This year marks a half-century since the discovery that a quantum field theory, now known as QCD (quantum chromodynamics), could be the underlying explanation for the strong nuclear force. That’s the force that holds quarks and gluons inside of protons and neutrons, and keeps protons and neutrons clumped together in atomic nuclei. This major step in theoretical physics occurred just a couple of years after it was discovered that a similar quantum field theory for the weak nuclear force (which includes W bosons, a Z boson and a Higgs boson) is mathematically consistent.
With these two breakthroughs came the sudden and unexpected triumph of quantum field theory, emerging as the basic mathematical and conceptual language for understanding the cosmos. It came after two decades in which most experts were convinced that quantum field theory was inconsistent, and only a stepping stone to something deeper.
This week I am in New York City attending two attached scientific meetings, both focused on QCD and other quantum field theories that share its key property, known as “confinement.” One meeting is hosted by New York University, and the other, the Annual Meeting of the Simons Collaboration on Confinement and QCD Strings, by the Simons Foundation. Many luminaries who have spent time on this subject are here together, ranging from David Gross, who co-invented the subject (and was a winner of the 2004 Nobel Prize), to brilliant graduate students who are hoping to reinvent it.
Why Does QCD Confine?
To say that “QCD confines” is to say that its force traps quarks and gluons inside particles such as protons and neutrons, collectively referred to as hadrons. [I have previously described this effect here, where I explain that it’s not quite true that a quark can’t ever come out of a proton, even though it can never be on its own.] But as I’ll describe in a moment, what’s really trapped, more fundamentally, are some of QCD’s own fields.
We physicists are all convinced that this is true. No quark or gluon has ever been observed outside a proton, neutron, or other hadron. Meanwhile, numerical simulations (known as “lattice QCD”, in which a tiny chunk of the cosmos is modeled in the computer as a small grid of points) show overwhelming evidence that quarks cannot exist on their own. Furthermore, we have a number of examples of theories with “supersymmetry” (more symmetry than that of QCD itself) in which there are multiple lines of mathematical evidence that quarks in those theories are confined. But no one can start with the math of QCD and prove, using mathematical and conceptual arguments alone without numerical simulation, that QCD predicts confinement.
Why The Electric Field and ELECTRIC Force Satisfy an Inverse Square Law
The mechanism of confinement is most easily explained by imagining a world slightly different from our own. It all goes back to an idea encountered already in first-year physics classes, known as Gauss’s law.
Gauss’s law explains why electrical forces satisfy an inverse square law — why, as the distance r between two electrically charged objects increases, the force between them declines as the square of the distance (F is proportional to 1/r2). The argument goes as follows: according to Gauss’s law, the electric field passing through a surface surrounding an electrically charged object, times the area of the surface where the field is non-zero, is proportional to the electric charge of that object. For instance, consider a ball of charge Q; its electric field will arrange itself in a spherically symmetric way, as illustrated in Figure 1. Now suppose we surround the ball with an imaginary sphere of radius R, and thus of surface area 4πR2. Then Gauss’s law states that the electric field passing through the imaginary surface is equal to the charge of the proton divided by the area of the imaginary sphere
- Electric field at the radius R equals Q / 4πR2
from which it immediately follows that if a charged object is at a distance r from the ball, the force exerted by the ball on that object falls as 1/r2.
If Electric Fields Confined
But now imagine that there were something about empty space that abhorred an non-zero electric field. More precisely, imagine that having a non-zero electric field anywhere in space would cost enormous amounts of energy. What would happen to the electric field around the ball? The field could not be zero, because then Gauss’s law would not be satisfied. On the other hand, its non-zero electric field would no longer spread out freely, because of the huge energy cost. What’s the poor field to do?
One possible answer: the field could form a narrow tube, as in Figure 2. We call this a flux tube. The existence of the tube still costs energy, but the cost is much smaller than if the non-zero field were spread out everywhere.
Let’s see what Gauss’s law says about this possibility. We can still draw an imaginary sphere as we did before, but across most of the sphere, the electric field is zero. Only where the tube intersects the sphere, in a region whose area equals the cross-sectional area A of the tube, would the electric field be non-zero. Thus Gauss’s law would then say
- Electric field within the tube at radius R equals Q / A
Since the cross-sectional area of the tube doesn’t depend on the size R of the imaginary sphere, the electric field in the tube doesn’t either — which means that an electrically-charged object inside the tube would feel a constant force, one that doesn’t decrease as the object moves away from the ball. Such an object could not ever escape from the ball, because no matter where it were placed, even many millions of miles away from the ball, an undiminished force would pull it back. This inability to escape is confinement.
Importantly, what you see from Fig. 2 is that it is not just charged objects that are confined. What is really confined, first and foremost, is the non-zero electric field itself, trapped within a flux tube. The confinement of electrically charged objects is an automatic secondary consequence of this primary fact.
You might ask why the tube doesn’t collapse to zero size or expand to infinite size. The tube can’t grow too wide because that would increase the cost for energy space from the electric field’s presence. But it can’t become too narrow either, because the energy inside the tube grows like the square of the electric field. That means that there will always be some A for which these effects balance.
Confinement HAPPENS IN NATURE, FOR SURE
The non-zero electric field found around electrons and protons most certainly isn’t confined. Nevertheless, confinement of other non-zero fields is observed in nature.
Non-zero magnetic field inside a Type II superconductor is confined; see Figure 3 for a rough illustration of such a flux tube inside a superconductor. These tubes, widely observed in experiments, are called “Abrikosov vortices”; Abrikosov was one of the awardees of the 2003 Nobel Prize. We do understand why a superconductor abhors the presence of a magnetic field; it’s a phenomenon known as the Meissner effect. [It is caused by the condensation of pairs of electrons known as Cooper pairs, which directly prevents a non-zero magnetic field from spreading out within the material of the superconductor.]
Meanwhile, the QCD analogue of the electric field, referred to as the chromo-electric field, is similarly confined into what are usually called QCD flux tubes or QCD strings. [I discussed the relation between QCD strings and string theory here; see especially Figures 3 and 4 of that post.] These tubes stretch between quarks, as in Figure 4, and that’s why quarks are confined. We can’t directly observe the flux tubes in experiments, as they’re far too short and thin, but there is considerable evidence for them. But we do not know exactly why they form — why empty space abhors a chromo-electric field. The fact that it does so has been dubbed a “Dual Meissner effect.” [Electric and magnetic fields are said to be related by “duality”, and so an effect that forces non-zero magnetic field into a tube is dual to one that has the same effect on non-zero electric field; the terminology has been extended to the chromo-electric field, as though empty space itself is a sort of dual chromo-superconductor.]
A proof that nonzero chromo-electric field must bind into tubes has remained elusive for fifty years. Not motivated? A million dollars goes to the clever person who figures it out. So get to work!