An Introduction to Supergravity

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 Teórica C{XI and Instituto de F´ısica Teórica C{XVI Universidad Autónoma 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. Cerdeño and C. Muñoz W W˜ ³ ° ° __^ ³ © ª ¨ « + 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 scattering 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. − − − − @ @ e e e e @ @ Á Á @ 1=2 1=2 @ ¨ «««¨«¨«¨«¨ ¨ ee___ GN GN ^^^ @ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¡ @ − − − − γ @ graviton @ e e Á Á e e @ @ Figure 3: Electron–electron scattering through the Figure 4: Electron–electron scattering through the exchange of a graviton. exchange of a photon (γ). 2 Corfu Summer Institute on Elementary Particle Physics, 1998 D.G. Cerdeño and C. Muñoz « « « © « © @ @ @ @ − « © Á Á e − @ @ ««¨«¨«¨«¨«¨ «¨«¨«¨«¨«¨«¨ ¨ + e ª ¨ ¢¡ ¢ ¡ ¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¡ @ − e @ ¨ g ª ª ¨ ª Sun γ ª Figure 5: Lowest–order gravitational light bending due to the Sun and a quantum correction. 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. Cerdeño and C. Muñoz φ φ h h £ @ @ ¡ ¢¡ £ @ @ ¡ ¡ h ¢ h @ @ ¨ «««¨«¨ ¨ 2 «¨«¨«¨ k k £ ¢ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢ ¡ ¡ ¢ ¡ £ ¡ ¢¡ 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.

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