Gravitinos and hidden Supersymmetry at the LHC - Universität ...

2.3. SUPERGRAVITY
Name Spin-2 Spin-3/2 SU(3) C , SU(2) L , U(1) Y
Graviton, Gravitino g µν ψ µ (1, 1, 0)
Table 2.3: The gravity supermultiplet present in all locally supersymmetric theories. Listed
are quantum numbers with respect to the Standard Model gauge group
supersymmetry at the TeV scale, since it would lead to problems with tree-level sum rules
which imply that some scalar partners of fermions (sfermions) must be lighter than fermions.
The MSSM is, therefore, regarded as a low energy effective theory to be derived from a
theory which incorporates supersymmetry breaking. Often, this theory is assumed to be a
supergravity theory.
Upon promoting SUSY to a local symmetry, one is forced to add a new supermultiplet
to the theory, the gravity multiplet, which consists of the spin-2 graviton and the spin-3/2
gravitino (see Table 2.3). Local transformations of the usual SUSY Lagrangian will require
the introduction of the spin-3/2 field, whose variation under local SUSY transformations
couples to the energy-momentum tensor and is canceled by the variation of the spin-2 field.
Supergravity is covered in detail in the Book by Wess and Bagger [78] and in the review
by Van Nieuwenhuizen [105], see references therein for the original works. Note that our
definition of the metric signature and sigma-matrices differs from the definition of Wess and
Bagger. The notational conventions of this chapter follow partly [103] (but also with different
signature).
2.3.1 The Supergravity Lagrangian
As stated above, supergravity is a non-renormalizable theory and therefore should in general
depend on the three functions defined in section on general actions in superspace. The
remarkable feature of the supergravity Lagrangian is that it depends on the gauge kinetic
function and just one combination:
G = K/M 2 P + ln ( |W | 2 /M 6 P
)
, (2.56)
of the Kähler potential and the superpotential. G is called Kähler function and is real and
dimensionless. We have maintained the dependence on the reduced Planck mass:
M P =
1
√ 8πGN
≃ 2.4 × 10 18 GeV. (2.57)
In what follows, derivatives of the Kähler function with respect to the chiral superfields are
denoted by:
G i = ∂G ∣ ∣∣∣Φi
, and G j = ∂G ∣ ∣∣∣Φj
∂Φ i →φ i
∂Φ ∗j . (2.58)
→φ j
Note that the superfields have been replaced by their scalar components after differentiation.
The position of the indices corresponds to the chirality of the superfield with respect to which
the quantity is differentiated. Raised (lowered) indices i correspond to derivatives with respect
to Φ i (Φ ∗i ). The Kähler metric
G j i =
∣ ∂2 G ∣∣∣Φi
∂Φ j ∂Φ ∗i = Ki j
→φ i
MP
2 , (2.59)
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