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Aspects of Supersymmetry and its Breaking
Thomas T. Dumitrescua! and Zohar Komargodskib†
aDepartment of Physics, Princeton University,Princeton, New Jersey, 08544, USA
bSchool of Natural Sciences, Institute for Advanced Study,Princeton, New Jersey, 08540, USA
We describe some basic aspects of supersymmetric field theories, emphasizing the structure of various supersym-metry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry currentand the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersym-metric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered atthe Cargese Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERNWinter School 2011 on “Supergravity, Strings, and Gauge Theory.”
1. Supersymmetric Theories
1.1. Supermultiplets and SuperfieldsA four-dimensional theory possesses N = 1
supersymmetry (SUSY) if it contains a con-served spin- 1
2 charge Q! which satisfies the anti-commutation relation
{Q!, Q!} = 2!µ!!Pµ . (1)
Here Q! is the Hermitian conjugate of Q!. (Wewill use bars throughout to denote Hermitian con-jugation.) Unless otherwise stated, we follow theconventions of [1]. In local quantum field theo-ries, the basic objects of interest are well-definedlocal operators. In SUSY field theories all suchoperators must be embedded in multiplets of thesupersymmetry algebra, or supermultiplets. Con-served currents furnish an important class of localoperators. Of particular interest is the supersym-metry current S!µ, which satisfies
"µS!µ = 0 , Q! =∫
d3xS!0 . (2)
∗T.D. is supported by a DOE Fellowship in High EnergyTheory, NSF grant PHY-0756966, and a Centennial Fel-lowship from Princeton University.†Z.K. is supported by DOE grant DE-FG02-90ER40542.Any opinions, findings, and conclusions or recommenda-tions expressed in this material are those of the authorsand do not necessarily reflect the views of the NationalScience Foundation.
In this review we will describe some basic aspectsof SUSY field theories, emphasizing the structureof various supermultiplets – especially those con-taining the supersymmetry current. In particu-lar, we will show how these supercurrents can beused to study the dynamics of supersymmetricfield theories, SUSY-breaking, and supergravity.
We begin by recalling basic facts about super-multiplets and superfields. A set of bosonic andfermionic operators {OB
i (x)} and {OFi (x)} fur-
nishes a supermultiplet if these operators satisfycommutation relations of the schematic form[Q!,OB
i (x)]! OF
j (x) + "OFk (x) + · · · ,
{Q!,OF
i (x)}! OB
j (x) + "OBk (x) + · · · , (3)
and likewise for the Q! commutators, such thatthe SUSY algebra (1) is satisfied. It is straightfor-ward to show that a supermultiplet must containequally many independent bosonic and fermionicoperators (see, for instance, [2]).
It is always possible (and very convenient) toembed the component fields OB,F
i (x) of a super-multiplet in a superfield S(x, #, #). Here #! is ananti-commuting superspace coordinate. (For nowwe suppress any Lorentz indices carried by S.)The component fields are identified with the x-dependent coe!cients when S(x, #, #) is expandedas a power series in #, #. The commutation rela-
2
tions (3) are succinctly encoded in the formula
[$Q + $Q, S
]= i
($Q+ $Q
)S , (4)
which is the defining property of a superfield.3Here $! is an arbitrary Grassmann parameterand Q!, Q! are the superspace di"erential oper-ators
Q! ="
"#!" i!µ
!!#!"µ ,
Q! = " "
"#!+ i!µ
!!"µ . (5)
Conversely, the defining property (4) implies thatthe components of any superfield furnish a su-permultiplet. Thus, supermultiplets and super-fields are in one-to-one correspondence, and wewill treat them synonymously.
To see this in a little more detail, consider thecomponent expansion of a general scalar super-field:
S(x, #, #) = C(x)+i#!%!(x)+· · ·+#2#2D(x) . (6)
As explained above, the defining property (4) de-termines the commutation relations of the super-charges Q!, Q! with the component fields. Thesecommutators show that the supercharges act asraising operators for the component fields.4 Thishas two important consequences:
• The superfield S(x, #, #) can be constructedfrom its bottom component C(x) by apply-ing the supercharges. Thus, any local oper-ator can be embedded in the bottom com-ponent of a superfield. However, it is notalways possible to embed an operator in ahigher component. This will play a crucialrole in our analysis of various supercurrents.
• The SUSY variation of the top compo-nent D(x) of any superfield is always a totalderivative. This fact will enable us to writesupersymmetric Lagrangians.
3 The factor of i in (4) is necessary for Hermiticity inMinkowski space.4For instance, [Qα, C] = −ψα.
A general superfield does not furnish an irre-ducible representation of supersymmetry. To re-duce a supermultiplet, we impose supersymmet-ric constraints. This is most conveniently done interms of the superspace di"erential operators
D! ="
"#!+ i!µ
!!#!"µ ,
D! = " "
"#!" i!µ
!!"µ . (7)
These anti-commute with the supersymmetrygenerators Q!, Q!, and thus map superfields tosuperfields. Hence, any constraint written interms of D!, D! is automatically supersymmet-ric. We are now ready to begin exploring variousimportant supermultiplets.
1.2. Chiral Multiplets and LagrangiansThe most familiar representation of supersym-
metry is the chiral multiplet. It is the basic build-ing block which enables us to write SUSY La-grangians describing only scalars and fermions.The bottom component of a chiral multiplet is an-nihilated by Q!. For example, the bottom com-ponent &(x) of a scalar chiral multiplet satisfies[Q!,&(x)
]= 0 . (8)
The multiplet obtained from &(x) by acting withthe supercharges is organized in a superfieldwhich satisfies the constraint D!# = 0. This con-straint can be solved in components:
# = &(y) +#
2#%(y) + #2F (y) , (9)
where yµ = xµ + #!µ#. We immediately note twokey properties of chiral superfields:
• Any function which depends on the chiralsuperfields #i, but not their complex con-jugates, is again a chiral superfield. Such afunction is said to be holomorphic in the #i.
• The SUSY variation of F (x) is a totalderivative.
From (9) we see that # contains a complexscalar &, a Weyl fermion %!, and another com-plex scalar F which will turn out to be a non-propagating auxilliary field. Among other things,
3
F ensures that the chiral multiplet has four (real)bosonic degrees of freedom to match the fourfermionic degrees of freedom coming from %!. Wewill abbreviate this by saying that # is a 4 + 4multiplet. As advertised, # has exactly the rightfield content to describe a theory of scalars andfermions, and we would like to write a Lagrangianfor such a theory.
Up to total derivatives, a SUSY Lagrangian Lmust be a real scalar whose variation under su-persymmetry is a total derivative. We can thustake
L = D + F + F , (10)
where D is the top component of a real scalarsuperfield K = K and F is the #2-component ofa chiral superfield W . This ensures that L isreal and supersymmetric. For reasons that willbe explained below, K is known as the Kahlerpotential, and W is called the superpotential. Aparticularly simple choice is to take K = ##and W = 0. It is standard to pick out di"erentcomponents of a superfield using Grassmann in-tegration. For example, we can use
∫d4# to pick
out the top component of a superfield, or∫
d2#to pick out the #2 component. We thus write
L =∫
d4# ## , (11)
up to total derivatives. In components:
L = ""µ&"µ& + i"µ%!µ% + FF . (12)
This describes a free complex scalar &, a freeWeyl fermion %!, and a non-propagating auxil-iary field F . In this example F vanishes by itsequations of motion. Because the choice K = ##gives rise to the usual kinetic terms for & and %!,it is called a canonical Kahler potential.
This trivial free theory can be readily general-ized: consider N chiral superfields #i and con-sider the Lagrangian
L =∫
d4# K(#, #) , (13)
where the Kahler potential K may now be a com-plicated real function of the #i and their com-plex conjugates. (The superpotential W is still
taken to vanish.) After solving for the auxiliaryfields F i, which are now non-zero, the componentLagrangian takes the form
L = "gij
(&, &
)"µ&j"µ&i + igij "µ%i!µ%j
+ igij$ikl
("µ&k
)%l!µ%j +
14Rijkl %
i%k %j%l .(14)
This theory is known as the supersymmetric non-linear !-model. The bosonic part is an ordi-nary !-model, whose target space is an N com-plex dimensional manifold with metric gij . Sincethe Lagrangian (14) does not have a potentialterm, any point &i on the target manifold is asupersymmetric vacuum. (Recall that it followsfrom the SUSY algebra (1) that supersymmetricvacua have zero energy; vacua with positive en-ergy spontaneously break SUSY.) In supersym-metric theories, it is customary to refer to suchmanifolds of vacua as moduli spaces. As usual,the scalars &i should be thought of as coordinateson the target space. This is consistent with thefact that the theory is invariant under holomor-phic field redefinitions of the form
#"i = f i(#j) . (15)
Such field redefinitions can be thought of as co-ordinate changes on the target manifold underwhich the metric transforms in the usual tenso-rial way.
In the supersymmetric !-model, the metric isdetermined by the Kahler potential:
gij = "i"jK . (16)
Thus, the target space is a Kahler manifold. Notethat gij must be positive definite so that the ki-netic terms in (14) have the correct sign requiredby unitarity. The fermions couple to the geome-try of this Kahler manifold through the connec-tion $i
jk and the curvature tensor Rijkl, whichdepend on the metric and its derivatives. Undera coordinate chage (15) the fermions transformas %"i = "#!i
"#j%j . They can thus be identified
with vectors in the tangent space of the targetmanifold.
The dynamics of the non-linear !-model is un-changed under redefinitions of the Kahler poten-
4
tial which take the form
K "(#, #) = K(#, #) + F (#) + F (#) , (17)
for some holomorphic function F of the #i. Thisis known as a Kahler transformation. Such atransformation only changes the Lagrangian by atotal derivative. This is because F (#i) is chiral sothat its top component is a total derivative. Al-ternatively, we see from (16) that the metric gij isuna"ected by Kahler transformations, so that thecomponent Lagrangian (14) is also unchanged.
It is important to note that while we are alwaysfree to perform Kahler transformations, there aresituations in which we are forced to do so. Thisis the case whenever the target manifold has non-trivial topology and must be covered with severalpatches which make it impossible to consistentlydefine a single-valued Kahler potential. In thiscase we are forced to use di"erent Kahler poten-tials in di"erent patches. These Kahler poten-tials must be related by a Kahler transformationwhenever two patches overlap.
As an example of a supersymmetric !-modelwhich illustrates this point, we consider a singlechiral superfield # with Kahler potential
K = f2$ log
(1 + |#|2
). (18)
In this normalization # is dimensionless, whilethe constant f$ has dimensions of mass.5 Themetric is g## = f2
π(1+|#|2)2 , which we recognize
as the familiar round metric on the two-spherewith radius ! f$. The coordinate values & = 0and & = $ correspond to antipodal points. Aswe explained above, the moduli space of SUSYvacua is just the two-sphere itself. This theory isusually referred to as the CP1-model. The Kahlerpotential (18) gives rise to the Fubini-Study met-ric on CP1, which is nothing but the round metricon the two-sphere.
To describe the point at infinity, we must per-form a change of variables # % 1/#. This in-duces a Kahler transformation (17) with F (#) =f2
$ log(#). It is, in fact, impossible to cover the5This notation is due to the fact that fπ is the ana-logue of the pion decay constant in the chiral Lagrangianfor QCD. Our example (18) describes the coset mani-fold SU(2)/U(1) = CP1.
two-sphere (or, as we will later explain, any othercompact manifold) with patches in such a waythat K is a globally well-defined scalar.
So far we have only discussed theories with-out a superpotential W , such as the SUSY !-model (13). These theories did not have anyscalar potential. The simplest example with asuperpotential is a single chiral superfield # withcanonical Kahler potential and with superpoten-tial W = 1
2m#2. After integrating out the aux-iliary field F , this leads to the component La-grangian
L =" "µ&"µ&" |m|2|&|2 + i"µ%!µ%
" 12m%2 " 1
2m!%2 . (19)
Thus, & and % both acquire a mass |m|.More generally, we can take the SUSY !-
model (13) and add a superpotential W (#i).Since W must be chiral, it must be holomorphicin the #i. The resulting theory is called a Wess-Zumino model:
L =∫
d4# K(#, #) +∫
d2# W (#) + c.c. . (20)
After integrating out the auxiliary fields, thisadds to the component Lagrangian (14) of the!-model a scalar potential and Yukawa-type in-teractions:
L &" gij"iW"jW
" 12
(("i"jW " $k
ij"kW)%i%j + c.c.
). (21)
The vacua of this theory are given by the min-ima of the scalar potential which appears in thefirst line of (21). Since supersymmetric vacuahave zero energy and the metric gij is positivedefinite, the space of SUSY vacua is specified bythe N complex equations "iW = 0 in the N com-plex unknowns &i. This implies that without anyfurther structure Wess-Zumino models generallyhave SUSY vacua. In section 3, we will exploresome simple theories whose vacua spontaneouslybreak supersymmetry.
1.3. Vector Multiplets and Gauge TheoriesIn this section we will describe the vector mul-
tiplet, which will enable us to write Lagrangians
5
for supersymmetric gauge theories. For simplicitywe will limit ourselves to the Abelian case. A vec-tor multiplet is a real superfield V = V , subjectto the gauge transformations
V % V + i(%" %
), (22)
where % is a chiral superfield. These gauge trans-formations allow us to fix Wess-Zumino gauge inwhich the low-lying components of V vanish:
V = "(#!µ#
)Aµ+i#2#'"i#2#'+
12#2#2D . (23)
Here Aµ is a real gauge field, the Weyl fermion '!
is its gaugino superpartner, and D is a real scalarauxiliary field. The residual gauge freedom whichremains in Wess-Zumino gauge just consists ofordinary gauge-transformations for Aµ.
To capture the gauge-invariant informationin V , we can define the superfield
W! = "14D2D!V , (24)
which is invariant under (22). From this defini-tion, we immediately see that W! satisfies theconstraints
D!W! = 0 , D!W! = D!W ! . (25)
In components:
W! =" i'!(y) + #%
((!
%D(y)
" i(!µ&)!%Fµ&(y)
)+ #2!µ
!!"µ'!(y) , (26)
where the closed, real two-form Fµ& = "µA& ""&Aµ is the field strength of Aµ. We thereforrefer to W! as a field strength superfield. Wesee that W! is a 4+4 multiplet, corresponding tothe 4+4 gauge-invariant degrees of freedom in V .
For future reference, we note that any spinorsuperfield W! which satisfies the constraints (25)has the component expansion (26) for someclosed, real two-form Fµ& . Locally, we can al-ways express such a two-form as Fµ& = "µA& ""&Aµ for some vector Aµ, and we can conse-quently write W! as in (24) for some vector
superfield V . However, in general Aµ and Vare not well-defined: they may undergo gauge-transformations.
The superfield W! has exactly the right fieldcontent to write supersymmetric kinetic terms forthe gauge field and the gaugino. Since W! is chi-ral, we can take
L =1
4e2
∫d2# W!W! + c.c.
= " 14e2
Fµ&Fµ& +i
e2"µ'!µ' +
12e2
D2 . (27)
Here e is the gauge coupling. In this simple freetheory the auxiliary field D vanishes by its equa-tions of motion.
In Abelian gauge theories, it is possible to adda so-called Fayet-Iliopoulos (FI) term to the La-grangian:
LFI = $
∫d4#V . (28)
Here the FI-parameter $ is real. The FI-term isgauge invariant because the top-component of achiral superfield is a total derivative. Note thatthe gauge-invariance of the FI-term resembles theinvariance of the Wess-Zumino Lagrangian (20)under Kahler transformations. This connectionwill resurface in later sections.
2. Global Symmetries
2.1. Global Current SupermultipletsIn local quantum field theories, the Noether
theorem guarantees that each continuous globalsymmetry gives rise to a conserved current jµ sat-isfying
"µjµ = 0 , Q =∫
d3x j0 . (29)
Here Q is the conserved charge correspondingto jµ. In this review, we will not discuss R-symmetries, so that Q is just an ordinary globalsymmetry.6 In SUSY theories, we must em-bed jµ in some supermultiplet. In order to de-cide what constraints this superfield will satisfy,we are guided by the following two observations:6An R-symmetry is a global symmetry which does notcommute with SUSY: [R, Qα] = −Qα.
6
• Because Q is an ordinary global symmetrycharge, it commutes with SUSY, [Q!, Q] =0. This implies the current algebra
[Q!, jµ] = (S.T.) . (30)
The right-hand-side of this equation is con-served when acting with "µ on both sides,and it is a total space-derivative when µ = 0so that it integrates to zero; it is known asa Schwinger Term (S.T.).
• We expect jµ to be the highest spin com-ponent in its supermultiplet. If this is notthe case, we could not gauge jµ without in-troducing higher-spin gauge fields. This isnot expected to be consistent in rigid fieldtheories without gravity.
We thus posit that jµ is embedded in a real, scalarsuperfield J which schematically takes the form
J = J + i#j" i#j +(#!µ#
)(jµ + · · · )+ · · · . (31)
Here J is a real scalar, and j! is a Weyl fermion.The dots are terms which are not uniquely fixedby the requirements outlined above. Di"erentchoices of Schwinger terms in (30) terms lead todi"erent ways of completing the multiplet. Al-though there is a constructive way of writing themost general solution, we will not follow this ap-proach here. Instead, we will simply write down aset of constraints for J and explore the resultingcomponent structure.
Conventionally, the global current supermulti-plet J is taken to satisfy
D2J = D2J = 0 . (32)
A real superfield obeying these constraints iscalled a linear multiplet. In components, sucha multiplet takes the form
J = J + i#j " i#j +(#!µ#
)jµ
+12#2#!µ"µj " 1
2#2#!µ"µj " 1
4#2#2"2J , (33)
where jµ is the conserved vector current. Weimmediately see that J contains no higher-spin
components. Moreover, we can use (4) on (33) toextract the commutation relation
[Q!, jµ] = "2i(!µ&)!%"&j% , (34)
whose right-hand-side is indeed a pure Schwingerterm. Having familiarized ourself with the struc-ture of the global current multiplet (33), we cannow explore how it arises in theories with globalsymmetries.
We begin by revisiting the free theory of a chi-ral superfield (11), which is invariant under U(1)phase rotations of #. This corresponds to the in-variance of the component theory (12) under U(1)phase rotations of &,%! and F . The equation ofmotion D2# = 0 implies that J = ## is a lin-ear multiplet, which in the normalization of (33)contains the conserved vector current
jµ = i(&"µ&" "µ&&
)+ %!µ% . (35)
This is the usual U(1) phase current which givescharge +1 to & and %!.7
More generally, we can consider continuousglobal symmetries of the SUSY !-model (13). As-sume that this theory is invariant under the in-finitesimal global symmetry transformation
(#i = )(a)X(a)i(#) . (36)
Here the X(a)i are holomorphic in the #i andthe )(a) are infinitesimal real parameters; we willuse indices a, b, . . . to label di"erent global sym-metries. The transformation (36) must leavethe metric gij invariant.8 However, the La-grangian (13) may pick up a Kahler transforma-tion:
(L =∫
d4# )(a)(F (a)(#) + F (a)(#)
), (37)
where the functions F (a) are holomorphic inthe #i. This only changes the Lagrangian by atotal derivative.
7In our convention, an operator O has charge q under thesymmetry generated by Q if it satisfies [Q,O] = qO.8Geometrically, this means that the X(a)i are the compo-nents of holomorphic Killing vector fields X(a) = X(a)i∂i.
7
To find the conserved currents correspondingto the symmetries (36), we perform the super-space analog of the usual Noether procedure: re-place each infinitesimal transformation parame-ter )(a) with an arbitrary infinitesimal chiral su-perfield %(a). (The fact that the %(a) are chi-ral ensures that the variations of the chiral su-perfields #i are still chiral.) To linear order inthe %(a), the change in the Lagrangian must nowtake the form
(L =∫
d4# %(a)(F (a) " iJ (a)
)+ c.c. , (38)
for some real superfields J (a). Note that (38)reduces to (37) if we set %(a) = )(a). However,using the explicit form (36) of the infinitesimaltransformation with )(a) replaced by %(a), we canalso write the change in the Lagrangian as
(L =∫
d4# %(a)X(a)i"iK + c.c. . (39)
Since (38) and (39) must agree for all chiral super-fields %(a), we can identify the Noether currents
J (a) = iX(a)i"iK " iF (a) . (40)
These are guaranteed to be real and conserved, ascan be checked using the sigma model equationsof motion D2"iK = 0. Note that the Noethercurrents (40) depend on the functions F (a) whichappear in the Kahler transformation (37). This isnot surprising: when the Lagrangian changes bya total derivative under a symmetry transforma-tion, then the Noether current acquires an addi-tional piece.
Such a situation can arise even in the triv-ial theory (11). This theory has a shift sym-metry (# = ) under which the Lagrangian un-dergoes a Kahler transformation with F = #.By (40), the Noether current for this symmetryis given by
J = i(#" #) . (41)
An interesting complication occurs if we compact-ify the target space of this model to a cylinder byidentifying # ! #+ i. As we go around the cylin-der once, the bottom component of J shifts by
a constant. It is thus not a good operator in thetheory. In this case, we cannot embed the vec-tor current corresponding to the shift symmetryin the linear multiplet J . This also means thatwe cannot gauge the shift symmetry in the usualway (see subsection 2.2). This di!cutly can beovercome by embedding jµ in a di"erent multipletwhich is well-defined. (The details of this proce-dure are beyond the scope of this review.) Con-ceptually, this situation has an exact analoguefor multiplets containing the supersymmetry cur-rent S!µ and the energy-momentum tensor Tµ& ,to be discussed in detail below.
2.2. Gauging Global SymmetriesIn order to couple matter to a gauge field Aµ,
we add to the Lagrangian a term
(L ! jµAµ + · · · . (42)
Here jµ is a conserved matter current, so that thisinteraction is invariant under gauge transforma-tions of Aµ. If the current jµ itself transformsunder gauge transformations, then (42) containsadditional terms (represented by the dots) withhigher powers of Aµ to ensure gauge invariance.They will not be important for us, and we willnot discuss them.
In SUSY theories, the analogue of (42) is givenby the coupling
(L =∫
d4#J V , (43)
where J is a linear multiplet and V is a vec-tor multiplet. Since D2J = 0, this interac-tion term is invariant under gauge transforma-tions (22) of V . In addition to the term in (42),the interaction (43) contains Yukawa-type inter-actions for the gaugino '!, as well as a couplingof the bottom component J of J to the auxiliaryfield D of the vector multiplet V . When D is in-tegrated out, it gives rise to a new contributionto the scalar potential:
VD =e2
8(J + $)2 , (44)
where e is the gauge coupling and $ is an FI-term,which may be present. We refer to (44) as a D-term potential, since it arises from integrating out
8
the D-component of a vector multiplet. This isto be contrasted with the scalar potentials dis-cussed at the end of subsection 1.2, which werethe result of integrating out the F -componentsof chiral superfields. These are known as F -termpotentials.
As a simple example, which will reappear inlater sections, consider a theory of two chiral su-perfields #1,2 with canonical Kahler potential andcharge +1 under U(1) phase rotations.9 The cor-responding conserved current is given by
J = #1#1 + #2#2 . (45)
This gives rise to the D-term potential
VD =e2
8(|&1|2 + |&2|2 + $
)2. (46)
If $ > 0, then the vacuum energy is positive:there are no SUSY vacua. If $ < 0, then themoduli space of vacua is given by the three-sphere |&1|2 + |&2|2 = "$ modulo U(1) gaugetransformations acting on &1,2, which is nothingbut CP1. The corresponding low-energy theorydescribing the massless moduli is the CP1-model,which we introduced at the end of subsection 1.2.
For future use, we now briefly describe howto gauge a general global symmetry of the non-linear !-model. As in the examples above, wesimply add to the Lagrangian (13) a term
L " =∫
d4#J (a)V (a) . (47)
Here the J (a) are the Noether currents (40).Under a gauge-transformation with chiral gaugeparameter %(a), the vector fields transform inthe usual way, while the change in the !-modelLagrangian L is given in (38). The completeLagrangian is now gauge-invariant up to totalderivatives:
( (L + L ") =∫
d4# %(a)F (a) + c.c. + · · · , (48)
where the dots represent unimportant higher-order terms and the F (a) are the holomorphic9The fact that this theory is quantum mechanicallyanomalous will not be important for us.
functions which appear in the Kahler transfor-mation (37). The D-term scalar potential in thistheory is given by
VD =18
∑
a
g2a
(J (a)
)2. (49)
Here the ga are the gauge coupling constantswhich arise from the kinetic terms of the di"er-ent V (a), and the J (a) are the bottom componentsof the Noether currents J (a).
3. SUSY-Breaking
3.1. Simple ExamplesIf supersymmetry is to play any role in describ-
ing nature, then it must be spontaneously bro-ken. As we already mentioned, SUSY is sponta-neously broken if the vacuum energy is positive.Moreover, broken supersymmetry always leads toa massless fermion, the Goldstino, which is theexact analogue of the Goldstone bosons which ap-pear when ordinary global symmetries are sponta-neously broken. We will now explore a few simpleexamples which break SUSY at tree-level.
The simplest SUSY-breaking theory consists ofone chiral superfield # with a linear term in thesuperpotential
L =∫
d4# ## +∫
d2# f# + c.c. . (50)
The scalar potential in this theory is simply aconstant V = |f |2. Thus, the spectrum con-sists of a massless fermion %! – the Goldstino– and a massless scalar &. Since there is nopotential for &, the theory has infinitely manySUSY-breaking vacua with nonzero energy |f |2,and since the theory is free these vacua are notlifted.
We can make this model more interesting byadding an additional term to the Kahler poten-tial:
L =∫
d4#
(##" 1
4%2#2#2
)
+∫
d2# f# + c.c. . (51)
9
This is an e"ective theory below the high cuto"scale %. The scalar potential is now given by
V =|f |2
1" 1!2 |&|2
= |f |2(
1 +|&|2
%2+ · · ·
). (52)
Now there is a single SUSY-breaking vacuumat & = 0. The boson & has acquired amass |f |2/%2, while the massless fermion %! isagain identified with the Goldstino.
The model (51) is not renormalizable. The sim-plest non-trivial, renormalizable model of spon-taneous SUSY-breaking is the O’Raifeartaighmodel [3]. The model has three chiral super-fields X, #, # with canonical Kahler potential andsuperpotential
W = X#2 + m## + fX . (53)
The conditions for the existence of a SUSY vac-uum are & = 0, &2 = f , and m& = 2x&,where we denote by x,&, & the bottom compo-nents of X, #, #. Clearly these conditions are in-consistent and there is no supersymmetric vac-uum. By studying minima of the full scalar po-tential V , we see that the theory has a SUSY-breaking vacuum with energy V = |f |2 at & =& = 0. (This vacuum is only stable if 2|f | < |m|2;if this is not the case, the structure of the vacuumis somewhat di"erent, but SUSY is still broken.)For & = 0, the scalar potential is independentof x. The field x is thus massless and can takeon any vacuum expectation value. Just like thetrivial theory (50), the O’Raifeartaigh model thushas infinitely many SUSY-breaking vacua at treelevel. (This is a very general property of suchtheories [4,5].) However, unlike the previous the-ory which was free, it is now possible for radiativecorrections to lift this vacuum degeneracy: x be-comes massive and is stabilized at the origin bythe one-loop correction to the scalar potential:
V (1) ! 116*2
|f |2
m2|x|2 + · · · . (54)
Note that after integrating out the heavyfields #, # of mass ! m, the low-energy dynam-ics of X is governed by an e"ective theory of theform (51) with the cuto" given by % ! m.
In the three examples above SUSY was brokenbecause the vacuum energy was positive due tonon-vanishing F -terms coming from chiral super-fields. It is also possible to break supersymmetrythrough non-vanishing D-terms coming from vec-tor superfields. The prototypical such example isthe FI-model:
L =1
4e2
∫d2# W!W! + c.c. +
∫$d4# V . (55)
This model is free, but has a vacuum energy V =e2$2/8. The free gauge field is necessarilymassles, while the massless gaugino is the Gold-stino.
3.2. Comments on Dynamical SUSY-Breaking
Unlike other symmetries, the spontaneousbreaking of supersymmetry is highly con-strained. It is a consequence of powerful non-renormalization theorems [6,7] that supersymmet-ric vacua cannot be lifted by radiative corrections:if there is a SUSY vacuum in the classical theory,then it cannot disappear in perturbation theory.Thus SUSY can only be broken in two ways:
• The classical theory has no SUSY vacua.In this case we say that it breaks SUSY attree-level.
• The classical theory has SUSY vacua, butthey are lifted by non-perturbative ef-fects. This is known as dynamical SUSY-breaking.
All theories discussed in the previous subsec-tion break SUSY at tree-level. As was empha-sized by Witten [8], such models are unappealingbecause they force us to introduce dimensionfulparameters by hand so that we still have to ex-plain the large hierarchy between the electroweakscale and the Planck or GUT scale.
On the other hand, the option of dynamicalSUSY-breaking is very appealing. The scale
#f
of supersymmetry breaking can now arise throughdimensional transmutation√
f ! %UV e# 8π2
g2 , (56)
with g ! 1 an asymtotically free gauge cou-pling, and thus
#f can naturally be exponentially
10
smaller than the cuto" %UV . This might explainwhy the weak scale is so much smaller than thePlanck or GUT scale. We should thus be study-ing theories with strong dynamics in the IR whichcan trigger SUSY-breaking. This is a vast sub-ject (see [9] and references therein), and we willrestrict ourselves to a few general comments.
Roughly speaking, dynamical theories fall intothree classes:
• Theories in which supersymmetry breakingis triggered by non-perturbative e"ects, butthe vacuum is in the semiclassical regime.
• Theories in which the vacuum is in thestrongly-coupled regime of the original de-grees of freedom, but can still be analyzedvia Seiberg duality [10].
• Strongly coupled theories in which little canbe said about the vacuum, but for whichthere are convincing (usually indirect) ar-guments that SUSY is broken.
The first two types of models are referred to ascalculable. At low energies, they are described byWess-Zumino models, possibly with IR-free gaugefields and e"ective FI-terms. By studying suchmodels we may hope to shed light on the dynam-ics of interesting dynamical models.
In the next section we will continue our study ofinteresting supermultiplets. We will then explainhow these multiplets can be used to constrain thedynamics and low-energy behavior of various the-ories – both with and without SUSY breaking.
4. Supercurrents
4.1. The Ferrara-Zumino MultipletFour-dimensional theories with N = 1 super-
symmetry possess a conserved supersymmetrycurrent S!µ satisfying (2). Just like any otherwell-defined local operator, we have to embed S!µ
in some multiplet. In order to write down sensi-ble constraints for such a supercurrent multiplet,we are guided by the following two observations:
• The SUSY-algebra (1) gives rise to the cur-rent algebra
{Q!, S!&} = 2!µ!!Tµ& + (S.T.) , (57)
where (S.T.) are Schwinger Terms, and Tµ&
is a conserved, symmetric energy momen-tum tensor:
"&Tµ& = 0 , Pµ =∫
d3xTµ0 . (58)
The supersymmetry currrent and theenergy-momentum tensor must thus belongto the same supermultiplet – a supercur-rent.
• We expect Tµ& to be the highest spin com-ponent in the supercurrent multiplet. If thisis not the case, then it might be problematicto couple the theory to supergravity.
We conclude that the supersymmetry current andthe energy-momentum tensor are embedded ina real vector superfield Jµ which schematicallytakes the form
Jµ = jµ " i# (Sµ + · · · ) + i#(Sµ + · · ·
)
+(#!& #
)(2T&µ + · · · ) + · · · . (59)
The real vector jµ is generally not conserved.As in the case of the global current multiplet,the dots are terms which are not uniquely fixedby the requirements outlined above, and di"erentchoices of Schwinger terms in in (57) give rise todi"erent ways of completing the multiplet. Againwe will not discuss the constructive approach towriting the most general solution [11]. Instead,we will begin by exploring the most conventionalset of constraints for the supercurrent multiplet,only amending them when we are forced to do so.
The simplest and most widely known super-current is called the Ferrara-Zumino (FZ) multi-plet [12]. It is defined by the equations10
D!J!! = D!X , D!X = 0 . (60)
10Our convention for switching between vectors and bi-spinors is #αα = −2σµ
αα#µ, #µ = 14 σαα
µ #αα .
11
In components, the FZ-multiplet takes the form
Jµ = jµ " i#
(Sµ +
13!µ!&S&
)+
i
2#2"µx
+ i#
(Sµ +
13!µ!& S&
)" i
2#2"µx
+(#!& #
) (2Tµ& "
23+µ&T " 1
2)&µ'("'j(
)
+ · · · . (61)
Here T = T)) is the trace of the energy-
momentum tensor. The higher componentsof J!! only contain derivatives of componentswhich we have already displayed. The chiral su-perfield X takes the form
X = x(y) +#
2#%(y) + #2F (y) , (62)
with
% =i#
23
!µ!!S!
µ ,
F =23T + i"µjµ . (63)
We see that the FZ-multiplet contains 12 =2 [x] + 4 [jµ] + 6 [Tµ& ] independent bosonic com-ponents, which match the 12 fermionic compo-nents of S!µ. Thus, the FZ-multiplet is a 12+12multiplet. It is straightforward to check that theexplicit component expressions (61) and (62) leadto a SUSY current algebra of the form (57):
{Q!, S!µ} = !&!!
(2T&µ + i"&jµ
" i+µ&")j) "12)&µ')"'j)
). (64)
When X vanishes, then we see from (63) that theenergy-momentum tensor is traceless, and that jµ
is now a conserved current. In this case, the the-ory is superconformal and the FZ-multiplet onlyhas 8 + 8 components.
As the simplest example, we again consider thefree theory (11) of a single chiral superfield #.Using the superspace equation of motion D2# =0 it is straightforward to check that the multiplet
J!! = 2D!#D!#" 23[D!, D!]## (65)
satisfies
D!J!! = 0 . (66)
As expected, the FZ-multiplet reduces to thesuperconformal multiplet for the free scalar.By expanding # in components and comparingwith (61), we see that the energy-momentum ten-sor embedded in J!! reads
Tµ& = "µ&"&& + (µ ' ,)" +µ&")&")&
" 13
("µ"& " +µ&"2
)|&|2 + (fermions) . (67)
The first line is the familiar energy-momentumtensor for a free, massless scalar. The second lineis an improvement term: it is automatically con-served (without using the equations of motion),and it does not a"ect the conserved charge Pµ
because setting , = 0 turns it into a total spacederivative. We are free to add these kind of im-provement terms to any conserved current. Inthis case the constraint D!J!! = 0 fixes theimprovement term in a particular way: it guar-antees that Tµ& is traceless. The relation be-tween supersymmetric constraints and improve-ment terms will play a pivotal role in what fol-lows.
More generally, we can consider the Wess-Zumino model (20). Using the equations of mo-tion D2"iK = 4"iW , we can check that
J!! = 2gijD!#iD!#j " 23[D!, D!]K ,
X = 4W " 13D2K (68)
satisfy the defining equation (60) of the FZ-multiplet. We will make use of this formula onseveral occasions.
Finally, the FZ-multiplet for the FI-model (55)readily follows from the equation of mo-tion D!W! = e2$. It is given by:
J!! = " 4e2
W!W! "23$[D!, D!]V ,
X = "$
3D2V . (69)
It is similarly straightforward to obtain the FZ-multiplet for gauge theories coupled to matter (we
12
will not write down the most general expressionhere).
Looking at (55) we make an unsettling obser-vation: in the presence of an FI-term $, nei-ther J!! nor X are invariant under the usualgauge-transformation (V = i(%" %), with % chi-ral [13]. (This conclusion is not changed by theinclusion of matter.) Thus, the FZ-multiplet isnot gauge-invariant, and consequently its compo-nents are not well-defined operators. From this,we might mistakenly conclude that models withFI-terms are inherently ill-defined, because theydo not have a well-defined supersymmetry currentor energy-momentum tensor.
To see why this is not so, we examine the gaugenon-invariance of the FZ-multiplet (69) in moredetail. Under a gauge transformation, this mul-tiplet transforms as follows:
(J!! = "23$"!!
(% + %
),
(X =i
3$D2% . (70)
By expanding these expressions in componentsand comparing with (61), we see that the bot-tom component jµ ! $Aµ + (fermions) explic-itly depends on the gauge field Aµ and is sim-ply not gauge-invariant. However, the gauge non-invariance of the energy-momentum tensor takesthe special form
(Tµ& =23$("µ"& " +µ&"2
)Im%| , (71)
where %| denotes the bottom component of thesuperfield %. We see that the transformationof Tµ& in (70) is a pure improvement term. As weexplained above, this means that it is automati-cally conserved and does not a"ect the conservedcharged Pµ. Completely analogously, it can bechecked that the supersymmetry current S!µ alsoonly shifts by a pure improvement term. TheFI-model thus has well-defined, gauge-invariantoperators Pµ, Q!, which make it a well-definedsupersymmetric field theory.
How, then, do we interpret the fact that the su-persymmetry current and the energy-momentumtensor embedded in the FZ-multiplet (61) are notgauge-invariant and do not exist as well-defined
operators? The answer is that S!µ and Tµ& arenot unique; they are only defined up to improve-ment terms. Di"erent choices of improvementterms lead to di"erent local currents, but do nota"ect the corresponding charges. It is usuallypossible to chose the improvement terms in sucha way that the currents are well-defined opera-tors. However, a problem could arise when at-tempting to embed these operators into super-multiplets. In this case there may be a clash be-tween the existence of certain supermultiplets andthe requirement that these multiplets be gaugeinvariant. This is precisely what happens inthe FI-model: forcing S!µ and Tµ& into an FZ-multiplet requires us to pick gauge non-invariantimprovement terms. Conversely, it is possible topick gauge-invariant improvement terms for S!µ
and Tµ& , but this makes it impossible to embedthese operators into an FZ-multiplet. In this caseit is necessary (and possible) to find di"erent su-permultiplets into which we can embed gauge-invariant choices for S!µ and Tµ& . Such multi-plets will be discussed in section 6.
In light of the preceding discussion, we are ledto carefully reconsider the existence of the FZ-multiplet (68) for the Wess-Zumino model. Wesee that the Kahler potential K formally appearsin the same way as the vector field V in theFZ-multiplet for the FI-model. Morover, Kahlertransformations of K are take the same form asgauge-transformations of V . We conclude thatthe FZ-multiplet of the Wess-Zumino model is notinvariant under Kahler transformations. Again,these Kahler transformations only change the su-persymmetry current and the energy-momentumtensor by improvement terms.
However, unlike gauge transformations, Kahlertransformations are not an absolute necessity. Itoften does not matter if the multiplet transformsunder Kahler transformations. The only becomeessential when the target space M of the sigmamodel needs to be covered with several patchesand Kahler transformations are needed to switchbetween the patches. This happens, for instance,in the CP1 model described in subsection 1.2.In such theories, the FZ-multiplet is not a well-defined operator. Equivalent ways of describingthe target space of the theory result in energy-
13
momentum tensors which di"er by improvementterms, even though they should be identical, andlikewise for the supersymmetry currents. In otherwords, there is no unambiguous way of fixingthese operators.
We would like to understand the precise mathe-matical conditions under which the FZ-multipletfor Wess-Zumino models is well-defined. Fromthe metric gij we can construct the Kahler form
- = igijd&i ( d&j . (72)
Since gij is locally derived from a Kahler poten-tial, - is closed: d- = 0. In every patch wecan thus find a one-form A such that - = dA.Locally, this Kahler connection is given by A !i"iKd&i +c.c. . In general, - is not exact and theKahler connection A is not globally well-defined.The obstruction to the global existence of A ismeasured by the cohomology class [-] ) H2(M).If - vanishes in H2(M), then A is globally well-defined and thus a good operator the theory.
The bottom component of (68) is given by
jµ =2i
3"iK"µ&i + c.c." 1
3gij%
j !µ%i . (73)
While the fermionic piece only depends on themetric and is thus invariant under Kahler trans-formations, we recognize the bosonic piece as thepull-back to space-time of the Kahler connec-tion A. This is is well-defined only if - is ex-act and vanishes in H2(M). If this condition isnot satisfied, then jµ is not a good local opera-tor in the theory, since it depends on the choiceof patch in target space. In this case, the entireFZ-multiplet (68) is not well-defined [14].
As a special case, note that the Kahler form of acompact manifold can never be exact. If this werethe case, then the volume form -dim(M)/2 wouldalso be exact and thus integrate to zero on thecompact manifold; this is a contradiction. Thus,a Wess-Zumino model with compact target man-ifold, such as the CP1-model (18), cannot have awell-defined FZ-multiplet.
In this section we have explored the FZ-multiplet, in which the supersymmetry currentand the energy-momentum tensor are convention-ally embedded. This multiplet exists in the vast
majority of SUSY theories, including asymptot-ically free gauge theories, such as supersymmet-ric QCD (SQCD). As far as we know, the FZ-multiplet only fails to exist when one of the fol-lowing conditions holds:
• Some U(1) gauge group has an FI-term.
• The target-space has non-trivial topology,so that the Kahler form is not exact.
In the first case, the FZ-multiplet is not gauge-invariant, while in the second case it is not glob-ally well-defined. These insights will enable us toderive exact results about the dynamics of SUSYfield theories, SUSY-breaking, and supergravity.
4.2. Consequences for SUSY TheoriesIt is a general fact about quantum field the-
ory that operator equations are invariant underrenormalization group (RG) flow as long as theoperators involved are well-defined and gauge-invariant. This means that if some operator rela-tion holds in the UV (where field theories are usu-ally defined), then it continues to hold along theentire RG-flow. Applying this reasoning to theFZ-multiplet immediately leads to constraints onthe RG-flow of supersymmetric field theories.
Consider a supersymmetric field theory whichhas no FI-terms in the UV. This means that thistheory has a well-defined, gauge invariant FZ-multiplet. This multiplet must therefore persistalong the entire RG-flow. We conclude that FI-terms cannot be generated along the flow; this istrue at any order in perturbation theory and evennon-perturbatively. For instance, it might hap-pen that the theory flows through a regime withstrong dynamics, and that a U(1) gauge symme-try emerges in some weakly-coupled dual descrip-tion. In this case the gauge-invariance of the FZ-multiplet prevents this emergent U(1) gauge the-ory from having an FI-term. It is even possibleto argue that if an FI-term is present in the UV,then its value is not renormalized. Other deriva-tions of these results can be found in [15–18].
Completely analogously, let us consider a the-ory which has a well-defined, gauge-invariant FZ-multiplet in the UV. For example, we couldconsider an ordinary SUSY gauge theory, with
14
canonical Kahler potential for the matter fieldsand no FI-terms. At low energies, such field the-ories are often described by a weakly-coupled !-model (perhaps with IR-free gauge fields). It isexpected that this will happen whenever the fieldtheory has a strong coupling scale %, below whichit is described by massless moduli. Since this the-ory has a well-defined FZ-multiplet in the UV,and this multiplet persists along the entire RG-flow, we conclude that the low-energy !-modelmust have have an exact Kahler form -. Thisseverely constrains the quantum moduli space ofthe low-energy theory: the integral
∫- ( - ( · · ·
over any compact, even-dimensional cycle mustvanish. As we already explained in the previoussubsection, the means in particular that the mod-uli space cannot be compact (of course, it can be aset of points). It is particularly interesting to testthis theorem in SQCD with the same number ofcolors and flavors. In this case the topology of themodul-space changes under RG-flow and becomesnon-trivial in the IR [19]. However, in accordancewith the general result above, the Kahler form ofthis deformed quantum moduli space is exact.
The formal similarity between the two theo-rems described above is not coincidental. Onthe one hand, they are both consequences of theRG-invariance of the FZ-multiplet. On the otherhand, it may be that if a theory does not havean FZ-multiplet in the UV due to the presence ofan FI-term, it may flow to a !-model which failsto have an FZ-multiplet because its target spacehas a non-exact Kahler form.
The simplest example of this phenomenon isthe theory discussed at the end of subsection 2.2.In the UV, this theory starts out as an abeliangauge theory with an FI-term $ and two chiralmatter fields of charge +1. When $ < 0, it followsfrom the scalar potential (46) that the low-energytheory which describes the moduli space of SUSY-vacua is given by the CP1-model, whose targetspace is compact.
5. Applications to SUSY-Breaking
In this section we will consider two applica-tions of the FZ-multiplet to SUSY-breaking. Thefirst application rests on the relation of the FZ-
multiplet to the Goldstino. This will enable us toderive exact results about such SUSY-breakingtheories. It will also lead to a useful formalismfor writing Lagrangians with non-linearly realizedsupersymmetry. The second application concernsthe interplay of F -terms and D-terms in SUSY-breaking theories. We show under very broad as-sumptions that there can be no SUSY-breakingvacua (even meta-stable ones) in which the D-terms are parametrically larger than the F -terms.
5.1. The FZ-Multiplet and the GoldstinoA universal prediction of spontaneous SUSY-
breaking is the existence of a massless Weylfermion, the Goldstino G!. We will not do jus-tice to the extensive literature on the Goldstino.Most of the references can be found in [20], whichis also the main reference for this subsection.
When a global symmetry is spontaneously bro-ken, the corresponding conserved charge is not awell-defined operator because its correlation func-tions are IR divergent. As a result, the physicalstates of the theory are not in linear representa-tions of the symmetry. However, the conservedcurrent and even commutators of local operatorswith the conserved charge do exist. In contrast tothe states, the local operators do in fact furnisha linear representation of the symmetry. Thesestatements carry over without modification whensupersymmetry is spontaneously broken in infi-nite volume: the supercharge Q! does not exist,but the supersymmtry current and (anti-) com-mutators with Q! do exist. (When SUSY is spon-taneously broken in finite volume, even Q! ex-ists.) This means that operators still reside in su-permultiplets, and that we can use the formalismof superspace and superfields without modifica-tion.
Recall the defining equation (60) of the FZ-multiplet:
D!J!! = D!X , (74)
where X is chiral. In this subsection we willonly discuss theories which admit a well-defined,gauge-invariant FZ-multiplet. This multiplet re-quires the existence of a complex scalar x, thebottom component of X, which is a well-definedoperator in the theory. If we are given a micro-
15
scopic description of the theory in the UV, we canexpress x in terms of elementary fields.
The chiral superfield X can be consistently fol-lowed along the RG-flow down to the IR, evenwhen SUSY is spontaneously broken. We can seethis in the simplest SUSY-breaking model (50),which only has a single chiral superfield #. Inthis theory, the F -term is non-zero, but & and %!
are free, massless fields. As was discussed insubsection 3.1, we identify %! with the Gold-stino G!. Since this theory is a Wess-Zuminomodel with K = ## and W = f#, we findfrom (20) that X = 8
3f#. In this simple exam-ple, the operator X is thus proportional to #, sothat the #-component of X is proportional to theGoldstino.
More generally, standard arguments aboutsymmetry breaking show that when supersymme-try is spontaneously broken, the low-energy su-persymmetry current is expressed in terms of theGoldstino G! as
S!µ =#
2f!µ!!G!+f "(!µ&)!%"&G% +· · · , (75)
for some constants f and f ". Note that the termwith f " is an improvement term and can be ig-nored. We see that at low energies, the spin- 3
2component of the supersymmetry current essen-tially decouples, while the spin- 1
2 component isproportional to the Goldstino. Since we knowfrom (63) that the #-component of X is given bythe spin- 1
2 projection of S!µ, we again concludethat at long distance the #-component of X isproportional to the Goldstino.
In the trivial example of a free chiral super-field #, the lowest component x of the superfieldX is proportional to the free scalar &. In general,the low-energy Goldstino is not accompanied bya massless scalar. Then, the bosonic operator xcannot create a-one particle state. Instead, thesimplest bosonic state it can create contains twoGoldstinos. When supersymmetry is broken infinite volume, the states in the Hilbert space arein linear supersymmetry multiplets. The super-symmetric partner of a one Goldstino state Q!|0*is a two Goldstino state Q1Q2|0*. At infinitevolume the supercharge does not exist and thezero momentum Goldstino state is not normaliz-able. However, the finite volume intuition is still
valid. The operator %# creates a one Goldstinostate and its superpartner & ! x creates a two-Goldstino state.
We denote this non-linear superfield which con-tains the Goldstino bilinear in the bottom com-ponent XNL, and xNL denotes the bottom com-ponent itself. Consistency of this superfield undersupersymmetry transformations (which, again,are legal even if SUSY is spontaneously broken)fixes it to take the form
XNL =G2
2F+#
2#G + #2F , (76)
where all the fields are functions of yµ.Clearly, (76) satisfies the interesting operatoridentity
X2NL = 0 . (77)
(Other realizations of nonlinear supersymmetrycan be found, for example, in [21,22])
We conclude that the Goldstino resides in a chi-ral superfield XNL which satisfies X2
NL = 0. Fur-thermore, this chiral superfield is the IR limit ofthe microscopic superfield X. Surprisingly, thisresult is true both for F -term and for D-termbreaking because it relies only on the existenceof the chiral operator X. (The only exception isthe situation with pure D-term breaking whichoccurs when a tree-level FI-term is present for anunbroken U(1) gauge theory. In this case the op-erator X is not gauge invariant and we do notdiscuss it here.11 Thus, our discussion is appli-cable in all the interesting models of dynamicalSUSY breaking.
We will momentarily see that this identificationof the low-energy limit of the operator X, whichexists in all field theories we are interested in,allows us to derive some non-perturbative results.
It is instructive to compare the situation herewith the theory of ordinary Goldstone bosons likepions. Clearly, the decay constant f here is anal-ogous to the decay constant f$ of pion physics.Both of them are well defined. However, in pion
11However, our discussion does apply in the Higgs phaseof the FI-model [23], where the low-energy limit of X canbe rendered gauge invariant by dressing it with fields thatobtain expectation values.
16
physics the order parameter +%%* for chiral sym-metry breaking does not have a universal def-inition with a precise normalization (it su"ersfrom wavefunction renormalization). In our casethe order parameter for supersymmetry breakingis the energy-momentum tensor which resides inthe same multiplet as the supersymmetry cur-rent. Hence, it has a well-defined normalization.Therefore, our X is completely well-defined. Cor-respondingly, the operator %% of pion physicsacts as an interpolating field for pions, but itsnormalization is not meaningful. In our case Xis used both as an order parameter for supersym-metry breaking and as an interpolating field forGoldstinos with well-defined normalization.
The analogy with pion physics also clarifiesthe meaning of our constraint (77). It arisesfrom removing the massless scalar in X. This isanalogous to describing pion physics by startingwith a linear sigma model with a !-field. Re-moving the !-field is implemented by imposing aconstraint UU† = 1, which is analogous to ourX2
NL = 0.Since in every microscopic theory we can iden-
tify X in the ultraviolet, and if SUSY is spon-taneously broken we also know its universal low-energy limit, we can calculate various correlationfunctions of operators at large separations evenin strongly coupled models. Hence even “incalcu-lable” models of SUSY breaking (like the SU(5)theory of [24] or the SO(10) theory of [25]) havea solvable sector at long distances. The oper-ator x interpolates between the vacuum and astate with two Goldstinos, with a universal nor-malization. This is because at long distances theleading behavior of x is proportional to that ofG2. Therefore,
lim|r|$%
+x(r)x(0)* =(
43*2
)2 1|r|6 . (78)
This is independent of the details of the micro-scopic theory and its coupling constants. In asimilar fashion we can calculate the long distancelimit of any correlation function of x and x.
This idea can be pushed further and many morenon-perturbative results of this form can be ob-tained. In fact, a pretty rich set of exact resultson SUSY-breaking theories arises with interesting
underlying mathematical structure, but we willnot elaborate on this any further here.
We will now explain how to use XNL to writesupersymmetric e"ective Lagrangians. We startwithout including derivatives. At that level, themost general supersymmetric Lagrangian subjectto the constraint X2
NL = 0 is∫
d4# XNLXNL +∫
d2# fXNL + c.c. , (79)
where without loss of generality we take f to bereal. This looks like the free chiral multiplet ex-cept that the superfield is constrained. This con-straint removes the massless scalar field and in-troduces nonlinearities.
More explicitly, the constraint can be solved asin (76). Substituting this in (79), we derive thecomponent Lagrangian
i"µG!µG + FF +G2
2F"2
(G2
2F
)+ (fF + c.c.) .
(80)
The equations of motion of the auxiliary fieldsF, F can be solved and upon substituting thisback in the component Lagrangian we find
L = "f2 + i"µG!µG + 14f2 G2"2G2
" 116f6 G2G2"2G2"2G2 . (81)
This is equivalent to the Akulov-Volkov La-grangian [26]. (See also [27–29].) Here we havegiven a fully o"-shell supersymmetric descriptionof this theory.
The real advantage of this approach is the sim-plifications in writing higher derivative correc-tions to (79) and, more interestingly, couplings topossibly light matter fields (such as MSSM fields,Goldstone bosons, ’t Hooft fermions etc.). Indeedsuch problems are easily solved in this formal-ism since we have an o"-shell description and sowe can use all the familiar superspace techniquesfor writing Lagrangians (the number of allowedoperators is smaller because of the various con-straints).
This toolbox can be put to use in many possiblecontexts and some recent examples include [30–37]
17
The last thing we would like to demonstrate be-fore closing this subsection is the way the nilpo-tent equation (77) arises from the dynamics in asimple example. In order to remove the masslessscalar we include a non-canonical Kahler poten-tial
K = ##" c
M2#2#2 , (82)
and
W = f# . (83)
Here c is a dimensionless positive number of orderone. Such a theory can arise as the low-energyLagrangian below some scale M after neglect-ing higher order terms in 1
M . (For example, thelow-energy limit of the standard O’Raifeartaighmodel looks like (82).) It is valid for√
f , E , M . (84)
Let us integrate out the massive bosons. Re-membering that the Lagrangian contains inter-action terms of the form " c
M2
∣∣2&F# " %2∣∣2, the
zero-momentum equation of motion of & sets itto
& =%2
2F#. (85)
Note that it is independent of c, or in other words,it is independent of the details of the high energyphysics (this is an example of our universality).
Upon substituting (85) back into #, we discoverthat in the #2 = 0. Hence, it satisfies the sameconstraint as XNL. To make the relation moreprecise, we can go through a careful calculationof the supercurrent. The main point, however,is that the nilpotentcy equation arises from inte-grating out heavy degrees of freedom.
5.2. Restrictions on Large D-TermsIn subsection 3.1 we encountered several simple
models of F -term and D-term SUSY-breaking.While models with FI-terms lead to many tree-level examples of D-term SUSY-breaking, mostknown calculable models of dynamical supersym-metry breaking are predominately F -term driven.We would like to understand whether this has
to hold in general, or whether there are in factdynamical models with large D-terms. In thissubsection, we will show that such models donot exist: under broad assumptions, the D-termsare necessarily parametrically smaller than the F -terms [36].
To get an intuitive picture, we can loosely iden-tify D-term breaking with the presence of FI-terms. In dynamical models, one usually startswith a well-behaved, asymptotically-free gaugetheory without FI-terms in the UV. Since suchtheories has a well-defined FZ-multiplet, the dis-cussion of subsection 4.2 show that they cannotacquire FI-terms at low energies. Roughly speak-ing, the absence of low-energy FI-terms impliesthe absence of large D-terms.
We can make this intuitive picture precise byusing the tools we have developed so far. Sincewe are interested in calculable models with para-metrically small F -terms, we consider theories inwhich the low-energy dynamics responsible forSUSY-breaking is described by a !-model (13)and the F -terms are set to zero in first approx-imation. In addition, we will include IR-freegauge fields by gauging some global symmetriesof the !-model. If the D-term potential whichresults from this gauging leads to SUSY-breakingvacua (even meta-stable ones), then these puta-tive vacua will have parametrically large D-termswhich are larger than the F -terms by inverse pow-ers of the small, IR-free gauge couplings. We willnow show that such vacua do not exist in the-ories which have a well-defined, gauge-invariantFZ-multiplet. (Therefore they cannot exist in in-teresting dynamical models.)
As was discussed at the end of subsection 2.2,the Lagrangian for the gauged !-model is givenby making the substitution
K % K + J (a)V (a) , (86)
and including conventional kinetic terms for thegauge fields. By making an identical substitutionin expression (68) for the FZ-multiplet of the !-model, we see that a gauge-transformation (48)changes this new FZ-multiplet J!! by an amount
(J!! ! i"!!%(a)F (a) + c.c. . (87)
18
Since we assumed that the UV theory had awell-defined FZ-multiplet, the same must be truefor the low-energy gauged !-model. Thus, allfunctions F (a) must vanish. The Noether cur-rents J (a) now take the simple form
J (a) = iX(a)i"iK . (88)
Using this expression, it is straightforward tocheck that the D-term potential (49) of thegauged !-model satisfies the identity
VD =12gij"iK"jVD . (89)
This identity immediately implies that a criticalpoint of the potential can only occur when VD =0. In other words, every critical point must bea supersymmetric minimum and this D-term po-tential does not admit SUSY-breaking vacua –not even metastable ones.
The argument we have given above explains theabsence of dynamical models with parametricallylarge D-terms. It is, however, possible to buildmodels of dynamical SUSY-breaking with com-parable D-terms and F -terms (see [36] and refer-ences therein).
6. More Supercurrent Multiplets
We have seen that there are well-defined su-persymmetric field theories which do not admitan FZ-multiplet. In these theories, we mustsearch for other, well-defined supercurrent mul-tiplets into which the energy-momentum tensorand the supersymmetry current can be embed-ded. Note that the existence of such alterna-tive multiplets has no e"ect on the results we ex-tracted in previous sections, which were based onthe existence of a well-defined FZ-multiplet.
There is indeed a supercurrent multiplet S!!
which exists in theories with FI-terms or non-exact Kahler form, which do not admit an FZ-multiplet. This supercurrent multiplet satisfiesthe defining equations
D!S!! = D!X + .! , D!X = 0 ,
D!.! = 0 , D!.! = D!.! . (90)
Note that X appears exactly as in the FZ-multiplet, while .! satisfies the same constraints
as the field-strength superfield W! discussed insubsection 1.3. In components, the S-multiplettakes the form
Sµ = jµ " i#
(Sµ "
i#2!µ%
)+
i
2#2"µx
+ i#
(Sµ "
i#2!µ%
)" i
2#2"µx +
(#!& #
) (2Tµ&
" +µ&Z +12/µ&'("'j( +
18/µ&'(F '(
)+ ... . (91)
The chiral superfields X and .! are given by
X = x(y) +#
2#%(y) + #2 (Z(y) + i"&j&(y)) ,
.! = "i'!(y) + #%
((!
%D(y)" i(!µ&)!%Fµ&(y)
)
+ #2!&!!"& '!(y) ,
(92)
with
D = "4T + 6Z ,
' = 2!µSµ + 3i#
2% . (93)
As before, the operator jµ is not in general con-served, while the supersymmetry current S!µ andthe symmetric energy-momentum tensor Tµ& areconserved. The S-multiplet has 4+4 more degreesof freedom than the FZ-multiplet. They consistsof the real scalar Z, the closed, real two-form Fµ&
and the Weyl fermion %!. The S-multiplet is thusa 16 + 16 multiplet.12
As in the case of the FZ-multiplet, the com-ponent expressions (91)and (92) lead to a SUSYcurrent algebra of the form (57):
{Q!, S!µ} = !&!!
(2Tµ& "
18/&µ'(F '( + i"&jµ
" i+&µ"'j' "12/&µ'("'j(
).
(94)
However, the Schwinger Terms are now mor com-plicated. The supersymmetry algebra (1) follows12Setting X = 0 in the S-multiplet results in the so-calledR-multiplet. It is a 12 + 12 multiplet whose bottom com-ponent jµ is a conserved R-current. We will not discussthe R-multiplet in this review, although we will occasion-ally refer to it in passing.
19
provided that∫
d3xFij vanishes for all spatial in-dices i, j.
All known supersymmetric field theories ad-mit a well-defined S-multiplet.13 Those theorieswhich do not have FI-terms or non-exact Kahlerform also admit a well-defined FZ-multiplet.14For instance, consider a Wess-Zumino model witharbitrary K and W . If the Kahler form is not ex-act, then the FZ-multiplet does not exist. How-ever, the S-multiplet exists, and takes the form
S!! = 2gijD!#iD!#j , (95)
with X = 4W and .! = D2D!K.The existence of the S-multiplet in theories
without an FZ-multiplet has important conse-quences for supergravity theories. In the nextsection we will explain how to construct super-gravity starting from the supercurrent, and thiswill lead to additional applications.
7. Elements of Linearized Supergravityand Constraints on Moduli
In this section we study the couplings of su-percurrent multiplets to supergravity theories.We are only interested in linearized supergrav-ity, namely the leading order in 1
Mp. This ap-
proximate analysis is su!cient to derive severalgeneral results.
This approach to supergravity is taken, for ex-ample, in [2]. We begin with a review of thecoupling of the FZ-multiplet to supergravity. Wethen explain the coupling of the S-multiplet tosupergravity.15
7.1. Gauging the FZ-MultipletWe start by reviewing the coupling of the FZ-
multiplet to linearized gravity. The FZ-multipletcontains a conserved energy-momentum tensorand supercurrent and can therefore be coupled to
13There are, however, theories in which DαX is well-defined, but X itself is not.14R-symmetric theories always admit a well-defined R-multiplet.15The case of the R-multiplet is very similar but will notbe discussed in detail because gravitational theories withglobal symmetries do not seem to be relevant according toour current understanding of quantum gravity.
supergravity. The supergravity multiplet is em-bedded in a real vector superfield H!!. The ##component of H!! contains the metric field, hµ& ,a two form field Bµ& , and a real scalar. The cou-pling of gravity to matter is dictated at leadingorder by∫
d4#J!!H!! . (96)
We should impose gauge invariance, namely,the invariance under coordinate transformationsand local supersymmetry transformations. Thegauge parameters are embedded in a complex su-perfield L!, which so far obey no constraints. Weassign a transformation law to the supergravityfields of the form
H "!! = H!! + D!L! " D!L! . (97)
where L! is the complex conjugate of L!, andthus this maintains the reality condition.
Requiring that (96) be invariant under thesecoordinate transformations, we get a constrainton the superfield L!. Indeed, invariance requiresthat
0 =∫
d4# D!J!!L! =∫
d4# XD!L! . (98)
Since X is an unconstrained chiral superfield weget the complex equation
D2D!L! = 0 . (99)
The analog of the Wess-Zumino gauge is that thelowest components of Hµ vanish, i.e.
Hµ
∣∣ = Hµ
∣∣*
= Hµ
∣∣*
= 0 , (100)
as well as the fact that Hµ
∣∣*(ν *
is symmetric inµ and ,.
There is also some residual gauge freedom:
• Hµ
∣∣*2 can be shifted by any complex di-
vergenceless vector. This leaves only onecomplex degree of freedom, "µHµ
∣∣*2 .
• The metric field hµ& transforms as
(hµ& = "&$µ + "µ$& , (101)
where $µ is a real vector.
20
• The gravitino transforms as
(&µ! = "µ-! . (102)
In this Wess-Zumino gauge the componentscontaining the gravitino and metric take the form
Hµ
∣∣*(ν *
= hµ& " +µ&h , (103)
and
Hµ
∣∣*2*
= &µ! + !µ!'&' . (104)
The top component of Hµ is a vector field whichsurvives in the Wess-Zumino gauge. The bosonico"-shell degrees of freedom in Hµ consist of thecomplex scalar "µHµ
∣∣*2 , six real degrees of free-
dom in the graviton and the four real degrees offreedom in the top component of Hµ, for a to-tal of 12 o"-shell bosons. For the fermions, wehave only the gravitino. It has 16 " 4 = 12 o"-shell degrees of freedom. This is the old minimalmultiplet of supergravity [38–40]. This is in ac-cordance with the 12 degrees of freedom in theFZ-multiplet.
A simple consistency check is to verify the lead-ing couplings of the graviton and gravitino tomatter. Recalling the formula for J!! (61) wefind
L ! hµ&Tµ& , (105)
as expected. Similarly, for the coupling of thegravitino to matter we get
L ! /!% (&µ + !µ!'&')!
(Sµ + 1
3!µ!'S'
)%
= &µ!S!µ . (106)
The last ingredient is the kinetic term for thegraviton and gravitino. We begin by constructinga real superfield EFZ
!! by covariantly di"erentiat-ing H!!
EFZ!%
= D+D2D+H!% + D+D2D%H +!
+D,D2D!H,% " 2"!%",+H,+ . (107)
The reader can easily check that this expressionis real. It is equivalent to a di"erent-looking ex-pression in [41]. The gauge transformations (97)act as
E"FZ!%
= EFZ!%
+[D!, D% ](D2D!L! + D2D%L%
).
(108)
We see that EFZ!%
is invariant if (99) is imposed.The superfield EFZ
!%satisfies another important
algebraic equation
D%EFZ!%
= D!
(D2[D, , D+ ]H,+
). (109)
Here, the expression in parenthesis is chiral. Notethe similarity of the equation above to the defin-ing property of the FZ-multiplet itself (60). Thefact that EFZ is invariant and satisfies an equa-tion identical to the supercurrent superfield guar-antees that the Lagrangian
Lkin ! M2P
∫d4# HµEFZ
µ (110)
is invariant. This contains in components thelinearized Einstein and Rarita-Schwinger terms.The six additional supergauge-invariant bosons,"µHµ
∣∣*2 , Hµ
∣∣*4 are auxiliary fields which are eas-
ily integrated out yielding "µHµ
∣∣*2 ! ix, Hµ
∣∣*4 !
jµ where x and jµ are the matter operators in thesupercurrent multiplet.
We conclude that theories which have a well-defined FZ-multiplet can be coupled to super-gravity in this fashion. The coupling to super-gravity adds to the original theory a propagat-ing graviton and gravitino. No other propagatingfields are present in theory other than the orig-inal ones and the graviton and gravitino. Thisis important to remember in order to appreciatethe point we will make soon. Note that if theFZ multiplet is not well defined (for example ifit is not gauge invariant or if it is not globallywell defined) the procedure above of constructingminimal supergravity cannot be carried out.16
7.2. Supergravity from the S-MultipletWe emphasized in the previous sections that
various supersymmetric field theories do not have
16If there is no FZ-multiplet but there is an R-symmetry,we can construct the R-multiplet and couple it to super-gravity. For example, a free supersymmetric U(1) theorywith an FI-term can be coupled to supergravity in thisfashion. For more comments on this case see also [42,43].This procedure, however, gives rise to a supergravity the-ory with a continuous global R-symmetry.
21
an FZ-multiplet and the energy-momentum ten-sor and the supersymmetry current must be em-bedded in a larger multiplet S!!. In such a casethe construction of the previous subsection can-not be accomplished and the only possible su-pergravity theory is the one in which the S!! isgauged.
In this section we analyze this theory and asin the previous subsection, we limit ourselves tothe analysis of the linearized theory. Since wehave already understood how to do such things inthe previous sections, we will be somewhat briefernow.
We begin from the coupling to matter∫
d4# S!!H!! . (111)
For this to be invariant under (97), we need toimpose the constraints
D2D!L! = 0 , D!D2L! = D!D2L! . (112)
The first of them already appeared in the gaug-ing of the FZ-multiplet but the second one is new.Since L! is more constrained here than in the pre-vious subsection, we will find more gauge invari-ant degrees of freedom. Some of them will evenpropagate.
Using an arbitrary L! subject to these con-straints we can choose the Wess-Zumino gauge
Hµ
∣∣ = Hµ
∣∣*
= Hµ
∣∣*
= 0 . (113)
The residual gauge transformations allow us totransform Hµ
∣∣*2 by any divergence-less vector
so we remain with one complex gauge invariantoperator "µHµ
∣∣*2 . Another important residual
transformation is
(Hµ
∣∣*(ν *
+ (H&
∣∣*(µ*
= "µ$& + "&$µ , (114)
with "&$& = 0. This means that the trace part ofthis symmetric tensor is invariant under the resid-ual symmetries and therefore, the ## componentcontains the usual graviton but also an additionalinvariant scalar. The antisymmetric piece enjoysthe usual gauge transformation of a two-form
(Bµ& = "&-& " "&-µ . (115)
We also note that the top component of Hµ isinvariant. Thus, we see that we have 16 o"-shellbosonic degrees of freedom. The fermion is in the#2# component (and its complex conjugate). Ithas residual gauge symmetry
(&µ! = i"µ-! , !µ!!"µ-! = 0 . (116)
Since -! satisfies the Dirac equation it cannotbe used to set any further components to zero.Therefore, our theory includes a gravitino as wellas an additional Weyl fermion. Thus, we have 16o"-shell fermionic degrees of freedom.
We conclude that the theory has 16+16 fields.This is in accord with the 16 + 16 operators inthe multiplet S!!. This supergravity multiplethas been recognized in the supergravity litera-ture [44–46].
It is easy to construct a kinetic term; in factEFZ
!%defined in (107) is still invariant because
the set of transformations here is smaller thanwhen the FZ-multiplet is gauged. However, thistheory has another invariant. It is easy to seethat [D% , D% ]H%% is invariant. We can use thisobservation to write an invariant kinetic term∫
d4#([D, D]H
)2, (117)
in addition to the one we have already includedin the discussion of the FZ-multiplet.
To summarize, we find that this theory admitstwo independent kinetic terms. Thus there is onefree real parameter r, and the most general ki-netic term is∫
d4#
(H!!EFZ
!!
+12r
H!![D!, D!][D% , D% ]H%%
). (118)
Our goal now is to identify the on-shell degreesof freedom in this theory and study their cou-plings to matter fields. There are many ways todo this, here we will choose a somewhat pecu-liar way that will make some very important factstransparent. We enlarge the gauge symmetry, re-laxing either one of the two constraints (112) orboth, and add compensator fields.
22
In order to contrast the situation with that inthe previous subsection we choose to keep the firstconstraint in (112) and relax the second one byadding a chiral compensator field '! which trans-forms as
('! =32D2L! . (119)
First, the non-invariance of the coupling to mat-ter
∫d4# H!!S!! can be corrected by a adding
to the Lagrangian the term " 16
∫d2# '!.! +c.c..
Next, we move to the kinetic terms (118). Thefirst term is invariant, but the second term is not.This is easily fixed by adding more terms to theLagrangian. We end up with the invariant La-grangian
L =∫
d4#(H!!EFZ
!! +12r
H!
+ H!!S!!
)"
(16
∫d2#'!.! + c.c.
)
" 1r
∫d4#
((D,', + D, ',
)[D, D]H
" 12(D,', + D, ',
)2)
.
(120)
The first term in the second line corrects the non-invariance of the coupling to matter and the othertwo terms fix the transformations of the kineticterms.
In order to read out the spectrum we denoteG = D,', + D, ', which is a real linear super-field (i.e. it satisfies D2G = 0). We also express.! = " 3
2D2D!U , with a real U . We should re-member that this U might not be well-defined;e.g. it might not be globally well-defined or mightnot be gauge invariant. In fact, the need of gaug-ing the S-multiplet arises precisely when this U isnot well-defined. The Lagrangian above becomes
L =∫
d4#(H!!EFZ
!! + 12r H!![D!, D!][D, D]H
+H!!S!!
)
" 1r
∫d4#
(G
([D, D]H " rU
)" 1
2G2
). (121)
Now we can dualize G. This is done by view-ing it as an arbitrary real superfield and imposing
the constraint D2G = 0 by a Lagrange multiplierterm
∫d4#
(# + #
)G where # is a chiral super-
field. This makes it easy to integrate out G usingits equation of motion G = r2
(# + #
)+ r2U +
[D, D]H to find the Lagrangian
L =∫
d4#
(H!!EFZ
!! +(# + # + U
)[D, D]H
" r2
(# + # + U
)2 + H!!S!!
). (122)
In this presentation the theory looks like astandard supergravity theory based on the FZ-multiplet which is coupled to a matter systemwhich includes the original matter as well as thechiral superfield #. This is consistent with thecounting of degrees of freedom (4 + 4 degrees offreedom in addition to ordinary supergravity) andwith the identification of the 16/16 supergravityas an ordinary supergravity coupled to a chiralsuperfield. Note that even though the new super-field # originated from the gravity multiplet, itscouplings are not completely determined. At thelinear order we have freedom in the dimensionlessparameter r and we expect additional freedom athigher orders.
The linear multiplet G, or equivalently the chi-ral superfield #, are easily recognized as the dila-ton multiplet in string theory. There the gravitonand the gravitino are accompanied by a dilaton,a two-form field and a fermion (dilatino). Theseare the degrees of freedom in G. After a dualitytransformation this multiplet turns into a chiralsuperfield #. Furthermore, as in string models,the second term in (122) mixes the dilaton andthe trace of the linearized graviton hµ
µ. Both thisterm and the term quadratic in # lead to the dila-ton kinetic term.
As we mentioned above, the need for the mul-tiplet S!! arises when the operator U is not agood operator in the theory. In this case the cur-rent J!! does not exist. The couplings in (122)explain how the chiral field # fixes this prob-lem. Even though U is not a good operator,U = # + # + U is a good operator. If U isnot gauge invariant, # transforms under gaugetransformations such that U is gauge invariant.And if U is not globally well-defined because itundergoes Kahler transformations, # has simi-
23
lar Kahler transformations such that U is well-defined.
The result of this discussion can be presentedin two di"erent ways. First, as we did here,we started with a rigid theory without an FZ-multiplet and we had to gauge the S-multiplet.This has led us to the Lagrangian (122). Alterna-tively, we could add the chiral superfield # to theoriginal rigid theory such that the combined the-ory does have an FZ-multiplet. Then, this newrigid theory can be coupled to standard super-gravity by gauging the FZ-multiplet.
Our discussion makes it clear that if we want tocouple the theory to supergravity, the additionalchiral superfield # is not an option – it must beadded, and it is propagating.
7.3. Summary and Constraints on ModuliOf particular interest to us in this section was
the coupling of theories without an FZ-multipletto supergravity. Here we have limited ourselves tosupersymmetric field theories in which all dimen-sionful parameters are fixed and we have studiedthe limit Mp % $. We have not studied theo-ries in which the matter couplings depend on Mp.These have been recently discussed in [47–51] butwe will not elaborate on such “intrinsically grav-itational” theories here.
In case where the FZ-multiplet does not exist,we have to gauge the S-multiplet. The upshot ofthe analysis of this gauging is the following. Weadd to the rigid theory a chiral superfield # whosecouplings are such that the combined system in-cluding # has an FZ-multiplet. This determinessome but not all of the couplings of # to the mat-ter fields. In the case of the FI-term # Higgsesthe symmetry and in the case of nontrivial tar-get space geometry of the rigid theory it createsa larger total space in which the topology is sim-pler. Now that we have an FZ-multiplet we cansimply gauge it using standard supergravity tech-niques. In particular, at the linearized level thecouplings of # depend on only one free parameter:the normalization of its kinetic term.
Our results fit nicely with the many known ex-amples of string vacua. We see that the ubiquityof moduli in string theory is a result of low energyconsistency conditions in supergravity. As we em-
phasized above, the chiral superfield # is similarto the dilaton superfield in four dimensional su-persymmetric string vacua. We often have fieldtheory limits without an FZ-multiplet. For ex-ample, we can have a theory on a brane with anFI-term. The field theory limit does not havean FZ-multiplet and correspondingly, U ! $Vis not gauge invariant. This problem is fixed,as in (122), by coupling the matter theory to# which is not gauge invariant (note the simi-larity to the way this is realized in string the-ory [52]). Similarly, we often consider field the-ory limits with a target space whose Kahler formis not exact. This happens, for instance, on D3-branes at a point in a Calabi-Yau manifold. If thelatter is non-compact we find a supersymmetricfield theory on the brane which typically does nothave an FZ-multiplet because U is not globallywell-defined. Coupling this system to supergrav-ity corresponds to making Mp finite. In this casethis is achieved by making the Calabi-Yau com-pact. Then in addition to the graviton, variousmoduli of the Calabi-Yau space become dynami-cal. They include fields like our # which coupleas in (122), thus avoiding the problems with theFZ-multiplet and making the supergravity theoryconsistent.
This discussion has direct implications for mod-uli stabilization. It is often desirable to stabilizesome moduli at energies above the supersymme-try breaking scale. In this case we have to makesure that the resulting supergravity theory is stillconsistent. In particular, it is impossible to stabi-lize # in a supersymmetric way and be left witha low energy theory without an FZ-multiplet.
For example, if the low energy theory includes aU(1) gauge field with an FI-term, this term mustbe # dependent. Furthermore, if the mass of #is above the scale of supersymmetry breaking, itmust be the same as the mass of the gauge fieldit Higgses. Consequently, there is no regime inwhich it is meaningful to say that there is an FI-term. Similar comments hold for theories with acompact target space. It is impossible to stabilizethe Kahler moduli while allowing moduli for thepositions of branes to remain massless withoutsupersymmetry breaking.
The comments above have applications to
24
many popular string constructions including D-inflation, flux compactifications, and sequester-ing.
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