In order to understand the Higgs mechanism, we need to first understand how it came to be that the Higgs field, unlike all the other fields corresponding to the other 18 elementary particles, came to have a non-zero average value in the vacuum. As I said in the previous post in this series, this is the key fact about the Higgs field that leads to it giving mass to the other particles. So how did that come about?
In general, the stable equilibrium values of quantities tend to be those that result in the lowest energy states of the system as a whole, so that once a system enters that state, it stays there. For example, a rolling ball will come to rest such that its position corresponds to a point in space that has the lowest energy for the ball. It will not move from there without an external impetus. The subsequent average value of the position will then correspond to the actual position of the ball.
But there can be states that are in equilibrium that are not stable. For example, a ball exactly on the peak of a mound will be in equilibrium, but it is not stable. As long as it is undisturbed, it will stay on top. But if displaced slightly, it will roll down until it reaches a location that has the lowest energy for the ball and then stay there. It will not, by itself, roll back up to the top of the mound.
There can also be situations where the equilibrium state need not be one in which the average value of a quantity gives the lowest energy of the system.
Consider a rod that is hinged at one end to a table, such that it can rotate freely in a vertical plane. If left to itself, the rod will end up in stable equilibrium lying flat on the table, either pointing towards the east (say) or pointing towards the west, because those are the lowest energy states of the rod. So we can say that the lowest energy state of the rod corresponds to the rod’s length as measured along the east-west axis having its maximum value, equal to the length of the rod.
But there is another equilibrium state, and that is with the rod vertical. In that position, the distance the rod’s length measures along the east-west axis is zero. If perfectly balanced, the rod could theoretically stay upright but this equilibrium state is at a higher energy than the other two and hence, like the ball at the top of the mound, is unstable and the slightest perturbation will cause the rod to go into one of the other two stable equilibrium states.
One way to make the rod vertical is to carefully set it up that way. But there is a more ‘natural’ way of having the average orientation of the rod be vertical that does not require careful setting up. If the table to which the rod is hinged is shaken very vigorously and randomly, the rod will flip from one side to the other very rapidly. If so, its average position would then be the vertical one. So in the case of high table agitation, the rod’s state will be such that on average it has zero length along the east-west axis, even though this is not the lowest energy state of the rod.
But as the level of table agitation drops, there will come a point when there is not enough energy to flip the rod from one side to another, and the rod will come to rest in one or other position. The new stable configuration of the rod will be that with the rod’s length along the east-west axis being equal to the full length of the rod.
So to sum up, in a state of high agitation, the rod will be in a state in which the average value of the length of the rod along the east-west axis is zero but as the agitation drops below a certain critical value, there will be a sudden switch to a state in which the average length of the rod along the east-west axis is the full length of the rod, i.e., non-zero.
The vertical state of the rod is symmetric with respect to east-west axis. But when the rod is lying flat on the table in one of the two orientations, we say the ‘symmetry is broken’ because the rod suddenly and unpredictably gets oriented in just one of the states it could have had, and is no longer symmetric about the vertical position.
The Higgs field acquires its non-zero average vacuum value in an analogous way, where we make the identification of the average value of the Higgs field as corresponding to the average length of the rod along the east-west axis.
In the very, very early stages of the universe immediately after the Big Bang, the temperature was extremely high, corresponding to a state of extremely high agitation, and the Higgs field had an average (i.e., ‘vacuum expectation’) value of zero. This was because it was varying rapidly in such a way that the equilibrium state corresponded to a zero average value of the field. But as the universe cooled, the level of agitation dropped and at one particular temperature (which happened about one-trillionth of a second (10-12s) after the Big Bang) the wildly varying Higgs field got ‘locked’ into the lower energy stable equilibrium state that corresponded to it having a non-zero value. This process is known as ‘spontaneous symmetry breaking’ corresponding to a symmetric situation about the value zero spontaneously becoming non-symmetric by acquiring a single non-zero value, one that could not be predicted in advance.
This process of a randomly ordered state locking itself into a particular orientation upon cooling is quite common in everyday life. It happens with water molecules with random motion freezing into ice when the temperature drops below the freezing point, and with the little domains of magnetism in ferromagnetic material suddenly becoming all aligned in one direction below the critical temperature, giving rise to large magnetic effects. Such changes are also referred to as ‘phase transitions’. The transition in the case of the Higgs field is referred to as the ‘electroweak phase transition’ because, as we will see in the next post in this series, it is this transition that results in the weak interaction force particles W+, W-, and Z acquiring mass.
Next: The Higgs mechanism