Supersymmetry

Supersymmetry is a theoretical framework in physics that suggests the existence of a symmetry between particles with integer spin (bosons) and particles with half-integer spin (fermions). It proposes that for every known particle, there exists a partner particle with different spin properties.^[1] There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in nature.^[2] If evidence is found, supersymmetry could help explain certain phenomena, such as the nature of dark matter and the hierarchy problem in particle physics.

A supersymmetric theory is a theory in which the equations for

In particle physics, the first realistic supersymmetric version of the Standard Model was proposed in 1977 by Pierre Fayet and is known as the Minimal Supersymmetric Standard Model or MSSM for short. It was proposed to solve, amongst other things, the hierarchy problem.

Supersymmetry was coined by Abdus Salam and John Strathdee in 1974 as a simplification of the term super-gauge symmetry used by Wess and Zumino, although Zumino also used the same term at around the same time.^[24][25] The term supergauge was in turn coined by Neveu and Schwarz in 1971 when they devised supersymmetry in the context of string theory.^[19][26]

Applications

Extension of possible symmetry groups

One reason that physicists explored supersymmetry is because it offers an extension to the more familiar symmetries of quantum field theory. These symmetries are grouped into the Poincaré group and internal symmetries and the Coleman–Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or if there is not any mass gap, the conformal group with a compact internal symmetry group. In 1971 Golfand and Likhtman were the first to show that the Poincaré algebra can be extended through introduction of four anticommuting spinor generators (in four dimensions), which later became known as supercharges. In 1975, the Haag–Łopuszański–Sohnius theorem analyzed all possible superalgebras in the general form, including those with an extended number of the supergenerators and central charges. This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.

The supersymmetry algebra

Traditional symmetries of physics are generated by objects that transform by the

and all other anti-commutation relations between the Qs and commutation relations between the Qs and Ps vanish. In the above expression P_μ = −i ∂_μ are the generators of translation and σ^μ are the Pauli matrices.

There are

Supersymmetric quantum mechanics

Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often becomes relevant when studying the dynamics of supersymmetric

SUSY quantum mechanics involves pairs of

In finance

In 2021, supersymmetric quantum mechanics was applied to

In quantum field theory, supersymmetry is motivated by solutions to several theoretical problems, for generally providing many desirable mathematical properties, and for ensuring sensible behavior at high energies. Supersymmetric quantum field theory is often much easier to analyze, as many more problems become mathematically tractable. When supersymmetry is imposed as a local symmetry, Einstein's theory of general relativity is included automatically, and the result is said to be a theory of supergravity. Another theoretically appealing property of supersymmetry is that it offers the only "loophole" to the Coleman–Mandula theorem, which prohibits spacetime and internal symmetries from being combined in any nontrivial way, for quantum field theories with very general assumptions. The Haag–Łopuszański–Sohnius theorem demonstrates that supersymmetry is the only way spacetime and internal symmetries can be combined consistently.^[29]

While supersymmetry has not been discovered at high energy, see Section Supersymmetry in particle physics, supersymmetry was found to be effectively realized at the intermediate energy of hadronic physics where baryons and mesons are superpartners. An exception is the pion that appears as a zero mode in the mass spectrum and thus protected by the supersymmetry: It has no baryonic partner.^[30][31] The realization of this effective supersymmetry is readily explained in quark–diquark models: Because two different color charges close together (e.g., blue and red) appear under coarse resolution as the corresponding anti-color (e.g. anti-green), a diquark cluster viewed with coarse resolution (i.e., at the energy-momentum scale used to study hadron structure) effectively appears as an antiquark. Therefore, a baryon containing 3 valence quarks, of which two tend to cluster together as a diquark, behaves likes a meson.

Supersymmetry in condensed matter physics

SUSY concepts have provided useful extensions to the WKB approximation. Additionally, SUSY has been applied to disorder averaged systems both quantum and non-quantum (through statistical mechanics), the Fokker–Planck equation being an example of a non-quantum theory. The 'supersymmetry' in all these systems arises from the fact that one is modelling one particle and as such the 'statistics' do not matter. The use of the supersymmetry method provides a mathematical rigorous alternative to the replica trick, but only in non-interacting systems, which attempts to address the so-called 'problem of the denominator' under disorder averaging. For more on the applications of supersymmetry in condensed matter physics see Efetov (1997).^[32]

In 2021, a group of researchers showed that, in theory, ${\displaystyle N=(0,1)}$ SUSY could be realised at the edge of a Moore–Read

All stochastic (partial) differential equations, the models for all types of continuous time dynamical systems, possess topological supersymmetry.

SUSY is also sometimes studied mathematically for its intrinsic properties. This is because it describes complex fields satisfying a property known as holomorphy, which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful "toy models" of more realistic theories. A prime example of this has been the demonstration of S-duality in four-dimensional gauge theories^[40] that interchanges particles and monopoles.

The proof of the

Supersymmetry in string theory

Supersymmetry is an integral part of string theory, a possible theory of everything. There are two types of string theory, supersymmetric string theory or superstring theory, and non-supersymmetric string theory. By definition of superstring theory, supersymmetry is required in superstring theory at some level. However, even in non-supersymmetric string theory, a type of supersymmetry called misaligned supersymmetry is still required in the theory in order to ensure no physical tachyons appear.^[41][42] Any string theories without some kind of supersymmetry, such as bosonic string theory and the ${\displaystyle E_{7}\times E_{7}}$, ${\displaystyle SU(16)}$, and ${\displaystyle E_{8}}$

Supersymmetry in particle physics

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence Hierarchy problem Dark matter Dark energy Quintessence Phantom energy Dark radiation Dark photon Cosmological constant problem Strong CP problem Neutrino oscillation
Theories Brans–Dicke theory Cosmic censorship hypothesis Fifth force F-theory Theory of everything Unified field theory Grand Unified Theory Technicolor Kaluza–Klein theory 6D (2,0) superconformal field theory Noncommutative quantum field theory Quantum cosmology Brane cosmology String theory Superstring theory M-theory Mathematical universe hypothesis Mirror matter Randall–Sundrum model N = 4 supersymmetric Yang–Mills theory Twistor string theory Dark fluid Doubly special relativity de Sitter invariant special relativity Causal fermion systems Black hole thermodynamics Unparticle physics Graviphoton Graviscalar Graviton Gravitino Massive gravity Gauge gravitation theory Gauge theory gravity CPT symmetry
Supersymmetry MSSM NMSSM Superstring theory M-theory Supergravity Supersymmetry breaking Extra dimensions Large extra dimensions
Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory
Experiments ANNIE Gran Sasso INO LHC SNO Super-K Tevatron NOvA
v t e

In particle physics, a supersymmetric extension of the Standard Model is a possible candidate for

elementary particles

would be new and undiscovered particles and supersymmetry is expected to be spontaneously broken.

There is no experimental evidence that a supersymmetric extension to the Standard Model is correct, or whether or not other extensions to current models might be more accurate. It is only since around 2010 that particle accelerators specifically designed to study physics beyond the Standard Model have become operational (i.e. the Large Hadron Collider (LHC)), and it is not known where exactly to look, nor the energies required for a successful search. However, the negative results from the LHC since 2010 have already ruled out some supersymmetric extensions to the Standard Model, and many physicists believe that the Minimal Supersymmetric Standard Model, while not ruled out, is no longer able to fully resolve the hierarchy problem.^[54]

Supersymmetric extensions of the Standard Model

Incorporating supersymmetry into the Standard Model requires doubling the number of particles since there is no way that any of the particles in the Standard Model can be superpartners of each other. With the addition of new particles, there are many possible new interactions. The simplest possible supersymmetric model consistent with the Standard Model is the Minimal Supersymmetric Standard Model (MSSM) which can include the necessary additional new particles that are able to be superpartners of those in the Standard Model.

Supersymmetry close to the electroweak scale, such as in the Minimal Supersymmetric Standard Model, would solve the hierarchy problem that afflicts the Standard Model.^[55] It would reduce the size of the quantum corrections by having automatic cancellations between fermionic and bosonic Higgs interactions, and Planck-scale quantum corrections cancel between partners and superpartners (owing to a minus sign associated with fermionic loops). The hierarchy between the electroweak scale and the Planck scale would be achieved in a natural manner, without extraordinary fine-tuning. If supersymmetry were restored at the weak scale, then the Higgs mass would be related to supersymmetry breaking which can be induced from small non-perturbative effects explaining the vastly different scales in the weak interactions and gravitational interactions.

Another motivation for the Minimal Supersymmetric Standard Model comes from

In many supersymmetric extensions of the Standard Model, such as the Minimal Supersymmetric Standard Model, there is a heavy stable particle (such as the neutralino) which could serve as a weakly interacting massive particle (WIMP) dark matter candidate. The existence of a supersymmetric dark matter candidate is related closely to R-parity. Supersymmetry at the electroweak scale (augmented with a discrete symmetry) typically provides a candidate dark matter particle at a mass scale consistent with thermal relic abundance calculations.^[58][59]

The standard paradigm for incorporating supersymmetry into a realistic theory is to have the underlying dynamics of the theory be supersymmetric, but the ground state of the theory does not respect the symmetry and supersymmetry is broken spontaneously. The supersymmetry break can not be done permanently by the particles of the MSSM as they currently appear. This means that there is a new sector of the theory that is responsible for the breaking. The only constraint on this new sector is that it must break supersymmetry permanently and must give superparticles TeV scale masses. There are many models that can do this and most of their details do not matter. In order to parameterize the relevant features of supersymmetry breaking, arbitrary soft SUSY breaking terms are added to the theory which temporarily break SUSY explicitly but could never arise from a complete theory of supersymmetry breaking.

Searches and constraints for supersymmetry

SUSY extensions of the

Historically, the tightest limits were from direct production at colliders. The first mass limits for squarks and gluinos were made at CERN by the UA1 experiment and the UA2 experiment at the Super Proton Synchrotron. LEP later set very strong limits,^[61] which in 2006 were extended by the D0 experiment at the Tevatron.^[62][63] From 2003 to 2015, WMAP's and Planck's dark matter density measurements have strongly constrained supersymmetric extensions of the Standard Model, which, if they explain dark matter, have to be tuned to invoke a particular mechanism to sufficiently reduce the neutralino density.

Prior to the beginning of the LHC, in 2009, fits of available data to CMSSM and NUHM1 indicated that squarks and gluinos were most likely to have masses in the 500 to 800 GeV range, though values as high as 2.5 TeV were allowed with low probabilities. Neutralinos and sleptons were expected to be quite light, with the lightest neutralino and the lightest stau most likely to be found between 100 and 150 GeV.^[64]

The first runs of the LHC surpassed existing experimental limits from the Large Electron–Positron Collider and Tevatron and partially excluded the aforementioned expected ranges.

In response to the so-called "naturalness crisis" in the Minimal Supersymmetric Standard Model, some researchers have abandoned naturalness and the original motivation to solve the hierarchy problem naturally with supersymmetry, while other researchers have moved on to other supersymmetric models such as split supersymmetry.^[54][73] Still others have moved to string theory as a result of the naturalness crisis.^{[74][47][48][50]} Former enthusiastic supporter Mikhail Shifman went as far as urging the theoretical community to search for new ideas and accept that supersymmetry was a failed theory in particle physics.^[75] However, some researchers suggested that this "naturalness" crisis was premature because various calculations were too optimistic about the limits of masses which would allow a supersymmetric extension of the Standard Model as a solution.^[76][77]

General supersymmetry

Supersymmetry appears in many related contexts of theoretical physics. It is possible to have multiple supersymmetries and also have supersymmetric extra dimensions.

Extended supersymmetry

It is possible to have more than one kind of supersymmetry transformation. Theories with more than one supersymmetry transformation are known as extended supersymmetric theories. The more supersymmetry a theory has, the more constrained are the field content and interactions. Typically the number of copies of a supersymmetry is a power of 2 (1, 2, 4, 8...). In four dimensions, a spinor has four degrees of freedom and thus the minimal number of supersymmetry generators is four in four dimensions and having eight copies of supersymmetry means that there are 32 supersymmetry generators.

The maximal number of supersymmetry generators possible is 32. Theories with more than 32 supersymmetry generators automatically have massless fields with spin greater than 2. It is not known how to make massless fields with spin greater than two interact, so the maximal number of supersymmetry generators considered is 32. This is due to the Weinberg–Witten theorem. This corresponds to an N = 8^{[clarification needed]} supersymmetry theory. Theories with 32 supersymmetries automatically have a graviton.

For four dimensions there are the following theories, with the corresponding multiplets^[78] (CPT adds a copy, whenever they are not invariant under such symmetry):

N = 1	Chiral multiplet	(0,	1/2)
	Vector multiplet	(1/2,	1)
	Gravitino multiplet	(1,	3/2)
	Graviton multiplet	(3/2,	2)
N = 2	Hypermultiplet	(−1/2,	02,	1/2)
	Vector multiplet	(0,	1/2 2,	1)
	Supergravity multiplet	(1,	3/2 2,	2)
N = 4	Vector multiplet	(−1,	−1/2 4,	06,	1/2 4,	1)
	Supergravity multiplet	(0,	1/2 4,	16,	3/2 4,	2)
N = 8	Supergravity multiplet	(−2,	−3/2 8,	−128,	−1/2 56,	070,	1/2 56,	128,	3/2 8,	2)

Supersymmetry in alternate numbers of dimensions

It is possible to have supersymmetry in dimensions other than four. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In d dimensions, the size of spinors is approximately 2^d/2 or 2^{(d − 1)/2}. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.^{[citation needed]}

Fractional supersymmetry

Fractional supersymmetry is a generalization of the notion of supersymmetry in which the minimal positive amount of spin does not have to be 1/2 but can be an arbitrary 1/N for integer value of N. Such a generalization is possible in two or fewer spacetime dimensions.

See also

References

Further reading

• Supersymmetry and Supergravity page in String Theory Wiki lists more books and reviews.

Theoretical introductions, free and online

• Arygres, P. (2001), An Introduction to Global Supersymmetry (PDF).
• Bilal, A. (2001). "Introduction to Supersymmetry".
  arXiv:hep-th/0101055
  .
• Manuel, D. (1996). "An Introduction to Supersymmetry".
  arXiv:hep-ph/9611409
  .
• Cooper, F.; Khare, A.; Sukhatme, U. (1995). "Supersymmetry and quantum mechanics". Physics Reports (Submitted manuscript). 251 (5–6): 267–385.
  S2CID 119379742
  ..
• arXiv:hep-th/9612114
  .
• Martin, S. (2011). "A Supersymmetry Primer". Perspectives on Supersymmetry. Advanced Series on Directions in High Energy Physics. Vol. 18. pp. 1–98.
  S2CID 118973381
  .
• Tong, D. (2021), Supersymmetric Field Theory (PDF).

Monographs

• Baer, H., and Tata, X., Weak Scale Supersymmetry, Cambridge University Press, Cambridge, (1999).
  ISBN 978-0813341194
  .
• Binetruy, P., Supersymmetry: Theory, Experiment, and Cosmology, Oxford University Press, Oxford, (2012).
  ISBN 978-0199652730
  .
• Cecotti, S., Supersymmetric Field Theories: Geometric Structures and Dualities, Cambridge University Press, Cambridge, (2015).
  ISBN 978-1107053816
  .
• Drees, M., Godbole, R., and Roy, P., Theory & Phenomenology of Sparticles, World Scientific, Singapore (2005).
  ISBN 9-810-23739-1
  .
• Dreiner, H.K.,
  ISBN 978-0521800884
  .
• Duplij, S., Siegel, W., and Bagger, J.,Concise Encyclopedia of Supersymmetry, Springer, (2003).
  ISBN 978-1-4020-1338-6
  .
• Freud, P.G.O., Introduction to Supersymmetry, Cambridge University Press, Cambridge, (1988).
  ISBN 978-0521356756
  .
• Junker, G., Supersymmetric Methods in Quantum and Statistical Physics, Springer, (2011).
  ISBN 978-3-540-61591-0
  .
• ISBN 0-7382-0489-7
  .
• Kane, G.L., and Shifman, M., eds. The Supersymmetric World: The Beginnings of the Theory, World Scientific, Singapore (2000).
  ISBN 981-02-4522-X
  .
• ISBN 978-981-4293-41-9
  .
• Nath, P., Supersymmetry, Supergravity and Unification, Cambridge University Press, Cambridge, (2016).
  ISBN 0-521-19702-3
  .
• Raby, S., Supersymmetric Grand Unified Theories, Springer, (2017).
  ISBN 978-3319552538
  .
• Tachikawa, Y., N=2 Supersymmetric Dynamics for Pedestrians, Springer, (2014).
  ISBN 978-3319088211
  .
• Terning, J., Modern Supersymmetry: Dynamics and Duality, Oxford University Press, Oxford, (2009).
  ISBN 978-0199559510
  .
• Wegner, F., Supermathematics and its Applications in Statistical Physics, Springer, (2016).
  ISBN 978-3662491683
  .
• ISBN 0-521-66000-9
  .
• ISBN 0-691-02530-4
  .

On experiments

• Bennett GW, et al. (Muon (g−2) Collaboration) (2004). "Measurement of the negative muon anomalous magnetic moment to 0.7 ppm". Physical Review Letters. 92 (16): 161802.
  S2CID 3183567
  .
• Brookhaven National Laboratory (Jan 8, 2004). New g−2 measurement deviates further from Standard Model. Press Release.
• Fermi National Accelerator Laboratory (Sept 25, 2006). Fermilab's CDF scientists have discovered the quick-change behavior of the B-sub-s meson. Press Release.

External links

Wikiquote has quotations related to Supersymmetry.

• Supersymmetry – European Organization for Nuclear Research (CERN)
• The status of supersymmetry – Symmetry Magazine (Fermilab/SLAC), January 12, 2021
• As Supersymmetry Fails Tests, Physicists Seek New Ideas – Quanta Magazine, November 20, 2012
• What is Supersymmetry? – Fermilab, May 21, 2013
• Why Supersymmetry? – Fermilab, May 31, 2013
• The Standard Model and Supersymmetry – World Science Festival, March 4, 2015
• SUSY running out of hiding places – BBC, December 11, 2012