View source for Supersymmetry

{{Short description|Symmetry between bosons and fermions}}
{{Redirect|SUSY||Susy (disambiguation)}}
{{For|the TV series episode|Supersymmetry (Angel)}}
'''Supersymmetry''' is a [[Theory|theoretical framework]] in [[physics]] that suggests the existence of a symmetry between [[Particle physics|particles]] with integer [[Spin (physics)|spin]] (''[[boson]]s'') and particles with half-integer spin (''[[fermion]]s''). It proposes that for every known particle, there exists a partner particle with different spin properties.<ref name="CERN Overview">{{Cite web|url=https://home.web.cern.ch/science/physics/supersymmetry|title=Supersymmetry|archive-url=https://web.archive.org/web/20230714210240/https://home.web.cern.ch/science/physics/supersymmetry|archive-date=2023-07-14|publisher=CERN|access-date=2023-09-11}}</ref> There have been multiple experiments on supersymmetry that have failed to provide evidence that it exists in [[nature]].<ref name="quantamagazine.org">{{cite journal |last=Wolchover |first=Natalie |title=What No New Particles Means for Physics |journal=Quanta Magazine |date=August 9, 2016 |url=https://www.quantamagazine.org/what-no-new-particles-means-for-physics-20160809}}</ref> 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 [[force]] and the equations for [[matter]] are identical. In [[theoretical physics|theoretical]] and [[mathematical physics]], any theory with this property has the ''principle of supersymmetry'' (SUSY). Dozens of supersymmetric theories exist.<ref>{{Citation |title=What is Supersymmetry? | date=21 May 2013 |url=https://www.youtube.com/watch?v=0CeLRrBAI60 |access-date=2023-09-30 |publisher=Fermilab |language=en}}</ref> In theory, supersymmetry is a type of [[Spacetime symmetries|spacetime symmetry]] between two basic classes of particles: [[boson]]s, which have an integer-valued [[Spin (physics)|spin]] and follow [[Bose–Einstein statistics]], and [[fermion]]s, which have a half-integer-valued spin and follow [[Fermi–Dirac statistics]].<ref>{{cite web|last1=Haber|first1=Howie|title=Supersymmetry, Part I (Theory)|url=http://pdg.lbl.gov/2014/reviews/rpp2014-rev-susy-1-theory.pdf | website=Reviews, Tables and Plots|publisher=Particle Data Group (PDG)|access-date=8 July 2015}}</ref> The names of bosonic partners of fermions are prefixed with ''s-'', because they are [[scalar particle]]s. For example, if the [[electron]] exists in a supersymmetric theory, then there would be a particle called a ''selectron'' (superpartner electron), a bosonic partner of the electron.<ref>{{Cite web |last=Beringer |first=J. |date=2012 |title=SEARCHES FOR MONOPOLES, SUPERSYMMETRY, TECHNICOLOR, COMPOSITENESS, EXTRA DIMENSIONS, etc. |url=https://pdg.lbl.gov/2012/tables/rpp2012-sum-searches.pdf |access-date=August 25, 2024 |website=pdg.lbl.gov}}</ref>

In supersymmetry, each particle from the class of fermions would have an associated particle in the class of bosons, and vice versa, known as a [[superpartner]]. The spin of a particle's superpartner is different by a half-integer. In the simplest supersymmetry theories, with perfectly "[[broken symmetry|unbroken]]" supersymmetry, each pair of superpartners would share the same [[mass]] and internal [[quantum number]]s besides spin. More complex supersymmetry theories have a [[spontaneously broken symmetry]], allowing superpartners to differ in mass.<ref>{{cite book |arxiv=hep-ph/9709356 |last1=Martin |first1=Stephen P. |title=Perspectives on Supersymmetry |year=1997 |doi=10.1142/9789812839657_0001 |volume=18 |pages=[https://archive.org/details/perspectivesonsu0000unse_y6n5/page/1 1–98] |series=Advanced Series on Directions in High Energy Physics |isbn=978-981-02-3553-6 |chapter=A Supersymmetry Primer |s2cid=118973381 |chapter-url=https://archive.org/details/perspectivesonsu0000unse_y6n5/page/1 }}</ref><ref>{{cite book |last1=Baer |first1=Howard |last2=Tata |first2=Xerxes |title=Weak scale supersymmetry: From superfields to scattering events |year=2006}}</ref><ref>{{cite book |last=Dine |first=Michael |title=Supersymmetry and String Theory: Beyond the Standard Model |url=https://archive.org/details/supersymmetrystr00dine |url-access=limited |page=[https://archive.org/details/supersymmetrystr00dine/page/n190 169] |year=2007|publisher=Cambridge University Press |isbn=9780521858410 }}</ref>

Supersymmetry has various applications to different areas of physics, such as [[quantum mechanics]], [[statistical mechanics]], [[quantum field theory]], [[condensed matter physics]], [[nuclear physics]], [[optics]], [[stochastic dynamics]], [[astrophysics]], [[quantum gravity]], and [[cosmology]]. Supersymmetry has also been applied to [[high energy physics]], where a supersymmetric extension of the [[Standard Model]] is a possible candidate for [[physics beyond the Standard Model]]. However, no supersymmetric extensions of the Standard Model have been experimentally verified, and some physicists are saying the theory is dead.<ref name="Wolchover">{{cite journal |last=Wolchover |first=Natalie |title=Supersymmetry Fails Test, Forcing Physics to Seek New Ideas |journal=Quanta Magazine |date=November 20, 2012 |url=https://www.quantamagazine.org/20121120-as-supersymmetry-fails-tests-physicists-seek-new-ideas/}}</ref><ref name="quantamagazine.org" />

== History ==
A supersymmetry relating [[mesons]] and [[baryon]]s was first proposed, in the context of hadronic physics, by [[Hironari Miyazawa]] in 1966. This supersymmetry did not involve spacetime, that is, it concerned internal symmetry, and was broken badly. Miyazawa's work was largely ignored at the time.<ref>{{Cite journal |author=Miyazawa |first=H. |year=1966 |title=Baryon Number Changing Currents |journal=Prog. Theor. Phys. |volume=36 |issue=6 |pages=1266–1276 |bibcode=1966PThPh..36.1266M |doi=10.1143/PTP.36.1266 |doi-access=free}}</ref><ref>{{Cite journal |author=Miyazawa |first=H. |year=1968 |title=Spinor Currents and Symmetries of Baryons and Mesons |journal=Physical Review |volume=170 |issue=5 |pages=1586–1590 |bibcode=1968PhRv..170.1586M |doi=10.1103/PhysRev.170.1586}}</ref><ref>{{cite book | last=Kaku |first=Michio |author-link=Michio Kaku |title=Quantum Field Theory |isbn=0-19-509158-2 |page=663|year=1993 |publisher=Oxford University Press }}</ref><ref>{{cite book | last=Freund |first=Peter |author-link=Peter Freund |title=Introduction to Supersymmetry |isbn=0-521-35675-X |pages=26–27, 138|date=1988-03-31 |publisher=Cambridge University Press }}</ref>

[[Jean-Loup Gervais|J. L. Gervais]] and [[Bunji Sakita|B. Sakita]] (in 1971),<ref>{{Cite journal | last1 = Gervais | first1 = J.-L. | last2 = Sakita | first2 = B. | doi = 10.1016/0550-3213(71)90351-8 | title = Field theory interpretation of supergauges in dual models | journal = Nuclear Physics B | volume = 34 | issue = 2 | pages = 632–639 | year = 1971 |bibcode = 1971NuPhB..34..632G }}</ref> [[Yuri Golfand|Yu. A. Golfand]] and E. P. Likhtman (also in 1971), and D. V. Volkov and V. P. Akulov (1972),<ref>{{cite book |chapter-url=https://www.researchgate.net/publication/226173648 |doi=10.1007/BFb0105270 |chapter=Possible universal neutrino interaction |title=Supersymmetry and Quantum Field Theory |series=Lecture Notes in Physics |date=1998 |last1=Volkov |first1=D. V. |last2=Akulov |first2=V. P. |volume=509 |pages=383–385 |isbn=978-3-540-64623-5 }}</ref><ref>{{cite journal | doi=10.1016/0370-2693(73)90490-5 | title=Is the neutrino a goldstone particle? | date=1973 | last1=Volkov | first1=D.V. | last2=Akulov | first2=V.P. | journal=Physics Letters B | volume=46 | issue=1 | pages=109–110 | bibcode=1973PhLB...46..109V }}</ref><ref>{{cite journal | doi=10.1007/BF01036922 | title=Goldstone fields with spin 1/2 | date=1974 | last1=Akulov | first1=V. P. | last2=Volkov | first2=D. V. | journal=Theoretical and Mathematical Physics | volume=18 | issue=1 | pages=28–35 | bibcode=1974TMP....18...28A | s2cid=123352238 }}</ref> independently rediscovered supersymmetry in the context of [[quantum field theory]], a radically new type of symmetry of spacetime and fundamental fields, which establishes a relationship between elementary particles of different quantum nature, bosons and fermions, and unifies spacetime and internal symmetries of microscopic phenomena. Supersymmetry with a consistent Lie-algebraic graded structure on which the Gervais−Sakita rediscovery was based directly first arose in 1971 in the context of an early version of [[string theory]] by [[Pierre Ramond]], [[John H. Schwarz]] and [[André Neveu]].<ref>{{Cite journal | last1 = Ramond | first1 = P. | title = Dual Theory for Free Fermions | doi = 10.1103/PhysRevD.3.2415 | journal = Physical Review D | volume = 3 | issue = 10 | pages = 2415–2418 | year = 1971 |bibcode = 1971PhRvD...3.2415R }}</ref><ref name="NS">{{cite journal |last1=Neveu |first1=A. |last2=Schwarz |first2=J. H. |date=1971 |title=Factorizable dual model of pions |journal=Nuclear Physics B |volume=31 |issue=1 |pages=86–112 |bibcode=1971NuPhB..31...86N |doi=10.1016/0550-3213(71)90448-2 |authorlink1=André Neveu |authorlink2=John Henry Schwarz}}</ref>

In 1974, [[Julius Wess]] and [[Bruno Zumino]]<ref>{{Cite journal | last1 = Wess | first1 = J. | last2 = Zumino | first2 = B. | doi = 10.1016/0550-3213(74)90355-1 | title = Supergauge transformations in four dimensions | journal = Nuclear Physics B | volume = 70 | issue = 1 | pages = 39–50 | year = 1974 |bibcode = 1974NuPhB..70...39W | url = https://cds.cern.ch/record/201649 | type = Submitted manuscript }}</ref> identified the characteristic renormalization features of four-dimensional supersymmetric field theories, which identified them as remarkable QFTs, and they and [[Abdus Salam]] and their fellow researchers introduced early particle physics applications. The mathematical structure of supersymmetry ([[graded Lie superalgebra]]s) has subsequently been applied successfully to other topics of physics, ranging from [[nuclear physics]],<ref>{{Cite web |title=Hagen Kleinert |url=http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=supersym |access-date=2023-09-30 |website=users.physik.fu-berlin.de |language=en}}</ref><ref>{{Cite journal | last1 = Iachello | first1 = F. | title = Dynamical Supersymmetries in Nuclei | doi = 10.1103/PhysRevLett.44.772 | journal = Physical Review Letters | volume = 44 | issue = 12 | pages = 772–775 | year = 1980 |bibcode = 1980PhRvL..44..772I | s2cid = 14130911 }}</ref> [[critical phenomena]],<ref>{{Cite journal | last1 = Friedan | first1 = D. | last2 = Qiu | first2 = Z. | last3 = Shenker | first3 = S. | doi = 10.1103/PhysRevLett.52.1575 | title = Conformal Invariance, Unitarity, and Critical Exponents in Two Dimensions | journal = Physical Review Letters | volume = 52 | issue = 18 | pages = 1575–1578 | year = 1984 |bibcode = 1984PhRvL..52.1575F | s2cid = 122320349 }}</ref> [[quantum mechanics]] to [[statistical mechanics|statistical physics]], and supersymmetry remains a vital part of many proposed theories in many branches of physics.

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.<ref>{{Citation |last=Shifman |first=M. |title=Introduction to the second edition of ''The Supersymmetric World'' |date=2024-01-19 |arxiv=2401.11027}}</ref><ref>{{cite journal |last1=Salam |first1=A. |last2=Strathdee |first2=J. A. |date=1974 |title=Supersymmetry and Nonabelian Gauges |url= |journal=Physics Letters B |volume=51 |issue=4 |pages=353–355 |bibcode=1974PhLB...51..353S |doi=10.1016/0370-2693(74)90226-3 |authorlink1=Abdus Salam}}</ref> The term supergauge was in turn coined by Neveu and Schwarz in 1971 when they devised supersymmetry in the context of string theory.<ref name="NS"/><ref>{{cite book|last1=Shifman|first1=M.|author-link1=Mikhail Shifman|last2=Kane|first2=G.|author-link2=Gordon L. Kane|date=2000|title=The Supersymmetric World:The Beginnings of the Theory|url=|doi=|location=|publisher=World Scientific Publishing|chapter=|page=19|isbn=978-9810245221}}</ref>

== 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 space|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 charge]]s. This extended super-Poincaré algebra paved the way for obtaining a very large and important class of supersymmetric field theories.

==== The supersymmetry algebra ====
{{Main|Supersymmetry algebra}}
Traditional symmetries of physics are generated by objects that transform by the [[tensor]] [[representations of Lie groups|representations]] of the [[Poincaré group]] and internal symmetries. Supersymmetries, however, are generated by objects that transform by the [[spin representation]]s. According to the [[spin-statistics theorem]], bosonic fields [[commutative operation|commute]] while fermionic fields [[anticommutativity|anticommute]]. Combining the two kinds of fields into a single [[Lie algebra|algebra]] requires the introduction of a [[graded algebra|'''Z'''<sub>2</sub>-grading]] under which the bosons are the even elements and the fermions are the odd elements. Such an algebra is called a [[Lie superalgebra]].

The simplest supersymmetric extension of the [[Poincaré algebra]] is the [[Super-Poincaré algebra]]. Expressed in terms of two [[Weyl spinor]]s, has the following [[commutator|anti-commutation]] relation:
:<math>\{ Q_{ \alpha }, \bar {Q}_{ \dot{ \beta }} \} = 2( \sigma^{\mu} )_{ \alpha \dot{ \beta }} P_{\mu} </math>
and all other anti-commutation relations between the ''Q''s and commutation relations between the ''Q''s and ''P''s vanish. In the above expression {{nowrap|''P<sub>μ</sub>'' {{=}} −''i'' ∂<sub>''μ''</sub>}} are the generators of translation and ''σ<sup>μ</sup>'' are the [[Pauli matrices]].

There are [[representation of a Lie superalgebra|representations of a Lie superalgebra]] that are analogous to representations of a Lie algebra. Each Lie algebra has an associated [[Lie group]] and a Lie superalgebra can sometimes be extended into representations of a [[Lie supergroup]].

=== Supersymmetric quantum mechanics ===
{{Main|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 [[solitons]], and due to the simplified nature of having fields which are only functions of time (rather than space-time), a great deal of progress has been made in this subject and it is now studied in its own right.

SUSY quantum mechanics involves pairs of [[Hamiltonian (quantum mechanics)|Hamiltonians]] which share a particular mathematical relationship, which are called ''partner Hamiltonians''. (The [[potential energy]] terms which occur in the Hamiltonians are then known as ''partner potentials''.) An introductory theorem shows that for every [[eigenstate]] of one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy. This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory's fermions. Each boson would have a fermionic partner of equal energy.

==== In finance ====

In 2021, supersymmetric quantum mechanics was applied to [[option pricing]] and the analysis of [[market (economics)|markets]] in [[finance]],<ref>{{cite SSRN|title=Non-Equilibrium Skewness, Market Crises, and Option Pricing: Non-Linear Langevin Model of Markets with Supersymmetry|last=Halperin|first=Igor|date=14 January 2021|ssrn=3724000}}</ref> and to [[financial networks]].{{Dubious|date=March 2024}}<ref>{{cite journal|title=The physics of financial networks|journal=Nature Reviews Physics|date=10 June 2021|doi=10.1038/s42254-021-00322-5|url=https://www.nature.com/articles/s42254-021-00322-5|first1=Marco|last1=Bardoscia|first2=Paolo|last2=Barucca|first3=Stefano|last3=Battiston|first4=Fabio|last4=Caccioli|first5=Giulio|last5=Cimini|first6=Diego|last6=Garlaschelli|first7=Fabio|last7=Saracco|first8=Tiziano|last8=Squartini|first9=Guido|last9=Caldarelli|volume=3|issue=7|pages=490–507|arxiv=2103.05623|bibcode=2021NatRP...3..490B|s2cid=232168335}}</ref>

=== Supersymmetry in quantum field theory ===
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 [[Symmetry (physics)|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.<ref>{{cite journal|doi=10.1016/0550-3213(75)90279-5|title=All possible generators of supersymmetries of the S-matrix|journal=Nuclear Physics B|volume=88|issue=2|pages=257–274|year=1975|last1=Haag|first1=Rudolf|last2=Łopuszański|first2=Jan T.|last3=Sohnius|first3=Martin|bibcode=1975NuPhB..88..257H}}</ref>

While supersymmetry has not been discovered at [[Particle physics|high energy]], see Section [[#Supersymmetry_in_particle_physics|Supersymmetry in particle physics]], supersymmetry was found to be effectively realized at the intermediate energy of [[hadron|hadronic physics]] where [[baryon]]s and [[meson]]s 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.<ref>
{{cite journal
 |last1=Dosch |first1=H. G.
 |last2=de Téramond |first2=G. F. 
 |last3=Brodsky |first3=S. J.
 |title=Supersymmetry Across the Light and Heavy-Light Hadronic Spectrum
 |journal=Phys. Rev. D
 |volume=92 |issue=74010
 |year=2015
 |page=074010
 |arxiv=1504.05112
  |doi=10.1103/PhysRevD.92.074010
 |bibcode=2015PhRvD..92g4010D
 |s2cid=118554130
 }}</ref><ref>
{{cite journal |last1=Brodsky |first1=S. J. |last2=de Téramond |first2=G. F. 
|last3=Dosch |first3=H. G. 
|year=2023 |pages=132–143
|title=In 50 Years of quantum chromodynamics 
|journal=European Physical Journal C
|volume=83 |issue=1125 |arxiv=2212.11107  |bibcode= 2023EPJC...83.1125G
|doi=10.1140/epjc/s10052-023-11949-2 
|s2cid=254926381}}</ref> The realization of this effective supersymmetry is readily explained in [[diquark|quark–diquark models]]: Because two different [[color charge]]s 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 [[Supersymmetric WKB approximation|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).<ref>{{cite book|title=Supersymmetry in Disorder and Chaos |first=Konstantin |last=Efetov |publisher=Cambridge University Press |date=1997}}</ref>

In 2021, a group of researchers showed that, in theory, <math>N=(0,1)</math> SUSY could be realised at the edge of a Moore–Read [[quantum Hall effect|quantum Hall]] state.<ref name="ma_wang_yang">{{cite journal |author=Ma |first1=Ken K. W. |last2=Wang |first2=Ruojun |last3=Yang |first3=Kun |date=19 May 2021 |title=Realization of Supersymmetry and Its Spontaneous Breaking in Quantum Hall Edges |journal=Physical Review Letters |volume=126 |issue=20 |page=206801 |arxiv=2101.05448 |bibcode=2021PhRvL.126t6801M |doi=10.1103/PhysRevLett.126.206801 |pmid=34110185 |s2cid=231603192}}</ref> However, to date, no experiments have been done yet to realise it at an edge of a Moore–Read state. In 2022, a different group of researchers created a computer simulation of atoms in 1 dimensions that had supersymmetric [[topological quantum field theory|topological]] [[quasiparticles]].<ref name="minar_van_voorden_schoutens">{{cite journal |author=Minář |first1=Jiří |last2=van Voorden |first2=Bart |last3=Schoutens |first3=Kareljan |date=4 February 2022 |title=Kink Dynamics and Quantum Simulation of Supersymmetric Lattice Hamiltonians |journal=Physical Review Letters |volume=128 |issue=5 |page=050504 |arxiv=2005.00607 |bibcode=2022PhRvL.128e0504M |doi=10.1103/PhysRevLett.128.050504 |pmid=35179932 |s2cid=218486961}}</ref>

=== Supersymmetry in optics ===
In 2013, [[integrated optics]] was found<ref name="miri13prl">
{{cite journal
| last1         = Miri
| first1        = M.-A.
| last2        = Heinrich
| first2       = M.
| last3        = El-Ganainy
| first3       = R.
| last4        = Christodoulides
| first4       = D. N.
| year         = 2013<!-- Deny Citation Bot-->
| title        = Superymmetric optical structures
| journal      = Physical Review Letters
| volume       = 110
| issue        = 23
| pages        = 233902
| doi          = 10.1103/PhysRevLett.110.233902
| pmid        = 25167493
|arxiv = 1304.6646 |bibcode = 2013PhRvL.110w3902M | s2cid = 15354588
}}
</ref> to provide a fertile ground on which certain ramifications of SUSY can be explored in readily-accessible laboratory settings. Making use of the analogous mathematical structure of the quantum-mechanical [[Schrödinger equation]] and the [[wave equation]] governing the evolution of light in one-dimensional settings, one may interpret the [[refractive index]] distribution of a structure as a potential landscape in which optical wave packets propagate. In this manner, a new class of functional optical structures with possible applications in [[phase matching]], mode conversion<ref name="heinrich14">
{{cite journal
| last1         = Heinrich
| first1        = M.
| last2        = Miri
| first2       = M.-A.
| last3        = Stützer
| first3       = S.
| last4        = El-Ganainy
| first4       = R.
| last5        = Nolte
| first5       = S.
| last6        = Szameit
| first6       = A.
| last7        = Christodoulides
| first7       = D. N.
| date         = 2014
| title        = Superymmetric mode converters
| journal      = Nature Communications
| volume       = 5
| pages        = 3698
| doi          = 10.1038/ncomms4698
| pmid        = 24739256
|arxiv = 1401.5734 |bibcode = 2014NatCo...5.3698H | s2cid = 2070325
}}
</ref> and [[space-division multiplexing]] becomes possible. SUSY transformations have been also proposed as a way to address inverse scattering problems in optics and as a one-dimensional [[transformation optics]].<ref name="miri14">
{{cite journal
| last1         = Miri
| first1        = M.-A.
| last2        = Heinrich
| first2        = Matthias
| last3        = Christodoulides
| first3       = D. N.
| date         = 2014
| title        = SUSY-inspired one-dimensional transformation optics
| journal      = Optica
| volume       = 1
| issue        = 2
| pages        = 89–95
| doi          = 10.1364/OPTICA.1.000089
|arxiv = 1408.0832 |bibcode = 2014Optic...1...89M| s2cid = 15561466
}}
</ref>

=== Supersymmetry in dynamical systems ===
{{Main|Supersymmetric theory of stochastic dynamics}}

All stochastic (partial) differential equations, the models for all types of continuous time dynamical systems, possess topological supersymmetry.<ref>{{cite journal |author=Ovchinnikov, Igor |title=Introduction to Supersymmetric Theory of Stochastics |journal=Entropy |volume=18 |issue=4 |pages=108 |date=March 2016 |doi=10.3390/e18040108|arxiv=1511.03393 |bibcode=2016Entrp..18..108O |s2cid=2388285 |doi-access=free }}</ref><ref>{{cite journal |last1=Ovchinnikov|first1=Igor |last2=Ensslin|first2=Torsten |title=Kinematic dynamo, supersymmetry breaking, and chaos |journal=Physical Review D |volume=93 |issue=8 |pages=085023 |date=April 2016 |doi=10.1103/PhysRevD.93.085023|arxiv=1512.01651 |bibcode=2016PhRvD..93h5023O |s2cid=59367815 }}</ref> In the operator representation of stochastic evolution, the topological supersymmetry is the [[exterior derivative]] which is commutative with the stochastic evolution operator defined as the stochastically averaged [[Pullback (differential geometry)|pullback]] induced on [[differential forms]] by SDE-defined [[diffeomorphism]]s of the [[phase space]]. The topological sector of the so-emerging [[supersymmetric theory of stochastic dynamics]] can be recognized as the [[Topological quantum field theory|Witten-type topological field theory]].

The meaning of the topological supersymmetry in dynamical systems is the preservation of the phase space continuity—infinitely close points will remain close during continuous time evolution even in the presence of noise. When the topological supersymmetry is broken spontaneously, this property is violated in the limit of the infinitely long temporal evolution and the model can be said to exhibit (the stochastic generalization of) the [[butterfly effect]]. From a more general perspective, spontaneous breakdown of the topological supersymmetry is the theoretical essence of the ubiquitous dynamical phenomenon variously known as [[Chaos theory|chaos]], [[turbulence]], [[self-organized criticality]] etc. The [[Goldstone boson|Goldstone theorem]] explains the associated emergence of the long-range dynamical behavior that manifests itself as [[Pink noise|{{sfrac|1|''f''}} noise]], [[butterfly effect]], and the scale-free statistics of sudden (instantonic) processes, such as earthquakes, neuroavalanches, and solar flares, known as the [[Zipf's law]] and the [[Richter magnitude scale|Richter scale]].

=== Supersymmetry in mathematics ===
SUSY is also sometimes studied mathematically for its intrinsic properties. This is because it describes complex fields satisfying a property known as [[holomorphic function|holomorphy]], which allows holomorphic quantities to be exactly computed. This makes supersymmetric models useful "[[toy model]]s" of more realistic theories. A prime example of this has been the demonstration of S-duality in four-dimensional gauge theories<ref name="krasnitz02">
{{cite book
 | last =Krasnitz
 | first =Michael
 | title =Correlation functions in supersymmetric gauge theories from supergravity fluctuations
 | publisher = Princeton University Department of Physics
 | year = 2003
 | location = Princeton University Department of Physics
 | pages =91
 | url =http://inspirehep.net/record/623429/files/MKrasnitzthesis.pdf
 }}
</ref> that interchanges particles and [[Magnetic monopole|monopoles]].

The proof of the [[Atiyah-Singer index theorem#Heat equation|Atiyah–Singer index theorem]] is much simplified by the use of supersymmetric quantum mechanics.

=== Supersymmetry in string theory ===
{{Main|Superstring theory|String theory|String theory landscape}}

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 [[Tachyonic field#Tachyons in string theory|tachyons]] appear.<ref>{{citation|title=Modular invariance, finiteness, and misaligned supersymmetry: New constraints on the numbers of physical string states|last=Dienes|first=Keith R.|journal=Nuclear Physics B|doi=10.1016/0550-3213(94)90153-8|date=7 November 1994|pages=533–588|volume=429|issue=3|arxiv=hep-th/9402006|bibcode=1994NuPhB.429..533D|s2cid=14794874}}</ref><ref>{{citation|title=Towards a Non-Supersymmetric String Phenomenology|last1=Abel|first1=Steven|last2=Dienes|first2=Keith R.|last3=Mavroudi|first3=Einni|journal=Physical Review D|volume=91|date=15 June 2015|issue=12|page=126014|doi=10.1103/PhysRevD.91.126014|arxiv=1502.03087|bibcode=2015PhRvD..91l6014A|s2cid=118655927}}</ref> Any string theories without some kind of supersymmetry, such as [[bosonic string theory]] and the <math>E_7 \times E_7</math>, <math>SU(16)</math>, and <math>E_8</math> [[heterotic string theory|heterotic string theories]], will have a tachyon and therefore the [[spacetime]] [[vacuum state|vacuum]] itself would be unstable and would decay into some tachyon-free string theory usually in a lower spacetime dimension.<ref>{{citation|title=Stable Vacua for Tachyonic Strings|last=Kaidi|first=Justin|journal=Physical Review D|volume=103|date=15 May 2021|issue=10|page=106026|doi=10.1103/PhysRevD.103.106026|arxiv=2010.10521|bibcode=2021PhRvD.103j6026K|s2cid=224814212}}</ref> There is no experimental evidence that either supersymmetry or misaligned supersymmetry holds in our universe, and many physicists have moved on from supersymmetry and string theory entirely due to the non-detection of supersymmetry at the LHC.<ref name="Economist1">{{cite news|title=Physics seeks the future|url=https://www.economist.com/science-and-technology/physics-seeks-the-future/21803916|date=28 August 2021|access-date=31 August 2021|newspaper=[[The Economist]]}}</ref><ref name="Economist2">{{cite news|title=Fundamental physics is humanity's most extraordinary achievement|url=https://www.economist.com/leaders/2021/08/28/fundamental-physics-is-humanitys-most-extraordinary-achievement|date=28 August 2021|access-date=31 August 2021|newspaper=[[The Economist]]}}</ref>

Despite the null results for supersymmetry at the LHC so far, some [[particle physicists]] have nevertheless moved to string theory in order to resolve the [[Naturalness (physics)|naturalness crisis]] for certain supersymmetric extensions of the Standard Model.<ref>See below at section [[Supersymmetry#Supersymmetry in particle physics]] for details.</ref> According to the particle physicists, there exists a concept of "stringy naturalness" in [[string theory]],<ref name="baer4">{{cite journal |author=Baer |first1=H. |last2=Barger |first2=V. |last3=Salam |first3=S. |date=June 2019 |title=Naturalness versus stringy naturalness (with implications for collider and dark matter searches) |journal=Physical Review Research |volume=1 |issue=2 |pages=023001 |arxiv=1906.07741 |bibcode=2019PhRvR...1b3001B |doi=10.1103/PhysRevResearch.1.023001 |s2cid=195068902}}</ref> where the [[string theory landscape]] could have a power law statistical pull on soft SUSY breaking terms to large values (depending on the number of hidden sector SUSY breaking fields contributing to the soft terms).<ref name="douglas1">{{cite arXiv |eprint=hep-th/0405279 |first=Michael R. |last=Douglas |title=Statistical analysis of the supersymmetry breaking scale |date=May 2004}}</ref> If this is coupled with an anthropic requirement that contributions to the weak scale not exceed a factor between 2 and 5 from its measured value (as argued by Agrawal et al.),<ref name="agrawal1">{{cite journal |last1=Agrawal |first1=V. |last2=Barr |first2=S. |last3=Donoghue |first3=J. F. |last4=Seckel |first4=D. |date=January 1998 |title=Anthropic considerations in multiple domain theories and the scale of electroweak symmetry breaking |journal=Physical Review Letters |volume=80 |issue=9 |pages=1822–1825 |arxiv=hep-ph/9801253 |bibcode=1998PhRvL..80.1822A |doi=10.1103/PhysRevLett.80.1822 |s2cid=14397884}}</ref> then the Higgs mass is pulled up to the vicinity of 125&nbsp;GeV while most sparticles are pulled to values beyond the current reach of LHC.<ref name="baer2">{{cite journal |author=Baer |first1=H. |last2=Barger |first2=V. |last3=Serce |first3=H. |last4=Sinha |first4=K. |date=December 2017 |title=Higgs and superparticle mass predictions from the landscape |journal=Journal of High Energy Physics |volume=1803 |issue=3 |pages=002 |arxiv=1712.01399 |doi=10.1007/JHEP03(2018)002 |s2cid=113404486}}</ref> (The Higgs was determined to have a mass of 125 GeV ±0.15 GeV in 2022.) An exception occurs for [[higgsino]]s which gain mass not from SUSY breaking but rather from whatever mechanism solves the SUSY mu problem. Light higgsino pair production in association with hard initial state jet radiation leads to a soft opposite-sign dilepton plus jet plus missing transverse energy signal.<ref name="baer3">{{cite journal |author=Baer |first1=H. |last2=Mustafayev |first2=A. |last3=Tata |first3=X. |date=September 2014 |title=Monojet plus soft dilepton signal from light higgsino pair production at LHC14 |journal=Physical Review D |volume=90 |issue=11 |pages=115007 |arxiv=1409.7058 |bibcode=2014PhRvD..90k5007B |doi=10.1103/PhysRevD.90.115007 |s2cid=119194219}}</ref>

== Supersymmetry in particle physics ==
{{Beyond the Standard Model|expanded=Supersymmetry}}
In particle physics, a '''supersymmetric extension of the Standard Model''' is a possible candidate for [[physics beyond the Standard Model|undiscovered particle physics]], and seen by some physicists as an elegant solution to many current [[List of unsolved problems in physics|problems in particle physics]] if confirmed correct, which could resolve various areas where current theories are believed to be incomplete and where limitations of current theories are well established.<ref name=ATLAS2017Sep/><ref name=CMS2017Sep/> In particular, one supersymmetric extension of the [[Standard Model]], the [[Minimal Supersymmetric Standard Model]] (MSSM), became popular in theoretical particle physics, as the Minimal Supersymmetric Standard Model is the simplest supersymmetric extension of the Standard Model that could resolve major hierarchy problems within the Standard Model, by guaranteeing that quadratic [[The problem of infinities|divergences]] of all orders will [[cancel out]] in [[perturbation theory]]. If a supersymmetric extension of the Standard Model is correct, superpartners of the existing [[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 accelerator]]s 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.<ref name="status-of-supersymmetry">{{cite web|title=The status of supersymmetry|url=https://www.symmetrymagazine.org/article/the-status-of-supersymmetry|website=Symmetry Magazine|last=Hershberger|first=Scott|date=12 January 2021|access-date=29 June 2021}}</ref>

=== Supersymmetric extensions of the Standard Model ===
{{Main|Minimal Supersymmetric 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.
[[Image:Hqmc-vector.svg|thumb|upright=1.3|right|Cancellation of the [[Higgs boson]] quadratic [[mass renormalization]] between fermionic [[top quark]] loop and [[scalar field|scalar]] stop [[squark]] [[tadpole (physics)|tadpole]] [[Feynman diagram]]s in a supersymmetric extension of the Standard Model]]

One of the original motivations for the Minimal Supersymmetric Standard Model came from the [[hierarchy problem]]. Due to the quadratically divergent contributions to the Higgs mass squared in the Standard Model, the quantum mechanical interactions of the Higgs boson causes a large renormalization of the Higgs mass and unless there is an accidental cancellation, the natural size of the Higgs mass is the greatest scale possible. Furthermore, the electroweak scale receives enormous [[Planck mass|Planck-scale]] quantum corrections. The observed hierarchy between the electroweak scale and the Planck scale must be achieved with extraordinary [[Fine-tuning (physics)|fine tuning]]. This problem is known as the hierarchy problem.

Supersymmetry close to the [[electroweak scale]], such as in the Minimal Supersymmetric Standard Model, would solve the [[hierarchy problem]] that afflicts the Standard Model.<ref>{{cite thesis|type=PhD thesis|last1=David|first1=Curtin|title=Model Building And Collider Physics Above The Weak Scale |date=August 2011|url=http://www.lepp.cornell.edu/rsrc/Home/Research/GradTheses/Curtin_David.pdf|publisher=Cornell University}}</ref> 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 [[naturalness (physics)|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 [[grand unified theory|grand unification]], the idea that the gauge symmetry groups should unify at high-energy. In the Standard Model, however, the [[weak interactions|weak]], [[strong interactions|strong]] and [[electromagnetism|electromagnetic]] gauge couplings fail to unify at high energy. In particular, the [[renormalization group]] evolution of the three gauge [[coupling constant]]s of the Standard Model is somewhat sensitive to the present particle content of the theory. These coupling constants do not quite meet together at a common energy scale if we run the renormalization group using the Standard Model.<ref name="GKane">{{cite journal|first=Gordon L. |last=Kane |title=The Dawn of Physics Beyond the Standard Model |journal=[[Scientific American]] |volume=288 |issue=6 |pages=68–75 |date=June 2003 |bibcode=2003SciAm.288f..68K |doi=10.1038/scientificamerican0603-68 |doi-broken-date=1 November 2024 |pmid=12764939 }}</ref><ref>{{cite journal |title=The Frontiers of Physics |url=https://www.docme.su/doc/1764248/-scientific-american-special------volume-15--number-3-200... |journal=Scientific American |edition=Special |volume=15 |issue=3 |page=8 |date=2005 |access-date=2021-11-05 |archive-date=2021-11-05 |archive-url=https://web.archive.org/web/20211105143920/https://www.docme.su/doc/1764248/-scientific-american-special------volume-15--number-3-200... |url-status=dead }}</ref> After incorporating minimal SUSY at the electroweak scale, the running of the gauge couplings are modified, and joint convergence of the gauge coupling constants is projected to occur at approximately 10<sup>16</sup>&nbsp;[[GeV]].<ref name="GKane"/> The modified running also provides a natural mechanism for radiative [[electroweak symmetry breaking]].

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.<ref>{{cite web |first=Jonathan |last=Feng |url=http://theory.fnal.gov/jetp/talks/feng.pdf |title=Supersymmetric Dark Matter |publisher=University of California, Irvine |date=11 May 2007 |access-date=16 February 2013 |archive-date=11 May 2013 |archive-url=https://web.archive.org/web/20130511104145/http://theory.fnal.gov/jetp/talks/feng.pdf |url-status=dead }}</ref><ref>{{cite web|first=Torsten |last=Bringmann |url=http://www.desy.de/~troms/teaching/SoSe11/DM_slides_05.pdf |title=The WIMP "Miracle" |archive-url=https://web.archive.org/web/20130301033728/http://www.desy.de/~troms/teaching/SoSe11/DM_slides_05.pdf |archive-date=1 March 2013 |publisher=University of Hamburg}}</ref>

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 [[spontaneous symmetry breaking|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.

{{Supersymmetry particles}}

===Searches and constraints for supersymmetry===
SUSY extensions of the [[standard model]] are constrained by a variety of experiments, including measurements of low-energy observables – for example, the [[Anomalous magnetic dipole moment|anomalous magnetic moment of the muon]] at [[Fermilab]]; the [[WMAP]] dark matter density measurement and direct detection experiments – for example, [[XENON]]-100 and [[Large Underground Xenon experiment|LUX]]; and by particle collider experiments, including [[B-physics]], Higgs phenomenology and direct searches for superpartners (sparticles), at the [[Large Electron–Positron Collider]], [[Tevatron]] and the [[Large Hadron Collider|LHC]]. In fact, CERN publicly states that if a supersymmetric model of the Standard Model "is correct, supersymmetric particles should appear in collisions at the LHC."<ref>{{cite web|work=CERN: Supersymmetry |title=Supersymmetry predicts a partner particle for each particle in the Standard Model, to help explain why particles have mass
|url=https://home.cern/science/physics/supersymmetry |access-date=5 September 2019}}</ref>

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,<ref>{{cite book |author=((LEP2 SUSY Working Group)) |title=ALEPH, DELPHI, L3 and OPAL experiments, charginos, large m0 |id=LEPSUSYWG/01-03.1 |date=2004}}<!-- Please check  --></ref> which in 2006 were extended by the D0 experiment at the Tevatron.<ref>{{cite journal |author=((The D0-Collaboration)) |title=Search for associated production of charginos and neutralinos in the trilepton final state using 2.3 fb<sup>−1</sup> of data |year=2009 |arxiv=0901.0646 |bibcode = 2009PhLB..680...34D |doi = 10.1016/j.physletb.2009.08.011 |volume=680 |issue=1 |journal=Physics Letters B |pages=34–43|hdl=10211.3/195394 |s2cid=54016374 }}</ref><ref>{{cite journal |author=((The D0 Collaboration)) |title=Search for squarks and gluinos in events with jets and missing transverse energy using 2.1 fb<sup>−1</sup> of pp<sup>−</sup> collision data at ''s'' = 1.96 TeV |arxiv=0712.3805 |year=2008 |bibcode = 2008PhLB..660..449D |doi = 10.1016/j.physletb.2008.01.042 |volume=660 |issue=5 |journal=Physics Letters B |pages=449–457|s2cid = 18574837}}</ref> From 2003 to 2015, WMAP's and [[Planck (spacecraft)|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&nbsp;GeV range, though values as high as 2.5&nbsp;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&nbsp;GeV.<ref>{{Cite journal|display-authors=etal|vauthors=[[Oliver Buchmueller|Buchmueller O]] |date=2009 |title=Likelihood Functions for Supersymmetric Observables in Frequentist Analyses of the CMSSM and NUHM1 |journal=The European Physical Journal C |volume=64 |issue=3 |pages=391–415 |arxiv=0907.5568 |bibcode=2009EPJC...64..391B |doi=10.1140/epjc/s10052-009-1159-z |s2cid=9430917}}</ref>

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.<ref>{{cite journal |last1=Roszkowski|first1=Leszek|last2=Sessolo|first2=Enrico Maria|last3=Williams|first3=Andrew J.|title=What next for the CMSSM and the NUHM: improved prospects for superpartner and dark matter detection|journal=Journal of High Energy Physics|date=11 August 2014|volume=2014|issue=8|pages=67|doi=10.1007/JHEP08(2014)067|arxiv = 1405.4289 |bibcode = 2014JHEP...08..067R |s2cid=53526400}}</ref> In 2011–12, the LHC discovered a [[Higgs boson]] with a mass of about 125&nbsp;GeV, and with couplings to fermions and bosons which are consistent with the Standard Model. The MSSM predicts that the mass of the lightest Higgs boson should not be much higher than the mass of the [[Z boson]], and, in the absence of [[Fine-tuning (physics)|fine tuning]] (with the supersymmetry breaking scale on the order of 1&nbsp;TeV), should not exceed 135&nbsp;GeV.<ref>{{cite journal |author=Carena |first1=Marcela |last2=Haber |first2=Howard E. |year=2003 |title=Higgs Boson Theory and Phenomenology |journal=Progress in Particle and Nuclear Physics |volume=50 |issue=1 |pages=63–152 |arxiv=hep-ph/0208209 |bibcode=2003PrPNP..50...63C |doi=10.1016/S0146-6410(02)00177-1 |s2cid=5163410}}</ref> The LHC found [[Desert (particle physics)|no previously unknown particles]] other than the Higgs boson which was already suspected to exist as part of the Standard Model, and therefore no evidence for any supersymmetric extension of the Standard Model.<ref name=ATLAS2017Sep>{{cite web |url=https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/CombinedSummaryPlots/SUSY/ |title=ATLAS Supersymmetry Public Results |department=ATLAS Collaboration |publisher=CERN |access-date=2017-09-24 |df=dmy-all}}</ref><ref name=CMS2017Sep>{{cite web |url=https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSUS|title=CMS Supersymmetry Public Results |department=CMS |publisher=CERN |access-date=2017-09-24 |df=dmy-all}}</ref>

Indirect methods include the search for a permanent electric dipole moment (EDM) in the known Standard Model particles, which can arise when the Standard Model particle interacts with the supersymmetric particles. The current best constraint on the [[electron electric dipole moment]] put it to be smaller than 10<sup>−28</sup> e·cm, equivalent to a sensitivity to new physics at the TeV scale and matching that of the current best particle colliders.<ref>{{cite journal |doi=10.1126/science.1248213 |pmid=24356114 |arxiv=1310.7534 |bibcode=2014Sci...343..269B |title=Order of Magnitude Smaller Limit on the Electric Dipole Moment of the Electron |journal=Science |volume=343 |issue=6168 |pages=269–272 |year=2014 |vauthors=Baron J, Campbell WC, Demille D, Doyle JM, Gabrielse G, Gurevich YV, Hess PW, Hutzler NR, Kirilov E, Kozyryev I, O'Leary BR, Panda CD, Parsons MF, Petrik ES, Spaun B, Vutha AC, West AD, West AD|s2cid=564518 |display-authors=5}}</ref> A permanent EDM in any fundamental particle points towards [[T-symmetry|time-reversal]] violating physics, and therefore also [[CP-symmetry]] violation via the [[CPT theorem]]. Such EDM experiments are also much more scalable than conventional particle accelerators and offer a practical alternative to detecting physics beyond the standard model as accelerator experiments become increasingly costly and complicated to maintain. The current best limit for the electron's EDM has already reached a sensitivity to rule out so called 'naive' versions of supersymmetric extensions of the Standard Model.<ref>{{cite journal |author=The ACME Collaboration |date=October 2018 |title=Improved Limit on the Electric Dipole Moment of the Electron |journal=[[Nature (journal)|Nature]] |volume=562 |issue=7727 |pages=355–360 |doi=10.1038/s41586-018-0599-8|pmid=30333583 |bibcode=2018Natur.562..355A |s2cid=52985540 |url=https://authors.library.caltech.edu/89472/3/41586_2018_599_MOESM1_ESM.pdf }}</ref>

Research in the late 2010s and early 2020s from experimental data on the [[cosmological constant]], [[LIGO]] [[noise]], and [[pulsar timing]], suggests it's very unlikely that there are any new particles with masses much higher than those which can be found in the standard model or the LHC.<ref name="cosmological-bounds">{{cite journal |last1=Afshordi |first1=Niayesh |last2=Nelson |first2=Elliot |title=Cosmological bounds on TeV-scale physics and beyond |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.93.083505 |journal=Physical Review D |access-date=20 February 2023 |pages=083505 |doi=10.1103/PhysRevD.93.083505 |date=7 April 2016|volume=93 |issue=8 |arxiv=1504.00012 |bibcode=2016PhRvD..93h3505A |s2cid=119110506 }}</ref><ref name="ligo-noise">{{cite arXiv |last1=Afshordi |first1=Niayesh |title=On the origin of the LIGO "mystery" noise and the high energy particle physics desert |date=21 November 2019|class=gr-qc |eprint=1911.09384 }}</ref><ref name="pulsar-timing">{{cite arXiv |last1=Afshordi |first1=Niayesh |last2=Kim |first2=Hyungjin |last3=Nelson |first3=Elliot |title=Pulsar Timing Constraints on Physics Beyond the Standard Model |date=15 March 2017|class=hep-th |eprint=1703.05331 }}</ref> However, this research has also indicated that [[quantum gravity]] or [[perturbative]] quantum field theory will become strongly coupled before 1 PeV, leading to other new physics in the TeVs.<ref name="cosmological-bounds" />

===Current status===
The negative findings in the experiments disappointed many physicists, who believed that supersymmetric extensions of the Standard Model (and other theories relying upon it) were by far the most promising theories for "new" physics beyond the Standard Model, and had hoped for signs of unexpected results from the experiments.<ref name="Wolchover"/><ref name="quantamagazine.org"/> In particular, the LHC result seems problematic for the Minimal Supersymmetric Standard Model, as the value of 125&nbsp;GeV is relatively large for the model and can only be achieved with large radiative loop corrections from top [[squarks]], which many theorists consider to be "unnatural" (see naturalness and fine tuning).<ref>{{cite journal|title=Implications of a 125 GeV Higgs for the MSSM and Low-Scale SUSY Breaking|journal=Physical Review D|volume=85|issue=9|pages=095007|last1=Draper|first1=Patrick|date=December 2011|arxiv=1112.3068|bibcode = 2012PhRvD..85i5007D |doi = 10.1103/PhysRevD.85.095007 |last2=Meade|first2=Patrick|last3=Reece|first3=Matthew|last4=Shih|first4=David|s2cid=118577425}}</ref>

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]].<ref name="status-of-supersymmetry"/><ref name="WittenCERN">{{cite web|title=Witten reflects|url=https://cerncourier.com/a/witten-reflects/|website=[[CERN Courier]]|date=21 December 2021|access-date=23 December 2021}}</ref> Still others have moved to string theory as a result of the naturalness crisis.<ref>See the section [[Supersymmetry#Supersymmetry in string theory]] above for details.</ref><ref name="baer4"/><ref name="douglas1"/><ref name="baer2"/> 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.<ref>{{cite conference |first=M. |last=Shifman |title=Reflections and Impressionistic Portrait |conference=Frontiers Beyond the Standard Model |place=FTPI |date=31 October 2012|arxiv=1211.0004v1 }}</ref> 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.<ref name="naturalness_crisis">{{cite journal |last1=Baer |first1=Howard |last2=Barger |first2=Vernon |last3=Mickelson |first3=Dan |date=September 2013 |title=How conventional measures overestimate electroweak fine-tuning in supersymmetric theory |journal=Physical Review D |volume=88 |issue=9 |pages=095013 |arxiv=1309.2984 |bibcode=2013PhRvD..88i5013B |doi=10.1103/PhysRevD.88.095013 |s2cid=119288477}}</ref><ref name="baer1">{{cite journal |last1=Baer |first1=Howard |last2=Barger |first2=Vernon |last3=Huang |first3=Peisi |last4=Mickelson |first4=Dan |last5=Mustafayev |first5=Azar |last6=Tata |first6=Xerxes |date=December 2012 |title=Radiative natural supersymmetry: Reconciling electroweak fine-tuning and the Higgs boson mass |journal=Physical Review D |volume=87 |issue=11 |pages=115028 |arxiv=1212.2655 |bibcode=2013PhRvD..87k5028B |doi=10.1103/PhysRevD.87.115028 |s2cid=73588737}}</ref>

== 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 supersymmetry|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''&nbsp;=&nbsp;8{{clarify|date=February 2021}} supersymmetry theory. Theories with 32 supersymmetries automatically have a [[graviton]].

For four dimensions there are the following theories, with the corresponding multiplets<ref>Polchinski, J. ''String Theory. Vol. 2: Superstring theory and beyond'', Appendix B
</ref> (CPT adds a copy, whenever they are not invariant under such symmetry):
:{|class="wikitable"
!rowspan=4|''N'' = 1
|Chiral multiplet||style="border-right:none"|(0,||style="border-left:none" colspan=8|{{sfrac|1|2}})
|-
|Vector multiplet||style="border-right:none"|({{sfrac|1|2}},||style="border-left:none" colspan=8|1)
|-
|Gravitino multiplet||style="border-right:none"|(1,||style="border-left:none" colspan=8|{{sfrac|3|2}})
|-
|Graviton multiplet||style="border-right:none"|({{sfrac|3|2}},||style="border-left:none" colspan=8|2)
|-
!rowspan=3|''N'' = 2
|Hypermultiplet||style="border-right:none"|(−{{sfrac|1|2}},||style="border-left:none;border-right:none"|0{{sup|2}},||style="border-left:none" colspan=7|{{sfrac|1|2}})
|-
|Vector multiplet||style="border-right:none"|(0,||style="border-left:none;border-right:none"|{{sfrac|1|2}}&thinsp;{{sup|2}},||style="border-left:none" colspan=7|1)
|-
|Supergravity multiplet||style="border-right:none"|(1,||style="border-left:none;border-right:none"|{{sfrac|3|2}}&thinsp;{{sup|2}},||style="border-left:none" colspan=7|2)
|-
!rowspan=2|''N'' = 4
|Vector multiplet||style="border-right:none"|(−1,||style="border-left:none;border-right:none"|−{{sfrac|1|2}}&thinsp;{{sup|4}},||style="border-left:none;border-right:none"|0{{sup|6}},||style="border-left:none;border-right:none"|{{sfrac|1|2}}&thinsp;{{sup|4}},||style="border-left:none" colspan=5|1)
|-
|Supergravity multiplet||style="border-right:none"|(0,||style="border-left:none;border-right:none"|{{sfrac|1|2}}&thinsp;{{sup|4}},||style="border-left:none;border-right:none"|1{{sup|6}},||style="border-left:none;border-right:none"|{{sfrac|3|2}}&thinsp;{{sup|4}},||style="border-left:none" colspan=5|2)
|-
!''N'' = 8
|Supergravity multiplet||style="border-right:none"|(−2,||style="border-left:none;border-right:none"|−{{sfrac|3|2}}&thinsp;{{sup|8}},||style="border-left:none;border-right:none"|−1{{sup|28}},||style="border-left:none;border-right:none"|−{{sfrac|1|2}}&thinsp;{{sup|56}},||style="border-left:none;border-right:none"|0{{sup|70}},||style="border-left:none;border-right:none"|{{sfrac|1|2}}&thinsp;{{sup|56}},||style="border-left:none;border-right:none"|1{{sup|28}},||style="border-left:none;border-right:none"|{{sfrac|3|2}}&thinsp;{{sup|8}},||style="border-left:none"|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<sup>''d''/2</sup> or 2<sup>(''d''&nbsp;−&nbsp;1)/2</sup>. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.{{citation needed|date=October 2018}}

===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 {{sfrac|1|2}} but can be an arbitrary {{sfrac|1|''N''}} for integer value of ''N''. Such a generalization is possible in two or fewer [[spacetime]] dimensions.

== See also ==
{{div col|colwidth=23em}}
* [[4D N = 1 global supersymmetry]]
* [[Anyon]]
* [[Next-to-Minimal Supersymmetric Standard Model]]
* [[Quantum group#Quantum groups and non-commutative geometry|Quantum group]]
* [[Split supersymmetry]]
* [[Supercharge]]
* [[Supermultiplet]]
* [[Supergeometry]]
* [[Supergravity]]
* [[Supergroup (physics)|Supergroup]]
* [[Superpartner]]
* [[Superspace]]
* [[Supersplit supersymmetry]]
* [[Supersymmetric gauge theory]]
* [[Supersymmetry nonrenormalization theorems]]
* [[Wess–Zumino model]]
{{div col end}}

== References ==
{{Reflist}}

== Further reading ==
{{Refbegin}}
* [http://www.stringwiki.org/wiki/Supersymmetry_and_Supergravity Supersymmetry and Supergravity] page in [http://www.stringwiki.org/wiki/String_Theory_Wiki String Theory Wiki] lists more books and reviews.

{{Refend}}

=== Theoretical introductions, free and online ===
{{Refbegin}}
* {{citation |url=http://homepages.uc.edu/~argyrepc/cu661-gr-SUSY/susy2001.pdf |title=An Introduction to Global Supersymmetry |last1=Arygres |first1=P. |year=2001}}.
* {{cite arXiv |eprint=hep-th/0101055 |title=Introduction to Supersymmetry |last1=Bilal |first1=A. |year=2001}}
* {{cite arXiv |eprint=hep-ph/9611409 |title=An Introduction to Supersymmetry |last1=Manuel |first1=D. |year=1996}}
* {{cite journal |doi=10.1016/0370-1573(94)00080-M |title=Supersymmetry and quantum mechanics |journal=Physics Reports |volume=251 |issue=5–6 |pages=267–385 |year=1995 |last1=Cooper |first1=F. |last2=Khare |first2=A. |last3=Sukhatme |first3=U. |arxiv=hep-th/9405029 |bibcode=1995PhR...251..267C |s2cid=119379742 |url=https://cds.cern.ch/record/262851 |type=Submitted manuscript}}.
* {{cite arXiv |eprint=hep-th/9612114 |title=Introduction to Supersymmetry |last1=Lykken |first1=J. D. |year=1996 |author-link=Joseph Lykken}}
* {{Cite book |arxiv=hep-ph/9709356 |last1=Martin |first1=S. |title=Perspectives on Supersymmetry |year=2011 |doi=10.1142/9789812839657_0001 |volume=18 |pages=[https://archive.org/details/perspectivesonsu0000unse_y6n5/page/1 1–98] |series=Advanced Series on Directions in High Energy Physics |isbn=978-981-02-3553-6 |chapter=A Supersymmetry Primer |s2cid=118973381 |chapter-url=https://archive.org/details/perspectivesonsu0000unse_y6n5/page/1}}
* {{citation |url=http://www.damtp.cam.ac.uk/user/tong/susy/susy.pdf |title=Supersymmetric Field Theory |last1=Tong |first1=D. |author-link1=David Tong (physicist) |year=2021}}.
{{refend}}

=== Monographs ===
{{Refbegin}}
* 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., [[Howard E. Haber|Haber]], H.E., Martin, S.P., ''From Spinors to Supersymmetry'', Cambridge University Press, Cambridge, (2023). {{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}}.
* [[Gordon L. Kane|Kane]], G.L., ''Supersymmetry: Unveiling the Ultimate Laws of Nature'', Basic Books, New York (2001). {{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}}.
* [[Harald J. W. Mueller-Kirsten|Müller-Kirsten]], H.J.W., and Wiedemann, A., ''Introduction to Supersymmetry'', 2nd ed., World Scientific, Singapore (2010). {{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}}.
* [[Steven Weinberg|Weinberg]], S., ''The Quantum Theory of Fields, Volume 3: Supersymmetry'', Cambridge University Press, Cambridge, (1999). {{ISBN|0-521-66000-9}}.
* [[Julius Wess|Wess]], J. and Bagger, J., ''Supersymmetry and Supergravity'', Princeton University Press, Princeton, (1992). {{ISBN|0-691-02530-4}}.
{{Refend}}

=== On experiments ===
{{Refbegin}}
* {{cite journal |collaboration=Muon (g−2) Collaboration | title=Measurement of the negative muon anomalous magnetic moment to 0.7 ppm | journal=Physical Review Letters | volume=92 | issue=16 | year=2004 | pages=161802 | pmid=15169217 | doi=10.1103/PhysRevLett.92.161802 | bibcode=2004PhRvL..92p1802B|arxiv = hep-ex/0401008 |vauthors=Bennett GW, Bousquet B, Brown HN, Bunce G, Carey RM, Cushman P, Danby GT, Debevec PT, Deile M, Deng H, Dhawan SK, Druzhinin VP, Duong L, ((Farley FJM)), Fedotovich GV, Gray FE, Grigoriev D, Grosse-Perdekamp M, Grossmann A, Hare MF, Hertzog DW, Huang X, Hughes VW, Iwasaki M, Jungmann K, Kawall D, Khazin BI, Krienen F, Kronkvist I, Lam A, Larsen R, Lee YY, Logashenko I, McNabb R, Meng W, Miller JP, Morse WM, Nikas D, ((Onderwater CJG)), Orlov Y, Özben CS, Paley JM, Peng Q, Polly CC, Pretz J, Prigl R, Qian T, Redin SI, Rind O, Roberts BL, Ryskulov N, Semertzidis YK, Shagin P, ((Shatunov YuM)), Sichtermann EP, Solodov E, Sossong M, Sulak LR, Trofimov A, von Walter P, Yamamoto A, zu Putlitz G | s2cid=3183567 | display-authors=1}}
* Brookhaven National Laboratory (Jan 8, 2004). ''[http://www.bnl.gov/bnlweb/pubaf/pr/2004/bnlpr010804.htm New g−2 measurement deviates further from Standard Model].'' Press Release.
* Fermi National Accelerator Laboratory (Sept 25, 2006). ''[http://www.fnal.gov/pub/presspass/press_releases/CDF_meson.html Fermilab's CDF scientists have discovered the quick-change behavior of the B-sub-s meson.]'' Press Release.
{{Refend}}

== External links ==
{{Wikiquote}}
* [https://home.cern/science/physics/supersymmetry Supersymmetry] – [[CERN|European Organization for Nuclear Research (CERN)]]
* [https://www.symmetrymagazine.org/article/the-status-of-supersymmetry The status of supersymmetry] – Symmetry Magazine (Fermilab/SLAC), January 12, 2021
* [https://www.quantamagazine.org/physicists-debate-future-of-supersymmetry-20121120/ As Supersymmetry Fails Tests, Physicists Seek New Ideas] – [[Quanta Magazine]], November 20, 2012
* [https://www.youtube.com/watch?v=0CeLRrBAI60 What is Supersymmetry?] – [[Fermilab]], May 21, 2013
* [https://www.youtube.com/watch?v=09VbAe9JZ8Y Why Supersymmetry?] – Fermilab, May 31, 2013
* [https://www.youtube.com/watch?v=zofMPB52Y3M The Standard Model and Supersymmetry] – [[World Science Festival]], March 4, 2015
* [https://www.bbc.co.uk/news/science-environment-20300100 SUSY running out of hiding places] – BBC, December 11, 2012

{{Industrial and applied mathematics}}
{{Particles}}
{{Supersymmetry topics |state=collapsed}}
{{String theory topics |state=collapsed}}
{{Standard model of physics}}
{{Authority control}}

[[Category:Supersymmetry| ]]
[[Category:Concepts in physics]]
[[Category:Concepts in the philosophy of science]]
[[Category:History of science]]
[[Category:Physics beyond the Standard Model]]
[[Category:Quantum field theory]]
[[Category:Theoretical physics]]