Peskin & Schroeder 1

Quantum Field Theory: Particle Collisions

At this point in the series on renormalization, we are going to switch from a high-level view into a deep dive into physical foundations. For this, I am going to become mostly unoriginal and follow Peskin & Schroeder’s popular text, An Introduction To Quantum Field Theory, which contains an entire section on renormalization. We still have a ways to go before getting there, however, and as we go there will be a lot of my own conceptual insertions as I, too, puzzle my way through the material. I intend to enjoy myself throughout the journey, but we can derive extra motivation from the basic desire to understand how properties of larger physical objects, e.g. electrical circuits, emerge from properties of their component physical objects, e.g. electrons. We remind ourselves that, though we take it for granted that we don’t need to think about subatomic particles to interact meaningfully with things like batteries, why this is is actually a profound and difficult question to investigate. Thanks to the hard work of physicists, we have some beautiful thinking tools like renormalization with which to approach it.

To refresh, renormalization requires as conceptual setting the combination of quantum, field, and relativity principles. The quantum theory of electromagnetism is a natural starting point from both physics and engineering perspectives, given our starting question of how it is that electrical circuits behave the way they do.

The first situation we’ll think about in the quantum realm is that of the collision of an electron with its antiparticle, the positron. A positron has the same mass and spin as an electron but charge equal to the negative of the charge of the electron. When these two particles collide, they “annihilate”, in the sense that, after the collision, those particles no longer exist in the same forms as when they entered; their collision produces other particles, such as muons.

Since there are multiple possible outcomes of an electron-positron collision (at least from the perspective of an observer), we can ask ourselves what the probability is that a given outcome happens. In physics, the term for this probability is called a ‘cross section’. A great deal of work has gone into developing computational frameworks and techniques for calculating cross sections for the full range of particle collisions. In general, the basic setup is to shoot a beam of particles of one type at a beam of another type. From there, measuring what was produced by the collision is accomplished by measuring the energy at the centre of mass of the collision and the relative angle between the incoming particles and outgoing particles. As a wrong but helpful analogy, you can think of billiard balls colliding such that the billiard balls can change color post-collision, and they’ll exit having produced a certain energy on impact and with an angled trajectory.

Once we have these energy and exit angle data, we need a recipe that tells us how to transform them into a probability of interest. The outline of this recipe is to: 1) Fix the frame of reference used for subsequent analysis of the physical situation, and 2) Apply a formula which technically answers the probability question but also invites more questions. To fill in 1), we choose to work in the center-of-mass (COM) frame in which we take the center of mass of the particle collision to be moving at constant velocity (not accelerating). Choosing to work in the COM frame allows us to make the helpful simplification that, in this frame, the total momentum of the system of colliding particles is zero. If we’re thinking about a particular electron and a particular positron colliding, this means that the momentum of the incoming electron has to be the negative of the incoming positron: m(electron) = – m(positron). This also has to be true of the muon and anti-muon produced by and exiting the collision: m(muon) = -m(anti-muon).

Before pushing around mathematical symbols, how should we think about the probabilities of particular outcomes of a particle collision? In the discrete formulation of probability, the way to compute the probability of a given event is to calculate this ratio: (# ways the event can happen) / (total # possible outcomes), where we’re only counting things within a pre-specified and valid probability space. The flavour of the cross section calculation is similar, but the ratio that we consider is instead the number of particles produced from the collision that occur within an infinitesimal angle of a particle detector’s extent.

Now to introduce some mathematical notation: Let $\sigma$ represent the cross section quantity that we’re aiming for, and let $\Omega$ represent the angle across a particle detector’s extent. Then, if we imagine chopping up $\Omega$ into ever finer and finer smaller angles and consider the infinitesimal ratios of particles to angles, we have $\frac{d \sigma}{d \Omega}$ . What must this be equal to? Well, it should involve a quantity that represents the probability that a particle produced in the collision arrives at the detector measuring the results, to begin with. This quantity is called the probability amplitude of a process. We formally define it as the probability that a particle arrives at the detector given that the particle came from a beam shooting particles into the collision. There are different accepted ways of shorthanding this. One is to write it at $< x | s >$ (read, “x given s”). We’ll refer to it as $M$ . In addition to the probability of landing at any location on a detector, we’ll also need to have a term that specifies the amount of infinitesimal surface area spanned by the detector, so that when we integrate the whole final formula, we get a quantity that accounts for the entire detector surface area. Sweeping some geometrical details under the rug, the whole differential cross section equation is this: $\frac{d \sigma}{d \Omega} = \frac{|M|^2}{64\pi^2E^2_{cm}}$ , where $E_{cm}$ is a term referring to the particle beam energy and $|M|^2$ is the norm of $M$ as a complex number.

So we have this beautiful formula. Huzzah! If we’re shooting particle beam guns, we can surely get the value of $E_{cm}$ from our experimental settings. But what about this quantum mechanical amplitude, $M$ ? Turns out that computing this quantity from theoretical physical principles is a tricky problem. Even in the simplest situations involving particle collisions, it’s not possible to compute it with complete exactness. In practice, there turns out to be a method “close enough” for the practical purposes of doing physics and at least understanding things better than when you started. The next post will introduce this method. (Hint: It is quite famous!).

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Peskin & Schroeder 1

Quantum Field Theory: Particle Collisions

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