Supersymmetry

In particle physics, supersymmetry (often abbreviated SUSY) is a physical theory which proposes a physical symmetry between bosons and fermions (also called supersymmetry). In supersymmetric theories, every fundamental fermion has a bosonic superpartner and vice versa. Assumption of supersymmetry mitigates and explains several problems in the Standard Model of particle physics; it also introduces complications of its own.

Supersymmetry was orginally proposed in 1973 by Julius Wess (Karlsruhe University) and Bruno Zumino (CERN). Earlier, the supersymmetry algebra had been discovered in the late 1960's by Soviet theorists Gol'fand and Likhtman, but not applied by them directly to the then outstanding problems in elementary particle physics. Supersymmetry first arose in the context of an early version of string theory, but the mathematical structure of supersymmetry has subsequently been applied successfully to other areas of physics; first by Wess, Zumino, and Abdus Salam (Imperial College) and their fellow researchers to particle physics, and later to a variety of fields, ranging from quantum mechanics to classical statistical physics. It remains a vital part of many proposed theories of physics.

As of 2006 there is no direct 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. By the year 2007 the Large Hadron Collider at CERN should be ready for use, producing high-energy particle collisions that may be sufficient to reveal superpartner particles.

Motivations

File:Hqmc600.png

Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar stop squark tadpole Feynman diagrams in a supersymmetric extension of the Standard Model

Scalar bosons

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 (the scale at which new physics appears).

Since the Higgs field in the Standard Model is a scalar field, this poses a problem if we assume that the cutoff scale is really high (as in most nonsupersymmetric GUT models, where there is a desert of many orders of magnitude between the electroweak unification scale and the GUT scale) or if we assume that there is no new physics beyond the Standard Model up to the Planck scale. However, if we set the cutoff scale to be around or slightly above the electroweak scale, then it is no surprise that the Higgs boson has the mass it has.

However, this creates a different problem: there are many nonrenormalizable terms that may be added to the Standard Model, but from an effective theory would expect all of them to be suppressed by suitable powers of the cutoff scale. If the cutoff scale is low, then those nonrenormalizable terms should not be small, conflicting with precision electroweak experiments, which have set very low bounds on the possible size of such terms.

Despite these constraints, there are several models with new physics at the TeV scale that stabilize the mass of the Higgs boson but do not induce large nonrenormalizable terms at that scale. One such model is the Minimal Supersymmetric Standard Model (MSSM), a supersymmetric theory augmented with soft SUSY breaking terms at the TeV scale (but see flavor changing neutral current). Supersymmetry cancels the quadratic divergences due to scalar-scalar couplings by couplings due to scalar-fermion couplings. (See hierarchy problem.) The soft SUSY breaking part of the theory does not induce most nonrenormalizable terms at the TeV scale.

Some other models with this property are the little Higgs models or some versions of technicolor models (the simplest versions have been ruled out because they induce large nonrenormalizable terms) or extra dimensional models. Recently, anthropic landscape arguments have been used to try to explain the fine-tuning problem, which potentially could obviate the need for SUSY. See split supersymmetry and supersplit supersymmetry.

Muon g−2 experiment

One piece of indirect evidence concerns muons. When suspended in a uniform magnetic field, the spin-axis of a muon precesses (wobbles) like a spinning top. The rate of this precession is proportional to the muon Landé g-factor. This difference between this g-factor and its classical value 2, called the anomalous magnetic dipole moment, is a quantity that may be calculated to extreme precision in the Standard Model. In experiment, muons have been found to precess slightly faster than is predicted by the Standard Model. Although this discrepancy is within the bounds expected due to uncertainty in the experiment, some physicists take it as evidence that supersymmetric partners are contributing to the muon g-factor.

Coupling constants

Another motivation is the coupling constants for QCD, electroweak interactions and hypercharge 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 current experimental bounds.

Symmetry groups

Yet another motivation stemmed 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. The Coleman-Mandula theorem states 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 1975, the Haag-Lopuszanski-Sohnius theorem showed that considering symmetry generators which satisfy anticommutation relations allows for such nontrivial extensions of space-time symmetry.

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 us to make exact calculations of quantities which can't be computed exactly in other quantum field theories. This makes supersymmetric models excellent toy models.

The supersymmetric Standard Model

To incorporate supersymmetry into particle physics, the Standard Model must be extended to include at least twice as many particles, since there is no way that any of the particles in the Standard Model can be superpartners of each other (they have incompatible masses and quantum numbers). 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). However, the MSSM appears to be unnatural in a number of ways, and many physicists doubt that it will be the correct theory.

A possibility in some supersymmetric models is the existence of very heavy stable particles (such as neutralinos) which would be WIMPs (weakly interacting massive particles). These would be candidates for dark matter.

Unsolved problem in physics:

Is supersymmetry a symmetry of Nature? If so, how is supersymmetry broken, and why? Can the new particles predicted by supersymmetry be detected?

(more unsolved problems in physics)

As mentioned above, in supersymmetric theories, every fundamental particle has a superpartner. If the vacuum state happens to be supersymmetric, this would mean superpartners would have the same mass as their ordinary partners, which is clearly ruled out by experiment. Hence, the vacuum must have broken supersymmetry. 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.

The supersymmetry algebra

Main article: 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:

\{Q_{\alpha },{\bar {Q}}_{\dot {\beta }}\}=2({\sigma ^{\mu }})_{\alpha {\dot {\beta }}}P_{\mu }

and all other anti-commutation relations between the Qs and Ps vanish. In the above expression $P_{\mu }=-i\partial _{\mu }$ are the generators of translation and $\sigma ^{\mu }$ 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.

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, that is, the failure to observe superpartners of equal mass to ordinary particles. 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.

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 extensions 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.

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.
Bennett GW, et al; Muon (g−2) Collaboration (2004). "Measurement of the negative muon anomalous magnetic moment to 0.7 ppm". Physical Review Letters. 92 (16): 161802. PMID 15169217.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
Brookhaven National Laboratory (Jan. 8, 2004). New g−2 measurement deviates further from Standard Model. Press Release.
A Supersymmetry Primer by S. Martin, 1999
Introduction to Supersymmetry By Joseph D. Lykken, 1996
An Introduction to Supersymmetry By Manuel Drees, 1996
An Introduction to Global Supersymmetry by Philip Arygres, 2001
Weak Scale Supersymmetry by Howard Baer and Xerxes Tata, 2006.