Intuitive

• Supersymmetry — What Is It? by Matt Strassler

Concrete

[Supersymmetry means] extending the Poincare algebra] of infinitesimal space-time symmetries by adding some new operators. The new, extended algebra was called a supersymmetry algebra. The new operators were in some sense square roots of translations: one could multiply two of them and get the momentum or energy operator that gives infinitesimal translations in space or time. Doing this required using the Clifford algebra that Dirac had rediscovered for his Dirac equation. If one tries to build these new operators in a quantum field theory, they have the peculiar feature that they relate bosons and fermions. […] The new operators that extend the Poincare' algebra transform fermions into bosons and vice versa, thus forcing a certain relation between the bosons and fermions of a quantum field theory with this new kind of symmetry. This gets around the Coleman-Mandula theorem […] The Coleman-Mandula theorem had implicitly assumed that the symmetry was taking bosons to bosons and fermions to fermions, not mixing the two the way supersymmetry does.

Not Even Wrong by P. Woit p. 108

Standard Textbooks and Lecture Notes:

• A Supersymmetry Primer by Stephen P. Martin
• Theory and phenomenology of sparticles by Manuel Drees et. al.
• Introduction to Supersymmetry by Joseph D. Lykken

Resources:

• Formulae for Supersymmetry — MSSM and more — by Toru Goto is a huge awesome collection of useful formulas like, for example, loop functions.
• Complete set of Feynman rules for the MSSM by Janusz Rosiek
• Introduction to Supersymmetry by Edward Witten with a very nice discussion round at the end

Abstract

Why is it interesting?

Supersymmetry is a possibility to solve the hierarchy problem, provides a dark matter candidate and helps to unify the gauge couplings.

History

According to the SLAC database, there were 322 papers on supersymmetry in 1979 and 446 in 1980. The subject then really took off, with 1,066 papers in 1982, so it was 1981 that saw a dramatic increase in the subject's popularity. In that year, Witten gave a series of lectures on the topic at the summer school for particle theorists held in Sicily at Erice. This summer school had a long tradition as a venue at which some of the top particle theorists gathered together with postdocs and graduate students to give survey lectures on the latest, hottest topics in the field.

Not Even Wrong by P. Woit

Criticism

Supersymmetry (SUSY) is decidedly not a new symmetry. It was devised and applied to hadronic physics nearly a half century ago. If nature were supersymmetric, every known fermion would have a spinless superpartner, every known boson a spin-1/2 superpartner. Particles and their partners would have the same mass. At best, SUSY is a badly broken symmetry because no superpartner has yet been seen. Even as a broken symmetry, SUSY is theoretically attractive and phenomenologically useful. Evidence for supersymmetric particles has been sought at accelerator laboratories throughout the world and by several generations of high-energy experimentalists. No trace of SUSY has been detected, not even at today’s most powerful particle collider, the LHC at CERN. It appears that the energy scale associated with any still viable SUSY scheme is likely too high for the theory to fulfil its several assigned tasks: stabilize the Higgs boson mass, provide a dark matter candidate, and enable the coupling-constant convergence predicted by grand unification. Perhaps the time has come for experimenters to peer beyond their supersymmetric chimera. Sheldon Glashow

Besides string theory, the other part of the standard orthodoxy of the last two decades has been the concept of a supersymmetric quantum field theory. Such theories have the huge virtue with respect to string theory of being relatively well-defined and capable of making some predictions. The problem is that their most characteristic predictions are in violent disagreement with experiment. Not a single experimentally observed particle shows any evidence of the existence of its “superpartner”. One can try and explain this away by claiming that an unknown mechanism for breaking the supersymmetry of the vacuum state exists and is precisely such that all superpartners happen to have uncalculable masses too large to have been observed. If one believes this, one is faced with the problem that the vacuum energy should then be of at least the scale of the supersymmetry breaking. Assuming that one’s theory is also supposed to be a theory of gravity, there seems to be no way around the prediction that the universe will be a lot smaller than the size of a proton. Supersymmetry has a complicated relationship with modern mathematics. The general formalism one gets by naively replacing vector spaces by “super vector spaces”, groups by “supergroups”, etc. produces new structures but does not obviously lead to much new insight into older mathematics.

Quantum Field Theory and Representation Theory: A Sketch by Peter Woit

[S]upersymmetry must be a spontaneously broken symmetry, because if it were a symmetry of the vacuum state, then one can show that each particle would have to have the same mass as its superpartner. The necessity for spontaneous breaking of supersymmetry is a disaster for the whole supersymmetric quantum field theory project. Supersymmetric extensions of the standard model are well enough understood that it is clear that their dynamics is such that they cannot by themselves dynamically break their own supersymmetry, and if one tries to break it in a similar way to the Higgs mechanism, one gets relations between particle masses that are incorrect. One can come up with ways of spontaneously breaking the supersymmetry, but these all involve conjecturing a vast array of new particles and new forces, on top of the new ones that come from supersymmetry itself. […] Supersymmetry advocates describe this situation with phrases such as 'the lack of a compelling mechanism for supersymmetry breaking'. Since one doesn't understand the supersymmetry breaking, to define the MSSM one must include not only an unobserved superpartner for each known particle, but also all possible terms that could arise from any kind of supersymmetry breaking. The end result is that the MSSM has at least 105 extra undetermined parameters that were not in the standard model

Not Even Wrong by P. Woit

“The Large Hadron Collider will either make a spectacular discovery or rule out supersymmetry entirely.” Michael Dine (2007)