In particle physics, supersymmetry (SUSY)
is a principle that proposes a relationship
between two basic classes of elementary particles:
bosons, which have an integer-valued spin,
and fermions, which have a half-integer spin.
A type of spacetime symmetry, supersymmetry
is a possible candidate for undiscovered particle
physics, and seen as an elegant solution to
many current problems in particle physics
if confirmed correct, which could resolve
various areas where current theories are believed
to be incomplete. A supersymmetrical extension
to the Standard Model would resolve major
hierarchy problems within gauge theory, by
guaranteeing that quadratic divergences of
all orders will cancel out in perturbation
theory.
In supersymmetry, each particle from one group
would have an associated particle in the other,
which is known as its superpartner, the spin
of which differs by a half-integer. These
superpartners would be new and undiscovered
particles. For example, there would be a particle
called a "selectron" (superpartner electron),
a bosonic partner of the electron. In the
simplest supersymmetry theories, with perfectly
"unbroken" supersymmetry, each pair of superpartners
would share the same mass and internal quantum
numbers besides spin. Since we expect to find
these "superpartners" using present-day equipment,
if supersymmetry exists then it consists of
a spontaneously broken symmetry allowing superpartners
to differ in mass. Spontaneously-broken supersymmetry
could solve many mysterious problems in particle
physics including the hierarchy problem.
There is no evidence at this time to show
whether or not supersymmetry is correct, or
what other extensions to current models might
be more accurate. In part this is because
it is only since around 2010 that particle
accelerators specifically designed to study
physics beyond the Standard Model have become
operational, and because it is not yet known
where exactly to look nor the energies required
for a successful search.
The main reasons for supersymmetry being supported
by physicists is that the current theories
are known to be incomplete and their limitations
are well established, and supersymmetry would
be an attractive solution to some of the major
concerns. Direct confirmation would entail
production of superpartners in collider experiments,
such as the Large Hadron Collider (LHC). The
first runs of the LHC found 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 supersymmetry. 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 10-28 e·cm, equivalent to a sensitivity
to new physics at the TeV scale and matching
that of the current best particle colliders.
A permanent EDM in any fundamental particle
points towards 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.
These findings disappointed many physicists,
who believed that supersymmetry (and other
theories relying upon it) were by far the
most promising theories for "new" physics,
and had hoped for signs of unexpected results
from these runs. 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. However it has also been argued that
this "naturalness" crisis was premature, because
various calculations were too optimistic about
the limits of masses which would allow a supersymmetry
based solution.
== Motivations ==
There are numerous phenomenological motivations
for supersymmetry close to the electroweak
scale, as well as technical motivations for
supersymmetry at any scale.
=== The hierarchy problem ===
Supersymmetry close to the electroweak scale
ameliorates the hierarchy problem that afflicts
the Standard Model. In the Standard Model,
the electroweak scale receives enormous Planck-scale
quantum corrections. The observed hierarchy
between the electroweak scale and the Planck
scale must be achieved with extraordinary
fine tuning. In a supersymmetric theory, on
the other hand, 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 is achieved in
a natural manner, without miraculous fine-tuning.
=== Gauge coupling unification ===
The idea that the gauge symmetry groups unify
at high-energy is called Grand unification
theory. In the Standard Model, however, the
weak, strong and electromagnetic couplings
fail to unify at high energy. In a supersymmetry
theory, the running of the gauge couplings
are modified, and precise high-energy unification
of the gauge couplings is achieved. The modified
running also provides a natural mechanism
for radiative electroweak symmetry breaking.
=== Dark matter ===
TeV-scale supersymmetry (augmented with a
discrete symmetry) typically provides a candidate
dark matter particle at a mass scale consistent
with thermal relic abundance calculations.
=== Other technical motivations ===
Supersymmetry is also 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. It is
also a necessary feature of the most popular
candidate for a theory of everything, superstring
theory, and a SUSY theory could explain the
issue of cosmological inflation.
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 like the Standard Model with
very general assumptions. The Haag–Łopuszański–Sohnius
theorem demonstrates that supersymmetry is
the only way spacetime and internal symmetries
can be combined consistently.
== History ==
=== 
Origination ===
The father of Supersymmetry is Kurt Gödel,
he theorized two things that support this
fact. First, he theorized that the universe
is not expanding but rotating. The centrifugal
force arising from the rotation was what kept
everything from collapsing under the force
of gravity. What makes this universe weird
is the way it mixes up space and time. Second,
that if the universe is truly suppersymmetric,
then by completing a sufficiently long round
trip in a sufficiently advanced spaceship
a resident of Gödel's theoretical Supersymmetric
universe could travel to any point in time.
Albert Einstein followed Godel's theory up
by stating: "This separation between past,
present, and future is only an illusion, if
a stubborn one." Jim Holt (2018). When Einstein
Walked with Gödel: Excursions to the Edge
of Thought.
=== Quantum Field Theory ===
J. L. Gervais and B. Sakita (in 1971), Yu.
A. Golfand and E. P. Likhtman (also in 1971),
and D. V. Volkov and V. P. Akulov (1972),
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.
=== Supersymmetric Field Theory ===
Julius Wess and Bruno Zumino (in 1974) 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 superalgebras) has
subsequently been applied successfully to
other topics of physics, ranging from nuclear
physics, critical phenomena, quantum mechanics
to statistical physics. It remains a vital
part of many proposed theories of 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.
== 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-Lopuszanski-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 tensor representations
of the Poincaré group and internal symmetries.
Supersymmetries, however, are generated by
objects that transform by the spin representations.
According to the spin-statistics theorem,
bosonic fields commute while fermionic fields
anticommute. Combining the two kinds of fields
into a single algebra requires the introduction
of a Z2-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 spinors, has
the following anti-commutation relation:
{
Q
α
,
Q
¯
β
˙
}
=
2
(
σ
μ
)
α
β
˙
P
μ
{\displaystyle \{Q_{\alpha },{\bar {Q}}_{\dot
{\beta }}\}=2(\sigma {}^{\mu })_{\alpha {\dot
{\beta }}}P_{\mu }}
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
{\displaystyle P_{\mu }=-i\partial {}_{\mu
}}
are the generators of translation and
σ
μ
{\displaystyle \sigma {}^{\mu }}
are the Pauli matrices.
There are 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.
=== The 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.
One of the main motivations for SUSY comes
from the quadratically divergent contributions
to the Higgs mass squared. 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. This problem is known as the hierarchy
problem. Supersymmetry reduces the size of
the quantum corrections by having automatic
cancellations between fermionic and bosonic
Higgs interactions. If supersymmetry is restored
at the weak scale, then the Higgs mass is
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.
In many supersymmetric Standard Models there
is a heavy stable particle (such as 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.
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.
==== Gauge-coupling unification ====
One piece of evidence for supersymmetry existing
is gauge coupling unification.
The renormalization group evolution of the
three gauge coupling constants 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. With the addition
of minimal SUSY joint convergence of the coupling
constants is projected at approximately 1016
GeV.
=== 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 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.
=== 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' don't 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 the book
=== 
Supersymmetry in optics ===
Integrated optics was recently found 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 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
=== 
Supersymmetry in dynamical systems ===
All stochastic (partial) differential equations,
the models for all types of continuous time
dynamical systems, possess topological supersymmetry.
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 induced
on differential forms by SDE-defined diffeomorphisms
of the phase space. The topological sector
of the so-emerging supersymmetric theory of
stochastic dynamics can be recognized as the
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,
turbulence, self-organized criticality etc.
The Goldstone theorem explains the associated
emergence of the long-range dynamical behavior
that manifests itself as 1/f noise, butterfly
effect, and the scale-free statistics of sudden
(instantonic) processes, e.g., earthquakes,
neuroavalanches, solar flares etc., known
as the Zipf's law and the 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 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 that interchanges
particles and monopoles.
The proof of the Atiyah-Singer index theorem
is much simplified by the use of supersymmetric
quantum mechanics.
=== Supersymmetry in quantum gravity ===
Supersymmetry is part of superstring theory,
a string theory of quantum gravity, although
it could in theory be a component of other
quantum gravity theories as well, such as
loop quantum gravity. For superstring theory
to be consistent, supersymmetry seems to be
required at some level (although it may be
a strongly broken symmetry). If experimental
evidence confirms supersymmetry in the form
of supersymmetric particles such as the neutralino
that is often believed to be the lightest
superpartner, some people believe this would
be a major boost to superstring theory. Since
supersymmetry is a required component of superstring
theory, any discovered supersymmetry would
be consistent with superstring theory. If
the Large Hadron Collider and other major
particle physics experiments fail to detect
supersymmetric partners, many versions of
superstring theory which had predicted certain
low mass superpartners to existing particles
may need to be significantly revised.
== 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, i.e. 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 supersymmetry
theory. Theories with 32 supersymmetries automatically
have a graviton.
For four dimensions there are the following
theories, with the corresponding multiplets
(CPT adds a copy, whenever they are not invariant
under such symmetry)
N = 1Chiral multiplet:
(0,​1⁄2)
Vector multiplet:
(​1⁄2,1)
Gravitino multiplet:
(1,​3⁄2)
Graviton multiplet:
(​3⁄2,2)
N = 2hypermultiplet:
(-​1⁄2,02,​1⁄2)
vector multiplet:
(0,​1⁄22,1)
supergravity multiplet:
(1,​3⁄22,2)
N = 4Vector multiplet:
(-1,-​1⁄24,06,​1⁄24,1)
Supergravity multiplet:
(0,​1⁄24,16,​3⁄24,2)
N = 8Supergravity multiplet:
(-2,-​3⁄28,-128,-​1⁄256,070,​1⁄256,128,​3⁄28,2)
=== 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
2d/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.
=== 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
{\displaystyle 1/2}
but can be an arbitrary
1
/
N
{\displaystyle 1/N}
for integer value of N. Such a generalization
is possible in two or less spacetime dimensions.
== Current status ==
Supersymmetric models are constrained by a
variety of experiments, including measurements
of low-energy observables – for example,
the anomalous magnetic moment of the muon
at Fermilab; the WMAP dark matter density
measurement and direct detection experiments
– for example, XENON-100 and 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 LHC.
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, which in
2006 were extended by the D0 experiment at
the Tevatron. From 2003-2015, WMAP's and Planck's
dark matter density measurements have strongly
constrained supersymmetry models, 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.The first run of the LHC found no
evidence for supersymmetry, and, as a result,
surpassed existing experimental limits from
the Large Electron–Positron Collider and
Tevatron and partially excluded the aforementioned
expected ranges.In 2011–12, the LHC discovered
a Higgs boson with a mass of about 125 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 (with the supersymmetry breaking
scale on the order of 1 TeV), should not exceed
135 GeV.The LHC result seemed problematic
for the minimal supersymmetric model, as the
value of 125 GeV is relatively large for the
model and can only be achieved with large
radiative loop corrections from top squarks,
which many theorists had considered to be
"unnatural" (see naturalness (physics) and
fine tuning).
== See also
