How String Theory Solved a Sticky Problem, or Hadrons, Hydra of the Strong Force

I’m in the midsts of playing paper ping-pong with collaborators in Europe. That is to say that we’re writing a paper together and it’s only efficient for one person to be working on the draft at any one time.

In the mornings here in Beijing I have been writing the paper and performing calculations. Then when my collaborators come online in Europe I send them the draft and my latest findings, we discuss changes and additions and then as I leave the office in the evening here they start working on their changes, e-mailing it to me when they leave the office. I then pick it up Beijing am. etc.

We’re getting near the end, I hope, but there’s still more to do.

———-

Anyway, while the paper is in the other court I have a few minutes to explain roughly what I do, having introduced myself at length in the previous post.

My primary interest is in using string theory (see several previous posts to find video material on string theory at all levels) to try and understand one of the four forces of nature – the strong force. A force with a peculiarly tricky set of properties.

I’ll explain a little about the strong force (which holds the nucleus of an atom together and is felt by particles called hadrons) in a bit and tell you why it’s so hard to understand its properties.

It’s rather strange that we have any problems with understanding this force because we can write down exactly its fundamental constituents and interactions. In fact we’ve been able to do this for about 30 years. The theory is described by QCD (quantum chromodynamics). QCD describes how the quarks (the constituent particles of hadrons) interact with each other by exchanging gluons (the QCD equivalent of photons in electromagnetism)

The form of this QCD theory doesn’t look vastly different from that of the electromagnetic force, described by QED (quantum electrodynamics) though there are some very important differences between the two.

We can perform calculations in QED to immensely high accuracy and can test these calculations against experiment. The answers come out time and again to be correct to many many decimal places. This theory and its applications are amongst the greatest triumphs of 20th century physics.

The method that we use to perform calculations in QED is called perturbation theory. QED describes photons and charged particles (for instance electrons) and how they interact. The lucky thing about QED is that they don’t feel each other’s effects very strongly. We say that they are interacting weakly.

In fact in the limit that they don’t interact at all we can calculate exactly what happens – it’s usually a completely solvable computation. Perturbation theory says that we start with the solution in the case where they don’t interact at all and ask what happens when we turn on the interactions a tiny bit. Let’s say we turn on the strength of the interaction to be some tiny value epsilon.

Well, we can calculate what the interaction looks like when we’ve turned the strength above zero in a series expansion in this value epsilon. First there will be terms which come with a single power of epsilon, then terms with two powers of epsilon, then three, etc. etc. This will go on indefinitely.

I’ll give you an example. Say my calculation gave me a series like this:

Answer = result if we had no interaction + epsilon O(1) +epsilon^2 O(1) + epsilon^3 O(1)+…

(The sort of question we might be asking is about the scattering amplitude of one particle off another. Clearly when there are no interactions these particles should pass each other as if there is nothing else about)

O(1) means that these terms are of order 1. They might be 0.5, or 3 say, but not some huge or tiny number which is dependent on epsilon (this is a simplification but for this line of argument it is enough)

So, if epsilon is really tiny, say 0.0001, then the third term is 10000 times smaller than the second and so it really shouldn’t be important to keep this, or other, ‘higher order’ terms. We can simplify everything by not calculating anything except the first two terms. The other infinite terms can be thrown away.

What if epsilon was 0.1? Well, then the third term would be 10 times smaller than the second, but maybe we are interested in getting our answer accurate to this precision (10%). The third term is 100 times smaller and we are not interested in 1% accuracy, for the sake of argument. If we wanted 1% accuracy then we would need to include the fourth term as well, but not the fifth.

How far we go is dependent on: How accurate we want our answer and also the strength of the interaction. For really weak interactions everything is very easy and we have no problems.

It’s lucky that in QED the value of this coupling, epsilon, is very small. We can calculate just a few terms in the series and get answers to extremely high accuracy.

But what if the value of epsilon wasn’t a small number, but a number, for instance, larger than 1? Well, we could still calculate the terms in the series in just the same way as before, but now the third term is more important than the second, and the fourth term more important still. This is clearly a problem because it looks like we have to calculate ALL THE TERMS and we so liked this method because we could stop when we’d had enough. Calculating all the terms one by one would be impossible.

We may hope that we can add up these terms in a very simple way to give us something which is a nice function, rather than an infinite, divergent series.

The reason QCD is difficult to deal with is exactly because the value of this coupling constant, epsilon, can be large and there is no simple way to resum the series. We seem to be stuck! Our tried and tested method has broken down on us!

It turns out that there are some techniques we can use to look at what is happened to QCD at low energies (it turns out that this coupling constant epsilon is dependent on energy and at high energy it is small again and we can happily use perturbation theory). We can use large computer simulations to try and calculate the partition function (a function which encapsulates many properties of the theory) for our system by brute force, by summing over a huge number of field configurations generated by clever algorithms. In order to do this we have to discretise spacetime, amongst other approximations. This is called lattice QCD and gives us a lot of very interesting results. However, there are limitations and the computer power and time that goes into this is enormous.

There are other methods to understand QCD at low energies. We can use the symmetries of the theory to construct an effective low energy theory which doesn’t have the quarks and gluons as fundamental constituents but rather bound states of these objects – called hadrons.

One of the important points about QCD is that we never seem to see quarks moving around on their own, like we see electrons moving about freely. When you have two quarks and you try and pull them apart you find that the force between them gets greater as you pull, until, when you’ve put in enough energy you find you have created another two quarks which are now attached to the ones you were first pulling apart.

The particles which we do see in our particle accelerators are hadrons, in which the quarks are confined to live within close proximity of one another, two or three at a time.

We’d really like to be able to prove from our relatively simple theory of QCD that confinement must be seen – that we never see quarks (at normal temperatures and pressures) living alone. It turns out that this is a really hard problem.

In fact, when string theory came along in the 60s it was first developed to try and understand what was happening when these hadrons interacted. It worked pretty well, but there were some places that the predictions didn’t agree with experiments. Then came along QCD and we realised that this was actually the correct theory to describe the strong force.

String theory then took off when people realised that it was a consistent, quantum mechanical theory which included gravity. It turns out that having a quantum theory of gravity is really very tricky and when we try and use point particles to construct this, we get into trouble very quickly. However, string theory automatically gave us gravity and removed the problems which had been there in the point particle case.

String theory has (with periods of quiet reflection) moved forwards to give us many insights into how amazing our universe might well be, but this isn’t what I’m going to talk about here.

In 1997 it was found that string theory when formulated in a particular geometry seemed to be telling us something about strongly coupled theories – theories where the parameter epsilon is too big to perform a perturbation expansion. In fact what it told us was that string theory in this 10 dimensional geometry was equivalent to a particular strongly coupled gauge theory in four dimensions.

First of all, what this equivalence meant wasn’t really understood but we have learnt more and more about this over the last decade and amazingly string theory has come back to where it originated as we try to use it to understand QCD.

There is a lot to say about the equivalence of these two theories, what it tells us about QCD and what it tells us about the nature of spacetime. I will leave this for the next installment…