Introduction to supergravity 2015: Exercise 8.
Institute for Theoretical Physics, Masaryk University,
611 37 Brno, Czech Republic
The bosonic sector of the generic coupling of a chiral superfield to supergravity (with no higher
derivatives) is [1]
e−1
L = −
1
2
R − KA ¯A∂A∂ ¯A − eK 1
KA ¯A
(WA + KAW)( ¯W ¯A + K ¯A
¯W) − 3W ¯W . (1)
The Polonyi model is given by
K = Φ¯Φ
W = µ(Φ + β).
(2)
If we set
β = 2 −
√
3 (3)
we saw that the model has a supersymmetry breaking vacuum with vanishing vacuum energy [2].
1. Find the scalar potential for the Polonyi model with generic β.
2. Verify that for β = 2 −
√
3 supersymmetry is broken and that the vacuum is Minkowski.
3. Investigate what happens for small deviations from β = 2−
√
3: What happens if β 2−
√
3
or β 2 −
√
3? Is supersymmetry broken? What is the vacuum energy?
References
[1] J. Wess and J. Bagger, “Supersymmetry and supergravity,” Princeton, USA: Univ. Pr. (1992).
[2] J. Polonyi, “Generalization of the massive scalar multiplet coupling to the supergravity,”
Hungary Central Inst. Res. - KFKI-1977-93.
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