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Phenomenological Aspects

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Supergravity

Part of the book series: Lecture Notes in Physics ((LNP,volume 991))

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Abstract

In previous chapters, we gave a complete picture of the Lagrangian and the supersymmetry rules for \(\mathcal {N}=1\) supergravity coupled to vector and chiral multiplets. In the present chapter, we now take a closer look at these theories from a phenomenological point of view. Our main focus will again be on those phenomenological aspects that differ in important ways between locally and globally supersymmetric theories.

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Notes

  1. 1.

    Our definition of a “vacuum” here should not be confused with the notion of a “vacuum solution” in general relativity, which is more general and refers to solutions to Einstein’s equation without contributions to the energy-momentum tensor T μν from any kind of matter excitation. Note that a constant scalar field ϕ(x) = ϕ 0 = const. does not count as an excitation here and hence should be allowed in such a vacuum solution, because its contribution to T μν arises only from its constant potential V (ϕ(x)) = V (ϕ 0), which is indistinguishable from a contribution due to a cosmological constant, Λ, in the Einstein equation. In other words, our maximally symmetric “vacua” are special cases of the “vacuum solutions” of general relativity, which, however, also encompass less symmetric matter-free solutions such as the Schwarzschild metric or a gravitational wave in empty space. Viewing the inhomogeneities of the metric in these solutions as gravitational excitations, one could also characterize our “vacua” as solutions that are free of matter and gravitational excitations and hence form the natural semi-classical analogues of the Poincaré-invariant vacua in conventional quantum field theories on Minkowski spacetime.

  2. 2.

    Effects such as gaugino condensation, where strongly coupled gauge dynamics induce a non-trivial vev for gaugino bilinears, \(\langle \overline {\lambda }\lambda \rangle \neq 0\), can, in an effective field theory below the condensation scale, often be described by an additional scalar field and contribute to spontaneous supersymmetry breaking in the effective scalar field sector. This effective scalar field dynamics would be contained in our analysis.

  3. 3.

    Note that what we call V F − V G here is usually called the “F-term potential” and would usually be denoted by V F.

  4. 4.

    The couplings of the goldstino are proportional to inverse powers of M susy and hence vanish in this limit.

  5. 5.

    In fact, it is easy to construct globally supersymmetric theories in which the goldstino completely decouples from the χ m⊥.

  6. 6.

    This name is a bit misleading, as it is not exactly gravity that induces the soft terms but other Planck-suppressed contact interactions that are (in part) related to gravity by supersymmetry; see Sect. 7.3.2.

  7. 7.

    The gaugino masses arise at one-loop, whereas the squares of the sfermion masses are due to a two-loop correction.

  8. 8.

    This is not a strong assumption for the MSSM, because if SU(3)c and U(1)em as well as R-parity are to be unbroken, only the neutral Higgs fields can get a vev in the visible sector, and this vev is around the electroweak scale. The only term in W v that can then get a vev is the quadratic term in the Higgs fields, the μ-term, which will then also be near the electroweak scale. The vev of W h in the Polonyi model, by contrast, is of order \(M_P^2 M_{3/2}\), which is many orders of magnitude higher. Similar remarks apply to the derivatives of the superpotentials.

  9. 9.

    The canonical normalization factor X is not parametrically large due to the logarithm, so unlike in case (i), the canonical normalization does not change the result by at most an order of magnitude.

  10. 10.

    In the original sense, moduli denote massless scalar fields, usually related to constant energy deformations of the internal space in string compactifications or to exactly marginal deformations of conformal quantum field theories. As described above, in phenomenologically realistic string compactifications, such scalar fields should have a certain mass, i.e., the corresponding deformations of the internal space should cost some energy (e.g., due to the presence of p-form fluxes in the internal space or as a consequence of non-perturbative quantum corrections). Nevertheless, these massive scalar fields are still called moduli.

  11. 11.

    This bound is an approximate estimate based on the following assumptions [11]: (1) Moduli couple with M P-suppressed interactions to other matter with a resulting decay width \(\varGamma \sim \mathcal {O}(M_{\mbox{Mod}}^3/M_{P}^2)\); (2) the Hubble scale after inflation is larger than M Mod, and when it drops to H ≈ M Mod, the modulus starts oscillating around its potential minimum; (3) the moduli so produced decay before Big Bang Nucleosynthesis (BBN).

  12. 12.

    This subsection is based on [12].

  13. 13.

    This subsection is based on [20].

  14. 14.

    This does of course not guarantee that there is really no tachyon, because (7.144) is just an upper bound.

  15. 15.

    As we have discussed in Chap. 4, stable AdS vacua would even be consistent with tachyonic scalars as long as they satisfy the Breitenlohner–Freedman bound.

  16. 16.

    Note that we have been a bit sloppy here with the fact that ϕ in supergravity is complex and that in single field inflation the inflaton is real. One thus has to go over to the diagonalized mass matrix of the real and imaginary part of ϕ first.

  17. 17.

    The inflaton mass term can alternatively be viewed as due to gravity-mediated spontaneous supersymmetry breaking during slow-roll inflation.

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Authors and Affiliations

  1. Physics and Astronomy, University of Padua, Padua, Italy

    Gianguido Dall’Agata

  2. Hamburg, Germany

    Marco Zagermann

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  1. Gianguido Dall’Agata
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Dall’Agata, G., Zagermann, M. (2021). Phenomenological Aspects. In: Supergravity. Lecture Notes in Physics, vol 991. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63980-1_7

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