Aspects of Supersymmetry and Its Breaking

Home , Field (physics), Goldstino, Moduli (physics), Supermultiplet, Symmetry breaking

Aspects of Supersymmetry and its Breaking

a b Thomas T. Dumitrescu ∗ and Zohar Komargodski † 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 supersymmetry multiplets. In particular, we discuss supercurrents – multiplets which contain the supersymmetry current and the energy-momentum tensor – and explain how they can be used to constrain the dynamics of supersymmetric field theories, supersymmetry breaking, and supergravity. These notes are based on lectures delivered at the Cargése Summer School 2010 on “String Theory: Formal Developments and Applications,” and the CERN Winter School 2011 on “Supergravity, Strings, and Gauge Theory.”

1. Supersymmetric Theories In this review we will describe some basic aspects of SUSY field theories, emphasizing the structure 1.1. Supermultiplets and Superfields of various supermultiplets – especially those con- A four-dimensional theory possesses = 1 taining the supersymmetry current. In particu- supersymmetry (SUSY) if it contains Na con- 1 lar, we will show how these supercurrents can be served spin- 2 charge Qα which satisfies the anti- used to study the dynamics of supersymmetric commutation relation field theories, SUSY-breaking, and supergravity. ¯ µ Qα, Qα˙ = 2σαα˙ Pµ . (1) We begin by recalling basic facts about super- { } multiplets and superfields. A set of bosonic and Here Q¯ is the Hermitian conjugate of Q . (We α˙ α fermionic operators B(x) and F (x) fur- will use bars throughout to denote Hermitian con- i i nishes a supermultiplet{O if these} operators{O satisfy} jugation.) Unless otherwise stated, we follow the commutation relations of the schematic form conventions of [1]. In local quantum field theo- B F F ries, the basic objects of interest are well-defined Qα, i (x) j (x) + ∂ k (x) + , local operators. In SUSY field theories all such O ∼ O O · · · Q , F (x) B(x) + ∂ B(x) + , (3) operators must be embedded in multiplets of the " α Oi # ∼ Oj Ok · · · supersymmetry algebra, or supermultiplets. Con- and$ likewise%for the Q¯α˙ commutators, such that served currents furnish an important class of local the SUSY algebra (1) is satisfied. It is straightfor- operators. Of particular interest is the supersym- ward to show that a supermultiplet must contain metry current Sαµ, which satisfies equally many independent bosonic and fermionic operators (see, for instance, [2]). µ 3 0 ∂ Sαµ = 0 , Qα = d x Sα . (2) It is always possible (and very convenient) to ! B,F embed the component fields i (x) of a super- ∗T.D. is supported by a DOE Fellowship in High Energy O ¯ Theory, NSF grant PHY-0756966, and a Centennial Fel- multiplet in a superfield S(x, θ, θ). Here θα is an lowship from Princeton University. anti-commuting superspace coordinate. (For now †Z.K. is supported by DOE grant DE-FG02-90ER40542. we suppress any Lorentz indices carried by S.) Any opinions, findings, and conclusions or recommenda- The component fields are identified with the x- tions expressed in this material are those of the authors ¯ and do not necessarily reflect the views of the National dependent coefficients when S(x, θ, θ) is expanded Science Foundation. as a power series in θ, θ¯. The commutation rela- 2 tions (3) are succinctly encoded in the formula A general superfield does not furnish an irre- ducible representation of supersymmetry. To re- ξQ + ξ¯Q,¯ S = i ξ + ξ¯ ¯ S , (4) duce a supermultiplet, we impose supersymmet- Q Q ric constraints. This is most conveniently done in " # & ' which is the defining property of a superfield.3 terms of the superspace differential operators Here ξα is an arbitrary Grassmann parameter ∂ µ α˙ and α, ¯α˙ are the superspace differential oper- ¯ Dα = α + iσαα˙ θ ∂µ , atorsQ Q ∂θ ∂ D¯ = iσµ ∂ . (7) α˙ α˙ αα˙ µ ∂ µ −∂θ¯ − = iσ θ¯α˙ ∂ , α ∂θα αα˙ µ Q − These anti-commute with the supersymmetry ∂ ¯ = + iσµ ∂ . (5) generators , ¯ , and thus map superfields to α˙ ¯α˙ αα˙ µ α α˙ Q −∂θ superfields.Q Hence,Q any constraint written in ¯ Conversely, the defining property (4) implies that terms of Dα, Dα˙ is automatically supersymmet- the components of any superfield furnish a su- ric. We are now ready to begin exploring various permultiplet. Thus, supermultiplets and super- important supermultiplets. fields are in one-to-one correspondence, and we 1.2. Chiral Multiplets and Lagrangians will treat them synonymously. The most familiar representation of supersym- To see this in a little more detail, consider the metry is the chiral multiplet. It is the basic build- component expansion of a general scalar super- ing block which enables us to write SUSY La- field: grangians describing only scalars and fermions. The bottom component of a chiral multiplet is an- S(x, θ, θ¯) = C(x)+iθαψ (x)+ +θ2θ¯2D(x) . (6) ¯ α · · · nihilated by Qα˙ . For example, the bottom component φ(x) of a scalar chiral multiplet satisfies As explained above, the defining property (4) determines the commutation relations of the super- Q¯α˙ , φ(x) = 0 . (8) charges Qα, Q¯α˙ with the component fields. These commutators show that the supercharges act as The" multiplet# obtained from φ(x) by acting with raising operators for the component fields.4 This the supercharges is organized in a superfield ¯ has two important consequences: which satisfies the constraint Dα˙ Φ = 0. This constraint can be solved in components: The superfield S(x, θ, θ¯) can be constructed 2 • from its bottom component C(x) by apply- Φ = φ(y) + √2θψ(y) + θ F (y) , (9) ing the supercharges. Thus, any local oper- where yµ = xµ + θσµθ¯. We immediately note two ator can be embedded in the bottom com- key properties of chiral superfields: ponent of a superfield. However, it is not always possible to embed an operator in a Any function which depends on the chiral higher component. This will play a crucial • superfields Φi, but not their complex con- role in our analysis of various supercurrents. jugates, is again a chiral superfield. Such a function is said to be holomorphic in the Φi. The SUSY variation of the top compo- • The SUSY variation of F (x) is a total nent D(x) of any superfield is always a total • derivative. This fact will enable us to write derivative. supersymmetric Lagrangians. From (9) we see that Φ contains a complex scalar φ, a Weyl fermion ψ , and another com- 3 The factor of i in (4) is necessary for Hermiticity in α Minkowski space. plex scalar F which will turn out to be a non- 4 For instance, [Qα, C] = ψα. propagating auxilliary field. Among other things, − 3

F ensures that the chiral multiplet has four (real) taken to vanish.) After solving for the auxiliary bosonic degrees of freedom to match the four fields F i, which are now non-zero, the component fermionic degrees of freedom coming from ψα. We Lagrangian takes the form will abbreviate this by saying that Φ is a 4 + 4 ¯j µ i i µ ¯j L = g ¯ φ, φ¯ ∂ φ¯ ∂ φ + ig ¯ ∂ ψ σ ψ¯ multiplet. As advertised, Φ has exactly the right − ij µ ij µ field content to describe a theory of scalars and i k l µ ¯j 1 i k ¯j ¯l + ig ¯Γ &∂ φ ' ψ σ ψ¯ + R ¯ ¯ψ ψ ψ¯ ψ¯ (14). fermions, and we would like to write a Lagrangian ij kl µ 4 ijkl for such a theory. & ' This theory is known as the supersymmetric non- Up to total derivatives, a SUSY Lagrangian L linear σ-model. The bosonic part is an ordi- must be a real scalar whose variation under su- nary σ-model, whose target space is an N com- persymmetry is a total derivative. We can thus plex dimensional manifold with metric g ¯. Since take ij the Lagrangian (14) does not have a potential L = D + F + F¯ , (10) term, any point φi on the target manifold is a supersymmetric vacuum. (Recall that it follows where D is the top component of a real scalar from the SUSY algebra (1) that supersymmetric 2 superfield K = K¯ and F is the θ -component of vacua have zero energy; vacua with positive en- a chiral superfield W . This ensures that L is ergy spontaneously break SUSY.) In supersym- real and supersymmetric. For reasons that will metric theories, it is customary to refer to such be explained below, K is known as the Kähler manifolds of vacua as moduli spaces. As usual, potential, and W is called the superpotential. A the scalars φi should be thought of as coordinates particularly simple choice is to take K = ΦΦ¯ on the target space. This is consistent with the and W = 0. It is standard to pick out different fact that the theory is invariant under holomor- components of a superfield using Grassmann in- phic field redefinitions of the form tegration. For example, we can use d4θ to pick 2 i i j out the top component of a superfield, or d θ Φ" = f (Φ ) . (15) to pick out the θ2 component. We th(us write ( Such field redefinitions can be thought of as co- L = d4θ ΦΦ¯ , (11) ordinate changes on the target manifold under which the metric transforms in the usual tenso- ! up to total derivatives. In components: rial way. In the supersymmetric σ-model, the metric is L = ∂µφ∂¯ φ + i∂ ψσµψ¯ + F¯F . (12) determined by the Kähler potential: − µ µ

This describes a free complex scalar φ, a free gi¯j = ∂i∂¯jK . (16) Weyl fermion ψα, and a non-propagating auxiliary field F . In this example F vanishes by its Thus, the target space is a Kähler manifold. Note equations of motion. Because the choice K = ΦΦ¯ that gi¯j must be positive definite so that the ki- gives rise to the usual kinetic terms for φ and ψα, netic terms in (14) have the correct sign required it is called a canonical Kähler potential. by unitarity. The fermions couple to the geome- This trivial free theory can be readily general- try of this Kähler manifold through the connec- i i ized: consider N chiral superfields Φ and con- tion Γjk and the curvature tensor Ri¯jk¯l, which sider the Lagrangian depend on the metric and its derivatives. Under a coordinate chage (15) the fermions transform i 4 i ∂φ! j = d θ K(Φ, Φ¯) , (13) as ψ" = ψ . They can thus be identified L ∂φj ! with vectors in the tangent space of the target where the Kähler potential K may now be a com- manifold. plicated real function of the Φi and their com- The dynamics of the non-linear σ-model is un- plex conjugates. (The superpotential W is still changed under redefinitions of the Kähler poten- 4 tial which take the form two-sphere (or, as we will later explain, any other compact manifold) with patches in such a way ¯ ¯ ¯ ¯ K"(Φ, Φ) = K(Φ, Φ) + F (Φ) + F (Φ) , (17) that K is a globally well-defined scalar. So far we have only discussed theories with- for some holomorphic function F of the Φi. This out a superpotential W , such as the SUSY σ- is known as a Kähler transformation. Such a model (13). These theories did not have any transformation only changes the Lagrangian by a scalar potential. The simplest example with a total derivative. This is because F (Φi) is chiral so superpotential is a single chiral superfield Φ with that its top component is a total derivative. Al- canonical Kähler potential and with superpoten- ternatively, we see from (16) that the metric g ¯ is ij tial W = 1 mΦ2. After integrating out the aux- unaffected by Kähler transformations, so that the 2 iliary field F , this leads to the component La- component Lagrangian (14) is also unchanged. grangian It is important to note that while we are always µ 2 2 µ free to perform Kähler transformations, there are L = ∂ φ∂¯ µφ m φ + i∂µψσ ψ¯ situations in which we are forced to do so. This − − | | | | 1 2 1 2 is the case whenever the target manifold has non- mψ m∗ψ¯ . (19) − 2 − 2 trivial topology and must be covered with several patches which make it impossible to consistently Thus, φ and ψ both acquire a mass m . | | define a single-valued Kähler potential. In this More generally, we can take the SUSY σ- case we are forced to use different Kähler poten- model (13) and add a superpotential W (Φi). tials in different patches. These Kähler poten- Since W must be chiral, it must be holomorphic tials must be related by a Kähler transformation in the Φi. The resulting theory is called a Wess- whenever two patches overlap. Zumino model: As an example of a supersymmetric σ-model 4 ¯ 2 which illustrates this point, we consider a single L = d θ K(Φ, Φ) + d θ W (Φ) + c.c. . (20) chiral superfield Φ with Kähler potential ! ! After integrating out the auxiliary fields, this 2 2 K = fπ log 1 + Φ . (18) adds to the component Lagrangian (14) of the | | σ-model a scalar potential and Yukawa-type in- In this normalization& ' Φ is dimensionless, while teractions: the constant f has dimensions of mass.5 The π ¯ f 2 ij ¯ π L g ∂iW ∂¯jW metric is gφφ¯ = (1+ φ 2)2 , which we recognize ⊃ − | | as the familiar round metric on the two-sphere 1 k i j ∂i∂jW Γij∂kW ψ ψ + c.c. . (21) with radius fπ. The coordinate values φ = 0 − 2 − and φ = ∼correspond to antipodal points. As && ' ' ∞ The vacua of this theory are given by the min- we explained above, the moduli space of SUSY ima of the scalar potential which appears in the vacua is just the two-sphere itself. This theory is first line of (21). Since supersymmetric vacua usually referred to as the 1-model. The Kähler CP have zero energy and the metric gi¯j is positive potential (18) gives rise to the Fubini-Study met- definite, the space of SUSY vacua is specified by 1 ric on CP , which is nothing but the round metric the N complex equations ∂iW = 0 in the N com- on the two-sphere. plex unknowns φi. This implies that without any To describe the point at infinity, we must per- further structure Wess-Zumino models generally form a change of variables Φ 1/Φ. This in- → have SUSY vacua. In section 3, we will explore duces a Kähler transformation (17) with F (Φ) = some simple theories whose vacua spontaneously 2 fπ log(Φ). It is, in fact, impossible to cover the break supersymmetry. 5 This notation is due to the fact that fπ is the ana- 1.3. Vector Multiplets and Gauge Theories logue of the pion decay constant in the chiral Lagrangian for QCD. Our example (18) describes the coset mani- In this section we will describe the vector mul- 1 fold SU(2)/U(1) = CP . tiplet, which will enable us to write Lagrangians 5

for supersymmetric gauge theories. For simplicity superfield V . However, in general Aµ and V we will limit ourselves to the Abelian case. A vec- are not well-defined: they may undergo gauge- tor multiplet is a real superfield V = V¯ , subject transformations. to the gauge transformations The superfield Wα has exactly the right field content to write supersymmetric kinetic terms for V V + i Λ Λ¯ , (22) the gauge field and the gaugino. Since W is chi- → − α ral, we can take where Λ is a& chiral'superfield. These gauge transformations allow us to fix Wess-Zumino gauge in 1 2 α L = d θ W Wα + c.c. which the low-lying components of V vanish: 4e2 ! 1 µν i µ ¯ 1 2 1 = F Fµν + ∂µλσ λ + D . (27) V = θσµθ¯ A +iθ2θ¯λ¯ iθ¯2θλ+ θ2θ¯2D . (23) −4e2 e2 2e2 − µ − 2 & ' Here e is the gauge coupling. In this simple free Here Aµ is a real gauge field, the Weyl fermion λα theory the auxiliary field D vanishes by its equa- is its gaugino superpartner, and D is a real scalar tions of motion. auxiliary field. The residual gauge freedom which In Abelian gauge theories, it is possible to add remains in Wess-Zumino gauge just consists of a so-called Fayet-Iliopoulos (FI) term to the La- ordinary gauge-transformations for Aµ. grangian: To capture the gauge-invariant information 4 in V , we can define the superfield LFI = ξ d θV . (28) ! 1 2 Wα = D¯ DαV , (24) Here the FI-parameter ξ is real. The FI-term is −4 gauge invariant because the top-component of a which is invariant under (22). From this defini- chiral superfield is a total derivative. Note that tion, we immediately see that Wα satisfies the the gauge-invariance of the FI-term resembles the constraints invariance of the Wess-Zumino Lagrangian (20) under Kähler transformations. This connection α α˙ D¯ α˙ Wα = 0 , D Wα = D¯ α˙ W¯ . (25) will resurface in later sections.

In components: 2. Global Symmetries 2.1. Global Current Supermultiplets W = iλ (y) + θ δ βD(y) α − α β α In local quantum field theories, the Noether ) theorem guarantees that each continuous global i(σµν ) βF (y) + θ2σµ ∂ λ¯α˙ (y) , (26) symmetry gives rise to a conserved current jµ sat- − α µν αα˙ µ * isfying where the closed, real two-form Fµν = ∂µAν − ∂µj = 0 , Q = d3x j0 . (29) ∂ν Aµ is the field strength of Aµ. We therefor µ refer to Wα as a field strength superfield. We ! see that Wα is a 4+4 multiplet, corresponding to Here Q is the conserved charge corresponding µ the 4+4 gauge-invariant degrees of freedom in V . to j . In this review, we will not discuss R- For future reference, we note that any spinor symmetries, so that Q is just an ordinary global 6 superfield Wα which satisfies the constraints (25) symmetry. In SUSY theories, we must em- µ has the component expansion (26) for some bed j in some supermultiplet. In order to de- closed, real two-form Fµν . Locally, we can al- cide what constraints this superfield will satisfy, ways express such a two-form as F = ∂ A we are guided by the following two observations: µν µ ν − ∂ν Aµ for some vector Aµ, and we can conse- 6An R-symmetry is a global symmetry which does not quently write W as in (24) for some vector commute with SUSY: [R, Qα] = Qα. α − 6

Because Q is an ordinary global symmetry components. Moreover, we can use (4) on (33) to • charge, it commutes with SUSY, [Qα, Q] = extract the commutation relation 0. This implies the current algebra [Q , j ] = 2i(σ ) β∂ν j , (34) α µ − µν α β [Qα, jµ] = (S.T.) . (30) whose right-hand-side is indeed a pure Schwinger The right-hand-side of this equation is con- term. Having familiarized ourself with the struc- µ served when acting with ∂ on both sides, ture of the global current multiplet (33), we can and it is a total space-derivative when µ = 0 now explore how it arises in theories with global so that it integrates to zero; it is known as symmetries. a Schwinger Term (S.T.). We begin by revisiting the free theory of a chiral superfield (11), which is invariant under U(1) We expect jµ to be the highest spin com- • phase rotations of Φ. This corresponds to the in- ponent in its supermultiplet. If this is not variance of the component theory (12) under U(1) the case, we could not gauge jµ without in- phase rotations of φ, ψα and F . The equation of troducing higher-spin gauge fields. This is motion D2Φ = 0 implies that = ΦΦ¯ is a lin- not expected to be consistent in rigid field ear multiplet, which in the normalizationJ of (33) theories without gravity. contains the conserved vector current We thus posit that jµ is embedded in a real, scalar j = i φ∂¯ φ ∂ φφ¯ + ψ¯σ¯ ψ . (35) superfield which schematically takes the form µ µ − µ µ J & ' = J + iθj iθ¯¯j + θσµθ¯ (j + ) + . (31) This is the usual U(1) phase current which gives µ 7 J − · · · · · · charge +1 to φ and ψα. & ' More generally, we can consider continuous Here J is a real scalar, and jα is a Weyl fermion. The dots are terms which are not uniquely fixed global symmetries of the SUSY σ-model (13). As- by the requirements outlined above. Different sume that this theory is invariant under the in- choices of Schwinger terms in (30) terms lead to finitesimal global symmetry transformation different ways of completing the multiplet. Al- i (a) (a)i though there is a constructive way of writing the δΦ = ε X (Φ) . (36) most general solution, we will not follow this ap- (a)i i proach here. Instead, we will simply write down a Here the X are holomorphic in the Φ and (a) set of constraints for and explore the resulting the ε are infinitesimal real parameters; we will component structure.J use indices a, b, . . . to label different global sym- Conventionally, the global current supermulti- metries. The transformation (36) must leave 8 plet is taken to satisfy the metric gi¯j invariant. However, the La- J grangian (13) may pick up a Kähler transforma- D2 = D¯ 2 = 0 . (32) tion: J J A real superfield obeying these constraints is δL = d4θ ε(a) F (a)(Φ) + F¯(a)(Φ¯) , (37) called a linear multiplet. In components, such ! + , a multiplet takes the form where the functions F (a) are holomorphic in µ i = J + iθj iθ¯¯j + θσ θ¯ jµ the Φ . This only changes the Lagrangian by a J − total derivative. 1 2 µ 1 2 µ 1 2 2 2 + θ θ¯σ¯ ∂µj θ¯& θσ ∂'µ¯j θ θ¯ ∂ J , (33) 2 − 2 − 4 7In our convention, an operator has charge q under the symmetry generated by Q if it satisfiesO [Q, ] = q . O O where jµ is the conserved vector current. We 8Geometrically, this means that the X(a)i are the compo- immediately see that contains no higher-spin nents of holomorphic Killing vector fields X(a) = X(a)i∂ . J i 7

To find the conserved currents corresponding a constant. It is thus not a good operator in the to the symmetries (36), we perform the super- theory. In this case, we cannot embed the vec- space analog of the usual Noether procedure: re- tor current corresponding to the shift symmetry place each infinitesimal transformation parame- in the linear multiplet . This also means that ter ε(a) with an arbitrary infinitesimal chiral su- we cannot gauge the shiftJ symmetry in the usual perfield Λ(a). (The fact that the Λ(a) are chi- way (see subsection 2.2). This difficutly can be ral ensures that the variations of the chiral su- overcome by embedding jµ in a different multiplet perfields Φi are still chiral.) To linear order in which is well-defined. (The details of this proce- the Λ(a), the change in the Lagrangian must now dure are beyond the scope of this review.) Con- take the form ceptually, this situation has an exact analogue for multiplets containing the supersymmetry cur- 4 (a) (a) (a) δL = d θ Λ F i + c.c. , (38) rent Sαµ and the energy-momentum tensor Tµν , − J ! + , to be discussed in detail below. (a) for some real superfields . Note that (38) 2.2. Gauging Global Symmetries reduces to (37) if we set ΛJ(a) = ε(a). However, In order to couple matter to a gauge field Aµ, using the explicit form (36) of the infinitesimal we add to the Lagrangian a term transformation with ε(a) replaced by Λ(a), we can µ also write the change in the Lagrangian as δL jµA + . (42) ∼ · · ·

4 (a) (a)i Here jµ is a conserved matter current, so that this δL = d θ Λ X ∂iK + c.c. . (39) interaction is invariant under gauge transforma- ! tions of Aµ. If the current jµ itself transforms Since (38) and (39) must agree for all chiral super- under gauge transformations, then (42) contains fields Λ(a), we can identify the Noether currents additional terms (represented by the dots) with higher powers of Aµ to ensure gauge invariance. (a) = iX(a)i∂ K iF (a) . (40) J i − They will not be important for us, and we will not discuss them. These are guaranteed to be real and conserved, as In SUSY theories, the analogue of (42) is given can be checked using the sigma model equations 2 by the coupling of motion D ∂¯iK = 0. Note that the Noether currents (40) depend on the functions F (a) which δL = d4θ V , (43) appear in the Kähler transformation (37). This is J not surprising: when the Lagrangian changes by ! a total derivative under a symmetry transforma- where is a linear multiplet and V is a vector multiplet.J Since D2 = 0, this interaction, then the Noether current acquires an addi- J tional piece. tion term is invariant under gauge transforma- Such a situation can arise even in the triv- tions (22) of V . In addition to the term in (42), ial theory (11). This theory has a shift sym- the interaction (43) contains Yukawa-type inter- metry δΦ = ε under which the Lagrangian un- actions for the gaugino λα, as well as a coupling of the bottom component J of to the auxiliary dergoes a Kähler transformation with F = Φ. J By (40), the Noether current for this symmetry field D of the vector multiplet V . When D is in- is given by tegrated out, it gives rise to a new contribution to the scalar potential:

= i(Φ¯ Φ) . (41) 2 J − e 2 VD = (J + ξ) , (44) An interesting complication occurs if we compact- 8 ify the target space of this model to a cylinder by where e is the gauge coupling and ξ is an FI-term, identifying Φ Φ+i. As we go around the cylin- which may be present. We refer to (44) as a D- der once, the∼bottom component of shifts by term potential, since it arises from integrating out J 8 the D-component of a vector multiplet. This is functions which appear in the Kähler transfor- to be contrasted with the scalar potentials dis- mation (37). The D-term scalar potential in this cussed at the end of subsection 1.2, which were theory is given by the result of integrating out the F -components 1 2 of chiral superfields. These are known as F -term V = g2 J (a) . (49) D 8 a potentials. a As a simple example, which will reappear in - + , later sections, consider a theory of two chiral su- Here the ga are the gauge coupling constants perfields Φ1,2 with canonical Kähler potential and which arise from the kinetic terms of the differ- charge +1 under U(1) phase rotations.9 The cor- ent V (a), and the J (a) are the bottom components responding conserved current is given by of the Noether currents (a). J = Φ¯ Φ + Φ¯ Φ . (45) J 1 1 2 2 3. SUSY-Breaking This gives rise to the D-term potential 3.1. Simple Examples 2 If supersymmetry is to play any role in describ- e 2 2 2 VD = φ1 + φ2 + ξ . (46) ing nature, then it must be spontaneously bro- 8 | | | | ken. As we already mentioned, SUSY is sponta- & ' If ξ > 0, then the vacuum energy is positive: neously broken if the vacuum energy is positive. there are no SUSY vacua. If ξ < 0, then the Moreover, broken supersymmetry always leads to moduli space of vacua is given by the three- a massless fermion, the Goldstino, which is the sphere φ 2 + φ 2 = ξ modulo U(1) gauge exact analogue of the Goldstone bosons which ap- | 1| | 2| − transformations acting on φ1,2, which is nothing pear when ordinary global symmetries are sponta- 1 but CP . The corresponding low-energy theory neously broken. We will now explore a few simple 1 describing the massless moduli is the CP -model, examples which break SUSY at tree-level. which we introduced at the end of subsection 1.2. The simplest SUSY-breaking theory consists of For future use, we now briefly describe how one chiral superfield Φ with a linear term in the to gauge a general global symmetry of the non- superpotential linear σ-model. As in the examples above, we simply add to the Lagrangian (13) a term L = d4θ ΦΦ¯ + d2θ fΦ + c.c. . (50)

4 (a) (a) ! ! L " = d θ V . (47) J The scalar potential in this theory is simply a ! constant V = f 2. Thus, the spectrum con- Here the (a) are the Noether currents (40). | | J sists of a massless fermion ψα – the Goldstino Under a gauge-transformation with chiral gauge – and a massless scalar φ. Since there is no (a) parameter Λ , the vector fields transform in potential for φ, the theory has infinitely many the usual way, while the change in the σ-model SUSY-breaking vacua with nonzero energy f 2, Lagrangian L is given in (38). The complete and since the theory is free these vacua are| not| Lagrangian is now gauge-invariant up to total lifted. derivatives: We can make this model more interesting by adding an additional term to the Kähler poten- 4 (a) (a) δ (L + L ") = d θ Λ F + c.c. + , (48) · · · tial: ! 1 where the dots represent unimportant higher- L = d4θ ΦΦ¯ Φ¯ 2Φ2 (a) − 4Λ2 order terms and the F are the holomorphic ! ) * 9The fact that this theory is quantum mechanically + d2θ fΦ + c.c. . (51) anomalous will not be important for us. ! 9

This is an effective theory below the high cutoff In the three examples above SUSY was broken scale Λ. The scalar potential is now given by because the vacuum energy was positive due to non-vanishing F -terms coming from chiral super- f 2 φ 2 V = = f 2 1 + + . (52) fields. It is also possible to break supersymmetry | 1| 2 | |2 1 2 φ | | Λ · · · through non-vanishing D-terms coming from vec- − Λ | | ) * tor superfields. The prototypical such example is Now there is a single SUSY-breaking vacuum the FI-model: at φ = 0. The boson φ has acquired a 2 2 1 2 α 4 mass f /Λ , while the massless fermion ψ is L = d θ W Wα + c.c. + ξd θ V . (55) | | α 4e2 again identified with the Goldstino. ! ! The model (51) is not renormalizable. The sim- This model is free, but has a vacuum energy V = plest non-trivial, renormalizable model of spon- e2ξ2/8. The free gauge field is necessarily taneous SUSY-breaking is the O’Raifeartaigh massles, while the massless gaugino is the Gold- model [3]. The model has three chiral super- stino. fields X, Φ, Φ˜ with canonical Kähler potential and 3.2. Comments on Dynamical SUSY- superpotential Breaking W = XΦ2 + mΦΦ˜ + fX . (53) Unlike other symmetries, the spontaneous breaking of supersymmetry is highly con- The conditions for the existence of a SUSY vac- strained. It is a consequence of powerful non- uum are φ = 0, φ2 = f, and mφ˜ = 2xφ, renormalization theorems [6,7] that supersymmet- where we denote by x, φ, φ˜ the bottom compo- ric vacua cannot be lifted by radiative corrections: nents of X, Φ, Φ˜. Clearly these conditions are in- if there is a SUSY vacuum in the classical theory, consistent and there is no supersymmetric vac- then it cannot disappear in perturbation theory. uum. By studying minima of the full scalar po- Thus SUSY can only be broken in two ways: tential V , we see that the theory has a SUSY- The classical theory has no SUSY vacua. 2 breaking vacuum with energy V = f at φ = • In this case we say that it breaks SUSY at | | 2 φ˜ = 0. (This vacuum is only stable if 2 f < m ; tree-level. if this is not the case, the structure of the| | vacuum| | The classical theory has SUSY vacua, but is somewhat different, but SUSY is still broken.) • For φ = 0, the scalar potential is independent they are lifted by non-perturbative ef- of x. The field x is thus massless and can take fects. This is known as dynamical SUSY- on any vacuum expectation value. Just like the breaking. trivial theory (50), the O’Raifeartaigh model thus All theories discussed in the previous subsec- has infinitely many SUSY-breaking vacua at tree tion break SUSY at tree-level. As was empha- level. (This is a very general property of such sized by Witten [8], such models are unappealing theories [4,5].) However, unlike the previous the- because they force us to introduce dimensionful ory which was free, it is now possible for radiative parameters by hand so that we still have to ex- corrections to lift this vacuum degeneracy: x be- plain the large hierarchy between the electroweak comes massive and is stabilized at the origin by scale and the Planck or GUT scale. the one-loop correction to the scalar potential: On the other hand, the option of dynamical

2 SUSY-breaking is very appealing. The scale √f (1) 1 f 2 of supersymmetry breaking can now arise through V 2 | |2 x + . (54) ∼ 16π m | | · · · dimensional transmutation Note that after integrating out the heavy 8π2 f Λ e− g2 , (56) fields Φ, Φ˜ of mass m, the low-energy dynam- ∼ UV ∼ ics of X is governed by an effective theory of the with. g ! 1 an asymtotically free gauge cou- form (51) with the cutoff given by Λ m. pling, and thus √f can naturally be exponentially ∼ 10

smaller than the cutoff ΛUV . This might explain where (S.T.) are Schwinger Terms, and Tµν why the weak scale is so much smaller than the is a conserved, symmetric energy momen- Planck or GUT scale. We should thus be study- tum tensor: ing theories with strong dynamics in the IR which can trigger SUSY-breaking. This is a vast sub- ν 3 0 ject (see [9] and references therein), and we will ∂ Tµν = 0 , Pµ = d x Tµ . (58) restrict ourselves to a few general comments. ! Roughly speaking, dynamical theories fall into The supersymmetry currrent and the three classes: energy-momentum tensor must thus belong Theories in which supersymmetry breaking to the same supermultiplet – a supercur- • is triggered by non-perturbative effects, but rent. the vacuum is in the semiclassical regime. Theories in which the vacuum is in the • We expect Tµν to be the highest spin com- strongly-coupled regime of the original de- • ponent in the supercurrent multiplet. If this grees of freedom, but can still be analyzed is not the case, then it might be problematic via Seiberg duality [10]. to couple the theory to supergravity. Strongly coupled theories in which little can • be said about the vacuum, but for which We conclude that the supersymmetry current and there are convincing (usually indirect) ar- the energy-momentum tensor are embedded in guments that SUSY is broken. a real vector superfield µ which schematically takes the form J The first two types of models are referred to as calculable. At low energies, they are described by Wess-Zumino models, possibly with IR-free gauge = j iθ (S + ) + iθ¯ S¯ + Jµ µ − µ · · · µ · · · fields and effective FI-terms. By studying such ν ¯ + θσ θ (2Tνµ + ) +& . ' (59) models we may hope to shed light on the dynam- · · · · · · ics of interesting dynamical models. & ' In the next section we will continue our study of The real vector jµ is generally not conserved. interesting supermultiplets. We will then explain As in the case of the global current multiplet, how these multiplets can be used to constrain the the dots are terms which are not uniquely fixed dynamics and low-energy behavior of various the- by the requirements outlined above, and different ories – both with and without SUSY breaking. choices of Schwinger terms in in (57) give rise to different ways of completing the multiplet. Again 4. Supercurrents we will not discuss the constructive approach to writing the most general solution [11]. Instead, 4.1. The Ferrara-Zumino Multiplet we will begin by exploring the most conventional Four-dimensional theories with = 1 super- set of constraints for the supercurrent multiplet, N symmetry possess a conserved supersymmetry only amending them when we are forced to do so. current Sαµ satisfying (2). Just like any other The simplest and most widely known super- well-defined local operator, we have to embed Sαµ current is called the Ferrara-Zumino (FZ) multi- in some multiplet. In order to write down sensi- plet [12]. It is defined by the equations10 ble constraints for such a supercurrent multiplet, we are guided by the following two observations: α˙ D¯ αα˙ = DαX , D¯ α˙ X = 0 . (60) The SUSY-algebra (1) gives rise to the cur- J • rent algebra 10Our convention for switching between vectors and bi- µ µ 1 αα˙ Q¯ , S = 2σ T + (S.T.) , (57) spinors is # = 2σ #µ, #µ = σ¯ # . { α˙ αν } αα˙ µν αα˙ − αα˙ 4 µ αα˙ 11

In components, the FZ-multiplet takes the form satisﬁes

α˙ 1 ν i 2 D¯ αα˙ = 0 . (66) µ = jµ iθ Sµ + σµσ¯ Sν + θ ∂µx¯ J J − 3 2 ) * As expected, the FZ-multiplet reduces to the 1 i + iθ¯ S¯ + σ¯ σν S¯ θ¯2∂ x superconformal multiplet for the free scalar. µ 3 µ ν − 2 µ ) * By expanding Φ in components and comparing ν 2 1 ρ σ with (61), we see that the energy-momentum ten- + θσ θ¯ 2Tµν ηµν T ενµρσ∂ j − 3 − 2 sor embedded in reads ) * αα˙ & ' J + . (61) λ · · · Tµν = ∂µφ∂¯ ν φ + (µ ν) ηµν ∂ φ∂¯ λφ ↔ − Here T = T λ is the trace of the energy- 1 λ ∂ ∂ η ∂2 φ 2 + (fermions) . (67) momentum tensor. The higher components − 3 µ ν − µν | | of only contain derivatives of components & ' Jαα˙ The first line is the familiar energy-momentum which we have already displayed. The chiral su- tensor for a free, massless scalar. The second line perfield X takes the form is an improvement term: it is automatically con- X = x(y) + √2θψ(y) + θ2F (y) , (62) served (without using the equations of motion), and it does not affect the conserved charge Pµ with because setting ν = 0 turns it into a total space derivative. We are free to add these kind of im- √ i 2 µ ¯α˙ provement terms to any conserved current. In ψ = σ Sµ , 3 αα˙ α˙ this case the constraint D¯ αα˙ = 0 fixes the 2 J F = T + i∂µj . (63) improvement term in a particular way: it guar- 3 µ antees that Tµν is traceless. The relation be- We see that the FZ-multiplet contains 12 = tween supersymmetric constraints and improve- 2 [x] + 4 [jµ] + 6 [Tµν ] independent bosonic com- ment terms will play a pivotal role in what fol- ponents, which match the 12 fermionic compo- lows. nents of Sαµ. Thus, the FZ-multiplet is a 12 + 12 More generally, we can consider the Wess- multiplet. It is straightforward to check that the Zumino model (20). Using the equations of mo- ¯ 2 explicit component expressions (61) and (62) lead tion D ∂iK = 4∂iW , we can check that to a SUSY current algebra of the form (57): i ¯j 2 = 2g ¯D Φ D¯ Φ¯ [D , D¯ ]K , Jαα˙ ij α α˙ − 3 α α˙ ¯ ν Qα˙ , Sαµ = σαα˙ 2Tνµ + i∂ν jµ 1 { } X = 4W D¯ 2K (68) ) − 3 1 iη ∂λj ε ∂ρjλ . (64) − µν λ − 2 νµρλ satisfy the defining equation (60) of the FZ- * multiplet. We will make use of this formula on When X vanishes, then we see from (63) that the several occasions. energy-momentum tensor is traceless, and that jµ Finally, the FZ-multiplet for the FI-model (55) is now a conserved current. In this case, the the- readily follows from the equation of mo- α 2 ory is superconformal and the FZ-multiplet only tion D Wα = e ξ. It is given by: has 8 + 8 components. 4 2 As the simplest example, we again consider the αα˙ = WαW¯ α˙ ξ[Dα, D¯ α˙ ]V , free theory (11) of a single chiral superfield Φ. J −e2 − 3 2 ξ Using the superspace equation of motion D Φ = X = D¯ 2V . (69) 0 it is straightforward to check that the multiplet −3 It is similarly straightforward to obtain the FZ- ¯ ¯ 2 ¯ ¯ αα˙ = 2DαΦDα˙ Φ [Dα, Dα˙ ]ΦΦ (65) multiplet for gauge theories coupled to matter (we J − 3 12

will not write down the most general expression operators? The answer is that Sαµ and Tµν are here). not unique; they are only defined up to improve- Looking at (55) we make an unsettling obser- ment terms. Different choices of improvement vation: in the presence of an FI-term ξ, nei- terms lead to different local currents, but do not ther αα˙ nor X are invariant under the usual affect the corresponding charges. It is usually gauge-transformationJ δV = i(Λ Λ¯), with Λ chi- possible to chose the improvement terms in such ral [13]. (This conclusion is not−changed by the a way that the currents are well-defined opera- inclusion of matter.) Thus, the FZ-multiplet is tors. However, a problem could arise when at- not gauge-invariant, and consequently its compo- tempting to embed these operators into super- nents are not well-defined operators. From this, multiplets. In this case there may be a clash be- we might mistakenly conclude that models with tween the existence of certain supermultiplets and FI-terms are inherently ill-defined, because they the requirement that these multiplets be gauge do not have a well-defined supersymmetry current invariant. This is precisely what happens in or energy-momentum tensor. the FI-model: forcing Sαµ and Tµν into an FZ- To see why this is not so, we examine the gauge multiplet requires us to pick gauge non-invariant non-invariance of the FZ-multiplet (69) in more improvement terms. Conversely, it is possible to detail. Under a gauge transformation, this mul- pick gauge-invariant improvement terms for Sαµ tiplet transforms as follows: and Tµν , but this makes it impossible to embed these operators into an FZ-multiplet. In this case 2 δ αα˙ = ξ∂αα˙ Λ + Λ¯ , it is necessary (and possible) to find different su- J −3 permultiplets into which we can embed gauge- i 2 & ' δX = ξD¯ Λ¯ . (70) invariant choices for Sαµ and Tµν . Such multi- 3 plets will be discussed in section 6. By expanding these expressions in components In light of the preceding discussion, we are led and comparing with (61), we see that the bot- to carefully reconsider the existence of the FZ- tom component j ξA + (fermions) explic- multiplet (68) for the Wess-Zumino model. We µ ∼ µ itly depends on the gauge field Aµ and is sim- see that the Kähler potential K formally appears ply not gauge-invariant. However, the gauge non- in the same way as the vector field V in the invariance of the energy-momentum tensor takes FZ-multiplet for the FI-model. Morover, Kähler the special form transformations of K are take the same form as gauge-transformations of V . We conclude that 2 2 δTµν = ξ ∂µ∂ν ηµν ∂ ImΛ , (71) the FZ-multiplet of the Wess-Zumino model is not 3 − | invariant under Kähler transformations. Again, where Λ denotes& the bottom' component of the these Kähler transformations only change the su- superfield| Λ. We see that the transformation persymmetry current and the energy-momentum of Tµν in (70) is a pure improvement term. As we tensor by improvement terms. explained above, this means that it is automati- However, unlike gauge transformations, Kähler cally conserved and does not affect the conserved transformations are not an absolute necessity. It charged Pµ. Completely analogously, it can be often does not matter if the multiplet transforms checked that the supersymmetry current Sαµ also under Kähler transformations. The only become only shifts by a pure improvement term. The essential when the target space of the sigma M FI-model thus has well-defined, gauge-invariant model needs to be covered with several patches operators Pµ, Qα, which make it a well-defined and Kähler transformations are needed to switch supersymmetric field theory. between the patches. This happens, for instance, 1 How, then, do we interpret the fact that the su- in the CP model described in subsection 1.2. persymmetry current and the energy-momentum In such theories, the FZ-multiplet is not a well- tensor embedded in the FZ-multiplet (61) are not defined operator. Equivalent ways of describing gauge-invariant and do not exist as well-defined the target space of the theory result in energy- 13 momentum tensors which differ by improvement majority of SUSY theories, including asymptot- terms, even though they should be identical, and ically free gauge theories, such as supersymmet- likewise for the supersymmetry currents. In other ric QCD (SQCD). As far as we know, the FZ- words, there is no unambiguous way of fixing multiplet only fails to exist when one of the fol- these operators. lowing conditions holds: We would like to understand the precise mathe- Some U(1) gauge group has an FI-term. matical conditions under which the FZ-multiplet • for Wess-Zumino models is well-defined. From The target-space has non-trivial topology, the metric g ¯ we can construct the Kähler form ij • so that the Kähler form is not exact. i ¯j ω = ig ¯dφ dφ¯ . (72) ij ∧ In the first case, the FZ-multiplet is not gauge- invariant, while in the second case it is not glob- Since g ¯ is locally derived from a Kähler poten- ij ally well-defined. These insights will enable us to tial, ω is closed: dω = 0. In every patch we derive exact results about the dynamics of SUSY can thus find a one-form such that ω = d . field theories, SUSY-breaking, and supergravity. Locally, this Kähler connectionA is given by A i A ∼ i∂iKdφ +c.c. . In general, ω is not exact and the 4.2. Consequences for SUSY Theories Kähler connection is not globally well-defined. A It is a general fact about quantum field the- The obstruction to the global existence of is ory that operator equations are invariant under 2 A measured by the cohomology class [ω] H ( ). renormalization group (RG) flow as long as the If ω vanishes in H2( ), then is globally∈ wMell- M A operators involved are well-defined and gauge- defined and thus a good operator the theory. invariant. This means that if some operator rela- The bottom component of (68) is given by tion holds in the UV (where field theories are usually defined), then it continues to hold along the 2i i 1 ¯j i j = ∂ K∂ φ + c.c. g ¯ψ¯ σ¯ ψ . (73) entire RG-flow. Applying this reasoning to the µ 3 i µ − 3 ij µ FZ-multiplet immediately leads to constraints on While the fermionic piece only depends on the the RG-flow of supersymmetric field theories. metric and is thus invariant under Kähler trans- Consider a supersymmetric field theory which formations, we recognize the bosonic piece as the has no FI-terms in the UV. This means that this pull-back to space-time of the Kähler connec- theory has a well-defined, gauge invariant FZ- tion . This is is well-defined only if ω is ex- multiplet. This multiplet must therefore persist act andA vanishes in H2( ). If this condition is along the entire RG-flow. We conclude that FI- M not satisfied, then jµ is not a good local opera- terms cannot be generated along the flow; this is tor in the theory, since it depends on the choice true at any order in perturbation theory and even of patch in target space. In this case, the entire non-perturbatively. For instance, it might hap- FZ-multiplet (68) is not well-defined [14]. pen that the theory flows through a regime with As a special case, note that the Kähler form of a strong dynamics, and that a U(1) gauge symme- compact manifold can never be exact. If this were try emerges in some weakly-coupled dual descrip- dim( )/2 the case, then the volume form ω M would tion. In this case the gauge-invariance of the FZ- also be exact and thus integrate to zero on the multiplet prevents this emergent U(1) gauge the- compact manifold; this is a contradiction. Thus, ory from having an FI-term. It is even possible a Wess-Zumino model with compact target man- to argue that if an FI-term is present in the UV, 1 ifold, such as the CP -model (18), cannot have a then its value is not renormalized. Other deriva- well-defined FZ-multiplet. tions of these results can be found in [15–18]. In this section we have explored the FZ- Completely analogously, let us consider a the- multiplet, in which the supersymmetry current ory which has a well-defined, gauge-invariant FZ- and the energy-momentum tensor are convention- multiplet in the UV. For example, we could ally embedded. This multiplet exists in the vast consider an ordinary SUSY gauge theory, with 14 canonical Kähler potential for the matter fields multiplet to the Goldstino. This will enable us to and no FI-terms. At low energies, such field the- derive exact results about such SUSY-breaking ories are often described by a weakly-coupled σ- theories. It will also lead to a useful formalism model (perhaps with IR-free gauge fields). It is for writing Lagrangians with non-linearly realized expected that this will happen whenever the field supersymmetry. The second application concerns theory has a strong coupling scale Λ, below which the interplay of F -terms and D-terms in SUSY- it is described by massless moduli. Since this the- breaking theories. We show under very broad as- ory has a well-defined FZ-multiplet in the UV, sumptions that there can be no SUSY-breaking and this multiplet persists along the entire RG- vacua (even meta-stable ones) in which the D- flow, we conclude that the low-energy σ-model terms are parametrically larger than the F -terms. must have have an exact Kähler form ω. This severely constrains the quantum moduli space of 5.1. The FZ-Multiplet and the Goldstino the low-energy theory: the integral ω ω A universal prediction of spontaneous SUSY- over any compact, even-dimensional cycle∧ ∧must· · · breaking is the existence of a massless Weyl vanish. As we already explained in(the previous fermion, the Goldstino Gα. We will not do jus- subsection, the means in particular that the mod- tice to the extensive literature on the Goldstino. uli space cannot be compact (of course, it can be a Most of the references can be found in [20], which set of points). It is particularly interesting to test is also the main reference for this subsection. this theorem in SQCD with the same number of When a global symmetry is spontaneously bro- colors and flavors. In this case the topology of the ken, the corresponding conserved charge is not a modul-space changes under RG-flow and becomes well-defined operator because its correlation func- non-trivial in the IR [19]. However, in accordance tions are IR divergent. As a result, the physical with the general result above, the Kähler form of states of the theory are not in linear representa- this deformed quantum moduli space is exact. tions of the symmetry. However, the conserved The formal similarity between the two theo- current and even commutators of local operators rems described above is not coincidental. On with the conserved charge do exist. In contrast to the one hand, they are both consequences of the the states, the local operators do in fact furnish RG-invariance of the FZ-multiplet. On the other a linear representation of the symmetry. These hand, it may be that if a theory does not have statements carry over without modification when an FZ-multiplet in the UV due to the presence of supersymmetry is spontaneously broken in infi- an FI-term, it may flow to a σ-model which fails nite volume: the supercharge Qα does not exist, to have an FZ-multiplet because its target space but the supersymmtry current and (anti-) com- has a non-exact Kähler form. mutators with Qα do exist. (When SUSY is spon- The simplest example of this phenomenon is taneously broken in finite volume, even Qα ex- the theory discussed at the end of subsection 2.2. ists.) This means that operators still reside in su- In the UV, this theory starts out as an abelian permultiplets, and that we can use the formalism gauge theory with an FI-term ξ and two chiral of superspace and superfields without modifica- matter fields of charge +1. When ξ < 0, it follows tion. from the scalar potential (46) that the low-energy Recall the defining equation (60) of the FZ- theory which describes the moduli space of SUSY- multiplet: 1 vacua is given by the -model, whose target α˙ CP D¯ αα˙ = DαX , (74) space is compact. J where X is chiral. In this subsection we will 5. Applications to SUSY-Breaking only discuss theories which admit a well-defined, gauge-invariant FZ-multiplet. This multiplet re- In this section we will consider two applica- quires the existence of a complex scalar x, the tions of the FZ-multiplet to SUSY-breaking. The bottom component of X, which is a well-defined first application rests on the relation of the FZ- operator in the theory. If we are given a micro- 15

scopic description of the theory in the UV, we can valid. The operator ψφ creates a one Goldstino express x in terms of elementary fields. state and its superpartner φ x creates a two- The chiral superfield X can be consistently fol- Goldstino state. ∼ lowed along the RG-flow down to the IR, even We denote this non-linear superfield which con- when SUSY is spontaneously broken. We can see tains the Goldstino bilinear in the bottom com- this in the simplest SUSY-breaking model (50), ponent XNL, and xNL denotes the bottom com- which only has a single chiral superfield Φ. In ponent itself. Consistency of this superfield under this theory, the F -term is non-zero, but φ and ψα supersymmetry transformations (which, again, are free, massless fields. As was discussed in are legal even if SUSY is spontaneously broken) subsection 3.1, we identify ψα with the Gold- fixes it to take the form stino Gα. Since this theory is a Wess-Zumino 2 ¯ G 2 model with K = ΦΦ and W = fΦ, we find XNL = + √2θG + θ F , (76) 8 2F from (20) that X = 3 fΦ. In this simple example, the operator X is thus proportional to Φ, so where all the fields are functions of yµ. that the θ-component of X is proportional to the Clearly, (76) satisfies the interesting operator Goldstino. identity More generally, standard arguments about symmetry breaking show that when supersymme- 2 XNL = 0 . (77) try is spontaneously broken, the low-energy supersymmetry current is expressed in terms of the (Other realizations of nonlinear supersymmetry Goldstino Gα as can be found, for example, in [21,22]) √ ¯α˙ β ν We conclude that the Goldstino resides in a chi- Sαµ = 2fσµαα˙ G +f "(σµν )α ∂ Gβ + , (75) 2 · · · ral superfield XNL which satisfies XNL = 0. Fur- for some constants f and f ". Note that the term thermore, this chiral superfield is the IR limit of with f " is an improvement term and can be ig- the microscopic superfield X. Surprisingly, this 3 nored. We see that at low energies, the spin- 2 result is true both for F -term and for D-term component of the supersymmetry current essen- breaking because it relies only on the existence 1 tially decouples, while the spin- 2 component is of the chiral operator X. (The only exception is proportional to the Goldstino. Since we know the situation with pure D-term breaking which from (63) that the θ-component of X is given by occurs when a tree-level FI-term is present for an 1 the spin- 2 projection of Sαµ, we again conclude unbroken U(1) gauge theory. In this case the op- that at long distance the θ-component of X is erator X is not gauge invariant and we do not proportional to the Goldstino. discuss it here.11 Thus, our discussion is appli- In the trivial example of a free chiral super- cable in all the interesting models of dynamical field Φ, the lowest component x of the superfield SUSY breaking. X is proportional to the free scalar φ. In general, We will momentarily see that this identification the low-energy Goldstino is not accompanied by of the low-energy limit of the operator X, which a massless scalar. Then, the bosonic operator x exists in all field theories we are interested in, cannot create a-one particle state. Instead, the allows us to derive some non-perturbative results. simplest bosonic state it can create contains two It is instructive to compare the situation here Goldstinos. When supersymmetry is broken in with the theory of ordinary Goldstone bosons like finite volume, the states in the Hilbert space are pions. Clearly, the decay constant f here is anal- in linear supersymmetry multiplets. The super- ogous to the decay constant fπ of pion physics. symmetric partner of a one Goldstino state Q¯ 0 Both of them are well defined. However, in pion α˙ | * is a two Goldstino state Q¯ ˙ Q¯ ˙ 0 . At infinite 1 2 11 volume the supercharge does not| *exist and the However, our discussion does apply in the Higgs phase of the FI-model [23], where the low-energy limit of X can zero momentum Goldstino state is not normaliz- be rendered gauge invariant by dressing it with fields that able. However, the finite volume intuition is still obtain expectation values. 16 physics the order parameter ψψ for chiral sym- underlying mathematical structure, but we will metry breaking does not ha+ve a* universal def- not elaborate on this any further here. inition with a precise normalization (it suffers We will now explain how to use XNL to write from wavefunction renormalization). In our case supersymmetric effective Lagrangians. We start the order parameter for supersymmetry breaking without including derivatives. At that level, the is the energy-momentum tensor which resides in most general supersymmetric Lagrangian subject 2 the same multiplet as the supersymmetry cur- to the constraint XNL = 0 is rent. Hence, it has a well-defined normalization. 4 2 Therefore, our X is completely well-defined. Cor- d θ X¯NLXNL + d θ fXNL + c.c. , (79) respondingly, the operator ψψ of pion physics ! ! acts as an interpolating field for pions, but its where without loss of generality we take f to be normalization is not meaningful. In our case X real. This looks like the free chiral multiplet ex- is used both as an order parameter for supersym- cept that the superfield is constrained. This con- metry breaking and as an interpolating field for straint removes the massless scalar field and in- Goldstinos with well-defined normalization. troduces nonlinearities. The analogy with pion physics also clarifies More explicitly, the constraint can be solved as the meaning of our constraint (77). It arises in (76). Substituting this in (79), we derive the from removing the massless scalar in X. This is component Lagrangian analogous to describing pion physics by starting with a linear sigma model with a σ-field. Re- ¯2 2 ¯ µ ¯ G 2 G moving the σ-field is implemented by imposing a i∂µGσ¯ G + F F + ∂ + (fF + c.c.) . 2F¯ 2F constraint UU † = 1, which is analogous to our ) * 2 (80) XNL = 0. Since in every microscopic theory we can iden- The equations of motion of the auxiliary fields tify X in the ultraviolet, and if SUSY is spon- F, F¯ can be solved and upon substituting this taneously broken we also know its universal low- back in the component Lagrangian we find energy limit, we can calculate various correlation 2 ¯ µ 1 ¯2 2 2 functions of operators at large separations even = f + i∂µGσ¯ G + 4f 2 G ∂ G in strongly coupled models. Hence even “incalcu- L − 1 G2G¯2∂2G2∂2G¯2 . (81) lable” models of SUSY breaking (like the SU(5) − 16f 6 theory of [24] or the SO(10) theory of [25]) have This is equivalent to the Akulov-Volkov La- a solvable sector at long distances. The oper- grangian [26]. (See also [27–29].) Here we have ator x interpolates between the vacuum and a given a fully off-shell supersymmetric description state with two Goldstinos, with a universal nor- of this theory. malization. This is because at long distances the The real advantage of this approach is the sim- leading behavior of x is proportional to that of plifications in writing higher derivative correc- G2. Therefore, tions to (79) and, more interestingly, couplings to 2 possibly light matter fields (such as MSSM fields, 4 1 lim x(r)x¯(0) = . (78) Goldstone bosons, ’t Hooft fermions etc.). Indeed r + * 3π2 r 6 | |→∞ ) * | | such problems are easily solved in this formal- This is independent of the details of the micro- ism since we have an off-shell description and so scopic theory and its coupling constants. In a we can use all the familiar superspace techniques similar fashion we can calculate the long distance for writing Lagrangians (the number of allowed limit of any correlation function of x and x¯. operators is smaller because of the various con- This idea can be pushed further and many more straints). non-perturbative results of this form can be ob- This toolbox can be put to use in many possible tained. In fact, a pretty rich set of exact results contexts and some recent examples include [30– on SUSY-breaking theories arises with interesting 37] 17

The last thing we would like to demonstrate be- to hold in general, or whether there are in fact fore closing this subsection is the way the nilpo- dynamical models with large D-terms. In this tent equation (77) arises from the dynamics in a subsection, we will show that such models do simple example. In order to remove the massless not exist: under broad assumptions, the D-terms scalar we include a non-canonical Kähler poten- are necessarily parametrically smaller than the F - tial terms [36]. c To get an intuitive picture, we can loosely iden- K = ΦΦ¯ Φ¯ 2Φ2 , (82) − M 2 tify D-term breaking with the presence of FI- terms. In dynamical models, one usually starts and with a well-behaved, asymptotically-free gauge theory without FI-terms in the UV. Since such W = fΦ . (83) theories has a well-defined FZ-multiplet, the dis- Here c is a dimensionless positive number of order cussion of subsection 4.2 show that they cannot one. Such a theory can arise as the low-energy acquire FI-terms at low energies. Roughly speak- Lagrangian below some scale M after neglect- ing, the absence of low-energy FI-terms implies 1 the absence of large D-terms. ing higher order terms in M . (For example, the low-energy limit of the standard O’Raifeartaigh We can make this intuitive picture precise by model looks like (82).) It is valid for using the tools we have developed so far. Since we are interested in calculable models with para- f E M . (84) metrically small F -terms, we consider theories in , , which the low-energy dynamics responsible for .Let us integrate out the massive bosons. Re- SUSY-breaking is described by a σ-model (13) membering that the Lagrangian contains inter- and the F -terms are set to zero in first approx- c 2 2 imation. In addition, we will include IR-free action terms of the form M 2 2φFφ ψ , the zero-momentum equation−of motion of− φ sets it gauge fields by gauging some global symmetries / / to / / of the σ-model. If the D-term potential which results from this gauging leads to SUSY-breaking ψ2 φ = . (85) vacua (even meta-stable ones), then these puta- 2Fφ tive vacua will have parametrically large D-terms which are larger than the F -terms by inverse pow- Note that it is independent of c, or in other words, ers of the small, IR-free gauge couplings. We will it is independent of the details of the high energy now show that such vacua do not exist in the- physics (this is an example of our universality). ories which have a well-defined, gauge-invariant Upon substituting (85) back into Φ, we discover FZ-multiplet. (Therefore they cannot exist in in- that in the Φ2 = 0. Hence, it satisfies the same teresting dynamical models.) constraint as X . To make the relation more NL As was discussed at the end of subsection 2.2, precise, we can go through a careful calculation the Lagrangian for the gauged σ-model is given of the supercurrent. The main point, however, by making the substitution is that the nilpotentcy equation arises from integrating out heavy degrees of freedom. K K + (a)V (a) , (86) → J 5.2. Restrictions on Large D-Terms In subsection 3.1 we encountered several simple and including conventional kinetic terms for the models of F -term and D-term SUSY-breaking. gauge fields. By making an identical substitution While models with FI-terms lead to many tree- in expression (68) for the FZ-multiplet of the σ- level examples of D-term SUSY-breaking, most model, we see that a gauge-transformation (48) known calculable models of dynamical supersym- changes this new FZ-multiplet by an amount Jαα˙ metry breaking are predominately F -term driven. We would like to understand whether this has δ i∂ Λ(a)F (a) + c.c. . (87) Jαα˙ ∼ αα˙ 18

Since we assumed that the UV theory had a as the field-strength superfield Wα discussed in well-defined FZ-multiplet, the same must be true subsection 1.3. In components, the -multiplet for the low-energy gauged σ-model. Thus, all takes the form S functions F (a) must vanish. The Noether cur- (a) i i 2 rents now take the simple form µ = jµ iθ Sµ σµψ¯ + θ ∂µx¯ J S − − √2 2 ) * (a) = iX(a)i∂ K . (88) i i i 2 ν J + iθ¯ S¯µ σ¯µψ θ¯ ∂µx + θσ θ¯ 2Tµν − √2 − 2 Using this expression, it is straightforward to ) * ) check that the D-term potential (49) of the 1 1 & ' η Z + / ∂ρjσ + / F ρσ + ... . (91) gauged σ-model satisfies the identity − µν 2 µνρσ 8 µνρσ * 1 i¯j V = g ∂ K∂¯V . (89) The chiral superfields X and χα are given by D 2 i j D √ 2 ν This identity immediately implies that a critical X = x(y) + 2θψ(y) + θ (Z(y) + i∂ jν (y)) , β point of the potential can only occur when VD = β µν χα = iλα(y) + θβ δα D(y) i(σ )α Fµν (y) 0. In other words, every critical point must be − − 2 ν ¯α˙ + , a supersymmetric minimum and this D-term po- + θ σαα˙ ∂ν λ (y) , tential does not admit SUSY-breaking vacua – (92) not even metastable ones. The argument we have given above explains the with absence of dynamical models with parametrically D = 4T + 6Z , large D-terms. It is, however, possible to build − µ models of dynamical SUSY-breaking with com- λ = 2σ S¯µ + 3i√2ψ . (93) parable D-terms and F -terms (see [36] and references therein). As before, the operator jµ is not in general conserved, while the supersymmetry current Sαµ and 6. More Supercurrent Multiplets the symmetric energy-momentum tensor Tµν are conserved. The -multiplet has 4+4 more degrees We have seen that there are well-defined su- of freedom thanSthe FZ-multiplet. They consists persymmetric field theories which do not admit of the real scalar Z, the closed, real two-form Fµν an FZ-multiplet. In these theories, we must and the Weyl fermion ψα. The -multiplet is thus search for other, well-defined supercurrent mul- a 16 + 16 multiplet.12 S tiplets into which the energy-momentum tensor As in the case of the FZ-multiplet, the com- and the supersymmetry current can be embed- ponent expressions (91)and (92) lead to a SUSY ded. Note that the existence of such alterna- current algebra of the form (57): tive multiplets has no effect on the results we ex- 1 tracted in previous sections, which were based on Q¯ , S = σν 2T / F ρσ + i∂ j { α˙ αµ} αα˙ µν − 8 νµρσ ν µ the existence of a well-defined FZ-multiplet. ) There is indeed a supercurrent multiplet αα˙ 1 S iη ∂ρj / ∂ρjσ . which exists in theories with FI-terms or non- − νµ ρ − 2 νµρσ * exact Kähler form, which do not admit an FZ- (94) multiplet. This supercurrent multiplet satisfies the defining equations However, the Schwinger Terms are now mor com-

α˙ plicated. The supersymmetry algebra (1) follows D¯ αα˙ = DαX + χα , D¯ α˙ X = 0 , S 12 ¯ α ¯ α˙ Setting X = 0 in the -multiplet results in the so-called Dα˙ χα = 0 , D χα = Dα˙ χ¯ . (90) -multiplet. It is a 12 +S 12 multiplet whose bottom com- R ponent jµ is a conserved R-current. We will not discuss Note that X appears exactly as in the FZ- the -multiplet in this review, although we will occasion- R multiplet, while χα satisﬁes the same constraints ally refer to it in passing. 19

3 provided that d x Fij vanishes for all spatial in- supergravity. The supergravity multiplet is em- dices i, j. bedded in a real vector superfield H . The θθ¯ ( αα˙ All known supersymmetric field theories ad- component of Hαα˙ contains the metric field, hµν , 13 mit a well-defined -multiplet. Those theories a two form field Bµν , and a real scalar. The cou- which do not have SFI-terms or non-exact Kähler pling of gravity to matter is dictated at leading form also admit a well-defined FZ-multiplet.14 order by For instance, consider a Wess-Zumino model with arbitrary K and W . If the Kähler form is not ex- d4θ Hαα˙ . (96) Jαα˙ act, then the FZ-multiplet does not exist. How- ! ever, the -multiplet exists, and takes the form We should impose gauge invariance, namely, S the invariance under coordinate transformations i ¯j = 2g ¯D Φ D¯ Φ¯ , (95) Sαα˙ ij α α˙ and local supersymmetry transformations. The gauge parameters are embedded in a complex su- ¯ 2 with X = 4W and χα = D DαK. perfield Lα, which so far obey no constraints. We The existence of the -multiplet in theories S assign a transformation law to the supergravity without an FZ-multiplet has important conse- fields of the form quences for supergravity theories. In the next ¯ ¯ section we will explain how to construct super- H" = Hαα˙ + DαLα˙ Dα˙ Lα . (97) αα˙ − gravity starting from the supercurrent, and this ¯ will lead to additional applications. where Lα˙ is the complex conjugate of Lα, and thus this maintains the reality condition. Requiring that (96) be invariant under these 7. Elements of Linearized Supergravity coordinate transformations, we get a constraint and Constraints on Moduli on the superfield Lα. Indeed, invariance requires In this section we study the couplings of su- that percurrent multiplets to supergravity theories. 0 = d4θ D¯ α˙ Lα = d4θ XDαL . (98) We are only interested in linearized supergrav- Jαα˙ α ity, namely the leading order in 1 . This ap- ! ! Mp proximate analysis is sufficient to derive several Since X is an unconstrained chiral superfield we general results. get the complex equation This approach to supergravity is taken, for ex- 2 α D¯ D Lα = 0 . (99) ample, in [2]. We begin with a review of the coupling of the FZ-multiplet to supergravity. We The analog of the Wess-Zumino gauge is that the then explain the coupling of the -multiplet to lowest components of Hµ vanish, i.e. supergravity.15 S Hµ = Hµ θ = Hµ θ¯ = 0 , (100) 7.1. Gauging the FZ-Multiplet / / / We start by reviewing the coupling of the FZ- as w/ ell as /the fact/ that Hµ θσν θ¯ is symmetric in µ and ν. multiplet to linearized gravity. The FZ-multiplet / contains a conserved energy-momentum tensor There is also some residual/ gauge freedom: and supercurrent and can therefore be coupled to H can be shifted by any complex di- • µ θ2 13 vergenceless vector. This leaves only one There are, however, theories in which DαX is well- / µ defined, but X itself is not. complex/ degree of freedom, ∂ Hµ θ2 . 14R-symmetric theories always admit a well-defined - R The metric field h transforms as/ multiplet. • µν / 15The case of the -multiplet is very similar but will not R be discussed in detail because gravitational theories with δhµν = ∂ν ξµ + ∂µξν , (101) global symmetries do not seem to be relevant according to our current understanding of quantum gravity. where ξµ is a real vector. 20

The gravitino transforms as (108) • δΨµα = ∂µωα . (102) We see that EF Z is invariant if (99) is imposed. αβ˙ The superﬁeld EF Z satisﬁes another important In this Wess-Zumino gauge the components αβ˙ containing the gravitino and metric take the form algebraic equation

β˙ F Z 2 γ τ˙ Hµ ν ¯ = hµν ηµν h , (103) D¯ E = D D¯ [D , D¯ ]H . (109) θσ θ − αβ˙ α γτ˙ and/ & ' / Here, the expression in parenthesis is chiral. Note ρ the similarity of the equation above to the defin- Hµ θ¯2θ = Ψµα + σµσ¯ Ψρ . (104) ing property of the FZ-multiplet itself (60). The The/ top component of H is a vector field which F Z / µ fact that E is invariant and satisfies an equa- survives in the Wess-Zumino gauge. The bosonic tion identical to the supercurrent superfield guar- off-shell degrees of freedom in Hµ consist of the antees that the Lagrangian µ complex scalar ∂ Hµ θ2 , six real degrees of freedom in the graviton and the four real degrees of 2 4 µ F Z / Lkin MP d θ H Eµ (110) freedom in the top comp/ onent of Hµ, for a to- ∼ tal of 12 off-shell bosons. For the fermions, we ! have only the gravitino. It has 16 4 = 12 off- is invariant. This contains in components the shell degrees of freedom. This is the−old minimal linearized Einstein and Rarita-Schwinger terms. multiplet of supergravity [38–40]. This is in ac- The six additional supergauge-invariant bosons, µ cordance with the 12 degrees of freedom in the ∂ Hµ θ2 , Hµ θ4 are auxiliary fields which are eas- FZ-multiplet. ily integrated out yielding ∂µH ix, H / / µ θ2 µ θ4 A simple consistency check is to verify the lead- j where/ x and/ j are the matter op∼erators in the∼ µ µ / / ing couplings of the graviton and gravitino to supercurrent multiplet. / / matter. Recalling the formula for (61) we We conclude that theories which have a well- Jαα˙ find defined FZ-multiplet can be coupled to supergravity in this fashion. The coupling to super- L h T µν , (105) ∼ µν gravity adds to the original theory a propagat- as expected. Similarly, for the coupling of the ing graviton and gravitino. No other propagating gravitino to matter we get fields are present in theory other than the original ones and the graviton and gravitino. This L /αβ (Ψ + σ σ¯ρΨ ) Sµ + 1 σµσ¯ρS ∼ µ µ ρ α 3 ρ β is important to remember in order to appreciate αµ = ΨµαS &. (106)' the point we will make soon. Note that if the FZ multiplet is not well defined (for example if The last ingredient is the kinetic term for the it is not gauge invariant or if it is not globally graviton and gravitino. We begin by constructing F Z well defined) the procedure above of constructing a real superfield Eαα˙ by covariantly differentiat- minimal supergravity cannot be carried out.16 ing Hαα˙ 7.2. Supergravity from the -Multiplet EF Z = D¯ D2D¯ τ˙ H + D¯ D2D¯ Hτ˙ αβ˙ τ˙ αβ˙ τ˙ β˙ α We emphasized in the previousS sections that γ 2 γτ˙ +D D¯ D H ˙ 2∂ ˙ ∂ H . (107) various supersymmetric field theories do not have α γβ − αβ γτ˙ The reader can easily check that this expression 16If there is no FZ-multiplet but there is an R-symmetry, is real. It is equivalent to a different-looking ex- we can construct the -multiplet and couple it to super- R pression in [41]. The gauge transformations (97) gravity. For example, a free supersymmetric U(1) theory with an FI-term can be coupled to supergravity in this act as fashion. For more comments on this case see also [42,43]. This procedure, however, gives rise to a supergravity the- F Z F Z ¯ 2 ¯ ¯α˙ ¯ 2 β Eα" β˙ = Eαβ˙ +[Dα, Dβ˙ ] D Dα˙ L + D D Lβ . ory with a continuous global R-symmetry. & ' 21

an FZ-multiplet and the energy-momentum ten- We also note that the top component of Hµ is sor and the supersymmetry current must be em- invariant. Thus, we see that we have 16 off-shell bedded in a larger multiplet αα˙ . In such a case bosonic degrees of freedom. The fermion is in the the construction of the previousS subsection can- θ2θ¯ component (and its complex conjugate). It not be accomplished and the only possible su- has residual gauge symmetry pergravity theory is the one in which the αα˙ is S µ α˙ gauged. δΨµα = i∂µωα , σαα˙ ∂µω¯ = 0 . (116) In this section we analyze this theory and as in the previous subsection, we limit ourselves to Since ωα satisfies the Dirac equation it cannot the analysis of the linearized theory. Since we be used to set any further components to zero. have already understood how to do such things in Therefore, our theory includes a gravitino as well the previous sections, we will be somewhat briefer as an additional Weyl fermion. Thus, we have 16 now. off-shell fermionic degrees of freedom. We begin from the coupling to matter We conclude that the theory has 16 + 16 fields. This is in accord with the 16 + 16 operators in 4 αα˙ the multiplet αα˙ . This supergravity multiplet d θ αα˙ H . (111) S S has been recognized in the supergravity litera- ! ture [44–46]. For this to be invariant under (97), we need to It is easy to construct a kinetic term; in fact impose the constraints EF Z defined in (107) is still invariant because αβ˙ 2 α 2 α˙ 2 α the set of transformations here is smaller than D¯ D Lα = 0 , D¯ α˙ D L¯ = DαD¯ L . (112) when the FZ-multiplet is gauged. However, this The first of them already appeared in the gaug- theory has another invariant. It is easy to see β ¯ β˙ ing of the FZ-multiplet but the second one is new. that [D , D ]Hββ˙ is invariant. We can use this Since Lα is more constrained here than in the pre- observation to write an invariant kinetic term vious subsection, we will find more gauge invari- 2 ant degrees of freedom. Some of them will even d4θ [D, D¯]H , (117) propagate. ! & ' Using an arbitrary Lα subject to these con- in addition to the one we have already included straints we can choose the Wess-Zumino gauge in the discussion of the FZ-multiplet. To summarize, we find that this theory admits H = H = H = 0 . (113) µ µ θ µ θ¯ two independent kinetic terms. Thus there is one The/ residual/ gauge/ transformations allow us to free real parameter r, and the most general ki- / / / netic term is transform Hµ θ2 by any divergence-less vector so we remain with one complex gauge invariant / 4 αα˙ F Z µ d θ H Eαα˙ operator ∂ Hµ/ θ2 . Another important residual transformation is ! ) / 1 αα˙ β β˙ / + H [Dα, D¯ α˙ ][D , D¯ ]H ˙ . (118) δH + δH = ∂ ξ + ∂ ξ , (114) 2r ββ µ θσν θ¯ ν θσµθ¯ µ ν ν µ * / ν / Our goal now is to identify the on-shell degrees with/∂ ξν = 0. This/ means that the trace part of this symmetric tensor is invariant under the resid- of freedom in this theory and study their cou- ual symmetries and therefore, the θθ¯ component plings to matter fields. There are many ways to contains the usual graviton but also an additional do this, here we will choose a somewhat pecu- invariant scalar. The antisymmetric piece enjoys liar way that will make some very important facts the usual gauge transformation of a two-form transparent. We enlarge the gauge symmetry, re- laxing either one of the two constraints (112) or δB = ∂ ω ∂ ω . (115) both, and add compensator fields. µν ν ν − ν µ 22

In order to contrast the situation with that in the constraint D2G = 0 by a Lagrange multiplier the previous subsection we choose to keep the first term d4θ Φ + Φ¯ G where Φ is a chiral super- constraint in (112) and relax the second one by field. This makes it easy to integrate out G using ( & ' 2 2 adding a chiral compensator field λα which trans- its equation of motion G = r Φ + Φ¯ + r U + forms as [D, D¯]H to find the Lagrangian & ' 3 2 4 αα˙ F Z ¯ ¯ δλα = D¯ Lα . (119) L = d θ H Eαα˙ + Φ + Φ + U [D, D]H 2 ) ( & ' First, the non-invariance of the coupling to mat- r ¯ 2 αα˙ 2 Φ + Φ + U + H αα˙ . (122) ter d4θ Hαα˙ can be corrected by a adding − S Sαα˙ * to the Lagrangian the term 1 d2θ λαχ + c.c.. In this presen& tation the' theory looks like a ( − 6 α Next, we move to the kinetic terms (118). The standard supergravity theory based on the FZ- ( first term is invariant, but the second term is not. multiplet which is coupled to a matter system This is easily fixed by adding more terms to the which includes the original matter as well as the Lagrangian. We end up with the invariant La- chiral superfield Φ. This is consistent with the grangian counting of degrees of freedom (4 + 4 degrees of freedom in addition to ordinary supergravity) and 1 L = d4θ Hαα˙ EF Z + Hαα˙ [D , D¯ ] [D, D¯]H with the identification of the 16/16 supergravity αα˙ 2r α α˙ ! as an ordinary supergravity coupled to a chiral αα˙ & 1 2 α & ' superfield. Note that even though the new super- + H αα˙ d θλ χα + c.c. S − 6 field Φ originated from the gravity multiplet, its ! 1 ' & ' couplings are not completely determined. At the d4θ Dγ λ + D¯ λ¯γ˙ [D, D¯]H − r γ γ˙ linear order we have freedom in the dimensionless ! ) parameter r and we expect additional freedom at 1 & 2 ' Dγ λ + D¯ λ¯γ˙ . higher orders. − 2 γ γ˙ * The linear multiplet G, or equivalently the chi- & ' (120) ral superfield Φ, are easily recognized as the dila- The first term in the second line corrects the non- ton multiplet in string theory. There the graviton invariance of the coupling to matter and the other and the gravitino are accompanied by a dilaton, two terms fix the transformations of the kinetic a two-form field and a fermion (dilatino). These terms. are the degrees of freedom in G. After a duality In order to read out the spectrum we denote transformation this multiplet turns into a chiral γ γ˙ superfield Φ. Furthermore, as in string models, G = D λγ + D¯ γ˙ λ¯ which is a real linear superfield (i.e. it satisfies D2G = 0). We also express the second term in (122) mixes the dilaton and µ 3 ¯ 2 the trace of the linearized graviton hµ. Both this χα = 2 D DαU, with a real U. We should remember− that this U might not be well-defined; term and the term quadratic in Φ lead to the dila- e.g. it might not be globally well-defined or might ton kinetic term. not be gauge invariant. In fact, the need of gaug- As we mentioned above, the need for the multiplet arises when the operator U is not a ing the -multiplet arises precisely when this U is Sαα˙ not well-defined.S The Lagrangian above becomes good operator in the theory. In this case the current αα˙ does not exist. The couplings in (122) 4 αα˙ F Z 1 αα˙ ¯ ¯ J L = d θ H Eαα˙ + 2r H [Dα, Dα˙ ][D, D]H explain how the chiral field Φ fixes this prob- αα˙ lem. Even though U is not a good operator, ( & +H αα˙ S Uˆ = Φ + Φ¯ + U is a good operator. If U is 1 d4θ G [D, D¯]H 'rU 1 G2 . (121) not gauge invariant, Φ transforms under gauge − r − − 2 ) * transformations such that Uˆ is gauge invariant. ( & ' Now we can dualize G. This is done by view- And if U is not globally well-defined because it ing it as an arbitrary real superfield and imposing undergoes Kähler transformations, Φ has simi- 23 lar Kähler transformations such that Uˆ is well- phasized above, the chiral superfield Φ is similar defined. to the dilaton superfield in four dimensional su- The result of this discussion can be presented persymmetric string vacua. We often have field in two different ways. First, as we did here, theory limits without an FZ-multiplet. For ex- we started with a rigid theory without an FZ- ample, we can have a theory on a brane with an multiplet and we had to gauge the -multiplet. FI-term. The field theory limit does not have This has led us to the Lagrangian (122).S Alterna- an FZ-multiplet and correspondingly, U ξV tively, we could add the chiral superfield Φ to the is not gauge invariant. This problem is ∼fixed, original rigid theory such that the combined the- as in (122), by coupling the matter theory to ory does have an FZ-multiplet. Then, this new Φ which is not gauge invariant (note the simi- rigid theory can be coupled to standard super- larity to the way this is realized in string the- gravity by gauging the FZ-multiplet. ory [52]). Similarly, we often consider field the- Our discussion makes it clear that if we want to ory limits with a target space whose Kähler form couple the theory to supergravity, the additional is not exact. This happens, for instance, on D3- chiral superfield Φ is not an option – it must be branes at a point in a Calabi-Yau manifold. If the added, and it is propagating. latter is non-compact we find a supersymmetric field theory on the brane which typically does not 7.3. Summary and Constraints on Moduli have an FZ-multiplet because U is not globally Of particular interest to us in this section was well-defined. Coupling this system to supergrav- the coupling of theories without an FZ-multiplet ity corresponds to making Mp finite. In this case to supergravity. Here we have limited ourselves to this is achieved by making the Calabi-Yau com- supersymmetric field theories in which all dimen- pact. Then in addition to the graviton, various sionful parameters are fixed and we have studied moduli of the Calabi-Yau space become dynami- the limit M . We have not studied theo- cal. They include fields like our Φ which couple p → ∞ ries in which the matter couplings depend on Mp. as in (122), thus avoiding the problems with the These have been recently discussed in [47–51] but FZ-multiplet and making the supergravity theory we will not elaborate on such “intrinsically grav- consistent. itational” theories here. This discussion has direct implications for mod- In case where the FZ-multiplet does not exist, uli stabilization. It is often desirable to stabilize we have to gauge the -multiplet. The upshot of some moduli at energies above the supersymme- the analysis of this gaugingS is the following. We try breaking scale. In this case we have to make add to the rigid theory a chiral superfield Φ whose sure that the resulting supergravity theory is still couplings are such that the combined system in- consistent. In particular, it is impossible to stabi- cluding Φ has an FZ-multiplet. This determines lize Φ in a supersymmetric way and be left with some but not all of the couplings of Φ to the mat- a low energy theory without an FZ-multiplet. ter fields. In the case of the FI-term Φ Higgses For example, if the low energy theory includes a the symmetry and in the case of nontrivial tar- U(1) gauge field with an FI-term, this term must get space geometry of the rigid theory it creates be Φ dependent. Furthermore, if the mass of Φ a larger total space in which the topology is sim- is above the scale of supersymmetry breaking, it pler. Now that we have an FZ-multiplet we can must be the same as the mass of the gauge field simply gauge it using standard supergravity tech- it Higgses. Consequently, there is no regime in niques. In particular, at the linearized level the which it is meaningful to say that there is an FI- couplings of Φ depend on only one free parameter: term. 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