Supersymmetry Breaking with Fields, Strings and Branes

TL;DR

This two-part review surveys how supersymmetry can be broken and realized in both low-energy field theory and high-energy string frameworks. It first develops bottom-up foundations in 4D N=1 SUSY, detailing multiplet structures, spontaneous breaking mechanisms, soft terms, and MSSM phenomenology, before turning to top-down string-inspired constructions with fluxes, Calabi–Yau compactifications, and KKLT scenarios. It also connects SUSY breaking to cosmology, inflationary tadpoles, and non-linear realizations via Volkov–Akulov and constrained superfields, highlighting the Goldstino/gravitino dynamics and the role of geometry (Kähler, moduli) in mediating breaking. Collectively, the work emphasizes how soft terms arise in supergravity and string contexts, the constraints from anomalies and GIM-like flavor structure, and the phenomenological implications for collider searches, dark matter, and early-universe cosmology, while acknowledging ongoing challenges in vacuum stability and experimental verification.

Abstract

The first part of this review tries to provide a self-contained view of supersymmetry breaking from the bottom-up perspective. We thus describe N=1 supersymmetry in four dimensions, the Standard Model and the MSSM, with emphasis on the ``soft terms'' that can link it to supergravity. The second part deals with the top-down perspective. It addresses, insofar as possible in a self-contained way, the basic setup provided by ten-dimensional strings and their links with supergravity, toroidal orbifolds, Scherk-Schwarz deformations and Calabi-Yau reductions, before focusing on a line of developments that is closely linked to our own research. Its key input is drawn from ten-dimensional non-tachyonic string models where supersymmetry is absent or non-linearly realized, and runaway ``tadpole potentials'' deform the ten-dimensional Minkowski vacua. We illustrate the perturbative stability of the resulting most symmetrical setups, which are the counterparts of circle reduction but involve internal intervals. We then turn to a discussion of fluxes in Calabi-Yau vacua and the KKLT setup, and conclude with some aspects of Cosmology, emphasizing some intriguing clues that the tadpole potentials can provide for the onset of inflation. The appendices collect some useful material on global and local N=1 supersymmetry, in components and in superspace, on string vacuum amplitudes, and on convenient tools used to examine the fluctuations of non-supersymmetric string vacua.

Figures (16)

• Figure 1: The four different cases for gauge and (global) supersymmetry breaking. Left panel: gauge symmetry unbroken and supersymmetry unbroken (solid) or broken (dashed). Right panel: gauge symmetry broken and supersymmetry unbroken (solid) or broken (dashed).
• Figure 2: The hidden sector breaks supersymmetry, which is mediated to the visible sector by gravity or other non--renormalizable interactions or via quantum loops.
• Figure 3: Fermi theory of weak interactions: the beta decay $n \rightarrow p e^- \,{\bar{\nu}}_e$ at low energies $E << M_W$ can be described via an effective four-fermion interaction, in line with Fermi's initial proposal. Here the arrows are entering (outgoing) for incoming particles (antiparticles) and outgoing (entering) for outgoing particles (antiparticles).
• Figure 4: $K^0$-${\bar{K}}^0$ mixing generated at loop level in the Standard Model, with quarks $u_i= u,c,t$ running in the loop. Here the arrows are again entering (outgoing) for incoming particles (antiparticles) and outgoing (entering) for outgoing particles (antiparticles).
• Figure 5: Adler-Bell-Jackiw triangle anomalies.