This learning material is part of the PhD work of Merten Dahlkemper. It is still work in progress. Any question, comment, remark, critique etc. is highly appreciated and should be sent to merten.dahlkemper@cern.ch
Alpacarticle Physics with Feynman diagrams
How particle interactions rule the world from animals to particle accelerators
start
Introduction
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If you click the button in the top right corner you will find all interactive elements on the slide.
Introduction
At the end of this course, you should have a basic understanding of how particles interact with each other and how particle physicists can make predictions about what they might observe.
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From fluorescent animals to cosmic rays, from simple things like touching something to astonishing things like northern lights: Everything around us is ruled by particles interacting with each other. In this course, you will learn how we can understand these particle interactions using very simple diagrams.
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Presentation
You can always come back to this slide by using this button.
These buttons will lead you directly to the respective chapter.
Fundamental rules of particle processes
You know which rules govern fundamental particle processes and how they are translated into simple diagrams.
Contents and Goals
Understand particle interactions
You understand what it means that "particles interact with each other" and get to know examples of these interactions.
This course contains four chapters, each with its particular goal. We recommend to go through it step by step. But you can always come back to this slide and skip to a previous chapter again.
Understand calculations with diagrams
You learn how the simple diagrams are used to make calculations in particle physics.
Understand particle discoveries
You will understand how these diagrams are used to discover and investigate new particles.
Fundamental rules
Fundamental rules of particle interactions
Feynman diagrams are are a combination of different lines and their intersection points, so-called vertices. One Feynman diagram is then any combination of at least two of these vertices, like these: There are just some rules about these vertices for the diagram to be valid.
Left vertex
Feynman diagram
Right vertex
Maybe you have heard that the whole world around us is made out of particles. But particles alone are not very interesting. The interesting thing are particle interactions. These particle interactions are described by a theory called the Standard Model of Particle Physics. But even though this theory is very mathematical, there are some graphical tools which can help us understand its basics. These graphical tools are called Feynman diagrams.
Charge Conservation
All particles have certain invariant properties called charges. In this course, we will look at two different ones, electric charge and weak charge. It is a fundamental law of physics that the total charge of a system, just like its energy, does not change over time. It is also said that the charges are conserved. In each vertex of a diagram, both the electric charge and the weak charge are conserved independently. This means that both the sum of the weak charge and the sum of the electric charge on the left-hand side are exactly the same as on the right-hand side.
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electric charge on the left side of the vertex is 0, but n the right side -1. The electric charge is therefore not conserved. The weak charge is on the left side 0 and on the right side also 0 (because -½ + ½ = 0).
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electric charge on the left side of the vertex is 0, but n the right side -1. The electric charge is therefore not conserved. The weak charge is on the left side 0 and on the right side also 0 (because -½ + ½ = 0).
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side of the vertex is 0, and on the right side also 0 (because 1 - 1 = 0). The weak charge on the left side is 0 and on the right side also 0 (because ½ - ½ = 0).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
Menu
Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
Menu
Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
On the right vertex none of the charges is conserved. The electric charge is -1 on the left side, but +1 on the right side. The weak charge is -1 on the left, but ½ + ½ = + 1 on the right side.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
Menu
Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
On the right vertex none of the charges is conserved. The electric charge is -1 on the left side, but +1 on the right side. The weak charge is -1 on the left, but ½ + ½ = + 1 on the right side.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
Therefore the whole diagram is not allowed.
Menu
In the following you will see seven short tasks on charge conservation. Answer each task by clicking on the correct answer. Note: Clicking on the correct answer will take you directly to the next task. You cannot go back and correct. So think before you click.
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. The vertex on the left is incomplete. Which of the three lines on the right complete the diagram correctly at the place of the question mark?
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. A line to complete the two vertices is missing. Which of the three lines on the right complete the diagram correctly at the place of the question mark?
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. The vertex on the bottom right is incomplete. Which of the three alternatives on the right complete the diagram correctly at the place of the question mark?
Below you see three vertices. The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three vertices is wrong?
Below you see three vertices. The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three vertices is wrong?
On this slide you see three diagrams The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three diagrams is wrong?
On this slide you see three diagrams The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three diagrams is wrong?
Fundamental rules
Fundamental rules of particle interactions
Feynman diagrams are are a combination of different lines and their intersection points, so-called vertices. One Feynman diagram is then any combination of at least two of these vertices, like these:
Left vertex
Feynman diagram
Right vertex
Maybe you have heard that the whole world around us is made out of particles. But particles alone are not very interesting. The interesting thing are particle interactions. These particle interactions are described by a theory called the Standard Model of Particle Physics. But even though this theory is very mathematical, there are some graphical tools which can help us understand its basics. These graphical tools are called Feynman diagrams.
Particle Interactions
But what exactly happens in these particle interactions?
Now you know how the lines in the Feynman diagrams go together. But all the lines have a meaning. Click on the lines in this diagram to find out what the lines actually stand for.
matter particle
matter particle
anti-particle
anti-particle
interaction particle
Particle Interactions
Particle interactions and interaction particles
On this slide you will find out about two fundamental interactions and how they govern our universe.
Most of the processes in our everyday life happen through the electromagnetic interaction.For example, when your hand touches the soft fur of an alpaca, it's electrons from your hand interacting with electrons from the atoms of the fur.
Read more
Whenever matter particles transform into other matter particles, it’s usually the weak interaction which is responsible for this process. For example, muons from cosmic radiation transform into other particles.
Read more
Accept
Particle Interactions
The weak interaction
Whenever matter particles transform into other matter particles, it’s usually the weak interaction which is responsible for this process.
Here you will learn about your first diagram of the weak interaction.
muon
muon-neutrino
W-Minus-particle
When a particle from outer space hits an atom in the upper atmosphere, it subsequently transforms into a so-called muon which we can observe on earth for example in a cloud chamber. But the muon itself transforms into other particles. And we can see in a Feynman diagram how that works. Click on the muon to find out!
electron-antineutrino
electron
Particle Interactions
The electromagnetic interaction
Most of the processes in our everyday life are happen through the electromagnetic interaction.
Here you will learn about your first diagram of the electromagnetic interaction.
Elektron
Elektron
Elektron
Photon
What we perceive as touching - for example, an alpaca, your computer mouse, or the chair you sit on - is just the electrons of the atoms of our hand interacting with electrons of the atoms of what we touch. Here you see an electron from the atoms of your hand and one electron from the atoms of the alpaca. Click on one of the electrons to find out what happens when they interact.
Elektron
Particle Interactions
Examples for particle interactions
In the following pages you will find out which particle interactions are involved in the following real world examples. The examples lead you to separate pages where you can test your knowledge by choosing which diagram and which interaction is involved. Click on the images to discover what they are about. A second click then opens a new slide where you can
Medical imaging
Cosmic particles
Fluorescent animals
Northern lights
The sun's interior
Archeological dating
Particle Interactions Quiz
Fluorescent animals
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Fluorescence can be found in many places in nature and everyday life, such as in certain plants and animals, like platypusses and flying squirrels, and even in some everyday objects. You may have encountered fluorescence while attending a blacklight party, or using a highlighter. Fluorescence is a process in which certain substances absorb light of a specific wavelength and then emit light of a different, usually longer, wavelength. This causes the substance to glow in a different color than the original light source.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! Fluorescence means that an electron absorbs a photon with high energy (meaning light with a low wavelength), gets in an excited state and then emits a photon with a lower energy (meaning light with a higher wavelength). If you want to know more, click the button below. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Positron-Emission-Tomography
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
An important method for medical imaging - that means, making visible what happens inside your body - is the Positron-Emission-Tomography, or PET for short. In a PET scan, a positron is emitted by a radioactive source. This positron then annihilates with a nearby electron. The particles which are created in this annihilation can then be easily detected.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! The positron and the electron annihilate each other and transform into two photons. The photons are then detected by so-called scinitillators. If you want to know more about why the diagram looks as it looks, click on "more info". Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Radiocarbon method
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
One of the most widely used methods for estimating the age of archeological excavations and other historic artefacts is the radiocarbon method, also known as C14-method. One naturally occuring isotope of carbon - the 14C-isotope - transforms into nitrogen while emitting an electron and an antineutrino. The radiocarbon method uses the fact that this transformation occurs with a certain prbability which means that from the amount which is observed today it can be inferred how long ago the transformation process started.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! This is an example for the so-called beta-transformation. To be more precise, it is a beta-minus-transformation, since the neutron transforms into a proton, an antineutrino, and a negatively charged particle, the electron. However, this transformation can only take place with the involvement of a W-particle. Click on "more info" to find out how exactly this works. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Muon production
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Before, you have seen how the muon transforms into an electron, a neutrino, and an anti-neutrino. However, the muon does not come from outer space, but is also only produced in the upper atmosphere. Since nothing comes from nothing, this also happens in some particle process. The muon is produced as the transformation product of another particle, called pion. The pion is not an elementary particle, but is made up of an up-quark and an anti-down-quark. These transform into a muon and an anti-neutrino. The muon subsequently travels toward the earth where we can detect it and measure how it transforms.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! This is a typical example for a particle transformation. However, the pion does not directly transform into the muon and the anti-neutrino, but there is an intermediate particle: It is the W-particle you already know from the muon transformation. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Hydrogen fusion
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
In the interior of the sun, atomic nuclei fuse into other nuclei. This process releases energy, because the resulting atomic nuclei have less mass than the original atomic nuclei combined. Because of Einstein's famous formula E = mc2 this extra mass is converted to energy. The starting point of this fusion process are two hydrogen nuclei transforming into a deuterium nucleus while emitting a positron and a neutrino.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! This is an example for the so-called beta-transformation. To be more precise, it is a beta-plus-transformation, since the proton transforms into a neutron, a neutrino, and a positively charged particle, the positron. However, this transformation can only take place with the involvement of a W-particle. Click on "more info" to find out how exactly this works. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Production of Northern lights
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Northern lights are among the most astonishing phenomena you can observe in nature. When they occur, you can observe a glimmer in the sky, mostly in green, but sometimes also in other colors, such as red, blue, or pink. Northern lights are produced when electrically charged particles from space such as electrons are accelerated towards the higher atmosphere due to solar activity. In the atmosphere these particles interact with the atoms of the air in such a way that they kick out the electrons. When the electrons then recombine with the atom, they are releasing energy in the form of light.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! The process consists of two sub-processes: First the electron from space transfers energy to the electron in the atom of the air molecule. This happens in the form of a photon. This electron can then escape the atom. Once it recombines with an ion, i.e., it forms a neutral atom. Since it requires the electron less energy to sit in the atom than to move freely, it releases its energy in the form of a photon. We can see this photon then as northern light. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions
Particle interactions - Summary
Now you have seen some examples of what particle interactions are and how they can be represented in terms of Feynman diagrams. Let's summarize briefly what we have learned so far. You can do so by clicking on the different elements of the Feynman diagram.
One diagram always consists of two or more so-called vertices. These vertices have to obey to some simple rules.
There are different types of fundamental interactions. Every type of interaction has their own interaction particle(s). The electromagnetic interaction is responsible for most everyday phenomena, such as touching things or everything that has to do with light, such as fluorescence. The interaction particle is the photon, written as γ. The weak interaction is responsible for interactions where matter particles transform into other matter particles, such as radioactive processes. It has three interaction particles: The W+-, the W--, and the Z-particle.
The lines have different meanings: Lines with an arrow to the right represent matter particles, lines with an arrow to the left represent anti-particles, and wavy lines represent interaction particles.
Calculations
Calculations with Feynman diagrams
It is currently being discussed if it’s feasible to build a successor to the current largest particle accelerator in the world, the LHC. This new accelerator would be called FCC and would be almost 100 km long. In this FCC, electrons and positrons will be accelerated. When these two particles collide, all kinds of particle can be created - and the ratio with which these particles are being created, tells us something about how particles interact. The probability with which a certain process takes place, is called cross-section. And this cross-section can be calculated using Feynman diagrams.
After we have seen where particle processes occur in the world, we can turn to a more systematic study of them which is done in particle accelerators. There are different types of accelerators used for different purposes and during this course you will encounter several of them. For now, let's turn to a very common type of accelerator which was important for CERN in the past but might also become important in the future.
Calculations
What are the diagrams used for?
The mathematical expression connected to a Feynman diagram is very complex and requires some mathematical foundations which are beyond the scope of this course. For us it's important that the order of magnitude of this contribution is determined by the number of vertices and the properties of the interaction involved. Click on the different parts of the diagram to see what they contribute to the mathematical expression.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
A simple (?) process
One process which happens in an electron-positron collider is the seemingly very simple process, where one electron and one positron interact and have one electron and one positron as output. We can write this in a reaction equation like this: e+ + e- ➔ e+ + e- It can be very precisely measured how often an electron and a positron are produced in the collisions of the accelerator. In order to be able to compare the measurements with theory we need to calculate the cross-section of this process as precisely as possible using the method described before. But we need to take every diagram into account which describes this process. Now there are different ways how this process can be written in a Feynman diagram. In the following, you can try out if you find more than one way how to draw the corresponding Feynman diagram.
You will be forwarded to a program called FeynGame. In this program you will see the initial state and the final state of the process (both one electron and one positron) and it is your task to complete this into one diagram using the particles at the bottom.
Calculations Quiz
Sesiones de aprendizaje / 02
Good Work!
Have you succeeded in drawing one or even two correct diagrams? Well done! If not: no worries, let's have a look now. For now, there are two diagrams which describe this process in the first order (we will see later what that means). Click on the two diagrams to learn how they are different from each other.
But which of these two processes will occur in the particle collider? And how are the diagrams of any use for calculations? That we will learn on the next slide.
Calculations
Superposition of diagrams
A fundamental principle in quantum physics is the superposition principle. This states that it's not one single possibility which is realised but a mix of all of them. In the case of Feynman diagrams, this means that to calculate the cross-section of a process (for example e+ + e- ➔e+ +e-, the one we are discussing here), we need to add up the mathematical expressions of all the diagrams contributing to that process. Once we added up all the diagrams, we need to square the sum to calculate the cross-section. This process is shown if you click the button.
2x
Calculations
Calculations with Feynman diagrams - Summary
- Draw all diagrams which contribute to the process.
Let's summarize what we have learned in this section.
With Feynman diagrams we can calculate how likely a certain process takes place. This is done in three steps. Click the buttons to reveal the process step by step:
A1 = (u(p1) * v(p2)) * gem * 1/((mγ)2-p2) * gem * (u(p3) * v(p4)
A2 = (u(p1) * v(p3)) * gem * 1/((mγ)2-p2) * gem * (u(p2) * v(p4)
2. Multiply the contributions from the different parts of the diagrams (outer lines, vertices, inner lines).
Can you explain this process in your own words?
σ ~ (A1 +A2)2 = A12 + A22 +2*A1*A2
3. Add up the products from all diagrams and square the sum.
Alpacarticle physics
Merten Dahlkemper
Created on April 18, 2023
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Transcript
This learning material is part of the PhD work of Merten Dahlkemper. It is still work in progress. Any question, comment, remark, critique etc. is highly appreciated and should be sent to merten.dahlkemper@cern.ch
Alpacarticle Physics with Feynman diagrams
How particle interactions rule the world from animals to particle accelerators
start
Introduction
You can always go back to the previous slide using this button.
If you click the button in the top right corner you will find all interactive elements on the slide.
Introduction
At the end of this course, you should have a basic understanding of how particles interact with each other and how particle physicists can make predictions about what they might observe.
Click me!
From fluorescent animals to cosmic rays, from simple things like touching something to astonishing things like northern lights: Everything around us is ruled by particles interacting with each other. In this course, you will learn how we can understand these particle interactions using very simple diagrams.
Hover your mouse over me!
Hover your mouse over me!
Click me if you want to continue to the next slide.
Presentation
You can always come back to this slide by using this button.
These buttons will lead you directly to the respective chapter.
Fundamental rules of particle processes
You know which rules govern fundamental particle processes and how they are translated into simple diagrams.
Contents and Goals
Understand particle interactions
You understand what it means that "particles interact with each other" and get to know examples of these interactions.
This course contains four chapters, each with its particular goal. We recommend to go through it step by step. But you can always come back to this slide and skip to a previous chapter again.
Understand calculations with diagrams
You learn how the simple diagrams are used to make calculations in particle physics.
Understand particle discoveries
You will understand how these diagrams are used to discover and investigate new particles.
Fundamental rules
Fundamental rules of particle interactions
Feynman diagrams are are a combination of different lines and their intersection points, so-called vertices. One Feynman diagram is then any combination of at least two of these vertices, like these: There are just some rules about these vertices for the diagram to be valid.
Left vertex
Feynman diagram
Right vertex
Maybe you have heard that the whole world around us is made out of particles. But particles alone are not very interesting. The interesting thing are particle interactions. These particle interactions are described by a theory called the Standard Model of Particle Physics. But even though this theory is very mathematical, there are some graphical tools which can help us understand its basics. These graphical tools are called Feynman diagrams.
Charge Conservation
All particles have certain invariant properties called charges. In this course, we will look at two different ones, electric charge and weak charge. It is a fundamental law of physics that the total charge of a system, just like its energy, does not change over time. It is also said that the charges are conserved. In each vertex of a diagram, both the electric charge and the weak charge are conserved independently. This means that both the sum of the weak charge and the sum of the electric charge on the left-hand side are exactly the same as on the right-hand side.
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
Menu
Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electric charge on the left side of the vertex is 0, but n the right side -1. The electric charge is therefore not conserved. The weak charge is on the left side 0 and on the right side also 0 (because -½ + ½ = 0).
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
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Fundamental rules
Charge Conservation
Let's look at some examples. In these vertices the charges are either conserved (which means they are allowed) or not conserved (in which case they are not allowed). In the following the number on the top signifies the electrical, the number on the bottom the weak charge.
The electric charge on the left side of the vertex is 0, but n the right side -1. The electric charge is therefore not conserved. The weak charge is on the left side 0 and on the right side also 0 (because -½ + ½ = 0).
The electrical charge on the left side is +1, on the right side also +1. The weak charge on the left side is +½ and on the right side also +½ (because +1 - ½ = +½).
The electric charge on the left side of the vertex is 0, and on the right side also 0 (because 1 - 1 = 0). The weak charge on the left side is 0 and on the right side also 0 (because ½ - ½ = 0).
The electric charge on the left side is +1, on the right side also +1. The weak charge is on the left side +½ and on the right side 3/2 (because 1 + ½ = 3/2). The weak charge is therefore not conserved.
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Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
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Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
On the right vertex none of the charges is conserved. The electric charge is -1 on the left side, but +1 on the right side. The weak charge is -1 on the left, but ½ + ½ = + 1 on the right side.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
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Fundamental rules
Charge conservation
Now those were single vertices. In a graph composed of two or more vertices, charge conservation must be satisfied at each vertex.Also here, the number on the top signifies the electrical, the number on the bottom the weak charge.
On the right vertex none of the charges is conserved. The electric charge is -1 on the left side, but +1 on the right side. The weak charge is -1 on the left, but ½ + ½ = + 1 on the right side.
At the left vertex, both charges are conserved. The electric charge is -1 on the left and -1 on the right. The weak charge is -½ -½ = -1 on the left and -1 on the right side of the vertex.
Therefore the whole diagram is not allowed.
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In the following you will see seven short tasks on charge conservation. Answer each task by clicking on the correct answer. Note: Clicking on the correct answer will take you directly to the next task. You cannot go back and correct. So think before you click.
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. The vertex on the left is incomplete. Which of the three lines on the right complete the diagram correctly at the place of the question mark?
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. A line to complete the two vertices is missing. Which of the three lines on the right complete the diagram correctly at the place of the question mark?
Below you see an unfinished diagram. The number on the top signifies the electrical, the number on the bottom the weak charge. The vertex on the bottom right is incomplete. Which of the three alternatives on the right complete the diagram correctly at the place of the question mark?
Below you see three vertices. The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three vertices is wrong?
Below you see three vertices. The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three vertices is wrong?
On this slide you see three diagrams The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three diagrams is wrong?
On this slide you see three diagrams The number on the top signifies the electrical, the number on the bottom the weak charge. Which of the three diagrams is wrong?
Fundamental rules
Fundamental rules of particle interactions
Feynman diagrams are are a combination of different lines and their intersection points, so-called vertices. One Feynman diagram is then any combination of at least two of these vertices, like these:
Left vertex
Feynman diagram
Right vertex
Maybe you have heard that the whole world around us is made out of particles. But particles alone are not very interesting. The interesting thing are particle interactions. These particle interactions are described by a theory called the Standard Model of Particle Physics. But even though this theory is very mathematical, there are some graphical tools which can help us understand its basics. These graphical tools are called Feynman diagrams.
Particle Interactions
But what exactly happens in these particle interactions?
Now you know how the lines in the Feynman diagrams go together. But all the lines have a meaning. Click on the lines in this diagram to find out what the lines actually stand for.
matter particle
matter particle
anti-particle
anti-particle
interaction particle
Particle Interactions
Particle interactions and interaction particles
On this slide you will find out about two fundamental interactions and how they govern our universe.
Most of the processes in our everyday life happen through the electromagnetic interaction.For example, when your hand touches the soft fur of an alpaca, it's electrons from your hand interacting with electrons from the atoms of the fur.
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Whenever matter particles transform into other matter particles, it’s usually the weak interaction which is responsible for this process. For example, muons from cosmic radiation transform into other particles.
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Particle Interactions
The weak interaction
Whenever matter particles transform into other matter particles, it’s usually the weak interaction which is responsible for this process.
Here you will learn about your first diagram of the weak interaction.
muon
muon-neutrino
W-Minus-particle
When a particle from outer space hits an atom in the upper atmosphere, it subsequently transforms into a so-called muon which we can observe on earth for example in a cloud chamber. But the muon itself transforms into other particles. And we can see in a Feynman diagram how that works. Click on the muon to find out!
electron-antineutrino
electron
Particle Interactions
The electromagnetic interaction
Most of the processes in our everyday life are happen through the electromagnetic interaction.
Here you will learn about your first diagram of the electromagnetic interaction.
Elektron
Elektron
Elektron
Photon
What we perceive as touching - for example, an alpaca, your computer mouse, or the chair you sit on - is just the electrons of the atoms of our hand interacting with electrons of the atoms of what we touch. Here you see an electron from the atoms of your hand and one electron from the atoms of the alpaca. Click on one of the electrons to find out what happens when they interact.
Elektron
Particle Interactions
Examples for particle interactions
In the following pages you will find out which particle interactions are involved in the following real world examples. The examples lead you to separate pages where you can test your knowledge by choosing which diagram and which interaction is involved. Click on the images to discover what they are about. A second click then opens a new slide where you can
Medical imaging
Cosmic particles
Fluorescent animals
Northern lights
The sun's interior
Archeological dating
Particle Interactions Quiz
Fluorescent animals
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Fluorescence can be found in many places in nature and everyday life, such as in certain plants and animals, like platypusses and flying squirrels, and even in some everyday objects. You may have encountered fluorescence while attending a blacklight party, or using a highlighter. Fluorescence is a process in which certain substances absorb light of a specific wavelength and then emit light of a different, usually longer, wavelength. This causes the substance to glow in a different color than the original light source.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle interactions Quiz
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Correct!
You are right! Fluorescence means that an electron absorbs a photon with high energy (meaning light with a low wavelength), gets in an excited state and then emits a photon with a lower energy (meaning light with a higher wavelength). If you want to know more, click the button below. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Positron-Emission-Tomography
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
An important method for medical imaging - that means, making visible what happens inside your body - is the Positron-Emission-Tomography, or PET for short. In a PET scan, a positron is emitted by a radioactive source. This positron then annihilates with a nearby electron. The particles which are created in this annihilation can then be easily detected.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
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Correct!
You are right! The positron and the electron annihilate each other and transform into two photons. The photons are then detected by so-called scinitillators. If you want to know more about why the diagram looks as it looks, click on "more info". Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Radiocarbon method
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
One of the most widely used methods for estimating the age of archeological excavations and other historic artefacts is the radiocarbon method, also known as C14-method. One naturally occuring isotope of carbon - the 14C-isotope - transforms into nitrogen while emitting an electron and an antineutrino. The radiocarbon method uses the fact that this transformation occurs with a certain prbability which means that from the amount which is observed today it can be inferred how long ago the transformation process started.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
Sesiones de aprendizaje / 02
Correct!
You are right! This is an example for the so-called beta-transformation. To be more precise, it is a beta-minus-transformation, since the neutron transforms into a proton, an antineutrino, and a negatively charged particle, the electron. However, this transformation can only take place with the involvement of a W-particle. Click on "more info" to find out how exactly this works. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Muon production
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Before, you have seen how the muon transforms into an electron, a neutrino, and an anti-neutrino. However, the muon does not come from outer space, but is also only produced in the upper atmosphere. Since nothing comes from nothing, this also happens in some particle process. The muon is produced as the transformation product of another particle, called pion. The pion is not an elementary particle, but is made up of an up-quark and an anti-down-quark. These transform into a muon and an anti-neutrino. The muon subsequently travels toward the earth where we can detect it and measure how it transforms.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
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Correct!
You are right! This is a typical example for a particle transformation. However, the pion does not directly transform into the muon and the anti-neutrino, but there is an intermediate particle: It is the W-particle you already know from the muon transformation. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Hydrogen fusion
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
In the interior of the sun, atomic nuclei fuse into other nuclei. This process releases energy, because the resulting atomic nuclei have less mass than the original atomic nuclei combined. Because of Einstein's famous formula E = mc2 this extra mass is converted to energy. The starting point of this fusion process are two hydrogen nuclei transforming into a deuterium nucleus while emitting a positron and a neutrino.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle Interactions Quiz
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Correct!
You are right! This is an example for the so-called beta-transformation. To be more precise, it is a beta-plus-transformation, since the proton transforms into a neutron, a neutrino, and a positively charged particle, the positron. However, this transformation can only take place with the involvement of a W-particle. Click on "more info" to find out how exactly this works. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions Quiz
Production of Northern lights
On the left you find a short explanation of the process. On the right you find several Feynman diagrams. Choose which one of them describes this particle process. If you're stuck, Pauline might help you.
Northern lights are among the most astonishing phenomena you can observe in nature. When they occur, you can observe a glimmer in the sky, mostly in green, but sometimes also in other colors, such as red, blue, or pink. Northern lights are produced when electrically charged particles from space such as electrons are accelerated towards the higher atmosphere due to solar activity. In the atmosphere these particles interact with the atoms of the air in such a way that they kick out the electrons. When the electrons then recombine with the atom, they are releasing energy in the form of light.
No, this diagram describes another process. You will learn about that one soon. You can ask Pauline for help.
Particle interactions Quiz
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Correct!
You are right! The process consists of two sub-processes: First the electron from space transfers energy to the electron in the atom of the air molecule. This happens in the form of a photon. This electron can then escape the atom. Once it recombines with an ion, i.e., it forms a neutral atom. Since it requires the electron less energy to sit in the atom than to move freely, it releases its energy in the form of a photon. We can see this photon then as northern light. Knowing that it should be easy for you to answer which interaction is involved in the process. If not, Pauline can help you again.
More info
Particle Interactions
Particle interactions - Summary
Now you have seen some examples of what particle interactions are and how they can be represented in terms of Feynman diagrams. Let's summarize briefly what we have learned so far. You can do so by clicking on the different elements of the Feynman diagram.
One diagram always consists of two or more so-called vertices. These vertices have to obey to some simple rules.
There are different types of fundamental interactions. Every type of interaction has their own interaction particle(s). The electromagnetic interaction is responsible for most everyday phenomena, such as touching things or everything that has to do with light, such as fluorescence. The interaction particle is the photon, written as γ. The weak interaction is responsible for interactions where matter particles transform into other matter particles, such as radioactive processes. It has three interaction particles: The W+-, the W--, and the Z-particle.
The lines have different meanings: Lines with an arrow to the right represent matter particles, lines with an arrow to the left represent anti-particles, and wavy lines represent interaction particles.
Calculations
Calculations with Feynman diagrams
It is currently being discussed if it’s feasible to build a successor to the current largest particle accelerator in the world, the LHC. This new accelerator would be called FCC and would be almost 100 km long. In this FCC, electrons and positrons will be accelerated. When these two particles collide, all kinds of particle can be created - and the ratio with which these particles are being created, tells us something about how particles interact. The probability with which a certain process takes place, is called cross-section. And this cross-section can be calculated using Feynman diagrams.
After we have seen where particle processes occur in the world, we can turn to a more systematic study of them which is done in particle accelerators. There are different types of accelerators used for different purposes and during this course you will encounter several of them. For now, let's turn to a very common type of accelerator which was important for CERN in the past but might also become important in the future.
Calculations
What are the diagrams used for?
The mathematical expression connected to a Feynman diagram is very complex and requires some mathematical foundations which are beyond the scope of this course. For us it's important that the order of magnitude of this contribution is determined by the number of vertices and the properties of the interaction involved. Click on the different parts of the diagram to see what they contribute to the mathematical expression.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
What are the different contributions of the diagram?
Now let's find out how these contributions look like mathematically. Down here are the verbal descriptions again.
The outer lines contribute a function that is dependent on the particle's momentum (i.e., velocity and rest mass, a quantity usually named p in formulas).
Each vertex contributes a number which is dependent on the interaction involved at that vertex. This is called the vertex factor g. For example, the factor for the weak interaction gw is slightly larger than the factor for the electromagnetic interaction gem.
The inner line contributes a factor that is depndent on the momentum and the rest mass m of the particle. The larger the difference, the smaller the contribution.
Calculations
A simple (?) process
One process which happens in an electron-positron collider is the seemingly very simple process, where one electron and one positron interact and have one electron and one positron as output. We can write this in a reaction equation like this: e+ + e- ➔ e+ + e- It can be very precisely measured how often an electron and a positron are produced in the collisions of the accelerator. In order to be able to compare the measurements with theory we need to calculate the cross-section of this process as precisely as possible using the method described before. But we need to take every diagram into account which describes this process. Now there are different ways how this process can be written in a Feynman diagram. In the following, you can try out if you find more than one way how to draw the corresponding Feynman diagram.
You will be forwarded to a program called FeynGame. In this program you will see the initial state and the final state of the process (both one electron and one positron) and it is your task to complete this into one diagram using the particles at the bottom.
Calculations Quiz
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Good Work!
Have you succeeded in drawing one or even two correct diagrams? Well done! If not: no worries, let's have a look now. For now, there are two diagrams which describe this process in the first order (we will see later what that means). Click on the two diagrams to learn how they are different from each other.
But which of these two processes will occur in the particle collider? And how are the diagrams of any use for calculations? That we will learn on the next slide.
Calculations
Superposition of diagrams
A fundamental principle in quantum physics is the superposition principle. This states that it's not one single possibility which is realised but a mix of all of them. In the case of Feynman diagrams, this means that to calculate the cross-section of a process (for example e+ + e- ➔e+ +e-, the one we are discussing here), we need to add up the mathematical expressions of all the diagrams contributing to that process. Once we added up all the diagrams, we need to square the sum to calculate the cross-section. This process is shown if you click the button.
2x
Calculations
Calculations with Feynman diagrams - Summary
Let's summarize what we have learned in this section.
With Feynman diagrams we can calculate how likely a certain process takes place. This is done in three steps. Click the buttons to reveal the process step by step:
A1 = (u(p1) * v(p2)) * gem * 1/((mγ)2-p2) * gem * (u(p3) * v(p4)
A2 = (u(p1) * v(p3)) * gem * 1/((mγ)2-p2) * gem * (u(p2) * v(p4)
2. Multiply the contributions from the different parts of the diagrams (outer lines, vertices, inner lines).
Can you explain this process in your own words?
σ ~ (A1 +A2)2 = A12 + A22 +2*A1*A2
3. Add up the products from all diagrams and square the sum.