Gaugino Condensation, Duality and Supersymmetry Breaking

Home , Gaugino

CERN-TH/95-308

hep{th/9511131

Gaugino Condensation, Duality and

Sup ersymmetry Breaking

Fernando Quevedo

Theory Division CERN

CH-1211 Geneva23

Switzerland.

e-mail:[email protected]

Abstract

The status of gaugino condensation in low-energy string theory is reviewed.

Emphasis is given to the determination of the efective action b elow conden-

sation scale in terms of the 2PI and Wilson actions. We illustrate how the

di erent p erturbative duality symmetries survive this simple nonp erturbative

phenomenon, providing evidence for the b elieve that these are exact nonp er-

turbative symmetries of string theory. Consistency with T duality lifts the

mo duli degeneracy. The B axion duality also survives in a nontrivial way

in which the degree of freedom corresp onding to B is replaced by a massive

H eld but duality is preserved. S dualitymay also b e implemented in

this pro cess. Some general problems of this mechanism are mentioned and

the p ossible nonp erturbative scenarios for sup ersymmetry breaking in string

theory are discussed.

CERN-TH/95-308

Novemb er 1995

Contribution to the Conference on S-duality and Mirror Symmetry,Trieste, June 1995 1

1 Intro duction

In the e orts to extract a relation b etween string theory and physics, we nd two

main problems, namely how the large vacuum degeneracy is lifted and how sup er-

symmetry is broken at low energies. These problems, when present at string tree

level, cannot b e solved at any order in string p erturbation theory. The reason is

the following: It is known that at tree-level, setting all the matter elds to zero

forces the sup erp otential to vanish, for any value of the mo duli and dilaton elds.

The corresp onding scalar p otential vanishes implying at directions for the mo duli

and dilaton. Also, the F and D auxiliary elds, which are the order parameters

for sup ersymmetry breaking, vanish in this situation, implying unbroken sup ersym-

metry. Since the sup erp otential do es not get renormalized in p erturbation theory,

if it vanish at tree level it will also vanish at all orders of string p erturbation the-

ory. Then the F-term part of the p otential also vanishes p erturbatively. The only

p erturbative correction that could alter this situation is the generation of a Fayet-

Iliop oulos D-term by an `anomalous' U (1), usually present in 4D strings. However,

in all the cases considered so far there are charged elds getting nonvanishing vev's

which cancel the D -term, breaking gauge symmetries instead of sup ersymmetry.

Therefore these problems are exact in p erturbation theory and the only hop e to

solve them is nonp erturbativephysics. This has a go o d and a bad side. The go o d

side is that nonp erturbative e ects represent the most natural way to generate large

hierarchies due to their exp onential suppression, this is precisely what is needed

to obtain the Weinb erg-Salam scale from the fundamental string or Planck scale.

The bad side is that despite many e orts, we do not yet have a nonp erturbative

formulation of string theory. At the moment, the only concrete nonp erturbative

information we can extract is from the purely eld theoretical nonp erturbative e ects

inside string theory. Probably the simplest and certainly the most studied of those

e ects is gaugino condensation in a hidden sector of the gauge group, since it has the

p otential of breaking sup ersymmetry as well as lifting some of the at directions, as

we will presently discuss.

2 Gaugino Condensation

The idea of breaking sup ersymmetry in a dynamical waywas rst presented in

refs. [1]. In those articles a general top ological argumentwas develop ed in terms of

the Witten index Tr() , showing that dynamical sup ersymmetry breaking cannot 2

be achieved unless there is chiral matter or we include sup ergravity e ects for which

the index argument do es not apply. This was subsequently veri ed by explicitly

studying gaugino condensation in pure sup ersymmetric Yang-Mills, a vector-like

theory, for which gauginos condense but do not break global sup ersymmetry [2] (for

a review see [3]). Breaking global sup ersymmetry with chiral matter was an op en

p ossibili ty in principle, but this approach ran into many problems when tried to b e

realized in practice.

The situation improved very much with the coupling to sup ergravity. The reason

was that simple gaugino condensation was argued to b e sucient to break sup er-

symmetry once the coupling to gravitywas included. This works in a hidden sector

mechanism where gravity is the messenger of sup ersymmetry breaking to the ob-

servable sector [4]. Furthermore, string theory provided a natural realization of

this mechanism [7, 6] byhaving naturally a hidden sector esp ecially in the E E

8 8

versions. Also, it gave another direction to the mechanism by the fact that gauge

couplings are eld dep endent (as anticipated for sup ergravity mo dels in ref. [5]).

This same fact raised the hop e that gaugino condensation could lift the mo duli and

dilaton at directions, but so on it was recognized that it only changed at to run-

away p otentials, thus destabilizing those elds in the `wrong' direction (zero gauge

coupling and in nite radius) .

A simple way to see this is by setting the gaugino condensate h i with

2 19

M exp(1=(bg )), the renormalization group invariant scale. Here M 10

Gev is the compacti cation scale, b the co ecient of the one-lo op b eta function of

the hidden sector group and g the corresp onding gauge coupling. In string theory

wehave that 4g hS+S iwhere S is the chiral dilaton eld (including also

the axion and fermionic partner). Also, M hT+Tiwith T b eing one of

the mo duli elds. Substituting naively h i into the lagrangia n induces a scalar

p otential for the real parts of S and T (S and T resp ectively), namely V (S ;T )

R R R R

exp(3S =4b). This p otential has a runaway b ehaviour for b oth S and T ,

R R R

S T

as advertized.

The T dep endence of the p otential was completely changed after the consider-

ation of target space or T duality. In its simplest form, this symmetry acts on the

eld T as an SL(2; Z) symmetry:

aT ib

T ! ; ad bc =1: (1)

icT + d

The p ossibility of a nonvanishing hH i stabilizing the p otential with vanishing cosmological

ij k

constant [6], was discarded after it was realized that this eld was always quantized, breaking

sup ersymmetry at the Planck scale, also its incorp oration do es not seem consistent with T -duality. 3

It was shown [8], that imp osing this symmetry changes the structure of the scalar

p otential for the mo duli elds in suchaway that it develops a minimum at T 1:2

(in string units), whereas the p otential blows-up at the decompacti cation limit

(T !1), as desired. The mo di cations due to imp osing T duality can b e traced

to the fact that the gauge couplings get mo duli dep endent threshold corrections

from lo ops of heavy string states [9]. This in turn generates a mo duli dep endence

on the sup erp otential induced by gaugino condensation of the form W (S; T )

(T ) exp(3S=8b) with (T ) the Dedekind function.

This mechanism however did not help in changing the runaway b ehaviour of

the p otential in the direction of S . For stabilizing S , the only prop osal was to

consider gaugino condensation of a nonsemisimple gauge group, inducing a sum of

exp onentials in the sup erp otential W (S ) exp(3S=8b ) which conspire to

i i

generate a lo cal minimum for S [10]. These have b een named `racetrack' mo dels in

the recent literature.

It was later found that combining the previous ideas together with the addition

of matter elds in the hidden sector (natural in many string mo dels)[11 , 12 ], was

sucient to nd a minimum with almost all the right prop erties, namely, S and T

xed at the desired value, S 25;T 1, sup ersymmetry broken at a small scale

R R

( 10 GeV) in the observable sector, etc. This lead to studies of the induced soft

breaking terms at low energies.

Besides that relative succes, there are at least ve problems that assures us that

we are far from a satisfactory treatment of these issues.

(i) Unlike the case for T , xing the vev of the dilaton eld S , at the phenomenologi-

cally interesting value, is not achieved in a satisfactory way. The conspiracy of

several condensates with hidden matter to generate a lo cal minimum at a go o d

value, requires certain amount of ne tunning and cannot b e called natural.

(ii) The cosmological constant turns out to b e always negative, which lo oks like

an unsourmountable problem at present. This also makes the analysis of soft

breaking terms less reliable, b ecause in order to talk ab out them, a constant

piece has to b e added to the lagrangian that cancels the cosmological constant.

It is then hard to b elieve that the unknown mechanism generating this term

would leave the results on soft breaking terms (such as small gaugino masses)

untouched.

(iii) The derivation of the e ective theory b elow condensation is not completely

understo o d. There are several approaches to this and the exact relation among 4

them is not completely clear.

(iv) There is an inherently stringy problem which is due to the fact that the S eld

is not stringy. S is only the dual of another eld, L which is the one created by

string vertex op erators, having the dilaton and the antisymmetric tensor eld

B (instead of the axion) as the b osonic comp onents. The problem resides in

the fact that, if there is not a Peccei-Quinn (PQ) symmetry S ! S + i constant,

as in the many condensates scenario, it is not clear if the theory in terms of S

is any longer dual to the L theory. This sets serious doubts on whether the S

approach mentioned ab oveisvalid at all. Another way to express this problem

is to ask if it is p ossible to formulate directly gaugino condensation in terms

of the stringy eld L.

(v) Finally,even if the previous problems were solved, there are at least two serious

cosmological problems for the gaugino condensation scenario. First, it was

found under very general grounds, that it was not p ossible to get in ation

with the typ e of dilaton p otentials obtained from gaugino condensation [13 ].

Second is the so-called `cosmological mo duli problem' which applies to any

(nonrenormalizble) hidden sector scenario including gaugino condensation [14 ].

In this case, it can b e shown that the mo duli and dilaton elds acquire masses

of the electroweak scale ( 10 GeV) after sup ersymmetry breaking. Therefore

if stable, they overclose the universe, if unstable, they destroynucleosynthesis

by their late decay, since they only have gravitational strength interactions.

In the next section, I will present a general description of the e ective theory

b elow condensation scale, addressing the issue of problem (iii) ab ove. Section 4 will

show the solution of problem (iv) whereas in section 5, I will discuss ideas towards

solving problems (i) and (v). The resolution of problem (ii) is left to the reader.

3 Wilson vs 2PI Actions

To study the e ects of gaugino condensation we should b e able to answer the fol-

lowing questions: Do gauginos condense? If so, is sup ersymmetry broken by this

e ect? What is the e ective theory b elow the scale of condensation? In order to

answer these questions, several ideas have b een put forward [2, 5 , 6, 15]. Let me

revise brie y the di erent approaches.

In ref. [2], a chiral sup er eld U was intro duced representing the condensate

W W . The e ective sup ersymmetric theory in terms of U was found by matching

the anomaly of an original R-symmetry of the underlying sup ersymmetric Yang-

Mills action.

In refs. [6], [16 ], the same anomalous symmetry was used to repro duce the ef-

fective action b elow condensation scale, without the need of intro ducing U . That

gave rise to the sup erp otential W (S ) exp(3S=8b) mentioned b efore. The ear-

lier approach of ref. [5] was based on the direct substitution of in the original

sup ergravity lagrangian. A more recent analysis of ref. [15 ], uses a Nambu-Jona-

Laisinio approach to describ e the condensation mechanism.

Even though some of these approaches gave similar results, there are imp ortant

di erences among them. In particular, following ref. [5], since they substitute

directly into the sup ersymmetric action in comp onents, the e ective lagrangian is

not explicitly sup ersymmetric unlike for instance the results of ref. [6]. Also,

the approach of [15], even though it repro duces the results in [2] at tree-level, by

including quantum corrections, they nd very di erent results, for instance, the

dilaton could b e stabilized with a single condensing group. Finally the formalisms

of [2] and [6] have b een compared in [11, 17 ]. They eliminate the eld U by assuming

it do es not break global sup ersymmetry, ie by using @ W =@ U = 0 and nd agreement

between the two metho ds. However this condition should not b e imp osed b eforehand

and it is not well justi ed in the sup ergravity case.

We can see there is no satisfactory understanding of the e ective theory b elow

condensation. Furthermore, the anomalous symmetry argument which is the most

solid description of the single condensing case, cannot b e used for the interesting

case of several condensing groups.

We will now present a self contained discussion which will at the end identify

the main approaches with known eld theory quantities, ie the 2PI and Wilsonian

e ective actions [18], and mention how these two approaches are actually related in

a consistent manner.

3.1 Sup ergravity Basics

Since the elds S and T are exp ected to havevery large vev's, it is more convenient

to work with lo cal sup ersymmetry without taking the Planck scale to 1. The most

general action for chiral matter sup ermultiplets coupled to sup ergravity can b e

These two approaches were shown to b e equivalent in ref. [17], once the sup erconformal struc-

ture of the original sup ergravity action is considered in detail, giving rise an explicit sup ersymmetric

action as in [6] 6

written as [19]:

4 K (; )=3

I = d x [S S e ] + (2)

0 D

3 a b

[S W ()] +[ f ()W W ] +cc

F ab F

where the Kahler p otential K (; ), the sup erp otential W () and the gauge kinetic

function f () de ne a particular theory. The eld S is an extra chiral sup er eld

ab 0

called `the comp ensator'. Its existence is due to the fact that action (2) is not

only invariant under sup er Poincare symmetries but under the full sup erconformal

symmetry. This simpli es the treatment of the theory in particular the calculation

of the action in comp onents. Sup er Poincare sup ergravity is easily obtained by

explicitly xing the eld S to a particular value, it is usually chosen in suchaway

that the co ecient of the Einstein term in the action is just Newton's constant.

Two symmetries of the sup erconformal algebra have a particular imp ortance for

us: Weyl and chiral U (1) transformations. These two symmetries do not commute

with sup ersymmetry. The chiral U (1) group is at the origin of the R-symmetry

of Poincare theories. Weyl and chiral transformations with parameters and

w +in =2

j j

resp ectively, act on comp onent elds with a factor e , w and n b eing the

j j

Weyl and chiral weights of the comp onent eld. For a left-handed chiral multiplet

(z; ; f ), one nds the following weights:

z : w; n = w;

; n ; : w +

f : w +1; n3: (3)

Chiral matter multiplets have w = n = 0, except for S which has w = n =1.

The chiral multiplet of gauge eld strength W has w = n =3=2. The U (1)

transformations of (left-handed) gauginos and chiral fermions are therefore:

a 3i =4 a 3i =4

! e ; ! e : (4)

These transformations generate a gauge-chiral U (1) mixed anomaly. This anomaly

can b e cancelled by the `Green-Schwarz' counterterm [20, 17 ] :

TrW W log S ] +cc : (5) d x [ I = c

0 F

where c = [C (G) C (R )] ; (C here represents the Casimir of the represen-

tation, for the case without matter wehave that c =8b). This counterterm is 7

claimed to cancel the anomaly to all orders in p erturbation theory [17] and plays an

imp ortant role in what follows.

The action (12) has also a symmetry under Kahler transformations: K ! K +

'()

'() + ' ( );W !e W since any such a transformation can b e absorb ed by

'=3

rede ning S : S ! e S .

0 0 0

3.2 The Wilson E ective Action

Let us now restrict to a simple case that has all the prop erties we need to discuss

gaugino condensation, ie a single chiral multiplet S coupled to sup ergravity and a

nonab elian gauge group with K = K (S + S ) arbitrary, W (S ) = 0 and f (S )=S.

This is the case for the dilaton in string theory at the p erturbative level. This de nes

the e ective (Wilson) action at scales M E . We are interested in the Wilson

action at scales E 10 GeV in whichwe exp ect that gauginos have condensed

and S is the only degree of freedom, that means wewanttointegrate out the full

gauge sup ermultiplet to obtain the e ective action for S at low energies. This is

precisely the approach of ref. [6] mentioned ab ove. We need to compute:

Z Z

i(S;S ) 4

e DV exp i d x f[(S c log S )

Tr W W ] +ccg (6)

First of all we can observe that (S; S ) dep ends on its arguments only through

the combination S exp(S=c). Second, since the result of the integration has

to b e sup erconformal invariant (b ecause the anomaly is cancelled), we know that

[S exp(S=c)] has to b e written in the form of equation (2) (plus higher derivative

terms) with f = 0 since there are no gauge elds. Since the p owers of S are exactly

given by (2) and S only app ears multiplying exp(S=c)we can just read the sup er

and Kahler p otentials to b e:

3S=c

W (S ) = we

K=3 K =3 (S+S )=c

e = e ke (7)

where w and k are arbitrary constants (k>0 to assure p ositive kinetic energy). The

sup erp otential is just the one found in [6]. The correction to the Kahler p otential

is new [18 ]. Notice that b oth are corrections of order exp 1=g as exp ected. A

word of caution is in order. Unlike the sup erp otential which has no corrections in

p erturbation theory, the Kahler p otential can b e corrected order by order in p ertur-

bation theory, therefore in practice the p erturbative part of the Kahler p otential K

p 8

is simply unknown and for weak coupling those corrections are bigger than the non-

p erturbative correction found here. Our result could b e useful, only after the exact

p erturbativeKahler p otential is known. It is still interesting to realize that sucha

simple symmetry argument can give us the exact expressions for the nonperturbative

sup er and Kahler p otentials, without the need of holomorphy!

3.3 The 2PI E ective Action

To answer the questions p osed at the b eginning of this chapter, ie whether gaugi-

nos condense and break sup ersymmetry, it is convenient to think ab out the case of

sp ontaneous breaking of gauge symmetries. In that case we minimize the e ective

p otential for a Higgs eld, obtained from the 1PI e ective action and see if the mini-

mum breaks or not the corresp onding gauge symmetry. In our case, we are interested

in the exp ectation value of a comp osite eld, namely or its sup ersymmetric

expression W W . Therefore we need the so-called two particle irreducible e ective

action.

We start then with the generating functional in the presence of an external

current J coupled to the op erator that wewant the exp ectation value of, namely,

W W :

Z Z

iW [S;S ;J ] 4

e DV exp i d x f[(S c log S

+J )TrW W ] +ccg (8)

From this wehave

= hW W iU (9)

and de ne the 2PI action as

^ ^

[S; S ; U ] W d x UJ (10)

To nd the explicit form of we use the fact that W dep ends on its three arguments

only thorugh the combination S + J c log S , therefore, we can see that = (S

c log S )== J = U .Integrating this equation determines the dep endence of

in S and S :

^ ^ ^

[S; S ; U ]= U(S clog S )+(U) (11)

0 0

where (U ) can b e determined using symmetry arguments as follows. First we

de ne a chiral sup er eld U by U S U . Therefore U is a standard chiral sup er eld

with vanishing chiral and conformal weight(w =n= 0). Then [S; S ;US ] can

b e writeen in the form (2) with chiral elds S and U . Again the fact that the S

0 9

dep endence of (2) is very restricted, allows us to just read again the corresp onding

Kahler and sup erp otential. We nd:

log U + ] W [S; U ] = U [S +

1=3

K=3 K =3

e = e a (UU ) (12)

Here is an arbitrary constant. We can see that the sup erp otential corresp onds to

the one found in [2]. The Kahler p otential is new, in [2] it was found for the global

case, to which this reduces in the global limit.

Notice that wehave identi ed the two main approaches to gaugino condensation

with the two relevant actions in eld theory, namely the Wilson and 2PI e ective

actions. Our approach to the 2PI action is a reinterpretation of the one in [2]. We

have to stress that in our treatment U is only a classical eld, not to b e integrated

out in any path integral. It also do es not make sense to consider lo op corrections

to its p otential, this solves the question raised in [15 ] where lo op corrections to the

U p otential could change the tree level results. Furthermore, since U is classical we

can eliminate it by just solving its eld equations:@ =@ U = 0. (Since this implies

J =0,itmakes equations (11) and (9) reduce to (7).) These equations cannot

b e solved explicitly but we nd the solution in an 1= expansion. We nd that

the solution of these equations repro duce the Wilson action derived in the previous

subsection (obtaining b oth W (S ) and K (S + S ) as in equation (9)) plus extra terms

suppressed byinverse p owers of the condensation scale. This shows explicitly the

relation b etween the two approaches.

We can also consider the case of several condensates. This case shows the p ower

of the techniques used previously. Following the original discussions of [6]itwas

needed to use the PQ symmetry of S to cancel the U (1) anomaly,however when

there are several condensing groups wewould have neede several S elds to cancel

the anomaly (see [18]) but there is only one S eld in string theory. In our approach

however, we use the counterterm (5) which in the case of several groups is a sum

of terms [17]. Therefore wehave one counterterm for each group and so the path

integrals just factorize into pro ducts for eachofthemany condensates, implying

K=3

that the total sup erp otential (W ) and e functions are the sum of the ones for

one single condensate. This is the rst real derivation of this well used result!

By studying the e ective p otential for U we recover the previously known results.

For one condensate and eld indep endent gauge couplings (no eld S ) the gauginos

condense (U 6= 0) but sup ersymmetry is unbroken. For eld dep endendt gauge

coupling, the minimum is for U =0 (S !1) so gauginos do not condense (this 10

is re ected in the runaway b ehaviour of the Wilsonian action for S ). For several

condensing groups we nd U 6= 0 and sup ersymmetry broken or not, dep ending on

the case [12 ].

4 Linear vs Chiral Formalisms

Here we rep ort on the resolution of question (iv) of section 2 [21] : p erturbative

4D string theory has in its sp ectrum a two-index antisymmetric tensor eld B .

Because it only has derivative couplings, B is dual to a pseudoscalar eld, the axion

a.We can transform back and forth from the B and a formulations as long as

the corresp onding shift symmetries are preserved. It is known that nonp erturbative

e ects break the PQ symmetry of a giving it a mass, then the puzzle is: what

happ ens to the stringy B eld in the presence of non-p erturbative e ects? Is the

duality symmetry also broken by those e ects? Is it then correct to forget ab out

the B eld, as it is usually done, and work only with a? (Since, unlike the axion,

B is the eld created by string vertex op erators). The answer to these questions

is very interesting: duality symmetry is not broken by the nonp erturbative e ects

but the B eld disapp ears from the propagating sp ectrum! Its place is taken bya

massive 3-index antisymmetric tensor eld H dual to the massive axion.

Here I will just sketch the main steps of the derivation and refer the reader

to [21 ] for further details. In 4D strings, the antisymmetric tensor b elongs to a

DD L = 0), together with the dilaton and the dilatino. For linear sup er eld L (

simplicitywe only consider the couplings of this eld to gauge sup er elds in global

sup ersymmetry (the sup ergravity extension is straightforward), the most general

^ ^

action is then the D -term of an arbitrary function , L = [(L)] , with L L

L D

and the Chern Simons sup er eld, satisfying DD =W W .

Since the gauginos app ear in the lagrangian through the arbitrary function ,

the analysis of gaugino condensation is far more complicated in the linear case than

in the chiral case. Furthermore, the Wilson action is not well de ned in this case,

b ecause the eld L is not gauge invariant, we cannot just integrate the gauge elds

out leaving an e ective action for L alone as we did for S . Therefore wehaveto

consider the 2PI action, and to nd it, wehavetowork in the rst order formalism

where the gauge elds app ear only through Tr W W as in the S case. This will also

allow us to p erform a duality transformation and show that the L and S approaches

are equivalent.

The duality transformation is obtained by starting with the rst order system 11

coupled to the external current J :

Z Z

iW (J ) 4

e = DV DS DY exp i d x (

L(Y; S)+2<[J TrW W ] ) (13)

Where V is the gauge sup er eld, Y an arbitrary vector sup er eld with the la-

DD (Y + )g , and S (the same S of of the grangian L(Y; S)= f(Y )g + fS

F D

previous section!) starting life as a Lagrange multiplier chiral sup er eld.

Integrating out S , implies DD (Y + )= 0 or Y = L L, giving back the

original theory. On the other hand integrating rst Y gives the dual theory in terms

of S and V . This is the situation ab ove the condensation scale. Below condensation,

however, wehavetointegrate rst the gauge elds, after that wehave the same two

options for getting the two dual theories, the di erence now is that the integration

over V breaks the PQ symmetry (if there are at least two condensing gauge groups)

and we are left with a duality without global symmetries.

To see this, we will concentrate on the 2PI e ective action (U; Y ; S ) obtained

in the standard way for U hTrW W i [18]. The imp ortant result is that since

W dep ends on S and J only through the combination S + J ,we can see as in

eq. (12) that (U; S; Y )= US +(U; Y ), where (U; Y ) is arbitrary, therefore

S app ears only linearly in the path integral and its integration gives again a -

DD Y = U instead of the constraint DD (Y + )=0 function, but imp osing now

ab ove condensation scale. We can then see that there is no linear multiplet implied

by this new constraint. This is an indication that the B eld is no longer in the

sp ectrum.

The new propagating b osonic degrees of freedom in Y are, a scalar comp onent,

the dilaton, b ecoming massive after gaugino condensation and a vector eld v dual

to a, the pseudoscalar comp onentofS. Instead of showing the details of this duality

in comp onents, I will describ e the following slightly simpli ed toy mo del which has

all the relevant prop erties:

2 2

L = v v a@ v m a

v ;a

If we solve for v we obtain v = @ a, substituting backwe nd

2 2

L = @ a@ a m a

describing the massive scalar a. On the other hand, solving for a we get a =

(@ v ) which gives

1 1

0 2

L = v v + (@ v ) :

2 4m 12

The lagrangian L also describ es a massive scalar given by the longitudinal, spin

zero, comp onentof v .We can see that the only comp onent that has time derivatives

is v , so the other three are auxiliary elds. Notice that for m =0,we recover the

standard duality among a massless axion and B eld. Therefore, after the gaugino

condensation pro cess, the original B eld of the linear multiplet is pro jected out

of the sp ectrum in favour of a massive scalar eld corresp onding to the longitudinal

comp onentofv or to the transverse comp onent of the antisymmetric tensor H

v .Thus solving the puzzle of the axion mass in the two dual formulations.

Other interesting discussions of gaugino condensation in the linear formalism can

b e found in [22 ].

5 Scenarios for SUSY Breaking

The results of the previous sections have shown us that the general results extracted

in the past years ab out gaugino condensation in string mo dels, in terms of the eld

S , are robust. Wehave seen how gaugino condensation can in principle lift the string

vacuum degeneracy and break sup ersymmetry at low energies (mo dulo de problems

mentioned b efore). But this is a very particular eld theoretical mechanism and it

would b e surprising that other nonp erturbative e ects at the Planck scale could b e

completely irrelevant for these issues. In general we should always consider the two

typ es of nonp erturbative e ects:stringy (at the Planck scale) and eld theoretical

(like gaugino condensation). Four di erent scenarios can b e considered dep ending

on which class of mechanism solves each of the two problems:lifting the vacuum

degeneracy and breaking sup ersymmetry.

For breaking sup ersymmetry at low energies, we exp ect that a eld theoretical

e ect should b e dominant in order to generate the hierarchy of scales. We are then

left with two preferred scenarios: either the dominant nonp erturbative e ects are

eld theoretical, solving b oth problems simultaneously, or there is a `two steps' sce-

nario in which stringy e ects dominate to lift vacuum degeneracy and eld theory

e ects dominate to break sup ersymmetry. The rst scenario has b een the only one

considered so far, the main reason is that we can control eld theoretical nonp er-

turbative e ects but not the stringy. In this scenario, indep endent of the particular

mechanism, wehave to face the cosmological mo duli problem.

In the two steps scenario the dilaton and mo duli elds are xed at high energies

with a mass M thus avoiding the cosmological mo duli problem. It is also

P lanck

reasonable to exp ect that Planck scale e ects can generate a p otential for S and T . 13

The problem resides in the implementaion of this scenario [23], mainly due to our

ignorance of nonp erturbative string e ects.

5.1 S Duality

To approach nonp erturbative string e ects wemay use the conjectured SL(2;Z)

S-dualityinN = 1 e ective lagrangian s [24] :

aS ib

; ad bc =1: (14) S !

icS + d

Even though there is mounting evidence for this symmetry in N =4;2 string back-

grounds, it is not yet clear how it will b e extended to N = 1 and if so most probably

the lagrangian is not invariant under this symmetry since it usually exchanges `elec-

tric' and `magnetic' degrees of freedom. However, similar to the case of T duality,if

we restrict to the part of the action that dep ends only on S , (which is the relevant

part when lo oking for vacuum con gurations) this is exp ected to b e invariant under

S duality. Recall that if we do the same for the classical action, the continuous

SL(2;R) transformation is a symmetry of the truncated action, so the argument

that quantum e ects break the continuous to the discrete S duality could actually

make sense in this case. As found in ref. [24], the sup erp otential should b e a mo dular

form of weight 1 and can b e written as:

W (S )=(S) Q[j(S)] (15)

where Q is an arbitrary rational function of the absolute mo dular invariant function

j (S ). Its arbitrariness forbids us to extract concrete conclusions, but there are

several general issues worth mentioning. Since the weightofW(S) is negative, it

necessarily has p oles [25 ]. If we further imp ose that the scalar p otential has to

vanish at S !1 (zero string coupling)[27 ] there should b e p oles at nite values

of S whichmay need interpretation. The functions (S ) and j (S ) can b e expressed

as in nite sums of q e ,thus encompassing the exp ected nonp erturbative

instanton-like expansion. The selfdual p oints S =1;exp i =6 are always extrema

of the p otential and very often are minima. For those p oints sup ersymmetry is

unbroken, thus making the two steps scenario very plausible at least for the S eld.

This way of xing the vev of S is much more elegant than the racetrack scenario

with several condensing gauge groups. It is similar to the waywe understo o d the

xing of T . A general question to b e addressed to this scenario is that usually the

vev of S is very close to S 1 b ecause the nontrivial structure of the p otentials is

always close to the selfdual p oints. This is far from the phenomenologicall y required 14

value where wewant4=g 25. However, as emphasized in [26 ] the gauge coupling

is S only at tree level, it is exp ected to get nonp erturbative corrections and wemay

have a situation with S = 1 but with a larger value of f (S ) at the minimum leading

to the desired gauge coupling at the string scale.

Let us mention as an aside that the gaugino condensation pro cess can b e made

consistent with S -duality [27 , 26 , 28 ]. A way to do it is to write the gaugino con-

as the rst term in an in nite expansion densation sup erp otential W exp

of the form (15). Another approach is to try to derive the e ective sup erp otential

from nonp erturbative corrections to the gauge kinetic function f (S ). The problem

with this approach is that we do not knowhowf(S) should transform under S

duality(we cannot forget the gauge elds as we did for nding W (S )). In ref. [26 ],

it was assumed that f is invariant, but then the gaugino condensation-induced su-

would also b e invariant instead of a weight 1 form as p erp otential W exp

required by S -duality. An extra factor (S ) Q[j (S )] has to b e put in by hand

without justi cation, losing the connection with the condensation pro cess.

A probably b etter way to derivean S dualityinvariant e ective theory after

gaugino condensation, may b e to assume a noninvariant f (S ) [29], after all that is

precisely what happ ens in T duality for which f (T ) log (T ). If for instance we

take,

o n

(C 12)=24C

(16) log (S )(j(S) 744) f (S )=

nonp erturbatively (here C is the Casimir of the corresp onding gauge group, see dis-

cussion b elow equation (5)), we can see that it has the right limit for large S (ie

f ! S ) and induces a gaugino condensation sup erp otential W (S ) (S ) (j (S )

(12C )=12C

744) which has the right transformation prop erties under S duality and

reduces to the gaugino condensation sup erp otential in the large S limit. The non-

invariance of f (S )may probably b e related with S -duality anomalies [29 ] as it

happ ened in the T duality case. A problem with this approach is that if we are

considering nonp erturbative corrections to the f function, we should also include

those corrections for W and K . This may diminish the imp ortance of the gaug-

ino condensation-induced sup erp otential ab ove, b ecause it would b e just an extra

contribution to the original nonp erturbative sup erp otential whichwe do not know.

There may still b e situations, as argued in [30], for which gaugino condensation

sup erp otentials could nevertheless b e dominant. 15

5.2 Two Steps Scenario

In the two steps scenario, after wehave xed the vev of the mo duli by stringy e ects,

it remains the question of how sup ersymmetry is broken at low energies. Notice that

wewould b e left with the situation present b efore the advent of string theory in which

the gauge coupling is eld independent. In that case we know from Witten's index

that gaugino condensation cannot break global sup ersymmetry. Since there are

no `mo duli' elds with large vev's, the sup ergravity correction should b e negligible

b ecause we are working at energies much smaller than M .

P lanck

In fact we can p erform a calculation by setting S to a constant in eq. (12), it

is straightforward to show that sup ersymmetry is still unbroken in that case [23 ],

as exp ected. A more general way to see this is computing explicitly the 1=M

P lanck

correction to a global sup ersymmetric solution W = 0, and see that it coincides

with the solution of W + WK =M = 0 whichisalways a sup ersymmetric extremum

of the sup ergravity scalar p otential.

As mentioned in section 2, there seems to b e however a counterexample in the

literature. In ref. [4] a mo di cation of the Kahler p otential (12) was considered:

1=3

K=3

e =1a (UU ) b (UU ) (17)

with the same sup erp otential. For a = 9b sup ersymmetry was found to b e broken

with vanishing cosmological constant. But also for this choice of parameters the

global limit is such that K vanishes, and so the kinetic energy for U . This

makes the corresp onding minimum in the global case ill de ned, since there may

b e other nonconstant eld con gurations with vanishing energy. This is then not a

counterexample, b ecause the global theory is not well de ned in the minimum. In

any case, in our general analysis, there are no such extra corrections to the Kahler

p otential for U .

We are then left with a situation that if global sup ersymmetry is unbroken, we

cannot break lo cal sup ersymmetry, unless there are mo duli like elds. This can

bring us further back to the past and reconsider mo dels with dynamical breaking of

global sup ersymmetry (for a recent discussion with new insights see [31] ).

6 Conclusions

(i) Gaugino condensation provides a simple example of how sup ersymmetry can

b e broken dynamically with partial succes. Some of the problems maybe 16

solved after having b etter control of the sup ergravity lagrangian . In partic-

ular, in the single hidden sector group case wehave seen that the gauginos

do not condense, but this situation maybechanged after p erturbative and

nonp erturbative corrections to the Kahler p otential are considered [30 ]. The

cosmological problems may b e more generic, however.

(ii) The gaugino condensation pro cess is also an interesting lab oratory to test non-

p erturbative prop erties of string and eld theories. In particular duality sym-

metries survive this simple, but nontrivial, nonp erturbative test.

(iii) The di erent approaches to describ e the e ective theory underlying the con-

densation pro cess corresp ond simply to the use of the Wilson or 2PI e ective

actions, therefore there is a well de ned relation among them. Even though

the Wilson action is usually simpler to work with, the 2PI action is more suit-

able to follow the condensation pro cess, it also is the only one that could b e

used to describ e the condensation of gauginos in the `linear formalism'. The

Wilson action cannot b e used without previously identifying the low energy

degrees of freedom. We needed the 2PI action to nd out that the axion degree

of freedom is represented by a massive H tensor.

(iv) The linear and chiral descriptions are equivalent, even in the absence of PQ

symmetries. Which formulation is more convenient dep ends on the situation.

In the linear description, the stringy B eld is replaced by the massive H

eld. We b elieve, this will also b e the case in more general nonp erturbative

e ects. Wemay conjecture that this result could b e related with the claims

that `stringy' nonp erturbative e ects are not well describ ed by strings but

b etter by membranes, which couple naturally to H or ve-branes, which

provide the 10D origin of the eld S . A (massless) eld H also app ears

naturally in 11D sup ergravity.

(v) There is not a comp elling scenario for sup ersymmetry breaking and the eld re-

mains op en, but wehavea much b etter p ersp ective on the relevant issues now.

The nonrenormalizable hidden sector mo dels of which the gaugino condensa-

tion is a particular case, may need a convincing solution of the cosmological

mo duli problem to still b e considered viable. Hop efully, this will lead to in-

teresting feedbackbetween cosmology and string theory [32]. Furthermore,

the recent progress in understanding sup ersymmetric gauge theories can b e

of much use for reconsidering gaugino condensation with hidden matter, the 17

discussion in the string literature is far from complete. The understanding

of mo dels with chiral matter could also provide new insights to global sup er-

symmetry breaking, relevant to the two steps scenario mentioned ab ove. In

any case the techniques found to b e useful in the simplest gaugino condensa-

tion approach discussed here, will certainly help in understanding those more

complicated mo dels.

I thank the organizers for the invitation to participate in such an exciting con-

ference.

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