Glossary

11.2 Supersymmetry and supergravity

5 min read•august 14, 2024

proposes a symmetry between bosons and fermions, extending the Standard Model with superpartners for each particle. This theory could solve the hierarchy problem, provide a dark matter candidate, and unify fundamental forces at high energies.

Supergravity combines supersymmetry with general relativity, introducing the gravitino as the superpartner of the graviton. It connects to string theory and offers a framework for addressing Standard Model shortcomings, including electroweak symmetry breaking and grand unification.

Supersymmetry Fundamentals

Symmetry between Bosons and Fermions

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  • Supersymmetry proposes a symmetry between bosons (particles with integer spin) and fermions (particles with half-integer spin)
    • Each particle in the Standard Model has a hypothetical superpartner with a spin differing by 1/2
      • For example, the electron (a fermion) would have a bosonic superpartner called the selectron
    • Supersymmetry transforms bosons into fermions and vice versa, extending the Poincaré algebra with anticommuting spinorial generators called supercharges

Broken Symmetry and the Hierarchy Problem

  • Supersymmetry must be a broken symmetry since superpartners have not been observed at the same masses as their Standard Model counterparts
    • If supersymmetry were an exact symmetry, the superpartners would have the same mass as their Standard Model counterparts
  • Supersymmetry can potentially solve the hierarchy problem in the Standard Model
    • The hierarchy problem arises from the vast difference between the weak scale (~100 GeV) and the Planck scale (~10^19 GeV)
    • Supersymmetry cancels quadratic divergences in the Higgs mass, stabilizing the weak scale against large quantum corrections

Dark Matter and Gauge Coupling Unification

  • The lightest supersymmetric particle (LSP) is a candidate for dark matter if it is stable and electrically neutral
    • Stability can be ensured by imposing a discrete symmetry called R-parity, which distinguishes between ordinary particles and their superpartners
  • Supersymmetry can lead to the unification of gauge couplings (strong, weak, and electromagnetic) at a high energy scale (~10^16 GeV)
    • This unification hints at the possibility of a grand unified theory (GUT) that describes all fundamental interactions, except gravity, within a single framework

Supersymmetric Field Theories

Construction and Properties

  • Supersymmetric field theories are constructed by extending the Poincaré algebra with anticommuting spinorial generators called supercharges
    • The number of supercharges determines the amount of supersymmetry (N=1, N=2, etc.)
  • The simplest supersymmetric field theory is the Wess-Zumino model, which consists of a complex scalar field and a Weyl fermion
    • The Wess-Zumino model is a toy model that demonstrates the basic features of supersymmetry
  • Supersymmetric theories have improved ultraviolet behavior compared to their non-supersymmetric counterparts
    • Supersymmetry leads to cancellations between bosonic and fermionic loop diagrams, resulting in more convergent loop integrals and a better-behaved perturbative expansion

Supersymmetric Extensions of the Standard Model

  • Supersymmetric gauge theories, such as the , incorporate supersymmetry into the Standard Model
    • In the MSSM, each Standard Model particle has a superpartner with a spin differing by 1/2 (squarks, sleptons, gauginos, and higgsinos)
  • The superpotential is a holomorphic function of the chiral superfields that determines the interactions and masses of the particles in a supersymmetric theory
    • The superpotential is constrained by the symmetries of the theory, such as gauge invariance and R-parity
  • R-parity is a discrete symmetry in supersymmetric models that distinguishes between ordinary particles (R-parity even) and their superpartners (R-parity odd)
    • R-parity conservation has important consequences for proton stability (preventing rapid proton decay) and dark matter (ensuring the stability of the LSP)

Supersymmetry and Supergravity

Combining Supersymmetry and General Relativity

  • Supergravity is a field theory that combines the principles of supersymmetry and general relativity
    • In supergravity, the graviton (the hypothetical quantum of the gravitational field) has a fermionic superpartner called the gravitino
  • The number of supersymmetry generators determines the type of supergravity theory
    • is the simplest extension of general relativity with a single gravitino
    • N=8 supergravity is the maximal supergravity theory in four dimensions, with eight gravitinos

Connection to String Theory

  • Supergravity theories can be constructed as the low-energy effective theories of string theory
    • String theory is a candidate for a quantum theory of gravity that describes fundamental particles as vibrating strings
  • The local supersymmetry transformations in supergravity theories lead to the emergence of the gravitino and the associated gauge field, the spin connection
    • The gravitino is the gauge field associated with local supersymmetry transformations, similar to how the photon is the gauge field associated with local U(1) gauge transformations in electromagnetism

Supersymmetry Beyond the Standard Model

Addressing Shortcomings of the Standard Model

  • Supersymmetry provides a framework for extending the Standard Model and addressing some of its shortcomings
    • The hierarchy problem: Supersymmetry stabilizes the weak scale against large quantum corrections by canceling quadratic divergences in the Higgs mass
    • The nature of dark matter: The lightest supersymmetric particle (LSP) can serve as a stable, weakly interacting massive particle (WIMP) dark matter candidate
  • The Minimal Supersymmetric Standard Model (MSSM) is a well-studied extension of the Standard Model that incorporates supersymmetry with minimal additional particle content
    • The MSSM predicts a rich spectrum of new particles, such as squarks, sleptons, gauginos, and higgsinos, which can be searched for in high-energy collider experiments (LHC, future colliders)

Electroweak Symmetry Breaking and Grand Unification

  • Supersymmetry can provide a mechanism for electroweak symmetry breaking through the Higgs sector
    • In the MSSM, the Higgs sector is extended to include two Higgs doublets, leading to five physical Higgs bosons (h, H, A, H±)
    • The lightest Higgs boson (h) in the MSSM can be compatible with the observed 125 GeV particle discovered at the LHC
  • Supersymmetric grand unified theories (SUSY GUTs) can potentially explain the unification of gauge couplings and the origin of matter-antimatter asymmetry in the universe
    • SUSY GUTs embed the Standard Model gauge groups (SU(3)xSU(2)xU(1)) into a larger symmetry group, such as SU(5) or SO(10), at a high energy scale (~10^16 GeV)
    • The unification of gauge couplings in SUSY GUTs is more precise than in non-supersymmetric GUTs due to the modified renormalization group running of the couplings

Supersymmetry in Theories Beyond the Standard Model

  • Supersymmetry is a key ingredient in many theories of physics beyond the Standard Model
    • String theory: Supersymmetry is necessary for the consistency of string theory, which aims to provide a unified description of all fundamental interactions, including gravity
    • Extra-dimensional models: Supersymmetry can be combined with the idea of extra spatial dimensions to address the hierarchy problem and provide novel (Kaluza-Klein states)
  • Supersymmetry provides a rich framework for exploring new physics and extending our understanding of the fundamental laws of nature
    • The discovery of supersymmetry would revolutionize our understanding of particle physics and have far-reaching implications for cosmology and the early universe

Key Terms to Review (19)

Conformal Symmetry: Conformal symmetry is a type of symmetry that preserves angles but not necessarily distances, meaning it allows for the transformation of spacetime in a way that keeps the shapes of objects intact. This symmetry is crucial in various physical theories, particularly in quantum field theory and string theory, as it connects the behavior of fields at different energy scales and plays a significant role in the formulation of supergravity theories.
Daniel Z. Freedman: Daniel Z. Freedman is a theoretical physicist known for his significant contributions to the fields of supersymmetry and supergravity. His work has played a crucial role in the development of models that unify quantum mechanics and general relativity, particularly in understanding the fundamental symmetries of nature.
Dark matter candidates: Dark matter candidates are hypothetical particles or entities proposed to explain the unseen mass in the universe that interacts gravitationally but does not emit or absorb light. They are crucial for understanding the structure and behavior of galaxies, as well as the overall dynamics of the cosmos, with certain models suggesting that supersymmetry could provide viable candidates for dark matter.
Duality: Duality refers to the concept that two seemingly different theories or frameworks can describe the same physical phenomena. In the context of advanced theoretical physics, this idea is particularly significant, as it implies a deeper underlying relationship between various physical models. The exploration of duality reveals how different perspectives can yield equivalent insights into the behavior of complex systems, especially in fields like supersymmetry and supergravity.
Extended supergravity: Extended supergravity refers to a class of supersymmetric theories that include additional bosonic and fermionic fields beyond the minimal content, often accommodating higher-dimensional spacetimes. This theory provides a rich framework for understanding gravity in the context of supersymmetry and plays a critical role in theoretical physics, particularly in unifying gravity with other fundamental forces.
Fermionic partners: Fermionic partners are particles that are related to bosons through the concept of supersymmetry, meaning each fermionic particle has a corresponding bosonic partner and vice versa. This relationship provides a way to unify different types of particles in theoretical physics, suggesting that for every fermion, which follows the Pauli exclusion principle, there exists a boson that can occupy the same quantum state. This symmetry plays a crucial role in the frameworks of advanced theories like supergravity, which extends general relativity to include quantum mechanical aspects.
Gauge coupling unification: Gauge coupling unification refers to the theoretical framework where the coupling constants of the fundamental forces converge to a single value at high energies. This concept is crucial in understanding how different forces, like electromagnetic, weak, and strong interactions, might be unified in a grander theory, especially within models that include supersymmetry and supergravity.
Grassmann numbers: Grassmann numbers are mathematical entities that extend the concept of numbers to include anti-commuting variables. They are crucial in the formulation of supersymmetry and supergravity, as they allow for the representation of fermionic fields within a bosonic framework. Their unique property of anti-commutation means that when two Grassmann numbers are multiplied, the order matters, leading to significant implications in calculations involving particle interactions and field theories.
Minimal supersymmetric standard model (MSSM): The minimal supersymmetric standard model (MSSM) is an extension of the Standard Model of particle physics that incorporates supersymmetry, a theoretical symmetry relating bosons and fermions. It introduces superpartners for each particle in the Standard Model and aims to resolve several issues like hierarchy problems and unification of forces, while being minimal in its structure by keeping the number of new parameters to a minimum.
N=1 supergravity: n=1 supergravity is a theoretical framework that combines the principles of supersymmetry with general relativity to describe gravity and its interactions with fermions and bosons in a unified manner. This model is a specific case of supergravity where there is one set of supercharges, leading to the introduction of a graviton and a gravitino as the spin-2 and spin-3/2 fields, respectively. n=1 supergravity plays a crucial role in understanding high-energy physics, including string theory and models beyond the Standard Model.
Scalar interactions: Scalar interactions refer to the types of particle interactions in quantum field theory that involve scalar fields, which are characterized by having a single value at each point in space and time, rather than direction. These interactions play a crucial role in various physical theories, especially in supersymmetry and supergravity, where scalar fields can represent the scalar partners of fermionic particles and are important for maintaining symmetry between different types of particles.
Sergio Ferrara: Sergio Ferrara is a prominent theoretical physicist known for his significant contributions to the fields of supersymmetry and supergravity. His work has been instrumental in developing the theoretical frameworks that connect particle physics and gravity, emphasizing the unification of fundamental forces. Ferrara's research plays a critical role in understanding how these concepts can provide deeper insights into the nature of spacetime and the structure of matter at a fundamental level.
Soft breaking: Soft breaking refers to a mechanism in supersymmetry where supersymmetry is broken in a way that preserves the structure of the theory without introducing large mass scales or fine-tuning issues. This concept is significant in contexts like supergravity, where the interactions between bosons and fermions remain controlled and lead to viable low-energy effective theories. It allows for the emergence of superpartners with masses that are not excessively high, keeping the framework theoretically appealing.
Spontaneous breaking: Spontaneous breaking occurs when a system that is symmetric in its fundamental laws exhibits an asymmetry in its lowest energy state. This concept is crucial in understanding how symmetries can be broken without external influence, leading to the emergence of mass for particles and fields, particularly in theories like supersymmetry and supergravity.
Supercharge: In theoretical physics, particularly in supersymmetry and supergravity, 'supercharge' refers to a generator of supersymmetry transformations that relates bosonic and fermionic states. It plays a crucial role in bridging the gap between these two types of particles, allowing for a more unified description of the fundamental forces and particles in the universe.
Supermultiplet: A supermultiplet is a collection of particles in supersymmetry that includes both bosons and fermions, with the total number of degrees of freedom being the same for each component. This concept connects various particle types in a single framework, demonstrating how particles can transform into one another through supersymmetric transformations. Supermultiplets play a crucial role in formulating theories like supergravity, where they help unify different particle species and their interactions.
Superspace: Superspace is an extension of conventional space-time that incorporates both bosonic and fermionic coordinates, enabling the formulation of theories with supersymmetry. It acts as a mathematical framework to unify these different types of particles, providing a more comprehensive description of physical laws. This concept is essential in theories of supergravity, where the interplay between gravity and supersymmetry is explored.
Superstring theory: Superstring theory is a theoretical framework that attempts to reconcile general relativity and quantum mechanics by positing that the fundamental constituents of the universe are one-dimensional strings rather than point-like particles. This approach introduces supersymmetry, which suggests a symmetry between fermions and bosons, leading to a richer structure in particle physics and offering potential solutions to various unsolved problems in theoretical physics.
Supersymmetry: Supersymmetry is a theoretical framework in particle physics that posits a symmetry relationship between fermions and bosons, suggesting that each particle has a superpartner with different spin characteristics. This concept aims to solve various problems in quantum field theory, such as the hierarchy problem and unification of forces, and plays a crucial role in models like supergravity and string theory.
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