Spontaneous Supersymmetry Breaking

• Spontaneous Supersymmetry Breaking is the phenomenon where a theory’s ground state does not respect supersymmetry, resulting in nonzero vacuum energy and the emergence of a massless Goldstino.
• It is realized through diverse mechanisms such as vacuum condensate-induced breaking, higher-dimensional operators, and spatially modulated vacua, each with distinctive experimental and theoretical signatures.
• Its practical implications include the generation of soft SUSY-breaking terms in effective models, benchmark tests in lattice simulations, and potential observable effects in particle physics and cosmology.

Spontaneous supersymmetry breaking (SSB) is a central concept in supersymmetric quantum field theory and string theory, referring to the phenomenon where the ground state of a manifestly supersymmetric theory does not respect the full supersymmetry of the action. Spontaneous breaking induces nonvanishing vacuum energy, gives rise to a massless Goldstino fermion (in global SUSY), and typically realizes soft breaking patterns important for low-energy phenomenology. The SSB order parameter in many formulations is the expectation value of an auxiliary field (F- or D-term), but more generally it is the nonvanishing vacuum energy tied by the SUSY algebra to broken invariance of the ground state. Diverse mechanisms for SSB have been constructively analyzed, spanning field-theoretic (O’Raifeartaigh, dynamical, gauge-mediated), geometric (topologically induced), and string-theoretic settings, with both qualitative and fully explicit quantitative realizations. Below, key aspects of spontaneous SUSY breaking are systematically organized, with emphasis on rigorous criteria, dynamical realizations, distinctive signatures, and implications for theory and phenomenology.

1. Defining Criteria and Order Parameters for Spontaneous SUSY Breaking

A system exhibits spontaneous SUSY breaking if its vacuum state is not annihilated by all the supercharges, or equivalently if the ground state energy is strictly positive. In global SUSY, the algebra reads $$\{Q_\alpha,Q_\beta^\dagger\}=2(\gamma^0)_{\alpha\beta}H$$, so a necessary and sufficient criterion is

$$\langle 0|H|0\rangle > 0 \implies Q_\alpha |0\rangle\neq 0.$$

This criterion is fully operative in field theory and on the lattice, and matches nonperturbative diagnostics such as the vanishing of the Witten index, the emergence of a massless Goldstino, or the violation of SUSY Ward identities. On the lattice, lattice super QCD and Wess-Zumino models demonstrate through numerical simulations the phase structure and transitions where vacuum energy becomes nonzero and massless Goldstino modes are observed, confirming the general order parameter picture (Baumgartner et al., 2013, Catterall et al., 2015, Steinhauer et al., 2014).

For local SUSY (supergravity), any nonvanishing cosmological constant from the vacuum state signals SSB, with the Goldstino being eaten by the gravitino, making it massive by the super-Higgs mechanism (Guendelman et al., 2014, Guendelman et al., 2015).

2. Dynamical and Geometric Mechanisms of SSB

Mechanisms of SSB fall into several typological classes, unified by the SSB order parameter but with distinct dynamical origins:

A. Vacuum Condensate–Induced Breaking

Vacuum states related by nontrivial Bogoliubov transformations to the Fock vacuum (e.g., due to temperature, external fields, flavor mixing, BCS pairing, or the Unruh effect) universally carry a positive energy density, which spoils the cancellation between bosonic and fermionic zero-point energies that is enforced by SUSY (Capolupo et al., 2012, Capolupo et al., 2013, Capolupo et al., 2010, Capolupo et al., 2013). In the Wess-Zumino model, both numerical and analytic computations demonstrate that any nonzero condensate component leads to nonvanishing expectation values for the Hamiltonian, directly breaking SUSY and generating a Goldstino. This mechanism generalizes to mixings in the flavor sector, with the flavor vacuum constructed by a mixing-induced Bogoliubov transformation. The positive vacuum energy is strictly proportional to the mixing parameters and mode occupation numbers (Capolupo et al., 2013, Capolupo et al., 2010).

B. Breaking by Higher-Dimensional Operators

Supersymmetry can be spontaneously broken in $N=1$ field theories by adding higher-dimensional supersymmetric operators, e.g., to complex linear multiplets. These trigger nonzero VEVs for auxiliary fields in the absence of a scalar sgoldstino partner, yielding nilpotent goldstino multiplets consistent with nonlinear realizations of SUSY (Volkov-Akulov framework) (Farakos et al., 2013). The dynamical goldstino emerges from an auxiliary sector, and the nonvanishing $F$-term is a direct signal of SSB.

C. Spatially Modulated and Inhomogeneous Vacua

In higher-derivative chiral models, supersymmetry can be broken in spatially modulated vacua where translation and $U(1)$ symmetries are also spontaneously broken. The vacuum energy can be positive (meta-stable), zero, or negative depending on parameter regions. In positive-energy modulated vacua, a healthy Goldstino mode emerges; in negative energy, this Goldstino becomes a ghost, and in zero energy vacua, it decouples (Nitta et al., 2017). For inhomogeneous deformations of couplings, the addition of spatial dependence can preserve at most half the supersymmetries, but—consistent with the Nelson-Seiberg argument—no true spontaneous SUSY breaking arises without spontaneous $R$-symmetry breaking (Kim et al., 2023).

D. Geometric/Topological Mechanisms: Non-Riemannian Measure

In the context of $N=1$ supergravity, modifying the spacetime volume form to be metric-independent and constructed from auxiliary 3-forms leads to a dynamically generated cosmological constant as an integration constant of the field equations. Its nonzero value triggers SSB, making the gravitino massive, purely through geometric means and without hidden sector fields or explicit $F$- or $D$-terms (Guendelman et al., 2014, Guendelman et al., 2015). This mechanism elegantly decouples the cosmological constant from the gravitino mass and exposes a pure topological/dynamical route to SSB in gravity.

3. String and Brane Realizations: The Anti-D3-Brane Goldstino

In string theory, the KKLT construction of de Sitter vacua involves the explicit inclusion of an anti-D3-brane for vacuum uplift. Detailed worldvolume analyses show that, upon orientifolding (introducing an O3-plane), the combined Dirac-Born-Infeld and Wess-Zumino actions on the anti-D3-brane reduce to the Volkov-Akulov goldstino action. The surviving worldvolume fermion—a goldstino—packages into a nilpotent chiral superfield $S$ with $S^2=0$, and the associated nonvanishing $F$-term precisely reproduces the uplift term in the 4d effective supergravity potential. This string-theoretic realization provides a manifest and explicit spontaneous SUSY breaking mechanism, firmly linking the string uplift sector with a microscopic nonlinearly realized supersymmetry (Kallosh et al., 2014).

4. Lattice and Low-Dimensional Field Theory Demonstrations

Nonperturbative lattice simulations provide a direct window into SSB, particularly in dimensions where continuum arguments may fail. In the 2d $N=1$ Wess-Zumino model with a cubic superpotential, both lattice and analytic approaches demonstrate a second-order phase transition at critical coupling $f_c$, separating a SUSY unbroken phase (zero vacuum energy, degenerate boson-fermion spectrum) from a broken phase (positive vacuum energy, massless Goldstino, bosonic gap) (Baumgartner et al., 2013, Steinhauer et al., 2014). The lattice super QCD model in $d=2$ shows that SSB correlates with the rank deficit in the matter sector: $N_f<N_c$ leads to unavoidable D-term induced SSB, robustly evidenced by order parameters, broken Ward identities, and emergent Goldstino correlators (Catterall et al., 2015). Majorana chain models further demonstrate that SSB in nonrelativistic, interacting systems yields distinct Goldstino dispersion relations—linear or cubic—providing a benchmark for emergent SUSY in cold atom or quantum spin-chain systems (Sannomiya, 15 Jan 2024).

5. Spontaneous Breaking in Gauge, GUT, and Supergravity Models

A variety of explicit models realize SSB in conjunction with other physical mechanisms:

• Gauge, R-symmetry, and SUSY Breaking in a Single Sector: O’Raifeartaigh-type models with added gauge symmetries exhibit flat directions lifted by D-terms and stabilized by loop corrections, ensuring gauge, $R$-symmetry, and SUSY breaking in the same sector, which is essential for generating soft SUSY-breaking terms such as gaugino masses (Kobayashi et al., 2017).
• Visible-Sector Breaking in Grand Unified Theories: Modified $SO(10)$ GUT models without a hidden sector (omitting a crucial singlet) spontaneously break SUSY in the visible sector. The resulting F-term VEV is non-zero and produces a high-scale mass spectrum with characteristic $SU(5)$-patterned sfermion masses and long-lived exotic particles, providing direct experimental signatures (Maekawa et al., 2019).
• Functional Renormalization and Random Field Systems: In statistical field theory, spontaneous breaking of Parisi–Sourlas supersymmetry (superrotational invariance) occurs below a critical dimension in the Random Field Ising Model, as revealed by the nonperturbative functional renormalization group flow equations. The SUSY-broken fixed point is associated with the breakdown of dimensional reduction and nonanalyticities (cusps) in the effective action (Tissier et al., 2011).

6. Signatures, Goldstino Properties, and Phenomenological Implications

The universal signature of SSB is the emergence of a massless Goldstino for global SUSY (a gapless NG fermion with characteristic dispersion). The Goldstino arises from the low-energy limit of the symmetry algebra and manifests directly in the physical spectrum (or becomes a longitudinal mode of the gravitino in supergravity). In models with higher-derivative terms or spatial modulation, Goldstino modes can exhibit unusual (cubic, linear, or zero-norm) dispersion behaviors depending on the vacuum energy and stability properties (Nitta et al., 2017, Sannomiya, 15 Jan 2024). SSB often leads to soft supersymmetry breaking terms, such as gaugino and sfermion masses, in effective low-energy theories. Signals include vacuum energy shifts accessible in cold atom systems, vacuum noise, or long-lived exotic particles at colliders, tightly linking SSB mechanisms to observable phenomena (Capolupo et al., 2012, Maekawa et al., 2019, Sannomiya, 15 Jan 2024).

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References

• "Emergence of Spontaneously Broken Supersymmetry on an Anti-D3-Brane in KKLT dS Vacua" (Kallosh et al., 2014)
• "Vacuum condensates, flavor mixing and spontaneous supersymmetry breaking" (Capolupo et al., 2013)
• "Mixing-induced Spontaneous Supersymmetry Breaking" (Capolupo et al., 2010)
• "A New Mechanism of Dynamical Spontaneous Breaking of Supersymmetry" (Guendelman et al., 2014)
• "Spontaneous supersymmetry breaking in the 2d N=1 Wess-Zumino model" (Baumgartner et al., 2013, Steinhauer et al., 2014)
• "Supersymmetry Breaking by Higher Dimension Operators" (Farakos et al., 2013)
• "Spontaneous Supersymmetry Breaking Induced by Vacuum Condensates" (Capolupo et al., 2012)
• "Spontaneous Supersymmetry Breaking and Nambu-Goldstone Modes in Interacting Majorana Chains" (Sannomiya, 15 Jan 2024)
• "Nonperturbative Functional Renormalization Group for Random Field Models. IV: Supersymmetry and its spontaneous breaking" (Tissier et al., 2011)
• "A New Venue of Spontaneous Supersymmetry Breaking in Supergravity" (Guendelman et al., 2015)
• "Spontaneous supersymmetry breaking in two dimensional lattice super QCD" (Catterall et al., 2015)
• "Spontaneous SUSY breaking in natural SO(10) grand unified theory" (Maekawa et al., 2019)
• "Supersymmetry Breaking in Spatially Modulated Vacua" (Nitta et al., 2017)
• "Realization of a spontaneous gauge and supersymmetry breaking vacuum" (Kobayashi et al., 2017)
• "Spontaneous supersymmetry breaking probed by geometric invariants" (Capolupo et al., 2013)
• "Spontaneous Supersymmetry Breaking in Inhomogeneous Supersymmetric Field Theories and BPS Vacua" (Kim et al., 2023)
• "Spontaneous Supersymmetry Breaking, Negative Metric and Vacuum Energy" (Kugo, 2017)