When I wrote my article last week about the relation between the Higgs and gravity, emphasizing that there really was no relation at all, I said that the Higgs field is not the universal giver of mass. I cited four reasons:
- The Higgs field does not give an atomic nucleus all of its mass, and since the nucleus is the vast majority of the mass of an atom, that means it does not provide all of the mass of ordinary matter.
- Black holes appear at the centers of galaxies, and they appear to be crucial to galaxy formation; but the Higgs field does not provide all of a black hole’s mass. In fact the Higgs field’s contribution to a black hole’s mass can even be zero, because black holes can in principle be formed from massless objects, such as photons.
- There is no reason to think that dark matter, which appears to make up the majority of the masses of galaxies and indeed of all matter in the universe, is made from particles that get all of their mass from the Higgs field.
- The Higgs field, though it provides the mass for all other known particles with masses, does not provide the Higgs particle with its mass.
Although it doesn’t matter too much to the main point of the Higgs-and-gravity article (since the first three points are not in question), the editor of a leading physics journal, Robert Garisto, took issue with the fourth point, arguing that I was making a statement that really wasn’t right, or at least is too strong. His argument has some merit, though in the end, I stick with my statement. I think it’s worth describing what he had in mind (as best I understand it) and why I feel strongly that one should think about it differently. There are some semantic aspects to the disagreement, but there are also some interesting and important subtle scientific points. I don’t want to suggest that this discussion is really that big a deal — the very fact that we can argue about whether the Higgs field does or doesn’t provide the Higgs particle with its mass distinguishes the Higgs particle from, say, the W particle, whose mass indisputably arises from the Higgs field. But there’s something to learn here about quantum field theory and how the Higgs mechanism works.
What Garisto said was basically this: within the Standard Model (the equations for the known particles and forces along with the simplest possible Higgs particle), the formula for the Higgs particle’s mass is simply (as explained in this article, for those who’ve got a little math background)
- mh = √2 (h/2πc2) b v
- v is the non-zero average value of the Higgs field, equal to 246 GeV
- b is a quantity that determines how strongly the Higgs field interacts with itself
- h is Planck’s constant, and
- c is the universal speed limit (often called “the speed of light.”)
This is to be compared with the formula for the W particle’s mass
- mW = ½ (gW/c2) v
where gW is the number that determines how strongly the Higgs field interacts with the W field. These formulas look very similar, in that both the W mass and the Higgs mass are proportional to v — and so, what’s the difference? It would appear that the Higgs field’s value v determines both masses.
That this argument has a flaw isn’t instantly obvious, and is subtle, especially if you don’t have some technical experience. I’ve been slow to answer because it isn’t so simple to answer this in a form that non-experts can follow. [Fine point: And then, there's an additional complication which I haven't yet explained on this website, involving the effect of the strong nuclear force on certain particle masses, which means that making precise statements requires some very tricky caveats.]
To explain my viewpoint, let me start by making an almost-true statement:
- If the Higgs field’s value were zero instead of v, nature would have a massless W particle, a massless Z particle, a massless electron, massless quarks, massless neutrinos, and so forth.
- However, there would be no massless Higgs particle.
This is a crucial indication that there is something fundamentally different about the Higgs particle, as far as its relation to the value v, compared to the others.
A Minor Complication, Requiring a Caveat
Now this statement, though simple and clear, isn’t exactly right, because it turns out the strong nuclear force does something complicated: it creates, spontaneously, a composite (i.e. not elementary) Higgs-like field [which I'll call the ``Sigma field'', or ``Σ'']. The non-zero value of Σ is far too small to play the role of the what we call the real “Higgs field” H, but it does mean that if H were zero, the W and Z particle and the others wouldn’t quite be massless. So to evade having to explain this subtlety every single time I make a remark that’s not quite true, I’m going to call this the Σ caveat, and whenever you see me refer to it, you should understand this to mean that the nearby statement in the text is almost true, and would be precisely true if somehow the strong nuclear force were turned off, in which case Σ would not exist at all.
[Fine point: The correct version of the statement above about the masslessness of other particles is this: in the presence of the strong nuclear force and its composite Σ field, made from combining quarks and antiquark fields to form a new one, the masses of the various known particles wouldn't quite be zero, but they would be extremely small compared to their actual masses --- smaller than we observe by at least a factor of a thousand and in many cases much more. The exception is the Higgs particle's mass -- the mass of the particle of the field H -- which would not be similarly reduced.]
What’s at stake here? The issue is connected to very deep issues in particle physics.
For certain types of particles, masses can be forbidden by fundamental obstructions, and there are mathematical theorems that for masses to arise, something has to come along and violate the conditions of those theorems, and remove the obstructions. In the case of W particles and electrons and the rest of the known particles, the theorems apply, and what violates them is “v”, the non-zero value of the Higgs field. (Σ caveat: or the non-zero value of Σ.) But there is no such theorem for the mass of a Higgs particle, or indeed for any similar type of particle. The absence of such a theorem is intimately related to the existence of the hierarchy problem.
It is these theorems, and their limitations, that assure that the Higgs field being zero would force many types of particles to be massless (Σ caveat), but not necessarily all of them… and among the massive ones would be the particles of the H field themselves. (Should we still call them “Higgs particles” even when the Higgs field’s average value is zero? I still would, but Garisto wants to call by a different name. However, this point is semantic: it doesn’t matter what we call them, the important point is that they aren’t massless.)
No matter how we word the debate, the conclusion is the same: the Higgs field is not, in principle, the universal giver of mass to all the elementary particles of nature.
Still, we’ve seen above that in the formulas of the Standard Model, it sure does look as though the value of the Higgs field determines both the W particle’s and the Higgs particle’s mass. Both masses are proportional to v. So how do I explain that?
It’s an accident. The Standard Model is so very simple that this accident is unavoidable. In other theories that are even just a little more complicated than the Standard Model, it isn’t true that the Higgs particle’s mass is proportional to v.
In any theory with a single Higgs field, the W particle’s mass must be proportional to v [even more precisely, must vanish when H=0 (Σ caveat)], because of a theorem; but the fact that the Standard Model Higgs particle’s mass is proportional to v is an accident [and in general it need not vanish when H=0 (and on this point there is no Σ caveat)].
So let me now prove this to you. Apologies to those of you who don’t do math; you’ll have to take my word for the final conclusion.
First let’s recall what happens in the Standard Model, and its simplest possible Higgs. The Higgs field’s constant value satisfies the equation of motion
- 0 = a2 H – b2 H3 = – b2 H (H2 – [a/b]2)
where a and b are positive constants. This has an unstable solution at H=0; the stable solution is at H = a/b, so we identify v, the equilibrium value of the Higgs field, as simply a/b. The formula for the Higgs mass mh, obtained in this article, was quoted above in terms of b and v, but we should write it in terms of the parameters a and b that actually appear in the equation of motion,
- v = a/b
- mh = √2 (h/2πc2) a
Now in this case, because v = a/b and therefore a = b v, we can choose to rewrite the second formula as
- mh = √2 (h/2πc2) b v
which seems to imply the Higgs field’s value determines the Higgs particle’s mass the same way it determines all the other masses.
However, suppose we move away from the Standard Model very slightly; we don’t add any additional fields or particles, but we change the equation of motion to read
- 0 = a2 H – b2 H3 – d2 H5
where d, like a and b, is a positive constant. If you now follow the same mathematical logic used in this article and this one, you will find that v is given by one formula in terms of a, b and d, while mh is given by a quite different formula — and in this case the formula for mh cannot be written as proportional to v. Despite this, the formulas for the masses of the W and Z particle, the electron, the top quark and the rest continue to be proportional to v. [(Σ caveat)]
Where does the Higgs mass come from, then? We can’t say it comes from the Higgs field’s value v; they’re related, but not closely enough for one to say that v is entirely responsible for the Higgs particle’s mass. All we can say is that it arises in a more complex way from the quantities a, b and d in the equation of motion, and so we have to figure out where they come from — which has not yet been done.
The example given above is far from the only one. A few comments about others.
In other similar variants, it can happen that mh is zero even when v is not, and it can happen that v is zero even when mh is not. This is never true of, say, the W particle’s mass [(Σ caveat).].
And there are plenty of other variants, including ones with one or more additional spin-zero fields that are unaffected by the electromagnetic, weak nuclear or strong nuclear forces, with similar features.
Also there are variants where the formula relating v to the mass of, say, the top quark, can become more complicated. Nevertheless, it always remains true that its mass is zero when the Higgs field is zero [(Σ caveat).] This is not true of the Higgs particle’s mass.
So the point is that while v and the mass of the Higgs particle (or masses of the various Higgs-like particles that may be in the theory) are always given by functions of the parameters (such as a, b and d) that appear in the equations, only in the Standard Model (and a few of its simple variants) can one can write the formula for the mass of the Higgs particle in a form that naively looks very similar to the formula for the mass of any other known particle, such as a W particle. And this is a consequence of the underlying logical point: for particles like the quanta of the W field, the Z field, the electron field, the muon field, the quark fields, etc., it is impossible for their particles to have masses unless there is a Higgs field around [(Σ caveat)]. But that’s simply not true for the Higgs field H and its particle.