (v) free of unobserved effective AB ~ 0 and AL ~ 0 interactions.
1. Introduction The main underlying problem in gauge theories is the generation of masses for elementary particles. In the last years, the traditional f r a m e w o r k ~ la Higgs has been questioned and other schemes with dynamical generation of masses have been favoured. The latter scheme in its two versions of technicolor, and extended technicolor, was able to predict masses for gauge bosons and fermions. Unfortunately, problems with F C N C darkened the validity of this scheme. Recently, a renewed interest for supersymmetric gauge theories has emerged. The idea is that theorems which provide criteria on non-renormalization for supersymmetric theories allow the existence of mass hierarchies, and the rather unnatural procedure of fine tuning is not needed. H o w e v e r the real world does not seem to be supersymmetric. H e n c e the first trouble of these theories: supersymmetry has to be broken in such a way that it is not observable at low energies, while the non-renormalization theorems should keep playing their role. This has led us to consider spontaneous SUSY breaking. Two kinds of spontaneous breakings have been proposed: (i) the O'Raifeartaigh method [1], where the true ground state cannot annihilate the F terms of the lagrangian, and (ii) the Fayet-Iliopoulos method [2], where the presence of a l~l(1) * Part•ysupp•rtedbytheC•misi6nAses•radeInvestigaci6nCienti•cayT6cnica•underc•ntract32•9.
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J. Le6n et al. / Spontaneously broken supersymmetric models
105
gauge group prevents the cancellation of the D terms. Nevertheless, it is very difficult to find a supersymmetric model with a non-supersymmetric vacuum being the most stable one, Hence, explicit soft breakings, that do not destroy the theorems of non-renormalization, have been proposed [3]. Unfortunately, these breakings show problems connected with the masses of the fermion partners of the Higgs scalars. There is another additional question which is common to all SUSY GUTs proposed up to now, namely [4]: the mass parameter, in the Higgs superpotentiaJ, has to be fine-tuned in order to keep massless the Higgs doublets needed for the breaking of SU(2)Lx U(1) Y. Some progress along this line can be achieved by the introduction of a singlet [5] decoupled from the rest of the fields of the theory, i.e. the v.e.v, of the singlet produces the same results as a mass parameter, or by considering further Higgs supermultiplets [6]. In this work we propose a class of SUSY GUTs related to the U(n) gauge groups with a canonical assignment of lJ(1) charges. This will allow us to understand the U(n)'s as grand unification groups at energies of the order of the Planck mass. Canonical U(n) theories share the following properties: (i) absence of anomalies; (ii) absence of quadratic renormalizations for the D term related to 1~(1); (iii) no fine tuned parameters; (iv) spontaneous SUSY breaking ~ la Fayet-Iliopoulos at a convenient scale; (v) no effective interactions with dB ~ 0 and AL ~ 0 and dimension lower than 6. By a fine-tuning problem we understand here an unnatural adjustment of the parameters in the superpotential in order to keep Weinberg-Salam higgses massless. In other words, once a particular model is given the fine-tuning problem does not exist if the WS Higgs doublets are massless without any need of fine adjustments of the parameters in the corresponding superpotential. The choice of a model amounts to the writing of a particular superpotential, eventually setting certain couplings allowed by gauge symmetry equal to zero, which is technically possible by non-renormalization theorems. We present the scheme step by step. We show in sect. 2 that U(5) (and not SU(5)) is the gauge group able to overcome the fine-tuning problem. In sect. 3, we consider a model invariant under SU(5 + n) which is similar to the one studied in sect. 2. We prove that the fine tuning problem remains unsolved in this case. In sect. 4 we propose canonical U(n) groups that satisfy properties 1, 2 and 3 already mentioned. The SO(4n + 2) groups are considered in sect. 5. They verify properties 1, 2 and 3 as well, but are unable to accommodate a large va Majorana mass and property 4 simultaneously. In sect. 6, we demonstrate that canonical U(n) theories can satisfy properties 4 and 5. We analyze several questions on mass splittings and the presence of heavy multiplets in the theory. Finally, we finish with our conclusion in sect. 7.
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2. Supersymmetric U($) gauge theory without fine-tuned parameters There are three fundamental problems in the minimal supersymmetric SU(5) model of Sakai [7] and Dimopoulos and Georgi [4, 8]. (a) An unnatural fine tuning is necessary to adjust to zero the masses of the Higgs doublet [4]. (b) There is not an invariance that prevents a violation of the baryon number [9, 10] at d < 6. (c) Supersymmetry has to be broken softly. Hence it is not possible to give a mass to the fermionic partners of the Higgs supermultiplet [3]. Problems (b) and (c) have been already considered by Weinberg [9] in the supersymmetric SU(3)× SU(2)× U(1) model. He comes to the conclusion of the necessity of introducing an additional 0(1) that, while breaking the supersymmetry la Fayet-Iliopoulos [2], prevents the presence, in the superpotential, of terms that violate the baryon number. Solutions to problem (a) have been considered by several authors recently [5]. They start from the fact that the SU(5) SUSY model needs the inclusion of two higgses H and/-I, in the 5 and 5 representation, to break SU(2)L×U(1), and also an adjoint ,~ to break SU(5). Furthermore, as 5 x 5 = 1 + 24, the introduction of the term/-I,~H in the superpotential is unavoidable. This term would provide masses for/-t and H of the order of the grand unification mass. In order to obtain massless Higgs doublets, avoiding at the same time the introduction of mass parameters that require fine tuning, Ibafiez et al. [5] proposed the introduction of a new singlet that allows the introduction of a term/-7~H in the superpotential. This singlet is not coupled to the rest of the fields, and therefore it can acquire an arbitrary v.e.v. without breaking supersymmetry and gauge symmetry. Obviously, a suitable choice of this v.e.v, seems to solve the problem. Nevertheless, the introduction of the singlet • is somewhat artificial. In short, a fine tuning of a mass parameter is transferred to a fine adjustment of an otherwise arbitrary v.e.v. Another solution to the fine-tuning problem has been advocated by Masiero et al. [6] in the SU(5) SUSY GUT. They use a Higgs supermultiplet in the 75 representation to break SU(5) and need the further introduction of higgses 50 + 50. However, apart from the fact that their solution is highly uneconomical from the point of view of the number of involved fields, they need to impose the existence of an R symmetry or an extra 1~I(1) gauge symmetry to avoid the appearance of disastrous dimension-five operators. We would like to present here what, we believe, is the most natural solution to the problem. Namely, a replacement of the adjoint of SU(5) by the adjoint of U(5)(,~). Now ~b(=tr ~) is associated with the new generator in the algebra and corresponds to the direction 0(1). In this way, 0 does not remain uncoupled from the rest of the fields in the superpotential and its v.e.v, is determined by the supersymmetry conditions.
3. Ledn et al. / Spontaneously broken supersymmetric models
107
We propose now a model invariant under SU(5) x 1~1(1)that involves H , / 4 and ,~: V = A1/-[x~H y + A 2 ~ x y + A 3 , ~ Y ~ z .
(2.1)
This is not the most general superpotential, but WS doublets will remain massless in this particular model without any need of fine tunings. Therefore the model does not need any unnatural adjustment of parameters in the sense we stated in sect. 1. In eq. (2.1) x
1
x
X~ = try + 76 rob,
(2.2)
~r~ being the adjoint of SU(5). The superpotential can be written in terms of the above fields as -x y x y V =AI(Hxo'yH + g1 H-- x H x ¢)+A2(o'yo'x +½~b2)+
x y ~ + 73¢ 0 , x (2.3)
The conditions for a supersymmetric vacuum to exist are given by V,,~ = V~ = Vu = Vg = 0.
(2.4)
For the breaking of SU(5) into SU(2)× S U ( 3 ) x U ( 1 ) [11] we take the v.e.v. cr = w diag (1, 1, 1, - 2 , - 2 ) and obtain two sets of solutions: (i)
H =/-I = 0
¢ '
(ii)
H=/-I=0,
2A2 A3
& ~ - - - - 4 -A2 3
A 3 '
4 /~2 15 h3 0.) ~ -
(2.5) 4 -A2 15
A 3 '
Now the mass term for the higgses is m3/~aH3 + rn2I~EHE,
(2.6)
where the mass of the triplet i s m 3 = (~b +to))tl and the mass for the doublet is m2 =
Using (2.5) we see that solutions (i) cancel m2 independently of the values A2 and ,~3, and m3 = ~eoh~ where w is the grand unification scale. On the other hand, solutions (ii) cancel m3, also independently of h2 and h3, being rn2=-~a~Aa. Obviously, the former solution is the one we are interested in and it solves the problem satisfactorily. Therefore there is no fine tuning involved in our model since the supersymmetric solution should be naturally preferred by nature. Also, we would like to stress two virtues of our model: the economy in the number of involved fields and the simplicity of the superpotential. As we have just seen, the natural solution to the problem of the massless Higgs doublet led us to the replacement of an adjoint of SU(5) by an adjoint of U(5). This poses new aspects we will be treating in the following sections. In the first place, as the charge 1~(1) of the adjoint is zero, the gauge symmetry will break to
108
Jr. Ledn et al. / Spontaneously broken supersymmetric models
SU(2)Lx SU(3)CxU(1) Y x0(1). A suitable choice of the charges 1:1(1)* should forbid possible baryon number violations as it was pointed out by Weinberg [9]. Also the presence of an additional 1~(1) could be associated with a breaking of the supersymmetry /~ la Fayet-Iliopoulos. Apart from these low-energy constraints, there are additional general restrictions due to the requirement of a U(5) anomaly free theory. Furthermore, being SU(5)xl~I(1) a non-semi-simple group, it seems logical to ask oneself whether it is possible to find a unification group at higher energies. On the other hand, let us stress that although SU(5) x l:l(1) is not semisimple, i.e. it has two different coupling constants gsu~5~ and go~l), strong and electro-weak interactions remain unified in SU(5) at the scale mx. The new hypercharge I7" obviously commutes with all SU(5) generators and does not affect all the good properties (electric charge, Weinberg angle,...) of the minimal DimopoulosGeorgi-Sakai SUSY SU(5) model. A separate study of all the above questions will be carried out in the following sections.
3. Discarding SU(n) SUSY GUTS Supersymmetric gauge models could also be considered as effective GUTs of a supergravity theory, and this is certainly a very attractive feature. This possibility has generated many works [12], but its analysis is not the subject of this paper. Nevertheless, we would like to emphasize here that N = 8 extended supergravity contains an effective SU(8) gauge theory at energies higher than Mp [13]. Furthermore, the possible breaking of supersymmetric theory was speculated recently [14]. The above ideas seem to suggest that the SU(n) groups would be possible unification groups. In this section, we will study the consequences of such statement. SU(n) groups (n > 5) have been somehow traditional models for grand unification. The idea is to reach the existence of three families, but not as a mere repetition of the structure 5+ 10. King [15] tabulated asymptotically free and anomaly free representations able to contain at least one family of light fermions without exotic states. Leaving aside asymptotic freedom that leads to different results for the supersymmetric case and is anyhow of rather dubious necessity, we find an ample variety of solutions in the form:
R =~.Ck[n,k],
(3.1)
k
where [n, k] is the k-fold antisymmetric tensor: [n, k] = 4~ac''~k. Apart from these complex representations, we need a gauge vector supermultiplet and a chiral Higgs supermultiplet, both in the adjoint of SU(n). The D terms of these representations
* T h e 1:1(1) factor was first introduced by Fayet in order to obtain a correct low-energy spectrum, otherwise scalar quarks or leptons would be lighter than their spinor partners.
J. Ledn et al. / Spontaneously broken supersymmetn'c models
109
can be written as -k6r,...,,,,To,
D,~[,~r,...,.,,]
,.~c~
= k6r'"'r"T~,r~,-,,~,.~...,.,,
,
,
D ~ [ e # ba ] = T~~c,[~b, - t#]~',
(3.2)
while the superpotential will depend on n. Now, we could write not only terms like ~ba~bab
(3.3)
For the first breaking we will need the Higgs supermultiplets ~b~1.... -, ~ba,. . . . , ~ba, that take non-zero v.e.v.'s for the components al . . . . . a , = 6 . . . . . 5 + n . Their ratios will be determined by the supersymmetry condition D~ = 0 [11, 16]. The introduction of an additional group G reflects the possibility that a certain linear combination of generators of the Cartan subalgebra, which are not contained in SU(5), annihilates each one of the v.e.v.'s. Therefore G would be of the form (U(1))", and a straightforward analysis shows that for our particular v.e.v.'s m = 0. As usual, the second breaking comes through a non-zero v.e.v, of an adjoint , ~ of SU(5) which is contained in the adjoint , ~ of SU(5 + n). Our notation goes as follows r, s, t = 1 . . . . . 5 + n ; x, y, z = 1 . . . . . 5; a, b, c = 6 . . . . . 5 + n . We consider a SU(5 + n) superpotential similar to the one we wrote in eq. (2.1). However, we will show that, unlike U(5), WS doublets will not be "natural" massless in the SU(5 + n) model. Our superpotential is V
= A 1/r'~VH +
A2 tr X 2 + A3 tr Z 3 ,
(3.4)
.~ contains, among others, (1, n2--1)(trb~,), (24, 1)(try) and (1, 1)(~b) with respect
110
J. Le6n et al. / Spontaneously broken supersymmetric models
tO SU(5)x SU(n). We have the relations: X~ = o'~ + 1 8 ~ b , n
(3.5a)
2yx =cryx - ~l ~ox ~ .t
(3.5b)
In terms of representations of SU(5)x SU(n), the superpotential (3.4) can be written in the form:
+A2 tr bO'a + 2 ~ , ~ r
a
b
c
+ o'ytrx + a
b
x
:"1-A3tO'bO'cOv a + 3trb.~,x,,~,a + r /1
1\
a x
1
+ a
x
y
x
y
z
3~O'y.,~a + Orytrztrx
,, b
1
~ y'~
1
1
a
The condition D~ = 0 allows us to diagonalize simultaneously the real and imaginary parts of ,~. We take the following set of v.e.v.'s: ,~, ~ = Z ~ = n '
,
trb=X
(li, 0
0
= ISl, = O,
--p/(n--p)l,,_p
o',=to 0
),
0
(3.7)
-312
and, using the conditions of the supersymmetric vacuum V ~ = V~; = 0, we obtain for x and to the relations: 4../h2 3 \ to
=
n-p
Notice that o-~ in (3.7) has been chosen in a way that will break SU(5) to SU(3) x SU(2)xU(1)(to # 0). The masses of the colored triplet and the SU(2) doublet contained in H are given by m3 = (--51-¢ +to)}[1 ,
m2 = (--51-¢- 3to)A1.
Using (3.8), together with the supersymmetric vacuum condition V, = 0, the results of the table 1 are obtained. It is easy to see that no couple of values (n, p) exists such that the mass of the Higgs doublet vanishes. We then deduce that the
111
J. Ledn et al. / Spontaneously broken supersymmetric models TABLE 1
Vacuum expectation values and doublet Higgs masses for the breaking SU(5 + n) ~ SU(5) ~ SU(3) x SU(2) x U(1) x 0
~
~o
m2
n n+l n 3A - n-1 A 2n-5p n-2p+l A 3- n --5 p n-2p-1
n+5 n+l 2 n +5 ~A n-1 ~a n+__.___~_5 n-2p+ l 2 ° n+5 _---3A n ---2p- 1
(n+3) n+l AAa 3-n-1 -,~,~1 n + 3 - p n-2p+l AAx- p+3 n-2p-1
-¼A(n -5)
~A(n +5)
-¼AAI(n+7)
-½hA
~A(n +5)
-½aa l(n +6)
2/~ - -
0 -a (n-p)(n+5) n ( n - 2 p + l) (n -p)(n+5) n(n-2p-1) ±A (n-p)(n+5)* 2n ±½A(n +5) **
2~
--/~ 1 -
The parameter A = 2A2/3A3. *The ± sign is valid for p =[(n ±1), n odd. **Is valid for p = ½n, n even. The rest holds for the other values 0 < p < n. s y m m e t r y b r e a k i n g p a t t e r n (3.3) does not solve the p r o p o s e d p r o b l e m without fine tuning. A m o r e general situation that the one outlined in (3.3) could be o b t a i n e d passing through SU(5+m)×G instead of S U ( 5 ) x G . In this case, the second b r e a k i n g would give a colour SU(3 + m) (an example could be the SU(4) of P a t i - S a l a m [18]). T h e r e f o r e , we will analyze the s y m m e t r y b r e a k i n g pattern given by SU(5+n)-->SU(5+rn)xG-->SU(3+m)xSU(2)xU(1)xG,
(3.9)
with n > m. W e use the index notation: r,s,t=l .... ,5+n, a,b,c =6+m .....
x,y,z=l
.....
5+m,
5+n.
A s before, the first s y m m e t r y b r e a k i n g needs Higgs supermultiplets $ a c a " - m , $al . . . . . $ . . . . that take n o n - z e r o v.e.v.'s, for the c o m p o n e n t s a l . . . . . a , _ , , = 6 + m . . . . . 5 + n , with suitable p r o p o r t i o n s [16]. T h e g r o u p G would be again (U(1)) s, and we have s = 0 with the p r o p o s e d v.e.v.'s. T h e second b r e a k i n g of (3.9) would p r o c e e d t h r o u g h the n o n - z e r o v.e.v, of an adjoint 2 ~ of S U ( 5 + m ) that is c o n t a i n e d in the adjoint , ~ of S U ( 5 + n ) . T h e relevant superpotential is given by (3.4). ,~ contains, w h e n d e c o m p o s e d with respect to S U ( 5 + m ) x S U ( n - m ) , (1, ( n - m ) 2 - 1 ) ( o ' ~ ) , ( ( 5 + m ) 2 - 1 , 1) (o"s) and (1, 1) ($). T h e y are related a m o n g themselves through: 1
" ~:~ = orb+ n-m
" 864',
1
-8#. ~ = o'y~ ' - -5+m
(3.10)
112
Y.Le6n et al. / Spontaneously broken supersymmetric models
In terms of SU(5 + m ) x S U ( n - m) representation the superpotential (3.4) can be written as V =/~1
(O.ba "4-
1
8~
n-m
)fib
1
+ Hx ( o ' , -x ___5+m 8~qb)HY
]
[ as / 1 1 ~ 2"] +A2 ort~ra+2~'a~"x + c r y - : +kn----~+5-V~m)~b J a
b c
a
b
x
a
+As o'w~o-a + 3cr~,~:.~ + + 3
4,((~__~__1 m
1 ) 5
1
m
x
y
1 .yay~+
1
n -m
+ or,~r~rx
~ b
OgbO'a
(3.1"1)
1 _ 5 _ +_m
. ,)
OryKFx
3
+ ( ( n - m ) 2 (5+m)2) ~b ]" The supersymmetric vacuum conditions V~g = V ~ = D e = 0 diagonalize X~ and obtain the following set of v.e.v.'s: xa
o(1
Crb=X 0
= .y, ~ ,~ =
o
H r =fir=O,
)
0
--p/(n--m--p)l .... p
o'y =to
-
permit us to
o)
(3.12)
to#O
I(3 +m)12
with to and x given by the relations 1 [4 A2 (n +5)_44~] ' to = (n + l ) ( n +5) L3 A3 2(A2+n3A m ~b)x+ 3A3 n --m--2Px2 = 0. -
n -m
(3.13)
-p
The masses of the coloured triplet and the SU(2) doublet contained in H are given by ma=(to-qb/(5+m))A1 and m 2 = ( - ~ ( 3 + m ) t o - d ~ / ( 5 + m ) ) a l . Using (3.13), together with the supersymmetric vacuum condition V~ = 0, we obtain the results of table 2. Notice again that is impossible to get m2 = 0 for any set of values (n, re, p). Summing up, the SU(n) SUSY G U T models considered above require a fine tuning or an arbitrary introduction of singlets in order to keep the Higgs doublet massless. It is amusing to verify that in the limit n ~ 0o, the rest of the parameters fixed, there are solutions with m2 = 0 in tables 1 and 2. This suggests that the problem proposed in this section might have a solution in the framework of a non-semi-simple group such as U(n). We will solve this question in sect. 4.
J. Le6n et al. / Spontaneously broken supersymmetric models
>(
I
~
4-
~.
t"~ l h-
÷
÷
v
.z
>(
113
v
7
÷
"t
~+
÷
÷
.=. t.q~'
r
1.:.
"0 ¢. t~
I
4-
I +1 II cO
¢q I t'q 44-
~
:> ..<
÷
ra
4-
t"4
~ea q'l
•z . . ~
I
[...,
114
3. Ledn et al. /Spontaneously broken supersymmetric models
4. Canonical U(n) SUSY GUTS In this section we will study a supersymmetric theory based on U(n) groups. In particular we will analyze the symmetry breaking pattern given by U(5 + n) ~ U(5) × G ~ SU(3) × SU(2) x U(1) × 1](1) x G ,
(4.1)
where we do not exclude, as in sect. 3, the existence of a G factor whose presence should be investigated in each particular case. The U(1) group could be the one, frequently proposed [9, 10, 19], that is able to depress the A B ~ 0 processes in a way required by experiment, and be the origin of a spontaneous supersymmetry breaking in a framework free of the problems exhibited by the theorem of Fayet et al. (see appendix of ref. [8]). As the U(n) groups are not semi-simple we move on to the old problem of obtaining an anomaly free theory. Moreover, the D term presents two additional problems: (i) it has to remain free of quadratic renormalizations [11] and (ii) supersymmetric vacuum conditions have to be enlarged with equation Do¢~)= 0. We will show further on that we have positive answers to all these problems, however we would like to start by showing that this framework solves the fine-tuning problem. Following a similar treatment to that of sect. 3, the superpotential relevant to our problem is given by (3.4). The adjoint of U(5 + n) contains, among others, the adjoint of U(5)(.~) and the adjoint of U(n)(,~). Performing now the decomposition: =o'y-5Oy~,
Z b =trb +
8
(4.2)
,
n
where d~ ~ d~ indicates that we are in U(5 + n) instead of SU(5 + n). The potential (3.4) decomposed with respect to U(5)× U(n) is written as
n [
a
b
c
a
b
"JI-A3[O'bO'cO'a . t _ 3 O r ~ , , ~ x , ~ '
3 * n
a
x
x a
3
J
a x~,~ y + O'r xO'~ yO"x z + 32~"
a b 3 x a +o'b~ra)--~(.V~.Vx +crycrx)
(4.3)
3". Ledn et al. / Spontaneously broken supersymmetric models
115
TABLE 3 Vacuum expectation values, doublet and triplet Higgs masses for the breaking U(5 + n) ~ U(5) ~ SU(3) x SU(2) x U(1) × l~l(1) t~
tO
2A
~A -~A
3A
m2
-AAI
0
0
-AAI
mp
m n_p
0 -AA1
0 -hA1
-AA1
0
0
-AA1
.
X
0 -nA
0 0 n-p -A--
-Ap
m3
n
n-p A- -
-A(n -p)
n
The parameter A = 2A2/3A3.
choosing those v.e.v.'s that lead us to SU(3)c x S U ( 2 ) L × U ( 1 ) Y, i.e. o'y=.,
-312
0
[-p/(n-p)]l._p
o(1 o'b=x
0)
0
'
0
) '
(4.4)
and imposing supersymmetric vacuum conditions V, = Vg = 0, we obtain the results of table 3. Notice that, in the case (U(n)), the sets of v.e.v.'s (~b,to) and (~, x) are decoupled. For the decomposition associated with (4.1) the Higgs field H is given by H r = H x ~ ) H a = ( H ~2)( ~ H ~3))~ ( H
¢p}~)H("-P)),
(4.5)
following a similar notation to the one used before./-It will be written analogously. The physically acceptable solution is the one for which only the H ~2) doublet is massless. Then, the solution to the proposed problem can be read from table 3. It will be given by the v.e.v.'s 4, = 3 A , = -nA,
m3 = -AA1,
to = -52-A,
m2=0,
x = 0,
mp = m,_ o = -hAl.
(4.6)
Next, we will concentrate on the other problem of the supersymmetric vacuum: the cancellation of D terms. The first breaking in (4.1) needs Higgs fields ~brc''r", ~b~1). . . . . ~b~") with non-zero v.e.v.'s ~b~i) =l~65+i.,, ~ 6 .... =tz. This assures D,, = 0, a being an index of the Lie algebra of SU(n) [16]. On the other hand, the D term for the higgses, associated
J. Le6n et al. / Spontaneouslybrokensupersymmetricmodels
116
to the group l~l(1), is given by --
I=1
i
1
with an obvious, but not unique, solution: I7"1. . . . .
I7",= ___1 17". n
(4.8)
Then, we propose a canonical assignment for the charges 0(1)(17") as a simple non-trivial solution for the problem of the cancellation of D terms. A canonical assignment means that the generator of 0(1) acts in a similar way to the rest of the generators of U(5 + n), i.e. 8,d~ r'''''m = T,~",~d~ r~'2"'''m + " • • + T , , " , ~ , ~ ' ' ' ' ' ' m - ' ' ; ~ ,
(4.9)
The U(1) generator is 7~ = I, therefore
g¢,c",. = m&rc"',,
t" = m.
(4.10)
As the complex conjugate representation ¢,, ..... transforms with - T t, we have g&,,...,, = -m&,, ......
I7"= - m .
(4.11)
Analogously, for a representation with m upper and m' lower indices the hypercharge 0 ( 1 ) is I7"= m - m'. Now, we have the necessary ingredients to build an anomaly-free U(n) theory. Let us assume, in general, that R of eq. (3.1) is a set of Nk supermultiplets & , c , k (k = 1 , . . . , n), i.e. Ck = Nk, where negative values of Nk will indicate the conjugate representation ~,,...,k. The anomalies of a U(n) theory can be classified into three classes: (0(1)) 3, U(1)(SU(n))2 and (SU(n))3. Their cancellation will be assured, respectively, by the equations:
~lk3N~(n)=o,
i klkNk=O
k
k
k=l
~ AkNk=O, '
(4.12)
k=l
where
(n-2k)(n-3)!
Ak=(n_k_l)!(k_l)!,
lk =
(n--2) k-1
(4.13)
"
The solution to the system of equations (4.12) is given by k=4
N2 =
k=4
(k - 3)
_ 2 Nk,
2/n-1\ N3 = -
k=4 k - 3
Nk.
(4.14)
3".Ledn et al. / Spontaneously broken supersymmetric models
117
Surprisingly, there are admissible solutions (Nk integer) for any group U(n). Before analyzing in detail the obtained representations, we will study the problem arisen from the/9 term. The problem for a supersymmetric gauge theory with a l~l(1) factor is well known: the /~ term of Fayet-Iliopoulos is generated perturbatively through quadratic divergences i.e./~ ocM~. This is a disaster for supersymmetric GUTs. Witten showed [11] that if a 1~1(1) factor gets unified in a semi-simple group G, then the term/~ is not generated perturbatively. More recently, Fischler et al. [20], obtained a less restrictive condition: It is sufficient to demand a cancellation of the l~l(1) hypercharge, tr I7"= 0, to protect the theory from a perturbative generation of the/~ term. With the canonical assignment (4.10) (4.11) tr I7" can be written as: tr f" =
k=l
k
k
Nk,
(4.15)
where N1, N2 and N3 are fixed by (4.14). We rewrite (4.15) in the form: tr f ' - - ~ d k N k . k~4 Remarkably enough, an explicit calculation shows dk=O,
(4.16)
k = 4 . . . . . n.
then tr I7"= 0. Now we can state the following theorem. Theorem. "Any supersymmetric gauge theory based on a gauge group U(n), with canonical assignment of charges and free of anomalies, is protected from quadratic renormalizations". The origin of this beautiful result can be traced back to the fact that eq. (4.15) is a linear combination of eqs. (4.14). Thus, we have generalized the results of Kaul and Majumdar [21] to the general case of arbitrary canonical U(n) groups. Still, two questions do remain: (i) supersymmetry breaking (ii) depression of AB ~ 0 processes. We postpone their discussion to sect. 6. Finally, we would like to discuss the relation between the supersymmetric GUTs of this section and the effective gauge theories contained in N supergravities [13]. Discarding N = 8 supergravity (with an effective SU(8) theory), N supergravities (N = 5, 6, 7), with effective U(N) gauge theories, .do remain as possible candidates for a superunification. Among these, N = 7 supergravity is the candidate that presents a greater degree of convergence in its perturbative expansion. Therefore, a supersymmetric gauge theory based on a U(7), and unified to N = 7 supergravity at the Planck mass, seems to be the most attractive candidate for the superunification. Work along this direction is now in progress by the authors. 5. Naturalness in SO(4n + 2) SUSY GUTS The well known properties of this series of groups have made them serious candidates for grand unification [22]. In this section we show that SO(4n + 2) groups
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J. Ledn et al. / Spontaneously broken supersymmetric models
are free from fine-tuning problems. However, if we want a very massive rightneutrino we will lose the U(1) invariance at low energies. Although our reasoning and results are completely general, we will centre our treatment on SO(10) for the sake of clearness. In SO(10), the content of our theory will be: (i) a gauge vector supermultiplet in the adjoint (45), (ii) matter supermultiplets in the spinor representation (16), (iii) Higgs supermultiplets: several in the 10 representation (vectors that break SU(2)L× U(1)Y), one in the 45 (antisymmetric second rank tensor) and another in the 54 (traceless, symmetric second rank tensor). They form a minimal set of higgses whose SU(5) decomposition is 10a = {H x,/-Iax},
a = 1. . . . . h ,
45 = {A Ex'yl,AE~.yl, .,~zxy, 4,},
(5.1)
54 = {S ~x'y~, S~x,y~, .,~l"y}, where H~ = 5 ,
I7Iax=5,
a=l ..... h,
A tx'yl = 10,
/~[x,y]= 1--0,
,,~2xy = 24,
S ~x'y~= 15,
go,.y~ = 1--5,
.2"lXy= 24.
4,=1 (5.2)
Notice that fine tuning demands the presence of more than one 10, due to the fact that the natural masslessness of the doublet is achieved through the coupling of the 10's with the antisymmetric 45. The more general superpotential compatible with supersymmetry and gauge symmetry is V = A 1(54) 2 + A 2(45) 2 + A 3(10 × 10)s × 5 4 + A4(10 × 1 0 ) a
× 45 + A5(54) 3 + A654 × (45 × 45)s
(5.3)
and the SU(5) decomposition of these terms is (54) 2 = 2 tr ($S+~V21), (45) 2 = 2 tr (AA + ~ + ~ 4 , 2 ) , (10 × 10)s × 54 = dab (HaSHb + 2I'7I~.Y,IHb + HaSHb ) , (10 × 10)A × 45 = fab (H, ug.Hb + I4aAI~b + I'tb (~,Z + kd~)Ha), (54) 3 = 2 tr (3Sg,~1 + ~ ] ) , 54 × (45 × 45)s = 2 tr (AASa +~1-~z~2 + 20-~z-~1),
where lab (dab) is an antisymmetric (symmetric) arbitary matrix.
(5.4)
J. Ledn et al. / Spontaneously broken supersymmetric models
119
The supersymmetric vacuum conditions give us: A = A = S = S = H a =/-I= = 0 , V~b= ~A2¢~"1"2A6 tr (X2X~) = O, Vx, = 4A 1,~1+ 6A s (XxX1 - ~ tr X~) + 2A64~$2 q.. ~6(,~2,~ 2 -- 1
tr $2) = 0,
V,~1 = 4 A 2 5 2 + 2 A 6 ( $ 2 5 1 -
51-tr $2X1) + 52-X6~b$1= 0 .
(5.5)
We are interested in the v.e.v.'s that break SU(5) to SU(3)c x SU(2)L × U(1)v:
.~l = 0)1( ~ __23_O2),
$2 = ~o2(10 _~O2}.
(5.6)
The solutions to eqs. (5.5) are given in table 4. Notice that A2 and A6 should be different from zero; nevertheless A1, A3 and A5 could be zero. Obviously, the only condition for •2 # 0 will be A1 or A5 different from zero. The mass term is rn~j-7~Hb, where the mass matrix rn~b can be expressed as mab = A 3dabS1 + A4fab ( $ 2 + l~b ) .
(5.7)
Let us consider the c a s e / ~ 3 = 0. Then, by using the relations between ~b and to2 given in table 4, we obtain for the doublet (triplet) mass m2(m3) the following values: ~b = -5to2, --'~~¢M2,
m3 = 0,
m2 = -5to2,
(5.8a)
m 3 = 5¢-D2,
m2 = O.
(5.8b)
Choosing, hence, solution (5.8b), the fine-tuning problem is solved. Another possibility, that will lead us to natural results as well, could be accomplished with the introduction of two higgses 10 and 120 through the couplings: 8V = A4(120 x 10 x 45) +~ ~ (120 x 120L × 54+A~(10 x 10L x 54
(5.9)
The SU(5) content of 120 is given by t~ [ijk] ~- {D[,,yz],
D[,,yz],Dzxy + 8 ~xD y - 8 ~ yD -
,l)yz+Sy1~-8715y},
TABLE 4 Vacuum expectation values for the breaking of SO(10).
4,2 3 A6
-5.2
2 15 \A 2 A6/\A6] A2
A6]\A6]
(5.10)
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where Dt~yzl = 1 - 0 ;
/~txy~l= 10 ;
D~Z+87DY-6~DX=45+5,
and Dy~ -x +8~/9z x - - 8 ~xD-y = 4 5 + 5 . The relevant mass terms in the superpotential, i.e. /~4 terms in eq. (5.9), are 1 0 x 5 x 1 0 + 1 0 x 5 x 1 0 + ( 4 5 + 5 ) x 5 x 1 0 + ( 4 5 + 5 ) x 5 x 1-0 + 2(45 + 5) × 5 x (2~2+~d,) + 2(45 + 5) x 5 x (2~2+~b).
(5.11)
Again, as the non-zero v.e.v.'s are in the adjoint Z and in the singlet ~b, the appearance of a structure (Z +½~b) assures us the repetition of results of the type (5.8) as requested. We are analyzing a symmetry breaking pattern that breaks SO(10) to SU(2)L× SU(3)c× U(1)Y × 0(1) directly, without an intermediate step through SU(5). This framework presents a disadvantage: The right-neutrino cannot acquire a large Majorana mass. In order to circumvent this problem Harvey et al. [23] proposed the introduction of a Higgs in the 126 with a non-zero v.e.v, for the SU(5) singlet. This procedure gives a Majorana mass to the right-neutrino, of the order of the grand unification scale, while the left-neutrino remains massless. Unfortunately, the singlet in the 126 is not invariant under 0(1). Thus, we lose the possibility of breaking supersymmetry, h la Fayet-Iliopoulos, at lower energies, along with the depression of AB ~ 0 terms. The symmetry breaking pattern is, in this case, SO(10)-, SU(5) ~ SU(3) × SU(2) × U(1),
(5.12)
where the first breaking is due to the non-zero v.e.v.'s of the 126 and 16. Both v.e.v.'s conspire to keep supersymmetry unbroken.
6. Spontaneous SUSY breaking and ~ B ~ 0 suppression Having in mind the results of sects. 3, 4 and 5 we would like to present here some general results on spontaneous supersymmetry breaking and on suppression of processes with AB ~ 0 for supersymmetric canonical U(n) theories. In the canonical U(n) theories, we can introduce a "small" spontaneous symmetry breaking of supersymmetry. This possibility comes from the absence of perturbative quadratic divergences as we showed in previous sections. On the other hand, as the supersymmetric vacuum is the most stable one, we should prevent the appearance of representations whose v.e.v.'s protect supersymmetry from breaking. We present here a study on the different patterns of gauge and supersymmetry breaking for a supersymmetric U(n) gauge theory. Supersymmetry breaking comes h la Fayet-Iliopoulos, i.e. from the non-cancellation
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121
of the L) term. In general, we write the D terms as /9 =• IT"(~l~b)+~,
D~ = Y <61T I >
(6.1)
for a theory with higgses (~b) of charge I7".Let us consider the case where supersymmetry breaking appears through D~ = 0 a n d / ~ # 0. The SU(n) gauge symmetry breaking with D,, = 0 has been classified in ref. [16] as follows. (i) Gauge invariance is not broken: tb = 0. (ii) U(n) gauge invariance breaks spontaneously to U(m) (m < n ) , through the higgses ~btrc''~-J, ~ b(1)r , i = l . . . . . m - n . (iii) U(n) gauge invariance breaks spontaneously through higgses in real representations. For example U(n) goes to U ( m ) x U ( n - m) (m ~ ½n) through a Higgs in the adjoint of U(n). (iv) U(2n) gauge invariance breaks spontaneously to Sp(2n) through a Higgs (v) U ( 2 n + 1) gauge invariance breaks spontaneously to Sp(2n) through higgses ¢~ [rl'r2] and
Notice that we are not including, in the U(n) theory, neither fields with an exotic colour content (i.e. symmetric tensors and so on) nor real reducible representations. The higgses conjugate to the above ones would lead to the same gauge symmetry breaking and then we do not consider them as separate ones. In cases (i), (ii) and (iii), condition D~ = 0 implies ~ I7"~<
(6.2)
where AT/is the number of fields 4~n-,,. Notice that Ni < 0 indicates conjugate fields. In case (v), supersymmetry is broken (LJ # 0) iff: N1 • N2 ~<0.
N2,-1 • N2, ~<0.
(6.3)
We would like now to present the symmetry breaking pattern that emerges from the above results. At the mass Planck scale, the effective gauge theory coming from supergravity would be U(n). At lower energies, we would have a series of successive steps described by the chain: SO(n) supergravity ~ U(n) x SUSY
A,
~ U(m) x SUSY
Am
SU(m - 2) x SU(2) x U(1) x l~l(1) x SUSY a~.~~ -~ SU(3) c x SU(2) x U(1) x I~l(1) x S U S Y -~ SU(3)c × SU(2) x U(1) x I~l(1)
) S U ( 3 ) C x U(1) .... Aw
(6.4)
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I. Le6n et al. / Spontaneously broken supersymmetric models
with n > m 1>5. The relevant cases would be m = 5 (Georgi-Glashow type) and m = 6 where an SU(4) colour group is obtained (Pati-Salam type). The first breaking, at scale An, is produced through a set of higgses ~b'l'~-m, 4~ ~) (i = 1 . . . . . n - m) with v.e.v.'s that guarantee D~ = 0. T h e n / 5 -- (, that points out the appearance of a spontaneous SUSY breaking which should not be observable up to energies of the order of (1/2((6.4)). With a canonical assignment of U(1) charges in U(n), the proposed v.e.v.'s lead to a canonical assignment of charges for the U(1) contained in U(m). Therefore, the 1~1(1) that appears later on corresponds to the canonical U(m). The second breaking, at a scale Am, happens through v.e.v.'s in the adjoint of U(n). The criteria, that we studied in sect. 4, assures the natural masslessness of the Higgs doublets. Now, by passing Pati-Salam breaking, about which we have nothing new to say, we could choose an energy scale (~/2 suitable for solving the mass hierarchy problem and making sfermions massive enough. Our guess is ~>A2w. Then, either SUSY and electroweak breakings occur simultaneously, or standard breaking of SU(2) L x l~I(1) x U(1) .... appears immediately after SUSY breaking. Regarding mass splittings and the existence of superheavy fermion families, there are certain constraints which are present in the models ~tla Fayet-Iliopoulos. Firstly, we consider the mass splitting formula for a supermultiplet [24]:
Y~(-1)2'(2J + 1) tr rn~ = 2g2gtr I7".
(6.5)
J
Eq. (6.5) shows that the order of magnitude of m a2 - r n ~ is determined by the breaking parameter ~ Moreover, either we have a light multiplet (mr2 <
(6.6)
Rp
from which we can easily see the joint presence of light and heavy multiplets. Building up realistic models where SUSY is spontaneously broken is a difficult task. This difficulty has been stressed particularly by Buceella et al. [16] in the context of the Dimopoulos-Georgi-Sakai model. The problem is that matter fields prefer to break colour and/or letpon number, in order to preserve a supersymmetric vacuum state, unless the couplings and fields present in the superpotential were able to prevent the above mechanism. However, there is not any no-go theorem that forbids a SUSY breaking ~ la Fayet-Iliopoulos and realistic standard SUSY models have been drawn by Barbieri et al. [25] and Hall and Hinchliffe [24]. Another possible mechanism of SUSY breaking is through N = 1 supergravity effects in SUSY Yang-Mills theories [26] although, in this case, R-invariance shQuld be imposed as deduced by Barbieri et al. [27].
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123
With the previous assignment of charges I7" (i.e. light multiplets have IT"'s with the same sign), there are not any terms (dimension <6) that violate the baryon number. This problem has been studied by Weinberg and we refer the reader to ref. [9] for further details.
7. Conclusions
The relevance of SUSY GUTs, as a description of the real world, has been darkened, up to now, by two annoying facts: (i) the need for an "unnatural" fine tuning of the mass parameters of the theory, and a more serious point, (ii) the difficulty to generate a SUSY breaking with a suitable mass hierarchy. In this work, we proved that both facts are indeed closely related and found a simultaneous solution for both of them. We began undertaking the problem of fine tuning, whose solution led us in a direct and natural way to models of the U(n) type. Specific difficulties connected with D terms, along with the anomalies present in U(n) theories, guided us to a canonical assignment for l~l(1) charges, recovering for these the character of grand unification at high energy scales. The absence of quadratic renormalizations for the D term permitted us to break supersymmetry spontaneously h la Fayet-Iliopoulos at an energy scale ~1/2. The value of ~ can be chosen conveniently in order to obtain realistic mass splittings for the supermultipiers. SO(4n +2) groups solved the fine-tuning problem. However, as we showed explicitly in our example SO(10), a large Majorana mass for UR is incompatible with l:l(1) invariance. Therefore we lose the possibility of spontaneous supersymmetry breaking. U(1) invariance is able to forbid the appearance of effective interactions of low dimension (d < 6) with A B ~ 0 and A L ~ O. Our treatment is connected with the one proposed by Weinberg [9] for a supersymmetric standard model. Summing up, the moral that follows from this work should be: canonical U(n) groups are the most suitable ones in order to build up a supersymmetric gauge theory, free of unnatural tunings, with a realistic spontaneous SUSY breaking and superunified at Mp with n supergravities. Now, an open question would be to find, within the proposed framework of this paper, a realistic model able to reproduce low-energy physics. Some work along this line is now in progress.
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