To the editors:

Édouard Brézin has written an elegant essay explaining how understanding the mechanism of spontaneous symmetry breaking has enabled physicists to construct the Standard Model of particle physics, an effective theory that describes our world surprisingly well. Yet, as he mentions, there are still major unanswered questions. Perhaps the most important of these is how a quantum theory of gravity can be included with the Standard Model. There is also the hierarchy problem. Furthermore, the Standard Model does not have any candidate for the dark matter of the universe. I would like to supplement Brézin’s essay by explaining how spontaneous symmetry breaking has also led us to possible solutions extending the Standard Model theory to include gravity and dark matter candidates.

Since the Standard Model was written in the 1970s, there has been major progress in ideas. First there was grand unification, such as the SU(5) symmetry of Howard Georgi and Sheldon Lee Glashow that incorporated the SU(2) and U(1) of the Standard Model into a single larger group. The breaking of SU(5) to the Standard Model is best described as a spontaneous breaking.

Then came supersymmetry, a remarkable hypothesis that proposed unifying bosons (integer spin particles) and fermions (half-integer spin particles) into a single theory. In the mid-1980s, string theory emerged and opened a path to relating the Standard Model and quantum gravity. Michael Greene and John Schwarz demonstrated that to have a mathematically consistent theory of quantum gravity, it is necessary to write it in nine, or more, space dimensions. This is a fact that cannot be ignored if we want to understand our world.

Yet we seem to live in three space dimensions. How can we make sense of this? After a while, researchers realized that another major spontaneous symmetry breaking occurred: spontaneous compactification, with a 10-dimensional world described as a product of a 4-D world and a 6-D compactified world. The 6-D world was curled up in a small region the size of the Planck scale. The Planck scales are the only natural scales that can be formed from the most fundamental constants available in the theory: Einstein’s speed of light c, Newton’s strength of the gravitational force G, and Planck’s constant measuring the size of the quantum of energy, h. The Planck length scale, for example, is L = (hG / c³)^1/2, with a tiny numerical value of about 10^–33cm. This is the natural size of a universe. We do not see the other six dimensions because they are so small.

Spontaneous compactification is said to occur when solutions to the combined system of Einstein’s equations and the Yang–Mills equations describing the Standard Model forces in 4-D space-time have space-time as the direct product of the usual 4-D Minkowski space and a manifold in the extra dimensions. This phenomenon is known to happen for a number of cases, and often generates masses on the order of the Planck mass.

The six dimensions are curled up in a region called a manifold, which has certain mathematical properties, such as the well-known Calabi–Yau manifold. Universes much larger than the Planck length, or much less than the Planck mass of about 10¹⁹ GeV require explanation. In particular, we do not yet understand masses much smaller than the Planck mass, especially the Higgs boson mass, about 10² GeV. The units do not really matter; what does is the large difference between the Planck mass physics and the physics at the Higgs boson mass. This is known as the hierarchy problem. If superpartners were to be discovered, they would provide a solution. 

Spontaneous compactification can also occur for 11-D M-theory, leaving a direct product of the 4-D space and a 7-D manifold called a G2 manifold. G2 manifolds have been poorly understood in comparison to Calabi–Yau manifolds, but a year ago the Simons Foundation awarded a four-year US$8.5 million grant to a group of mathematicians and mathematical physicists to study G2 manifolds and a few other related manifolds. We will soon know much more about G2 manifolds and the physics of compactifying M-theory. Luckily, many physics questions, such as solving the hierarchy problem, can be answered from a general knowledge of the theory, without detailed knowledge of the manifolds.

Theorists have been able to write a theory at the Planck scale consistent with the requirements of being a quantum theory of gravity and spontaneously relaxing to 4-D, where the 4-D theory is like the Standard Model. In a number of cases, that theory is supersymmetric. We want to know the theory at scales of hundreds of GeV and smaller than at the huge Planck scale. Brézin describes in his essay how the Wilson renormalization group is a powerful tool that can be applied here. We begin with a theory in the ultraviolet, at or near the Planck energy scale, and calculate how it flows to our energies. Two remarkable things happen, both forms of spontaneous symmetry breaking.

First, some sectors of the theory have large gauge groups and therefore large charges. They flow to strong forces as the energy scale decreases by a few orders of magnitude, giving condensation of particle-antiparticle pairs into bound systems and breaking the supersymmetry spontaneously. The resulting spectrum of superpartners is calculable and leads to the prediction that some are detectable by the LHC, perhaps requiring high luminosity or even an energy upgrade, but nonetheless detectable during the next few years. Also implied is that the expected superpartners are heavy enough that we would have been lucky if any were detected so far.

Second, the flow of the Higgs potential energy leads to it having a minimum away from the origin, giving the vacuum a non-zero value and determining the electroweak scale. That explains the spontaneous symmetry breaking that leads to the Higgs mechanism, breaking the electroweak symmetry and giving the gauge bosons mass. This spontaneous symmetry breaking becomes derived rather than assumed, as in the Standard Model. 

The renormalization group allows contact between the low and high energy scales, so one can make low-scale predictions from the high-scale theory, or calculate what the low-scale data implies for the high-scale theory.

It turns out that the extra dimensions can play an unanticipated role. The argument is a little technical, but one can see the general idea without understanding the details. A century ago, theorists started thinking about extra dimensions. Theodor Kaluza wondered what would happen if he imagined that there was just one extra dimension. He was working shortly after Einstein introduced general relativity. In general relativity, one first writes the metric, which describes the geometry of space-time. Let us call the metric of the five-dimensional world g_ab, a, b = 0, 1, 2, 3, 4, and the metric of our four-dimensional world g_μν, μ, ν = 0, 1, 2, 3. 

Kaluza showed that one could write the 5-D metric in the form

$g_{ab}=\left(\begin{array}{cc}{g}_{\mu \nu }& A_{\mu }\\ A_{\mu }& \phi \end{array}\right)$,

where A_μ can be interpreted as the vector potential of electromagnetism, and φ is a scalar we can ignore. The exciting thing is that the extra dimension can be interpreted as a force—5-D gravity contains 4-D gravity plus electromagnetism! A few years later, Oskar Klein, then an assistant professor in my department, suggested that the extra space dimension could not be seen because it was curled up too small for us to see it. The Kaluza–Klein theory did not work in detail for describing our world, but it pointed us in the right direction.

To describe our world, we have to take account of the need for a theory with 10 space-time dimensions while we only experience four space-time dimensions. Just as an atom falls into its ground state, our world will fall into its ground state, which is usually called the vacuum. The remarkable thing is that a compactified 10-dimensional world can contain our world, including gravity and all the forces of the Standard Model, in a way analogous to how the Kaluza–Klein theory generates a force like the electromagnetic one described above. We live in the ground state of a compactified string/M-theory. Because we have included gravity in the theory, it has an ultraviolet completion; it is UV complete.

The Kaluza–Klein idea that the extra dimensions turn effectively into forces in the compactified theory encourages us to think that compactified string/M-theories are the way to make a UV complete theory of our world and hopefully solve the hierarchy problem. It is exciting that 10, or 11, space-time dimensions are what is needed to accommodate the entire Standard Model.

It turned out that there were several ways to compactify. Early string theorists, who had hoped for a unique theory when gravity was included, were disappointed. The situation is analogous to how the Standard Model emerged in quantum field theory. Quantum field theories are the rules for any set of particles and forces. In the 1960s and 1970s, a number of sets of particles and forces were suggested. Each led to some predictions, which were then tested against existing knowledge and new data from several experiments. The Standard Model is the one that turned out to work. It might have happened that none worked, but clever physicists thought about the clues and found the right approach.

The current situation is similar. We can compactify M-theory, heterotic string theory, Type II string theories, and a few more. The resulting theories are very much like our world, with quarks and leptons and Yang–Mills forces. The different compactified string theories seem to have some different testable predictions, though that is not yet certain. That such theories emerge is encouraging. Just as several models were tried and the Standard Model emerged, now several different compactifications are being examined to find one that is a quantum theory of gravity and contains the Standard Model so it has a UV completion. The resulting theory generically has electroweak symmetry breaking, so it has an electroweak scale less than or of order 1 TeV, and it has the Planck scale, so it necessarily also has a hierarchy problem. To understand our world, we need to solve this hierarchy problem.

An important property of compactified string/M-theories is that they generically have a large number of sectors. We live on the visible sector, and others are called hidden sectors. It is known that supersymmetry is broken spontaneously in a hidden sector and the breaking transmitted to the visible one. It is also known that the lightest superpartner will generically decay to hidden sector matter, so it is not a good candidate for the dark matter of the universe, contrary to common lore. Rather, stable hidden sector matter provides candidates for the dark matter.

There are some problems that are very interesting, but that we do not need to solve in order to understand our world. The black hole information paradox is one. Understanding it does not help us understand Higgs physics, the hierarchy problem, or much else, and not understanding it does not stop us from understanding Higgs physics, the hierarchy problem, etc. The situation is similar for the cosmological constant problem, black holes in spaces of other than four dimensions, physical mathematics, moonshine, and a number of other issues.

You might wonder about needing to use string theory when many people have claimed that it is not testable. Some say that because string theories are naturally formulated at Planck scale high energies, far above what can be achieved at colliders, or at distances too small to probe directly, they cannot be tested. But one does not have to be somewhere to test there. We cannot go to other stars, but we have learned that stars throughout the universe are made of the same chemical elements. No one was at the Big Bang, but three strong pieces of evidence convince us that it happened: the expanding universe, helium abundance and nucleosynthesis, and the cosmic microwave background. No one was present when dinosaurs became extinct 65 million years ago, but we can test whether an asteroid impact was a major cause of dinosaur extinction. Compactified string/M-theories make a number of testable predictions.

As an example, it is worthwhile to describe the results from a study by Bobby Acharya, myself, and collaborators. We compactify M-theory on a G2 manifold plus our large 4-D world. We chose this direction for well-motivated reasons, one being that the resulting 4-D quantum field theory is automatically supersymmetric. We also demonstrated that the resulting theory automatically has spontaneous breaking of the supersymmetry in a hidden sector, and the supersymmetry breaking is necessarily gravity-mediated to our sector. The resulting theory has particles like those of the Standard Model, can accommodate parity violation, and probably has three families. It solves both the strong CP problem and the hierarchy problem. It has dark matter candidates, possible baryogenesis mechanisms, electroweak symmetry breaking, and a Higgs boson of the right mass. This is one theory, with no free parameters. The study of such compactified theories should be very fruitful and may lead us to candidate theories describing and explaining our world and gravity. Several spontaneous symmetry breakings have been essential to lead to theories taking us beyond the Standard Model.

Gordon Kane

Édouard Brézin replies:

I would like to thank our colleagues for their comments, which indeed point to other relevant, significant contributions to the topic of spontaneous symmetry breaking. In my short article, I wanted to illustrate how much this concept was ubiquitous; from condensed matter to cosmology and particle physics, I have tried to point out how our understanding of many phenomena is based on spontaneous symmetry breaking.

Gordon Kane’s letter describes clearly how much the same line of ideas is required to understand the extensions of the Standard Model. His discussion of grand unification, supersymmetry, string theory, space compactification, and quantum gravity, constitutes a welcome supplement to the topic of spontaneous symmetry breaking.