Spontaneous symmetry baking

After the discovery of a new boson (most probably the Higgs) at CERN I feel somewhat compelled to explain what the Higgs is and what it does. What better way to do it than with an analogy! We are going to try and recreate the Higgs mechanism with food. This is going to be a rather long post. First I'll explain what the Higgs field is and why it was theorised. After that I will give you a kitchen example of how we can view the Higgs field.



Mass

In physics there are various ways to theoretically explain why something has a mass. First of all we have to understand what we mean with mass. Perhaps surprisingly there are two ways of doing this. The first is maybe the most obvious, since it provides with the strongest net force we humans experience in our everyday lives. This force is of course gravity. We say that the more an object is attracted to the earth the greater its mass is. This definition of mass is something we use everyday and we even confuse mass with the force itself namely the weight. This is gravitating mass. There is another way to define mass and it is the inertia that every material object has. It is an resistance to changes in velocity. The easier the object is changed in its motion, the smaller its inertial mass. 

Are these two types of masses the same, are they perhaps equal or equivalent to each other? Well, we can do a simple experiment to find out. This experiment entails dropping two objects of different mass from a building and seeing what happens. If gravitating mass and inertial mass are different then the two different masses should fall to the ground at different rates. However, we know from experiments that all objects accelerate towards the earth with an equal change of velocity. Hence gravitating mass and inertial mass are the same (up to a scale factor which is like an exchange rate for currency, still the value remains the same).

Now we know what mass is! Or do we? Because why do objects with mass resist a change of motion? This question has been long asked by physicists. In fact it turns out when doing calculations of elementary particles, it can only be done in a consistent way when we assume that all elementary particles have zero mass. Of course we know that particles do have mass, so what is the mechanism that makes them massive? The theory gives an answer to that. In the sixties, three research groups independently came up with the same possible answer to that question. Englert and Brout in Belgium, Higgs in Scotland, and Kibble, Guralnik and Hagen in England all published their research around the same time.

Symmetry

The answer to the question has to do with symmetries. We all know what symmetries are in everyday life, for example a symmetric face is a face which has two similar sides. An object is generally said to be symmetric when parts of it resemble other parts of the same object. To define a symmetry one can say: A symmetry is a change in the configuration that doesn't change how the object behaves or looks. More explicitly, in the case of the face if we could mirror the two sides and interchange them we wouldn't be able to tell the difference from the original. Another example. A ball has many symmetries. One can rotate it by an arbitrary angle in an arbitrary direction, and you wouldn't be able to tell if it was rotated or the original. We can mirror the ball, and it would look the same. now we now what symmetries are. Physicists say symmetries are transformations that leave an object invariant under that transformation.

Symmetries can be broken. No, that doesn't mean you take a rock and smash the mirror you are looking at. It means that you can change something about the object to make it asymmetrical. If you do this yourself it is an explicit symmetry breaking, for example drawing a figure on the ball. However, some objects have hidden symmetries. These are symmetries which are not realised by the object because of external causes so it cannot exhibit its natural solution. What does this mean? What is a natural solution and what are external causes? Think of it this way, if I have a soft rubber ball which is normally round, and hence symmetric under rotations, and I place a heavy paperweight on it it becomes oblate, and it loses part of its rotational symmetry. The rubber ball no longer looks symmetric, but it is only because of the heavy paperweight, once we remove the paperweight the ball regains its original symmetry again. It is an external influence which has hidden the symmetry from us. 

Symmetries can also be broken spontaneously rather than explicitly. An unstable but symmetric object can decay to a stable but asymmetric solution. The symmetry is hidden in the realisation of the stable configuration. An amount of supercooled water has perfect rotational and translational symmetry. Wherever you look it looks the same. Because it is supercooled a small perturbation, or fluctuation, can cause the start of a nucleation site around which the water will start to crystalise. The ice crystals have a symmetry which is less than the liquid water. With the water crystalised one has only certain configurations that look the same. The symmetry of the supercooled water has been broken and a hexagonal discrete rotational symmetry remains.

Now what does all this spontaneous symmetry breaking have to do with the mass of objects? The so called electroweak theory predicts that there are three particles of which one is massless (the photon) and the other two have a mass (the W and Z particles). Since these particles are intrinsically massless in a self-consistent theory, the mass has to come from a new type of interaction that acts like mass. Another way to look at it is to say that the symmetry between the particles is broken and one remains massless and the other acquire a mass. The configuration in which they all look the same (massless) is now broken or hidden by a particular configuration of an external influence, the Higgs field. Extending this mechanism also allows other particles like the quarks and leptons to become massive.

Spontaneous symmetry breaking occurs at a phase transition in which one phase has a hidden symmetry and the other does not. The phase transition can be of two types: first-order or second order transition. In the first order case there is a discontinuous transition in the derivative of the free energy with respect to some thermodynamic quantity. It is associated with latent heat in which the system absorbs or releases energy but the temperature remains the same. It is not yet clear whether the electroweak phase transition is first order but if it were it could possibly be important to baryogenesis. In the early twentieth century Lev Landau explored the physics of phase transitions and the theory now bares his name.

How does spontaneous symmetry breaking work in a more ordinary example? There are many places where spontaneous symmetry breaking can be shown. Let us consider what I shall call spontaneous symmetry baking as an example. In the kitchen first order phase transitions happen all the time. Freezing and boiling are examples of phase transitions where the temperature remains constant while energy is transferred. So to tell you now what the Higgs mechanism does with a food example we must cook something that is symmetric at a high temperature and loses that symmetry at a lower temperature. Ideally we could have a field that acquires a mass and one that does not in the less symmetric phase. Unfortunately cooking is non-relativistic so I cannot turn enough energy into mass to actually create matter. Still to give you an edible idea I'm going to describe a tasty phase transition which is the tempering of chocolate.

Chocolate

What better way to show the Brout-Englert-Higgs mechanism than with chocolate, preferably Belgian, since the first two gentlemen who invented it are Belgian. Everyone who has melted chocolate and made some deserts with molten chocolate knows that, after you melt the chocolate and pour it into your mold or over your desert, the result is often a dull brown looking. Not anymore like those beautiful shiny chocolates you can buy at the patissier. The solution to this conundrum lies in the fact that something like a 'Higgs field' has not broken the symmetry correctly, and we end up with a dull chocolate universe 'in a wrong vacuum'. Let me explain what this means.

Tempered chocolate is shiny and hard

Chocolate consists mainly of cocoa butter, which contains triglycerides. The triglycerides can form crystals when cooled, in fact there exist six different crystalline states with different melting points. To each of these different crystals belongs a different phase. The crystal structure known as beta 2 or V (five) has the shiny crunchy properties we look for in good chocolate, it also has the second to largest melting point (33.6 C). When all the chocolate melts just below 33.6 C only the beta 2 crystals survive. To let the rest of the liquid phase make the transition to the beta 2 crystal phase it needs to be cooled slowly, this is called tempering. When the cooling is done too quickly one also gets spontaneous production of other forms and the symmetry will be broken in a different phase. When no more beta 2 crystals are left, one can seed the chocolate with some external beta 2 crystals from cocoa butter.

The reduction of the symmetry to the crystal state can be seen as the electroweak breaking where the Higgs field can be viewed as the temperature which lowers and causes a tempering process that causes the symmetry to be broken into the beta 2 form. The symmetry that is now hidden from us, was in the cocoa butter crystals all along but at high temperature. The realisation of the stable, less symmetric, lower energy configuration (which crystal form was chosen, what physicists would call a "vacuum state") was in this case seeded by the beta 2 crystals in the chocolate.