Supersymmetry

In particle physics, supersymmetry is a hypothetical symmetry that relates bosons and fermions. In supersymmetric theories, every fundamental fermion has a superpartner which is a boson and vice versa. Although supersymmetry has yet to be observed in the real world it remains a vital part of many proposed theories of physics, including various extensions to the Standard Model as well as modern superstring theories. The mathematical structure of supersymmetry, invented in a particle-physics context, has been applied with useful results in other areas, ranging from quantum mechanics to classical statistical physics. SUSY is the acronym preferred for whichever grammatical variation of supersymmetry occurs in a sentence. Experimentalists have not yet found any superpartners for known particles, possibly because they are too massive to be created in our current particle accelerators. Hopefully, by the year 2007 the Large Hadron Collider at CERN should be ready for use, producing collisions at sufficiently high energies to detect the superpartners many theorists expect to see.

The supersymmetric standard model

Under the Standard Model all fundamental particles can be broken down into two groups, fermions that make up matter, and bosons that exchange the forces acting on matter. Due to the physics of the theory, almost all of the behaviour of the universe can be explained based on this handful of particles.

Fermions themselves further break down into three generations; that is, each fermion comes in a variety of three subtypes of increasing mass. For instance, one of the most commonly known fermions is the electron, which also has two other less-well-known subtypes, the muon and tau. Fermions also come in two versions for each generation, with differing electric charge. A graph of all the fermions in the Standard Model is quite small. It contains the three generations of quarks and leptons, each broken down into two partners with differing charge.

On the other hand the bosons come in groupings that are nowhere near as "neat", including four distinct types, with subgroups containing anywhere from one to sixteen members. In addition, there appears to be no generational structure; the photon only comes in one type, and although it has partners in the W and Z particles, they don't really match up with anything in the fermion side.

The discrepancy between the "clean" fermion side and "messy" boson side has long been considered one of the most unaesthetic points of the Standard Model.

It turns out that none of the particles in the Standard Model can be superpartners of each other, so if supersymmetry is correct there must be at least as many extra particles to discover as there are in the Standard Model. The simplest possibility consistent with the Standard Model is the Minimal Supersymmetric Standard Model (MSSM).

A possibility in some supersymmetric models is the existence of very heavy stable particles called WIMPs (weakly interacting massive particles), neutralinos or photinos which would interact very weakly with normal matter. These would be possible candidates for dark matter.

As mentioned above, in supersymmetric theories, every fundamental fermion has a boson superpartner and vice versa. If the vacuum happens to be supersymmetric, this would mean superpartners would have the same mass, which is not what we observe. However, there are two ways out of this. Either we assume the vacuum is degenerate and SUSY is broken spontaneously, or we add soft SUSY breaking terms which break SUSY explicitly, making it an approximate symmetry. The latter approach is often preferred.

Motivations

One of the main motivations for SUSY comes from the quadratic divergence of the mass squared of scalar bosons. Put more simply, it means most quantum field theories predict that the mass of a scalar boson, when run down the renormalization group, is of the order of the cutoff scale. Since the Higgs field in the Standard Model is a scalar field, this poses a problem if we assume there is a desert of many orders of magnitude between the electroweak unification scale and the GUT scale. SUSY solves this problem by cancelling the quadratic divergences due to scalar-scalar couplings by couplings due to scalar-fermion couplings. See hierarchy problem. However, it could be possible there is new physics (other than SUSY) above the TeV scale or there is no scalar Higgs field as in top quark condensate and technicolor models.

Another motivation is the coupling constants for QCD, weak interactions and hypercharge do not quite meet together at a common energy scale if we run the renormalization group backwards if we do not include SUSY. With SUSY, the match is within current experimental bounds.

Yet another motivation stems from the desire of some physicists to find a symmetry group which includes the Poincaré group and internal symmetries but is not a direct product of the two. A theorem by Coleman and Mandula states that if we make certain assumptions about the S-matrix, the only possible extensions are 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. There is, however, a loophole in this theorem because symmetries need not necessarily be described by groups. If we relax the condition from symmetry groups to symmetry supergroups, we arrive at SUSY. However, there is no compelling reason why a symmetry ought to unify Poincaré symmetry with some other symmetry nontrivially.

SUSY is also sometimes studied mathematically for its own intrinsic properties.

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History

SUSY was discovered during the 70's while studying a string theory which includes fermions. SUSY, however, is independent of superstring theory, and applicable also to point particle theories.

Experimental searches

At present, there is no experimental evidence that supersymmetry exists in the real world. However, there is some indirect evidence which suggests that supersymmetry may be found at energies not too far above those accessible by today's particle accelerators. The search for supersymmetry is one of the primary goals of the Large Hadron Collider (LHC) at the CERN laboratory which is due to open in 2007. One of the detectors to be used in this search is ATLAS.

The supersymmetry algebra

Traditional symmetries in physics are generated by objects that transform under the various tensor representations of the Poincaré group. 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:

<math>\{Q_{\alpha}, \bar Q_{\dot{\beta}}\} = 2{\sigma^\mu}_{\alpha\dot{\beta}}P_\mu <math>

and all other anti-commutation relations between the Qs and Ps vanish. In the above expression <math>P_\mu<math> are the generators of translation and <math>\sigma^\mu<math> are the Pauli matrices.

Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra. For each Lie algebra, there exists an associated Lie group which is connected and simply connected. Unique up to isomorphism, this Lie group is canonically associated with the Lie algebra, and the algebra's representations can be extended to create group representations. In the same way, representations of a Lie superalgebra can sometimes be extended into representations of a Lie supergroup.

See supersymmetry algebra for a more detailed discussion, including a description of SUSY in Minkowski spacetime.

Supersymmetric quantum mechanics

Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.e., the lack of observed partner particles of equal mass. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped that studying SUSY's consequences in this simpler setting would lead to new understanding; remarkably, the effort created new areas of research in quantum mechanics itself.

For example, as of 2004 students are typically taught to "solve" the hydrogen atom by a laborious process which begins by inserting the Coulomb potential into the Schrödinger equation. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The final outcome is the set of hydrogen-atom energy states (labeled by quantum numbers n and l). Using ideas drawn from SUSY, the final result can be derived with significantly greater ease, in much the same way that operator methods are used to solve the harmonic oscillator. Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom. Of course, he did not call his solution supersymmetric, as SUSY was thirty years in the future—but it is still remarkable that the SUSY approach, both older and more elegant, is taught in so few universities.

The SUSY solution of the hydrogen atom is only one example of the very general class of solutions which SUSY provides to shape-invariant potentials, a category which includes most potentials taught in introductory quantum mechanics courses.

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—but, in the relativistic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass.

SUSY concepts have provided useful extentions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits.

See supersymmetric quantum mechanics for a more detailed discussion, including the SUSY QM superalgebra and an explicit example worked in two dimensions.

Supersymmetry, quantum gravity theories, and extra dimensions

Loop quantum gravity (LQG) in its current formulation predicts no additional spatial dimensions, nor anything else about particle physics. Lee Smolin, one of the originators of LQG, has proposed that loop quantum gravity incorporating either supersymmetry or extra dimensions, or both, be called loop quantum gravity II, in light of experimental evidence.

For string theory to be consistent, supersymmetry appears to be required at some level (although it may be a strongly broken symmetry). In contrast, LQG is a theory of quantum gravity which does not require supersymmetry. In particle theory, supersymmetry is recognized as a way to stabilize the hierarchy between the unification scale and the electroweak scale (or the Higgs particle mass), and can also provide a natural dark matter candidate; it may also introduce additional theoretical problems.

To date, no supersymmetric partner particles have been experimentally observed. If experimental evidence confirms supersymmetry in the form of supersymmetric particles such as the neutralino that is often believed to be the lightest superpartner, (the superpartner of the photon, Z boson, or Higgs boson), quite possibly as early as 2007 when Europe's Large Hadron Collider (LHC) will be in operation with sufficient energies to produce such particles, it may be possible to modify LQG's spin networks to accommodate these discoveries by requiring the spin networks to carry more quantum numbers.

String theory also requires extra spatial dimensions which have to be "hidden" somewhat as in Kaluza-Klein theory. By contrast, the minimal LQG is formulated in 3 spatial dimensions and one dimension of time. Again, as yet there is no experimental evidence for extra spatial dimensions, however it is possible (though seen as unlikely) that Kaluza-Klein modes may be seen at the LHC, or that there may be other future signals of extra dimensions.

See also

• Goldstone boson
• Majoron

Related topics

• Supergroup
• Lie superalgebra
• Representation of a Lie superalgebra
• Superspace
• String theory
• Quantum group

References

• Cooper, F., A. Khare and U. Sukhatme. "Supersymmetry in Quantum Mechanics." Phys. Rep. 251 (1995) 267-85 (arXiv:hep-th/9405029).
• Junker, G. Supersymmetric Methods in Quantum and Statistical Physics, Springer-Verlag (1996).
• Kane, G. L. and Shifman, M., eds. The Supersymmetric World: The Beginnings of the Theory, World Scientific, Singapore (2000). ISBN 981-02-4522-X.
• Weinberg, Steven, The Quantum Theory of Fields, Volume 3: Supersymmetry, Cambridge University Press, Cambridge, (1999). ISBN 0-521-66000-9.
• Wess, Julius, and Jonathan Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, (1992). ISBN 0-691-02530-4.ca:Supersimetria

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