Here’s a nice loophole: not all of the predictions from string theory take place at the unreachable Planck scale. Supersymmetry could give us a window on the Planck scale using currently available technology.
One troubling aspect of spontaneously broken gauge field theories based on the Higgs mechanism of giving mass to gauge bosons is that it’s not only the coupling constants, but also the masses, that get renormalized by quantum corrections from taking into account all possible virtual processes at all possible momentum scales.
Suppose there is new physics at some scale L, so that the Standard Model of particle physics is no longer adequate to describe physics at higher momentum scales. The quantum corrections to fermion masses would depend on that cutoff scale L only logarithmically
whereas the scalar Higgs particles would exhibit a quadratic dependence on the cutoff scale
This means that the masses of Higgs particles are very sensitive to the scale at which new physics emerges.
This sensitivity is called the gauge hierarchy problem, because the Higgs mass is related to the masses of the gauge bosons in the spontaneously broken gauge theory. The original question “How do the gauge bosons get mass without spoiling gauge invariance?” was only partially answered by the Higgs mechanism. In a way, the question wasn’t answered by the Higgs mechanism, it was just transferred up to a new level, to the question: “Why does the Higgs mass remain stable against large quantum corrections from high energy scales?“
The interesting thing about scalar mass divergences from virtual particle loops is that virtual fermions and virtual bosons contribute with opposite signs and could cancel each other completely if for every boson, there were a fermion of the same mass and charge.
At the level of quantum mechanics, this type of Fermi-Bose symmetry would entail some quantum operator, let’s call it Q, whose action would be to transform bosons into fermions, and vice versa. In operator language this would be written
And since this is a symmetry, this operator must commute with the Hamiltonian
Such a theory is called a supersymmetric theory, and the operator Q is called the supercharge. Since the supercharge corresponds to an operator that changes a particle with spin one half to a particle with spin one or zero, the supercharge itself must be a spinor that carries one half unit of spin of its own.
Supersymmetry is such a powerful idea because it is a symmetry under the exchange of classical and quantum physics. Bosons are particles that obey Bose statistics, meaning that any number of them can occupy the same quantum state at the same time. Fermions obey Fermi statistics and only one fermion can occupy any given quantum state at one time. But the classical limit of quantum physics is approached when the occupation numbers of available states are very high. For example, in this limit, the quantum photon field behaves like the classical electromagnetic field as described by Maxwell’s equations. But then the conclusion for fermions is that there is no classical limit for fermions. Fermionic fields are inherently quantum relativistic phenomena.
Therefore, any symmetry that exchanges fermions and bosons is a symmetry that exchanges physics that has a classical limit with physics that has no classical limit. So such a symmetry should have very powerful consequences indeed.
One big problem with supersymmetry: in the particle physics that is observed in today’s accelerators, every boson most definitely does NOT have a matching fermion with the same mass and charge. So if supersymmetry is a symmetry of Nature, it must somehow be broken. It’s easy enough for an expert to construct a supersymmetric theory. It’s breaking the symmetry, without destroying the beneficial effects of that symmetry, that has been the hardest part of the program to fulfill.
But would a broken supersymmetric theory still be able to solve the gauge hierarchy problem? That depends on the scale at which the supersymmetry is broken, and the method by which it is broken. In other words, it’s still an open question. Stay tuned.
How was supersymmetry developed?
Supersymmetry was not developed originally as a means of solving the gauge hierarchy problem. Supersymmetry was first developed independently by two different groups of theorists separated by the Cold War back in the 1970s. One group in the USSR was exploring the mathematics of space-time symmetry, and the other group in the West was trying to add fermions to bosonic string theory.
In the USSR, mathematicians Gol’fand and Likhtman wanted to do something exotic with the group theory of spacetime symmetries. The usual group of spacetime symmetries in relativistic quantum field theory is called the Poincaré group. This group includes symmetries under spatial rotations, spacetime boosts and translations in space and time.
The action of the group can be described by the algebra of the group, which is defined by a set of commutation relations between the generators of infinitesimal group transformations. The algebra of the Poincaré group looks like:
The momentum generator Pm generates space and time translations. The Lorentz matrices Jmn generate rotations in space and Lorentz boosts in spacetime. These are all bosonic symmetries, which ought to be true because momentum conservation and Lorentz invariance are present in classical physics.
But the Poincaré group also has representations that describe fermions. Since spin 1/2 particles arise as solutions to a relativistically invariant equation — the Dirac equation — this is to be expected. If there are spin 1/2 particles, could there be spin 1/2 symmetry generators in a spacetime symmetry algebra? Yes! One way to add them is shown below:
What are the new symmetry generators labeled by Q? These are the supercharges mentioned above.
What Gol’fand and Likhtman ended up with was the group theory of supersymmetric transformations in four spacetime dimensions, and using this new type of symmetry, they constructed the first supersymmetric quantum field theory.
Unfortunately for them, their work was ignored, both in the Soviet Union and in the West, until years later when supersymmetry finally mushroomed into a major topic of investigation in particle physics. In 1972, Gol’fand was judged one of the least important researchers in his group at FIAN in Moscow, and so he was let go in a cost reduction drive in 1973. He remained unemployed for seven years, until pressure from the world physics community led to his rehiring in 1980.
From strings to superstrings
In the West, physicists working on dual resonance models were beginning to understand them in terms of vibrating strings whose modes of vibration were solutions to the wave equation on the worldsheet swept out by the string as it propagated in spacetime. The modes of oscillation all have integer spin, and so these dual resonance models described bosonic string theory. In bosonic string theory, negative and zero norm states are eliminated from the spectrum by fixing the spacetime dimension D=26, which gives a conformally invariant worldsheet theory and a Poincaré invariant theory in spacetime. The infinite-dimensional Virasoro algebra
was discovered to be the string worldsheet analog of the Poincaré algebra in spacetime, except for the last term, called the “central extension”, which arises from quantum effects and is canceled when the dimension of spacetime is 26.
But in order to describe Nature, a theory must contain fermions. Physicist Pierre Ramond began to investigate solutions to the Dirac equation on the string worldsheet, and found that this led to a much larger symmetry algebra than the Virasoro algebra, one that included anticommuting operators Fn (the worldsheet analog of the supercharge Q). Ramond discovered the super-Virasoro algebra
or the algebra of a supersymmetric version of conformal invariance, which is called superconformal invariance.
At the same time, John Schwarz and André Neveu were working on a new bosonic string theory that had an anticommuting field with half integral boundary conditions on the world sheet. They also found a super-Virasoro algebra, but one that looked slightly different from what Ramond had found. It was soon realized that the theories developed by Ramond and by Neveu and Schwarz fit together into two sectors of the same theory, called the RNS model after the initials of the founders. In this case, the central extension cancels for d=10.
Physicists Gervais and Sakita put the two pictures together into a theory described by a two-dimensional worldsheet action and noted that this action was invariant under a global (that is, independent of position) symmetry that transformed bosons into fermions and vice versa. In other words, string theories with fermions were supersymmetric theories.
But the supersymmetry they uncovered was confined to the two dimensional surface swept out by the string as it propagated through spacetime. The super-Virasoro algebra represents an extension of the worldsheet symmetry of the theory from conformal invariance to superconformal invariance. What wasn’t understood yet was whether this worldsheet supersymmetry led to supersymmetry in the spacetime in which the string propagates. Or in other words, whether there was an analogous extension of the spacetime symmetry of the theory from Poincaré invariance to super-Poincaré invariance.
The biggest problem with bosonic string theory (aside from the lack of fermions) is that the lowest energy state was a tachyon, or a particle mode with negative mass squared. This means the vacuum state of the theory is unstable.
In the mid-seventies Gliozzi, Scherk and Olive realized that they could implement a rule to consistently discard certain states from the RNS model, and after this truncation, known as the GSO projection, was made on the string spectrum in ten spacetime dimensions, the ground state was massless, and the theory was tachyon free.
But string theory was out of favor by the mid-seventies, and as the number of physicists working in the field dropped, the pace of work on the theory slowed. It took another five years for John Schwarz and Mike Green to get together to reformulate the RNS description in a way such that the spacetime supersymmetry of the theory is visible and obvious. So in 1981 superstring theory was born.
From supersymmetry to supergravity
One of the complicating factors in string theory is that one cannot avoid gravity. And gravity complicates supersymmetry. It changes the supersymmetry from a global to a local symmetry.
First let’s discuss ordinary spacetime supersymmetry in a bit more detail. Remember that the supercharge Q acts on bosonic and fermionic states as
The operator Q is a spinor with spin 1/2. A supersymmetric field theory can be constructed by studying the variation of some field f by an infinitesimal spinor x in the Q direction such that
Then the appropriate terms in an action for the field can be constructed by demanding that the action be invariant under a variation by x.
If x is a constant spinor, i.e. not a function of spacetime position x(x), then the supersymmetry is a global symmetry. One can take the usual scalar, spinor and gauge fields, such as those present in the Standard Model, add some number of supercharges QI, figure out how each field in the action transforms under a variation by x, and then figure out what terms to add to the action to cancel the overall variation variation by x and make the theory globally supersymmetric. For one supercharge, the theory is called N=1 supersymmetry. If there are two superchargers, it is N=2 supersymmetry, etc.
The result of this exercise for a single supercharge is called the Minimal Supersymmetric Standard Model, or MSSM, and this will be discussed in the next section. The new fields in the MSSM have funny names. Higgsinos and gauginos are the names of the fermionic superpartners of the Higgs scalars and gauge bosons respectively. The scalar superpartners of quarks and electrons are called squarks and selectrons. Grand Unified Theories can also be turned into supersymmetric theories, and this will also be discussed in the next section.
If x is not a constant spinor, in order words x = x(x), then the picture changes. The loss of global Poincaré invariance means there is a dynamic spacetime geometry, i.e. gravity, rather than the rigid flat spacetime upon which the Standard Model is based. In this case, instead of mere supersymmetry, we have supergravity. There is a new gauge field for this new local symmetry, although since x(x) is a spinor, the new gauge field has spin 3/2. It’s called the gravitino because it is the superpartner of the graviton. The infinitesimal variation of the gravitino under the spinor x(x) can be written
Superstring theories invariably contain gravity. Therefore the low energy effective field theory that one gets when looking at a string theory at an energy scale so low that the strings look just like their massless particle modes is generally a supergravity theory. However, the topic of supergravity was developed independently from string theory, because eventually particle theorists began to look for quantum field theories that had larger symmetry groups than the Standard Model or Grand Unified Theories.
By the time Green and Schwarz realized that their GSO-projected, tachyon-free fermionic string theories had spacetime supersymmetry as well as the worldsheet variety, there was already a community at work understanding the implications of supersymmetry for particle physics. In 1984, when Green and Schwarz discovered the anomaly cancellation for Type I superstrings based on the gauge group SO(32), the most talked-about candidate for a unified field theory was a quantum field theory based on N=1 supergravity in eleven spacetime dimensions. Now both theories are a part of a larger framework that some people call M-theory.
If supersymmetry is a prediction of superstring theory, and whatever larger theory that may encompass it, then it is important to know:
a. How is supersymmetry broken to give the non-supersymmetric world we see so far?
b. What are the signs of supersymmetry that might show up in particle physics experiments?
Constructing Supersymmetric Models
A supersymmetric particle model consists of a collection of particle supermultiplets and a set of potentials that describe the interactions between the particles. The three potentials relevant to supersymmetry are: the superpotential W, the Kähler potential K, and the potential V for the scalar fields in the theory, derived from W and K.
For N=1 supersymmetry in four spacetime dimensions, the two possible types of supersymmetric particle multiplets are: the chiral multiplet, with a complex scalar field f with spin 0 and a chiral (that is, either right or left handed) fermion y with spin 1/2, and the vector multiplet, composed of a real (non chiral) fermion l with spin 1/2 and a vector field Am with spin 1.
Local gauge symmetry can be combined with global supersymmetry relatively easily. If a gauge field transforms according to the rule
where La is an infinitesimal gauge parameter, and the coefficients fabc are the structure constants of the group, then the spin 1/2 superpartner partner for the gauge field, called the gaugino, transforms as
The chiral and vector multiplets by themselves describe massless noninteracting particles. A mass matrix Mij for fermions, and Yukawa couplings yijk between fermions and scalars, can be added to theory as long as the action remains invariant under both gauge transformations and supersymmetry. The chiral multiplet contains an auxiliary field (with no kinetic term in the action) Fi, but the equations of motion equate it to a derivative of the superpotential, with no dynamic evolution of its own.
So in the end, in a model with several generations of chiral multiplets (fi,yi), the action with superpotential looks like
where the terms
are derivatives of the superpotential
with respect to the scalar fields.
The situation for gauge fields is a little more complicated, but similar. The supersymmetry transformation rules for the gauge field and the gaugino require an auxiliary field Da, where a labels a generator in the gauge algebra.
The resulting scalar potential of the theory, which is important for understanding the ground state of the full theory, can be written
The D-term in this potential, from the gauge multiplet auxiliary field Da, depends on the gauge coupling g and the gauge group generators Ta.
The Kähler potential will come in later in the section on supergravity.
The Minimal Supersymmetric Standard Model
If supersymmetry is to solve the gauge hierarchy problem in the Standard Model, then the Standard Model has to be derivable as a theory with supersymmetry. When all of the Standard Model fields are expressed in terms of chiral and gauge multiplets, and interactions terms are added, the resulting particle theory is called the Minimal Supersymmetric Standard Model, or MSSM for short.
The particles predicted by the Minimal Supersymmetric Standard Model are all of the particles that are already observed in the Standard Model, plus one extra Higgs doublet, and the supersymmetry partners of those particles.
Every chiral fermion in the Standard Model has a scalar superpartner; collectively these scalars are referred to as the sfermions, which divide like quarks and leptons into squarks and sleptons. The complex scalar Higgs SU(2) doublet from the Standard Model has a spin 1/2 superpartner called the Higgsino, as does the extra Higgs doublet that was made necessary by the supersymmetrization. The gauge bosons in the Standard Model have fermionic superpartners in the MSSM called gauginos.
At this stage of the discussion, we’re still in an imaginary supersymmetric world. The world we observe does not feature bosons and fermions all neatly paired up together as if they were ready to board Noah’s Ark. So a realistic Minimal Supersymmetric Standard Model requires a realistic method of breaking supersymmetry while still preserving the effects of supersymmetry in stabilizing the sensitivity of the Higgs mass to quantum corrections.
Supersymmetry breaking and unification
As was learned in the case of spontaneous symmetry breaking in the electroweak interactions, a theory can “have its symmetry cake and eat it too” by having a ground state that does not feature the full symmetry of the action.
The superpotential for a supersymmetric theory yields a scalar potential V(f,f*) = |F|2 + |D|2/2, which is either positive or zero. This means any ground state in the theory must have positive or zero energy. A supersymmetric vacuum has zero supercharge. But because the supersymmetry algebra relates the supercharge to the energy, so that the ground state energy can be rewritten as products of the supercharges, a vacuum with zero supercharges must also have zero energy. Therefore one can break supersymmetry spontaneously by adding terms to the action such that either |F| or |D| or both are nonzero.
To break supersymmetry using the D-term from the gauge sector of the theory, a gauge term is added to the superpotential that is invariant under supersymmetry up to a total derivative. This turns out to require an extra unbroken U(1) gauge symmetry which is not present in the MSSM (and not observed in Nature). So this method requires looking for a theory beyond the Standard Model in which this extra U(1) field can live.
To break supersymmetry using the F-term, one can add chiral multiplets that transform as singlets under the gauge symmetries in the theory. This method also requires extra fields not observed in Nature.
It is also possible to break supersymmetry non-spontaneously, or explicitly, by directly adding so-called “soft terms” to the superpotential that give mass to the gauginos and scalars. “Soft” in this context means terms having mass dimension 2 or 3, to avoid quadratic divergences in the quantum corrections.
Note that this is the only way to break global supersymmetry that is consistent with Standard Model physics is to add soft terms explicitly.
But this is hardly a satisfactory way of resolving the gauge hierarchy problem, because instead of having to fine tune the theory to tame large quantum corrections to the Higgs mass, new arbitrary supersymmetry breaking parameters have to be added to the physics by hand. That is in effect passing the gauge hierarchy problem upstairs.
If global supersymmetry doesn’t work, then what about local supersymmetry, i.e. supergravity?
Going from global to local supersymmetry means adding gravity to the theory. So supersymmetry starts to expand upward and involve unification very naturally. One desires to have a spontaneously broken supersymmetric theory, and then unification of elementary particle physics with gravity appears as a necessary ingredient.
When supersymmetry is a local symmetry, in addition to chiral and vector multiplets, there is another multiplet with the graviton and its supersymmetry partner, the gravitino. Since the graviton has spin 2, the gravitino Yam, has spin 3/2, and can be seen in some way as the gauge field of local supersymmetry. Breaking supersymmetry means giving mass to the gravitino.
As with a gauge boson, the gravitino can gain mass when the ground state of the scalar potential breaks the symmetry of the action. In the bosonic Higgs scenario, the massless Goldstone modes of the scalar field end up as the extra longitudinal components that make the massless gauge boson massive. In the supersymmetric case, in addition to Goldstone bosons, there are massless fermionic states called Goldstinos, and they provide the longitudinal modes that give mass to the gravitino and break supersymmetry.
With supergravity, we have the interesting possibility of breaking supersymmetry through gravitational couplings. For simple N=1 supergravity with a chiral multiplet, the Kähler potential looks like
with MP is the Planck mass and W is the superpotential of the theory. The resulting scalar potential for this theory is
In this model, the gravitino acquires a mass by eating a massless Goldstino, but because of the minus sign in the scalar potential, the total vacuum energy can be tuned to be zero. This is important because the total vacuum energy gives the cosmological constant of the theory, and the one that has been measured is extremely small.
How to test supersymmetry
One experimental and theoretical result that is very encouraging evidence for supersymmetry is the high energy behavior of three Standard Model coupling constants (two electroweak and one strong). As stated on a previous page, the search for a Grand Unified Theory with all Standard Model fields gathered into representations of one big Lie group was encouraged by projections that the three Standard Model coupling constants meet at a single value at some energy scale M = MGUT.
However, when quantum corrections are included, this agreement does not occur precisely at a single value. The three coupling constants come much closer to a single value when the model in which they are being calculated is the Minimal Supersymmetric Standard Model.
So supersymmetry suggests unification, and unification suggests supersymmetry.
None of this is proof, but it adds a lot of excitement to the search for proof.
One thing that a supersymmetric theory should NOT do is violate any of the observed conservation laws of particle interactions. One important observed conservation law that is easily violated by unified theories and supersymmetric theories is the conservation of baryon number.
The proton is the lightest baryon and hence, if baryon number is conserved, the proton should be extremely stable. The observed lifetime of the proton is currently measured to be
Grand unified theories (GUT for short) have gauge bosons that can mediate interactions that change quarks into leptons and hence allow the proton to decay by various interactions, including
where a proton, with baryon number 1, decays into a positron, which is a lepton and has baryon number 0, and a neutral pion, which is made of a quark and an antiquark and has baryon number 0. There are three quarks on the left hand side of the equation and two quarks and a lepton on the right hand side. If baryon number is not conserved, then the stable proton becomes unstable. The estimate for the proton lifetime in a GUT without supersymmetry is
So this is bad for unification.
In a GUT with supersymmetry there can also be baryon and lepton number violation, but for many reasons, the rate ends up being smaller so that
which is still an experimentally viable number, and a region that is close enough to the observed rate for future measurements, for example, at at the Super-Kamiokande experiment in Japan, to be able to tell us something meaningful about supersymmetry.
Dark matter and SUSY
Because of the way stars move inside galaxies, astronomers and astrophysicists have calculated that there is a huge amount of mass in the Universe that we can’t see with telescopes or other instruments because it’s not giving off light the way stars do. That’s why they call it dark matter.
The presence of this dark matter can be detected by seeing how it interacts gravitationally, but it’s been hard to figure out what it could be made of. One of the leading candidates for dark matter is a supersymmetry particle called the LSP, for Lightest Supersymmetric Particle.
The success of this idea depends on the stability of the LSP. The LSP is stable in supersymmetric theories with a symmetry called R-parity, which guarantees that supersymmetric particles are produced only in pairs. This means that a supersymmetric particle can only decay into another supersymmetric particle. Hence the lightest one is stable because it can’t decay into anything.
The LSP that could make up dark matter has to be massive and electrically neutral, therefore, it could only be the supersymmetry partner of a neutral particle. The three candidates are: a gravitino (fermionic superpartner to the graviton), a sneutrino (scalar superpartner to the neutrino) or a neutralino (fermionic superpartner to a neutral gauge boson or neutral Higgs scalar).
So far the most promising candidate for dark matter is the neutralino, because they interact weakly. Therefore they would decouple from thermal equilibrium at some early age of the universe and produce a stable residual density that could be large enough to provide the large amount of dark matter that is believed to be out there.
There are a lot of hints that supersymmetry could be out there, because it offers ways to solve many puzzling issues in particle physics and cosmology at once.
This is another arrow pointing to string theory, the only theory of elementary particles that requires both supersymmetry and gravity to exist in Nature.
1. Supersymmetric Grand Unified Theories: From Quarks to Strings via SUSY GUTs (Lecture Notes in Physics (939))
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2. Beyond Standard Model Phenomenology at the LHC
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3. Supersymmetric and Extradimensional Beyond the Standard Model Theories: Aspects and Probes
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