Corfu Summer Institute on Elementary Particle Physics, 1998

PROCEEDINGS

An Introduction to Supergravity

D.G. Cerde˜no and C. Mu˜noz Departamento de F´ısica Te´orica C–XI and Instituto de F´ısica Te´orica C–XVI Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain E-mail: [email protected], [email protected]

Abstract: In these lectures we give an elementary introduction to supergravity. The role it plays in the context of superstring theory is emphasized. The level is suitable for postgraduate students and non–specialist researches in the subject. The outline is the following: 1 From general relativity to higher dimensional supergravity 1.1 Quantum gravity 1.2 Supergravity 1.3 Extended supergravity 1.4 Discussion: supergravity as the low–energy limit of superstring theory 1.5 Higher dimensional supergravity 2 D =4,N= 1 Supergravity 2.1 The Lagrangian 2.2 Spontaneous supersymmetry breaking 2.3 Soft supersymmetry–breaking terms 3 Supergravity from superstrings 4 Conclusions A. Appendix: Superspace formalism in supergravity References

1. From general relativity to higher relates bosons and fermions introducing new par- dimensional Supergravity ticles, the so–called supersymmetry. The scalar masses and the masses of their superpartners, the fermions, are related and as a consequence, only xperimental E observations confirm that the a logarithmic divergence in scalar masses is left. standard model of particle physics works out- In diagrammatic language, the dangerous dia- standingly well. However, if we believe that it grams of standard model particles are canceled should be embedded within a more fundamental with new ones which are present due to the ex- theory, we are faced with grave theoretical prob- istence of the additional partners and couplings. lems. For example, in the context of a grand This is shown schematically in Fig. 1. unified theory the characteristic scale is of or- 15 der MGUT ≈ 10 GeV which gives rise to the On the other hand, gravity, the fourth inter- gauge hierarchy problem. There is no symmetry action in nature, is not included in the standard protecting the masses of the scalar particles, Hig- model. Nowadays we know that the correct de- gses, against quadratic divergences in perturba- scription of nature involves quantum field theo- tion theory. Therefore they will be proportional ries. This is of course the case of the standard to the huge scale MGUT . This problem of nat- model, whose finishing touches were put around uralness, to stabilize the electroweak scale MW 1974. It describes strong, weak and electromag- against quantum corrections MW << MGUT , netic interactions using the internal (gauge) sym- may be solved postulating a new symmetry which metry SU(3) × SU(2) × U(1). However, the the- Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

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« + ª HH© HH H˜ Figure 1: The quadratic divergence due to the loop of standard–model bosons is cancel with the loop of fermionic superpartners which has opposite sign. ory describing the gravitational interaction, gen- shown in Fig. 3. The problem with this approach eral relativity, which was completed in 1915, is a is that the gravitational interaction is extremely classical theory. In this sense both theories seem weak since the coupling constant, the Newtonian to be completely disconnected as is schematically gravitational constant GN ,is shown in Fig. 2. The question then is to know if 1 −38 −2 GN = 2 ≈ 10 (GeV ) . (1.1) MP lanck

Thus experimental verification of quantum grav- THE STANDARD MODEL GENERAL RELATIVITY ity at low energies (≈ MW ) in accelerators seems (quantum theory) (classical theory) to be unlikely. For example, the contribution of the diagram of Fig. 3 to electron–electron scat- tering is negligible with respect to the one of the diagram shown in Fig. 4, since the relation be- Figure 2: The standard model and general relativ- tween the electromagnetic and effective gravita- ity are disconnected. The former is a quantum field tional coupling constants is theory whereas the latter is still a classical theory. e2 E2 α = ≈ 10−2 >> α = ≈ 10−34 . we will be able to quantize general relativity and e.m. G 2 4π MP lanck to unify it with the standard model. As we will (1.2) see in the rest of the paper, an important role Note that the energy E must be included to have in the answer to this question is played again by a dimensionless scattering amplitude. supersymmetry, in particular by its local version, Another process that we might study with supergravity. the hope of detecting quantum corrections ex- To carry out this project a basic ingredient is perimentally is the gravitational light bending. the graviton. This is an elementary particle with The first diagram in Fig. 5 reproduces the classi- spin 2 which is an excitation of the gravitational cal result for the deflected photon. As mentioned field similarly to the photon which is an exci- above, mass and energy are equivalent in relativ- tation of the electromagnetic field. The charge ity and since all particles have energy, gravity associated with gravity is mass and therefore en- couples with everything, and in particular with ergy since mass and energy are equivalent in rel- photons. Unfortunately, the lowest–order quan- ativity. It is then natural to extend the approach tum correction with a loop of electrons shown in of quantum field theory to gravity. In particular, the second diagram, is too small to be detected the use of Feynman diagrams as in the example in present solar light–bending experiments.

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Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

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In conclusion, even if a consistent quantum where ε now depends on the space–time coordi- theory of gravity is ever built, an issue that we nates, the action is invariant only if we add a will discuss in this section below, its experimental spin 1 gauge field Aµ. This gives rise precisely verification is apparently unlikely. Fortunately, a to Quantum electrodynamics (QED). Therefore, possible candidate to quantize gravity, supergrav- spin 1 fields correspond to generalizing internal ity, gives rise to low–energy signals which could (i.e. non–Lorentz) symmetries. be detected. This will be discussed in section 2. Similarly, a spin 2 appears when space–time Before entering into details, a few words about symmetries, global Poincare invariance, bibliography. Although we discussed quantum µ → µ ν µ gravity above and we will continue discussing it x Λν x + a , (1.6) in the next subsection, it is not the main topic of are made local in space–time, i.e. general coordi- these lectures. A simple and interesting introduc- nate transformations: tion can be found in refs. [1]–[3]. There are excel- lent books and reviews of supergravity. We quote 0 xµ → x µ(x) . (1.7) several of them in the references. In particular, refs. [4]–[8] focus mainly on theory and refs. [9]– Let us consider for example the action of a scalar [11] focus mainly on phenomenology. Refs. [12] field in flat space–time: and [13] although are not so exhaustive as the Z   above mentioned, are quite recent and introduce 4 1 µν S = d x η ∂µφ∂ν φ − V (φ) , (1.8) supergravity from a modern perspective. Other 2 references interesting to understand specific is- where η is the Minkowski metric. This action sues will be mentioned in the text. µν is invariant under (1.7) only if we add a spin 2 field, the graviton: 1.1 Quantum gravity Z   p 1 Let us recall first the relation between global and S = d4x −detg gµν ∂ φ∂ φ − V (φ) . µν 2 µ ν local symmetries in quantum field theory using (1.9) the Noether procedure. Consider for instance a Adding the usual Einstein piece, free massless spin 1/2 field. Its action Z Z 1 p S = − d4x −detg R, (1.10) 4 µ 2 µν S = i d x ψγ¯ ∂µψ, (1.3) 2k one obtains the complete action. Here k is the ≡ ∂ where ∂µ ∂xµ , is clearly invariant under the gravitational coupling constant global transformation √ p 8π ≡ 1 → −iε k = 8πGN = (1.11) ψ e ψ (1.4) MP lanck MP × 18 with ε a constant phase. However when the trans- with MP =2.4 10 GeV the so–called reduced formation is local Planck mass. Once we know the action, we may proceed to ψ → e−iε(x) ψ, (1.5) compute various processes. However, as is well

3 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

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φ φ h h Figure 6: Graviton–scalar interactions Figure 7: Graviton–graviton vertex. known, field equations are non linear. One pos- gravitons and other quanta interact and prop- sible way to solve the problem consists of intro- agate within a fixed space–time background. In ducing the expansion this sense, the language of general relativity where the geometry is crucial tends to get lost.Inany gµν (x)=ηµν + khµν (x) , (1.12) case, if using perturbation theory we are able where hµν (x)measuresthedeviation of the space– to compute different gravitational processes, the time from flat Minkowski space. We have to analysis is worthwhile. Unfortunately, this is not introduce the constant k in front of it since its what happens. For example, the computation of quanta should have mass dimensions 1 as ap- photon–photon scattering in the Maxwell–Einstein propriate to a bosonic field describing the gravi- theory shown in Fig. 8, turns out to give a result ton. Recalling that perturbation theory works which is divergent. Of course, this is not a prob- outstandingly well in the standard model, and lem if the theory is renormalizable. However, taking into account that gravity is much weaker, we know from quantum field theory that theories the use of perturbations should be appropriate. with negative coupling constant are non renor- For example, using (1.12) the free part of the ac- malizable, and this is precisely the case of gravity tion (1.9) can be written as where     Z 2 −2 p k = M = −2 . (1.14) 4 1 µν P d x −detgµν g ∂µφ∂ν φ = Z 2 So quantum gravity contains an infinite variety of 4 1 µν infinities. One can use simple dimensional argu- d x η ∂µφ∂ν φ Z 2 ments to arrive to this conclusion. A dimension- 2 n 4 1 µν less probability amplitude of order (k ) must −k d x h ∂µφ∂ν φ Z 2 diverge as Z 1 2 4 1 µν σ ρ 2n−1 −k d x η ∂ φ∂ φh h¯ 2 p dp , (1.15) µ ν ρ σ M n 8  P 1 where p is the momentum. For instance for n = − hµρh¯ν ∂ φ∂ φ + ... , (1.13) 2 ρ µ ν 2, which is the case of the diagram in Fig. 8, quartic divergences will appear. where the “bar” operation on an arbitrary second 1 σ Therefore a consistent theory of gravity must rank tensor is defined by X¯µν = Xµν − ηµν X . 2 σ be finite order by order in perturbation theory. The first term in the right–hand side of (1.13) is the usual action for free scalar fields in flat space–

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4 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

We already know from the above discussion of su- Local persymmetry that theories with more symmetry THE STANDARD MODEL GENERAL RELATIVITY are more convergent. In supersymmetry quadratic Supersymmetry divergences are canceled (see Fig. 1). Then, why not to apply the ideas of supersymmetry to grav- Figure 9: The standard model and general relativity ity to solve the problem of divergences ?. This is are connected through local supersymmetry. what we will carry out in the next subsection. It is instructive to see explicitly the need for 1.2 Supergravity gravity in local supersymmmetry. Let us con- We showed above that gravity is the ‘gauge the- sider the simple case of a scalar field φ together ory’ of global space–time transformations. Like- with its supersymmetric partner the spin 1/2fermion wise we will see in this section that supergravity ψ. The Lagrangian is the gauge theory of global supersymmetry. µ ∗ 1 µ Let us consider schematically two consecu- L = −(∂ φ )(∂µφ) − ψγ¯ ∂µψ (1.20) tive infinitesimal global supersymmetric trans- 2 formations of a boson field B is invariant under global supersymmetry:

δ1B ∼ ε¯1 F, (1.16) δφ = εψ (1.21) µ δ2F ∼ ε2 ∂B , (1.17) δψ = −iσ ε∂¯ µφ. (1.22) where F denote a fermion field. Using dimen- However L is not invariant under local supersym- sional arguments one can deduce from (1.16) that metry since ε → ε(x) implies the dimension of the anticommuting fermionic L α µ parameter ε must be [ε]=−1/2 in mass unit δ = ∂µε Kα + h.c. (1.23) since [B]=1and[F]=3/2. Thus in the second with transformation (1.17) we must include a deriva- tive to obtain the correct dimension. This im- α ≡− ∗ α− i β ν α ∗ Kµ ∂µφ ψ ψ (σµσ¯ )β∂νφ (1.24) plies that two internal supersymmetric transfor- 2 mations have led us to a space–time translation, To keep the action invariant, a gauge field has to

µ µ µ be introduced (similarly to the case of an ordi- {δ1,δ2}B ∼a ∂µB ; a =¯ε2γ ε1 (1.18) nary gauge symmetry where Aµ is introduced as and therefore supersymmetry is an extension of we mentioned in the previous section) with the the Poincare space–time symmetry Noether coupling

µ L α µ {Q, Q¯} =2γ Pµ . (1.19) N = kKµΨα , (1.25)

Clearly, the generator Q is not an internal sym- where k is introduced to give LN the correct di- metry generator like the ones of the standard mension, [LN ] = 4, and Ψ is a Majorana vector model symmetries, SU(3) × SU(2) × U(1), since spinor field with spin 3/2, the so–called gravitino, it is related to the generator of space–time trans- transforming as lations Pµ. 1 Ψµ → Ψµ + ∂µε . (1.26) Promoting global supersymmetry to local, ε = α α k α ε(x), space–time dependent translations aµ∂ that µ L L differ from point to point are generated, i.e. gen- However, + N is not still invariant since eral coordinate transformations. Therefore lo- µν δ(L + LN )=kΨ¯µγνεT , (1.27) cal supersymmetry necessarily implies gravity as shown schematically in Fig. 9. This situation where T µν is the energy–momentum tensor. This is to be compared with the one summarized in contribution can only be canceled adding a new Fig. 2. By obvious reasons local supersymmetry term µν is also called supergravity. Lg = −gµν T (1.28)

5 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

Figure 10: Contributions to the photon–photon scattering in the supersymmetric Maxwell–Einstein theory. Ψandλdenote gravitino and photino respectively.

provided the tensor field gµν transforms as Fig. 8 those with supersymmetric particles, grav- itino and photino, shown in Fig. 10. The contri- ¯ δgµν = kΨµγν ε. (1.29) bution from each diagram is equal to an infinite quantity multiplied by the coefficient written be- Thus any locally supersymmetric theory has to include gravity. low the figure. The infinite quantity is the same for all figures. Unfortunately, adding the coeffi- In particular, the standard model which global cients shows that the sum is 25/12, i.e. non zero, supersymmetry contains the chiral supermulti- and therefore the divergence is not canceled. plets (ψ, φ) studied above with ψ denoting quarks, We have shown then that simple supersym- leptons, Higgsinos and φ denoting squarks, slep- tons, Higgses, plus vector supermultiplets (V , λ) metry is not enough to solve the problem of in- finities in quantum gravity. In the next subsec- with gauge bosons and gauginos (spin 1/2Ma- tion we will try to answer the following question: jorana fermions) respectively. In the presence of Is it possible to extended supersymmetry to a local supersymmetry we must include also the α bigger symmetry solving the problem?. gravity supermultiplet (gµν ,Ψµ) with graviton and gravitino respectively. The gravitino plays 1.3 Extended supergravity the role of the gauge field of local supersymme- try. It is natural to wonder what would be the con- In conclusion we can say that supergravity sequences of the introduction of more than one is a quantum theory of gravity. Since we have supersymmetry generator, i.e. now more symmetry than in pure gravity we can QA (A =1,2, ... ,N) , (1.30) expect that the high–energy (short distance) be- havior will improve. Although this is basically where true, still the (super)symmetry is not enough to QA|λi = |λ − 1/2i (1.31) cancel all divergences in the theory. To see this with λ the helicity of a massless state. In this diagrammatically we have to include the super- case the gravity supermultiplet will contain N symmetric partners in the graphs. For example, gravitinos since in the Maxwell–Einstein theory discussed in the previous subsection, we have to add to graphs in Q1|λ =2i=|λ=3/2i, ...

6 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

Figure 11: Contributions to the photon–photon scattering in the Maxwell–Einstein theory with N =2 supersymmetry. Ψ and Ψ0 denote the two gravitinos.

... ,QN|λ=2i=|λ=3/2i. (1.32) Although this result seems very promising, actually beyond one loop photon–photon scatter- On the other hand, since ing is not finite. Including matter the situation is worse, e.g. scalar–scalar scattering, etc. So Q1Q2 ...Q |λ=2i=|λ=2−N/2i (1.33) N again we have come back to our original prob- we have the bound lem in quantum gravity. Is there any way out? In principle we have not yet exhausted all possi- N ≤ 8 (1.34) bilities. N = 8 is the maximum number of su- persymmetries we can use. Since theories with to avoid massless particles with λ>2. Recall more symmetries are more convergent we should that not consistent coupling between massless analyze that case. In fact N = 8 case is also spin 2 particles and spin 5/2, 3,...seems to exist. interesting because gravity, Yang–Mills, matter The simplest extension of N =1supersym- multiplets cannot exist in isolation. They belong metry discussed in the previous section is N =2 to the only supermultiplet of the theory supersymmetry. Precisely this case realizes Ein- 3 1 1 3 stein’s dream of unifying electromagnetism and λ =(2, ,1, ,0,− ,−1,− ,−2) . 2 2 2 2 gravity since (1.36)

Q1,2|λ =2i= |λ=3/2i For example the multiplicity of states with helic-

Q1Q2|λ=2i= |λ=1i (1.35) ity 1/2 is 56 since we can do the following com- binations: giving rise to only one supermultiplet with gravi- Q1Q2Q3|λ =2i=|λ=1/2i ton, two gravitinos and one photon (2, 3/2, 1). | i | i This model was the first one where finite quan- Q1Q2Q4 λ=2 = λ=1/2 tum corrections were found. This is shown in ... (1.37) Fig. 11 where the photino of Fig. 10 correspond- In general, multiplicity of states with helicity λ− ing to N = 1 supersymmetry is substituted by m/2is   a second gravitino. Now adding the coefficients N N! = . (1.38) the sum is zero. m m!(N − m)!

7 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

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@ 1 1 Figure 13: Weak interaction in Weinberg–Salam λ =(− 0+ ) 2 2 theory. (1.39) left and right handed fields. Since in the case of similar to the one of the old Fermi theory. The the standard model e.g. eL ∈ SU(2) but eR is weak interaction was described as an interaction a singlet, necessarily unobserved fields ER must of four fermions with a dimensionfull coupling − belong to the same supermultiplet as eL.These constant [GF ]= 2. This is shown in Fig. 12. are the so–called mirror partners. They must Now we know that the correct theory involves in have a mass beyond the current experimental fact the graph shown in Fig. 13. For p ≈ MW bounds. On the one hand, it is extremely in- this is the correct way to carry out any compu- volved to build realistic models which generate tation. However, for p<anomaly cancels within each g /MW . Thus although the Fermi theory is non generation, why such mirrors should exist?. All renormalizable, in some particular energy regime these arguments make unlikely that extended su- one can trust the results. pergravities be realistic four–dimensional theo- Although for p ≈ MP one must use the the- ries. ory behind supergravity, for p<1 supergravity is not only non renormaliz- only one consistent theory of quantum gravity able but also non interesting from phenomeno- with matter: string theory. There the solution to logical viewpoint (at least in four dimensions). the problem of divergences in quantum field the- Given these pessimistic conclusions, one won- ory consists of considering the elementary par- ders whether to work at low energies with the ticles to be not points but one-dimensional ex- physically relevant N = 1 supergravity is con- tended objects, strings, as shown in Fig. 14. sistent. The answer is yes if we are consider- In fact, the consistency of the theory need ing the supergravity Lagrangian as an effective the presence of supersymmetry and that is the phenomenological Lagrangian which comes from reason why it is called superstring theory. Re- a (finite) bigger structure. This is a situation markably, the low–energy limit (massless modes)

8 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

Divergent Finite

Figure 14: Exchange of a graviton between two elementary particles in quantum field theory and superstring theory. of superstring theory is supergravity. The picture To build the pure N = 1 supergravity one is then the following. Around the Planck scale, has to realize that the number of bosonic and fermionic degrees of freedom must be equal. These SUPERSTRING THEORY E > M P are shown in Table 1. For example, to deduce the dimension of Dirac spinors in D space–time dimensions one can con-

SUPERGRAVITY E < M P struct the Dirac gamma matrices obeying the Clifford algebra {Γµ, Γν } =2ηµν . The result is

D/2 DΓ =2 ; Deven, SUPERSYMMETRIC E << M P (D−1)/2 STANDARD MODEL DΓ =2 ; D odd . (1.40)

Figure 15: Supergravity is the intermediate step In our case D = 10 implies that λ has 32 com- between the possible final theory of elementary par- plex components. Taking into account Majorana ticles and the supersymmetric standard model ob- condition we reduce this number to 16. With servable at low energy. the Weyl condition it is further reduced to 8 and finally field equations reduce it to 8 real compo- superstring theory might be the correct theory of nents. elementary particles, but below that scale super- The Lagrangian can be explicitly obtained gravity can be used as an effective theory. This by the Noether’s method or by a formal dimen- is schematically shown in Fig. 15. The last step, sional reduction from higher dimensional theo- around the electroweak scale, corresponds to the ries, in particular from D = 11, N =1super- supersymmetric standard model arising from the gravity. The result is spontaneous breaking of supergravity as men- tioned above. −1L − 1 − 3 −3/2 µνρ e = 2 R φ HµνρH From the above arguments we conclude that 2k 4 9 ∂ φ∂µφ 1 the study of supergravity is crucial. It is the − µ − Ψ¯ ΓµνρD Ψ 16k2 φ2 2 µ ν ρ connection between the possible final theory of √ 1 µ 3 2 ∂ν φ ν µ elementary particles and the low–energy effective − λ¯Γ Dµλ − Ψ¯ µΓ Γ λ theory which might be tested experimentally. √2 8 φ 2k −3/4 ¯ µνρστ 1.5 Higher dimensional supergravity + φ Hνρσ[ΨµΓ Ψτ 16 √ ν ρ σ νρσ µ Since superstring theory is only consistent in a +6Ψ¯ Γ Ψ − 2Ψ¯ µΓ Γ λ] ten–dimensional (D = 10) space–time, to build + four−fermion terms , (1.41) supergravities in extra space–time dimensions is important. For example, the coupled D = 10, where Γµ1...µn stands for the completely antisym-

N = 1 supergravity super Yang–Mills system is metrized product of Γ matrices and Hµνρ is the the massless sector of the type I superstring the- field strength of the antisymmetric tensor Bµν . m ory and heterotic string theory. The vierbein eµ with m a local Lorentz index

9 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

FIELD CONTENT IN D = 10, N = 1 SUPERGRAVITY

graviton (gµν ) = 35 components dilaton (φ)= 1 ”

antisymmetric tensor (Bµν )=28 ” 64 real field components

µ Majorana–Weyl gravitino (Ψα) = 56 components Majorana–Weyl fermion (λα)= 8 ” 64 real field components

Table 1: Bosonic and fermionic degrees of freedom in ten–dimensional pure supergravity

FIELD CONTENT IN D = 11, N = 1 SUPERGRAVITY

graviton (gµν ) = 44 components 3rd. rank antisymmetric tensor (Aµνρ)= 84 ” 128 real field components

µ Majorana gravitino (Ψα) = 128 components 128 real field components

Table 2: Bosonic and fermionic degrees of freedom in eleven–dimensional pure supergravity

must be used instead of the metric gµν when time in supersymmetry is constrained to be fermions are present. Their relationp is gµν = m n ≡ m − D ≤ 11 . (1.47) eµ eν ηmn and therefore e det eµ = det gµν . This Lagrangian is invariant under the local Otherwise, counting the number of degrees of supertransformations freedom as above, massless particles with spin higher than 2 would appear. This is extremely k δem = ε¯ΓmΨ , (1.42) interesting since D = 11, N = 1 supergravity is µ 2 µ √ the low–energy limit of so–called M–theory [14]. 2 k δφ = − φελ¯ , (1.43) There is the proposal that M–theory, from which 3 √ the five existent superstring theories can be de- 2 δB = φ3/4(¯εΓ Ψ − ε¯Γ Ψ rived, is a consistent quantum theory contain- µν 4 µ ν ν µ √ ing extended objects. In this sense the study of 2 − ε¯Γ λ) , (1.44) D = 11 supergravity may be important. 2 µν √ In fact, D = 11 supergravity is a very attrac- 3 2 1 δλ = − (Γ ∂µφ)ε tive theory by its own since supergravity equa- 8 φ µ tions look very simple and natural. Besides, this 1 + φ−3/4ΓµνρεH theory is unique. The field content of the theory 8 µνρ together with their degrees of freedom are shown + two − fermion terms , (1.45) √ in Table 2. By brute force the Lagrangian was 1 2 δΨ = D ε + φ−3/4(Γνρσ built [15] with the following relatively simple re- µ k µ 32 µ − ν ρσ sult: 9δµΓ )εHνρσ − L − 1 − 1 µνρσ + two fermion terms . (1.46) = 2 eR eFµνρσ F 2k 48 √ 1 2k − eΨ¯ ΓµνρD Ψ − e(Ψ¯ ΓµνρσδλΨ On the other hand, the dimension of space– 2 µ ν ρ 384 µ λ

10 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

+12Ψ¯ νΓρσΨδ)(F + Fˆ) available. In fact, although various matter cou- √ νρσδ 2k plings had previously been constructed using the − εµ1...µ4ν1...ν4ρσδ F F A . 3456 µ1...µ4 ν1...ν4 ρσδ Noether procedure, the complete Lagrangian in- (1.48) cluding vector supermultiplets [16] was construc- ted using the more efficient local tensor calculus It is invariant under method. For a review of the latter see [6], where

m k m also the references of the authors who have con- δeµ = ε¯Γ Ψµ , (1.49) 2 √ tributed to the subject can be found. In Ap- 2 pendix A we sketch another formulation, the su- δAµνσ = − ε¯Γ[µν Ψσ] , (1.50) 8 √ perspace formalism which is the most elegant one. 1 2 In what follows we present the final result δΨ = D ε + (Γνρσλ (1.51) µ k µ 288 µ (with up to two derivatives), obtained using any − ν ρσλ ˆ 8δµΓ )εFνρσλ . (1.52) of the available formulations. Let us concentrate first on the chiral supergravity Lagrangian. It 2. D =4,N=1Supergravity turns out to depend only on a single arbitrary ∗ real function of the scalar fields φi and φj with Although some higher dimensional theory will i, j =1, ..., n,theK¨ahler function probably be the unified theory of particle physics ∗ ∗ 2 as discussed above, needless to say to connect it G(φ ,φ)=K(φ ,φ)+ln|W(φ)| , (2.1) with the observable world one has to compactify which is a combination of a real function, the so the extra dimensions. At the end of the day, the called K¨ahler potential K, and an analytic func- theory which is left in four dimensions is N =1 tion, the so called superpotential W . This ex- supergravity. Thus the study of D =4,N=1 presses the fact that the scalar–field space in su- supergravity is crucial. We will carry it out in persymmetry is a K¨ahler manifold, i.e. a special this section in a completely general way with- type of analytic Riemann manifold (see above out assuming any particular underlying higher (A.15) in Appendix A for more details). The dimensional theory. After analyzing the Lagrang- scalar fields should be thought of as the coordi- ian, we will study the spontaneous breaking of nates of the K¨ahler manifold and, in particular, supergravity. This gives rise to the so–called soft the metric K ∗ is given by supersymmetry–breaking terms which determine ij the spectrum of supersymmetric particles. The ∂2K theory of soft terms provided by this computa- Kij∗ = ∗ . (2.2) ∂φi∂φj tion enable us to interpret the (future) experi- mental results on supersymmetric spectra. This An important property of G (and therefore of the would be an (indirect) test of supergravity. Lagrangian) is its invariance under the transfor- In section 3 we will apply these general re- mations sults to the particular case of D = 4 supergravity → ∗ ∗ arising from compactifications of D =10super- K K + F (φ)+F (φ ), −F(φ) strings. W → e W. (2.3)

2.1 The Lagrangian where F is an arbitrary function. This property is known as K¨ahler invariance. In section 1.2, starting with the global supersym- We split the (tree–level) Lagrangian into terms metry Lagrangian of free chiral supermultiplets, as we were able to obtain their couplings to su- LC = LC + LC +LC , (2.4) pergravity using the Noether procedure. In the B FK F LC LC general case, with chiral supermultiplets in in- where B contains only bosonic fields, FK con- teraction, we can also follow the same approach. tains fermionic fields and covariant derivatives However, it is worth noticing that other formu- with respect to gravity (i.e. including the su- LC lations of D =4,N= 1 supergravity are also persymmetric spin connection) and F contains

11 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz fermionic fields but no covariant derivatives. Then, Finally, the third term in (2.5), which arises when the auxiliary fields Fi appearing in the chiral su- −1 C 1 i µ ∗j e L = − R − G ∗ ∂ φ ∂ φ B 2 ij µ permultiplets are eliminated by their equations G −1 ij∗ of motion (an extra gaugino bilinear piece given −e (Gi(G ) Gj∗ − 3) , (2.5) ∗ by − 1 ∂fab (G−1)ik λaλb should be added when 4 ∂φk where repeated indices are summed in our no- vector supermultiplets be considered below) tation and e was already defined below (1.41). 2 −1 ∗ −1 ∗ G/ ij ∗ − ik ∗ j l m Fi = e (G ) Gj (G ) Gjlk ψ ψ It is worth recalling that eµ gives rise to inter- 1 j actions of the graviton with all other particles + ψ (G ∗ ψ ), (2.11) 2 i j using the expansion studied in section 1.2. Note that we have set the reduced Planck mass MP contributes to the (tree–level) scalar potential. defined in (1.11) equal to 1 for convenience (see It is a fundamental piece for model building.In e.g. the usual Hilbert–Einstein piece in (2.5)). particular, as we will discuss in the next sub- It can easily be inserted using dimensional argu- sections, it is crucial to analyze the breaking of ments as we will do below in some examples. We supersymmetry as well as the so called soft terms also follow the notation which determine the supersymmetric spectrum. Note the exponential factor eG which obviously ∂G G ≡ (2.6) indicates the non renormalizability of the theory. i ∂φ i The piece and 2 ∂ G −1LC −1 −1 µνρσ ¯ ∗ ≡ ∗ e = e ε Ψµγ5γν DρΨσ Gij ∗ = Gj i , (2.7) FK 2 ∂φi∂φj 1 −1 µνρσ i −1 ij∗ + e ε Ψ¯ µγν Ψρ(GiDσφ with the matrix (G ) the inverse of the Gj∗k 4  ∗ ∗ i j −1 ij i i ∗ ¯i (G ) G ∗ = δ . (2.8) −GiDσφ )+ Gij ψ D/ψ kj k 2 L L i 1 From (2.1) and (2.2) we deduce ¯i j k − ∗ ∗ + ψLD/φ ψL( Gijk + Gik Gj ) 2 2  ∗ ∗ Gij = Kij (2.9) 1 µ i∗ j + √ Gij∗ Ψ¯ D/φ γµψ + h.c. 2 L R and therefore the K¨ahler metric K ∗ determines ij (2.12) the kinetic terms for the scalars φi (see the sec- ond term in (2.5)). This is also the case for contains the kinetic terms for the fermions (i.e. the spin 1/2fermionsψi in (2.12) below since for the spin 3/2 gravitino Ψ and the spin 1/2 both belong to the same chiral multiplets. Thus fermions ψi)andsomenon–renormalizable inter- in general we will have non–renormalizable ki- action terms. For example, even assuming canon- netic terms as a consequence of the non renor- ical kinetic terms Gij∗ = δij , the last term in malizability of supergravity (see (A.13) in Ap- (2.12) has at least mass dimension 5 and there- pendix A for more details). As we will see in fore must be suppressed by a power of 1/MP . the next subsection, some scalar fields φi may This interaction term is shown in Fig. 16. Fi- acquire dynamically vacuum expectation values nally, h implying Kij∗ =6 0. This will give rise in general −1LC G/2 ¯ µ ν G/2 ¯ µ i to non–canonical kinetic terms and therefore we e F = e Ψ σµν Ψ + e GiΨLγµψL will have to normalize the fields to obtain, at the 1 + eG/2 − G − G G end of the day, canonical kinetic terms. A sim- 2 ij i j
i −1 ∗ j ple form of K leading to canonical kinetic terms ∗ k l ¯i + Gijk (G ) Gl ψLψR + h.c. Kij∗ = δij (the so called minimal supergravity  1 1 −1 model), which will be used in the next subsec- + G ∗ ∗ − G ∗ (G ) ∗ G ∗ ∗ 2 ijk l 2 ijm nm nk l tion as a toy model, is given by  1 k l ∗i − Gik∗ Gjl∗ ψ¯LiψLjψ¯ ψ K = φiφ . (2.10) 4 R R

12

Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

@ @

ψψ

@ @

ψφ

@ @

@

@ 1/MP ) ( V ) ( Ψ Figure 18: Gauge boson–fermion-fermion interac- Figure 16: gravitino–fermion–scalar non– tion renormalizable interaction also rise to interactions between gauge bosons 1 − ∗ ¯i abcd ¯ and fermions. This is shown in Fig. 18 for the Gi j ψRγdψLj ε ΨaγbΨc 8
minimal supergravity model. The other pieces in − ¯ a d Ψ γ5γ Ψa , (2.13) (2.14) are written below. LV The piece B is given by where σµν stands for the antisymmetrized prod- uct of γ matrices, contains the fermion Yukawa −1LV −1 a b µν e B = (Refab)(F )µν (F ) couplings and several non–renormalizable terms. 4 i a b µν The former are due to the third piece in (2.13) + (Im fab)(F )µν (F˜ ) ∂2W 4 since Gij is proportional to , and therefore ∂φi∂φj 1  ab for a trilinear superpotential, W = Y φ φ φ , − g2 (Re f)−1 G (T )ij φ G (T )klφ ijk i j k 2 i a j k b l terms of the type shown in Fig. 17 will arise. It (2.15) is worth noticing that the first term in (2.13) is a potential mass for the gravitino (local super- where g denotes the gauge coupling constant, a symmetry breaking) if some of the scalar fields denotes the gauge group index, T a the group gen- φi develop expectation values in such a way that erators in the same representation as the chiral 2 a eG/ =6 0. We will discuss this possibility in de- matter and (F )µν the gauge field strength. tail in the next subsection. Note that the supergravity Lagrangian de- To obtain the complete supergravity Lagrang- pends not only on G but also on an arbitrary ian which couples pure supergravity to supersym- analytic function of the scalar fields φi, metric chiral matter and Yang–Mills we still have fab(φ) . (2.16) to include the vector supermultiplets in the for- mulation. The result is: It must transform as a symmetric product of ad- joint representations of the gauge group G to ren- L LC LC LC = B + FK + F der the Lagrangian invariant. The function f, LV LV LV which appears due to the non renormalizability + B + FK + F , (2.14) of the theory (see (A.24) in Appendix A for more LC LC LC where B, FK and F are as in (2.5), (2.12) and details), is called the gauge kinetic function since (2.13), but with the derivatives covariantized also it is multiplying the usual gauge kinetic terms. with respect to the gauge group in the usual way. It is remarkable that this fact provide us with For example, the term which gives rise to the a mechanism to determine dynamically the gauge 0 kinetic energies for the fermions ψi in (2.12) gives coupling constant. By defining gVµ = Vµ in quantum field theory, g is removed from the field

strength covariant derivative and appears only

@

ψψ in an overall 1/g2 in the kinetic terms. There-

@

@ fore if some scalar fields φ acquire dynamically

@ i vacuum expectation values we may obtain expec- φ tation values for fab and this may play the part of the coupling constant. In particular, Figure 17: scalar–fermion–fermion Yukawa cou- 1 pling Re fab = 2 . (2.17) gab

13 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

λ

This is in fact the case of supergravity models

deriving from superstring theory as we will see

1 __ /MP

in section 3. In the formulae of this section we ^^ @

V @ @

keep for completeness both g and f. @

@@

Ψ @ The third term in (2.15), which arises when @ the auxiliary fields Da appearing in the vector Figure 19: Gravitino–gauge boson–gaugino non– supermultiplets are eliminated by their equations renormalizable interaction of motion, LV  The piece FK is given by 1 ∂f  D = i [(Ref)−1]ab gG (T )ij φ + i bc ψ λc 1 1 a i b j 2 ∂φ i e−1LV = (Re f ) λ¯aD/λb  i FK ab ∗ 2 2 1 ∂fbc i 1 1 − i ψ λ + λ (G ∗ ψ ), (2.18) ¯a µ νρ b 2 ∂φ∗i c 2 a i i + λ γ σ Ψµ(F )νρ 2  1 + G Dµφiλ¯a γ λb contributes to the (tree–level) scalar potential to- 2 i L µ L gether with the third term in (2.5) which was i a µ b − (Im f )D (eλ¯ γ5γ λ ) studied above. The complete scalar potential is 8 ab µ then 1 ∂f − ab ψ¯ σµν (F a) λb + h.c. .   2 ∂φi iR µν L G −1 ij∗ V = e Gi(G ) Gj∗ − 3 (2.22)   1 2 −1 ab ij kl + g (Re f) Gi(Ta) φj Gk(Tb) φl , The first term determines the kinetic energies 2 for gauginos λa. They will be non canonical in (2.19) general due to the contribution of the gauge ki- netic function f. A simple form of f leading to where obviously the first piece is due to the F – canonical kinetic terms is term contribution and the second piece is due to the D–term contribution. Note that F and D fab = δab . (2.23) terms in (2.11) and (2.18) are supergravity gen- eralizations of the ones in global supersymmetry, The other terms in (2.22) are in general non renor- where malizable. For example, the second one, even assuming f as in (2.23), has at least mass di- ∂W mension 5. This interaction is shown in Fig. 19. Fi = − , ∂φ LV i Finally, the piece F is given by ∗ ij D = −gφ (T ) φ , (2.20) ∗ a i a j ∗ −1LV 1 G/2 ∂fab −1 j k a b e F = e ∗j (G ) Gkλ λ ∗ 1 4 ∂φ and the scalar potential V = F F i + D Da global i 2 a i is given by − gG(Ta)ij φ Ψ¯ γµλ 2 i j µL aL 2 a ik +2ig Gij∗ (T ) φkλ¯aRψiL ∂W 1 2 ∗ ij ∗ kl Vglobal = + g φ (Ta) φj φ (Ta) φl .   i k i −1 ab ∂fbc ij ∂φi 2 − g (Re f) Gi(Ta) φj ψ¯kRλcL 2 ∂φk (2.21) ∗ ∗ − 1 −1 lk ∂fab ∂fcd ¯a b ¯c d (G ) ∗k λLλLλRλR It is worth mentioning the striking difference be- 32 ∂φl ∂φ
2 tween the scalar potentials (2.21) and (2.19). Whe- 3 ¯a b + RefabλLγmλR reas the global one (2.21) is positive semi–definite, 32 1 the local one (2.19) may be negative (see e.g. the + Ref λ¯aγµσρσΨ Ψ¯ γ λb 8 ab µ ρ σ piece −3eG). This fact will be crucial to break  1 ∂fab ¯ µν a ¯ b supergravity being able to fine tune the vacuum + ψLiσ λ ΨνLγµλ 2 ∂φ L R i  energy density (cosmological constant) to zero, 1 + Ψ¯ γµψ λ¯a λb as we will discuss in the next subsection. 4 µR Li L L

14 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

 1 j ¯ µ ¯d c ∗ We finally give the explicit form of the local– + ψLiγ ψRλRγµλL 2Gij Refcd 16  supersymmetry transformations: ∗  ab ∂f ∂f + (Re f)−1 ac bd ∂φ ∂φ∗j δem = −ikεγ¯ mΨ , (2.26) i µ µ 1 c d −1 lk∗ ∂fcd 2 ∗i + ψ¯ ψ λ¯ λ − 4G ∗ (G ) δΨ = D ε + kε(G D φ − G ∗ D φ ) 16 Li Lj L L ijk ∂φ µ k µ i µ i i µ  l 2 G/2 ∂f  ab ∂f ∂f + ie γµε +4 cd − (Re f)−1 ac bd ∂φ ∂φ ∂φ ∂φ 1 i ν i j i j − σ ε G ∗ ψ¯ γ ψ   2 µν L i j R Lj − 1 ¯ ¯c µν d −1 ab ψLiσµν ψLjλLσ λL (Re f) 1
16 − − ∗ i ΨµL Giε¯LψLi Gi ε¯RψR ∂f ∂f 4 × ac bd + h.c. (2.24) 1 a ν b ∂φi ∂φj + (g − σ )ε λ¯ γ λ Ref ,(2.27) 4 µν µν L L R ab µ − µ It is remarkable that if some of the scalar fields δVa = ε¯Lγ λaL + h.c. , (2.28) µν φi acquire vacuum expectation values, gaugino δλaL = σ (Fa)µν εL masses may appear through the first term in (2.24). i −1 ab + g[(Re f) ] Gi(Tb)ij φj εL This is an indication of supersymmetry breaking 2  which will be discussed in detail in subsection 2.3. 1 −1 ∂fbc + iε [(Re f) ]ab i ψ¯ λc 4 R ∂φ Li L The third term is a typical supersymmetric in-  i teraction. It is shown in Fig. 20 for the minimal ∂f∗ − i bc ψ¯i λc supergravity model. Note finally that (2.24) con- ∂φ∗i R R tains numerous four–fermion terms. 1
− α − ∗ i λR Giε¯LψLi Gi ε¯RψR , (2.29) In summary, the D =4,N= 1 supergravity √4 Lagrangian (2.14) depends only on two functions δφi = 2¯εψi , (2.30) √ of the scalar fields. the K¨ahler function and the 1 G/2 −1 ij∗ δψ = D/φ ε − 2e (G ) G ∗ ε gauge kinetic function i 2 i R j ∗ 1 ∗ ∂f + ε λ¯a λb (G−1)ik ab G(φ∗,φ)=K(φ∗,φ)+ln|W(φ)|2 , 8 L R R ∂φ∗k f (φ) . (2.25) 1 −1 ik∗ ab + εL(G ) Gjlk∗ ψ¯LjψLl 2   1 j The K¨ahler potential K is a real gauge–invariant − ∗ − ψLi Gj ε¯RψR Gjε¯LψLj .(2.31) function and f and the superpotential W are an- 4 alytic functions. Once G and f are given, the 2.2 Spontaneous supersymmetry breaking full supergravity Lagrangian is specified. Unfor- tunately for the predictivity of the theory, both In section 1.2 we arrived to the conclusion that functions are arbitrary. However, as we will see supergravity is the gauge theory of global super- in section 3, in supergravity models deriving from symmetry with the gravitino as the gauge field. superstring theory they are more constrained and Now that we know the supergravity Lagrangian explicit computations can be carried out. our next step is to ask whether the analog of the Higgs mechanism exists in this context. We

will see in this subsection that this is indeed the

@

case. The process is the following: scalar fields

ψφ@

acquire dynamically vacuum expectation values

@

@ giving rise to spontaneous breaking of supergrav- ity. The goldstino, which is a combination of the fermionic partners of those fields, is swallowed by λ the massless gravitino to give a massive spin 3/2 Figure 20: fermion–scalar–gaugino supersymmetric particle. This is the so–called super–Higgs effect. interaction Let us study it in detail.

15 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

As usual in gauge theories the condition for Whether or not scalar fields acquire expecta- (super)symmetry breaking is <δχm >=0,where6 tion values producing supersymmetry breaking χm is at least one of the fields in the theory. by F and/or D terms is a dynamical question The only local-supersymmetry transformations which must be answered minimizing the scalar in (2.26–2.31) which may acquire non–vanishing potential (2.19). Note that using (2.34) and (2.35) expectation values without breaking Lorentz in- this can also be written as variance are (2.29) and (2.31). Then, for non–
1 ∗ ∗ − G spatially varying expectation values one obtains V = Fi Gi j Fj 3e   1 i −1 ab + (Re f ) D D , (2.38) <δλ >= g (Re f) G (T ) φ ε , 2 ab a b a 2 i b ij j L (2.32) √ The form of the scalar potential allows in princi- G/2 −1 ij∗ <δψi >=− 2e (G ) Gj∗ ε, (2.33) ple the possibility of local supersymmetry break- ing with V = 0 (at tree level) unlike global su- where the scalar fields φi (present also through persymmetry breaking where the scalar poten- G and f in (2.32) and (2.33)) are being used to tial (2.21) is always positive definite. The former denote their vacuum expectation values. Note possibility seems to be preferred experimentally that (2.33) corresponds to vacuum expectation since the upper bound on the cosmological con- −45 4 values for auxiliary fields Fi in (2.11) stant is extremely close to zero V ≤ 10 (GeV ) . G/2 −1 ij∗ Of course this is not a solution to the cosmo- Fi = e (G ) Gj∗ . (2.34) logical constant problem. We do not know why Likewise, (2.32) corresponds to expectation val- the terms in the scalar potential conspire to pro- ues for auxiliary fields Da in (2.18) duce V = 0, but at least we can fine tune them 2 ≈ −1 ab ij to obtain the value we want . Otherwise, V Da = i [(Ref) ] gGi(Tb) φj . (2.35) 2 2 ≈ 40 4 m3/2MP 10 (GeV ) as we will see below. There are then two ways of breaking supersym- Let us now study a consequence of local su- metry, the so–called F –term and D–term super- persymmetry breaking, the super–Higgs effect. symmetry breaking. For example, if gauge sin- Discussing first F –term supersymmetry break- glets scalar fields acquire expectation values, the ing, we know that in global supersymmetry a lin- right–hand side of (2.32) is zero and supersym- ear combination of the spinors in the supermul- metry may be broken only by F terms (2.33). tiplets of the auxiliary fields Fi is the Goldstone Clearly, in the case of gauge non–singlet scalar fermion (Goldstino). In supergravity, where the fields the two possibilities, F –term and D–term mass terms from (2.13) are given by (assuming breaking, are allowed. for simplicity the minimal model of (2.10)) From the above discussion, we deduce that
−1 C i G/2 ¯ µν the relevant quantity for the study of supersym- e L = e Ψµσ Ψν F m 2 metry breaking is Gi. We need i G/2 µ i + √ e GiΨ¯ µγ ψ 2 Gi =06 , (2.36) 1 − eG/2(G + G G )ψ¯iψj , for at least one value of i, if we want to break 2 ij i j supersymmetry. For example, for the minimal (2.39) supergravity model of (2.10) this means the Goldstino ∗ 1 ∂W 6 η = G ψi , (2.40) φi + =0. (2.37) i W ∂φi 1An expectation value of bilinear fermion–antifermion as in ordinary gauge theory, gets mixed with states may occur in presence of a strongly interacting the gravitino as shown in Fig. 21 due to the sec- gauge force. For example, the third piece in (2.31) must ond term in (2.39). Its two degrees of freedom, be taken into account if one wants to study supersymme- try breaking by the mechanism of a gaugino condensate 2In fact higher–order corrections to the scalar potential [17]. will spoil this cancellation.

16

Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

and (2.42), we can deduce that the crucial func-

tion regarding this issue is the superpotential of

the theory. The most useful form for this consists

Ψµ @ of a sum of two functions @ ψi

@ W (Cα,hm)=WO(Cα)+WH(hm), (2.46) @ Figure 21: Gravitino–goldstino mixing interaction where Cα denote the usual scalar fields of the the- ory, squarks, sleptons, Higgses (and other possi- 1 ble particles in case we are working with a grand helicity  2 states, are eaten by the gravitino 3 1 unified model), which constitute the so–called which then gets helicity  2 ,  2 states. Redefin- ing fields we are left with a gravitino observable sector, and hm denote additional scalar √ fields responsible for the spontaneous breaking of 0 i 2 −G/2 supersymmetry. The latter are assumed to have Ψ =Ψµ− √ γµη− e ∂µη, (2.41) µ 3 2 3 only gravitational interactions with the observ- able sector. This means that they are singlets un- with mass der the observable gauge group, constituting the | | G/2 K/2 W so–called hidden sector. Therefore the hidden– m3/2 = e MP = e 2 , (2.42) MP sector fields do not have neither gauge interac- tions nor Yukawa couplings with the observable where we have inserted the reduced Planck mass sector. In principle, other very weak interactions to obtain the correct mass unit, using G dimen- between both sectors might be present without sionless and W with mass dimension 3. In par- being in conflict with experimental observations, ticular, however we prefer not to consider this complica- 
i 1 tion. e−1 LC = eG/2Ψ¯ 0 σµν Ψ0 − eG/2 G F m µ ν ij The simplest model, proposed in an unpub- 2  2 1 lished paper by Polonyi [18], consists of a gauge– + G G ψ¯iψj . (2.43) 3 i j singlet scalar field h with the following superpo- tential: 2 InthecaseofD–term supersymmetry break- WH (h)=m (h+β). (2.47) ing, the term in the Lagrangian mixing the grav- The superpotential must have mass dimension itino with the Goldstino is not only given by the 3 and therefore m and β are parameters with second term in (2.39) but also the second one in dimension of mass. Since the hidden–sector field (2.24) contributes is a gauge singlet, the scalar potential that we have to minimize is given only by the F part of −1L √i G/2 ¯ µ e mixing = e GiΨµLγ ψiL (2.19) 2   ∗ i µ 4 G −1 hh − gG (T ) φ Ψ¯ γ λ + h.c. V = M e Gh(G ) Gh∗ − 3 , (2.48) 2 i a ij j µL aL P (2.44) where G is dimensionless to obtain the correct mass unit for the potential. Note that if there At the end of the day, with the Goldstino given is no any cancellation, at the minimum V ≈ by 2 2 m3/2MP giving rise to a huge cosmological con- i g −G/2 ij stant. Considering for simplicity the minimal su- η = Giψ − √ e Gi(Ta) φj λa (2.45) 2 pergravity model of (2.10) ∗ | |2 one obtains similar results to the previous ones. h h WH G = 2 + ln 6 , (2.49) Now we know that spontaneous supersym- MP MP metry breaking in the context of supergravity is one obtains ∗ possible, but still we have to specify the mecha- h 1 ∂WH nism that generates it. Clearly, e.g. from (2.37) Gh = 2 + , (2.50) MP WH ∂h

17 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz and For example, in the Polonyi model studied above −1 hh∗ 2 (G ) = MP , (2.51) results (2.51) and (2.55) imply √ with the following result for the scalar potential 2 M = 3m3 2M . (2.59) (2.48): S / P

∗   h h ∗ | − |2 3 2 ≈ 4 M2 h 2 h β If, as discussed above, m / MW one obtains V = m e P | (h + β)+1| −3 . M 2 M2 q p P P 10 (2.52) MS ≈ m3/2MP ≈ MW MP ≈ 10 GeV . Choosing  √  (2.60) β = 2 − 3 MP , (2.53) This is in fact a generic result that must be ful- one can show that V has an absolute minimum filled by any supersymmetric model. Apart from that, let us finally remark that the Polonyi mech- at √  anism must be considered as a toy model. It is = 3−1 MP , (2.54) rather ad hoc. There is no a special reason why with vanishing cosmological constant V =0.Note W should be given as in (2.47). For example, in a that at this minimum supersymmetry is broken fundamental theory like superstring theory such since Gh in (2.50) is different from zero kind of superpotentials are not present. A more √ 3 realistic mechanism is gaugino condensation in Gh = . (2.55) some hidden–sector gauge group [17]. For exam- MP ple in superstring theory, besides the gauge group The gravitino acquires a mass eating the Gold- where the standard model is embedded, other ex- stino which in this case is the fermionic partner tra gauge groups are usually present providing a of h. Using the result (2.42), this is given by hidden sector. Due to non–perturbative effects a √ 2 1 2 m 2 ( 3−1) superpotential breaking supersymmetry is gener- m3/2 = e 2 MP . (2.56) MP ated dynamically. Note that the gravitino mass can be much smaller 2.3 Soft Supersymmetry–Breaking Terms than the Planck mass if m is small. For exam- MP ple, to obtain the gravitino mass of order the On experimental grounds supersymmetry cannot ≈ ≈ √electroweak scale, m3/2 MW , we need m be an exact symmetry of Nature: supersymmet- 10 MW MP ≈ 10 GeV. This implies ≈ ric partners of the known particles with the same 20 38 3 10 MP ≈ 10 (GeV ) . We will discuss in the masses (e.g. scalar electrons with masses of 0.5 next subsection that this possibility is in fact MeV) has not been detected. Let us recall a crucial if we want to avoid a hierarchy problem. mechanism to avoid this problem in the context The soft parameters which determine the super- of global supersymmetry. Simply one introduces symmetric spectrum and in particular contribute terms in the Lagrangian which explicitly break to the Higgs potential are of order the gravitino supersymmetry. The only constraint they must mass. Therefore we need this mass to be of or- fulfil is not to induce quadratic divergences in or- der MW to obtain a correct electroweak symme- der not to spoil the supersymmetric solution to try breaking. A value of m3/2 larger than 1 TeV the gauge hierarchy problem. This is the reason would reintroduce the hierarchy problem solved why these terms are called soft supersymmetry– by supersymmetry. breaking terms, soft terms in short. The sim- Defining the supersymmetry–breaking scale plest supersymmetric model is the so–called min- MS by imal supersymmetric standard model (MSSM),

2 G/2 −1 ij∗ where the matter consists of three generations of M ==M , S i P j quark and lepton superfields plus two Higgs dou- (2.57) blets superfields (supersymmetry demands the where we have inserted M in (2.34) to have the P presence of two Higgs doublets unlike the stan- correct mass unit, one obtains using (2.42) dard model where only one is needed) of oppo- 2 −1 ij∗ MS = m3/2 < (G ) Gj∗ >. (2.58) site hypercharge, and the gauge sector consists

18 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

of SU(3)C × SU(2)L × U(1)Y vector superfields. spontaneously broken in a hidden sector, the soft The associated superpotential, in an obvious no- terms for the observable fields are generated. Let tation, is given by us discuss this in some detail. X h In the previous subsection we studied mech- ˜ ˜ c ˜ ˜ ˜c W = Yu QLH2u˜L + Yd QLH1dL anisms in order to break local supersymmetry, generations i where the presence of fields in a hidden sector ˜ ˜ c was crucial to achieve it. Let us divide the super- + Ye LLH1e˜L + µH1H2 , (2.61) potential as in (2.46) and consider for simplicity where the first piece is related with the Yukawa the form of the K¨ahler potential K that leads couplings (we have taken for simplicity diago- to canonical kinetic terms (2.10) for the chiral nal couplings), which eventually determine the supermultiplets fermion masses, and the second piece, the so– ! 1 X X |W|2 called µ term, is necessary in order to break the G = C∗ C + h∗ h + ln . M 2 α α m m M6 electroweak symmetry. P α m P The soft terms can be parameterized by the (2.63) following parameters: gaugino masses Ma,scalar Then, assuming that supersymmetry is broken masses mα, trilinear parameters (associated with by F terms, only the first piece of the scalar po- the Yukawa couplings) Au,d,e and a bilinear pa- tential in (2.19) will contribute P P rameter (associated with the µ term) B.Thus 1 ∗ ∗ α+ m M2 ( α CαC m hmh ) the form of the soft Lagrangian is given by V = e P   X ∂W C∗ 1 × | O α |2 L ˆ ˆ − 2 ∗ + 2 (WH + WO) soft = Ma λaλa + h.c. mα CαCα ∂Cα M 2 α P X ∗ X h |∂WH hm |2 −  ˆ ˜ ˜ c + + 2 (WH + WO) Au Yu QLH2u˜L ∂hm M m P generations 2 i |WH + WO| c c − 3 2 , (2.64) ˆ ˜ ˜1 ˜ ˆ ˜ ˜1 + Ad Yd QLH dL + Ae Ye LLH e˜L MP + B µHˆ 1H2+h.c.) , (2.62) where we are using for the moment a generic hidden–sector superpotential WH (hm). The ob- where Cα denote all the observable scalars, i.e. c c c servable sector superpotential WO(Cα)mightbe Q˜ , u˜ , d˜ , L˜ , e˜ ,H1,H2,andλ with a corre- L L L L L a for example the one of the MSSM in (2.61) or a sponding to SU(3) ,SU(2) ,U(1) , denote all C L Y GUT generalization of it. Note that non renor- the observable gauginos, i.e.g, ˜ W˜ 1 2 3, B˜.The , , malizable terms can in principle be ignored for hat on Yukawa couplings and µ parameter de- analyses far below the Planck scale, MP →∞, note that they are rescaled. We will discuss these 1 since they are suppressed by powers of 2 .For MP points in (2.66) and (2.73) below. 2 −3|WO | example 2 → 0. Thus apparently one These soft terms are crucial not only because MP they determine the supersymmetric spectrum, like is left in the observable sector with the usual global–supersymmetry scalar potential | ∂WO |2 and gaugino, Higgsino, squark and slepton masses, ∂Cα but also because they contribute to the Higgs po- nothing new arises from the breaking of super- tential (together with the quartic terms coming gravity. However, if some fields acquire large from D terms) generating the radiative break- vacuum expectation values, the new gravitation- down of the electroweak symmetry. Let us recall ally induced terms may be important. We saw that in the standard model the whole Higgs po- in the previous subsection that although the first tential has to be postulated ad hoc. term in (2.13) is non renormalizable it gives rise Although in principle the breaking of super- to a sizeable contribution to the gravitino mass |WH | symmetry explicitly may look arbitrary, remark- m3/2 ≈ 2 (see (2.42)). For example, using the MP ably in the context of local supersymmetry it Polonyi mechanism we obtained ≈MP im- 38 3 happens in a natural way: when supergravity is plying WH ≈ 10 (GeV ) and therefore m3/2 ≈

19 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

100 GeV. From this discussionP we conclude that 1 ∗ SUPERSYMMETRY 2 ( hmhm) GRAVITATIONAL MSSM 2M m |WH (hm)| BREAKING ORIGIN 3 2 P 2 we must hold m / = e M (observable sector) P (hidden sector) INTERACTIONS fixed when we take the limit MP →∞.Here hmare being used to denote their vacuum ex- pectation values at the minimum of the potential Figure 22: Supersymmetry breaking occurs in a hidden sector and is transmitted to the observable (2.64). This limit where MP →∞but m3/2 is sector by gravitational interactions giving rise to soft held fixed is the so–called flat limit. Then, al- terms. though the observable sector does not contribute to give mass to the gravitino and therefore does Therefore at the end of the day we are left with not feel directly the breaking of supersymmetry, the usual global supersymmetric potential (with it feels the breaking indirectly due to the appear- superpotential Wˆ ) plus soft terms. The proce- ance of soft terms through gravitational inter- O dure studied in this section and the previous one actions in (2.64). For example, the term pro- ∗ −3WH to break supersymmetry is schematically sum- portional to 2 WO will give rise to couplings MP marized in Fig. 22. We can apply this result −3m3 2W . Also from the term proportional to / O to the case of the MSSM (2.62). For example, | WH ∗|2 2 ∗ 2 Cα , masses m3 2CαCα for the scalars will MP / absorbing the above rescaling (2.66) in Yukawa arise. In total, from (2.64) one obtains in the couplings and µ parameter as usual, low–energy limit P ∗ 1 ∗ m ( W 2M2 m hmh P ˆ H P 1 ∗ X Yu,d,e = Yu,d,e e , h hm 2 M2 m m ∂WO |WH | V = e P P ∗ 1 ∗ ∂Cα h hm α W 2M2 m m " H P µˆ = µ | | e , (2.69) X |W |2 W∗ X ∂W WH + H C∗C + H C O M4 α α M2 α ∂C one obtains the following soft parameters: α P P α α   ! X ∗ m = m3 2 , ∗ 1 ∂WH hm − α / + hm ∗ ∗ + 2 3 WO X WH ∂hm MP ∗ Fm #)m Au,d,e = m3/2 hm 2 , m3 2M m / P + c.c . (2.65) X ∗ Fm − B = m3/2 hm 2 m3/2(2.70). m3 2M m / P Rescaling the observable superpotential P It is worth noticing that the gravitino mass set ∗ 1 ∗ m W 2M2 m hmh ˆ ≡ H P the overall scale of the soft parameters. This WO WO | |e , (2.66) WH was expected since the F part of the scalar po- and taking into account that the expectation val- tential in (2.19) has an overall factor eG and ues Fm of the auxiliary fields associated with the therefore it is a general result valid for any su- scalars hm are given by (see (2.57)) pergravity model. This implies that the effective   ∗ supersymmetry–breaking scale in the observable 2 1 ∂WH hm Fm = m3/2MP ∗ ∗ + 2 , (2.67) sector is of order m3/2 as already mentioned above WH ∂hm MP and not the supersymmetry–breaking scale Fm (2.65) may be rewritten as which is of order 1010 GeV. Note that the scalar masses are automati- X ∂Wˆ 2 X O 2 ∗ ≡ V = + m3/2CαCα cally universal mα m. In particular, in this ∂Cα α " α model they are all equal to the gravitino mass, X ∂Wˆ O m = m3/2. Actually, universality of scalar masses + m3/2 Cα ∂Cα is a desirable property not only to reduce the α ! # X number of independent parameters in supersym- ∗ Fm − ˆ metric models, but also for phenomenological rea- + hm 2 3 WO + c.c . m3 2M m / P sons, particularly to avoid flavour changing neu- (2.68) tral currents. Besides, the A parameters are also

20 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

universal, Au,d,e ≡ A, and they are related to the same dependence on the hidden–sector fields, i.e. 3 B parameter fa(hm)=caf(hm), for the different gauge group factors. This is in fact the case of supergrav- − B = A m3/2 . (2.71) ity models deriving from (tree–level) perturba- tive superstring theory as we will see in the next These results show us that it is possible to learn section. Let us use the general formula (2.74) things about soft terms without knowing the de- to consider the simple case f = h/M with tails of supersymmetry breaking, i.e. the ex- a P thesameK¨ahler function (2.63) studied above, plicit form of the function WH (hm). We are where some hm ≡ h. Then, one obtains the uni- left in fact with only two free parameters, m3 2 / versal mass and A ≡ Au,d,e. Of course once this function is known we can compute these quantities since 1 −1 M = (Re h) Fh , (2.75) m3/2 and Fm depend on WH (hm). For example, 2 for the Polonyi mechanism studied in the previ- where F is given by (2.67) with h = h.For ous subsection one obtains straightforwardly h m example, for the Polonyi mechanism one obtains  √  √ A = 3 − 3 m3 2 . (2.72) / 3 M = m3 2 √ . (2.76) / − The above results (2.70) should be under- 2( 3 1) stood as being valid at some high scale O(M ) P As in the case of scalar soft parameters (2.70), and the standard renormalization group equa- the gravitino mass set the overall scale of the soft tions must be used to obtain the low–energy (≈ gaugino masses. Note to this respect the overall M ) values. 2 W factor eG/ in the first term of (2.24). On the other hand, from the fermionic part Let us finally remark that with the inclusion of the supergravity Lagrangian, (tree–level) soft of the other unsuppressed terms in (2.14) we are gaugino masses may also be obtained. Let us left finally with the usual global–supersymmetry assume for simplicity f = δ f . After canon- ab ab a Lagrangian plus the soft terms. This effective La- ically normalizing the gaugino fields in the first grangian is obviously renormalizable and there- term of (2.22), fore perfectly consistent in order to study phe- 1/2 nomenology. For example, the third term in (2.13) λˆa =(Refa) λa , (2.73) gives rise to the usual Yukawa couplings and Hig- the first term in (2.24) gives rise to a mass term gsino masses. Note that Gij give rises to the 2 as in (2.62) with observable–sector piece 1 ∂ WO and therefore WH ∂Cα∂Cβ 2 ˆ eG/2G induces the contribution ∂ WO with 1 −1 ∂fa ij ∂Cα∂Cβ Ma = (Refa) Fm , (2.74) 2 ∂hm Yukawa couplings and the µ parameter rescaled as in (2.69), i.e. with Fm as in (2.57). Note that the gauge ki- netic function f is dimensionless and therefore X 2 ˆ −1 ∂ WO ¯ the mass unit above is correct. For this to be ψαLψβR + h.c. (2.77) 2 ∂Cα∂Cβ non–vanishing at tree level, which is phenomeno- α,β logically interesting (experimental bounds on glu- The Higgsino masses given byµ ˆ are obviously ino masses imply M3 > 50 GeV), it is necessary supersymmetric masses. The same contribution a non–canonical choice of the vector supermulti- will appear in the Higgs masses through the first 6 plets fa = const. For example, universal gaug- term in (2.70) ino masses can be obtained if all the fa have the In conclusion, we have shown in this subsec- 3 In fact, this relation depends on the particular mech- tion that supergravity models are interesting and µ anism which is used to generate the parameter. Its give rise to concrete predictions for the soft pa- generation is the so–called µ problem and several solu- tions have been proposed in the context of supergravity rameters. However, one can think of many possi- [20]. ble supergravity models (with different K, W and

21 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz f) leading to different results for the soft terms4 of the perturbative heterotic string, one obtains For example, if instead of assuming the form of K L −1 µν a for the observable sector given by (2.63) we take = (Re S) Fa Fµν ∗ ∗ 4 K = Kα(hm,hm)CαCα it is straightforward to 1 + M 2 ∂ S∂µS∗ see that soft scalar masses are no longer univer- (S + S∗)2 P µ sal (we also must canonically normalize the scalar 3 ˆ 1/2 + M2 ∂ T∂µT∗ fields in (2.62) Cα = Kα Cα similarly to the case (T + T∗)2 P µ of gaugino fields (2.73)). Also, the Polonyi su- ∗ nα µ ∗ +(T + T ) ∂µCα∂ Cα + ... , (3.1) perpotential WH in (2.47) give rise to soft terms which are different from those obtained using where nα are negative integer numbers called a gaugino–condensation mechanism. This arbi- modular weights of the matter fields Cα.Acom- trariness, as we will see in the next section, can be parison of this result with the general supergrav- ameliorated in supergravity models deriving from ity Lagrangian (2.14), in particular with the sec- superstring theory, where K, f, and the hidden ond term in (2.5) and the first term in (2.15), al- sector are more constrained. We can already an- lows us to deduce the K¨ahler potential and gauge ticipate, for example, that in such a context the kinetic function kinetic terms are generically not canonical. K = − ln(S + S∗) − 3 ln(T + T ∗) ∗ ∗ nα CαCα +(T+T ) 2 , 3. Supergravity from superstrings MP fab = Sδab . (3.2) Recently there have been studies of supergravity models obtained in particularly simple classes of Note from our discussion in (2.17) that = 2 superstring compactifications. Such models have 1/ga and therefore the gauge coupling constants a natural hidden sector built-in: the complex are unified even in the absence of a grand uni- dilaton field S and the complex overall modulus fied theory. Thus grand unification groups, as field T . These gauge singlet fields are generically e.g. SU(5) or SO(10), are not mandatory in or- present in four-dimensional models: the dilaton der to have unification in the context of super- arises from the gravitational sector of the theory strings. Recall that S and T fields are dimension- and the modulus parameterizes the size of the less unlike all other scalars which have dimension ≈ 2 2 1. This implies dimension 1 for the F terms of compactified space R Mstring,whereR denotes the overall radius. Both fields are taken S and T whereas the F terms of all other scalars dimensionless. Assuming that the auxiliary fields have dimension 2 as studied in section 2.2. of those multiplets are the seed of supersymmetry As we learnt in the previous section, we can breaking, interesting predictions for this simple compute with information (3.2) the soft terms class of models are obtained. Here we will ana- (e.g. gaugino masses Ma are trivially obtained lyze very briefly this issue. More details can be from (2.74)): found in [20]. n m2 = m2 + α |F |2 , Once we choose the compact space, K and f α 3/2 (T + T ∗)2 T are calculable. Starting with the D =10super- F M = S , gravity Lagrangian, obtained as the low–energy a (S + S∗) limit of superstring theory, and expanding in D = F S F T A = − − (3 + n + n 4 fields, the D =4,N= 1 supergravity La- αβγ (S + S∗) (T + T ∗) α β grangian can be computed. In particular, work-  ∗ 1 ∂Yαβγ ing with orbifold compactifications in the context +nγ − (T + T ) . (3.3) Yαβγ ∂T 4 General formulae for tree–level soft parameters were Note that to obtain this result we did not assume computed in [19]. See also [20] for a review with super- gravity and superstring examples. One–loop corrections any specific supersymmetry–breaking mechanism, to the soft parameters were computed recently in [21]. i.e. a particular value of WH (S, T ). Due to

22 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz the form of the gauge kinetic function gaugino is therefore worthwhile. Besides, there are still masses turn out to be universal. However, due to open problems whose solutions are crucial for the the modular weight dependence the scalar masses consistency of the theory. The cosmological con- (and A parameters) show a lack of universal- stant problem and the mechanism of supersym- ity. Universality can be obtained in the so–called metry breaking are the most important ones. dilaton dominated limit, i.e. if we assume that the mechanism of supersymmetry breaking is in such a way that only the F term associated with Acknowledgments the dilaton acquires a vacuum expectation value, The work of D.G. Cerde˜no has been supported by F = 0. Then, T a Universidad Aut´onoma de Madrid grant. The

m = m3 2 , work of C. Mu˜noz has been supported in part α √ / by the CICYT, under contract AEN97–1678–E, Ma = 3 m3 2 , / and the European Union, under TMR contract − Aαβγ = Ma , (3.4) ERBFMRX–CT96–0090. where to obtain the value of Ma we have assumed a vanishing cosmological constant, i.e. V =0in A. Appendix (2.38) Superspace formalism in supergravity |F |2 3|F |2 S + T =3m2 . (3.5) (S + S∗)2 (T + T ∗)2 3/2 We sketch in this Appendix the derivation of Recall that only the F part of (2.38) contributes the most general D =4,N=1supersymmet- to supersymmetry breaking since the hidden sec- ric gauge theory coupled to supergravity, using tor fields, dilaton and modulus, are gauge sin- the superspace formalism. For a review of this glets. method see [5], where also the references of the It is worth noticing that the above results authors who have contributed to the subject can (3.4) are independent of the compactification spa- be found. ce since the dilaton couples in a universal man- Let us recall first how this approach works ner to all particles, i.e. f and the dilaton part in the context of global supersymmetry. The su- of K are not modified by the particular choice of perspace is defined as the space created by compact space. Because of the simplicity of this scenario, predictions are quite precise. For exam- xµ,θα,θ¯α˙ , (A.1) ple at high energies the relation between√ scalar masses and gaugino masses is Ma = 3 mα.Us- where xµ are the usual four space–time dimen- ¯ ing the renormalization group equations one ob- sions and the anticommuting parameters θα, θα˙ tains at low–energies (α =1,2), which are elements of a Grassmann algebra, are introduced as supersymmetric part- ≈ Mg˜ mq˜ >> m˜l . (3.6) ners of the x–coordinate. The components of the chiral supermultiplets (φi, ψi, Fi)withi= 4. Conclusions 1, ..., n,whereφi are complex scalar fields, ψi are Weyl spinor fields and Fi are auxiliary complex Supergravity cannot be the final theory of ele- scalar fields, arise as the coefficients in an expan- mentary particles since it is non renormalizable. sion in powers of θ and θ¯ of the so called chiral

However, it might be the effective theory of the superfields Φi(x, θ, θ¯), which are functions of the final theory (perhaps superstrings). In that case, superspace coordinates. Then, working in the su- supergravity might be subject to experimental perspace, the most general renormalizable super- test through the prediction of the soft supersym- symmetric Lagrangian involving only chiral su- metry breaking terms which determine the super- perfields (barring linear contributions which are symmetric spectrum. The study of supergravity forbidden, unless the superfields are neutral, once

23 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz gauge invariance is included) is given by supersymmetry (setting k equal to one): Z Z  Z  2 2 + 2 1 1
L = d θd θ¯Φ Φ + d θ m Φ Φ L = d2Θ2E −3R− D¯D−¯ 8R Φ+Φ global i i 2 ij i j local i i   8  1 1 1 + YijkΦiΦj Φk + h.c. , (A.2) + m Φ Φ + Y Φ Φ Φ + h.c. 3 2 ij i j 3 ijk i j k (A.6) where repited indices are summed in our notation and the couplings mij and Yijk are symmetric in It is interesting to write this equation as their indices. Z  Its generalization to the local–supersymmetry 1
L = d2Θ2E − D¯D−¯ 8R Ω(Φ , Φ+) case parallels the construction of a Lagrangian local i i  8 including gravity in the non–supersymmetric case +W (Φ ) + h.c. , (A.7) discussed in section 1.1. First, to obtain the su- i persymmetric gravitational action we need su- persymmetric generalizations of the measure ed4x, where + + m Ω(Φ , Φ )=Φ Φ −3 (A.8) where e = det eµ , and Ricci scalar R.These i i i i E are given by the chiral density superfield and is the superspace kinetic energy, and superspace curvature superfield R. Their com- ponents, (e,Ψµ,...) and (R,Ψµ,...) respectively, 1 1 W (Φi)= mij ΦiΦj + YijkΦiΦj Φk (A.9) where Ψµ is the gravitino and the dots denote 2 3 auxiliary fields, arise as the coefficients in an ex- is the superspace potential (also called superpo- pansion in powers of the superspace parameters tential). Θ. The latter are generalized θ parameters which After eliminating the auxiliary fields by their now carry local Lorentz indices. The pure super- Euler equations, and rescaling and redefining the gravity Lagrangian is then other fields, one arrives at the component La- Z grangian in terms of the physical fields, φ, ψ, em L − 6 2 ER µ sg = 2 d Θ , (A.3) k and Ψµ. Remarkably, its expression is written in terms of the real function 2 where k =8πGN is the gravitational coupling.   The Lagrangian (A.2) is now easily extended to ∗ − −Ω K(φi,φi)= 3ln , (A.10) the local case. We first write it in chiral form 3 Z  1 1 with L = d2θ − D¯D¯Φ+Φ + m Φ Φ global 8 i i 2 ij i j Ω(φ ,φ∗)=φ+φ −3. (A.11)  i i i i 1 + Y Φ Φ Φ + h.c. , (A.4) For example, the kinetic terms of the scalars are 3 ijk i j k proportional to where D and D¯ are the usual supersymmetric 2 ∂ K µ ∗ derivatives and we are using the property ∂µφi∂ φ . (A.12) ∂φ ∂φ∗ j Z Z i j

2 2 −1 2 F x, θ, θ¯ d θd θ¯ = D¯DF¯ x, θ, θ¯ d θ They have properly normalized kinetic energies 4 2 Z ∂ K
since ∗ = δij + .... The Lagrangian also −1 ∂φi∂φj = DDF x, θ, θ¯ d2θ.¯ 4 has higher–order interaction terms, such as non– (A.5) renormalizable four–fermion couplings, which are suppressed by powers of Newton’s constant. Be- We then add the supergravity action (A.3) and low we shall see that K is the so called K¨ahler po- 2 2 1¯¯ replace θ → Θ, d θ → d Θ2E,and−4DD→ tential and that the component Lagrangian has a 1 ¯ −4(DD − 8R), where D is the covariant deriva- natural interpretation in the language of K¨ahler tive. This gives the Lagrangian (A.2) in local geometry.

24 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

Since supergravity is a non–renormalizable The Lagrangian is invariant under coordinate tra- theory, we are in fact interested in the most gen- nsformations, where the scalar fields should be eral Lagrangian that can be built from chiral su- thought of as the coordinates of the K¨ahler man- perfields. This is ifold and the fermions as tensors in the tangent Z
space. From the superspace viewpoint, the source L 2 2 ¯ + global = d θd θK Φi ,Φj of this K¨ahler geometry is the invariance of (A.13) Z  under the superfield K¨ahler transformation: 2 + dθW(Φ )+h.c. . (A.13) + + ∗ + i K(Φ, Φ ) → K(Φ, Φ )+F(Φ) + F (Φ ) . (A.16) Here K and W are vector and chiral superfields Now, following the same steps as in the case respectively, with power series expansion in terms of the renormalizable chiral Lagrangian (A.2), of the chiral superfields Φ , i we generalize (A.13) to the local–supersymmetry X + case K(Φi, Φj )= ai1...iN ,j1...jM Φi1 ...ΦiN Z  1 3
× + + L = d2Θ2E D¯D−¯ 8R Φj1 ...ΦjM , local 2 X k 8  2 W(Φi)= bi1...iN Φi1 ...ΦiN . (A.14) −k Φ Φ+ 3 K( i, j ) 2 × e +k W(Φi) The component expansion of K implies that ki- + h.c. , (A.17) netic terms are obviously non renormalizable. On the other hand, only kinetic terms with two space– where K(Φ, Φ+) is a general hermitian function time derivatives are present. A higher–derivative and W (Φ) is the superpotential. The exponential theory might be obtained if supersymmetric deriva- form is suggested by the relation (A.10). Note tives of superfields were allowed in K.Weex- that expanding in k2, Z Z clude this problematic possibility in what follows. L − 6 2 ER 2 E To study the matter couplings of chiral multiplets local = 2 d Θ + d Θ2  k  it is convenient to use the language of K¨ahler ge- 1
× − (D¯D−¯ 8R)K Φ+,Φ +W(Φ ) ometry. 8 i j i A K¨ahler manifold is a special type of ana- + ... + h.c. , (A.18) lytic Riemann manifold, subject to certain con- ditions. These conditions imply that the metric we may recover the global–supersymmetry La- and the connection are determined by the deriva- grangian. After a straightforward computation, tives of a scalar function K called the K¨ahler po- the component form of (A.17) turns out to be the tential. E.g. the metric gij∗ is given by gij∗ = same as that of (A.7), where now K is an arbi- ∂2K(c,c∗) ∗ ∗ ≡ Kij∗ ,whereci and c are the com- trary real function of the scalar fields, the lowest ∂ci∂cj i + plex coordinates parameterizing the K¨ahler man- component of the superfield K(Φ, Φ ). Using a ifold. Under an analytic coordinate transforma- super–Weyl transformation one can simplify the → 0 ∗ → ∗0 ∗ component expression in such a way that it de- tion, ci ci(c)andci ci (c). Note that the metric is invariant under analytic shifts of pends only on the real function ∗ ∗ ∗ ∗ K, K(c, c ) → K(c, c )+F(c)+F (c ). They G(φ∗,φ)=K(φ∗,φ)+ln|W(φ)|2 . (A.19) are called K¨ahler transformations of the K¨ahler potential. For example, the scalar potential is given by  !  Then, using the K¨ahler notation, it is possi- −1 ∂G ∂2G ∂G ble to see that the component Lagrangian from V = eG  − 3 . (A.20) ∂φ ∂φ φ∗ ∂φ∗ (A.13) can be written in terms of the real func- i i j j ∗ tion K(φ, φ ). For example, the scalar potential This is different from the global–supersymmetry is given by Lagrangian, where K and W enter in an indepen- ∂W ∂W∗ dent way (see e.g. (A.15)). The final Lagrangian V = K ∗ . (A.15) ij ∂φi ∂φ∗j can be found in the text in eq. (2.4).

25 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

To obtain the complete supergravity Lagrang- Under supersymmetry it transforms as chiral su- ian which couples pure supergravity to supersym- perfield. Under a gauge transformation, it must metric chiral matter and Yang–Mills we need now transform as a symmetric product of adjoint rep- the complete gauge–invariant global supersym- resentations of the gauge group. metry Lagrangian in superspace. Let us recall Then, from (A.23) and (A.24), we deduce the first how to obtain the renormalizable gauge– local–supersymmetry Lagrangian: invariant Lagrangian. One imposes gauge in- Z  variance to the renormalizable chiral Lagrangian 2 3 Llocal = d Θ2E (D¯D−¯ 8R) (A.2) with the result that a vector superfield must  8  1   be introduced: × exp − K(Φ+, Φ )+Γ(Φ+,Φ ,V) Z 3 i j i j 2 2 + V  Lglobal = d θd θ¯ Φ e Φ 1 a b Z   + fab(Φi)W W + W (Φi) 1 1 16 + d2θ m Φ Φ + Y Φ Φ Φ 2 ij i j 3 ijk i j k + h.c. , (A.25)

+ h.c.] , (A.21) 1 ¯ ¯ −V V where Wα = − 4 (DD−8R)e Dαe is the curved– where now the chiral superfields Φ transforms as space generalization of the Yang–Mills field stren- a representation of a gauge group G and V ≡ gth superfield. After eliminating the auxiliary 2gT aV a with T a the group generators in that fields, and rescaling and redefining the other fields representation, V a the vector superfields and g as in the case of chiral models, one arrives at the the gauge coupling constant. The components of component Lagrangian in terms of the physical a a a m V are the vector bosons belonging to the ad- fields, φ, ψ, V ,λ ,eµ ,Ψµ. It depends on the two joint representation of G, their Majorana spinor arbitrary functions partners λa and the auxiliary real scalar fields ∗ ∗ | |2 Da. Adding the (gauge–invariant) kinetic term G(φ ,φ)=K(φ ,φ)+ln W(φ) , for the vector supermultiplet, fab(φi) , (A.26) Z 1 d2θWaαW a + h.c. , (A.22) and can be found in the text in eq. (2.14). 16 α

a we obtain the complete Lagrangian. Here Wα References is the gauge field strength (chiral spinor) super- field with spinor index α. In particular, Wα = [1] C. Isham, Quantum gravity, in the book ‘The 1 ¯ ¯ −V V − 4 DDe Dαe . We follow the same procedure New Physics’, Cambridge University Press to obtain the most general gauge–invariant La- (1989) 70, reprinted (1993), edited by P. grangian. From (A.13) we deduce Davies. Z   [2] R.P. Feynman, Feynman lectures on gravita- L 2 2 ¯ + + global = d θd θ K(Φi , Φj )+Γ(Φi ,Φj,V) tion, Addison–Wesley Publishing Co. (1995). Z  [3] M.D. Scadron, Advanced quantum theory, + d2θW(Φ )+h.c. , (A.23) i Springer–Verlag (1979), extended second edi- tion (1991). where Γ is a counterterm which is necessary for gauge invariance. As above we have to add the [4] P. van Nieuwenhuizen, Supergravity,Phys. Rep. 68 (1981) 189. kinetic term for the vector supermultiplet: Z [5] J. Wess and J. Bagger, Supersymmetry and su- 1 2 a b d θfab(Φi)W W , (A.24) pergravity, Princeton University Press (1983), 16 second edition revised and expanded (1992). but now an arbitrary analytic function of the [6] P. West, Introduction to supersymmetry and chiral superfields fab(Φi), which would be just supergravity, World Scientific Publishing Co. δab. in the renormalizable case, may be included. (1986), extended second edition (1990).

26 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerde˜no and C. Mu˜noz

[7] I. Buchbinder and S. Kuzenko, Ideas and meth- ods of supersymmetry and supergravity (or a walk through Superspace), Institute of Physics Publishing (1995). [8] L. Castellani, R. D’Auria and P. Fr´e, Super- gravity and superstrings (a geometric perspec- tive), World Scientific Publishing Co. (1991). [9] H. P. Nilles, Supersymmetry, supergravity and particle physics,Phys.Rep.110 (1984) 1. [10] R. N. Mohapatra, Unification and supersym- metry, Springer-Verlag (1986), second edition revised and expanded (1992). [11] D. Bailin and A. Love, Supersymmetric gauge field theory and string theory, Institute of Physics Publishing (1994), reprinted with cor- rections (1996). [12] J. D. Lykken, Introduction to supersymmetry, lectures given at TASI–96, hep-th/9612114. [13] Y. Tanii, Introduction to supergravities in di- verse dimensions, review talk given at YITP– 96, hep-th/9802138. [14] For a review, see: J.H. Schwarz, From super- strings to M theory,Phys.Rep.315 (1999) 107-121, hep-th/9807135. [15] E. Cremmer, B. Julia and J. Scherk, Super- gravity theory in 11 dimensions, Phys. Lett. 76 (1978) 409. [16] E. Cremmer, S. Ferrara, L. Girardello and A. van Proeyen, Yang–Mills theories with lo- cal supersymmetry: Lagrangian, transforma- tion laws and super–Higgs effect,Nucl.Phys. 212 (1983) 413. [17] For a review, see: H.P. Nilles, Gaugino con- densation and supersymmetry breakdown,Int. J. Mod. Phys. A5 (1990) 4199. [18] J. Polonyi, Budapest preprint KFKI–1977–93 (1977). [19] S.K. Soni and H.A. Weldon, Analysis of the supersymmetry breaking induced by N=1 super- gravity theories, Phys. Lett. 126 (1983) 215. [20] For a review, see: A. Brignole, L.E. Iba˜nez and C. Mu˜noz, Soft supersymmetry–breaking terms from supergravity and superstring mod- els, in the book ‘Perspectives on supersym- metry’, World Scientific Publishing Co. (1998) 125, edited by G. Kane, hep-ph/9707209. [21] K. Choi, J.S. Lee and C. Mu˜noz, Supergrav- ity radiative effects on soft terms and the µ term, Phys. Rev. Lett. 80 (1998) 3686, hep-ph/9709250.

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