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Perturbation theory: are we covering up new physics?

A timely award of the J. J. Sakurai Prize acknowledges how hard it can be sometimes to pin down what the Standard Model really thinks

A graphical representation of a proton-proton collision. Loosely speaking, the red, yellow and some blue bits are the skeleton, and the green stuff is squishy. Credit: Frank Krauss, Sherpa.

We're measuring all kinds of stuff at the Large Hadron Collider right now. The question we're addressing could be summed up as

Does the Standard Model of particle physics work at LHC energies or not?

If it works, there is a Higgs boson but not much else new. If it doesn't, there might not be a Higgs but there must be something weird and new going on. As I have said before, the energy range of the LHC is special.

This begs the question (of me at least)

How well do we really understand the predictions of the Standard Model at these energies?

This isn't an easy one. In general we can't solve the Standard Model exactly. We use approximations. Most of these rely on the fact that the "coupling", that is the strength of the fundamental forces, is not very large.

The strength of a force can be expressed as a number. If it was 0.1, say, then the chances of two particles interacting would be proportional to 0.1 x 0.1 = 0.01. But for three to interact it would be 0.1 x 0.1 x 0.1 = 0.001, four would be 0.0001 and so on. This means when the coupling is small, you can ignore the contributions which involve more than say four particles - they are just a small perturbation on the main result, because they are multiplied by 0.1 x 0.1 x 0.1 x 0.1 x 0.1 = 0.00001. They don't change the result much. This is "perturbation theory". It is accurate if the coupling is small, that is if the force is weak.

This is mostly true at LHC energies, except for when it isn't.

The bits when isn't mostly involve the strong nuclear force, Quantum Chromodynamics. That's why it's called the strong force. (We don't intentionally obfuscate, it's tough enough as it is.)

For example, aspects of how quarks and gluons are distributed inside the protons we collide can't be calculated from first principles. Neither can the way the quarks and gluons turn in to new hadrons in the end. We have some constraints from our theory, we have basic stuff like the conservation of energy and momentum, and we have a lot of data from other places. But we can't use perturbation theory. The coupling number gets near to one, and 1 x 1 x 1 x ... = 1. This means no matter how many particles you include in your calculation, you don't converge on a solid answer. In the end we have to make educated guesses, or models. And these are always adjustable.

A long time ago Lily wrote a piece, where she, and commenters, worried that we might be adjusting these models in such a way that we actually covered up exciting new physics. This is a real worry. To avoid it, you need to have calculations of what you know, done with perturbation theory, linked up to models of what you don't know very well. I think of this rather gruesomely as a skeleton of hard predictions inside and squidgy body of best guesses. The body can change shape. You can push in its stomach quite painlessly, but you really know about it if you break a bone.

Anyway, marrying the squidgy models to the rigid perturbation theory is mostly done using Monte Carlo event generators.

A timely award of the J. J. Sakurai Prize acknowledges how hard it can be sometimes to pin down what the Standard Model really thinks

A graphical representation of a proton-proton collision. Loosely speaking, the red, yellow and some blue bits are the skeleton, and the green stuff is squishy. Credit: Frank Krauss, Sherpa.

We're measuring all kinds of stuff at the Large Hadron Collider right now. The question we're addressing could be summed up as

Does the Standard Model of particle physics work at LHC energies or not?

If it works, there is a Higgs boson but not much else new. If it doesn't, there might not be a Higgs but there must be something weird and new going on. As I have said before, the energy range of the LHC is special.

This begs the question (of me at least)

How well do we really understand the predictions of the Standard Model at these energies?

This isn't an easy one. In general we can't solve the Standard Model exactly. We use approximations. Most of these rely on the fact that the "coupling", that is the strength of the fundamental forces, is not very large.

The strength of a force can be expressed as a number. If it was 0.1, say, then the chances of two particles interacting would be proportional to 0.1 x 0.1 = 0.01. But for three to interact it would be 0.1 x 0.1 x 0.1 = 0.001, four would be 0.0001 and so on. This means when the coupling is small, you can ignore the contributions which involve more than say four particles - they are just a small perturbation on the main result, because they are multiplied by 0.1 x 0.1 x 0.1 x 0.1 x 0.1 = 0.00001. They don't change the result much. This is "perturbation theory". It is accurate if the coupling is small, that is if the force is weak.

This is mostly true at LHC energies, except for when it isn't.

The bits when isn't mostly involve the strong nuclear force, Quantum Chromodynamics. That's why it's called the strong force. (We don't intentionally obfuscate, it's tough enough as it is.)

For example, aspects of how quarks and gluons are distributed inside the protons we collide can't be calculated from first principles. Neither can the way the quarks and gluons turn in to new hadrons in the end. We have some constraints from our theory, we have basic stuff like the conservation of energy and momentum, and we have a lot of data from other places. But we can't use perturbation theory. The coupling number gets near to one, and 1 x 1 x 1 x ... = 1. This means no matter how many particles you include in your calculation, you don't converge on a solid answer. In the end we have to make educated guesses, or models. And these are always adjustable.

A long time ago Lily wrote a piece, where she, and commenters, worried that we might be adjusting these models in such a way that we actually covered up exciting new physics. This is a real worry. To avoid it, you need to have calculations of what you know, done with perturbation theory, linked up to models of what you don't know very well. I think of this rather gruesomely as a skeleton of hard predictions inside and squidgy body of best guesses. The body can change shape. You can push in its stomach quite painlessly, but you really know about it if you break a bone.

Anyway, marrying the squidgy models to the rigid perturbation theory is mostly done using Monte Carlo event generators.

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