Supersymmetry

Prerequisites: Poincare Group, Canonical Quantum Field Theory. Full path (23 pages)

Best explanations

Intuitive Supersymmetry: What Is It? by Matt Strassler
Concrete A Supersymmetry Primer

Intuitive
Concrete
Abstract
Why is it interesting?
History
Criticism

Intuitive

Elementary particles come in two kinds. Fermions, such as electrons and quarks, make up matter; no two identical fermions can occupy the same state. Bosons, such as photons and gluons, carry forces and can pile up in the same state. All symmetries of the Standard Model turn a particle into another particle of the same kind: a boson into a boson, a fermion into a fermion.

Supersymmetry is a proposed symmetry that turns bosons into fermions and fermions into bosons. If nature had this symmetry, every known particle would have a partner whose spin differs by one half: the electron would have a spin-0 partner (the selectron), the photon a spin one-half partner (the photino), and so on. In an exactly supersymmetric world the partners would have the same mass as the known particles. No such partners have been observed, including in searches at the Large Hadron Collider, so if supersymmetry exists in nature it must be broken and the partners must be heavier than the energies probed so far.

The symmetry is unusual in that it is tied to spacetime. Doing a supersymmetry transformation twice in a row amounts to moving the system in space and time. For this reason supersymmetry is an extension of the symmetries of special relativity. In an interacting quantum field theory with particles it is the only way to combine them with other symmetries in a nontrivial way.

Physicists studied supersymmetry for several reasons. In the Standard Model the mass of the Higgs boson is very sensitive to quantum corrections from heavy particles, and in a supersymmetric theory the contributions of bosons and fermions cancel each other. The strengths of the three forces of the Standard Model come closer to meeting at a single value at high energies in supersymmetric extensions. The lightest partner particle could make up dark matter. Supersymmetry is also a basic ingredient of superstring theory.

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

Supersymmetry is a symmetry whose generators form a Lie superalgebra, a $\mathbb{Z}_2$ -graded extension of the Poincaré algebra. We use two-component spinors, the signature $(+,-,-,-)$ and $\sigma^\mu = (1,\vec\sigma)$ . For $\mathcal{N}$ supersymmetries the odd generators $Q^I_\alpha$ and $\bar Q_{\dot\alpha I} = (Q^I_\alpha)^\dagger$ , $I = 1,\ldots,\mathcal{N}$ , satisfy

$\{Q^I_\alpha, \bar Q_{\dot\beta J}\} = 2\,\delta^I_J\,\sigma^\mu_{\alpha\dot\beta}P_\mu, \qquad \{Q^I_\alpha, Q^J_\beta\} = \epsilon_{\alpha\beta}Z^{IJ}, \qquad [P_\mu, Q^I_\alpha] = 0,$

where the central charges $Z^{IJ} = -Z^{JI}$ commute with everything and vanish for $\mathcal{N} = 1$ . The $Q^I_\alpha$ transform as left-handed Weyl spinors under the Lorentz group, and the internal R-symmetry group (up to $U(\mathcal{N})$ ) rotates the index $I$ . The Coleman-Mandula theorem forbids a nontrivial mixing of spacetime and internal symmetries for Lie algebras of symmetries of the S-matrix; the Haag-Łopuszański-Sohnius theorem shows that, once graded algebras are allowed, the algebras above are the most general possibility for an interacting theory with massive particles in four dimensions (with superconformal extensions when there are only massless particles).

Consequences of the algebra. Taking the trace over $\alpha = \dot\beta$ in the first relation for fixed $I$ gives

$\sum_{\alpha}\{Q^I_\alpha, (Q^I_\alpha)^\dagger\} = 4P^0 ,$

so in a theory with a positive-definite Hilbert space $H \geq 0$ , and a state has zero energy if and only if it is annihilated by all supercharges. Supersymmetry is therefore unbroken precisely when the vacuum energy vanishes. The Witten index $\text{Tr}(-1)^F e^{-\beta H}$ receives contributions only from zero-energy states and is invariant under smooth deformations of the parameters, so a nonzero index shows that supersymmetry is not broken. Since $P^2$ commutes with all generators, every irreducible multiplet contains bosons and fermions of the same mass, and in a multiplet with $P^0 > 0$ the numbers of bosonic and fermionic states are equal. Massless $\mathcal{N}=1$ multiplets contain helicities $\lambda$ and $\lambda + \frac12$ (plus the CPT conjugates). Requiring helicities $|\lambda| \leq 1$ limits renormalizable gauge theories to $\mathcal{N} \leq 4$ , and $|\lambda| \leq 2$ limits supergravity to $\mathcal{N} \leq 8$ . For massive multiplets with nonzero central charge the mass is bounded from below by the moduli of the skew-eigenvalues of $Z$ (the BPS bound, with a normalization-dependent factor), and multiplets that saturate the bound are shorter than generic massive multiplets.

Superspace. Supersymmetric field theories are formulated on superspace with coordinates $(x^\mu, \theta^\alpha, \bar\theta^{\dot\alpha})$ , where $\theta, \bar\theta$ are Grassmann variables, more precisely on a supermanifold of dimension $(4|4)$ for $\mathcal{N}=1$ . Supercharges act as differential operators, and supersymmetric covariant derivatives $D_\alpha$ , $\bar D_{\dot\alpha}$ anticommute with them. A chiral superfield satisfies $\bar D_{\dot\alpha}\Phi = 0$ and contains a complex scalar, a Weyl fermion and an auxiliary field. Lagrangians are built from integrals $\int d^4\theta\, K(\Phi,\bar\Phi)$ and $\int d^2\theta\, W(\Phi) + \text{h.c.}$ , with a Kähler potential $K$ and a holomorphic superpotential $W$ . Holomorphy of $W$ and non-renormalization theorems make many quantities exactly computable, for example in Seiberg-Witten theory for $\mathcal{N}=2$ .

Local supersymmetry. Making the supersymmetry parameter spacetime dependent forces the presence of a spin- $\frac32$ gauge field, the gravitino, whose superpartner is the graviton: local supersymmetry implies gravity, and the resulting theories are supergravities. Spontaneous breaking of local supersymmetry gives the gravitino a mass by the super-Higgs mechanism.

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)

Path to this page

Leads to: Supergravity.