*This post is a continuation of three previous posts: #1, #2 and #3.*

**When the Strong Nuclear Force is Truly Strong**

Although I’ve already told you a lot about how we make predictions using the Standard Model of particle physics, there’s more to the story. The tricky quantum field theory that we run into in real-world particle physics is the one that describes the strong nuclear force, and the gluons and quarks (and anti-quarks) that participate in that force. In particular, for processes that involve

- distances comparable to or larger than the proton‘s size, 100,000 times smaller than an atom, and/or
- low-energy processes, with energies at or below the mass-energy (i.e. E=mc² energy) of a proton, about 1 GeV,

the force between quarks, gluons and anti-quarks becomes so “strong” (in a technical sense: strong enough that it makes these particles rush around at nearly the speed of light) that the methods I described previously do not work at all.

That’s bad, because how can one be sure our equations for the quarks and gluons — the quantum field theory equations of the strong nuclear force — are the correct ones, if we can’t check that these equations correctly predict the *existence* and the *masses* of the proton and neutron and other hadrons ( a general term referring to any particles made from quarks, anti-quarks and gluons)?

Fortunately, there is a way to check our equations, by brute force. *We simulate the behavior of the quark and gluon fields on a computer.* Sounds simple enough, but you should not get the idea that this is easy. Even figuring out how to do this requires a lot of cleverness, and making the calculations fast and practical requires even more cleverness. Only expert theoretical physicists can carry out these calculations, and make predictions that are relevant directly for the real world. Don’t try this at home.

The first step is to simplify the problem, and consider an imaginary world, an idealized world that is simpler than the real world. Since the strong nuclear force is extremely strong inside a proton, the electromagnetic and weak nuclear forces are small effects by comparison. So it makes sense to do the calculation in an imaginary world where* the strong nuclear force is present but all other forces are turned off*. If you put those unimportant forces in, you’d have a much more complicated computer problem and yet the answers would barely change. So including the other forces would be a big waste of time and effort.

Here we use an imaginary world as an **idealization** — a bit like treating the earth as a perfect sphere. Obviously the earth is not a sphere — it has mountains and valleys and tides and a slight bulge at the equator — but if you’re computing some simple properties of the earth’s effect on the moon, including these details will waste a lot of your time without affecting your calculation very much. The art of being a scientist requires knowing what you need to include in your calculations, and knowing what not to include because it makes no difference. In fact we do this all the time in particle physics; gravity’s effect on measurements at the Large Hadron Collider [LHC] is tiny, so we do our calculations in an imaginary world without gravity, a harmless simplification.

Here’s another idealization: although there are **six** types (often called “flavors”) of quarks — up, down, strange, charm, bottom and top — the last **three** are heavier than a proton and consequently don’t play much of a role in the proton, or in the other low-mass hadrons that I’ll focus on here. So the imaginary, idealized, simplified world in which the calculations are carried out has (see Figure 1)

- Three “flavors” of
**quark fields**: up, down and strange, each with its own mass, and each with a charge (analogous to electric charge in the case of the electric force) which is whimsically called “color”. Color can take three values, whimsically called “red”, “green” or “blue”. These fields give rise to both the quark particles and their antiparticles, called anti-quarks, which carry anti-color (anti-red, anti-blue, anti-green);
- Eight
**gluon fields** (each carrying a “color” and an “anti-color”.) *[You might have guessed there’d be nine; but when color and anti-color are the same there are some little subtleties which aren’t relevant today, so I ask you to just accept this for now.]*

So now we have a quantum field theory of three flavors of quarks with three possible colors, along with corresponding anti-quarks, and eight gluons which generate the strong nuclear force among the quarks, antiquarks and gluons. This isn’t the real world, but it is close enough to give us very accurate answers about the real world. And this is the one the experts actually put on a computer, to see if our equations do indeed predict that quarks, antiquarks and gluons form protons and other hadrons.

Fig. 1: The fields of the stripped-down world in which calculations of the proton mass and other hadron masses are done. Up, down and strange quark fields (responsible for both quarks and anti-quarks) interact with gluon fields (responsible for gluon particles.) Each of the eight quark fields has a “charge” (named, whimsically, red, green or blue) and each gluon field has a color and an anti-color.

Does it work? Yes! In Figure 2 is a plot showing the experimentally measured and computer-calculated values of the masses of various hadrons found in nature. Each hadron’s measured mass is the vertical location of a horizontal black line; the hadron’s symbol appears below that line at the bottom of the plot. I’ve written the names of a few of the most famous hadrons on the plot:

- the spin-zero pions,
- the spin-1 rho mesons and omega meson,
- the spin-1/2 “nucleons”, meaning the proton and the neutron, and
- the spin-3/2 Delta particles.

The colored dots represent different computer calculations of the masses of these hadrons; the vertical colored bars show how uncertain each calculation is. You can see that, within the uncertainties of the calculations, the measurements and calculations agree. And thus we learn that indeed the quantum field theory of this idealized world

- predicts that hadrons such as protons do exist
- predicts the ones we observe, without a lot of extra ones or missing ones
- predicts correctly the masses of these hadrons

from which we conclude that

- the quantum field theory with the fields shown in Figure 1 has something to do with the real world
- we were wise to choose the imaginary world of Figure 1 for our study, because clearly the idealizations we made didn’t affect our final results to an extent that they caused disagreements with the real world
Fig. 2: The masses of various hadrons, whose names appear at bottom and whose measured masses appear as grey horizontal lines, as calculated by computer: each colored dot is a particular calculation, whose uncertainty is shown by a vertical bar. I have written the names of some famous hadrons.

All looks great! And it is. However, I’ve lied to you. I haven’t actually told you how hard it is to obtain these answers. So let me give you a little more insight into what you have to do to obtain these calculations. You have to go off into even more imaginary worlds.

**How the Calculation is Really Done: Off In Imaginary Worlds**

The imaginary world I’ve described so far is still not simple enough for the calculation to be possible. The actual calculations **require that we make predictions in worlds very different from our own**. Two simplifications have to do with something you’d think would be essential: space itself. In order to do the calculation, we have to imagine

- that the world, rather than being enormous, is made of just a tiny little box — a box only large enough to hold a single proton or other hadron;
- that space itself, rather than being continuous, forms a discrete grid, or
*lattice*, in which the distances between points on the grid are somewhat but not enormously smaller than the distance across a proton.

This is schematically illustrated in Figure 3, though the grids used today are denser and the boxes a bit larger. The size of a proton, relative to the finite grid of points, is indicated by the round circle.

Fig. 3: The calculations are done in a world whose space is a small grid. Note, however, that this picture of a 4 x 4 x 4 grid is a cartoon to make the idea clear; with modern computers, grids of 32 x 32 x 32 are not unusual.

Advances in computer technology are certainly helping avoid this problem… the better and faster are your computers, the denser you can take your grid and the larger you can take your box. But simulating a large chunk of the world, with space that is essentially continuous, is way out of reach right now. So this is something we have to accept, and deal with. *Unlike the idealizations that led us to study the quantum field theory in Figure 1, choosing to study the world on a finite grid ***does** change the calculations substantially, and experts have to correct their answers after they’ve calculated them.

And there’s one more simplification necessary. **The smaller are the up and down and strange quark masses, the harder the calculation becomes.** If these masses were zero, the calculation just would be impossible. Even with the real world’s quark masses (the up quark mass is about 1/300 of a proton’s mass, the down quark 1/150, and the strange quark about 1/12) calculations *still* aren’t really possible — and they weren’t even close to possible until rather recently. So *calculations have to be done in an imaginary world with much larger quark masses*, especially for the up and down quark, than are present in the real world.

Fig. 4: Two types of imaginary worlds arise here. First, the real world is stripped down, with all irrelevant particles and forces dropped, giving the red imaginary world. Then this world’s space is made into the grid of Figure 3, and the up, down and strange quark masses are raised. In this purple imaginary world, calculations become practical, but they give incorrect answers; only by extrapolating (Figure 5) are useful predictions extracted.

So since we can’t calculate in the real world, but have to calculate in a world with a small spatial grid and heavier quarks, how can we hope to get reasonable answers for the hadron masses? Well, this is another place where the experts earn our respect. The trick is to learn how to *extrapolate*. For example:

- Do the calculation for fields in a small box.
- Then do the calculation again in a medium-sized box (which takes a lot longer.)
- Then do the calculation in a larger box (still small, but big enough that it uses about as much computer time as you can spare.)

Now, **if** you know how going from a small to medium to larger box should change your answer, **then** you can infer, from the answers you obtain, what the answer would be in a huge box where the walls are so far away they don’t matter.

The experts do this, and they do the same thing for the space grid, computing with denser grids and extrapolating to a world where space is continuous. And they do the same thing for the quark masses: they start with moderately large quark masses, and they shrink them in several steps. And *knowing from theoretical arguments what should happen to the hadron masses as the quark masses change, they can extrapolate from the ones they calculate to the ones that would be predicted if the quark masses were the real-world ones*. You can see this in Figure 4. As the up and down quark masses are reduced, the pion mass gets smaller, and the “nucleon” (i.e. proton and neutron) masses becomes smaller too.* (Also shown is the Omega hadron; this has three unpaired strange quarks, and you can see its mass doesn’t depend much on the up and down quark masses.)* The experts take the actual calculations (colored dots), and draw a properly-shaped curve through all the dots. Then they go to the point on the horizontal axis where the quark masses equal their real-world values and the pion mass comes out agreeing with experiment, and they draw a vertical black line upward. The intersection of the black vertical and blue curved line (the black X mark) is then the prediction for what the proton and neutron mass should be in the real world. Well, you can see that the black X is pretty close, within about 0.030 GeV/c², to what we find in experiments: 0.938 and 0.939 GeV/c² for the proton and neutron mass. And this is how all of the results shown in Figure 2 are obtained: **extrapolating to the real world by calculating in a few imaginary ones**.

Fig. 5: Calculations (colored dots) are done with larger quark masses than in the real world, and the results are as much as 50% too large. One must extrapolate to the smaller quark masses of the “real” or “physical” world (black dotted vertical line) to make predictions (black X’s). “N” stands for “nucleon”, meaning both protons and neutrons.

**The Importance of Such Calculations**

This is a tremendous success story. The equations of the strong nuclear force were first written down correctly in 1973. Calculations like this were just becoming possible in the mid-1980s. Only in the 1990s did the agreement start to become impressive. And now, with modern computer power, it’s become almost routine to see results like this.

More than that, these methods have become essential tools. There are many important predictions made for experiments which are partly made with the methods I described in my previous post and partly using these computer calculations. For example, they are extremely important for precise predictions of the decays of hadrons with one heavy quark, such as B and D mesons, which I have written about here and here. *If we didn’t have such precise predictions, we couldn’t use measurements of these decays to check for unknown phenomena that are absent from the Standard Model.*

**But There’s So Still Much That We Can’t Compute**

Despite all this success, the limitations of the method are profound. Although computers are fine for learning the masses of hadrons, and some of their other properties, and quite a few other interesting things, they are *terrible* for understanding everything that can happen when two protons (or other hadrons) bump into each other. Basically, computer techniques can’t handle things that change rapidly over time.

For example, the data in Figure 6 show two of the simplest things you’d like to know:

*how does the probability that two protons will collide change, if you increase the energy of the collision?*
*what is the probability, if they collide, that they will remain intact, rather than breaking apart into a spray of other hadrons?*

We can *measure* the answer (the black points are data, the black curve is an attempt to fit a smooth curve to the data.) But no one can *predict* this curve by starting with the quantum field theory of the strong nuclear force — not using successive approximation, fancy math, brute force computer simulation, string theory, or any other method currently available. *[Experts: there are plenty of attempts to model these curves (look up “pomeron”.) But the models involve independent equations that can’t actually be derived from or clearly related to the quantum field theory equations for quarks and gluons.]*

Fig. 6: The probability for two protons to collide (upper data points, “total”) and to collide without breaking (lower data points, “elastic”), as a function of the energy of one proton as viewed by the other proton. Data are taken from many experiments, including the LHC at the far right. The curve shows an attempt to fit the data, but this data cannot currently be predicted starting from the equations for quarks and gluons.

At the LHC, when a quark from one proton hits a quark from another proton, we can predict, using the successive approximation (“perturbative”) methods described in my previous post, what happens to the quarks. But what happens to the *other parts* of the two protons when the two quarks strike each other? We can’t even begin to predict that, either with successive approximation or with computers.

My point? The quantum field theory of the strong nuclear force allows us to make many predictions. But still, *many very basic natural phenomena for which the strong nuclear force is responsible cannot currently be predicted using any known method.*

Stay Tuned. It’s going to get worse.

*Continued here…*