SU(2)×U(1) breaking in supersymmetric guts

Nuclear Physics B227 (1983) 477-493 © North-Holland Publishing Company

S U ( 2 ) x U ( 1 ) BREAKING IN SUPERSYMMETRIC GUTs* Michael DINE The Institute for Advanced Study, Princeton, New Jersey 08540, USA

Willy FISCHLER Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Received 11 February 1983 (Revised 17 May 1983) We consider the problem of SU(2)x U(1) breaking in supersymmetric GUTs. In particular, we focus on models in which supersymmetry is broken at an intermediate scale. We review the arguments that light singlet fields coupled to Higgs spoil the hierarchy. We show explicitly how auxiliary fields associated with heavy particles drop out of the low-energy theory, as expected from decoupling. Theories with only quarks, leptons and Higgs thus have directions in which all quartic couplings vanish. Conditions on the Higgs mass matrix for obtaining the correct pattern of breaking are stated, and an explicit model satisfying these conditions is constructed. The relevance of these considerations to supergravity theories is discussed.

1. Introduction Recently, a great deal of attention has been d e v o t e d to the p r o b l e m of constructing models of particle physics based on s u p e r s y m m e t r y [1]. In particular, several s u p e r s y m m e t r i c grand-unified models have been constructed in which sypersymm e t r y is s p o n t a n e o u s l y b r o k e n at a large scale, d [2-7]. These include models based on both global s u p e r s y m m e t r y and on N = 1 supergravity coupled to matter. Typically, in the globally s u p e r s y m m e t r i c theories, only some very heavy fields couple to the G o l d s t o n e fermion at the tree level. Because the light fields (quarks, leptons, and Higgs) are d e c o u p l e d f r o m these h e a v y fields, the low-energy world is " n e a r l y " supersymmetric. It can be described, as emphasized by Banks [8] and by Susskind and Polchinski [9] by an effective lagrangian which is supersymmetric except for soft, explicit s u p e r s y m m e t r y - b r e a k i n g terms. These theories thus appear, effectively, like the theories discussed by D i m o p o u l o s and G e o r g i and by Sakai [10], with soft s u p e r s y m m e t r y breaking, except that the soft-breaking terms m a y now be calculated explicitly in terms of the underlying dynamics. In the case of N = 1 supergravity coupled to matter, the situation is s o m e w h a t m o r e complicated. Since one is dealing with n o n - r e n o r m a l i z a b l e theories, it is not clear in what sense one can apply such notions as decoupling. M o r e generally, it is unclear how one should treat q u a n t u m effects in these theories, or in what sense the tree level lagrangian can be viewed as providing a g o o d a p p r o x i m a t i o n to the 477

478

M. Dine, W. Fischler / SU(2) × U(1) breaking in supersymmetric GUTs

whole theory. Nevertheless, the theories, even at tree level, are of some interest for they suggest ways in which gravity may enter our understanding of low-energy physics. Moreover, as several authors have noted [6, 7], when supersymmetry is broken in these theories at a large scale, A, the low-energy theory (at tree level) is again described by an effective lagrangian which is globally supersymmetric except for soft-breaking terms. Thus in their low-energy structure, these models are similar to the globally symmetric theories discussed above. In all of these theories, the soft-breaking terms are low-dimension operators like scalar and fermion masses. Their coefficients involve powers of A2/MGuT (or A2/Mp), and powers of couplings. The precise form of the coefficient depends on the mechanism of sypersymmetry breaking and on the details of the model. These coefficients determine the masses of scalar quarks and leptons, and the masses of Higgs bosons. Generally one tries to arrange the model so that masses of scalar quarks and leptons are positive, and are comparable to the negative masses-squared of some Higgs bosons. In this way, one hopes to obtain a phenomenologically correct theory: unobserved scalars are heavy, while SU(2)× U(1) is broken in the correct fashion. However, some care is required in actually implementing this program. As we stated earlier, supersymmetry restricts tightly the form of the effective low-energy theory. In fact, in many of the theories which have been discussed to date, it is impossible to obtain the correct pattern of SU(2)× U(1) breaking! This results, in some instances, because only the quartic couplings in the low-energy theory leave over a fiat direction, i.e. they vanish if various scalar expectation values are aligned in an appropriate way, for arbitrarily large values of these v.e.v.'s. Moreover, as a result of certain discrete symmetries, the Higgs mass matrix is negative in these directions, for any values of the model parameters. Thus these models tend to break SU(2) × U(1) at the unification scale (or Planck mass). The model of ref. [3], in fact, suffers from this disease. This model, a globally supersymmetric theory, was shown to have many desirable properties: the correct pattern of SU(5) breaking, positive masses-squared for scalar quarks and leptons, and negative mass-squared for Higgs bosons, among others. However, SU(2) × U(1) breaking does not arise in the way claimed for that theory. The Higgs fields of this theory couple to some superheavy fields. We assumed that the IFI 2 terms in the potential (F denotes the auxiliary field associated with a chiral multiplet; here and throughout, we use the notation of Wess and Bagger [11]) associated with these heavy fields eliminated the fiat directions. However, if the low-energy theory is described by a supersymmetric theory with soft-breaking terms, this cannot be correct. Since these F ' s are partners of some decoupled heavy fields, they should not appear in the low-energy theory. In fact, as we will show in sect. 2, they do not, and thus the Higgs fields obtain expectation values of order MGUT. In some models [4, 6, 12], there are appropriate quartic couplings, because a light singlet field is added which couples to the ordinary Higgs (this is often referred

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric GUTs

479

to as the "sliding singlet"). Since the fields are all light, supersymmetry permits the needed extra couplings. These couplings are also supposed to prove a mechanism which renders the colored Higgs field very heavy, while leaving the doublet light. However, it has been pointed out that this mechanism does not work in models with a large-breaking scale [8, 9, 13, 14]. Instead, either SU(2) x U ( 1 ) is broken at the scale, or the singlet obtains an expectation value such that all of the Higgs particles are superheavy. In the latter case, S U ( 2 ) x U ( 1 ) does not break at all! These models can usually be repaired, as we will see, by omitting the singlet field and by admitting a certain amount of fine-tuning, to render the doublets light. However, in this case one usually recovers the problem mentioned in the previous paragraph. All is not lost, however. As we will show below with an explicit example, it is frequently possible to break the troublesome discrete symmetry by adding some new, superheavy fields to the model. While the addition of these new fields is not particularly aesthetic in the cases under consideration, it provides an "existence proof" that the problems of SU(2) x U(1) breaking can be solved. Moreover additional fields may well be required in these theories for other reasons: to obtain adequate production of baryons and to avoid undersirable relations among quark and lepton masses. This paper is organized as follows. In sect. 2, we show explicitly how heavy fields decouple from the low-energy potential. We discuss (following Georgi and Dimopoulos) the operators which may appear in the effective low-energy theory, and given the conditions under which the correct pattern of SU(2) x U(1) breaking may be obtained. In sect. 3, we review the problems associated with adding an additional singlet field to the theory. In sect. 4, we illustrate how the model of ref. [3] may be modified to obtain an acceptable low-energy theory. We also discuss SU(2) x U(1) breaking in inverted hierarchy schemes. We conclude with a brief discussion of the situation in some simple supergravity theories.

2. Decoupling and the form of the low-energy theory For definiteness, we will focus in this paper on models based on SU(5), in which SU(5) is broken to SU(3) x SU(2) x U(1) by the expectation value of a 24, Xij, and in which SU(2)x U(1) is broken by a Higgs 5 and 5,/-Ii, H i. The generalization to other cases will be immediate. The models we are considering typically include couplings W H = m H I - I H + AH/-?~X']-/i .

(2.1)

The expectation value of X has the form (X) = o- diag (2, 2, 2, - 3 , - 3 ) .

(2.2)

480

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric GUTs

In order that the Higgs field be light, one requires mH =

3AHtr.

(2.3)

We suppose that the X field is heavy, with a mass of at least order A. We wish to study the theory at scales near the weak interaction scale, much below Mcu-r and A. It should be possible, then, to "integrate out" the heavy fields, such as X and the superheavy gauge fields. It is easy to see that the heavy fermions, scalars, and gauge bosons must decouple from the low-energy theory. But what of the auxiliary components of these heavy superfields? Clearly if the low-energy theory is to look approximately supersymmetric, these do not belong in the effective lagrangian. In other words, the potential at low energies should have the form

v = E IF~,12+ E ~D ~ ~ . light

(2.4)

light

Here the ¢i are the light chiral fields of the theory, and the F6, are the corresponding auxiliary fields. D " are the auxiliary components of the light gauge fields. But how does the cancellation of the IF] z and D 2 terms involving the heavy fields come about? The cancellation of the F - t e r m s is the simplest to illustrate, and can easily be proven in some generality. Let us consider a general superpotential, in which all fields have been shifted about their vacuum expectation values and the mass matrix has been diagonalized W = ~ lmklAkAt k.l

(2.5)

+ ~ ,~yk¢idt)iAk -t-. • • . i,j,k,

Here the A fields are the heavy fields whose decoupling we wish to illustrate, ~bl are the light fields, and we have exhibited only the couplings relevant to our present considerations. Then the auxiliary fields of interest are (note that we are using the same symbol to denote a chiral field and its scalar component) (2.6)

FAk = m k l A l + A i j k l ~ i ~ j .

The corresponding terms in the potential are (diagonalizing mkl) ]FAk 12= ,~,kA ,mk4~,4,~, * *¢,~* + m ~lAk 12 +mkAijkA*qb,Oi + m ' A *iikl'Xktt~ a .¢*.~* i tl.~ i

.

(2.7)

Now consider the graph contributing to the potential shown in fig. 1. This graph is obtained when we integrate out the heavy fields in order to obtain the effective low-energy theory. Its value can be read off immediately from our potential, eq. (2.7). It is .3 -

ig m

=

--l 2 mk

Ai;kA,l,.km 2k ¢ i ¢ 1 ¢

,

,

t ¢ ,, ,

(2.8)

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric GUTs

(~i\ \

\

/

/

/

A

N

/

~L

\

/

/

/

481

N

4,j

\,

~m

Fig. 1. Diagramwhichgivesriset•cance••ati•n•f•F•2termsf•rheavy•e•ds.Dashed•inesden•tesca•ars.

or

E(1)= --&ikA *lmk¢,¢j¢ •t ¢ m. •

(2.9)

This precisely cancels the would-be quartic coupling! Similarly, we can illustrate the cancellation of D-terms associated with heavy gauge fields. Let us first consider a U(1) theory, with four charged fields, A ÷, A , ¢ +, ¢ - and a singlet field, Z. The superpotential is given by

W =AZ(A+A--Iz2).

(2.10)

Then we may take, at the minimum (A +) = (A-) = tz.

(2.11)

The U(1) symmetry is broken; the gauge field acquires a mass m2w = 4g2/x 2 ,

(2.12)

where g is the gauge coupling. Expanding the term 1D2 about these expectation values gives

1-D2=~g2(2u2Re(A+-A-)+W¢+I2-1¢-12)2+O(A3).

(2.13)

The graph of fig. 2 then gives

(i)394.

- i E (2)

411. +12

2-~--~w'+/x tlq~ I -kb-[2) 2,

¢

qb \

\

/ / \ \ / /

/

¢

>-

< A

/

\

/

\

\

\

Fig. 2. Diagram which gives rise to cancellation of 1D2 for heavy gauge fields.

(2.14)

482

M. Dine, IV. Fischler /

$U(2) x U(1)

breaking in supersymmetric GUTs

or

E~2~ = _lg2([~ +12_ ]q5-[2)2,

(2.15)

and again we obtain the desired decoupling. The generalization to the non-abelian case is straightforward. These simple calculations have served to illustrate the obvious: the auxiliary components of heavy fields must decouple if the effective low-energy theory is to be supersymmetric (up to soft breaking terms). What, then, is the form of the effective low-energy theory in a model with only a pair of Higgs doublets,/-I and H, the usual quarks and leptons and gauge fields? In particular, what is the form of the Higgs potential? Apart from Yukawa couplings to quarks and leptons, the only quartic couplings come from the terms l~Da2

.

(2.16)

a eSU(2)×U(1J

These vanish for doublet expectation values aligned in the following way:

Apart from these supersymmetric terms, soft-breaking terms are allowed. For the Higgs fields, these are just the quadratic terms:

V = m][ISIl2+m2lHl2+m2(tzlg + c.c).

(2.18)

Without loss of generality, we can take m 2 real and positive. In order that this potential be bounded below, we require

mZ + m 2 - 2 m 2 > O .

(2.19)

In order that the potential should have a non-zero minimum with the correct alignment of Higgs v.e.v.'s, we also need m l /2~ 2

2 -- m 4 < 0

(2.20)

(It is straightforward to show that these conditions are necessary and sufficient). Unfortunately, most of the models which have been considered to date have approximate discrete symmetries which exchange H and/-}, and therefore yield 2

2

ml -~m2.

(2.21)

Thus these models cannot achieve the correct pattern of gauge symmetry breaking, without some modification. In sect. 4, we will show, by means of a specific example, that the symmetry H ~--~/-I of the underlying theory may be broken so as to obtain a suitable Higgs mass matrix and the correct breaking of SU(2) x U(1). One can also try to generate additional quartic Higgs couplings by adding further light fields to the theory, and this approach has been employed by many authors.

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric G U T s

483

In sect. 3 we show that most models with additional singlet fields also cannot obtain the desired breaking of SU(2)× U(1), at least in theories with a large breaking of supersymmetry.

3. Models

with a singlet field

It has frequently been suggested that one should replace the mass, MH, appearing in eq. (2.1) by a singlet field. In other words, one should include in the superpotential a term AZIqH, (3.1) where Z is a gauge singlet. This yields an additional quartic coupling of the Higgs fields in the low-energy theory. Moreover, one might hope, as first suggested by Witten [2], that inclusion of this field would provide a mechanism to make the color-triplet Higgs superheavy automatically. For, if one examines the classical equations 0W O H i - AZICIi + A HITIi~,Ji = O , OW - - AZH

0Hi

j

+AHX/iH i = 0 ,

(3.2)

one sees that if Hs

= ISIS= v ,

(3.3)

AZ -- 3hHo-,

(3.4)

then

and the color triplet Higgs are rendered superheavy. However, at the quantum level, this is not what happens. In general, as we will shortly see, Z obtains a quite different expectation value than that of eq. (3.4) and a l l of the Higgs fields are superheavy. SU(2)× U(1) remains unbroken. Suppose, in particular, that at some order in perturbation theory (as in the model of ref. [3]) we obtain a correction to the Higgs potential of the form v `2' = - i z 2(IHI 2 +

I/-712)+ m 2(/~H + H + / ~ + ) ,

(3.5)

where/z 2> 0. Schematically, a diagram contributing to such terms is shown in fig. 3. The chiral field, Y, emerging from the diagram is a field whose F - c o m p o n e n t has a non-vanishing v.e.v., i.e. (Fy> = A : .

(3.6)

The masses in eq. (3.5) are expected to be of order 2 m

2 ,tz

IF, f2 -

2

MGUT

•

(3.7)

484

M. Dine, W. Fischler/ SU(2) x U(1) breaking in supersyrnmetric GUTs y

H

y+

y

H+

y+

H

Fig. 3. Typical supergaphs contributing quadratic terms to Higgs potential. Solid lines denote chiral superfields. X's denote expectation values and mass insertions.

A potential of this type would seem to lead to the correct pattern of symmetry breakdown. However, in the next order of perturbation theory, two problems will inevitably arise. The first problem has been noted by Polchinski and Susskind [9]. In theories in which there is no R-invariance below graphs such as that of fig. 4 will give rise to the operator

MGuT,

a I d4OY+Z=aA2f d2OZ.

(3.8)

H e r e a is a numerical coefficient, depending on coupling constants. If we now use the equations of motion which follow from the effective action, we find that this term corresponds to a mass term for Higgs fields a d 2(/~'H + c . c ) .

(3.9)

Such a term clearly spoils the hierarchy. This term might be suppressed if the theory possesses an R-invariance which survives to scales well below and under which Z does not carry R = 2. It is diffcult to see how such an invariance can be reconciled with the need for a large gluino mass, for the operators which give rise to gluino mass will tend to be similarly

MGUT

y+

Fig. 4. Diagrams contributing F~Fz term to the potential.

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric G U T s ¥

¥

H

485

+

H

Z

Z

+

Fig. 5. Diagrams contributing to the potential for the field z.

suppressed. However, the second problem, which is equally devastating, exists even in theories with an R - i n v a r i a n c e [8, 13, 14]. Consider the diagrams of fig. 5. These are the leading quantum corrections to the potential for the scalar field Z. They are computed by calculating the Higgs-loop contributions to the vacuum energy, with Higgs masses appropriate to a given expectation value of Z. The functional form of the potential depends on the details of the model, and may be rather complicated. To illustrate the problem, however, it is sufficient to consider a potential of the form

Vz=p

2a21zl 2 16rr 2 .

(3.10)

The coefficient, 0 2 is of order tz 2, the Higgs mass. For simplicity we assume p 2 > 0. Compare, then, the energy of the state with AZ = 3AHX; H - - H - v with that with Z = 0, H =/-] = 0. The energy difference is p

2 (3Act) 2 16rr 2 2/~2V 2 .

(3.11)

In other words, since v 2 = O(iz 2), the desired vacuum has much higher energy than the vacuum with Z = 0 and all Higgs fields superheavy. One can easily convince oneself that the desired vacuum is not even a local minimum. The general result, that we do not obtain anything resembling the desired ground state, is not special to the flavor of the potential which we assumed. For example., if p 2 were negative, we would expect Z to obtain some value of order MGUT, unrelated to the desired value. In such a case, it might be possible to obtain the desired value by fine tuning of the parameters of the model, i.e. one could seek a set of couplings and masses, a~ and M;, which solve the equation OV

(a,, m,)]~.z=3~ = 0

aV ~-~(o~,,

m,)[,-o = 0

OV ~(a,,

m,)lH-o = O.

(3.12)

486

M. Dine, W. Fischler / SU(2) x U(1) breaking in supersymmetric GUTs

Of course, the fine tuning involved would be at least as severe as that of conventional grand-unified theories. It should be noted that both of the problems exist whether the expectation value of X is of order MGuT, as in ref. [3], or of order A, as in ref. [4]. 4. A solution

In many cases, the problem of SU(2) x U(1) breaking can be solved by the addition of superheavy fields to the model. The coupling of these fields break the (generally approximate) symmetry H~--~/-I, and thus one can obtain low-energy potentials which break SU(2) × U(1) in the desired fashion. We will illustrate this idea in some detail in the model of ref. [3]. We will demonstrate explicitly that one can obtain, there, a Higgs mass matrix of the desired form, and thus achieve SU(2)× U(1) breakdown at a small scale (d2/MGuT). This result will be achieved at the price of adding some additional heavy (--McuT) fields. We will also discuss the situation in the inverted hierarchy model of Witten [3], as elaborated by Dimopoulos and Raby [4], and Banks and Kaplunovsky [8]. There we will see that the problem of SU(2)× U(1) breaking can also be solved, at the price of some fine tuning (of order one part in 101°) to make the Higgs doublet light. In sect. 5, we will remark on the situation in supergravity theories. The model of ref. [3] had two fundamental scales, put into the lagrangian "by hand": the grand unification scale and the scale, A, of supersymmetry breaking. In the model, SU(5) breaking and supersymmetry breaking were accomplished by different sets of fields. This was necessary to obtain positive mass-squared for scalar quarks and leptons. The degeneracy among different SU(5) vacua was lifted by quantum effects. In order to obtain the correct pattern of SU(5) breaking, it was necessary to consider a rather complicated model; the simplest theories which one writes down tend to prefer the vacuum with SU(5) symmetry*. Here we will consider a somewhat simpler model, in order to illustrate the method. The generalization to the model of ref. [3] is immediate. Let us review, briefly, the basic model and illustrate the origin of the approximate discrete symmetry. The model contains three 24's, Aij, B~j, and X~, a singlet Y, and the Higgs 5 and 5, H and/-I. We omit the quarks and leptons; they are not relevant to the issue at hand. We will assume that the X fields are heavy, and that they obtain, through their interactions, a v.e.v., X =or diag (2, 2, 2, - 3 , - 3 ) .

(4.1)

We will assume, also, that they do not couple to the Goldstone fermion at the tree level. For the rest of the superpotential we take W = WI + I4/"2, * As Weinberg (ref. [15]) has noted, this degeneracy will in fact be lifted by gravitational effects.

(4.2)

M. Dine, W. Fischler / SU(2) × U(1) breaking in supersymmetric GUTs W1 = m A B + Y ( k y A 2 + A 2 ) + k

tr ( A 2 X ) ,

W2 = k H/-I2H + m HI-IH.

487 (4.3) (4.4)

We take k ~tz 2 <
(4.5)

( F y ) = A2 ,

(4.6)

MH is chosen so that the Higgs doublet is massless at the tree level. It was shown in ref. [3] that this theory has many desirable features*. In particular, it was shown that at the quantum level, scalar quarks and leptons obtain positive mass-squared through their gauge interactions. Higgs mesons, in addition to these positive contributions, obtain negative contributions through their interactions with the X field. For a large range of parameters, the Higgs potential in fact turns down near the origin. However, this model clearly has a symmetry which interchanges H and H, and thus the low-energy theory will have the problems discussed in sect. 2: SU(2) × U(1) will be broken at the unification scale. This discrete symmetry will be broken explicitly once quarks and leptons are added to the model. However, this breaking is too small to improve the situation. The problem can, however, be solved by the introduction of additional superheavy fields, with interactions chosen so as to break the left-right symmetry. Perhaps the simplest possibility is to double the number of Higgs particles, and take (with an obvious notation) W2 = ~,aJ-I~,XHb + m a t ~ a H b .

(4.7)

If )tab and mab a r e not left-right symmetric, the low-energy theory will not be. We must, of course, adjust these parameters so that there is one pair of light Higgs doublets; this fine tuning is similar to that of eq. (2.3). For this theory, it is reasonable to expect that the correct pattern of SU(2)× U(1) breaking can be obtained. As the calculations required are a bit involved, however, we study below a slightly different model, in which we can readily demonstrate that the low-energy Higgs potential is of the type required, for a reasonable range of parameters. Starting with the original model with a single Higgs 5 and 5, we add a 10 and a 10. For W2 we take W2 = k H/-IZH + m H/~H + m 101010 + y leiiklm 10 ii l O k t H " ~ 10 ]LI +yEe i j k l m xuiilUkl, lm + A ' ~ i i A i k 10 kt

(4.8)

* The representation content of the theory has been somewhat altered from that of ref. [3]. In particular, the fields in the supersymmetry-breakingpotential, W1, were previously taken to be 5's and 5's. That model had certain naturalness problems associated with D-terms in the low-energy theory. The present model does not. The explicit calculations of ref. [3] are altered only in that certain Casimirs are different; none of the qualitative features of the results are changed.

488

M. Dine, W. Fischler / $ U ( 2 ) x U(1) breaking in supersymmetric GUTs y

y+

10 H

H

H

H

¥

,I0

I

IH+

H

H

y+

10

H

10

H

H+

Fig. 6. Diagrams contributing to the Higgs potential for the model of sect. 3. Wavy lines denote gauge fields.

we have indicated SU(5) indices explicitly where there may be ambiguity. If y l ~ y2, this model clearly has no symmetry which interchanges H and/-I. Moreover, with this condition, the vacuum has* (10) = (10) = 0 ,

(4.9)

and these fields are quite heavy. Quadratic terms in the low-energy Higgs potential arise at two loops in this model**. They arise from graphs involving exchange of gauge fields, X fields, and the 10 and 10 fields. We will denote corresponding contributions by the subscripts g, ~, 10 and 10, respectively. Typical graphs contributing to the masses are shown in fig. 6. The corresponding terms in the potential have the form V=(~x 2 +

2+tz210)lHi2+ ( 2 +

+ (A 2 +A210)(I~H + / ~ * H * ) .

2+

2)1/~r2

(4.10)

In ref. [3], it was shown that the quantity/Xg2 is positive, while/x 2 is negative for a range of parameters. A similar computation demonstrates that/Z2o and tZ~o are *We have searched exhaustively for other minima of the potential, and found none. However, we do not have a proof that there are no others. If there are, one must check, as in ref. [3, 15] that at the q u a n t u m level the desired m i n i m u m is preferred. ** Actually there are certain terms which arise already at 1-100p as a result of shifts in the I; field, they are almost certainly non-vanishing, they are of the form (A 2/167r2)(A ~.A2/rn)2(ITIH + c.c) (it is easy to show by an analysis similar to that of sect. 2 that terms of O(A 2) cancel). T h u s it is necessary to take A to be very small.

M. Dine, W. Fischler/ SU(2) x U(1) breakingin supersymmetric GUTs

489

also negative. Explicitly, these are given by, for M l o , M:~, )t~ << M , = - 1 8 4 { A y A 2 ~ 2 [ A 2 0 , ( y2 )lnA_~_2_M 2 , /X~o 5 \ M ] \167r 2] ~ ~,.1o

h

__oM

5 \ M ] \16~'2}\16zr 2] l n M ~ 0 '

_8(,y12 /z~=

2(

15\ M ] k l 6 r r 2 ] \ 1 6 r r 2]

M2

In 775,2 . Mz~

(4.11)

The quantities A 2 and A10 2 are non-vanishing, and proportional to h 2 and yly2 respectively. Thus it is clear that we can satisfy all of the conditions discussed in sect. 2 for obtaining the correct pattern of SU(2)x U(1) breaking. For example, this can be achieved if we take yl to be of order the gauge couplings, while y2 and hH are much smaller. This has been accomplished only at the price of introducing some additional superheavy fields into the model. More elegant versions of this idea may be possible, but the above discussion provides a proof that a solution to the problem of low-energy symmetry breaking exists. Can this solution be applied to other types of theories? In particular, can it be applied to the "inverted hierarchy" of Witten? The answer appears to be yes. In this theory, there is one fundamental scale, /z --,/M~uTV. The superpotential for the model contains a term W = h tr ( B A 2 ) - g X ( t r ( A 2 ) - / . t 2 ) .

(4.12)

Here A and B are 24's of SU(5), while X is a singlet. At the tree level, supersymmetry is broken and A obtains an expectation value of order A, while a linear combination of B and X is undetermined. At the one-loop level, B obtains an expectation value of order M~tJT, providing the needed breaking of SU(5). We focus here on the model as developed by Dimopoulos and Raby [4]. These authors noted that, in order to obtain the correct breaking of SU(5), while at the same time avoiding too-rapid proton decay, it was necessary for the model to include two 5's and two 5's of Higgs. These coupled to the 24, A, through couplings of the form hablQaaHb .

(4.13)

These authors attempted to keep the doublet components of these Higgs light while rendering the triplets heavy by coupling the Higgs to a "sliding singlet", as described in sect. 3. However, as we saw in that section, such an approach has little or no chance of success. Instead, all of the components of the Higgs field typically become heavy (this point has also been noted for these models by Banks and Kaplunovsky [8] and by Witten [14]). Thus it is necessary to introduce a mass term, mabIq~Hb and fine tune so as to the render the one set of doublets light. If the couplings and masses are not left-right symmetric, it is likely that one can again obtain the correct

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pattern of SU(2) x U(1) breaking, in the manner discussed above. Detailed calculations of the scalar potentials in this model would clearly be desirable. It is not certain, for example, that scalar quark and lepton masses are positive in these theories. Other problems with the inverted hierarchy model have been discussed by various authors. Nevertheless, the idea remains sufficiently intriguing as to merit further study.

5. SU(2) × U(1) breaking in supergravity Clearly, when one contemplates a large breaking of supersymmetry, it is no longer reasonable to ignore gravity [15]. Much effort has been devoted recently to the study of N = 1 supergravity coupled to matter [16, 17, 18]. Such models cannot provide an ultimate understanding of nature. To name only two difficulties, it is unclear how to include quantum effects, and the cosmological term must be treated in a very peculiar manner [6, 15, 16]. Nevertheless, these theories are interesting, for they provide some clues to the role which gravity might play in physics below the Planck scale. In particular, a number of models have been considered in which the low energy world (at least at tree level) has precisely the structure discussed in this paper: an N = 1 supersymmetric theory with soft-breaking terms. Obviously then, many of the considerations described above should be relevant to these theories. Once again, supersymmetry tightly restricts the form of the Higgs potential. For definiteness, we will again consider unified models based on the gauge group SU(5). If the low-energy theory contains only a pair of Higgs doublets, quarks, leptons, and S U ( 3 ) x S U ( 2 ) x U ( 1 ) gauge fields, the Higgs potential must have the form discussed in sect. 2. At the tree level, it is possible to obtain negative mass-squared for some Higgs fields, through appropriate fine tuning. For example, if the superpotential contains a term g =/-Ia (Xa~ + rn,b)Hb + ~M~ 2 + ½A~ 3 ,

(5.1)

one can show that if rnab is chosen carefully, and if the supersymmetry-breaking potential satisfies certain conditions (these latter conditions are similar to those discussed by Nilles et al. for low-energy theories [19]) then the Higgs fields obtain small negative mass-squared (of order the gravitino mass-squared). However, if the theory contains only a single pair of light doublets, one can show that the mass matrix is always symmetric under H ~--~/--I, even if the underlying theory has no left-right symmetry. Thus it is impossible to satisfy the conditions discussed in sect. 2 for SU(2) × U(1) breaking. Alternatively, one can consider models such as those discussed in sect. 4, now coupled to supergravity. It is likely that in these models, SU(2) × U(1) can be broken in the correct fashion, since these theories will reduce to their globally supersym-

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491

metric counterparts when Mp ~ co. However, since the Planck mass is not so much larger than the unification scale of these models, some care will be required in deciding when gravitational effects can and cannot be neglected. This is, of course, especially delicate since these theories are not renormalizable and must be cut off at some scale. Some of these issues will be discussed in a subsequent publication. Some authors have recently considered an alternative approach. They have attempted to break S U ( 2 ) x U(1) by adding additional light fields to the theory. For example, the authors of ref. [6] add a singlet field, U, and take the Higgs couplings to be (mH + U ) I ~ H + I ~ H ,

(5.2)

(we ignore, here, the couplings to quarks and leptons), mH is chosen so as to make the Higgs doublets light. The authors argued that, in this theory, with appropriate supersymmetry-breaking terms included, SU(2)× U(1) would break as desired at the tree level. As we mentioned earlier, it is important to consider in what sense theories such as this can be viewed as effective low-energy theories, since they are not renormalizable. Notions such as decoupling will require careful consideration. Let us just point out that, in a model such as this one, our considerations in sect. 2 on the sliding singlet are almost certainly relevant. At the quantum level, one might expect terms in the U and Higgs potential of the form A4

C~pp IU - M I 2 + bA2(ISIH + h.c).

(5.3)

Such terms will certainly appear if the supersymmetry-breaking fields (i.e. the fields which couple to the Goldstone fermion at tree level) are charged, or can couple through chiral fields to the Higgs fields. In that case, we expect (U) ~ cM,

( H ) = (/~) = 0 ,

(5.4)

(H), (/--I) - A,

(5.5)

or

(U)~0,

depending on the numerical value of c and b. Either result is completely unacceptable. In the model of ref. [6], the supersymmetry-breaking fields have only gravitational couplings, so it may be difficult to make statements about the U potential. However it would be surprising if such terms did not appear. This question is currently under investigation. It thus appears that, in constructing theories with super-gravity coupled to matter, one may have to adopt strategies such as those considered in sect. 4, and balance gravitational and non-gravitational effects.

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6. Summary M u c h of the current interest in s u p e r s y m m e t r y stems f r o m a desire to solve the so-called hierarchy problem. This p r o b l e m is basically one of u n d e r s t a n d i n g why the scale of w e a k interactions is so m u c h smaller than the unification scale or the Planck mass. As we have seen, even in supersymmetric theories which g e n e r a t e a small mass scale, the scale of weak interactions m a y well end up at the Planck mass. S o m e care is required to obtain models in which in the low-energy theory, the b o t t o m does not fall out of the Higgs potential. Clearly this is a p r o b l e m which must be kept in mind when model-building, and in considering the role of supersymmerry in l o w - e n e r g y physics. W e thank Joe Polchinski for worrying us a b o u t these questions, and L e n n y Susskind for helpful conversations a b o u t decoupling.

Note added A f t e r completion of this paper, we learned that some of the points considered here have also been noted by other authors. T h a t the results of refs. [8, 9, 13, 14] extend to superparity theories has also been r e m a r k e d by A l v a r e z - G a u m e et al. [20], Nilles et al. [21], F e r r a r a et al. [22], and H e l a y e l - N e t o et al. [23]. T h e p r o b l e m of obtaining the correct breaking of S U ( 2 ) × U ( 1 ) t h r o u g h radiative effects in supergravity theories has been considered by A l v a r e z - G a u m 6 et al. [20], by Ibanez, by Ellis et al. [24] and by Dine [25]. T h e first three papers obtain the n e e d e d breaking of the left-right s y m m e t r y by d e m a n d i n g that the top q u a r k be extremely heavy. T h e last achieves the desired breaking, using additional fields at MGUT, as in the text.

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