QCD and perturbation theory...

Factual Questions
1

I’m probably asking for trouble here, but what the hell:

Why does standard perturbation theory not work well for quantum chromodynamics? And how do lattice theories help clean this up?

Here’s the deal: All I know about perturbation theory I’ve gotten from laymens’ popularizations (pup-culti physics-for-poets, basically); probably the most I’ve struggled with the subject is Feynman’s QED, and I dutifully performed every one of his excercises. I think I have the basic idea, using Feynman’s path-integral formalization, though I know I’m doing chimapnzee-level calculations of “Lagrangians” by simple graphical vector additions of arrows pointing in various directions, and subsequently finding areas to get amplitudes.

Feynman says in his book QED, basically, that while the interactions of quarks and gluons look pretty much the same in a Feynman diagram as the interaction of an electron and a photon, the value of the color force coupling constant (pretty close to 1, instead of 1/137.xxx for electromagnetism) makes every subsequent diagram (every interaction requiring more arrows, I guess) about as important as the last. Hence your calculations barely converge on the final answer, and the higher you go in energy (the more loops in the diagram, and the closer to the particle in space, I assume this means), the more wooly the problem becomes, until the calculations get completely out of control. This problem, according to Feynman, makes QCD a pathetic quantitative failure next to the incredible precision of QED.

So along come lattice theories, like Wilson’s loops. I get the basic idea: Only let quarks live at the vertices of the lattice, and only let gluons live on the sides of each cell in the lattice. You never even approach infinite energy (arbitrarily close distance to the quark) because you quantize space. Maybe this means you can chuck some diagrams, I don’t know. Thing is, not only do I have no real concept of how this improves things, within (or without) a standard QFT framework, I was under the impression that these lattice theories tend to lead to nonsensical answers, like probabilities that don’t add up to 1. Does anybody use this for QED? Would you bother, since QED is so much easier? Come to think of it, it’s not at all clear to me how anyone calculates anything in QCD reliably these days without just putting big-ass computers on the task of calculating huge numbers of diagrams. Also, is the use of these lattice theories just an act of desperation in the face of trying to do perturbative calculations in QCD, hence physicists learn to live with their limitations (like overall nonsensical probabilities)?

Can anybody help a poor, dumb, but curious biologist make some sense of the current state of QCD and the challenges it faces, both in the perturbative and nonperturbative realm? Everybody loves to talk about the next big TOE, but I’m getting bored and frustrated hearing people ramble on about 26 dimensions that you can never observe. Seems to me there’s still plenty of tough, worthwhile physics out there left to be done, and while not as mind-blowing, at least people can test their hypotheses with experiments. I wanna talk about that stuff for a change.

Thanks so much!

2

I’m so dumb I can’t even spell chimpanzee correctly.

If somebody can calculate the magnetic moment of the muon to twelve decimal places, why in the HELL can’t the SDMB let you edit your posts!

Sorry, carry on.

3

There’s a bit of a myth here. Both QED and QCD are divergent. In both cases, you can calculate a perturbation series for many quantities and get definite numbers. In both cases, neither series will mathematically converge to a particular number.
Historically, the thing about QED was that lower terms of this series gave a good approximation to reality at low energies. By contrast, it was really only in the 1980s that QCD perturbation series began to match experiments at all. (The difference is tied up with the concept of asymptotic freedom, which QCD has, but QED doesn’t, so how good their estimates are depends on the energy involved.) Since then there have actually been many experiments that have tested QCD perturbation theory. Feynman was writing in a period that’s now rather dated, though much of the change has been due to the fact that experiments now probe higher energies.

Yes, people do do it for QED, though it is a particularly esoteric field. How QED behaves nonperturbatively is an enormously controversial issue. The best guess is that it’s actually totally trivial and hence that electromagnetism only makes physical sense when coupled to other forces. That’s an important question to answer. Hence people do explore QED in as many ways as possible.

There’s a whole range of techniques that can be brought to bear. Getting computers to churn away at the integrals involved is an important part of the field. Lattice gauge theory is another activity. But a lot of QCD calculations are still done by hand, others involve heuristics and some involve explicitely clever insights.

Sort of, sort of not. As someone who has worked alongside them, but has never been a lattice gauge theorist myself, I tend to regard their approach as mainly, but not entirely, brute force. All calculations in quantum field theory usually involve the notion of regularisation. Superficially, the theory only makes sense if you impose some sort of limitation on it. It doesn’t apply at high energies, it only works if stuff couples to some weird species of particle … Whatever. Lattice theories are just an example of one of these limitations. The theory only applies above some arbitrary length scale, so you can approximate space at shorter distances with this grid. As these things go, that’s actually arguably possibly the most sensible limitation to make. It’s hardly weirder than any of the others. The underlying point is that you impose an approximation, extract results and then (the subtle bit) understand how those results depend on the original approximation.
Even beyond this, lattice gauge ideas and results have been enormously important throughout physics. Wilson would have got that Nobel even if he’d never gone anywhere near QCD.

Having devoted a chunk of my life to this stuff, I’m biased, but I couldn’t agree more.

4

I still remember the day I found out that QED diverged. It made me so sad I cried. All my friends told me, “For heaven’s sake, even Sterling’s approximation diverges.” That just made me cry harder. QED is tame in a way, in that the divergence does not show up until you have summed up a lot of terms. For a long time the series seems to be converging. At low energies, QCD series diverge right off the bat, with certain exceptions.

5

QED doesn’t converge? None of them do?

Geez, does this mean we’ve known perturbation theory was an approximation before people got all hung up on quantum gravity?

I’m curious about this…

I seem to remember reading about the various GUTs and about how, if you plotted the strengths of the three non-gravitational forces (I guess there’s really only two, but you know what I mean) vs. increasing energies, EM and weak appear to get a bit stronger, while the color force gets weaker.

Now I may be totally wrong about this, but it seems to me if they’re saying the force gets stronger/weaker, they’re saying the coupling constant isn’t “constant”. Coupling constants are, so far as I know, related to the probability that a fermion that feels a particular force will emit or absorb a force-carrying boson. If the color force gets weaker, I assume it must mean that quarks are less likely to absorb or emit gluons. And if the EM force gets stronger, then electrons (for example) must be more likely to absorb or emit photons. Right? So in different energy realms, the additional terms in a perturbation series must be better or worse (depending on the force) to deal with, and it all has to do with the “effective” values of the coupling constant. Is this correct?

Interesting what semantics will do to one’s brain…coupling constant?

6

OK, now I’m really boggled…

Does all this mean one ought to use perturbative methods or non-perturbative methods depending on both the force and the energy realm? So perturbative to non-perturbative for EM with increasing energy, and the opposite for color with increasing energy?

And, what the heck, now that I’m more baffled then before, what is non-perturbative QFT, if not a lattice theory? Are lattice theories both perturbative and non-perturbative depending on the particular theory or method of calculating interactions?

7

I’m not sure what you mean by:

Peturbation theory is by defintion approximate and it’s used when either there is no method of obtaining an exact solution or an exact solution requires too much mathematical analysis (it has a variety of applications, not just QFTs; for example it’s used in non-relativistic quantum mechanics too). Therfore a non-peturbative QFT is one that does not utilize peturbation theory which does not necessarily make it a lattice theory, that would obviously depend on the method used to obatin the solution non-peturabatively.

8

I think I know what you mean. It could just be my misunderstanding of the popular literature on the subject. Usually when one encounters a layman’s book on, say, quantum gravity, one first gets this intro-to-GR-and-QM section, so that one can maybe comprehend what comes later. In the typical intro, one gets told how marvelously precise QED is. One gets the impression that while, yes, perturbative calculations in QED are approximate in practice, in principle, one could just keep calculating and calculating. Then they say, “well, this doesn’t work at all in quantum gravity because of graviton self-interactions blah-blah can’t be renormalized blah-blah-blah”, and cap it all off by saying “so QED must be a low-energy approximation, becuase at high energies, these forces all become one, and the final TOE should work in any energy realm”. It sseems like gtbiehle is saying it’s been known all along that you can’t keep calculating in perturbative QED as far as you like, because even though it looks like it converges at first, it eventually does diverge.

Add to this that QCD seems to get easier and more amenable to perturbation theory at higher energies, and it’s an interesting picture…kind of a mixed message about the nature of high-energy physics, if you ask me. I mean, what’s the deal with this “low-energy-approximation” stuff if the color force calculations get more precise the higher you go?

9

Lots of follow-up questions, so here goes …

Yes. Being very pedantic, we’ve always known that perturbation theory is an approximation. More to what I think you mean, we haven’t expected it to be a convergent approximation since 1952. And that expectation had nothing to do with quantum gravity. Historically, the original attempts to calculate higher orders of perturbation series in QED had made no sense whatsoever. As you probably know, that problem was addressed by the procedure of renormalisation as developed in the late 1940s and early 1950s. Once we could calculate definite numbers for the higher terms in the series, the mathematical properties of this series immediately became an obvious issue to think about. This question was addressed in a famous short paper in that year by Freeman Dyson, who came up with a plausible argument for why QED perturbation theory had to be divergent.
In the decades since, the behaviour of such series has been a much studied issue. Various other ways of approaching the question have been developed and the consensus is very much that any expansion in a coupling constant in quantum field theory is likely not to converge. QED and QCD thus have a very similar status.

gtbiehle is correct. Calculate enough higher orders in QED and your answers will start to get worst. The thing is that QED is still a stunningly good approximation in practice and will continue to be for the foreseeable future. Because we’re nowhere near calculating enough of those higher orders for us to run into this problem.
Where quantum gravity tends to get dragged into the issue here is the (probably correct) observation that its effects are likely to become “big” long before we have to worry about this problem with perturbation theory. Rather than deal with tiny corrections that we can’t estimate perturbatively in QED, by then we’ll hopefully have to be worrying about electromagnetism as part of some Theory of Everything. Which is a problem we’ll gladly grapple with when we get to, but which doesn’t necessarily have anything to do with our current ways of extracting numbers from QED.
[Actually, that argument is rather old-fashioned, but this isn’t the post to expand on the nuance involved.]

Roughly, yes. What one naively identifies as the “coupling constant” turns out not to be constant at all.
As always in QFT, there are various ways of thinking about what’s happening. One, very rough, picture is the following. Suppose one has an electron sitting in space and one wants to measure its charge. One way of doing this is to take a test charge and fire it at this electron. From which it’ll scatter off. Exactly how it scatters from the electron gives a measure of what the electron’s charge is. But in QED one never has an isolated electron. It’s surrounded by a writhing sea of virtual particles popping out of the vacuum. Let’s simplify this surrounding “cloud” of virtual particles to electrons and positrons. So one has electron-positron virtual pairs briefly emerging from the space around the electron in question. The key fact is that this electron has an effect on how these virtual pairs behave. It’s tending to attract the virtual positrons and repel their electron partners. The cloud is thus polarised. This tends to mask the charge of the central electron. In other words, observed at large distances the electron appears to have less of a charge than it does at short distances. Translated into the effect on the test charge fired at it, the electron appears to be interacting with a larger coupling constant (which is related to the electron charge) when the test particle has greater energy and so penetrates deeper into the cloud. The effective strength of the force thus appears to depend on energy, with the electromagnetic force getting stronger with higher energies.
Though it doesn’t look like it in simplified accounts, it’s actually easier to consider this running coupling as the “real” coupling constant. Including in perturbation theory. Sure, there’s an abuse of language here, but it’s no more than an historical accident and it’s just one of those quirks of usage that the professionals get used to.

QCD indeed behaves in the opposite way to QED here. So in QCD the running coupling constant is getting smaller as you go to higher energies. Over the range of energies that have been explored experimentally, this change makes quite some difference. At low energies, perturbation theory breaks down entirely and is basically useless. But go to currently accessable energies and the coupling constant falls to values where, in conjunction with the first few terms of the perturbation series, the perturbative approximation is sufficiently good that you’ve got a reasonable test of the theory. To put some numbers on it, good tests of perturbative QCD these days are accurate to within a few percent. That’s crude compared to what’s usual in QED, but it’s excellent compared to, say, 20 years ago. Perturbative calculations therefore have a major role in testing QCD these days.

Ideally, a non-perturbative QFT is just, well, defined in some more-or-less mathematically precise fashion. That said, there are usually a number of definitions of what one might mean by a particular QFT and it’s usually non-trivial to prove whether these give the same results. Gliding over that complication, a QFT is invariably a horribly complicated mathematical structure/object/device, whatever the definition. To get any definite results out of it, you usually have to come up with some approximate version of it and calculate it. Perturbation theory is one such approximation and lattice theory another. In these instances, they happen to give their best results in QCD at high and low energies respectively, so they tend to be seen as complementary.
Additionally, lattice theories in practice involve all sorts of perturbative messyness. Admittedly, this is largely so that what they’re talking about can match up to the (more traditional) definitions that are used in perturbation theory.
I should add that there are several other approaches to approximating non-perturbative theories in QFT, all with their appropriate strengths and weaknesses. There are also some very important results that can be derived entirely non-perturbatively and so apply without any approximations.

10

Thanks for the lengthy and thoughtful reply. I think I understand some of this, more or less. I’ve come across this idea of “bare” charge in Feynman’s “QED” and other places, and I kind of gathered from Feynman’s explanation that it was this “bare” charge that was causing all the infinities. If I understand correctly, it looks like it’s infinite because if the electron is a mathematical point, you can get as close as you like to it. Since the charge is related to the probability that the electron will interact with itself, wich is in turn related to the strength of the field around the elctron, which can increase to infinity (since you can get infinitely close to a mathematical point), the probability of self-interaction is infinite, and hence the charge is infinite, according to un-renormalized QED. So, if you want to cacluate anything sensible, go make an experimental measurement of the electron’s charge, and subtract from your calculation (infinity-measured charge) and you get back (measured charge). This is all there is to “renormalization”, as I understood it.

I guess I never made the connection, but now it makes some sense: If you make an experimental measurement of the electron’s charge, and you really whip another charge at it with high energy, the less quantum haze is between the electron and the test charge, and the charge appears greater. So, when you take this effective charge, subtract infinity-it, and you get the charge, this charge indicates a value that is larger than 1/137. So, I guess what you’re saying here is that, if this QFT model of the “clothing” of the bare mass is accurate, the more “unclothed” it gets, the bigger the coupling constant gets. The bigger the coupling constant, the more important higher order perturbations get to the final result. If this coupling constant gets too big, the values never converge, because, just like in QCD at low energies, the next calculation in the perturbation series is too big and the final result makes no sense.

OK - IF I got all that right (and that’s a big freakin’ if), here’s what I don’t understand at all: According to all the GUTs, at higher energies, the strengths of the color and electromagnetic force should converge, and hence they become one force. So, EM is getting stronger at higher energies (I guess this must have something to do with the “unclothing” of the electron) and the color force gets weaker (I guess unclothing quarks makes them less, not more, likely to emit and absorb a gluon). Now, if the EM force and color force can be extrapolated to higher energies to have equal strength in the GUT realm, and in this realm QCD becomes amenable to perturbation theory, WHY does QED diverge? Seems like it should still be manageable all the way up to GUT energies, if all this is true. Or does this mean to say that only at certain energies does QCD become tractable, but if you go high enough (maybe higher than GUT energies), it starts to diverge again?

11

I’ve been deliberately avoiding dragging renormalization into this. Opinions differ, but there’s no necessary reason to regard running couplings as a consequence of renormalization, thought the two issues are terribly entangled in practice. Hence, to simplify matters, I’m going to ignore renormalization.

It may however help to explicitly introduce the idea of an asymptotic series.
A series like

1/(1-x) = 1 + x + x[sup]2[/sup] + x[sup]3[/sup] + …

is convergent, at least for |x|<1. Not only are the terms in the series are getting smaller and smaller, if you add them up then the approximation to 1/(1-x) gets closer and closer to the actual value of the function. In a loose sense, knowing the power series is just as good as knowing the actual function.
However, there are also cases where functions have power series expansions where the series doesn’t converge for any value of x. Try to add all the terms of these series up and the result just blows up. Asymptotic series behave like this, but also have the property that if you just calculate the first few terms you’ll get a decent approximation to the function you started from. Only the first few terms, mind you - continue to add terms in the hope that you can keep improving this estimate and eventually everything gets out of control. By that stage, adding extra terms in the series just makes things worse.
The neat thing is that just because the approximation goes rotten when you add lots of terms doesn’t necessarily mean that it’s rotten if you just use the first few. Just taking those first few terms can often give a pretty reasonable estimate of the function.In fact, there are lots of such asymptotic series throughout mathematics. Some of them are astonishingly good approximations - provided, of course, if you don’t push them too far.
Now it’s a general feature of such asymptotic series that they give a better approximation for smaller values of x. But, no matter how small x gets, they’re always an inexact approximation. They may be extremely good, but you still can’t take that extremely good approximation and get it to converge to the exact answer. You’ll always have to be content with merely getting close.

Both QED and QCD have perturbation series that are believed to be examples of just such behaviour. Neither converges, but both can give good approximations when x is small. (In fact, the series in both are believed to be about “equally” divergent, so it’s not as if QCD is actually much worse than QED in any mathematical sense.)
Where running couplings come into the matter is that, in both theories, the appropriate value of x depends on the energies involved. In other words, the coupling constant x runs with energy. Now if the series give good approximations when x is small, this means that perturbation theory works well when it’s applied at energies where this “x” is small. Because the series is still divergent, it won’t work perfectly. But the first few terms will still give a better answer than you’ll get applying it, by contrast, when x is large.
At currently explored energies, the x in QED is very small and so the theory can give stonkingly good answers. This doesn’t mean that the QED perturbation series converge. It’s just a consequence of them being asymptotic and so the beginning of the series being able to give very good answers for small x. As energies increase, in QED the relevant value of x increases, but it’s still small for all the energies explored in experiments to date.
The situation with QCD is a little messier. Historically, the energies it was explored at were sufficiently low that x was pretty large and so the asymptotic series were giving rotten results. Hence the need to rely on alternate approximations like lattice theory to describe what’s happening at low energies. But in QCD the value of x drops as the energy is increased. Thus through the 1980s and beyond the values of x dropped to the point where good comparisons between theory and experiment became possible, as noted in my previous posts.
Both theories thus suffer from the “problem” that their perturbation series diverge. In practice, at current energies both still give good practical predictions.

Okay, so what happens when we ramp up to GUT energies? Not a lot, in a sense. As you understand, the QED coupling constant gets larger and the QCD one gets smaller. The latter getting smaller will mean that QCD perturbation theory is getting better and better. True, QED perturbation theory will be getting worse and worse, but not by enough to be particularly noticable. Then at the GUT scale everything, except gravity, is unified into one theory.
Which theory will have exactly the same problem.
You can calculate perturbation series for this GUT, but they’ll still be divergent. Though they’ll presumably be asymptotic and so you’ll again be able to get sensible - though inexact - numbers out of them.

To be slightly flip, why should it be any other way? In all these cases, perturbation theory is an approximation we use to try to extract reasonable numbers from an exact theory. It’s not as if Nature exactly owes us one to ensure that our approximations are convergent. That they’re actually bloody good in practice seems fair enough as it is.

Bottom line: for the most part, perturbative QED and QCD are mathematically really rather similarly good approximations. One just happens to be more exact in the circumstances we can currently apply them to experiments.

12

Bonzer, thanks again for taking the time. I hope my questions don’t seem inane. It’s really a wonderful thing when folks take the time to try to make complicated and technical subjects like this comprehensible to mere mortals. I guess for some reason I always thought perturbation theory (at least when applied to QED) worked like the Taylor series you gave as an example, and it’s quite a surprise to find out the divergence is really a math issue, and not something related intrinsically to the physics of high energies. I guess (though I can’t put my finger on it), it seems perturbative QFT must be based somehow on an incorrect assumption, but the fact that it looks like it converges for a while suited everyone (as you suggest) well enough that it’s not worth worrying about.

I guess I’ve been going around with this idea in my head (at least as far as QED was concerned), that perturbation theory worked like a really complicated Taylor expansion, and it’s just a matter of doing the calculations; the only reason nobody has done them is because there aren’t computers big and fast enough yet. It seems like the only alternative to this misconception is that, thinking graphically, perturbative QFT is a very, VERY good parametric, but nothing more. I guess that should have been obvious, since you can get results using other methods, like lattice theories.

Hm. Somehow I thought, since you can derive a physical picture from the equations of QED (positron-electron dipoles, and shielding), this somehow represented the “real” equation. Hm.

13

Perturbation theory in QFT is a really complicated Taylor expansion. It’s just that a particular Taylor series needn’t converge. Even for much simpler functions, it may just be asymptotic.

You can. The thing is that the perturbation series are derived as an approximation from the “real” equations rather than being those equations themselves. As is lattice theory another approximation derived from those same equations.