PHYSICAL REVIEW D VOLUME 45, NUMBER 10 15 MAY 1992

Nonlinear o models and supersymmetry-breaking effects in supergravity

Toru Goto and Yasuhiro Okada Department of Physics, Tohoku University, Sendai 980, Japan

(Received 27 January 1992)

Supersymmetry-breaking effects on U( m + n )/U( m 1 X U(n ) nonlinear a models are studied in the framework of minimal supergravity. It is shown that, in a total doubling realization model, full U(m t n ) nonlinearly realized symmetry is retained even after supersymmetry breaking in the hidden sector, while coset symmetry is explicitly broken in the hidden sector coupling in a pure realization mod- el. In a total doubling case, effective superpotential terms as well as soft breaking terms are induced, re- sulting in superpartners of Nambu-Goldstone bosons (quasi Nambu-Goldstone bosons and fermions) with masses which are determined by the gravitino mass. An application to the Higgs sector in the su- persymmetric standard model is also discussed.

PACS number(s1: 04.65. + e, 1 l.lO.Lm, 11.30.Pb, 11.30.Q~

I. INTRODUCTION

Supersymmetric nonlinear a models have been con- sidered in particle physics. As effective Lagrangians, they were used to investigate dynamical properties of su- persymmetric gauge theories [I]. They were also studied to explain the existence of light particles including both fermions and bosons [2,3]. So far most of these works have been considered in the context of global supersym- metry. On the other hand, in a unified model based on supersymmetry, the origin of supersymmetry breaking lies in supergravity [4]. The model contains a so-called hidden sector which interacts with other fields (observ- able sector) only gravitationally. The local supersym- metry is assumed to be spontaneously broken in the hid- den sector, resulting in soft supersymmetry-breaking terms that are induced in the observable sector. The soft breaking scale in the observable sector is determined by the gravitino mass m3/,. In order to use a nonlinear a model in a phenomenological supersymmetry model, we have to discuss it in the framework of supergravity. Al- though it is known that a supersymmetric nonlinear n model can be coupled to supergravity without matter fields [ 5 ] , a coupling to the hidden sector has not been considered yet.

In this paper, we study the supersymmetry-breaking effects on a nonlinear a model in the context of super- gravity. We take a U(m +n ) /U( m ) X U ( n )-super- symmetric nonlinear o model as the observable sector and investigate the effects of supersymmetry breaking in the hidden sector. We find that, in the case of so-called pure realization where all bosonic degrees of freedom are assigned as Nambu-Goldstone bosons [6], the nonlinearly realized U( m + n ) /U(m ) X U ( n ) symmetry is explicitly broken in the hidden sector coupling, and hence all Nambu-Goldstone bosons get a mass of m3/,. On the other hand, in a class of different realizations with extra bosonic degrees of freedom, which are called quasi Nambu-Goldstone bosons [2], the full

Nambu-Goldstone bosons remaining massless while quasi Nambu-Goldstone bosons become massive.

We also apply this model to the "p problem" in a su- persymmetric standard model [7]. In the model based on a radiative breaking scenario of the electroweak gauge symmetry [4], a supersymmetric mass parameter p is in- troduced by hand in the Higgs sector as an initial condi- tion at the grand unification scale. It is known that p - m 3 / 2 -m is plausible in order to realize S U ( 2 ) X U ( 1) electroweak symmetry breaking at the weak scale - m ,, though there is no theoretical reason why p and m3/, are in the same order. This problem can be solved if Higgs bosons are supposed to be (quasi) Nambu-Goldstone bosons which associate with a spon- taneous breakdown of a certain global symmetry at a high-energy scale such as a grand unification scale or the Planck scale [8]. They are massless as long as supersym- metry is exact and hence all mass terms are induced by supersymmetry breaking. Then it is natural that all mass parameters in the Higgs sector are of the order of m,/,. We argue that our model is applicable to this aspect and a tree-level Higgs potential is obtained.

This paper is organized as follows. In the next section, we summarize the general properties of a nonlinear CT model in the case of global supersymmetry and discuss important changes that occur when the model is coupled to supergravity. In Sec. 111, the supersymmetric U( m + n ) /U( m X U ( n ) nonlinear o model is considered in pure realization and it is explicitly shown that all Nambu-Goldstone bosons get a mass of m,,, as a result of supersymmetry breaking in the hidden sector. The case of total doubling is dealt with in Sec. IV where a general Lagrangian and mass terms for the Nambu- Goldstone supermultiplets are obtained. A summary of the results and discussions on an application to the p problem are given in Sec. V.

11. GENERAL FEATURES OF A SUPERSYMMETRIC NONLINEAR a MODEL IN SUPERGRAVITY

U ( m + n ) /U( m ) X U ( n ) symmetry is kept even after be- In this section, we first introduce a nonlinear o model ing coupled to the hidden sector, resulting in the true in global supersymmetry, and then discuss general

45 3636 - @ 1992 The American Physical Society

45 - NONLINEAR u MODELS A N D SUPERSYMMETRY-BREAKING . . .

features of the model when it is coupled to supergravity. A globally supersymmetric nonlinear a model is gen-

erally described in terms of several chiral superfields @' which contain Nambu-Goldstone bosons in their bosonic degrees of freedom. The Lagrangian is written as

where K is a real function which is called a Kahler poten- tial. We adopt the notation used in Ref. [9]. The La- grangian (2.1) has a Kahler invariance, that is, an invari- ance under a transformation on K:

The action is unchanged under (2.2) since a D term of a chiral superfield is a total derivative. Therefore, in order to obtain a supersymmetric Lagrangian with a nonlinear- ly realized global symmetry, we have to assign a transfor- mation law of @' under a group G, and then construct a Kahler potential which is transformed according to (2.2):

When the action is written down in terms of component fields, it is found that the action depends on a Kahler metric K/- a2K( A ', A: ) /a A 'a A f and its derivatives ( A i is a scalar component of the superfield @'). Hence the Kahler invariance is manifest.

Now, we would like to obtain a supergravity Lagrang- ian which contains the nonlinear a model corresponding to (2.1) and a hidden sector. The hidden sector is as- sumed to be made of a G-invariant chiral superfield Z and couples to other fields only gravitationally. A super- gravity Lagrangian of the nonlinear a model (2.1) with a minimal hidden sector coupling is written in terms of a total Kahler potential

and its derivatives [lo]. Here, z is the scalar component of Z and h (z ) is a superpotential. The vacuum is defined as a minimum of the scalar potential V,

and the vacuum energy (cosmological constant) is as- sumed to vanish:

where angular brackets stand for the vacuum expectation value. Supersymmetry is spontaneously broken in the hidden sector if a quantity

has a nonvanishing vacuum expectation value. In that case, the gravitino gets a mass m 3 / 2 = ( e K + ' z ' 2 ~ h ( z ) ~ ) and the supersymmetry-breaking effects in the observable sector are characterized by m 312 .

Since G itself appears in the scalar potential V and the fermion mass term contrary to the global supersymmetry case, the global symmetry based on the Kahler transfor- mation (2.2) is explicitly broken in the coupling to the hidden sector if f#O. In other words, K has to be com- pletely invariant in order to keep the global symmetry [ I l l .

Note that if h (z )=0, the supergravity Lagrangian con- tains first derivatives Ki -- aK ( A ', A: )/a A ' and K i r a K ( A ', A: )/a A: but does not contain K itself. In this case, the changes of those terms under the transfor- mation K(Ai ,A: ) -K+f(A)+f*(A*) are absorbed by a chiral rotation of the fermion fields, so that Kahler invariance is maintained. '

111. THE PURE REALIZATION OF U( m + n )/U( m )XU( n NONLINEAR a MODEL

We would like to consider a U( m + n ) /U( m ) XU( n ) supersymmetric nonlinear a model. As mentioned in the previous sections, it is known that a nonlinear realization of a coset G/H is not unique in supersymmetric theory [2]. In a class of realizations, there exist several extra bo- sonic degrees of freedom since scalar fields are necessarily complex due to supersymmetry. These extra scalar fields are called quasi Nambu-Goldstone bosons and each chiral supermultiplet contains either two true Nambu- Goldstone bosons or one true Nambu-Goldstone and one quasi Nambu-Goldstone boson. In this section, we con- struct a "pure" realization which contains no quasi Nambu-Goldstone bosons and study supersymmetry- breaking effects. In the next section, we consider a total doubling realization where each Nambu-Goldstone boson corresponds to a chiral superfield; hence, there are the same number of quasi Nambu-Goldstone bosons as the true Nambu-Goldstone bosons.

First we parametrize an element of U(m +n ), i.e., an ( m + n ) X ( m + n ) unitary matrix as follows:

wherea, A , a , a n d h a r e m X m , n X n , n X m , a n d m X n submatrices, respectively. An element of the subgroup U( m ) X U( n ) is represented as a block-diagonalized form

'1n Ref. [ 5 ] , it is shown that a ratio between the Newtonian constant and the spontaneous symmetry-breaking scale 1/F: has to be quantized in order that the nonlinear a model can be coupled to supergravity with a global consistency. However, it is not relevant in the following discussions.

3638 TORU GOT0 AND YASUHIRO OKADA !!?

in this parametrization. A globally supersymmetric Lagrangian of the pure

realization is constructed by a general method described in Ref. [ 6 ] . Let be a representative of U(m + n ) / U ( m ) X U ( n ):

where 1 means an m X m unit matrix and Y is an n X m matrix chiral superfield whose scalar components are all Nambu-Goldstone bosons. The transformation law of 5 under the U( m + n ) group is defined as

where u is an m Xm matrix chiral superfield which de- pends on U and Y, i.e.,

The Nambu-Goldstone superfield Y is transformed as

The Kahler potential is written in terms of 5 as

which is transformed to

Note that a nonvanishing f ( U, Y ) = -In de t (a +ZY) is necessary in the pure realization.

Let us now discuss the supersymmetry-breaking effects on the Nambu-Goldstone superfield Y when the non- linear a model (3.7) is coupled to a hidden sector minimally. The total Kahler potential is

where r is a scalar component of the chiral superfield Y. Since f ( U,Y)#0, the U ( m + n ) / U ( m ) X U ( n ) symmetry is explicitly broken in the coupling with the hidden sec- tor, while U(m ) XU( n ) symmetry is preserved. Conse- quently, the mass terms of the Nambu-Goldstone bosons should be induced according to supersymmetry breaking. In fact, the scalar potential term for the Nambu- Goldstone bosons which is obtained by freezing z to its vacuum value is

achieved when some quasi Nambu-Goldstone bosons are i n t r ~ d u c e d . ~

IV. THE TOTAL DOUBLING REALIZATION OF U( m + n ) /U( m ) X U( n NONLINEAR o MODEL

In this section, we construct a supersymmetric U( m + n )/U( m ) XU( n nonlinear a model with a total doubling realization, where a completely invariant Kahler potential is obtained.

In addition to 5, another representative 5 is introduced in order to obtain a total doubling realization:

c = ( l , q ) , (4.1)

where v is an m Xn matrix superfield. Here, we take m I n without loss of generality. The transformation law o f f is assigned as

and hence

q L . \ ~ ' = ( a ~ + q H ~ ) - ' ( a ~ + v ~ ~ ) . (4.3)

Then, four U( m ) X U ( n )-covariant quantities l t t , f e , 55, and ite are transformed as

u l + ( - ~ ~ - ' { ~ ~ u - ~ ,

u --1- -t t 1 fQ-u g 5 a - ,

U

E&-a-'Z{u -' , U t-l t - t t I ctQ-u g < v - ,

where v and iJ appeared in (3.4) and (4.2). Thus, traces of the products of matrices

~ , , ~ = t r [ ~ ~ ( ( ~ ~ ) - ' f e ( { ~ ~ ) - ' ] ~ , N=1,2, . . . (4.5)

are all completely invariant under the U( m + n ) transfor- mation (3.4) and (4.2). However, only m traces X I X , . . . , X are independent variables since ct{(f[)-' f f ig t l t ) - ' is an m X m m a t r i ~ , ~ and there- fore, an arbitrary function of X I ,X2, . . . ,X, is the most general form of an invariant quantity in this realization. Consequently, a completely invariant Kahler potential of the total doubling realization is written generally as

~ d o u b l r ( ~ , T ; ~ + , T + ) = F ( X ~ , . . . ,xm) , (4.6)

where F is an arbitrary real function of rn variables. Note

V' "I = :/2 det( ' + r'r)trr'ni; :I2 l r r t . (3. lo) 2 ,4 invanant K&ler potential is given for a U( rn + n )/SU( rn ) XU( n ) case with one quasi Nambu-Goldstone

Thus, all Nambu-Goldstone bosons get a common mass boson [11,12]. of m3,,2. A completely invariant (f = O ) Kahler potential 3We would like to thank T. Awaji and M. Hotta for discus- is needed in order to preserve the coset symmetry. It is sions on this point.

45 NONLINEAR u MODELS AND SUPERSYMMETRY-BREAKING . . . 3639 -

that in the case of global supersymmetry, two terms ~ ( a , ? ~ , z ; a + , ~ + , z * )=F(x, , . . . ,x, )+ lz 12+lnlh(z)12 , lndetgt< and lndetf 2 can be added to the KBhler poten- tial. (4.7)

As in the pure realization case, the nonlinear a model where X I , . . . ,x, denote the scalar components of which has a Kahler potential (4.6) is coupled to the X, , . . . , X,. Particularly in the m = 1 case, the scalar minimal hidden sector. The total Kahler potential is potential V is written explicitly as

where

x = ( l+ r t r ) ( l+? i ? i t )

( l+77a)( l+rt?r t ) '

The unbroken U( 1 ) XU( n ) symmetry provides ( a ) = O and (77) =O. Thus, the vacuum expectation values of x and its derivatives are

( x ) = l ,

and, therefore, the normalization condition of the kinetic terms,

fixes the value of F'( 1 ) to one. Consequently, the scalar potential term induced by supersymmetry breaking be- comes

and hence true Nambo-Goldstone bosons rt+?i remain massless since the whole U(n + 1) symmetry is preserved, while quasi Nambu-Goldstone bosons at-77 get a mass according to supersymmetry breaking. At the same time, a fermion mass term is also induced:

where F and ?? are fermionic components of the superfields \I, and \I/. In general m > 1 cases, the mass terms are

effective superpotential is induced by the hidden sector coupling. The scalar masses which come from this in- duced superpotential and one from soft supersymmetry breaking terms cancel each other, resulting in the Nambu-Goldstone bosons remaining massless and hence the nonlinearly realized U( m + n ) /U( m ) X U( n ) is main- tained.

V. CONCLUSION

We have studied supersymmetric U( m +n ) / U(m ) XU( n ) nonlinear u models. The models in a re- stricted class of realizations can be coupled to the hidden sector without an explicit symmetry breaking. That is, the Kahler potential has to be completely invariant in or- der to retain the nonlinearly realized global symmetry. Especially in pure realization, it is impossible to con- struct a completely invariant Kahler potential; hence, ex- plicit breaking occurs necessarily. As a result, all Nambu-Goldstone bosons get a mass of m 3 l z . On the other hand, we find a completely invariant Kahler poten- tial in the total doubling realization. In this case, true Nambo-Goldstone bosons remain massless, while quasi Nambu-Goldstone bosons and fermions get masses pro- portional to the gravitino mass.

Note that the true Nambu-Goldstone bosons rt+?i are 1:l mixtures of rt and 77, as seen in (4.1 1) and (4.13). This is a general result of the assumption that the Kahler potential K is completely invariant. Indeed, this fact can be shown by constructing K as a power series of a's im- posing the order-by-order invariance under the infinitesimal tran~formation.~

As mentioned in the introduction, the total doubling case can be applied to the Higgs sector in order to solve the p problem. There are two Higgs doublets Hand H,

in the minimal supersymmetric standard model. Now we would like to suppose them as Nambu-Goldstone super- multiplets in a certain U( m + n )/U( m ) XU( n ) nonlinear a model described in the above sections. The minimal choice is U(3) /U(2)XU(1) ( m = l , n = 2 ) in the total doubling realization. The Higgs doublets H and H are

although the full expression of the potential term is so - complicated that a compact form such as (4.8) is not 4 ~ n the global supersymmetry case, the mixtures with arbitrary given explicitly. ratio can be true Nambu-Goldstone bosons since a Kahler po-

The fact that a fermion mass term arises means an tential with f #O is allowed [2].

3640 TORU GOTO AND YASUHIRO OKADA 45

identified with ^ and ^ in Eq. (4.6). Using Eqs. (4.11) and (4.12), the mass terms of Higgs bosons and their su-perpartners (Higgsinos) h and h are determined as

F(Higgs-boson mass) = 2ml/2(h*-h)(h ~P) (5.2)

X(Higgsino mass )= — m 3 / 2 ^ + H.c . (5.3)

Thus, an effective \i term ~m3/2HH is induced. This is the same mass spectrum as the one given in Ref. [8]. They obtained it in the context of a linear a model.5

5A similar observation was made when an axion model was considered in supergravity [13].

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[2] W. Buchmiiller, R. D. Peccei, and T. Yanagida, Nucl. Phys. 8227,503(1983).

[3] T. Kugo and T. Yanagida, Phys. Lett. 134B, 313 (1984). [4] H. P. Nilles, Phys. Rep. 110, 1 (1984), and references

therein. [5] E. Witten and J. Bagger, Phys. Lett. 115B, 202 (1982). [6]B. Zumino, Phys. Lett. 87B, 203 (1979); M. Bando, T.

Kuramoto, T. Maskawa, and S. Uehara, ibid. 138B, 94 (1984); Prog. Theor. Phys. 72, 313 (1984).

[7] J. E. Kim and H. P. Nilles, Phys. Lett. 138B, 150 (1984). [8] K. Inoue, A. Kakuto, and H. Takano, Prog. Theor. Phys.

Since we have used a nonlinear realization, our results (5.2) and (5.3) are independent of the details of the model, such as an explicit form of a potential in the case of a linear realization. We hope that the nonlinear realization considered here has wide applications in the model build­ing of a supersymmetric unified theory.

ACKNOWLEDGMENTS

The authors would like to thank K. Inoue and T. Yanagida for helpful discussions. The work of T.G. was supported in part by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture of Japan No. 00030446.

75, 664 (1986); A. A. Anselm and A. A. Johansen, Phys. Lett. B 200, 331 (1988).

[9] J. Wess and J. Bagger, Princeton Series in Physics (Prince­ton University Press, Princeton, 1983).

[10] E. Cremmer, B. Julia, J. Scherk, S. Ferrara, L. Girardello, and P. van Nieuwenhuizen, Nucl. Phys. B147, 105 (1979); E. Cremmer, S. Ferrara, L. Girardello, and A. van Proeyen, ibid. B212, 413 (1983).

[11] T. Goto and T. Yanagida, Prog. Theor. Phys. 83, 1076 (1990).

[12] T. Kugo, I. Ojima, and T. Yanagida, Phys. Lett. 135B, 402 (1984); T. Kugo, S. Uehara, and T. Yanagida, ibid. 147B, 321 (1984); S. Uehara and T. Yanagida, ibid. 165B, 94 (1985).

[13] T. Goto and M. Yamaguchi, Phys. Lett. B 276, 103 (1992).