© Matt Strassler [November 27, 2012]
This is article 4 in the sequence entitled How the Higgs Field Works: with Math. Here is the previous article.
Up to this point in this series of articles on How the Higgs Field Works, I’ve explained to you the basic idea behind how the Higgs field works, and I’ve described how the Higgs field becomes non-zero and how the Higgs particle arises, at least for the simplest type of Higgs field and particle (that of the “Standard Model.” But I haven’t explained to you why there’s no alternative to introducing something like a Higgs field — why there’s a fundamental impediment to introducing masses for the known particles in the absence of this field. That’s the goal of this article.
- d/dt (d Z(x,t)/dt) – c2 d/dx (d Z(x,t)/dx) = – (2π c2/h) 2 m2 Z(x,t)
where Z(x,t) is a field, m is the corresponding particle’s mass, c is the speed of light, and h is Planck’s quantum mechanics constant. If the particle is massless, then the corresponding field satisfies the same equation with m=0, which I called a Class 0 equation.
Cases with m=0 include photons and gluons and gravitons, which are the quanta of the electric, chromoelectric (or “gluon”) and gravitational fields; they are all massless quanta (“particles”) traveling at the universal speed limit c. For electrons, muons, taus, all the quarks, all the neutrinos, and the W, Z and Higgs bosons, each one with its own mass, the corresponding field satisfies a Class 1 equation with the corresponding mass inserted.
Unfortunately, this isn’t the full story. You see, for all of the known elementary fields of nature that correspond to massive quanta, the violet equation that I’ve written above is illegal, at least as I’ve written it so far. Why? The problem is that we haven’t put in the weak nuclear force into our equations. And when we do, as we’ll see now, these simple equations can’t be used. More clever equations that lead to the same physical result are needed instead.
The problem is this; the equation we have just written is necessary but not sufficient. We need it to be true, but it’s not the only thing we need to be true. We’re leaving something out: the weak nuclear force. And the weak nuclear force and the violet equation above are not going to get along.
Now this could get rather technical if I went into it in detail, so I won’t. I’m going to explain this using equations that are similar to the ones that are actually used, but without giving the full story.
The Electron’s More Elaborate Equations
To see the problem, let’s consider it in the context of a particular field — we will take the electron field as an example. The problem is that the electron field doesn’t quite satisfy the equation I wrote above. An electron is a spin-1/2 particle, which means it not only moves but also incessantly rotates, in a way that is impossible to visualize — and it turns out the previous equation is only enough to describe how its position is changing, but not enough to describe what can happen to its spin. In the end it turns out the electron is actually formed from two fields ψ(x,t) and χ(x,t) satisfying two equations, of the form
- dψ/dt – c dψ/dx = μ χ
- dχ/dt + c dχ/dx = – μ ψ
where I’ve introduced the constant μ = 2π mc²/h to keep the equations short. Again, I’m lying to you slightly, because this is the equation for motion only along one direction of space, the x direction; the full form of the equations is more complicated. But the point is right; we’ll check in a moment that these two equations imply the violet equation at the beginning of this article. [Note: ψ and χ are often called the “left-handed electron” and “right-handed electron” fields, but without more math, this nomenclature is more confusing than illuminating, so I’m avoiding it for now. See this link.]
Now the sense in which these two fields jointly make up the electron field is that in an electron wave, the amplitude of χ and ψ have to be proportional to one another. In fact you can check that if you make a wave in both of them
- ψ = ψ0 cos (2π [ν t + x/λ])
- χ = χ0 sin (2π [ν t + x/λ])
where ψ0 and χ0 are the amplitudes of the waves, and ν and λ are their frequency and wavelength (which I’ve assumed are equal), we get the equations
- (2π) (ν -c/λ) ψ0 sin(2π [ν t + x/λ]) = μχ0 sin(2π [ν t + x/λ]) 0
- -(2π) (ν +c/λ) χ0 cos(2π [ν t + x/λ]) = -μψ0 cos(2π [ν t + x/λ])
- (ν + c/λ) ψ0 = (μ/2π) χ0
- (ν – c/λ) χ0 = (μ/2π) ψ0
These equations show ψ0 and χ0 are proportional; generally, if one is non-zero, so the other must be too, and if you make one larger, the other has to become larger too.
But look carefully: there are two equations, giving two relations that could easily contradict each other. The only way the two equations can be consistent is if there is an additional relation between ν, -c/λ and μ. What is that relation? Multiply the two equations together and divide by ψ0χ0 (which we’re allowed to do as long as both ψ0 and χ0 are non-zero — let’s assume that here) and we find
- ν2 – (c/λ)2 = (μ/2π)2
What is the implication of this equation? Suppose that we have a single quantum of a wave in the ψ and χ fields — a wave of minimal amplitude — in other words, an electron! Then the energy E = hν and momentum p = h/λ of that quantum can be obtained by multiplying this equation by h² and substituting μ = 2π mc²/h, giving
- E2 – (pc)2 = (mc2)2
which is Einstein’s relation between an object’s energy, momentum and mass, which of course an electron of mass m should satisfy.
This is no accident, because Einstein’s relation is true for a quantum of a wave that satisfies a Class 1 equation, and the two green equations for ψ and χ secretly imply that both ψ and χ satisfy a Class 1 equation! To see this, multiply the first equation by -μ and substitute the second equation
- -μ(dψ/dt – c dψ/dx) = (d/dt- c d/dx)(dχ/dt + c dχ/dx) = -μ² χ
which gives [using d/dx(dχ/dt) = d/dt(dχ/dx)] a Class 1 equation for χ (and a similar trick gives a Class 1 equation for ψ):
- d/dt(dχ/dt) – c² d/dx (dχ/dx) = – μ² χ
Having two equations instead of one is a clever way (invented by Dirac) of having spin-1/2 particles that satisfy Einstein’s relation for energy and momentum and mass. An electron is a quantum of a wave in the ψ and χ fields, which jointly make up the electron field, and that quantum acts as a particle with mass m and spin 1/2. The same is true for the muon, the tau, and the six quarks.
Naive Electron Mass and Weak Nuclear Force are Inconsistent
But unfortunately, this beautiful set of equations, set up in the 1930s, turns out to be inconsistent with experimental data. What we learned in the 1950s and 1960s is that the weak nuclear force affects only χ and not ψ! So that means the equation
- dχ/dt – dχ/dx = -μ ψ
makes no sense; the change in time of a field χ that is affected by the weak nuclear force cannot be proportional to a field ψ that is not affected by the weak nuclear force. A different way to say this is that the W field can convert the field χ(x,t) into the neutrino field ν(x,t), but it can’t convert ψ(x,t) into anything, so the version of this equation that arises when a W field is combined with it is undefined and meaningless:
- dχ/dt – dχ/dx = -μ ψ
W ↓ field
- dν/dt – dν/dx = ???
So this failing of the equations when combined with the weak nuclear force tells you (as it told physicists of the 1960s) that a different set of equations is needed… and the solution to this problem is going to require a new idea. That idea is the Higgs field.
Enter the Higgs Field: The Right Equations for the Electron’s Mass
At this point, the equations are going to become a little more tricky (which is why I didn’t explain this stuff right at the beginning.) You may want to read my non-technical article on what the world would be like if the Higgs field were zero. The structure described there is going to appear within the equations I’m about to write down.
We need to have equations for electrons and for neutrinos that allow for the possibility that a W particle turns an electron into a neutrino or vice versa… but only by interacting with χ (the so-called “left-handed electron field”) and not with ψ.
To do this we need to recall a subtle point: that before the Higgs field is non-zero, there are actually four Higgs fields, not one — three of them disappear in the end. What’s a bit confusing is that there are several ways to name them, each naming convention being useful in different contexts. In my post about the world with a Higgs field that’s zero, I called the four fields, each of which is a real number at each point in space and time, by the names H°, A°, H+, and H–; the Higgs field H(x,t) that I’ve referred to throughout previous articles in this series is H0(x,t). Here I’m going to name them as two complex fields — i.e., functions that have a real and an imaginary value at each point in space and time. I’ll call these two complex fields H+ and H0; the Higgs field H(x,t) that I’ve referred to throughout previous articles in this series is the real part of H0(x,t). After the Higgs field becomes non-zero, H+ gets absorbed into what we call the W+ field, and the imaginary part of H0 gets absorbed into what we call the Z field. [The complex conjugate of H+ is called H–; and since W+ absorbs H+, its complex conjugate W– absorbs H–.]
Now here’s a fact about the weak nuclear force: the particles in nature, and the equations they satisfy, have to be symmetric under the exchange of some of the fields with each other. The full symmetry is a bit complicated, but the part of the symmetry we need is the following:
- ψ is unchanged
- χ ⇆ ν
- H+ ⇆ H0
- H– ⇆ H0* (just the complex conjugate of the previous line)
- W+⇆ W–
The fact that χ ⇆ ν reflects the fact that these fields are affected by the weak nuclear force, while the fact that ψ is unchanged reflects the fact that it is not affected by this force. [Without this symmetry (and without the larger one of which it is a part) the quantum versions of the equations for the weak nuclear force simply don’t make sense: they lead to predictions that the probabilities for certain events to occur are greater than one, or negative.]
The required equations turn out to be (here y is the Yukawa coupling for the electron, and g is a constant which determines how strong the weak nuclear force is)
- dψ/dt – dψ/dx = (2π c2/h) y (H0* χ + H– ν)
- dχ/dt + dχ/dx + g W– ν = – (2π c2/h) y H0 ψ
- dν/dt + dν/dx + g W+ χ = – (2π c2/h) y H+ ψ
Notice that these equations satisfy the symmetry listed above. [Experts will note I have slightly over-simplified in multiple ways; but I hope they will agree that the essence of the issues is captured by these equations.] Note again that t and x are time and space (though I’m simplifying, since I’m only keeping track of one of our three space dimensions); c, h, y, and g are constants that don’t depend on space or time; and ψ, χ, W, H etc. are fields — functions of space and time.
Now, what happens when the Higgs field becomes non-zero? The H– field and the imaginary part of H0 disappear (in a way that I won’t explain here) from the equations, absorbed into other fields. The real part of H0 becomes non-zero, with an average value v ; as described in my overview of how the Higgs field works, we write
- Real[H0(x,t)] = H(x,t) = v + h(x,t),
where h(x,t) is the field whose quanta are the physical Higgs particles we observe in nature. And the equations then become
- dψ/dt – dψ/dx = (2π c2/h) y (v + h) χ
- dχ/dt + dχ/dx + g W– ν = – (2π c2/h) y (v + h) ψ
- dν/dt + dν/dx + g W+ χ = 0
These are the equations that, after the Higgs field takes a non-zero value v, describe the interactions among
- the electron field, whose quanta are electrons with a mass me = y v;
- one of the three neutrino fields, whose quanta are neutrinos (which are massless in these equations — putting in their masses requires small modifications that I won’t describe here);
- W fields, whose quanta are W particles, and whose presence implies the involvement of the weak nuclear force
- Higgs fields h(x,t), whose quanta are Higgs particles
Notice the equations no longer appear to satisfy the symmetry shown above in the red equations. This symmetry is “hidden”, or “broken”; its presence is no longer obvious once the Higgs field is non-zero. And yet, everything works the way it must to match what is observed in experiments:
- if the fields h and W and ν are zero in some region of space and time, the equations become the original green equations for the electron field, built as a combination of ψ and χ;
- if the W field is zero in some region, the terms involving h show that the interaction between electrons and Higgs particles are proportional to y, and therefore proportional to the electron’s mass
- if the h field is zero in some region, the terms involving W– and W+ indicate the weak nuclear force can convert electrons to neutrinos and vice versa, specifically by converting χ to ν while leaving ψ unaffected.
Let me bring this to a close through a quick summary. For spin-1/2 particles, the simple Class 1 equations
- d/dt (d Z(x,t)/dt) – c2 d/dx (d Z(x,t)/dx) = – (2π c2/h) 2 m2 Z(x,t)
that we studied up to now have to be made more elaborate, as Dirac realized; describing the electron and its mass requires multiple equations that imply the Class 1 equation but have more going on. Unfortunately Dirac’s simple equations aren’t enough, because their structure is inconsistent with the behavior of the weak nuclear force. The solution is to make the equations more complicated and introduce a Higgs field, which, once it is non-zero on average, can give the electron its mass without messing up the workings of the weak nuclear force.
We’ve seen how this works for the mass of the electron, as far as the equations for the electron field. Similar equations work for the electron’s cousins, the muon and the tau, and for all of the quark fields; a slight modification works for the neutrino fields. The masses of the W and Z particles arise through different equations, but some of the same concerns — the need to maintain certain symmetries in order that the weak nuclear force can make sense — play a role there too.
In any case, the behavior of the weak nuclear force, as we observe it in experiments, and the masses of the known apparently-elementary particles, as we observe them in experiments, would be completely inconsistent with each other if it weren’t for something like the Higgs field. Recent experiments at the Large Hadron Collider have provided what appears to be good evidence that the equations that I have described to you here, and the concepts that go with them, are more or less correct. We await further experimental study of the newly found Higgs-like particle, to see if there are more Higgs fields, and/or whether the Higgs field is more complicated, than I’ve described here.