A Supergravity Primer

The following sections are included:

• Preface
• Acknowledgments
• Contents
• List of Tables

Because of the homogeneity and the isotropy of spacetime, elementary particles are classified according to irreducible representations of the Poincaré algebra, and are therefore characterised by the eigenvalue of their corresponding Casimir operators, i.e., mass and spin (or helicity in the massless case) [Wigner (1939); Bargmann and Wigner (1948); Weinberg (1995)]. Internal interactions are associated to a compact Lie groups describing gauge interactions [Yang and Mills (1954)]. These two concepts have allowed the construction of many particle physics models. In particular, the so-called Standard Model [Glashow (1961); Salam and Ward (1964); Weinberg (1967); Glashow et al. (1970); Weinberg (1972b); Gross and Wilczek (1973, 1974); Politzer (1974)] seems to be able to describe almost all the currently available experimental high-energy physics data [Tanabashi et al. (2018)]. Nevertheless, it has been realised that it is possible to circumvent a strong no-go theorem [Coleman and Mandula (1967); Haag et al. (1975)], and to extend the Poincaré symmetry in a non-trivial way [Golfand and Likhtman (1971)]. Introducing anticommutators, i.e., symmetric counterparts of commutators, for a fermionic class of generators, the fermionic symmetries can be incorporated in an extended version of Lie algebras, referred as Lie superalgebras, which feature an associated supersymmetry. This superalgebra has the ℤ₂-graded structure…

Supersymmetric quantum field theories can be built naturally using the superspace formalism. In this chapter, we recall the basic features of the N = 1 superspace that are relevant for this book. Since details can be found in many textbooks and reviews, we omit them from the present document and refer instead to the existing literature [Salam and Strathdee (1974b, 1975); Ferrara et al. (1974); Salam and Strathdee (1978); Wess and Zumino (1974b); Ferrara and Zumino (1974); Derendinger (1989); Fuks and Rausch de Traubenberg (2011)].

This chapter is devoted to the introduction of an ad hoc curved superspace which extends the flat superspace of the previous chapter. In particular, the supergravity algebra, including local Lorentz transformations and superdiffeomorphims, will be defined. Dynamical superfields, i.e., supervierbein and spin superconnection will be introduced together with their associated curvature and tensor superfields. Since these superfields have too many degrees of freedom, some constraints upon the torsion tensors need to be imposed. The reduction of the Bianchi identities will enable us to express all curvature and torsion superfields in terms of three superfields only…

Similarly to the case of global supersymmetry, the degrees of freedom included in an N = 1 chiral and vector supermultiplet can be embedded into chiral and vector superfields, respectively. In this chapter, we present a detailed calculation of the component fields associated to chiral and vector superfields and their superderivatives, together with their transformation properties under supergravity transformations. Projectors mapping antichiral (chiral) scalar superfields to chiral (antichiral) superfields are introduced. Gauge interactions of chiral superfields, i.e., the way to couple vector and chiral superfields in a gauge and supergravity invariant fashion is also addressed.

The formalism introduced in the previous chapters allows for the construction of invariant actions. To this end, we adopt the standard procedure of simply substituting regular flat superderivatives with covariant derivatives acting on curved superspace. In Secs 4.1 and 4.2, we have defined such derivatives for the various components of chiral and vector superfields needed in our requisite substitution procedure. Our expressions will exhibit the strong correlation between curved superspace through their link to the zeroth order components of the vielbein, the connection and the ℜ and G_µ superfields (see Secs.3.1 and 3.3). However, the construction of an invariant action in curved superspace requires one additional ingredient, an invariant measure of integration. Its definition is the object of Sec. 5.1. Later in the present chapter, we will see that this leads to (at least) two different ways of building actions, either performing calculations in the full eight-dimensional superspace or in the six-dimensional chiral superspace, after having integrated out the antichiral variables.

In this chapter, we will expand the Lagrangian (5.35) for a generic supersymmetric theory. We will hence derive the most general expression of the supergravity Lagrangian in terms of the components of the gravitation multiplet and of those of the chiral and vector superfield of the theory. We specify the matter sector of the model by a set of chiral superfields denoted by Φⁱ which are assumed to lie in a representation ℜ of a gauge group G. Subsequently, the antichiral superfields ${\Phi }_{i^{*}}^{†}$ lie in the $\overline{\Re }$ representation of G (the complex conjugate representation). To the gauge group we associate a vector superfield V, lying in the adjoint representation of G, and we adopt the Wess-Zumino gauge (4.40). In order to facilitate the computation of the different terms appearing in the Lagrangian, we decompose the latter into different sectors, i.e., the pure gravitation sector, the gravitational and gauge interactions of the matter content and those of the gauge superfields. This approach is complementary to historical approaches obtained by means of the tensor calculus [Cremmer et al. (1979, 1982, 1978b, 1983b)].

This chapter is devoted to some applications of supergravity. Supergravity cannot be an exact symmetry and has to be broken. Indeed, since P_µP^µ is a Casimir operator in a given multiplet, matter, i.e. chiral superfields, or gauge, i.e. vector superfields, the fermions and bosons are degenerated in mass. In particular, this means that if supergravity is a symmetry in particle physics we should have observed experimentally such degenerate pairs, i.e., a scalar and a fermion with the same mass and quantum numbers, e.g., a neutral massless Majorana fermion (associated to the photon) or a light charged scalar (associated to the electron). This is obviously not the case. Consequently, supergravity has to be spontaneously broken. This feature, is not only a property of supergravity but also of supersymmetry. The first mechanisms of supersymmetry breaking involved chiral super-fields [O’Raifeartaigh (1975)] or a vector superfield associated to a U(1)-gauge symmetry [Fayet and Iliopoulos (1974); Fayet (1975)]. However, neither are realistic since after supersymmetry breaking the spectrum contains supersymmetric particles lighter than the standard particles. Then it was realised that it is more natural to break supersymmetry but in the context of supergravity. There are several ways to break supergravity, but all share a common feature, that is, supergravity is broken in a hidden sector. The interaction (gravitational, gauge, etc) of the hidden sector induces supergravity breaking within the observable sector. In the first types of mechanisms, called gravity mediated supersymmetry breaking [Chamseddine et al. (1982); Barbieri et al. (1982b); Ibañez (1982); Ohta (1983); Ellis et al. (1983); Alvarez-Gaumé et al. (1983); Polonyi (1977)], the supersymmetry breaking is communicated to the observable sector via gravitational interactions. In this mechanism the Kähler potential and the scalar potential play an essential rôle. In the second types of mechanisms, called gauge mediated supersymmetry breaking, supersymmetry is broken by a singlet field (called the spurion) which couples (via the superpotential) to a field called the messenger which is in a non-trivial representation of the gauge group. Quantum loop corrections involving the messenger break supersymmetry in the observable sector [Nilles (1984); Dimopoulos and Raby (1981); Dine et al. (1981); Derendinger and Savoy (1982); Fayet (1978); Dine and Fischler (1982); Nappi and Ovrut (1982); Alvarez-Gaumé et al. (1982); Dine and Nelson (1993); Dine et al. (1995, 1996); Giudice and Rattazzi (1999); Arkani-Hamed et al. (1998)]. In the third type of mechanisms, called anomaly mediated supersymmetry breaking, the introduction of compensating fields having a conformal (super-Weyl) anomaly induces supersymmetry breaking by purely quantum effects [Randall and Sundrum (1999); Arkani-Hamed et al. (1998); Bagger et al. (2000)]. Several other mechanisms to break supergravity have been proposed, as in string theory for instance etc…

In this chapter we investigate two applications of supergravity in Physics.

One application which has been intensively studied, for many decades, by any means of formal, theoretical, phenomenological or experimental aspects, is the application to particle physics. There are several ways to embed the Standard Model (or one of its Grand Unification extensions) in supergravity. Further, as we have seen in Chapter 7, supergravity or supersymmetry has to be broken for phenomenological reasons. Also there are many ways to break supergravity. One extension which figures most prominently in the literature is the Minimal Supersymmetric Standard Model (MSSM). This model, or at least the model were supersymmetry is broken in a universal way (see Sec. 7.2.1), is almost excluded experimentally. Nevertheless in this chapter the MSSM will be introduced, and in particular it will be established that considering specific hidden sectors it is possible to naturally reproduce the different models studied in the literature (such as the cMSSM, or any possible choice for the soft supersymmetric breaking terms considered so far in any phenomenological analysis). Since this study is generic it can be easily extended to different specific models…

The following sections are included:

• Appendix A Conventions
• Appendix B Some properties of the Lorentz group
• Appendix C Useful identities
• Appendix D Supermultiplets: conventions and main results
• Appendix E How to construct an action in supergravity
• Appendix F Kähler manifolds
  • General properties
  • Reparametrisation
  • Killing vectors and symmetries of the Kähler manifold
• Appendix G Two-loop renormalisation group equations
• Appendix H Lie superalgebras associated to supergravity and supersymmetry

The following sections are included:

• Bibliography
• Index