© Matt Strassler [October 14, 2012]
How does it happen that the Higgs field has a non-zero average value in nature, while the other (apparently-)elementary fields of nature that we know about so far do not? [Very fine point: other fields excepting the lowest-level gravitational field, called the "metric", that helps establish the very existence of space and time.]
First, it is impossible for fermion fields to have a large constant non-zero average value in nature. This is related to the difference between fermions and bosons; bosons can be non-zero on average, but fermions really can’t. So we can forget about electrons (and their cousins the muons and the taus), about neutrinos, and about the quarks. [Fine point: Fermions can pair with each other or with anti-fermions to make composite bosons, and those composite bosons can be non-zero on average. In fact this is true of the up and down quarks and their antiquarks, and it is true of electrons in a superconductor. But that's a long story, and not our immediate concern.]
What about the photon field, the gluon field, the W and the Z field? These are all bosons. In principle these fields could have a constant non-zero value on average throughout the universe. It is experiment, not theory, that says this isn’t the case. A large non-zero value for the electric field would have all sorts of effects that we do not observe, including most significantly an apparent violation of rotational invariance at large distance scales. The electric field is a vector (spin-1) — it points in a particular direction — so if it were non-zero, the direction in which its non-zero value points would be different from the other directions. See Figure 1, lower left.
By contrast the Higgs field is a scalar (spin-0) — it does not point anywhere. Other (non-elementary and non-relativistic) scalar fields include the density field of the air, the pressure field inside the earth, the temperature inside the ocean; at every point in space and in time, the density or pressure or temperature is just a number, whereas the electric field is a number and a pointing direction. So if the Higgs field has a non-zero value, it does not result in a preferred direction; see Figure 1, lower right. More bizarre is the fact that (because it is a relativistic field) it produces no preferred frame at all. For the density of the air, there is a preferred frame, because one is either at rest with respect to the air or moving through it. But this isn’t true for the Higgs field; all observers are at rest with respect to the Higgs field. Therefore the success of Einstein’s special relativity in describing all sorts of phenomena is not inconsistent with the presence of a non-zero value for a relativistic scalar field, such as the Higgs. In short the non-zero value of the Higgs field leaves the vacuum behaving much the way it does even when H=0; remarkably, you can only tells it’s there through its effects on the masses of particles (or by doing something dramatic, such as using the Large Hadron Collider to make Higgs particles.)
The simplest way for the Higgs field to end up with a non-zero value throughout the universe would be if has a non-zero equilibrium value H0 that appears in its Class 1 equation of motion:
- d2H/dt2 – c2 d2H/dx2 = -(2 π νmin)2 (H – H0)
(It has to be Class 1, not Class 0, for reasons that we’ll see when we discuss the Higgs particle.) In fact, the situation is a bit more complicated. The correct equation turns out to be
- d2H/dt2 – c2 d2H/dx2 = a2 H – b2 H3
where a and b are constants (whose squares are positive! notice the plus sign in front of a2H, and compare it with the minus sign in the previous equation) which we’ll learn something about in a minute. We can rewrite this as
- d2H/dt2 – c2 d2H/dx2 = – b2 H (H2 – [a/b]2)
Now if H(x,t) is a constant over space and time, then dH/dt=dH/dx=0, and so
- 0 = – b2 H (H2 – [a/b]2) (when H(x,t) is constant in x and t)
which has the solutions (slightly oversimplifying for now)
- H = 0
- H = + a/b
- H = – a/b
In other words, it has three equilibria rather than one. [Fine point: I'm oversimplifying here for the moment, but harmlessly.]
It isn’t instantly obvious, but the solution at H = 0 is unstable. The situation is analogous to the equation of motion for a ball in a bowl shaped like the one shown in Figure 2 — like the bottom of a wine bottle. This also has three equilibria, one at 0 and ones at ±x0. But clearly the equilibrium at 0 is unstable, in that even a tiny push will cause the red ball to roll far from x=0, a dramatic change. By contrast the equilibrium at x=x0 is stable, in that any little push will just cause the green ball to oscillate with small amplitude around the point x=x0, a not very dramatic change. (The same is true for the light green ball at x=-x0.) In a similar way, although H = 0 is a solution to the Higgs field’s equation, our universe’s history has been complex enough to assure that the Higgs field has been knocked around a bit, and so it can’t possibly be sitting there. Instead, the Higgs field ends up in a solution with a non-zero value, a situation which is stable.
We have known for decades, from a combination of experiment and theory, that the Higgs field’s value (which we traditionally call “v”) is 246 GeV. That tells us something about those two constants a and b: in particular
- a = v b = (246 GeV) b
So that determines a in terms of b, and we can rewrite the Higgs equation of motion as
- d2H/dt2 – c2 d2H/dx2 = – b2 H (H2 – v2)
But it doesn’t tell us what b itself is. We’ll learn more about the quantity b in the next article.
Now, although I’ve set things up so that H could be either v or -v, it doesn’t matter whether the value of the Higgs field is positive or negative (actually there are even more possibilities, see below); the world comes out looking the same, with the same physics, because nothing depends on the overall sign of H. This isn’t instantly obvious, but it’s true; one hint is that wherever you find H in the equations described in my overview of how the Higgs field works, it’s always H2 that appears, not H alone — and H2 doesn’t depend on whether H = v or H = -v. [Fine point: In fact, H is a complex field, with a real and an imaginary part, so H can be v times any complex number z with |z|=1; and in fact it's always H*H = |H|2 that appears in the equations, which is independent of z. Even that's not the whole story! but it's good enough for today.]
If you find a way (perhaps using the proton-proton collisions at the Large Hadron Collider) to push or disturb the Higgs field a little bit somehow, it will wiggle back and forth — i.e., waves will develop, of the form
- H = v + A cos[2 π (ν t - x / λ)]
where A is the amplitude of the wave, ν and λ are the frequency and wavelength, and the relationship between ν and λ depends on the precise form of the equation of motion, in particular on b and v. Since the Higgs field is a quantum field, these waves have a quantized amplitude, and quanta of these waves are what we call Higgs particles. We’ll look at the properties of these particles next.