Supersymmetry

In particle physics, supersymmetry (often abbreviated SUSY) is a symmetry that relates elementary particles of one spin to other particles that differ by half a unit of spin and are known as superpartner. In a theory with unbroken supersymmetry, for every type of boson there exists a corresponding type of fermion with the same mass and internal quantum numbers (other than spin), and vice-versa.

There is no direct evidence for the existence of supersymmetry. It is motivated by possible solutions to several theoretical problems. Since the superpartners of the Standard Model particles have not been observed, supersymmetry must be a broken symmetry if it is a true symmetry of nature. This would allow the superparticles to be heavier than the corresponding Standard Model particles.

If supersymmetry exists close to the TeV energy scale, it allows for a solution of the hierarchy problem of the Standard Model, i.e., the fact that the Higgs boson mass is subject to quantum corrections which — barring extremely fine-tuned cancellations among independent contributions — would make it so large as to undermine the internal consistency of the theory. In supersymmetric theories, on the other hand, the contributions to the quantum corrections coming from Standard Model are naturally canceled by the contributions of the corresponding superpartners. Other attractive features of TeV-scale supersymmetry are the fact that it allows for the high-energy unification of the weak interactions, the strong interactions and electromagnetism, and the fact that it provides a candidate for dark matter and a natural mechanism for electroweak symmetry breaking. Therefore, scenarios where supersymmetric partners appear with masses not much greater than 1 TeV are considered the most well-motivated by theorists. These scenarios would imply that experimental traces of the superpartners should begin to emerge in high-energy collisions at the LHC relatively soon. As of September 2011, no meaningful signs of the superpartners have been observed, which is beginning to significantly constrain the most popular incarnations of supersymmetry. However, the total parameter space of consistent supersymmetric extensions of the Standard Model is extremely diverse and can not be definitively ruled out at the LHC. According to a 2012 article in Scientific American, the failure of the LHC to find evidence for supersymmetry has led some physicists to suggest that the theory should be abandoned.

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 under very general assumptions. The Haag-Lopuszanski-Sohnius theorem demonstrates that supersymmetry is the only way spacetime and internal symmetries can be consistently combined.

In general, supersymmetric quantum field theory is often much easier to work with, as many more problems become exactly solvable. Supersymmetry is also a feature of most versions of string theory, though it may exist in nature even if string theories are not experimentally detected.

The Minimal Supersymmetric Standard Model is one of the best studied candidates for physics beyond the Standard Model. Theories of gravity that are also invariant under supersymmetry are known as supergravity theories.

History

A supersymmetry relating mesons and baryons was first proposed, in the context of hadronic physics, by Hironari Miyazawa in 1966, but his work was ignored at the time. In the early 1970s, J. L. Gervais and B. Sakita (in 1971), Yu. A. Golfand and E. P. Likhtman (also in 1971), D.V. Volkov and V.P. Akulov (in 1972) and J. Wess and B. Zumino (in 1974) independently rediscovered supersymmetry, 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 the microscopic world. Supersymmetry first arose in 1971 in the context of an early version of string theory by Pierre Ramond, John H. Schwarz and André 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 September 2011, no meaningful signs of the superpartners have been observed,. The Large Hadron Collider at CERN is currently producing the world's highest energy collisions and offers the best chance at discovering superparticles for the foreseeable future.

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), 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

Main article: 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. 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:

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

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

Main article: 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.

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

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 Weakly interacting massive particle (WIMP) 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 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 currently 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

Main article: Minimal_Supersymmetric_Standard_Model § 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

Main article: 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.

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

Main article: 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, with the corresponding multiplets(CPT adds a copy, whenever they are not invariant under such symmetry)

• N = 1

Chiral multiplet: (0,1⁄2) Vector multiplet: (1⁄2,1) Gravitino multiplet: (1,3⁄2) Graviton multiplet: (3⁄2,2)

• N = 2

hypermultiplet: (-1⁄2,0²,1⁄2) vector multiplet: (0,1⁄2²,1) supergravity multiplet: (1,3⁄2²,2)

• N = 4

Vector multiplet: (-1,-1⁄2⁴,0⁶,1⁄2⁴,1) Supergravity multiplet: (0,1⁄2⁴,1⁶,3⁄2⁴,2)

• N = 8

Supergravity 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 roughly 2^d/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.

Supersymmetry as a quantum group

Main article: 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 2009, has encouraged loop quantum gravity researchers.

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 Brookhaven; the WMAP dark matter density measurement and direct detection experiments, for example XENON-100; and by particle collider experiments, including B-physics, Higgs phenomenology and direct searches for 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. In 2006 these limits were extended by the D0 experiment

From 2003, WMAP's dark matter density measurements have strongly constrained supersymmetry models, which have to be tuned to invoke a particular mechansim to sufficiently reduce the neutralino density.

Prior to the launch 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 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 to 150 GeV.

As of 2012, the LHC has found no evidence for supersymmetry, and, as a result, has surpassed existing experimental limits from Large Electron–Positron Collider and Tevatron and partially excluded the aforementioned expected ranges. Based on the data sample collected by the CMS detector at the LHC through the summer of 2011, CMSSM squarks have been excluded up to the mass of 1.1 TeV and gluinos have been excluded up to 500 GeV.Searches are only applicable for a finite set of tested points because simulation using the Monte Carlo method must be made so that limits for that particular model can be calculated. This complicates matters because different experiments have looked at different sets of points. Some extrapolation between points can be made within particular models but it is difficult to set general limits even for the Minimal Supersymmetric Standard Model.

In 2011 and 2012, the LHC discovered a Higgs boson 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 130 GeV. Furthermore, for values of the MSSM parameter tan β ≤ 3, it predicts Higgs mass below 114 GeV over most of the parameter space. This region of Higgs mass was excluded by LEP by 2000. The LHC result is somewhat 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 consider to be "unnatural" (see naturalness and fine tuning). Furthermore, in 2012, the LHC measured deviations from Standard Model predicted Higgs couplings, particularly in their gamma-gamma final state, which, if they persist, could severely constrain the MSSM.

In spite of the null searches and the heavy Higgs, a recent analysis of the most constrained supersymmetry model, the CMSSM, suggests that the model is still compatible with all present experimental constraints. The preferred masses for squarks and gluinos is about 2 TeV. The resulting fine-tuning of the Higgs boson mass (see little hierarchy problem) and Z-boson mass (see mu problem), however, is considered "unnatural," and some theorists now favor extended supersymmetry models, for example, the NMSSM.

See also

• Wess–Zumino model
• Minimal Supersymmetric Standard Model
• Supersymmetry as a quantum group
• Quantum group
• Supercharge
• Superfield
• Supergeometry
• Supergravity
• Supergroup
• Superspace

Further reading

• A Supersymmetry Primer by S. Martin, 2011
• Introduction to Supersymmetry By Joseph D. Lykken, 1996
• An Introduction to Supersymmetry By Manuel Drees, 1996
• Introduction to Supersymmetry By Adel Bilal, 2001
• An Introduction to Global Supersymmetry by Philip Arygres, 2001
• Weak Scale Supersymmetry by Howard Baer and Xerxes Tata, 2006.
• 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).
• Gordon L. Kane.Supersymmetry: Unveiling the Ultimate Laws of Nature Basic Books, New York (2001). ISBN 0-7382-0489-7.
• Gordon L. Kane and Shifman, M., eds. The Supersymmetric World: The Beginnings of the Theory, World Scientific, Singapore (2000). ISBN 981-02-4522-X.
• D.V. Volkov, V.P. Akulov, Pisma Zh.Eksp.Teor.Fiz. 16 (1972) 621; Phys.Lett. B46 (1973) 109.
• V.P. Akulov, D.V. Volkov, Teor.Mat.Fiz. 18 (1974) 39.
• 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. arXiv:hep-ex/0401008. Bibcode:2004PhRvL..92p1802B. doi:10.1103/PhysRevLett.92.161802. 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.
• Fermi National Accelerator Laboratory (Sept 25, 2006). Fermilab's CDF scientists have discovered the quick-change behavior of the B-sub-s meson. Press Release.

External links

• What do current LHC results (mid-August 2011) imply about supersymmetry? Matt Strassler
• ATLAS Experiment Supersymmetry search documents
• CMS Experiment Supersymmetry search documents
• "Particle wobble shakes up supersymmetry", Cosmos magazine, September 2006
• LHC results put supersymmetry theory 'on the spot' BBC news 27/8/2011
• SUSY running out of hiding places BBC news 12/11/2012