When you step on a scales and look down at the number, what that number is telling you is how much force the mass of your body exerts on the ground. It is Newtonian mechanics saying just that force equals mass times acceleration: gravity exerts the acceleration on the mass of your body, which creates a force on the scales plate. That’s your “weight”. “Mass” however seems a little more nebulous a concept. If you were an astronaut floating in space far away from the Earth you would have essentially no weight, but you would still have the same mass. Where does the mass come from? Certainly, from the atoms that make up your body, each of which has a given mass. But that’s just displacing the problem. Where does each atom’s mass come from? That’s an even more interesting question, because the “mass” of the atom at the very last is really “energy”. From the famous Einstein’s equation, we know that mass is equivalent to energy. It turns out the about 99% of the mass of an atom is actually made up by the binding energy of protons and neutrons within the nucleus. This energy is the result of the strong nuclear force. In fact, by summing up the mass of each three quarks we could never get the masses of the fermion particles: by summing 2 “up” quarks of 2.3 MeV and 1 “down” quark of 4.8 MeV, we get 9.4 MeV, while the mass of the proton is 938 MeV. So, strong-force interaction energy is where 99% of our body’s mass is found. As far as we know, protons and neutrons are made up by quarks. Again, then, where the (little) mass of quarks (and electrons) come from? And this is where the Higgs field comes into the picture. A field that is dispersed everywhere in the spacetime.
According to the Standard Model of elementary particles, all fundamental particles in the universe are excitations in a quantum field: the excitation of the electromagnetic field is a photon, the excitation of the electron field is just an electron, an excitation in any one of the six quark fields is a quark. In this view, a field is a wave, but when it is “localized” as in an experimental measurement, it appears to us as a particle. Quantum fields span all of the spacetime, in all directions (and Higgs’ is not special, in this respect). Every type of matter particle and force-transmitting particle has its own quantum field, which in its ground state has oscillatory modes even if no excitations are present. In fact, because of Heisenberg’s principle, pairs of particles and antiparticles in each field are continuously being created and annihilated, for a time shorter than the inverse of their energy (times Planck’s h). In this process of creation, the field “borrows” energy from the vacuum state, and when the virtual pair is annihilated this energy is given back to the vacuum. All these energy fluctuations add up to zero net energy, and zero real particles actually observable.
Real particles appear only when enough energy is transferred to the quantum field from interacting with some other field, thus producing an “excitation”. The excitation is the real particle, and since the quantum field is quantized, excitations can only occur in discrete quantities (we never observe a half of a particle). What gives that exact value, of which each field must be an integer multiple, such as the value 0.511 MeV for the electron field or 938.27 MeV for the proton field, is its interaction with the Higgs field. Without such an interaction their mass would be zero. For example, if we suppose the electron would be materialized when its field interacts, e.g., with a quark field, in this (hypothetical) case the electron would only have momentum, and its energy would be E=cp, like a sort of “charged photon” traveling at the speed of light (just because its mass is zero). In fact, without the Higgs field all the particles in the universe would have zero mass. So, again, where this mass comes? We must go back to the concept of “vacuum expectation value” of the various particle-wave fields. You can imagine the electron field at any point in space as a harmonic potential centered in zero: fluctuations can move right or left of the harmonic parabola, but the average remains zero. This is its vacuum expectation value. This is true for any one of the quantum fields, with only one exception: you guessed it right… the Higgs field.
This field is special because, unlike every other quantum field, it has a net positive mass in empty space. That is, its vacuum expectation value is not zero, as for any other quantum field. The predicted value of the minimum of this potential is 246 GeV, and you should imagine this as a double-well potential (the so-called “Mexican hat” potential) with two symmetric minima at -246 GeV and a zero (higher) value at the midpoint. This represents the vacuum expectation value, or ground state, of the Higgs field. Now, any field that interacts with the Higgs field, finds it in this non-zero expectation value and as a result it will gain some energy. The energy associated with this interaction is practically indistinguishable from an energy associated with a rest mass. Whenever a fundamental particle interacts with the Higgs field, it gains an energy, that is an intrinsic mass. Since the Higgs field has a positive value everywhere in the universe, all particles are constantly interacting with it. Such an interaction effectively appears to “slow down” the particle (that would be otherwise constantly traveling at v=c). If we imagine applying a force to move an electron, a reaction from its interaction with the Higgs field causes the electron to resist acceleration. This is what we would call macroscopically its “inertial mass”. So, for any practical purpose, an electron in vacuum behaves as a particle with a finite rest mass of 0.511 MeV, whose precise value is determined by the strength of the coupling (interaction) between the electron’s and Higgs’ fields. The fields of all massive particles are coupled to the Higgs field to some degree, the larger the coupling the higher the mass.
I see the question coming: but where the Higgs particle gets its mass from? Well, the simple answer is, the Higgs gets its mass of by interacting with its own field… I also understand that many could think that all this theoretical construction looks somewhat artificial, a sort of gigantic, sand-based, “ad hoc” explanation. Honestly, it could be. The one firm nail in the wood that really keeps the whole story together is the extremely precise prediction of the Higgs mass at 125 GeV, which was formulated years before the discovery of the particle at CERN (although, supporters of the incompleteness the Standard Model note that such mass value fits also with predictions from other theoretical models, such as supersymmetry). However, as we will see later, such a value is exceedingly small for the hard job that the Higgs particle has to attend, namely to supply mass to all the Universe.
The mechanism by which the Higgs field gives out mass to other particle fields is another form of spontaneous symmetry breaking. The Standard Model works with three fundamental symmetry groups, the U1, the SU2 and the SU3. Each one of these symmetry groups is associated with one of the fundamental forces, namely the electromagnetic, the weak, and the strong force, respectively. In the basic theory, all fermions, hence all particles like protons, neutrons, electrons, should be massless, which is clearly not the case in reality. However, this is the most natural solution of the model (ahem…). And, as you may already know, a physical system with degenerate ground state displays a spontaneous breaking of the underlying symmetry. Spontaneous symmetry breaking seems sometimes a difficult concept to grasp. Here is an explanation for the rest of us.
Consider a thin cylindrical bar sitting on its vertical. In the absence of any force, it stays still. If now you push on the top of the bar, it will bend in a curve (that is Euler instability). However, you cannot tell a priori in which direction in the transverse plane it will bend. The free rod has only one stable state and is not degenerate: its potential energy is a parabola with a single minimum corresponding to the straight position. Under the extra force applied the rod is slightly compressed in the vertical direction, and upon increasing the compression it may have a different ground state: the bent rod could find itself with lower energy than the straight, but compressed rod. If it is so, the rod “prefers” to go into the bent shape. However, this bent shape has an extra rotational symmetry: it could bend in any direction in the cross-section plane. Now its potential is a “Mexican hat”, with equal low values all around the central axis (pretty much like the Higgs potential), while the central value (the rod remaining straight) is somewhat higher. An interesting point then arises. If we neglect friction, an infinitesimal kick in the cross-section plane can set the rod in motion: it will continue to spin indefinitely about the vertical at constant energy, in its bent state for which any rotation angle has the same energy in the Mexican hat potential well. In quantum mechanics such an excitation would be a Goldstone boson, necessarily massless (quantum equivalent of “neglect friction”). As it turns out, each broken symmetry in quantum mechanics is associated with the appearance of a Goldstone bosonic excitation, for example magnons (spin waves) in a polarized magnet. Therefore, in the “naked” Standard Model everything is without mass, both the fermion particles and the bosons that mediate the forces between them.
In 1962 (just three years before being crowned with a Nobel prize he shared with Feynman and Tomonaga), Julian Schwinger derived in two subsequent papers (Gauge invariance and mass I, and II) an interesting toy model of quantum electrodynamics with just one spatial and one time dimension, exactly solvable. Being a theory of electrons, the model had necessarily to maintain charge conservation, and therefore display some form of gauge invariance of the Lagrangian. Schwinger’s main result was that gauge invariance of the electron quantum field does not necessarily imply a zero mass for the associated particle. Even more interesting, this toy model exhibits fermion confinement, since the derived interaction potential increases with the distance r (instead of falling off as 1/r , as in the Coulomb interaction). Hence, while it is a poor model for electrons, it can be a relevant toy model for quark dynamics. Second weird-interesting point, the model exhibits spontaneous symmetry breaking, and as a result it gives also a photon with a finite mass.
In the summer of 1962, Phil Anderson was visiting at the Churchill College in Cambridge, where he met with John Taylor. Over a coffee, Taylor told him that the problem of massless Goldstone bosons was something that people in elementary particle theory were actively worrying about, and indicated him the two papers by Schwinger that had just appeared a few months earlier in the Physical Review.
Anderson started thinking about the problem…