- Quote: Shrinking the electron’s rest mass slowly down to zero, we’d find that even in the cold of outer space, all electrons would escape from their nuclei. As particles of zero rest mass, they’d go sailing off into the universe at the cosmic speed limit, much like starlight. . .We are truly dependent upon the electron’s rest mass, small as it is; were it to disappear, nothing in or on Earth would last an eyeblink. This, in turn, highlights our secret reliance upon the Higgs field. If it didn’t exist, or if it hadn’t switched on so it could play its important role, atoms would never have formed.
- Endnote: I’m cheating very slightly here. Depending on exactly how the Higgs field’s effect is removed, electrons’ rest masses might become zero or might instead merely drop by a factor of several billion. Either way, our atoms would disintegrate even in the void of deep space, ripped apart by the (suddenly ferocious) photons of the CMB.
If the Higgs field were somehow discarded from the universe altogether, then electrons’ rest masses would definitely fall to zero. But if the Higgs field were simply switched off (i.e., in the language of chapter 15, if its value were set to zero), the result would depend on some details.
If the mechanism that switched off the Higgs field left the Higgs boson’s rest mass infinitely large (i.e., in the language of chapter 20, it the Higgs field were to be made infinitely stiff as it was switched off), then, again, electron’s rest masses would become zero.
However, if the switching-off mechanism left the Higgs boson’s rest mass finite, then this would not quite be the case; the electron’s mass would be greatly reduced, but not zero. The reason is quite subtle, and involves a feature of the strong nuclear force that I did not discuss in the book. [You will probably want to have read through most of the book, or at least through chapter 15, before trying to follow the reasoning.]
Very briefly, in chapter 15, I mentioned the possibility of composite cosmic fields: fields formed by combining other fields together, an effect made possible only by quantum physics. The strong nuclear force actually makes such a field. It combines quark fields together to make what is called a quark/anti-quark bilinear field, which I’ll call the 
Though this field is made from combining fermionic fields, it is a bosonic field, like the Higgs field itself. Not only that, this bosonic
As covered in chapter 21-23, we don’t know why the Higgs field is switched on; we just observe that it is. But we do know why the
Now, before understanding what would happen if the Higgs field were switched off, let’s first look at how the electron, up quark and down quark get their masses from the Higgs field. There are interactions among all the relevant fields, which I will indicate only schematically, with imprecise mathematics that captures the basic ideas:
where E, U, D and H are the electron field, the up quark field, the down quark field and the Higgs field, and ye is the interaction strength of the Higgs field with the electron field, etc. When the Higgs field is switched on, then the Higgs field H becomes a constant, usually called “v“, giving
which is what provides masses for electrons, up quarks and down quarks, where ye v is the electron mass, etc.
But if the Higgs field is switched off, this last step doesn’t happen, and you might think nothing at all would happen. However, the fact that the Higgs field interacts with both the electron field and the quark fields leads to an additional effect, through quantum physics, in which the electron field interacts indirectly with the quark fields. That effect comes from the first equation, and looks like
where mH is the mass of the Higgs boson. (You see that if the Higgs boson mass were infinite, this term would be zero and we could forget about it.) Remembering that the strong interaction turns the combination of quark fields into the quark/anti-quark bilinear field, this becomes
Notice this expression has a bunch of constants out front, followed by three fields: two copies of the electron field, and a single copy of the quark/anti-quark bilinear field, the composite field created by the strong nuclear force. What happens when the latter switches on? Well, it becomes a constant, which we’ll call w. The result is that we find
which is a mass for the electron.
We know roughly how large w is from experiment: it is a little smaller than the cube of the proton mass.
Experiment also tells us that
In nature, the Higgs boson’s rest mass is about 133 times the proton’s rest mass, and if that rest mass were left unchanged as the Higgs field were switched off, then the electron’s rest mass come out to be approximately
This is less than a trillionth of an electron’s current rest mass, and so would lead to atoms a trillion times larger than they are now.
To ionize a hydrogen atom [i.e. to pop the electron away from the proton] requires 13.6 electron-Volts of energy; this means that hydrogen gas must be heated to many thousands of degrees before the atoms fall apart into separate electrons and protons. But if electrons’ rest masses were reduced to a trillionth of what they are, the temperature needed to ionize hydrogen gas would similarly drop — down to a few billionths of a degree (Kelvin) above absolute zero. Since the cosmos as a whole has a temperature, left over from the Big Bang, of nearly 3 degrees above absolute zero, it could host no atoms in it — at least, not until it had expanded for billions of billions of years (much longer than its current age of mere billions), to the point that the cosmic temperature dropped to a few billionths of a degree.
If the mass of the Higgs boson were left even larger after the Higgs field were switched off, then the electron’s mass would be even smaller, atoms even larger and more fragile, and an even longer wait would be required before the universe would be cool enough for atoms to form.
