Higgs Mechanism in a nutshell
Welcome back to a new edition of our monthly newsletter which is envisioned as a venue where a broader audience can get exposed to calculations in Theoretical Particle Physics. Today we will talk about the Higgs mechanism which is the generation mechanism of the property called "mass" for gauge bosons [1]. This mechanism also "gives" mass to the elementary particles called fermions. Last time we were discussing elementary particle generations in the Standard Model, wondering why there are two more generations each containing copies of the particles we see in the first generation and, strangely, these two generations are not at all part of the world we experience. To refresh your memory the world around us is just made of the first generation of elementary particles: electrons, neutrinos, and two types of quarks: up and down.?Also, it is useful to always keep in mind that the deep underlying reality is the quantum field and elementary particles are viewed as excitations of the respective underlying quantum field.?
And just a gentle reminder, in the Standard Model, we have the following fields:
Bosons are mediators of forces, so photons are mediators of the electromagnetic force, gluons are mediators of the strong force and the?W+, W-, and Z bosons are mediators of the weak force. For example, charged leptons like electrons, exchange photons. Precisely, when two electrons interact via the electromagnetic force, what is happening is two quantum excitations of the electron field exchange virtual photons. Two quantum excitations of the quark fields exchange virtual gluons. They can also exchange virtual W+, W-, and Z bosons, etc.
But, Higgs bosons are special! They are not mediators of a force. Higgs bosons [2] are quantum excitations of the Higgs field. It is a field that?gives mass to other subatomic particles, such as quarks and electrons or W+, W-, and Z bosons. In other words, Higgs bosons fill the space and supply all the elementary particles with mass when these particles interplay with them via the Higgs mechanism.
Let's now dive gently into the guts of this mechanism!
Spontaneous Symmetry Breaking
The guts of the Higgs mechanism is spontaneous symmetry breaking. We mentioned Lie groups as a natural model for the concept of continuous symmetry in our very first post. If a symmetry depends on where you are in space-time it is called "local", otherwise is "global". It turns out, locally gauge-invariant theories play a crucial role in particle physics, so-called Yang-Mills theories.
Now, these theories require those gauge bosons, like the?W+, W-, and Z bosons - mediators of the weak force, to be massless. But for a realistic description of the short-range interactions, which is the weak force, massive gauge fields are needed.
On the other hand, we want to keep the gauge symmetry since it allows well-defined calculations of physical quantities, without infinities and divergences. So we have a problem: How to keep the gauge symmetry and at the same time give the mass to the short-range interacting massive gauge fields, like the?W+, W-, and Z bosons?
This problem is resolved by spontaneous symmetry breaking.
The spontaneous breaking of a symmetry example
We have a very important concept in physics called the Lagrangian, usually denoted by L. This is essentially a function that contains all the physics we want to deal with. Consider this Lagrangian
that describes the self-interacting field
The first term in the Lagrangian describes the kinetic energy of the field and the other two terms are for the potential energy of the field. If you watch carefully the Lagrangian, you can see that the Lagrangian stays the same if you do this replacement
We say the Lagrangian is symmetric under group
Now, we assume that
The next step is to look at the potential which is given by
and to find out for which field value this potential has a minimum. This involves some math, calculating the first and second derivatives of the potential, determining the value of the field for which the first derivative vanishes, etc
There are two solutions
The first solution where the value of the field is zero corresponds actually to a maximum since
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while the second solution is a real minimum, actually two of them
It is possible to choose either one of these 2 minimums as the ground state, we call them vacuum states, but not both at the same time. To emphasize, different vacuum states correspond to different worlds.
If you watch carefully, you notice that each one of the two vacuum states breaks the symmetry
This Lagrangian is a model for Buridan's jackass. Essentially, if the symmetry is preserved the poor jackass will starve. Choosing one pile of hay to feed on is a spontaneous breaking of the symmetry.
So let's say we pick
as the ground state. It is convenient now to shift the field and introduce a new physically relevant variable
and rewrite the Lagrangian in terms of this new variable
And now something miraculous happened here!
The second term is a mass term for the new variable
It has the correct signature for the mass term while
It also appears that due to the third term
the Lagrangian violates the symmetry
But that is not the case since we can always go back to our old field
so the symmetry is still present - hidden, but not lost.
This is in a nutshell the Higgs mechanism. In the Standard Model, the role of the above fields is played by the Higgs field.
Stay well!
References
[1] Higgs Mechanism
[2] Higgs bosons