Dynamical supersymmetry breaking

40 Pages·2000·0.378 MB·English

by  Shadmi, Shirman.

Dynamical supersymmetry breaking
Yael Shadmi
DepartmentofParticlePhysics,WeizmannInstituteofScience,Rehovot76100,Israel
andPhysicsDepartment,PrincetonUniversity,Princeton,NewJersey08544
Yuri Shirman
PhysicsDepartment,PrincetonUniversity,Princeton,NewJersey08544
Supersymmetryisoneofthemostplausibleandtheoreticallymotivatedframeworksforextendingthe
standard model. However, any supersymmetry in Nature must be a broken symmetry. Dynamical
supersymmetrybreaking(DSB)isanattractiveideaforincorporatingsupersymmetryintoasuccessful
descriptionofNature.ThestudyofDSBhasrecentlyenjoyeddramaticprogress,fueledbyadvances
inourunderstandingofthedynamicsofsupersymmetricfieldtheories.Theseadvanceshaveallowed
fordirectanalysisofDSBinstronglycoupledtheories,andforthediscoveryofnewDSBtheories,
some of which contradict early criteria for DSB. The authors review these criteria, emphasizing
recently discovered exceptions. They also describe, through many examples, various techniques for
directlyestablishingDSBbystudyingtheinfraredtheory,includingbotholdertechniquesinregions
of weak coupling and new techniques in regions of strong coupling. Finally, they present a list of
representativeDSBmodels,theirmainproperties,andtherelationsamongthem.
CONTENTS Appendix:SomeResultsonSUSYGaugeTheories 57
1.NotationsandsuperspaceLagrangian 57
2.D-flatdirections 59
I. Introduction 25 3.PuresupersymmetricSU(N ) theory 59
c
II. Generalities 28 4.N,N : Affleck-Dine-Seibergsuperpotential 59
f c
A. Vacuumenergy—theorderparameterofSUSY 5.N5N : quantummodulispace 60
f c
breaking,andFandDflatness 28 6.N5N 11 61
f c
B. TheGoldstino 30 7.N.N 11: dualdescriptionsoftheinfrared
f c
C. Tree-levelbreaking 31 physics 61
1.O’Raifeartaighmodels 31 8. Othermodels 38
2.Fayet-Iliopoulosbreaking 32 References 62
III. IndirectCriteriaforDSB 33
A. TheWittenindex 33
B. Globalsymmetriesandsupersymmetrybreaking 34
I. INTRODUCTION
C. Gauginocondensation 36
D. Examples 36
Supersymmetry (SUSY), which rotates bosons into
1.Spontaneouslybrokenglobalsymmetry:the
fermions and vice versa, is a beautiful theoretical idea.
SU(5) model 36
But Nature is certainly not supersymmetric. If it were,
2.Gauginocondensation:theSU(5) model 37
3.RsymmetryandSUSYbreaking 37 wewouldseeafermionicpartnerforeachknowngauge
4.GeneralizationsoftheSU(5) model 37 boson, and a scalar partner for each known fermion,
IV. DirectAnalysis:CalculableModels 38 withdegeneratemasses.Butexperimentalistshavebeen
A. The3–2model 38 lookingfor‘‘superpartners’’longandhard,andsofarin
B. The4–1model 41 vain, pushing the limits on superpartner masses to
V. DirectAnalysis:StronglyCoupledTheories 41 roughlyabovea100GeV.Thusanydiscussionofsuper-
A. Supersymmetrybreakingthroughconfinement 41
symmetryinNatureisnecessarilyadiscussionofbroken
B. Establishingsupersymmetrybreakingthrougha
supersymmetry.
dualtheory 43
Still, even broken supersymmetry is theoretically
C. Integratingmatterinandout 45
more appealing than no supersymmetry at all. First, su-
VI. ViolationsofIndirectCriteriaforDSB 46
A. Nonchiralmodels 46 persymmetry provides a solution to the gauge hierarchy
1.SUSYQCDwithsinglets 46 problem.Withoutsupersymmetry,thescalarHiggsmass
2.TheIntriligator-Thomas-Izawa-Yanagida is quadratically divergent, so that the natural scale for it
model 47 is the fundamental scale of the theory, e.g., the Planck
B. Quantumremovalofflatdirections 49 scale, many orders of magnitude above the electroweak
1.ASUSY-breakingreducedtheory 50 scale.Inasupersymmetrictheory,themassofthescalar
2.Asupersymmetricreducedtheory 50
Higgs is tied to the mass of its fermionic superpartner.
C. SupersymmetrybreakingwithnoRsymmetry 51
Since fermion masses are protected by chiral symme-
VII. DSBModelsandModel-BuildingTools 52
tries, the Higgs mass can naturally be around the elec-
A. Discardedgeneratormodels 52
troweak scale, and radiative corrections do not destabi-
B. Supersymmetrybreakingfromananomalous
lizethishierarchy.Thissuccessisnotspoiledevenwhen
U(1) 54
C. Listofmodelsandliteratureguide 54 explicit supersymmetry-breaking terms are added to the
Acknowledgments 57 Lagrangian of the theory, as long as these terms are
ReviewsofModernPhysics,Vol.72,No.1,January2000 0034-6861/2000/72(1)/25(40)/$23.00 ©2000TheAmericanPhysicalSociety 25
26 Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking
‘‘soft,’’ that is, they introduce only logarithmic diver- model. Moreover, theories with N.1 supersymmetry
gences,butnoquadraticdivergences,intoscalarmasses. arebelievedtohaveanexactmodulispaceandthusare
The appearance of explicit supersymmetry-breaking not expected to exhibit dynamical supersymmetry
termsinthelow-energyeffectivetheorycanbetheoreti- breaking.
callyjustifiediftheunderlyingtheoryissupersymmetric, Therestrictiontoglobalsupersymmetrystillallowsus
yet the vacuum state breaks supersymmetry spontane- to answer most of the questions we would be interested
ously. in. This situation is quite analogous to studying the
While spontaneously broken supersymmetry would
breakingofagaugedbosonicsymmetryin,say,atheory
explainthestabilityofthegaugehierarchyagainstradia-
with scalar matter. In that case we can determine the
tivecorrections,itstilldoesnotexplaintheoriginofthe
patternofsymmetrybreakingjustbystudyingthescalar
hierarchy, that is, the origin of the small mass ratios in
potential. Similarly, we shall be able to determine
thetheory.Indeed,ifsupersymmetrywerebrokenatthe
whether supersymmetry is broken and, if the theory is
classical level (tree level), the scale of the soft terms
weakly coupled, what the vacuum energy and the light
would be determined by explicit mass parameters in the
spectrum are. From our perspective, the most relevant
supersymmetric Lagrangian, and one would still have to
consequence of ‘‘gauging’’ supersymmetry is the ana-
understand why such parameters are so much smaller
logue of the Higgs mechanism by which the massless
than the Planck scale. However, the origin of the hier-
fermion accompanying supersymmetry breaking, the
archycanbeunderstoodifsupersymmetryisbrokendy-
Goldstino, is eaten by the gravitino.
namically (Witten, 1981a). By ‘‘dynamical supersymme-
try breaking’’ (DSB) we mean that supersymmetry is As we shall see, supersymmetry is broken if and only
broken spontaneously in a theory that possesses super- if the vacuum energy is nonzero. Furthermore, as we
symmetricvacuaatthetreelevel,withthebreakingtrig- mentioned above, if supersymmetry is unbroken at tree
geredbydynamicaleffects.ThecrucialpointaboutDSB level, it can only be broken by nonperturbative effects.
isthatifsupersymmetryisunbrokenattreelevel,super- Thus studying supersymmetry breaking requires under-
symmetric nonrenormalization theorems (Ferrara, standing the nonperturbative dynamics of gauge theo-
Iliopoulos, and Zumino, 1974; Wess and Zumino, 1974; riesintheinfrared.Fortunatelyinrecentyearstherehas
Grisaru, Rocek, and Siegel, 1979) imply that it remains been tremendous progress in understanding the dynam-
unbroken to all orders in perturbation theory, and can ics of supersymmetric field theories.
therefore only be broken by nonperturbative effects, The potential of a supersymmetric theory is deter-
whicharesuppressedbyroughlye28p2/g2, whereg isthe mined by two quantities: the Ka¨hler potential, which
coupling. The electroweak scale is related to the size of contains the kinetic terms for the matter fields, and the
the soft supersymmetry-breaking terms, and thus it is superpotential, a holomorphic function of the matter
proportional to the supersymmetry-breaking scale. The fieldsthatcontrolstheirYukawainteractions.Holomor-
latter is suppressed by the exponential above, and can phy,togetherwiththesymmetriesofthetheory,maybe
easily be of the correct size, about 17 orders of magni- used to determine the physical degrees of freedom and
tude below the Planck scale. the superpotential of the infrared theory (Seiberg, 1994,
In addition, supersymmetry, or more precisely, local 1995). Since the latter two are precisely the ingredients
supersymmetry,providestheonlyknownframeworkfor needed for studying supersymmetry breaking, this
a consistent description of gravity, in the context of progress has fueled the discovery of many new
string theory. If indeed the underlying fundamental supersymmetry-breaking theories, as well as new tech-
physics is described by string theory, one can contem- niques for establishing supersymmetry breaking.
plate two qualitatively different scenarios. One is that The structure of this article is as follows. We start by
SUSY is directly broken by stringy effects. Then, how- describing general properties of supersymmetry break-
ever, the SUSY-breaking scale is generically around the ing and studying examples of tree-level breaking in Sec.
string scale (barring new and better understanding of II. In Sec. III, we discuss indirect methods for finding
string vacua), and thus the gauge hierarchy problem is theories with dynamical supersymmetry breaking, and
not solved by supersymmetry. Therefore we shall focus for establishing supersymmetry breaking. As we shall
here on a second possible scenario, namely, that in the see, these methods direct the search for supersymmetry
low-energy limit, string theory gives rise to an effective breaking towards chiral theories with no flat directions,
field theory, and supersymmetry is spontaneously bro- preferably possessing an anomaly-free R symmetry.
ken by the dynamics of this low-energy effective theory. Thesecriteriadonotamounttonecessaryconditionsfor
Theaimofthisreviewistodescribethephenomenon supersymmetry breaking, and we shall point out ‘‘loop-
of dynamical supersymmetry breaking in field theories holes’’ in the indirect methods that allow the possibility
withN51 globalsupersymmetry.(Ncountsthenumber ofsupersymmetrybreakingintheoriesthatviolateallof
of supersymmetries. For N51 there are four supersym- the above requirements. Recent developments have led
metry charges, and this is the smallest amount of super- to the discovery of such SUSY-breaking theories, and
symmetry allowed in four dimensions.) we shall postpone the discussion of representative ex-
The restriction on N comes from the fact that only amplestoSec.VI.Still,someoftheindirectmethodswe
N51 supersymmetry has chiral matter, which we need shall describe, most notably, the breaking of a global
in the low-energy theory if it is to contain the standard symmetryinatheorywithnoflatdirections,providethe
Rev.Mod.Phys.,Vol.72,No.1,January2000
Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking 27
most convincing evidence for supersymmetry breaking ingisthatitstillhasnotyieldedanyorganizingprinciple
in theories that cannot be directly analyzed. to the study and classification of supersymmetry-
In Secs. IV and V we turn to theories that can be breaking theories. In Sec. VII we shall describe one
directlyanalyzedintheinfrared.Intheearly1980s,such method for generating new supersymmetry-breaking
studieswerelimitedtoasemiclassicalanalysisinregions theories from known theories. However, this is far from
ofweakcoupling,andweshalldescribesuchanalysesin a full, systematic classification. Nor can we tell immedi-
Sec.IV.Themaindevelopmentinrecentyearshasbeen ately, without detailed analysis, whether a specific
the better understanding of supersymmetric theories in theory breaks supersymmetry or not. For these reasons
regionsofstrongcoupling,andweshallstudysupersym- we find it useful to present a rough survey of known
metrybreakinginsuchtheoriesinSec.V.Inbothcases, modelsinSec.VII,pointingouttheirmainfeatures,the
theanalysisofsupersymmetrybreakinginvolvestwoin- relationsbetweenthem,and,whereapplicable,theirrel-
gredients. The first is identifying the correct degrees of evant properties for model-building purposes. While we
freedom of the theory, in terms of which the Ka¨hler shall see many different mechanisms by which super-
potential is nonsingular. In all the theories we shall symmetry is broken in these examples, the breaking is
study, in the interior of the moduli space, these are ei- almostalwaystheconsequenceoftheinterplaybetween
ther the confined variables or the variables of a dual instanton effects and a tree-level superpotential.
theory.1 Indeed, we shall see that duality (Seiberg, Anotherarearequiringfurtherstudyistheanalysisof
1995)—the fact that different UV theories may lead to supersymmetry-breaking vacua, and their symmetries
the same infrared physics—can be a useful tool for es- and light spectra, in strongly coupled theories. One may
tablishing supersymmetry breaking, as one can some- hope that recently discovered realizations of supersym-
times pick a more convenient theory in which to study metric gauge theories as extended brane configurations
whether supersymmetry is broken or not. A second, re- in string theory will lead to further progress in this di-
lated ingredient is finding the exact superpotential in rection,aswellastosomeorganizingprincipleforDSB.
terms of the light, physical degrees of freedom. Indeed, several DSB models have been realized as
Havingestablishedthesetwoingredients,atlowener- D-brane configurations in string/M theory; see for ex-
giesonethentypicallyhasatheoryofchiralsuperfields, ample de Boer etal. (1998) and Lykken, Poppitz, and
with all gauge dynamics integrated out, with a known Trivedi (1999). Moreover, in some cases the dynamical
superpotential. The problem of establishing supersym- effects leading to supersymmetry breaking were under-
metry breaking is then reduced to solving a system of stood in stringy language (de Boer etal., 1998). How-
equations to check whether or not the superpotential ever,thisapproachhasyettoleadtoresultsthatcannot
can be extremized. be directly obtained in a field-theory analysis.
Although the results we shall use on the infrared de- We limit ourselves in this review to the theoretical
grees of freedom and the exact superpotential apply to analysis of supersymmetry breaking in different models.
supersymmetrictheories,theymaystillbeusedtoargue We do not discuss the questions of whether, and how,
for supersymmetry breaking, since the theories we shall thisbreakingcanfeeddowntothestandardmodel.Ide-
studyinthiswayareobtainedbyperturbingasupersym- ally, a simple extension of the standard model would
metric theory. For a sufficiently small perturbation, the breaksupersymmetrybyitself,generatinganacceptable
scale of supersymmetry breaking can be made suffi- superpartner spectrum. Unfortunately this is not the
ciently small, so that we can work above this scale and case. In simple supersymmetry-breaking extensions of
still use known results on the infrared supersymmetric the standard model without new gauge interactions,
theory.Ifsupersymmetryisindeedbrokeninthetheory, nonperturbative effects would probably be too small to
the breaking should persist even as the perturbation is generate soft terms of the correct size (Affleck, Dine,
increased. Otherwise, the theory undergoes a phase andSeiberg,1985).Moreover,unlesssomeofthescalars
transition as some coupling is varied, being supersym- obtained their masses either radiatively, or from non-
metricinoneregionandnonsupersymmetricinanother. renormalizable operators, some superpartners would be
However, one does not expect supersymmetric theories lighter than the lightest lepton or quark (Dimopoulos
toundergophasetransitionsascouplingsarevaried(In- andGeorgi,1981).Thussupersymmetrymustbebroken
triligator and Seiberg, 1994; Seiberg and Witten, 1994a, by a new, strongly interacting sector, and then commu-
1994b). nicated to the standard model either by supergravity ef-
Having learned various techniques for the analysis of fects, in which case the soft terms are generated by
DSB,weusetheseinSec.VItostudyafewexamplesof higher-dimensionoperatorsoroccuratthelooplevel,or
theories that break supersymmetry dynamically even bygaugeinteractions,inwhichcasethesofttermsoccur
thoughtheyviolatesomeofthecriteriadescribedinSec. at the loop level. These different possibilities introduce
III. different requirements on the SUSY-breaking sector.
Perhaps the most disappointing aspect of the recent For example, gravity mediation often requires singlet
progress in our understanding of supersymmetry break- fields that participate in the SUSY-breaking. Simple
models of gauge mediation require a large unbroken
globalsymmetryattheminimumoftheSUSY-breaking
1Attheboundaryofmodulispace,themicroscopicdegreesof theory,inwhichthestandard-modelgaugegroupcanbe
freedomwilloftenbemoreappropriate. embedded. Several of the supersymmetry-breaking
Rev.Mod.Phys.,Vol.72,No.1,January2000
28 Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking
models discovered recently have some of these desired Finally, we note that the interested reader can find
properties,andthusallowforimprovedphenomenologi- several useful reviews of dynamical supersymmetry
cal models for the communication of supersymmetry breaking which have appeared in the past couple of
breaking. years. Short introductions to recent developments can
Anotherissuethatisimportantforphenomenological befoundinSkiba(1997),Nelson(1998),Poppitz(1998),
applicationsofDSBthatweshallnotaddressisthecos- andThomas(1998).Mostnotably,thereviewbyPoppitz
and Trivedi (1998), although smaller in scope than the
mological constant problem. In globally supersymmetric
present review, emphasizes recent developments in the
theories, fermionic and bosonic contributions to the
field. It also contains a discussion of supersymmetry
vacuum energy cancel each other and the cosmological
breaking in quantum-mechanical systems. Shifman and
constantvanishes.Uponsupersymmetrybreakingthisis
Vainshtein (1999) give an excellent introduction to in-
no longer true, and the cosmological constant is compa-
stanton techniques and discuss their application for su-
rabletothescaleofsupersymmetrybreaking.Whileina
persymmetry breaking. The review by Giudice and Rat-
framework of local supersymmetry a further cancella-
tazzi (1998) focuses on applications of DSB to building
tion is possible, significant fine tuning is required. Even-
models of gauge-mediated supersymmetry breaking.
tually,amicroscopicunderstandingofsuchfinetuningis
needed in any successful phenomenological application
of dynamical supersymmetry breaking.
Inthebodyofthisreviewweassumethatthereaderis II. GENERALITIES
familiar with the general properties of supersymmetric
A. Vacuumenergy—theorderparameterofSUSY
fieldtheoriesandrelyheavilyonsymmetriesandanum-
breaking,andFandDflatness
ber of exact nonperturbative results obtained in recent
years. The reader who is just beginning the study of su-
Apositivevacuumenergyisanecessaryandsufficient
persymmetry should first consult the Appendix, where
condition for spontaneous SUSY breaking. This follows
we briefly present basic facts about supersymmetry and
from the fact that the Hamiltonian of the theory is re-
relevantresultstomakeourpresentationself-contained.
lated to the absolute square of the SUSY generators
For a more complete introduction to SUSY see, for ex-
(see Appendix Sec. 1):
ample, Bagger and Wess (1991) and Nilles (1984). Sev-
eralexcellentreviewsoftherecentprogressinthestudy
1
of strongly coupled SUSY gauge theories exist; see, for H5 ~Q¯ Q 1Q Q¯ 1Q¯ Q 1Q Q¯ !. (1)
example, Intriligator and Seiberg (1996), Peskin (1997), 4 1 1 1 1 2 2 2 2
and Shifman (1997).
In Appendix Sec. 1 we introduce notations and basic The energy is then either positive or zero. Furthermore,
formulas for the Lagrangians of supersymmetric theo- a state that is annihilated by Qahas zero energy, and
ries.Theknowledgeoftheseresultsisnecessaryinevery conversely, a zero-energy state is annihilated by Qa.
section of the review. In Sec. 2 we discuss a method for Thus the vacuum energy serves as an order parameter
finding the D-flat directions of a SUSY gauge theory for supersymmetry breaking.
(directions along which the gauge-interaction terms in Therefore the study of supersymmetry breaking re-
the scalar potential vanish), and the parametrization of quiresknowledgeofthescalarpotentialofthetheory.It
D-flat directions in terms of gauge-invariant operators. is convenient to formulate the theory in N51 super-
These results are necessary for the study of supersym- space, where space-time (bosonic) coordinates are
metrybreakinginnon-Abeliangaugetheories,whichwe supplemented by anticommuting (fermionic)
discuss starting in Sec. III.D. In Appendix Secs. 3–7 we coordinates.2 In this formulation fields of different spins
turn to SUSY QCD with different numbers of flavors. related by supersymmetry are combined in the super-
Whiletheresultswepresentareuseddirectlyinvarious symmetry multiplets, superfields. Matter fields form chi-
places in Secs. IV–VII, the discussion in the Appendix ral superfields, while gauge bosons and their spin 1/2
alsoillustratestechniquesintheanalysisofthedynamics superpartners form (real) vector superfields. In the su-
of general SUSY gauge theories and are applicable to perspaceformulation,physics,andinparticular,thesca-
models with matter transforming in general representa- lar potential, is determined by two functions of the su-
tions of the gauge group. The discussion also provides perfields, the superpotential and the Ka¨hler potential.
simpleexamplesofphenomenasuchastheorieswithno ThesuperpotentialencodesYukawa-typeinteractionsin
quantum moduli spaces, deformed quantum moduli the theory; in particular it contributes to the scalar po-
spaces, confinement without chiral symmetry breaking, tential. The superpotential is an analytic function of the
and duality. We shall encounter these phenomena in superfields. This fact, together with symmetries and
various theories throughout the review. References to known(weaklycoupled)limits,oftenallowsustodeter-
analyses of the dynamics of theories other than SU(N) mine the exact nonperturbative superpotential of the
are collected in Appendix Sec. 8. theory.TheKa¨hlerpotential,ontheotherhand,isareal
Wealsonotethatthereaderwhoisinterestedonlyin function of superfields, and can be reliably calculated
a general knowledge of DSB can skip Secs. VI and VII.
Section VII.C can be used independently of the rest of
the article as a guide to DSB models. 2FormoredetailseeAppendixSec.1.
Rev.Mod.Phys.,Vol.72,No.1,January2000
Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking 29
only when a weakly coupled description of the theory As mentioned in the Introduction, a key point in the
exists. From our perspective, the Ka¨hler potential is im- study of SUSY breaking is the fact that, due to the su-
portant in two respects. First, it gives rise to gauge- persymmetricnonrenormalizationtheorems(Ferrara,Il-
interaction terms in the scalar potential. Second, it de- iopoulos, and Zumino, 1974; Wess and Zumino, 1974;
termines the kinetic terms of the matter fields and thus
Grisaru, Rocek, and Siegel, 1979; Seiberg, 1993), the
modifiesscalarinteractionsarisingfromthesuperpoten-
moduli space remains unmodified in perturbation
tial. Assuming a canonical (quadratic in the fields)
theory.Iftheclassicalpotentialvanishesforsomechoice
Ka¨hler potential, the scalar potential is
ofVEV’s,itremainsexactlyzerotoallordersinpertur-
bation theory. Thus only nonperturbative effects may
( (
V5 ~Da!21 F2, (2) generate a nonzero potential and lift the classical zeros.
i
a i Indeed, nonperturbative effects can modify the moduli
space; they can lift the moduli space completely; or, fi-
wherethesumrunsoverallgaugeindicesa andallmat-
nally, the quantum moduli space may coincide with the
ter fields f. In Eq. (2), the D terms and F terms are
i classical one.
auxiliary components of vector and chiral superfields,
There are numerous possibilities, then, for the behav-
respectively. D terms and F terms are not dynamical
ior of the theory. If a theory breaks supersymmetry, it
and one should solve their equations of motion. In par-
has some ground state of positive energy at some point
ticular, F terms are given by derivatives of the superpo-
tential (for more details, and the analogous expressions in field space (or it may, in principle, have several
for D terms, see Appendix Sec. 1): ground states at different points).5 Alternatively, the
theory may remain supersymmetric, with either one
]W groundstateofzeroenergyatsomepointinfieldspace,
F 5 . or a few ground states at isolated points, or with a con-
i ]f
i tinuum of ground states, corresponding to completely
flatdirectionsthatarenotliftedeitherclassicallyornon-
Forsupersymmetrytoremainunbroken,therehastobe
perturbatively. It is also possible that the theory does
somefieldconfigurationforwhichboththeF termsand
not have a stable vacuum state. In such a case, while a
the D terms vanish.3 In fact, generically such configura-
supersymmetric vacuum does not exist, the energy can
tions exist not only at isolated points but on a subspace
become arbitrarily small along some direction on the
of the field space. This subspace is often referred to as
moduli space.6 While such a theory can still be given a
the moduli space of the theory.
cosmological interpretation (Affleck, Dine, and Seiberg,
Classically one could set all superpotential couplings
1984a),weshallnotconsideritaSUSY-breakingtheory
tozero.Thenthemodulispaceofthetheoryisthesetof
for our purposes.7
‘‘D-flat directions,’’ along which the D terms vanish. A
Note that supersymmetric theories are very different,
particularly useful parametrization of D-flat directions,
in this respect, from other theories. In a nonsupersym-
which we discuss in Appendix Sec. 2, can be given in
metrictheory,multiplegroundstatesareusuallyrelated
terms of the gauge-invariant operators of the theory
by a symmetry and are therefore physically equivalent.
(Luty and Taylor, 1996). Even when small tree-level su-
On the other hand, different ground states of a super-
perpotential couplings are turned on, the vacua will lie
symmetric theory may describe completely different
near the D-flat directions. It is convenient therefore to
physics. For example, classically, one flat direction of an
analyze SUSY gauge theories in two stages. First find
SU(3) gauge theory with two flavors is [see Appendix
the D-flat directions, then analyze the F terms along
Sec. 4 for details]
these directions. The latter have classical contributions
from the tree-level superpotential, and may ‘‘lift’’ some,
or all, of the D-flat directions. Since a classical superpo-
5Inthislattercasethegroundstatesarenondegenerateeven
tential is a polynomial in the fields, F terms typically
if they appear to have the same energy in a certain approxi-
growforlargescalarvacuumexpectationvalues(VEV),
mation. This is because low-energy physics is nonsupersym-
and vanish at the origin.4
metric, and the vacuum energy receives quantum corrections
(ontopofthenonperturbativeeffectsthatledtothenonvan-
ishing energy in the first place). Since different nonsupersym-
3Here we implicitly assumed that the Ka¨hler potential is a metricvacuaarenonequivalent,thesequantumcorrectionslift
regular function of the fields and has no singularities. For ex- thedegeneracy.
ample, if the derivatives of the Ka¨hler potential vanish, the 6Weshallcallsuchdirectionsinthemodulispace‘‘runaway’’
scalarpotentialcanbenonzeroevenifallFandDtermsvan- directions,andstudythemcarefullyinSecs.VI.A–VI.B.
ish;seeEq.(A6). 7Wealsonotethattherunawaymodulimaybestabilized,and
4Superpotential terms that are linear in the fields are an im- supersymmetry broken, due to Ka¨hler potential effects when
portant exception. They lead to potentials that are nonzero thetheoryiscoupledtogravity(DvaliandKakushadze,1998).
even at the origin. Such terms necessarily involve gauge sin- However,sincethetypicalVEV’sinthiscasewillbeofPlanck
glets and require the introduction of some mass scale. How- size, the vacuum will be determined by the details of the mi-
ever, a linear term can be generated dynamically. We shall croscopic theory at M , and the dynamical supersymmetry
P
encounter examples of trilinear, or higher, superpotential breaking is not calculable in the low-energy effective-field
termsthatbecomelinearafterconfinement. theory.
Rev.Mod.Phys.,Vol.72,No.1,January2000
30 Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking
S D
v 0
If indeed there is a massless fermion coupling to the
currentasinEq.(4),thentheleft-handsideofEq.(5)is
f5¯f5 0 0 , (3)
equal to
0 0 E
where f and¯f are the scalar components of the SU(3) f2shaa˙ snbb˙ d4x]h^0uTcaG˙~x!cbG~0!u0&
fundamentals and antifundamentals, respectively. For
SanUy(2g)ivgeanugcehothiceeoroyf, wvh,ostheegaluogwe-ecnoeurpglyingthdeeopreyndiss oann 5f2shaa˙ snbb˙@2iphG~p!a˙b#p!05f2snab˙ , (6)
v.8 Note also that the flat directions of SUSY theories
may extend to infinity, unlike the customary compact where G(p)a˙b˙ is the Goldstino propagator. So indeed f
flat directions of bosonic global symmetries. Further- isnonzero.Notethatafermionwithderivativecoupling
more, in the case of other global symmetries, the exis- to the current would not contribute to the right-hand
tence of a flat direction is usually associated with spon- side of Eq. (6) because of the additional factors of the
taneous breaking of the symmetry, with the massless momentum in the numerator. In fact, since from the
Goldstone bosons corresponding to motions along the SUSY algebra^0u$Qa,Jnb˙(0)%u0&52Esnab˙ , with E the
flat direction. This is not the case with SUSY. The rea- energy, we have f252E.
son, of course, is that the SUSY generators do not cor- To see the appearance of the Goldstino more con-
respond to motions in field space. Theories with unbro- cretely, consider the SUSY current
kenSUSYmayhavedegeneratevacuapreciselybecause
SUSY is unbroken. And, as we shall see in the next dL
(
section,theorieswithspontaneouslybrokenSUSYhave Jam; d~] f!~df!a, (7)
the analog of Goldstone particles even when they only f m
have a single ground state.
wherethesumisoverallfields,and(df)aistheshiftof
the field funder a SUSY transformation. Because of
Lorentzsymmetry,theonlylineartermsinEq.(7)come
B. TheGoldstino from vacuum expectation values of (df). Examining
the SUSY transformations of the chiral and vector mul-
The breaking of any bosonic global symmetry is ac- tiplets of N51 SUSY (see for example Bagger and
companied by the appearance of massless Goldstone Wess, 1991), we see from Lorentz invariance that the
bosons that couple linearly to the symmetry current. only fields whose SUSY transformations contain
Similarly a theory with broken supersymmetry contains Lorentz-invariantobjects,whichcandevelopVEV’s,are
a massless fermion, which is usually referred to as a the matter fermion c, whose SUSY transformation
i
‘‘Goldstone fermion,’’ or, in short, ‘‘Goldstino,’’ that gives F , and the gauge fermion la, whose transforma-
i
coupleslinearlytotheSUSYcurrent(SalamandStrath- tion gives Da. One then finds
dee, 1974, Witten, 1981a).
The Goldstino coupling to the SUSY current can be
( 1 (
expressed as cG; ^F &c1 ^Da&la, (8)
i i &
Jm5fsmb˙ cG1{{{, (4)
a a b˙ so that the Goldstino is a linear combination of the chi-
ral and gauge fermions whose auxiliary fields F and D
where cG is the Goldstino and, as we shall see momen- acquire VEV’s. Note that Eq. (8) actually holds only
b˙
tarily, f is a constant that is nonzero when SUSY is bro- with a canonical (quadratic) Ka¨hler potential, otherwise
ken. The ellipsis in Eq. (4) stands for terms quadratic in derivativesoftheKa¨hlerpotentialenteraswell.Wecan
the fields and for potential derivative terms. Conserva- usethistoarguethatanonvanishingF VEVorD VEV
tion of the SUSY current then implies that the is a necessary condition for SUSY breaking. When
Goldstino is massless. To justify Eq. (4), note that, for SUSY is broken, there is a massless fermion, the Gold-
broken SUSY (Witten, 1981a), stino,thattransformsinhomogeneouslyundertheaction
of the SUSY generators. But the only Lorentz-invariant
E
objects that appear in the SUSY variations of the N51
d4x]h^0uTJah~x!Jnb˙~0!u0&5^0u$Qa,Jnb˙~0!%u0&(cid:222)0. multiplets,andthatthereforemayobtainVEV’s,arethe
(5) auxiliary fields F and D. Thus for SUSY breaking to
occur, some F or D fields should develop VEV’s.
In light of the above, it would first seem that if SUSY
8Hereweconsidertheclassicalvacuaofthetheory.Quantum is relevant to nature, we should observe the massless
Goldstino. However, we ultimately need to promote
mechanically, the flat directions (3) are lifted, and the theory
does not have a stable vacuum. Yet in many models non- SUSY to a local symmetry to incorporate gravity into
equivalentquantumvacuaexist. the full theory. In the framework of local supersymme-
Rev.Mod.Phys.,Vol.72,No.1,January2000
Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking 31
try, the massless Goldstino becomes the longitudinal k
(
component of the gravitino, much like the case of the W5 fcf ~fn!, (9)
i i a
Higgs mechanism. The gravitino then has a coupling to i51
ordinary matter other than the gravitational interaction,
and,forsupersymmetrytobreak,atleastoneofthef ’s,
by virtue of its Goldstino component. It should come as i
no surprise, then, that the gravitino mass, as well as its say f1, contains a constant term, independent of the
coupling to matter fields, are related to the SUSY- fields. The equations of motion for the R-charged fields
breaking scale. This has important phenomenological ]W/]fc5f (fn)50 give k equations for l unknowns
i i a
implications. In particular, in models with low-scale fn. If k.l there are no solutions for generic functions
a
SUSYbreaking,thegravitinoisverylight,andthedecay f , the F-term conditions cannot all be satisfied, and su-
i
of other superpartners into the gravitino may be ob- persymmetry is broken.
served in collider searches for supersymmetry (Stump, We can modify these models by adding fields with R
Wiest, and Yuan, 1996; Dimopoulos etal., 1998). charges0,Q ,2.Sincesuchfieldscannotcoupletothe
R
fields fc while preserving the R symmetry, they will not
i
change the above discussion, and supersymmetry re-
C. Tree-levelbreaking mainsbroken.If,ontheotherhand,fieldswithnegative
R charges are added to the model, the total number of
variablesonwhichthef ’sdependincreases,andingen-
1. O’Raifeartaighmodels i
eralsupersymmetryisunbroken.Finally,weshouldnote
Oneofthesimplestmodelsofspontaneoussupersym- that adding to the superpotential explicit
metry breaking was proposed by O’Raifeartaigh (1975) R-symmetry-violating couplings that do not involve
andisbasedonatheoryofchiralsuperfields.Supersym- fieldsofR charge2willnotmodifytheabovediscussion.
metry in the model is broken at tree level: while the However, R-symmetry-violating terms that include
Lagrangian of the model is supersymmetric, even the fields of R charge 2 will generically lead to the restora-
classicalpotentialissuchthatasupersymmetricvacuum tion of supersymmetry.
state does not exist. It is also useful to look at the equations of motion for
In addition to giving the simplest example of sponta- the R-neutral fields. First, note that their VEV’s are
neously broken supersymmetry, the study of fixed by minimizing the part of the scalar potential aris-
O’Raifeartaigh models will be useful for our later stud- ing from the F terms of the R-charged fields (the re-
ies of DSB, as the low-energy description of many dy- maining terms in the scalar potential vanish at least
namical models we encounter will be given by an when ^fc&50 for all i). Therefore there are l F-term
i
O’Raifeartaigh-type model. equations depending on k independent variables fc. In
i
Before writing down the simplest example of an a SUSY-breaking model k.l, so there are k2l linear
O’Raifeartaigh model, let us describe the general prop- combinationsofthefieldsfc thatareleftundetermined.
i
erties of such models. First, we shall restrict our atten-
Thus O’Raifeartaigh models necessarily possess direc-
tion to superpotentials with only positive exponents of
tions of flat (nonzero) potential in the tree-level ap-
the fields. We shall later analyze a number of models in
proximation (Zumino, 1981; Einhorn and Jones, 1983;
which the low-energy description involves superpoten-
Polchinski, 1983). As we shall discuss later, this is not a
tials containing negative exponents of the fields. Such
generic situation in models of DSB.
terms, however, are generated by the nonperturbative
The simplest example of an O’Raifeartaigh model re-
dynamics of the underlying (strongly coupled) micro-
quires two fields with R charge 2, one field with R
scopic theory, and are not appropriate in the tree-level charge 0 (k52, l51), and has the superpotential
superpotential we consider here.
Second,sincethesuperpotentialisapolynomialinthe
W5f~M22l f2!1m ff . (10)
fields, at least one of the fields in this model needs to 1 1 1 2 2
appear linearly in the superpotential, or there will be a
ItiseasytoseethattheF-termconditionsforf andf
supersymmetric vacuum at the origin of the moduli 1 2
are incompatible. We can directly minimize the scalar
space.
potential
Itwouldproveuseful,forfuturepurposes,topayspe-
cial attention to the R symmetry of the model. [For the
definition of an R symmetry, see Appendix Sec. 1]. If V5uM22l f2u21um fu21um f22l ffu2. (11)
1 1 2 2 2 1 1
this symmetry is unbroken by the superpotential, there
is then at least one field of R charge 2. More generally, ThefirsttwotermsinEq.(11)determinethevalueoff
consider a model containing fields fc,i51,...,k with R at the minimum. In the limit m2/(l M2)@1 the mini-
i 2 1 1
charge2,andfieldsfn, a51,...,l withR charge0.(For mum is found at f50, while for small m2/(l M2) we
a 2 1 1
convenience,weshallcallthemchargedandneutral,re- find f5(M22m2/2)1/2/l . Note that f and f appear
1 2 1 1 2
spectively, even though various components of the su- onlyinthelastterminEq.(11).Thistermshouldbeset
perfields transform differently under R symmetry.) The to zero for the potential to be extremal with respect to
most general superpotential respecting the R symmetry f and f . This is achieved when f522l ff /m ,
1 2 2 1 1 2
can be written as and therefore at tree level the linear combination
Rev.Mod.Phys.,Vol.72,No.1,January2000
32 Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking
m f22l ^f&f is arbitrary, as was expected from the symmetry breaking, we recall that the Ka¨hler potential
2 1 1 2
previous discussion. Equivalently, we can parametrize can be written as a function of the gauge-invariant com-
different vacua by the expectation values of f . Note bination of fields
1
thatdifferentvacuaarephysicallynonequivalent;inpar-
ticular the spectrum depends on ^f&. K5f~f†eVf,W†W,S!, (13)
1
It is easy to find the tree-level spectrum of the model.
For any choice of parameters it contains a massless fer- where frepresents matter fields transforming in some
mion,theGoldstino.Thespectrumalsocontainsthesca- representation of the gauge group, V is a vector super-
lar field associated with the flat direction whose mass field whose supersymmetric field strength is W, and S
arises entirely due to radiative corrections (however, represents gauge-singlet fields. In a non-Abelian theory
there are no quadratic divergences since the action of this is the only possible form of field dependence in the
the theory is supersymmetric, although supersymmetry Ka¨hler potential. In an Abelian theory, however, the D
is not realized linearly). All other states are massive. term of the vector superfield V is invariant under the
Another important feature of this spectrum is that the gauge and supersymmetry transformations by itself.
supertrace of the mass matrix squared, STrm2, van- Thus, if one does not require parity invariance, the
i lowest-order Ka¨hler potential of a U(1) gauge theory
ishes.Thispropertyofthespectrumholdsforanymodel
can be written as9
with tree-level supersymmetry breaking (Ferrara, Gi-
rardello, and Palumbo, 1979).
Since supersymmetry is broken, the vacuum degen- K5Q†eVQ1Q¯ †e2VQ¯ 1j V. (14)
FI
eracy is lifted in perturbation theory. Huq (1976) calcu-
lated the one-loop corrections to the potential of this This Ka¨hler potential, together with superpotential
model. They are given by mass terms for the matter fields, leads to the following
scalar potential:
S D
( ~21!F m ~f!2
DV~f1!5 i 64p2 mi~f1!4ln im21 , (12) V5g2~uQu22uQ¯u21j!21m2~uQu21uQ¯u2!. (15)
2
where the sum is over all massive fields (and the masses
depend on the f VEV). Huq (1976) found that the It is easy to see that the vacuum energy determined by
corrections genera1te a positive mass for f and that the this potential is necessarily positive and supersymmetry
nonsupersymmetricvacuumislocatedatf1 50 withun- is broken. When g2j,m2 both scalar fields have posi-
1 tivemassandtheirVEV’svanish.Thepositivecontribu-
broken R symmetry. He also analyzed a model with an
tion to the vacuum energy comes entirely from the D
SU(3) global symmetry and a model constructed by
Fayet (1975) with SU(2)3U(1) symmetry, and in both term in the potential. The scalar mass matrix has eigen-
valuesm25m26g2j. Thegaugesymmetryisunbroken,
casesfoundthatthetree-levelmodulusacquirespositive 6
and thus the gauge boson remains massless. The matter
mass due to one-loop corrections to the Ka¨hler poten-
fermionsretaintheirmassm, whilethegauginoremains
tial, leading to the unique vacuum with unbroken R
massless and plays the role of the Goldstino. [In accord
symmetry. In fact, this conclusion is not surprising. In
withthefactthathere^F &50 and^D&(cid:222)0; seeEq.(8).]
the model discussed above, quantum corrections to the i
vacuum energy come from the renormalization of the When g2j.m2, the field Q¯ has negative mass and
mass parameter M2. Due to the holomorphy of the su- acquires a VEV. At the minimum of the potential Q
1
perpotential, these are completely determined by the 50 and Q¯ 5v, where v5(2j24m2/g2)1/2. We see that
wave function renormalization of f and, since the both the gauge symmetry and supersymmetry are bro-
1
model is infrared free, necessarily generate a positive ken. Moreover, both the D term and the F term are
contribution to the scalar potential. It is important to nonvanishing and supersymmetry breaking is of the
note that in modifications of the model that include mixedtype.Onecaneasilyfindthatthespectrumofthe
gaugefields,theremaybeanegativecontributiontothe model contains one vector field and one real scalar field
potential. The balance of the two perturbative effects of mass squared 1g2v2, one complex scalar of mass
2
may produce a stable minimum at large values of the squared2m2, twofermionsofmass(m211g2v2)1/2, and
2
modulus VEV (Witten, 1981b). a massless Goldstone fermion that is a linear combina-
tionoftheGoldstinoandthepositivelychargedfermion
2. Fayet-Iliopoulosbreaking S D
Another useful example of tree-level supersymmetry 1 igv
breaking is given by a model with U(1) gauge interac- ˜l5A ml1 A cQ . (16)
1 2
tions (Fayet and Illiopoulos, 1974). In this model super- m21 g2v2
symmetrybreakingisdrivenbyD-termcontributionsto 2
the potential, but depending on the parameters of the
Lagrangian, the nonzero vacuum energy comes either
entirely from D-term contributions or from both D and 9We restrict our attention to two matter multiplets with
F terms. To understand how D terms can drive super- charges61.
Rev.Mod.Phys.,Vol.72,No.1,January2000
Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking 33
III. INDIRECTCRITERIAFORDSB of the theory is set to zero, or is turned on. In that case,
states may ‘‘come in’’ from infinity or ‘‘move out’’ to
AswehaveseeninSec.II.A,thefactthatthevacuum infinity.
energy is the relevant order parameter immediately The index of several theories was calculated by Wit-
points the way in our quest for SUSY breaking: we ten (1982). In particular, Witten found that the index of
should study the zeros of the scalar potential. This in- a pure supersymmetric Yang-Mills theory is nonzero.11
deed is what we shall undertake to do in Secs. IV and Thus these theories do not break supersymmetry spon-
onward. Unfortunately, directly studying the zeros of taneously. An important corollary is that supersymmet-
the potential will not always be possible, or easy. In this
ric Yang-Mills theories with massive matter (and no
section we review several alternate ‘‘indirect’’ methods
massless matter) do not break supersymmetry either.
that are useful in the search for supersymmetry break-
The reason is that, at least in weak coupling, one can
ing.
takeallmassestobelarge,sothattherearenomassless
states in these theories beyond those of the pure super-
A. TheWittenindex
symmetric Yang-Mills theory, and so the value of the
Supersymmetrybreakingisrelatedtotheexistenceof index is the same as in the pure supersymmetric Yang-
zero-energy states. Rather than looking at the total Mills theory.
number of zero-energy states, it is often useful to con- What happens when the mass of the matter fields is
sider the Witten index (Witten, 1982), which measures taken to zero? The theory with zero mass has flat direc-
the difference between the number of bosonic and fer- tions, along which the potential is classically zero (away
mionic states of zero energy, from these flat directions the potential behaves as the
fourth power of the field strength). In contrast, the
Tr~21!F[n02n0. (17) theorywithmassforallmatterfieldshasnoclassicalflat
B F directions, with the potential growing at least quadrati-
cally for large fields. Thus, as the mass is taken to zero,
If the Witten index is nonzero, there is at least one
the asymptotic behavior of the potential changes, and
state of zero energy, and supersymmetry is unbroken. If
theWittenindexmaychangetoo.Infact,theindexisill
the index vanishes, supersymmetry may either be bro-
defined in the presence of flat directions, since zero
ken, with no states of zero energy, or it may be unbro-
modes associated with the flat directions lead to a con-
ken, with identical numbers of fermionic and bosonic
tinuous spectrum of states. (Indeed, to calculate the in-
states of zero energy.
dex of any theory one needs to consider the theory in a
The Witten index is a topological invariant of the
finite volume so that the resulting spectrum is discrete.)
theory.Inthisliesitsusefulness.Itmaybecalculatedfor
Wethereforecannotsayanythingaboutsupersymmetry
someconvenientchoiceoftheparametersofthetheory,
breaking in massless, nonchiral theories based on the
and in particular for weak coupling, but the result is
Witten index of the pure supersymmetric Yang-Mills
valid generally. To see this, note that in a finite volume,
theory.
fermionic and bosonic states of positive energy are
Consider for example supersymmetric QCD with N
paired by the action of the SUSY generator:
colors and F flavors of mass m, which we discuss in
A A AppendixSecs.4–7.Asexplainedthere,inthepresence
Qub &; Euf & Quf &; Eub &, (18)
E E E E of mass terms miQ Q¯ j, the theory has N vacua at
j i(cid:149)
where ub & (uf &) is a bosonic (fermionic) state of en-
ergy E.10E(RecEall that states of zero energy are annihi- Mj[Q Q¯ j5L(3N2F)/N~detm!1/Nm21j, (19)
i i(cid:149) i
lated by Q and are therefore not paired.) Thus, under
‘‘mild’’variationsoftheparametersofthetheory,states
maymovetozeroenergyandfromzeroenergy,butthey 11ForSU andSP groupstheindexisequaltor11,wherer is
always do so in Bose-Fermi pairs, leaving the Witten
the rank of the group. This is the same as the number of
index unchanged. gauginocondensatesforthesegroups.Moregenerally,thein-
Letusbeabitmoreprecisenowaboutwhatismeant dex equals the dual coxeter number of the group, which is
by ‘‘mild’’ variations. As long as a parameter of the different from r11 for some groups, notably some of the SO
theory, which is originally nonzero, is varied to a differ- groups (Witten, 1998). The fact that the number of gaugino
entnonzerovalue,wedonotexpecttheWittenindexto condensatesdoesnotalwaysequalr11,whichwasbelievedto
change, since different states can only move between be the value of the index, remained a puzzle until its recent
resolution by Witten (1998). This puzzle partly motivated the
different energy levels in pairs. The danger lies in the
conjecture that supersymmetric Yang-Mills theories have a
appearance of new states of zero energy. This can hap-
vacuum with no gaugino condensate (Kovner and Shifman,
pen if the asymptotic (in field space) behavior of the
1997). This possibility would have far-reaching consequences
potentialchanges,whichmayhappenifsomeparameter
forsupersymmetrybreaking.Thevastmajorityoftheoriesthat
break SUSY do so by virtue of a superpotential generated by
gauginocondensation.Avacuumwithnogauginocondensate
10Thisinfactjustifiesincludingonlyzero-energystatesinEq. would mean an extra ground state, or an entire branch of
(17).Statesofnonzeroenergydonotcontribute. groundstates,withzeroenergyandunbrokenSUSY.
Rev.Mod.Phys.,Vol.72,No.1,January2000
34 Y.ShadmiandY.Shirman: Dynamicalsupersymmetrybreaking
asymptotic behavior of the potential changes, and it be-
corresponding to the N roots of unity. This is in agree-
comes a nonzero constant asymptotically far along the
ment with the Witten index N of pure SU(N) gauge
flatdirection,sosupersymmetryisbroken.However,be-
theory. Consider now the massless limit mi!0. For F
j cause the potential is flat, running effects cannot be ne-
,N, the vacua (19) all tend to infinity. The theory has
glected. Indeed, as we shall argue in Sec. VI.A.2, such
no ground state at finite field VEV’s. The potential is
effects may lift the vacuum degeneracy and determine
nonzero in any finite region of field space and slopes to
the true nonsupersymmetric vacuum.
zeroatinfinity.Themasslesslimitofthetheoryisthere-
Tosummarize,puresupersymmetricYang-Millstheo-
fore not well defined. For F.N, by taking mi!0 in
j ries, as well as vectorlike theories with masses for all
different ways, any value of Mj may be attained. The
i matter fields, have a nonzero index and do not break
massless theory has an entire moduli space of vacua,
supersymmetry. When some masses are taken to zero,
parametrized by Mij. The N5F case is more subtle, but the resulting theories have classical flat directions, and
in this case too, the theory has a moduli space of vacua. thereforetheasymptoticbehaviorofthepotentialisdif-
Thus the ground states of the massless SU(N) theory
ferentfromthatofthemassivetheory.TheWittenindex
with F flavors are drastically different from those of the
may then change discontinuously and differ, if it is well
massive theory. In these examples we explicitly see how
defined, from that of the massive theory.
zero-energy states can disappear to infinity, or come in
What about chiral (parity-violating) theories? In such
from infinity. Again, this is possible because the
theories, at least some of the matter fields cannot be
asymptoticbehaviorofthepotentialchangesasthemass
given mass. Thus these theories cannot be obtained by
tends to zero. The theory including mass terms has no
deformingamassivevectorliketheory,andthereisnoa
flatdirections.Asymptoticallythepotentialrisesatleast
priori reason to expect, based on existing computations
quadratically. The massless theory has classical flat di-
of the Witten index, that these theories are supersym-
rections. Quantum mechanically, they are completely
lifted for F,N, and the potential asymptotes to zero as metric.Indeed,mostknownexamplesofsupersymmetry
a fractional power of the field. For F>N flat directions breaking are chiral.
remain even quantum mechanically. In any case, adding
mass terms changes the asymptotic behavior of the po-
tential.
In the examples above, the massive theory was super-
B. Globalsymmetriesandsupersymmetrybreaking
symmetric (with zero-energy states at finite fields
VEV’s) and the massless theory was either supersym-
In this section we shall discuss the connection be-
metric (with a continuum of vacua) or not well defined
tween global symmetries and supersymmetry breaking,
(withnogroundstate).Itisnaturaltoaskwhetherthere
which motivates two criteria for supersymmetry break-
exist vectorlike (parity-conserving) theories that break
ing. While these are useful guidelines for finding
SUSY as the relevant masses are taken to zero. The
supersymmetry-breaking theories, they are not strict
answer to this question is affirmative, as we shall see in
rules, and we shall encounter several exceptions in the
an explicit example in Sec. VI.A.2. It is useful to under-
following.
stand the general properties of the potential in such a
theory. As before, we expect the fully massive theory to Consider first a theory with an exact, nonanomalous
have a nonzero Witten index. The only way to obtain global symmetry, and no flat directions. If the global
supersymmetrybreakingasthemassesaretakentozero symmetry is spontaneously broken, there is a massless
is if the masses change the asymptotic behavior of the scalar field, the Goldstone boson, with no potential.
potential. Suppose that for any finite value of the mass With unbroken supersymmetry, the Goldstone boson is
parameter m, the theory possesses a supersymmetric part of a chiral supermultiplet that contains an addi-
vacuum at some VEV v (m), which moves away to in- tional massless scalar, again with no potential. This sca-
0
finity as some of the masses are taken to zero. Clearly, lar describes motions along a flat direction of zero po-
for small finite masses, the directional derivative of the tential. But this contradicts our initial assumption that
potential with respect to the modulus is negative for there are no flat directions. To avoid the contradiction
largevaluesofv,v (m). (Thetheorymayhavevarious we should drop the assumption of unbroken supersym-
0
minima for finite values of v, but we are interested in metry. This gives a powerful tool for establishing super-
the asymptotic behavior of the potential at large v.) In symmetry breaking (Affleck, Dine, and Seiberg, 1984b,
the absence of a phase transition at zero mass, such a 1985): If a theory has a spontaneously broken global
directional derivative will remain nonpositive in the symmetry and no flat directions, the theory breaks su-
limit m!0. However, there are still two possibilities. persymmetry.
First,itispossiblethatthedirectionalderivativeisnega- We have assumed here that the additional massless
tive for any finite VEV and only vanishes in the double scalar corresponds to motions along a noncompact flat
limit, m!0,v!‘. In such a case the theory does not direction.Thisisoftenthecasesince,insupersymmetric
have a stable vacuum. However, it is also possible that theories, the superpotential is invariant under the com-
the derivative vanishes in the limit m!0 for sufficiently plexified global symmetry, with the Goldstone boson
large but otherwise arbitrary v. If this is the case, the corresponding to the imaginary part of the relevant
Rev.Mod.Phys.,Vol.72,No.1,January2000