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Symmetry and Symmetry Breaking

Symmetry has proven to be an extremely powerful way of organizing our physics thoughts about Nature. Spacetime possesses some symmetries, referred to by physicists as Poincaré symmetry. The first set of symmetries possessed by (flat) spacetime is called translations. These just correspond to displacement in any of the spatial directions or in the time direction. In four spacetime dimensions there are four translations. The next set is called rotations. These correspond to what they sound like: rotation about any of the available axes. In four spacetime dimensions there are three rotations. The final set of symmetries is called boosts. These correspond to changing your frame of reference by changing your velocity. There are three of them in four spacetime dimensions.

Some symmetries of Nature act not on spacetime but on the particle or string itself. A notable example is the story of gauge symmetry, which is always transmitted by spin-one messenger bosons, which start life with zero mass. Symmetry can exist at high temperature but be spontaneously broken at low temperature. This process is involved in the Higgs mechanism which gives mass to the W+, W- and Z bosons but not to the photon.

Gauge Symmetry

Remember our old friend the electric charge? Force charge is like a handle for an interaction messenger boson (like the photon) to grab onto. A fancy phrase that theoretical physicists use to describe how the gauge symmetry acts on the particle/string is to talk about its internal space.

For electromagnetism, there is a gauge symmetry called U(1) that rotates the internal space of wavefunctions - in the case of particles/strings with electric charge. If a particle/string has no charge then this U(1) gauge symmetry would leave it alone.

In general, gauge forces can be in one of three possible phases:- Coulomb phase, where there is an inverse-square law in four spacetime dimensions; confined phase which works like QCD in which colour is confined; Higgs phase, in which the gauge symmetry is broken spontaneously.

Symmetry Breaking

Suppose that we take an iron bar at room temperature and then heat it up. When the iron becomes molten (at and above its melting point), the arrangement of the iron atoms changes. When the iron is solid, each iron atom has a spin that points in a particular direction. If the spins are (largely) aligned then we have a magnet. But when the iron is hot enough to become liquid, the iron atoms are free to move and so the average direction of the iron spins is zero. Each iron atom is a tiny magnet, but on average a blob of molten iron has no overall magnetism. We say that the iron liquid is rotationally symmetric.

Suppose that now we let the iron cool. Spins will freeze into small aligned domains, and alignment of the domains can be encouraged with another magnet. The groundstates of a block of solid iron happen when all the spins point in one unified direction: they have the lowest energy (and all groundstates have identical energy regardless of which direction the magnet points in). The funny thing about one of these groundstates is that it has broken the rotational symmetry: it has picked a direction in which to point! 

Our iron magnet could equally well have chosen any direction in which to point. Any direction is just as good as any other: all of them cost the same amount of energy. There is a continuous set of possibilities - 360 degrees' worth - and this is what matters for spontaneous symmetry breaking.

Another example of a system with spontaneous symmetry breaking is a perfectly cylindrical pencil balanced right on its very tip. The system in this phase has rotational symmetry about the axis of the pencil. Suppose that we wait. Eventually a small breath of wind will make the pencil fall down. When it falls down it can pick any direction in which to point. Once the pencil falls, the system stops being rotationally symmetric.

ElectroWeak Unification

Shelley Glashow, Abdus Salam and Steven Weinberg won a Nobel Prize for explaining how to unify $SU(2)$ weak and $U(1)$ electromagnetic gauge forces together from a theoretical perspective. It was a tour de force: they managed in one fell swoop to explain (a) generation of masses for leptons and quarks (b) generation of masses for W+,W-,Z boson and (c) unify two forces. The technical underpinnings of the Higgs mechanism in the Standard Model are complicated, and I am currently teaching them in my PHY2404S graduate course "Quantum Field Theory II". But the basic concepts can be distilled down to the essential bits pretty easily. Here is the approximate picture.

At high temperature, such as would have been available right after the Big Bang, the idea is that the W+, W-, Z and photon messenger bosons are all massless and are unified into one combination gauge field for the combination $SU(2)\times U(1)$ symmetry. By contrast, at low temperature (such as the ambient temperature of our Universe now) the W+,W- and Z get fat but the photon does not. The W+,W-,Z weak messenger bosons get fat by eating one mode of the Higgs boson. In the process, the $SU(2)$ force is broken away from the $U(1)$ force.

Discovery of the W+,W-,Z bosons in 1983 at the LEP experiment at CERN provided striking confirmation of the Glashow-Salam-Weinberg theory. The LEP machine used to live in the same tunnels currently used now for the LHC. At the time, the collider was running counterrotating beams of electrons and positrons to create the most forceful collisions available in the lab.

The Higgs Mechanism

You might wonder how the symmetry breaking mechanism works for electroweak physics. The name physicists give this is the Higgs mechanism. How does it work? The answer to this is complicated technically, in that I will be teaching it in my graduate PHY2404S course in a week or two! But the basic idea behind Higgsing can be visualized with the aid of a theoretical tool called the Mexican hat (sombrero) potential. This draws a picture of the (potential) energy tied up in the Higgs field, drawn as a function of how big the field is. The figure below graphs this partially: what it describes explicitly is the lowest-energy part of the potential energy story; the brim of the hat actually continues upwards and upwards but this is suppressed in the figure to make the bump visible. In other words, this is a cutaway. The height of the surface tells you the potential energy; how far away the surface is from the middle tells you how big the Higgs field is. 

At very high temperatures, the size of the bump in the sombrero is insignificant. The Higgs field basically stays close to the centre, in order to minimize the energy cost. But when the universe cools down, eventually the bump becomes really important compared to the size of your energy budget. As the universe cools more, the Higgs ends up balanced at the top of the bump, and eventually, the only way the energy can be decreased is for the Higgs to fall off the bump down into the valley that corresponds to the brim of the hat. The key point is that to lower its energy, the Higgs has to switch from being in a rotationally symmetric position (at the middle of the bump) to a rotationally asymmetric position (somewhere in the valley). This breaks the symmetry. This choosing of a particular direction in which to fall is what lets the Higgs break the $U(1)$ of electromagnetism away from the $SU(2)$ of the weak nuclear force.

The overall lesson is that a system possessing symmetry at high energy may not possess that symmetry at low energy.

Supersymmetry and Unification

One question you might have had about unification and string theory is whether it manages to unify matter and force together. I mean, string theory is impressive and all, unifying spin-one messenger bosons (like photons) with spin-two messenger bosons (gravitons), but can it unify bosons and fermions together? This is something the Standard Model cannot do.

It turns out that, in order to unify force and matter, physicists had to expand the concept of symmetries to include ones with fermionic parameters, not just bosonic parameters like rotation angles. The resulting symmetry was dubbed supersymmetry. I gather that the origin of this word is in the French, à la supermarché. It refers to a market that has more items than just a regular marché. By analogy, supersymmetry has more moving parts to it than regular symmetry. The neat thing about supersymmetry is that is is the unique exension of Poincaré symmetry that provides a 1-1 pairing between bosons and fermions.

SUSY Nomenclature

Physicists are lazy. All other things being equal, we don't go for a baroque solution over a simple one. We also love to abbreviate stuff to make it quicker to write - which is why we make such heavy use of math. Physicists working on supersymmetry got quickly bored of writing out such a long word and decided to abbreviate it as SUSY. This is pronounced like the girl's name Suzy.

Supersymmetry does not change the mass of a particle. So every particle's superpartner has the same mass as it does. The superpartner always has spin-half less than the original, unless the original had spin zero in which case its superpartner gets spin-half (not minus-half, because spin always has to be a non-negative number for it to make sense as a physics concept).

Superpartners of fermions get an s- prefix, for example the superpartner of the electron is the selectron. Superpartners of bosons, on the other hand, have a different nomenclature: they are designated by an -ino suffix.

While SUSY does change the spin by half, it does not change the mass. But... where are the stops, the Winos, etc? Shouldn't we have seen them already? In particular, shouldn't there be a particle called the selectron which has a spin of zero and also has a mass of 511keV? Well, if SUSY were a preserved symmetry, then yes, these particles should have been measurable already. But we certainly have not seen any kind of supersymmetric particle evidence yet, although physicists smarter than I have been looking for years. The conclusion is that SUSY must be a broken symmetry today, if it was ever a symmetry. It is not known whether SUSY is even a high-energy symmetry of nature. We hope to be lucky enough to find out by using LHC and cosmological data, but it is not clear if this wish will be realized. There is no compelling theoretical or experimental reason why SUSY has to be true. SUSY may be helpful for understanding the nature of dark matter (to be discussed in the next two weeks in our Cosmology section) and maybe even dark energy. We just do not know yet.

The Super in Superstring

Superstrings have an absolutely beautiful property that bosonic strings do not: they automatically take care of cancelling out the dangerous tachyon that might destabilize the whole edifice. SUSY for strings ensures that the mass of all string states, including the groundstate, is non-negative. Superstrings therefore come in bosonic and fermionic types. You can imagine electrons and its other lepton friends as coming from fermionic open string modes, while photons could be bosonic open string modes. And so forth.

Superstrings can be shown to be anomaly-free. This was a gigantic discovery by Green and Schwarz in 1984 that kicked off the First Superstring Revolution, in which theoretical particle physicists learned how to do quantum mechanics with superstrings. What does being anomaly-free mean? Basically, it means that your theory is free of nasty mathematical inconsistencies which are so problematic that they render a theory useless for practical applications. So getting your theory to be anomaly-free is a big bonus. It gives you street cred with other physicists.

Superstring anomaly cancellation, a highly technical topic I can only teach graduate students once they have had two semesters of technical Masters-level coursework, is a very complicated beast. But the essence of it distills down to one very simple equation: $$ D_{\rm superstring}-2=8 $$ This tells us that the dimension of spacetime must be $D_{\rm superstring}=10$. In other words, one time dimension and nine space dimensions. For bosonic strings, the equation is $$ D_{\rm bosonic\ string}-2=24 $$ which means $D_{\rm bosonic\ string}=26$.

The Type I superstring is an open superstring, and it has eight fermionic modes balancing the eight bosonic ones describing the spacetime coordinates. All of the other four superstring theories are closed superstring theories. The Type IIA superstring has eight left-handed fermionic modes and eight right-handed ones. It is nonchiral (has no handedness). The Type IIB superstring has sixteen left-handed fermionic modes and is chiral (has a handedness). The heterotic superstring theories are hybrids, which have left-moving bosonic string hybridized with right-moving superstring. The difficulty with making hybrids is that you have 26 left-moving bits of the spacetime coordinates but 10 right-moving bits. This seems silly because we know that there can only be 10 sensible dimensions built out of these bits. What the heterotic string theory discoverers found was that they could take the 16 unbalanced bits and form gauge symmetries out of them. Anomaly cancellation demands that the gauge group you build this way has to have 496 generators. This is a rather large number, and there are only two solutions: $SO(32)$ and $E8\times E8$ for the gauge symmetry. These two hybrids fill out the total of five known superstring theories.

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Author: Prof. A.W. Peet, University of Toronto
Creative Commons licence: BY-NC-SA 4.0 Int'l
Last updated: 2014-06-14