Definitions

# Supersymmetry

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In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to another particle that differs by half a unit of spin and are known as superpartners. In other words, in a supersymmetric theory, for every type of boson there exists a corresponding type of fermion, and vice-versa.

As of 2008 there is no direct evidence that supersymmetry is a symmetry of nature. Since superpartners of the particles of the Standard Model have not been observed, supersymmetry, if it exists, must be a broken symmetry allowing the 'sparticles' to be heavy.

If supersymmetry exists close to the TeV energy scale, it allows the solution of two major puzzles in particle physics. One is the hierarchy problem - on theoretical grounds there are huge expected corrections to the particles' masses, which without fine-tuning will make them much larger than they are in nature. Another problem is the unification of the weak interactions, the strong interactions and electromagnetism. Another advantage of supersymmetry is that supersymmetric quantum field theory can sometimes be solved. Supersymmetry is also a feature of most versions of string theory, though it can exist in nature even if string theory is wrong.

The Minimal Supersymmetric Standard Model is one of the best studied candidates for physics beyond the Standard Model.

## 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 no 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), 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 in physics are generated by objects that transform under the tensor representations of the Poincaré group and internal symmetries. Supersymmetries, on the other hand, are generated by objects that transform under the spinor representations. According to the spin-statistics theorem, bosonic fields commute while fermionic fields anticommute. In order to combine 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 contains two Weyl spinors with the following anti-commutation relation:

$\left\{Q_\left\{alpha\right\}, bar Q_\left\{dot\left\{beta\right\}\right\}\right\} = 2\left(\left\{sigma^mu\right\}\right)_\left\{alphadot\left\{beta\right\}\right\}P_mu$
and all other anti-commutation relations between the Qs and Ps vanish. In the above expression $P_mu=-ipartial_mu$ are the generators of translation and $sigma^mu$ 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 the 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).

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 highest 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 WIMPs (weakly interacting massive particles) dark matter candidate. The existence of a supersymmetric dark matter candidate is closely tied 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 by the particles of the MSSM. 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 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, soft SUSY breaking terms are added to the theory which break SUSY explicitly but could arise from a complete theory of supersymmetry breaking

#### Gauge Coupling Unification

One piece of evidence for supersymmetry existing at the weak scale is gauge coupling unification. The renormalization group evolution of the three gauge coupling constants of the Standard Model is sensitive to the 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 SUSY, the match is within the ability that theory is able to predict the values.

### Supersymmetric quantum mechanics

Supersymmetric quantum mechanics adds the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. Supersymmetric quantum mechanics often comes up when studying the dynamics of supersymmetric solitons and due to the simplified nature of having fields only functions of time (rather than space-time), a great deal of progress has been made in this subject and 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 called 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.

SUSY concepts have provided useful extensions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation.

See supersymmetric quantum mechanics for a more detailed discussion.

### 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.

## General Supersymmetry

Supersymmetry appears in many different contexts in theoretical physics that are closely related. 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 the field content and interactions are. 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 corresponds to an N=8 supersymmetry theory. Theories with 32 supersymmetries automatically have a graviton.

In four dimensions there are the following theories

• N=1 with Chiral, Vector, and Gravity multiplets
• N=2 with Hyper, Vector and Gravity multiplets
• N=4 with Vector and Gravity multiplets
• N=8 with only a Gravity multiplet

### Supersymmetry in Alternate Numbers of Dimensions

It is possible to have supersymmetry in alternate dimensions. Because the properties of spinors change drastically between different dimensions, each dimension has its characteristic. In $d$ dimensions, the size of spinors is roughly $2^\left\{d/2\right\}$ or $2^\left\{\left(d-1\right)/2\right\}$. Since the maximum number of supersymmetries is 32, the greatest number of dimensions in which a supersymmetric theory can exist is eleven.

## Supersymmetry as a quantum group

Supersymmetry can be reinterpreted in the language of noncommutative geometry and quantum groups. In particular, it involves a mild form of noncommutativity, namely supercommutativity. See the main article for more details.

## Supersymmetry in quantum gravity

Supersymmetry is part of a larger enterprise of theoretical physics to unify everything we know about the physical world into a single fundamental framework of physical laws, known as the quest for a Theory of Everything (TOE). A significant part of this larger enterprise is the quest for a theory of quantum gravity, which would unify the classical theory of general relativity and the Standard Model, which explains the other three basic forces in physics (electromagnetism, the strong interaction, and the weak interaction), and provides a palette of fundamental particles upon which all four forces act. Two of the most active approaches to forming a theory of quantum gravity are string theory and loop quantum gravity (LQG), although in theory, supersymmetry could be a component of other theoretical approaches as well.

For string theory to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In particle theory, supersymmetry is recognized as a way to stabilize the hierarchy between the unification scale and the electroweak scale (or the Higgs boson mass), and can also provide a natural dark matter candidate. String theory also requires extra spatial dimensions which have to be compactified as in Kaluza-Klein theory.

Loop quantum gravity (LQG), in its current formulation, predicts no additional spatial dimensions, nor anything else about particle physics. These theories can be formulated in three spatial dimensions and one dimension of time, although in some LQG theories dimensionality is an emergent property of the theory, rather than a fundamental assumption of the theory. Also, LQG is a theory of quantum gravity which does not require supersymmetry. Lee Smolin, one of the originators of LQG, has proposed that a loop quantum gravity theory incorporating either supersymmetry or extra dimensions, or both, be called "loop quantum gravity II".

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 string theory. Since supersymmetry is a required component of string theory, any discovered supersymmetry would be consistent with string theory. If the Large Hadron Collider and other major particle physics experiments fail to detect supersymmetric partners or evidence of extra dimensions, many versions of string theory which had predicted certain low mass superpartners to existing particles may need to be significantly revised. The failure of experiments to discover either supersymmetric partners or extra spatial dimensions, as of 2008, has encouraged loop quantum gravity researchers.

## History of supersymmetry

In the early 1970s Yu. A. Golfand and E.P. Likhtman in Moscow (in 1971), D.V. Volkov and V.P. Akulov in Kharkov (in 1972) and J. Wess and B. Zumino in USA (in 1974) independently discovered supersymmetry, a radically new type of symmetry of space-time and fundamental fields. It has allowed one to establish a relationship between elementary particles of different quantum nature, bosons and fermions, and to non-trivially unify space-time and internal symmetries of the microscopic World. Supersymmetry first arose in the context of an early version of string theory by Ramond, John H. Schwarz and Andre Neveu, but the mathematical structure of supersymmetry has subsequently been applied successfully to other areas of physics; firstly by Wess, Zumino, and Abdus Salam and their fellow researchers to particle physics, and later to a variety of fields, ranging from 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 1981 by Howard Georgi and Savas Dimopoulos and is called the Minimal Supersymmetric Standard Model or MSSM for short. It was proposed to solve the hierarchy problem and predicts superpartners with masses between 100 GeV and 1 TeV. As of 2008 there is no irrefutable experimental evidence that supersymmetry is a symmetry of nature. In 2008 the Large Hadron Collider at CERN is scheduled to produce the world's highest energy collisions and offers the best chance at discovering superparticles for the foreseeable future.

## References

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