Video Transcript
Feynman diagrams are simple graphical representations of particle interactions that underlie deep calculations about fundamental physics. In this lesson, we’re going to learn how Feynman diagrams represent particle interactions and also draw diagrams corresponding to some common scenarios.
Before defining formal rules, let’s understand the basics using the Feynman diagram for electron–positron annihilation. This diagram is one of several diagrams, all of which have the same basic shape that form the basis of quantum electrodynamics. Written as a nuclear equation, we have that an electron interacting with a positron can form a photon. Here, the Greek letter lowercase 𝛾 is used to represent photons.
To borrow language from the study of chemistry, we can think of this equation as a set of reactants interact to form a set of products. This equation also carries with it some implicit information that we’ll need to spell out to properly draw a Feynman diagram. When we say our reactants interact, we are implicitly saying that our reactants get close enough together in space and in time for the interaction to occur. How close is close? It depends on the scale of the interaction. For example, weak interactions tend to take place on much shorter length scales than electromagnetic interactions.
What’s really important is the qualitative statement that we’re making. An electron and a positron that are flying apart from each other won’t annihilate. Nor will they annihilate if one passes through a position in space and the other passes through that same position but only several seconds later.
The second piece of information implicit in this equation is that the reactants actually interact to form the products. This implicitly tells us that there is at least one valid mechanism that actually forms the products from the reactants. As an example of the alternative, the equation electron interacts with positron to form proton and antimuon has no valid mechanism, since this equation actually violates three conservation laws: charge, lepton number, and baryon number. It turns out that for any valid nuclear equation, there are actually infinitely many valid mechanisms that form the products from the reactants. The function of an individual Feynman diagram is to show one such valid mechanism.
Okay, so to actually draw a Feynman diagram for this interaction, we need to be able to represent the particles, their spatial and temporal locations, and verify that the mechanism pictured is a valid one. This verification is especially easy for electron–positron annihilation because the interaction is so simple. And that’s why we’re using this interaction to introduce the rules for Feynman diagrams. Since we’re interested in the spatial and temporal locations of our particles, we’ll represent them on a graph with axes of position and time. To show a particle on such a graph, we simply need to draw a line that represents that particle’s position at each time.
At this point, it’s worth stressing again that we only care about the qualitative relationships between the various positions and the various times. This is why we haven’t put scales on our axes. And in fact, many times Feynman diagrams are represented without axes at all. Furthermore, we usually use straight connections because what we care about is the qualitative relationship between the endpoints and not the particulars of how a particle got from one endpoint to the other.
Okay, so let’s say in this abstract sense that the line that we’ve drawn represents an electron moving through space. Let’s also add another line to represent a positron. Now, we’ve drawn the reactants from our nuclear equation. And as we can see, they initially start out separated in space. But as time progresses, they’re coming closer together. The simplest mechanism by which these particles could interact would be to collide by occupying the same position at the same time. On the diagram, we simply extend the two lines so that they meet at a point, which represents the electron and the positron occupying the same position at the same time.
Before we add the photon, we need to do one more thing to these two lines to make them valid parts of a Feynman diagram. Observe that this collision between an electron and a positron can be stated generally as a collision between a particle or matter and its corresponding antiparticle or antimatter. To encode this generality into the diagram so that we can use the same shape for any particle–antiparticle interaction, we add arrows pointing in different directions to each line. The directions are chosen such that if we follow the direction of the matter arrow, we move forward in time. And if we follow the direction of the antimatter arrow, we move backward in time.
This choice is very deliberate. Representing antimatter as matter going backward in time is actually a reflection of the deep symmetries underlying our universe. What this tells us essentially is that if we viewed this situation with time flowing in the usual direction, we would see a positron coming down and an electron coming up to meet in the middle. If we reverse time, we would see two particles flying apart instead of coming together. However, the arrow of the upper particle would now point in the same direction as time. And the arrow of the lower particle would now point in the opposite direction of time. So we would see an electron flying up and a positron flying down.
The fact that changing the direction of time also changes the charge of our particles is a reflection of a fundamental symmetry in our universe. At any rate, after this collision, the electron and positron disappear and a photon is produced. In the Feynman diagram, we represent this photon with a squiggly line, since it’s an uncharged boson and therefore its own antiparticle and doesn’t change when we reverse time. And that’s it.
Our Feynman diagram begins in time with our two reactants and ends in time with our single product. Everything in between is the mechanism by which our particles come together in space and time to produce the products. Observe that the three lines representing our particles meet at a similar point, which we call a vertex.
Now that we’ve successfully drawn a Feynman diagram, let’s go over some rules that apply to all valid diagrams. First, the arrows on lines representing particles point forward in time, while the arrows on lines representing antiparticles point backward in time. We’ll reserve this rule specifically for leptons, quarks, and also hadrons, which are just composite particles made of quarks. On the other hand, we’ll represent bosons by other styles of line, for example, the photon by a squiggly line like we’ve already drawn and the W and Z weak bosons by dashed lines. Marking every particle and antiparticle line with an arrow allows us to be very general with our diagrams and see, for example, that this diagram actually corresponds to the annihilation of any lepton and its corresponding antilepton.
Our second rule, which we can easily see holds true in our graph, is that every vertex is the junction of exactly three lines. One of those lines has an arrow pointing in, one of those lines has an arrow pointing out, and one of those lines is a boson. It’s important to note that rules one and two do not imply that every vertex is a meeting place of a particle and antiparticle and a boson, as we will see clearly later when we consider an electron absorbing or emitting a photon.
Finally, every Feynman diagram must obey all conservation laws both on a vertex-by-vertex basis and also for the overall interaction. This includes, among other things, the conservation of charge, conservation of lepton number, and conservation of baryon number. Let’s verify this for the single vertex in our diagram. None of the particles we have are baryons, so clearly baryon number is conserved. As for lepton number, we have an electron with a lepton number of positive one, a positron with a lepton number of negative one. So the total lepton number going into our vertex is one minus one, which is zero. And coming out of the vertex we have a photon with a lepton number of zero, so lepton number is conserved. The same is true for charge.
Electrons and positrons have charges that are equal in magnitude but opposite in sign. So the total charge going into the vertex is zero, and the photon is uncharged, so charge is in fact conserved. There is one more situation we need to consider before we have a full picture of Feynman diagrams. But first, let’s consider the reverse of our current example where a photon spontaneously creates a pair of an electron and a positron. We can actually use the Feynman diagram we’ve already drawn to figure out the Feynman diagram for this new situation called pair production.
In this interaction, the photon is the single reactant and the electron–positron pair are the two products. Looking at our diagram, we see that we have an electron–positron pair entering on the left and a single photon exiting on the right. If we imagine reversing time, we get a photon entering on the right and an electron and a positron exiting on the left, where their positions would be switched relative to the forward direction of time. But a photon entering the interaction and an electron–positron pair leaving the interaction is precisely the situation described by pair production. So all we need to do is just flip our diagram in time. And here’s the Feynman diagram for pair production.
The mechanism is valid as long as the electron and positron move in opposite directions. But we’ve chosen to exactly mirror the diagram for annihilation to illustrate how reversing time changes matter to antimatter and antimatter into matter. So as we can see, to find the Feynman diagram of a reversed particle interaction, we simply had to reverse the Feynman diagram.
Alright, to learn the last aspect we need for a full picture of Feynman diagrams, let’s examine the diagram for electrostatic repulsion. Electrostatic repulsion occurs when two objects with the same sign of charge are pushed apart by what we classically call the Coulomb force. What we’ve drawn here is the Feynman diagram representing that interaction for the specific case of two electrons. First things first, observe that this is indeed a valid Feynman diagram. We only have particles, not antiparticles, so all of the arrows point forward in time.
Secondly, at each vertex, there’s a particle line with an arrow pointing in, a particle line with an arrow pointing out, and a boson. In fact, both of these vertices are examples of how we can have the arrow pointing into a vertex and the arrow pointing out of a vertex both corresponding to particles, not antiparticles. Finally, since at both vertices we have an electron going in and an electron going out and the third particle is an uncharged boson, lepton number, baryon number, and charge are all conserved.
Now that we know we have a valid diagram, let’s actually confirm that this diagram shows electrostatic repulsion. On the left-hand side of the diagram, we have two electrons. And as we can see, they’re moving in space towards each other. On the right-hand side of the diagram, we again have two electrons. But this time they’ve changed directions and they’re moving away from each other. But this is exactly a qualitative picture of electrostatic repulsion. Two particles with the same sign of charge get close enough in space that they repel each other and move away. So because this diagram follows all the rules and the beginning and end agree with what we’d expect, by definition, this diagram shows a valid mechanism for the interaction.
There is an important subtlety here that we need to address. The actual interaction, represented by the Feynman diagram between the two dotted lines, is not observable. That is, no matter what experiment we set up, we won’t see a photon traveling between two electrons as they repel each other. All that we can observe is that the initial state of the electrons is moving towards each other and the final state of the electrons is moving away from each other and then conclude that these initial and final states agree with electrostatic repulsion.
This leaves us in a somewhat surprising position of having the photon, which plays a key role in this interaction, completely contained in a nonobservable region. Because this photon is never observable and in some sense only exists in the context of our diagram, we call it a virtual photon. More generally, a virtual particle is any particle whose representation line begins and ends at a vertex of the diagram. Since such a particle’s line has no open end, it can’t enter the interaction from the observable initial state nor exit the interaction into the observable final state. As a result, to the extent that we can think of such particles as existing, they only exist in the context of a particular mechanism of interaction.
Moreover, in part because these particles are never observable, they only need to follow the conservation laws but not other laws of physics. For example, in the diagram we’ve drawn, the virtual photon could be traveling forward in time from the bottom electron to the top electron but just as validly could be traveling backwards in time from the top electron to the bottom electron. Furthermore, even though the two electrons appear to emit and absorb the photon at different times, nothing about the photon is ever observable. So all of our observations will still agree with electrostatic repulsion.
Now that we have a notion of virtual particles, we’re ready to move on to some more complicated Feynman diagrams, like those representing beta decay. As its name suggests, beta decay is a decay process. Such processes are usually mediated by the weak force, so we’ll expect to find W and Z bosons in our Feynman diagrams.
There are two types of beta decay, beta plus and beta minus, named after the sign of the charged lepton released in the process. In beta minus decay, a down quark becomes an up quark and releases an electron and an electron antineutrino. In an atomic nucleus, this is observed when a neutron which consists of two down quarks and one up quark becomes a proton which consists of one down quark and two up quarks. In the Feynman diagram of this interaction, we have a down quark coming in and an electron antineutrino, an electron, and an up quark leaving.
Note the presence of the virtual W minus boson represented by the dashed line. The presence of this virtual boson ensures that both vertices follow all of the relevant rules. For practice, let’s verify that these rules are in fact followed. The down quark arrow, the up quark arrow, and the electron arrow all point forward in time, which is correct because down quarks, up quarks, and electrons are all particles. The arrow for the electron antineutrino points backward in time, which is also correct because an electron antineutrino is an antiparticle.
At the down up W minus vertex, the down arrow points in, the up arrow points out, and W minus is a boson. At the electron–electron antineutrino–W minus vertex, the electron arrow points out, the electron antineutrino arrow points in, and again W minus is a boson. So both vertices follow the rule of having an arrow pointing out, an arrow pointing in, and a boson line.
Okay, under conservation laws, let’s start by checking baryon number and lepton number. At the down up W minus vertex, we can easily see that lepton number is conserved since there are no leptons associated with this vertex. As for baryon number, that is also conserved because there’re no antiquarks and exactly one quark enters and one quark leaves the vertex. At the electron–electron antineutrino–W minus vertex, there are only leptons and bosons. So clearly, baryon number is conserved. As for lepton number, the boson entering the vertex has a lepton number of zero. Leaving the vertex is an electron with a lepton number of one and an electron antineutrino with a lepton number of negative one. One minus one is exactly zero, so lepton number is indeed conserved.
Finally, to double-check conservation of charge, we’ve marked the charges of each particle on the diagram. A down quark has a relative charge of negative one-third. An up quark has a relative charge of positive two-thirds. A W minus boson and an electron both have a relative charge of negative one. And an electron antineutrino is neutral. Quickly doing the sum for each vertex, at the bottommost vertex, negative one-third is the charge that enters and negative one plus two-thirds or negative one-third is the charge that leaves. At the top vertex, negative one is the charge that enters and at negative one plus zero, which is the same as negative one, is the charge that leaves. So we see that total charge is indeed conserved on a vertex-by-vertex basis.
Alright, let’s now draw the diagram for beta plus decay and use that as practice to work backwards from the Feynman diagram to the particle equation. Here’s the diagram for beta plus decay. Note that one of the products is a positron instead of an electron and the interaction is mediated by a W plus weak boson. To find the particle equation corresponding to this diagram, we start by finding all of the particles that enter the interaction. In this case, it’s only the up quark. The particles that leave the interaction are an electron neutrino, a positron, and a down quark. So the overall interaction is an up quark becomes a down quark and emits a positron and an electron neutrino.
In atomic nuclei, this interaction would result in a proton transforming into a neutron. Note that the W plus boson doesn’t appear in our overall equation because it’s a virtual particle. So it isn’t an observable part of the interaction. Alright, let’s now see the Feynman diagrams that correspond to the decay of the higher-mass 𝜇 and 𝜏 leptons.
One of the incredibly powerful features of Feynman diagrams is that this one diagram here can be used to represent all of the possible decay products of the 𝜇 and 𝜏 leptons by appropriately labeling the lines. Say we start with the muon. Since this is a decay, at the first vertex, charge is conserved with a W minus weak boson and lepton number is conserved with an outgoing 𝜇 neutrino. For a muon, the only possible further decay products are the lower-mass electron, which conserves the charge from the W minus boson. And then to conserve lepton number from the electron, we also need an electron antineutrino. Thus, we have a Feynman diagram for the decay of a muon into a 𝜇 neutrino, an electron, and an electron antineutrino.
By simply changing the 𝜇 to a 𝜏, we now have the decay of a 𝜏 into a 𝜏 neutrino, an electron, and an electron antineutrino. Since the 𝜏 also has a larger mass than the muon, it can decay into a muon and a 𝜇 antineutrino. This conserves charge and lepton number just like the electron. And as we can see, the shape of the underlying Feynman diagram is the same. Because of its mass, a 𝜏 lepton can also decay into a 𝜋 minus meson, which is composed of one down quark and one up antiquark. In this case, total charge is conserved due to the negative one-third charge of the down and negative two-thirds charge of the anti-up. Total baryon number is conserved because down is a quark and anti-up is an antiquark, so the total baryon number is still zero. Here, too, the shape of the underlying Feynman diagram is the same. We’ve just made the esthetic switch to a blue line to reflect the negative charge of the up antiquark.
Alright, now that we’ve seen a range of different interactions represented in Feynman diagrams, let’s review what we’ve learned in this lesson. In this lesson, we learned that Feynman diagrams show mechanisms for particle interaction on a graph with axes of time and position. Lines have arrows pointing in different directions in time, depending on whether they represent a particle or an antiparticle. Lines meet in groups of three called a vertex, where one line has an arrow pointing in, one has an arrow pointing out from the vertex, and the last line represents a boson.
Finally, the particles entering and leaving a vertex in time must obey the conservation laws. Following these simple rules, we drew the Feynman diagrams for electron–positron annihilation and a pair production. From the diagram for electrostatic repulsion, we introduced the notion of a virtual particle, which is an unobservable part of the interaction. Finally, we saw that all types of beta and lepton decay can be represented by different labelings of the same basic Feynman diagram.