Matt Strassler [May 31, 2013]
In this article I want to discuss basic properties of the forces that we know about — four that we’ve actually observed, and a fifth — a new one — whose existence we infer from the discovery of the Higgs particle.
Specifically, I want to discuss what particle physicists mean in describing forces as being weak or strong. It’s terminology that you’ll often see, but unless it’s been explained to you, there’s no way to guess what’s actually meant. So here’s an explanation… a long one, but I hope it will give you many insights into how nature works, as well as raising more questions I’ll have to answer later…
“Weak” versus “Strong”
What do these terms mean? While in ordinary life you and I would think of a strong force as one that can pull us off our feet and a weak force is one that we can counter by stiffening our muscles a little, that’s not what particle physicists mean at all.
By “strong” and “weak”, particle physicists are not talking about whether a force is absolutely strong, or absolutely weak. It’s not about whether the force could break a window, or hold up a bar of gold. “Strong” and “weak” in this context are semi-absolute terms, in a sense that we never use in daily life, or even in undergraduate physics classes. It’s a way of talking that only emerged from a deep understanding of quantum field theory, the modern mathematical language used to describe the known elementary particles and forces. But it’s fundamental to the way particle physicists think about these issues today. So I’m going to start by explaining the rationale behind this way of talking.
Take two objects of some type, perhaps elementary particles, and place them a distance r apart. Suppose each exerts a force F on the other. Then we will say this force is weak if
- F is much less than (h c / 2π r² )
where h is Planck’s quantum mechanics constant and c is the speed of light. Often it is convenient in physics to use not h but
- ℏ = h / (2π)
In short, for particle physicists,
- a weak force has F r² much less than ℏ c
- a strong force has F r² about as big as ℏ c
[We don't generally encounter forces, even in our theoretical studies, that are much stronger than (ℏ c / r² ); typically such strength makes them so complicated that we end up thinking about them in a different way. Long story.] This is a measure not of whether the force is weak in an absolute sense, but whether it is weak or strong compared to typical forces that are to be found at a distance r. The question isn’t about the force; it’s about the force times the distance-squared, and whether that is smaller than or comparable to ℏ c.
To explain why this notion of strength is useful, I’ll illustrate the concept in the case of electromagnetic forces acting on simple charged particles, such as electrons, anti-electrons ["positrons"] and protons. Electrons have electric charge -e; protons and positrons have charge +e.
First, imagine two stationary protons, each with mass m and electric charge +e, a distance r apart. The electric force between them pushes them apart and has a strength given by the following formula
- F = ke2 / r2
[What is k? see below...] The same formula would apply for two electrons, which both have electric charge -e. For an electron and a positron [an anti-electron, with charge +e], the force would have the same size, but would pull them together instead of pushing them apart.
Now — what is k? It’s called Coulomb’s constant, and what its value is depends on how we define e, the basic unit of charge. But it won’t matter, because in discussing electric forces involving elementary particles, we’ll always see ke2 appearing as a group, together. So we don’t need to know how big k is; we only need to know, how big is ke2?
It turns out that if r is larger than a millionth of a millionth of a meter, then ke2 is approximately 0.007 times (h c/2π), where h is Planck’s quantum mechanics constant and c is the speed of light. So we may write the electric force, times r² as (approximately)
- F r² = 0.007 ℏ c
Since 0.007 is much less than 1, electromagnetism is a weak force, and remains so at all distances, down as far as we have measured.
It’s very mportant not to get confused here! Just because electromagnetism is a weak force in this sense doesn’t mean that the force between two protons is weak in an absolute sense. In fact, the electric force trying to push the two protons in a helium nucleus apart is comparable to the weight of a truck! All that force acting on two tiny tiny particles!!! But still, this is, for such a small distance, a rather weak force, and indeed a somewhat stronger force (the “residual strong nuclear force”) counteracts that electromagnetic repulsion, and holds the protons and neutrons of the helium nucleus together.
By the way, there’s a historical name for this number 0.007; it’s called the “fine structure constant” (because it sets the size of little differences in the energies of various configurations of atoms) and it’s normally called “α”:
- α = ke2/ ℏ c = 0.007 297 352 57
It’s one of the most precisely measured quantities in nature. Often people write it as approximately 1/137 (and many decades ago various people thought maybe there was something special about the number 137), but if you’re going to do that really it should be written as 1/137.0359990…
Ok — so why is the fact that α is much less than 1 an indication that this force should be thought of as weak rather than as strong?
Why α«1 means the electromagnetic force is weak
This is best illustrated in the case where the force is attractive, as it is for the electron and positron, or the electron and proton. The electron and positron are a little easier to start with, because they have equal mass m; they form an atom-like state called positronium, analogous to a hydrogen atom formed by an electron and proton, but more symmetric, with the two particles orbiting each other, rather than in hydrogen, where the electron orbits the nearly stationary proton. In fact, the formulas for hydrogen, for those who know them, apply to positronium too, with some minor changes (factors of 2) in a few places. (Yes, the electron and positron in positronium eventually annihilate and turn into two or three photons, but only after the particles have orbited each other many billions of times — which admittedly only takes a tiny fraction of a second.) For positronium, in its lowest-energy state,
- the typical speed of either particle is α/2 × c
- the typical motion-energy (i.e. “kinetic” energy) of either particle is mc² × α²/8 .
- the interaction-energy (or “potential” energy) of the two particles is -mc² × α²/2.
- the binding-energy B of the positronium state (the sum of the motion-energy and the interaction-energy) is mc² × α²/4.
- the mass-energy of the positronium state 2 mc² – B; and since the latter is much smaller than the former, the mass of the atom is just a tiny bit smaller than the sum of the mass of the electron and of the positron.
In short, because α is much less than 1, there are three essential and related facts
- The electron and positron move at speeds slow compared to the speed of light c.
- Kinetic, potential and binding energy B are all small compared to the electron’s and positron’s mass-energy, E = mc2.
- The mass of positronium is very close to the sum of the masses of the electron and positron.
All of these statements would be true no matter how large or small were the electron’s mass; they depend only on α being small.
Together these facts mean that in describing this atom-like state, Einstein’s theory of special relativity is not important; Newton’s laws of motion are good enough to make predictions, up to details that are no larger than α, i.e. to about the 1% level or better. And as we’ll see in the next section, this means the system is relatively simple. It can be described using quantum mechanics, which has rather simple mathematics, without need for quantum field theory, which would be necessary if Einstein’s relativity were important. The math for the hydrogen atom is the same as for positronium, and it is so simple that physicists learn about it as undergraduates, early in their first quantum mechanics class.
There’s another useful, though a bit less familiar, way to think about this. We should remember that electrons, like all elementary “particles”, are really “quanta”, tiny ripples in quantum fields. They are more like waves than they are like little balls. Consequently they vibrate, as all waves do: they have a “frequency” of vibration. And the time it takes between one vibration and the next — which I like, poetically, to call a particle’s “heartbeat” — is equal to hc/m. If α is small, then the time it takes light to cross the atom-like state is much larger, by 1/α, than the heartbeat of the particles that it contains. In this sense, positronium is relatively big. And since the particles themselves travel much slower than light, it takes even longer for the particles themselves to cross the atom-like state, something like 1/α² heartbeats.
Other things that could have been complicated are rather simple too, when α is small. For instance, the force exerted by the positron on the electron can cause the electron to become, sometimes and briefly, a virtual electron and a virtual photon. (Virtual “particles” aren’t particles; a real particle is a well-behaved ripple in quantum fields, but a virtual particle is a more generalized disturbance of those fields.) But that’s rare, if α is small. Even rarer, the virtual photon will itself be disturbed into becoming a virtual electron and a positron. Since there isn’t nearly enough energy around to make another real electron and positron, which would require energy of 2mc2 to come from somewhere (recall the motion and interaction energy are much less than this), it is very rare to make a virtual electron and positron. The fact that virtual particles are rare is why we can say so simply that “a positronium atom consists of an electron and a positron” — that’s indeed what it is, most of the time. Only in high-precision calculations do we need to be more careful, and remember that’s not always quite true. The same is true of a hydrogen atom: it is (almost all of the time) just one electron and one proton, held together by a simple electric force.
You can read about atoms here, where my descriptions are relatively simple. In this article I partially explained how the hydrogen atom’s size can be inferred from quantum mechanics principles, and you can use that result to see why the speed comes out to be αc, and why the motion-energy and interaction-energy come out to be ½ mc² α².
What would happen if α were closer to 1?
But now imagine making α gradually larger and larger, increasing toward 1. What happens to positronium? [I must warn you what I'm about to describe isn't a rigorous discussion! Those of you who are planning to be experts someday need to be more careful than I'm about to be.]
As α increases, the force (at any given distance) between the electron and positron becomes stronger, and since the force pulls harder, it pulls the particles in the atom-like state tighter together. The particles move faster, approaching the speed of light. The motion-energy of each particle becomes larger; and the magnitude of the interaction-energy becomes larger, so the binding-energy becomes larger — becoming comparable to 2m itself. Conequently, the mass of the atom-like state is no longer approximately equal to 2 m. The size of the atom-like state becomes smaller, so that the time it takes light to cross the state, the time it takes the particles to move from one side of the state to the other, and the time between heartbeats of the particles all become about the same.
The stronger force between the electron and positron makes it more common for virtual photons to be present; and the larger amount of energy flying around the atom makes it easier for the virtual photon to itself become a virtual electron and positron. And in fact, when that happens, it can become difficult to say which electron is real and which is virtual, because there are also powerful forces between the two electrons, and between the electron and either positron, and these can cause a particle that was real to become one that is virtual, and make the virtual one real, and back again. Meanwhile, the virtual electrons and positrons can also release or absorb photons, which may be virtual but may sometimes be real.
In fact the very distinction between real and virtual particles starts to become difficult to decide. Real particles are supposed to be well-behaved ripples in a quantum field. But with the atom-like-state so small, it really only takes a single heartbeat, more or less, for an electron or positron to cross this atom, at which point powerful forces will already cause it to change direction. How can we say that such a particle is a “well-behaved ripple”? A well-behaved ripple should, well, ripple for quite a while — for many heartbeats — before being affected by external influences. Here, our electron, while more like a real particle than the generic virtual particle, is still highly distorted, and doesn’t fit the definition of “real particle” anymore either. And the electron may not even be around for long. The production of a virtual electron-positron pair can be followed by the annihilation of the formerly-real electron with the newly formed positron, leaving the maybe-virtual maybe-real electron behind.
So instead of what we had at small α — a simple system of mass just below 2m, consisting of an electron and a positron moving at speeds well below the speed of light — we find, as α approaches 1, an exceedingly complex system, with multiple particles moving at or near light-speed, with a mass that is very different from 2m. (See Figure 1) It’s impossible to say how many particles are inside; do you count the real ones only? and if so, how do you precisely distinguish the almost real ones from the mostly virtual ones? The number of real particles can be constantly changing.
Those of you who’ve read about the proton may note some similarity. There are important differences too, but yes, the similarities are not accidental.
Now this is what is characteristic of a truly strong force; the objects that it forms are vastly more complex than atoms. Scientists were lucky, in a sense, that the first objects that were encountered on the road to the theory of quantum field theory were atoms. These are held together by a weak force — the electromagnetic force — and were easy to understand using the simpler mathematics of quantum mechanics, in which the number of particles is held fixed. Protons, by contrast, are held together by a strong force — the “strong nuclear force.” So it’s not that surprising that protons are much, much more complex internally than atoms. (And I haven’t even attempted here to tell you about some of the additional complications that arise there; those are covered in this article.) Inside a proton, the number of particles is continuously changing — which requires the much more complex mathematics of quantum field theory.
By the way, the electric force between two electrons is weak because α is small. The same is true for forces between any two elementary particles, because the all of the known particles have electric charges that lie between -e and e — for instance, top quarks have charge (2/3) e. You might, however, wonder about the force between an electron and a uranium nucleus, since the uranium nucleus has charge 92 e. Well, in that case the force does get pretty strong! But this only has part of the effect that I’ve described for strong forces, because changing the charge on only one of the objects involved (and in particular the heavy one) doesn’t increase the probability for finding virtual electron-positron pairs. That would only change if the electron itself got a much larger charge than e! So even a uranium atom remains rather simple compared to a proton…
How Weak is the Weak Nuclear Force? It’s Tricky…
How strong are the other known forces of nature? We’ve seen that electric forces have a strength α, at least at macroscopic and even atomic and subatomic distances. And over those distances, down to a millionth of a millionth of a meter, α is a constant; it doesn’t depend on r, which is part of why it’s such a convenient quantity. But in fact, the strength of a force can change with distance, which complicates matters. For electromagnetism, this isn’t so important; the effect is very small. However, for other forces, it is a big deal.
The so-called weak nuclear force is, of course, weak. Well, it is weak at macroscopic, atomic, and even nuclear distances. But in fact its strength isn’t constant. For distances large compared with ℏ c/MW ~ 3 × 10-18 meters, i.e. (about 1/300 the radius of an proton), where MW = 80 GeV/c² is the mass of the W particle, its strength αweak is (roughly)
- Fweak r² /ℏ c = αweak(r) = 0.02 e-Mwr/ℏc
The exponential e-Mwr/ℏc makes the force remarkably weak! Even at distances the size of a proton, this factor gives a suppression of e-300, which means the force has already decreased by a number so spectacular I can’t write it out here: it is a 1 with 130 zeroes after it. (That’s bigger than a `googol’, a 1 with 100 zeros after it.) And the force becomes rapidly weaker from there. Why? The same effect that gives the W particle (a ripple in the W field) a mass makes it impossible for a particle to distort the W field over long distances, in contrast to the effect of an electron or proton on the electric field. Consequently, the force generated by the W field is completely ineffectual at long distances.
But for even shorter distances,
- αweak(r) = 0.02
Notice this is several times stronger than the electromagnetic force! The weak force is not intrinsically weak at all. See Figure 2. (Minor Caution: I’m leaving out a subtlety involving the interplay of the weak and electromagnetic forces at such short distances, and a very slow change in the force that becomes noticeable at vastly shorter distances.)
What makes the weak force so weak, when we observe it in the physics of nuclei, atoms, and daily life is the large mass of the W particle. If the W particle were massless, effects of the “weak” nuclear force would be stronger than those of the electric force! [This is another context in which the Higgs field, which gives the W its mass, is really important to our lives!]
The Strong Nuclear Force
The strong nuclear force, which pulls and pushes quarks and gluons (but not electrons), does something quite different. At the distances we were just discussing for the weak force — 3 × 10-18 meters — the strong nuclear force is quite a bit stronger than both the weak nuclear force and the electromagnetic force:
- Fstrong r² /ℏ c= αstrong = 0.11 (at r ~ 3 × 10-18 meters)
That’s really not so strong; it’s about a tenth as strong as a really strong force, and only about ten times stronger than electromagnetism. In fact, although they differ enormously at macroscopic distances, the strong nuclear, weak nuclear, and electromagnetic forces differ in strength by only about a factor of 10 at distances shorter than about 3 × 10-18 meters. This is remarkable, and perhaps not accidental. It’s a small step from here to the notion of the “Grand Unification” of these three forces — the idea that at much shorter distances, all three forces end up with the same strength, and become part of a more universal force.
But at longer distances, the strong nuclear force gradually becomes (relatively!) stronger. [Again, remember what we mean by ``weak'' and ``strong'' here; the force is actually becoming weaker in absolute terms as r increases, but relative to, say, electromagnetic forces at the same distance r, it's becoming stronger.]
- αstrong = 0.3 (at r ~ 10-16 meters)
That’s quite strong indeed! And by the time r reaches 10-15 meters, the radius of a proton, αstrong is bigger than 1, and becomes impossible to define uniquely.
In short, the strong nuclear force, which is only moderately strong at distances far smaller than the radius of a proton, grows (in relative terms) at larger distances, and becomes a truly strong force at a distance of 10-15 meters. (This is shown in Figure 2.) It is this truly strong force that creates the proton and the neutron, and a weaker residual effect of this strong force that combines these objects into atomic nuclei. Other important effects of this force becoming so strong are the conversion of high-energy quarks and gluons to jets (sprays of hadrons).
Why does the strong force become gradually stronger as r increases? That’s a story for another day, but essentially it is a very subtle effect due to disturbances (“virtual particles”) in the very gluon and quark fields that are affected by the strong force. Similar effects impact the weak and electromagnetic forces, but have much less dramatic effects on those two forces, which is why I haven’t mentioned them before. (For instance, at the distance of 3 × 10-18 meters, the electromagnetic α is closer to 1/128 than its long-distance value of about 1/137.)
Given how strong the strong nuclear force is, why don’t we encounter it on a daily basis? It has to do with important details of how the strong force binds quarks and gluons and anti-quarks so tightly into protons and neutrons that we never observe them separately. This is in sharp contrast to how the weaker electromagnetic force allows electrons to escape from atoms rather easily, making the phenomena of static electricity (including lightning) and electric currents (including those in electrical wires) possible.
The Strength of Gravity
What about gravity? Well, for the particles we know about, gravity is amazingly weak. For two (stationary) particles of mass m, the force of gravity has a strength
- αgravity = GN m2 / ℏ c
where GN is Newton’s gravitational constant. Compare this with the case of the electric force, where α = ke2 / ℏ c; the role of k and e in electric forces is played by GN and m. (Here I am using Newton’s formula for gravity, but as long as αgravity is small compared to 1, Einstein’s formula for gravity between two objects is essentially identical.)
Now we can rewrite this in terms of what is called the Planck mass Mpl = 1019 GeV/c2, or about the mass of 10 million million million protons, or of 20 thousand million million million electrons. (This is about the mass of a
paper clip. [OOPS! Proofread Failure. This is obviously wrong. Speaking approximately, Mpl = 1019 GeV/c2 = the mass of 1019 protons = the mass of 1019 Hydrogen atoms = 0.00002 moles of Hydrogen = 0.02 milligrams = about a tenth of the weight of a typical grain of salt. Thanks to reader Al Schwartz for pointing it out.]
- αgravity = (m / Mpl )2
So for two protons, each of which has a mass of about 1 GeV/c2the gravitational force between them has a relative strength of the square of (1/10 million million million), or (10-19)
- αgravity = (10-19)2 = 10-38
which is a 1 preceded by 37 zeroes and a decimal point! Meanwhile for two electrons
- αgravity = (10-19)2 = 3 × 10-46
which, since an electron has a mass almost 2000 times smaller than a proton, corresponds to a force four million times weaker. Even for a pair of top quarks, nearly 200 times heavier than a proton and with the largest masses of any known particles, the gravitational force has a strength of only
- αgravity = 10-34
That’s about 100,000,000,000,000,000,000,000,000,000,000 times smaller than the electric force between two top quarks. That’s why gravity doesn’t show up in Figure 2.
If you think about it, this incredible weakness of gravity explains why you (using electric forces that power your muscles and keep your body intact) can move so freely despite being pulled on by the gravitational pull from the entire, enormous earth. In fact it even explains why the earth can be so much larger than a single atom; gravity wants to crush the earth, but the integrity of atoms, whose electrical forces resist this crushing, prevents this. If gravitational forces were much stronger, or electric forces much weaker, gravity would crush the earth down to a much smaller size and a much greater density.
Gravity is so weak that it’s amazing that we could discover it at all. So why was it the first force we humans knew about? The reason is that it is the only force that survives to very large distances in ordinary matter.
- The weak nuclear force becomes extremely weak at long distances. (We’ll see in a moment that the same is true of the Higgs force.)
- Electromagnetism survives to larger distances, but though not very strong is still strong enough to bind up most electrons and atomic nuclei into electrically-neutral combinations, whose electric forces on other objects cancel. [For instance, a hydrogen atom does not have pull on a distant electron, because the electron in the hydrogen atom pushes and the proton in the hydrogen atom pulls that electron, with forces that essentially cancel.]
- The strong nuclear force is so strong that it binds quarks and gluons and anti-quarks together into combinations that similarly have cancelling effects.
- But gravity cannot be arranged to cancel in this way. There are no particles that generate gravitational forces that push things apart, so you can’t combine two particles so that their gravitational forces on all distant things cancel.
The Higgs Force!?
As of 2012, we have a new force to think about: the force between two particles induced by the Higgs field! This is not to be confused with the effect by which the Higgs field gives the known elementary particles their masses; the Higgs field can do this to a single, isolated particle. That’s not a force; it doesn’t push or pull. But the Higgs field can also induce a force between two particles; this happens in much the same way that electromagnetic forces are created. However, as far as its effect on ordinary matter, this force is very, very hard to detect. At short distances, for particles like electrons and the up and down quarks that dominate the proton, the Higgs force is very weak (much weaker than electromagnetism, but much much stronger than gravity). At long distances, like the weak nuclear force, the Higgs force becomes extremely weak, because the Higgs particle, like the W particle, has a mass.
The Higgs field induces a force similar to the weak nuclear force in that it has a very short range, becoming ineffectual at distances long compared to ℏ c / Mh ~ 2 × 10-18 meters (1/500 of a proton’s radius), where Mh ≈ 125 GeV/c² is the Higgs particle’s mass. And at first glance the formula is similar to that of gravity, in that it is an attractive force proportional to the masses m of the two elementary particles being attracted.
- αHiggs = (mc2 /4 π v)2 × e-Mhr/ℏc (for r » 2 × 10-18 meters)
- αHiggs = (mc2 /4 π v)2 (for r « 2 × 10-18 meters)
where v = 246 GeV is the constant value of the Higgs field found throughout the universe. [Actually, if one is careful, there is an extra square root of 2 in there, but let's keep the formulas simple-looking.]
Be cautious! The resemblance to gravity is misleading. This formula is only precise for the known elementary particles — those objects that get their mass from the Higgs field. It works for electrons and muons and quarks. It is not correct for protons, neutrons, atoms, or you! That’s because a proton’s mass (and a neutrons, and therefore an atom’s, and therefore yours) does not entirely come from the Higgs field. This is in contrast to the formula for gravity, which is correct for all slow objects! Instead, for ordinary atomic matter, we’d have to replace the formula with one that looks similar but has a different factor in front, slightly different for each atom. Qualitatively, however, the dependence on the distance would remain similar.
Also, the formula I’ve written also assumes there is only one Higgs field and one Higgs particle (which we don’t yet know to be true, but is the simplest possibility consistent with current data.) If that’s not the case the formula will become more complicated, while remaining of a similar form.
How strong is this force? Well, at very short distances, shorter than 2 × 10-18 meters, the Higgs force between two top quarks is comparable to the strong nuclear force at that distance (see Figure 2)! But for electrons, which have a low mass because their interaction with the Higgs field is small, the force even at short distances would be much weaker than electric forces — more than a thousand million times weaker — though still thousands of millions of millions of times stronger than gravitational forces between electrons. Yet if you consider two electrons in an atom, which are about ten million or so times further apart than 2 × 10-18 meters, then the Higgs force between them is much, much smaller even than the tiny gravitational force between them, as it is suppressed by e-10,000,000. [Even if the Higgs field did give protons and neutrons all their masses, the Higgs forces inside a nucleus would still be vastly smaller than those of gravity, which in turn are incredibly small compared to the residual strong nuclear force that holds the nuclei together.]
It is the incredible weakness of the Higgs force in the context of ordinary matter that makes it so hard to discover. On the one hand, the Higgs force, like gravity, is always attractive and can’t be cancelled. But on the other hand, that’s irrelevant, because, like the weak nuclear force, the Higgs force does not survive to long distance, because the Higgs particle, like the W particle, has a mass. The Higgs force at ultra-short distances is much stronger than gravity, but at nuclear and atomic distances, it is much weaker, because of the Higgs particle’s mass. And for the low-mass particles out of which we’re made, which interact weakly with the Higgs field, the Higgs force is always thousands of millions of times weaker than electric forces, even at very short distances. So even though every atom in the Earth exerts a Higgs force on every other atom in the Earth, that force is so incredibly minuscule, even for neighboring atoms, and especially for distant ones, that it has no detectable effect. This is why we had to go find the Higgs particle directly to confirm the Higgs field exists; we couldn’t just go looking for the force it creates, the way we can use the observation of electric and magnetic forces to confirm that the world has electric and magnetic fields.
When might we actually observe this new force? It’s effects will first be observed either in the scattering of W and Z particles off each other (which will eventually be done, indirectly, within the proton-proton collisions at the Large Hadron Collider) or in the interaction between a top quark and a top anti-quark (which can be observed at an electron-positron collider — in fact I wrote my first particle physics paper [see in particular Figure 11 of the paper] about this very phenomenon.)