I don’t plan on filling my second post with a lot of mathematical symbols (although some of my posts may be more technical in the future). Instead, I want to scratch an itch.
The itch is an allergic reaction to statements I often hear about strongly-coupled quantum chromodynamics (usually called QCD). Basically these statements are that it’s all solved and anything remaining is a minor detail. I’ve tried to scratch this itch before (on Peter Woit’s blog, Not Even Wrong), but I never get satisfaction.
QCD is the theory of how quarks and gluons interact. In doing so, they produce all kinds of phenomena, in particular:
- the very existence of hadrons, the strongly-interacting particles (like the proton and neutron). Leaving out the details, these are bound states of quarks, held together by glue.
- how everything from electrons to nuclei behave in collisions, at high energies.
We understand the behavior of QCD at short distances (or high transverse momentum) very well. Experiments probing short distances are pretty convincing that the theory is right. This is because of the property called asymptotic freedom, which tells us that quarks and gluons interact very weakly at short distances. This is the weakly-coupled regime. On the other hand very little is understood about why quarks are confined into hadrons or why the glue is massive (it is 99% of the hadrons’ mass!). This is the strongly-coupled regime.
Many physicists have tried to understand how confinement of quarks and the mass of glue (called the mass gap) follows from QCD. Even the Clay Mathematics Institute has gotten into the game, offering the weekly salary of someone who quits science to work on Wall Street.
My problem is with the claim that the strongly-coupled regime is understood, or nearly understood, a mere pimple on the beautiful wart of current theoretical ideas. Usually this claim is justified by arguing it is all just a string theory on a product of anti-DeSitter space and a five-sphere, with a few bells and whistles. But it’s a wrong claim.
The subject has a fascinating history, and I’m not going to summarize all of it here. Ken Wilson was the first person who saw how the strongly-coupled regime could be understood. It became clear that confinement and the mass gap could be true. Wilson also showed there was a strong-coupling expansion in which these phenomena were there. What is recovered is a kind of quark model, where hadrons form as color singlets. Unfortunately, extending this to genuine QCD is an open problem. The reason (as Wilson understood) is that this strong-coupling expansion has to be taken to many many orders to get the right strong-coupling description, where both asymptotic freedom and confinement are evident. And even that may not be good enough… Masses are multiples of an artificial scale, the lattice spacing. This scale has nothing to do with the QCD scale, emerging from dimensional transmutation.
The stringy models have the same trouble as Wilson’s. They are not guaranteed to describe real quarks or gluons. At best, they are phenomenological models. Just as with Wilson’s approach, the scale has nothing to do with the QCD scale.
Now there is a right strongly-coupled description of QCD, but we don’t know what it is. Wilson tells us how to find it. We start with QCD with a very large, nearly infinite ultraviolet cut-off. Then we integrate out all the short-wavelength degrees of freedom from the theory to get the strongly-coupled theory with a much smaller cut-off (say a few GeV). I wish I knew how to do this – it would solve the problem. This correct strongly-coupled description will be very complicated (with lots of features, called non-renormalizable operators). The probability of guessing it is zero.
Anyway, the message is this: We don’t yet understand strongly-coupled QCD.