Quantum Diaries

Posts Tagged ‘QED’

Quarks: Yeah, They Exist

Monday, April 16th, 2012

Physics Fact: 58 years ago, quarks were independently proposed by Murray Gell-Mann & George Zweig [1,2]. M.G.M. called them “quarks” and Zweig called them “aces.”

Hi All,

A question I often get, like really often, especially from other physicists, is “How do we know quarks exist?” In particular,

If (light) quarks cannot be directly observed, due to the phenomenon known as color confinement (or infrared slavery as I like calling it), then how do we know quarks exist?

This is a really good question and it has a number of different answers. To a physicist, being able to directly observe an object means being able to isolate it and subsequently measure its properties, for example: electric charge. Due to effects associated with the strong nuclear force, quarks lighter than the top quark will nucleate into other objects (hadrons) in about 3×10^-25 seconds. This is pretty fast, much faster than any piece of modern electronics. Consequentially, light quarks cannot be directly observed with present technology. However, this inability to isolate quarks does not imply we cannot directly measure their properties (like electric charge!).

This brings me to today’s post: How physicists measure quarks’ electric charges!

R

Fig. 1: An electron (e-) and positron (e+) annihilate to produce a virtual photon (γ*) that subsequently decays into a muon (μ-) and anti-muon (μ+). Click for full size.

A very typical calculation done by any student in a course on particle physics (undergraduate or graduate) is to calculate the likelihood (called cross section) of an electron and positron annihilating into a virtual photon, which then decays into a muon and anti-muon. (See the diagram to the right.). Since electrons, muons, and their anti-matter partners all have so little mass, it is pretty reasonable to just pretend they are all massless. The calculation becomes considerably easier, trust me on this. When all is said and done, we find that the cross section is equal to a bunch of constants (which I am just going to collectively call σ₀), times the square of the electron’s electric charge (Q²_e), times the square of the muon’s electric charge (Q²_μ):

Likelihood of e⁺e^– → μ⁺μ^– = σ₀ × Q²_e × Q²_μ

However, the electric charges of electrons and muons are both 1 (in elementary units) so the likelihood reduces to just σ₀. Convenient, right?

Now, if we replace muons with quarks, then he find that the cross section is this:

Likelihood of e⁺e^– → qq = 3 × σ₀ × Q²_q

That’s right: the probability of producing quarks with electrons & positrons is simply three times that for producing muons, scaled by the square of the quarks’ electric charge. This amazing result allows us to then define the quantity “R“, which is just the ratio of the likelihoods:

R = (Likelihood of e⁺e^– → qq) / (Likelihood of e⁺e^– → μ⁺μ^–) = 3 × Q²_q

In other words, by measuring the ratio of how likely it is to produce a particular set of quarks to how likely it is to produce muons, we can directly measure quarks’ electric charge! (BOOYA!)

Measuring R

As far as measuring R goes, it is pretty straightforward. However, there has to be some caveat or complication since this is physics we are talking about. Sure enough there are a few and I am just going to ignore them all, all but one.

In order to determine the probability of producing a particular pair of quarks using electron-positron collisions, experimentalists have to make sure the total energy of the collision is large enough. Simply put, no particle can ever be generated if there is not enough energy to make it. It is an example of the Conservation of Energy. The problem is this: if there is enough energy to make a particular set of quarks, then there is sufficient energy to produce any quark pair lighter than the original set. In addition, it is very difficult to isolate different quark-anti-quark pairs (see the top of this post for why that is).

The solution to this issue is to simply measure the likelihood of producing ALL types of quarks for a particular energy. To do so, all we need is to add up all the individual cross sections for each set of quarks. The total cross section simplifies to this:

Likelihood of e⁺e^– → ALL qq = 3 × σ₀ × Q²_e × Sum Q²_q

That is to say, the probability of producing ALL quark-anti-quark pairs in electron-positron collisions is equal to a bunch of constants (σ₀) times the square of the electron’s electric charge (Q²_e), times the sum of the square of each quark’s electric charge (Q²_q). Consequently, R becomes

R = (Likelihood of e⁺e^– → ALL qq) / (Likelihood of e⁺e^– → μ⁺μ^–) = 3 × Sum of all Q²_q

R may no longer be a direct measurement of a single quark’s electric charge, but it is still a direct measurement of the electric charge of all the quarks. Without further ado, here are the predictions:

Table 1: R-values for energies below 200 MeV (0.1 GeV) and above 9 GeV. Click for full size.

Here are the data. This plot is taken from my favorite particle physics books, Quarks & Leptons:

Fig. 2: The R value of light quarks versus energy of quark-anti-quark pair. Click for full size. Credit: F. Halzen and D. Martin, "Quarks and Leptons: An Introductory Course in Modern Particle Physics", Wiley 1984.

That Disagreement Near 5-8 GeV is Not Really a Disagreement

Time for a little extra credit. If you look closely at figure 2, you may notice that between 5 GeV and 8 GeV all the data points are uniformly above the R=10/3 line. This feature is actual the result of two things: the first is that quarks really do have masses and cannot be ignored at these energies; the second is that the strong nuclear force surprisingly contributes to this process. I will not say much about the first point other than mention that, in our quick calculation above, we pretended to ignore all masses because electrons and muons were so light. The mass (in natural units) of the charm quark is about 1.3 GeV, and that is hardly small compared to 5 GeV.

Taking a closer look at where the virtual photon produces a quark and anti-quar k pair, we realize that quark and anti-quark are pretty close together. They are actually close enough to emit and absorb gluons, the particle that mediates the strong nuclear force. This has a very important consequence. Previously, the quark and anti-quark pair could only be produced in such a way that the total momentum of the system was conserved. However, if we consider the fact that the quarks can exchange gluons, and hence exchange momenta, then the quark and anti-quark pair can be produced an infinite number of different ways that violate the conservation of total momentum, so long as at least one gluon is exchanged between the two in order to restore total momentum. This amplification in likelihood is highly sensitive to energy but it causes about a 20% increase in R between 5 and 8 GeV. This 20% increase in R is precisely the difference between all the data points and the R = 10/3 line.

Fig. 3: A Feynman diagram representing the annihilation of an electron (e-) and positron (e+) into a virtual photon (γ*) that decays into a quark (q) and anti-quark (q) pair. The photon-quark-quark vertex is enlarged to highlight the ability for nearby quarks to exchange gluons. Click for full size.

Happy Colliding.

– richard (@bravelittlemuon)

P.S. #PhysicsFact should totally be a trend today. Go! Make it trend!

Tags: @bravelittlemuon, Color Confinement, Electric Charge, Infrared Slavery, QED, Quarks, R Value
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Everytime a Belle Rings, A Hadron Gets Its Wings

Friday, January 27th, 2012

Richard Ruiz

Fun post for everyone today. In response to last week’s post on describing KEK Laboratory’s discovery of additional exotic hadrons, I got an absolutely terrific question from a QD reader:

Surprisingly, the answer to “How does an electron-positron collider produce quarks if neither particle contains any?” all begins with the inconspicuous photon.

No Firefox, I Swear “Hadronization” is a Real Word.

As far as the history of quantum physics is concerned, the discovery that all light is fundamentally composed of very small particles called photons is a pretty big deal. The discovery allows us to have a very real and tangible description of how light and electrons actually interact, i.e., through the absorption or emission of photon by electrons.

Figure 1: Feynman diagrams demonstrating how electrons (denoted by e^–) can accelerate (change direction of motion) by (a) absorbing or (b) emitting a photon (denoted by the Greek letter gamma: γ).

The usefulness of recognizing light as being made up many, many photons is kicked up a few notches with the discovery of anti-particles during the 1930s, and in particular the anti-electron, or positron as it is popularly called. In summary, a particle’s anti-particle partner is an identical copy of the particle but all of its charges (like electric, weak, & color!) are the opposite. Consequentially, since positrons (e⁺) are so similar to electrons (e^–) their interactions with light are described just as easily.

Figure 2: Feynman diagrams demonstrating how positrons (e⁺) can accelerate (change direction of motion) by (a) absorbing or (b) emitting a photon (γ). Note: positrons are moving from left to right; the arrow’s direction simply implies that the positron is an anti-particle.

Then came Quantum Electrodynamics, a.k.a. QED, which gives us the rules for flipping, twisting, and combining these diagrams in order to describe all kinds of other real, physical phenomena. Instead of electrons interacting with photons (or positrons with photons), what if we wanted to describe electrons interacting with positrons? Well, one way is if an electron exchanges a photon with a positron.

Figure 3: A Feynman diagram demonstrating the exchange of a photon (γ) between an electrons (e^–) and a positron (e⁺). Both the electron and positron are traveling from the left to the right. Additionally, not explicitly distinguishing between whether the electron is emitting or absorbing is intentional.

And now for the grand process that is the basis of all particle colliders throughout the entire brief* history of the Universe. According to electrodynamics, there is another way electrons and positrons can both interact with a photon. Namely, an electron and positron can annihilate into a photon and the photon can then pair-produce into a new electron and positron pair!

Figure 4: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) that then produces an e⁺e^– pair. Note: All particles depicted travel from left to right.

However, electrons and positrons is not the only particle-anti-particle pair that can annihilate into photons, and hence be pair-produced by photons. You also have muons, which are identical to electrons in every way except that it is 200 times heavier than the electron. Given enough energy, a photon can pair-produce a muon and anti-muon just as easily as it can an electron and positron.

Figure 5: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) that then produces a muon (μ^–) and anti-muon(μ⁺) pair.

But there is no reason why we need to limit ourselves only to particles that have no color charge, i.e., not charged under the Strong nuclear force. Take a bottom-type quark for example. A bottom quark has an electric charge of -1/3 elementary units; a weak (isospin) charge of -1/2; and its color charge can be red, blue, or green. The anti-bottom quark therefore has an electric charge of +1/3 elementary units; a weak (isospin) charge of +1/2; and its color charge can be anti-red, anti-blue, or anti-green. Since the two have non-zero electric charges, it can be pair-produced by a photon, too.

Figure 6: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) that then produces a bottom quark (b) and anti-bottom quark (b) pair.

On top of that, since the Strong nuclear force is, well, really strong, either the bottom quark or the anti-bottom quark can very easily emit or absorb a gluon!

Figure 7: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) that produces a bottom quark (b) and anti-bottom quark (b) pair, which then radiate gluons (blue).

In electrodynamics, photons (γ) are emitted or absorbed whenever an electrically charged particle changes it direction of motion. And since the gluon in chromodynamics plays the same role as the photon in electrodynamics, a gluon is emitted or absorbed whenever a “colorfully” charged particle changes its direction of motion. We can absolutely take this analogy a step further: gluons are able to pair-produce, just like photons.

Figure 8: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) that produces a bottom quark (b) and anti-bottom quark (b) pair. These quarks then radiate gluons (blue), which finally pair-produce into quarks.

At the end of the day, however, we have to include the effects of the Weak nuclear force. This is because electrons and quarks have what are called “weak (isospin) charges”. Firstly, there is the massive Z boson (Z), which acts and behaves much like the photon; that is to say, an electron and positron can annihilate into a Z boson. Secondly, there is the slightly lighter but still very massive W boson (W), which can be radiated from quarks much like gluons, just to a lesser extent. Phenomenally, both Weak bosons can decay into quarks and form semi-stable, multi-quark systems called hadrons. The formation of hadrons is, unsurprisingly, called hadronization. Two such examples are the the π meson (pronounced: pie mez-on) or the J/ψ meson (pronounced: jay-sigh mezon). (See this other QD article for more about hadrons.)

Figure 9: A Feynman diagram demonstrating an annihilation of an electrons (e^–) and a positron (e⁺) into a photon (γ) or a Z boson (Z) that produces a bottom quark (b) and anti-bottom quark (b) pair. These quarks then radiate gluons (blue) and a W boson (W), both of which finally pair-produce into semi-stable multi-quark systems known as hadrons (J/ψ and π).

In summary, when electrons and positrons annihilate, they will produce a photon or a Z boson. In either case, the resultant particle is allowed to decay into quarks, which can radiate additional gluons and W bosons. The gluons and W boson will then form hadrons. My friend Geoffry, that is how how you can produce quarks and hadrons from electron-positron colliders.

Now go! Discuss and ask questions.

Happy Colliding

– richard (@bravelittlemuon)

* The Universe’s age is measured to be about 13.69 billion years. The mean life of a proton is longer than 2.1 x 10²⁹ years, which is more than 15,000,000,000,000,000,000 times the age of the Universe. Yeah, I know it sounds absurd but it is true.

Tags: @bravelittlemuon, annihilation, b quark, Belle, electron, ewk, Feynman Diagrams, Hadrons, photons, positron, QCD, QED, Quarks, Twitter, w boson, Z boson
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QED + μ: introducing the muon

Sunday, April 4th, 2010

Flip Tanedo

It’s time to return to our ongoing exploration of the structure of the Standard Model. Our primary tools are Feynman diagrams, which we introduced in previous posts (part 1, part 2). By now we’ve already familiarized ourselves with quantum electrodynamics (QED): the theory of electrons, positrons, and photons. Now we’re going to start adding on pieces to build up the Standard Model. We’ll start with the muon, portrayed below by Los Angeles artist Julie Peasley. (These handmade plushes can be found at her website, The Particle Zoo.)

We’re all familiar with the electron. Allow me to introduce its heavier cousin, the muon (μ). Where did this muon come from? Or, as Nobel Prize winner I. I. Rabi once asked, “Who ordered that?” (This is still an unanswered question!) Besides its mass, the muon has the same fundamental properties as the electron: it has the same charge, feels the same forces, and—like the electron—has an anti-particle partner.

Feynman rules for QED+μ

This makes it really easy to extend our Feynman rules. We’ll call our theory “QED+μ,” quantum electrodynamics with an additional particle. We just have to write the rules for two copies of QED:

QEDmu

Let’s recall how to interpret this. The three lines tell us that we have three kinds or particles in the theory: electrons (e), muons (μ), and photons (γ). Recall that the matter particles, the ones whose lines have an arrow, also have antiparticles. We indicate antiparticles by arrows pointing in the wrong direction when we read the diagrams from left-to-right. The vertex rules tell us that we have two kinds of interactions: a photon can either interact with two electrons or two muons.

It’s important to note that we cannot have photon couplings that mix electrons and muons. In terms of conservation laws, we say that electron and muon number are each conserved. For example, in the theory we’ve developed so far, you cannot have a muon decay into an electron and a photon. (We’ll introduce these sorts of interactions next time when we discuss electroweak theory.)

Exercise: Is the following diagram allowed in QED + μ?

eemumu

Answer: Yes! But doesn’t this violate conservation of electron and muon number? You start out with two e‘s on the left and end up with two μ’s. Hint: what are the arrows telling you?

Once you’ve convinced yourself that the above diagram doesn’t violate electron or muon conservation, let me remark that this is an easy way to produce muons at low energy electron colliders. You just smash an electron against a positron and sometimes you’ll end up with a muon-antimuon pair which you can detect experimentally.

Exercise: when we previously did electron-positron to electron-positron scattering, we had to include two diagrams. Why is there only one diagram for eμ to eμ? Hint: draw the two diagrams for ee to ee and check if the Feynman rules still allow both diagrams if we convert the final states to muons.

Detecting muons, some collider physics

If you think about this a little, you might wonder: if electrons and muons are so similar, how can experimentalists distinguish between them at a collider? Seth and Mike might scold me for skipping over some information about the interaction of charged particles through matter, but one simple way to distinguish muons from electrons is to measure their energy and momenta. We know that (away from a potential) a particle’s energy is the sum of its kinetic energy plus it’s mass energy added in quadrature E²=m²c⁴+p²c² (this is the “real” version of E=mc²). Since muons are heavier than electrons, we can just check the mass of the particle by plugging in the measured energy and momentum.

Actually, this is an oversimplified picture. In order not to annoy the other US/LHC bloggers, I’d better provide a slightly less oversimplified “cartoon.” Electrons are light, so let’s imagine that they’re ping pong balls. On the other hand, muons are heavy, so let’s imagine them as bowling balls. As you probably know, the LHC detectors are big and full of stuff… by that I mean atoms, which in turn are made up of a nucleus and a cloud of electrons. We can thus imagine a sea of ping-pong balls (think of a Chuck-E-Cheese ball pit). When electrons hit this ball pit, they end up distributing all of their energy into the other balls. This happens in the electromagnetic calorimeter, or ECAL. “Calor” is Latin for heat, so you can guess that the ECAL is really just a big fancy thermometer that measures the energy that the electron dissipates. Muons on the other hand, are bowling balls that are so massive that they just barrel straight through the ball pit to get to the other side. Here’s a very scientific illustration:

electronandmuonecal

I hope we don’t get any comments saying, “oh man, muons are jerks.” In fact, they’re quite the opposite: muons are the only Standard Model particles that make it all the way to the outside of the detector, making it easy for us to identify them. In fact, the big distinctive toroidal magnets on the ATLAS detector below are there to bend the path of muons to help the outermost detector determine the muon momentum by measuring the curvature of their trail.

Exercise: [for those who want to do some actual calculations, requires a high school physics background] Convince yourself that this heuristic picture is correct by calculating the final momenta of a ball colliding elastically with (a) a ball of the same mass and (b) a ball of much lighter mass.

ATLAS toroidal magnets. Image from the Interactions.org Image Bank

Neat things that muons can do

Let me make a few more semi-historical remarks: our QED+μ model is just a theoretical toy. Historically, scientists knew immediately that something was weird about the muon: unlike electrons, it decayed into other particles and seemed to interact with mesons in unusual ways. In fact, for a while people thought that muons were a kind of meson. These differences ended up being a harbinger of something more interesting: the weak force.

Exercise: convince yourself that our Feynman rules for QED+μ do not allow muon decay, i.e. μ turning into non-μ stuff.

Muons are generated in the sky when cosmic rays hit atoms of the upper atmosphere. These rain down onto the Earth and force us to put our dark matter experiments deep underground to avoid their ‘noise.’ What’s really neat, however, is that the fact that muons make it to the surface of the Earth is a rousing experimental check of relativity. We know that muons at rest decay in microseconds. In this time, it seems like there’s no way for them to traverse the kilometers (about 4 km) between the Earth and its upper atmosphere; even if they were traveling at the speed of light! (c ~ 3. 10⁸ m/s). What’s happening is the phenomenon of time dilation!

Introducing the tau (via the Socratic method)

Exercise: the Standard Model actually has another cousin of the electron, the tau (τ), leading to three charged leptons in total. Write down the Feynman rules for the theory QED+μ+τ, i.e. the theory of electrons, muons, and taus interacting via photons. Make sure that electron, muon, and tau number are all conserved. Draw the diagram for tau production in an electron-positron collider.

Exercise: Above we argued that muons are special because they barrel right through our detectors like bowling balls through an array of ping pong balls. Taus are even heavier, shouldn’t they also make it to the outside of the detector?

Answer: This was a bit of a trick question. The logic is correct that sufficiently energetic taus should make it all the way to the outside of the detector in our QED+μ+τ theory. However, this is not the full story for electrons, muons, and taus (collectively known as leptons) in the Standard Model. Like muons, taus are unstable and will decay. In fact, they decay much more quickly than muons because they have more mass and can decay into stuff (they have more “phase space”). While muons are like bowling balls barreling through the detector, taus are more like grenades that burst into hadronic “shrapnel” inside the calorimeters. They are usually very difficult to reconstruct from the data.

A preview of things to come:

Now we’re very familiar with putting together multiple copies of QED. For now, there are only three copies we have to worry about. It is an open question why this is the case. The existence of at least three copies, however, turns out to be significant for the imbalance of matter and anti-matter in the universe. In the next post we’ll introduce the weak force and really see what we can do with these leptons.

I’m currently in the middle of my “Advancement to Candidacy” exam, so my posts might be a little more delayed than usual this month. By the end of it, however, I hope to be blogging as an official PhD candidate. 🙂

Erratum: virtual particles

I wanted to correct a misleading statement I made in my previous QED post: I discussed the visualization of virtual particles as balls that two kids toss back and forth while standing on frictionless ice. Conservation of momentum causes the two kids to slide apart as they throw and catch the ball, generating what we observe macroscopically as a repulsive force. We mentioned that it’s more difficult to see how this could give rise to an attractive force. I suggested that this is a phenomenon coming from the accumulated effect of many quantum exchanges. While this is true, there is a simpler way to understand this: pretend the ball has negative momentum! Since the particle is virtual, it is inherently quantum mechanical and needn’t have ‘on-shell’ (physical) momentum. Thus one could imagine tossing the ball with negative momentum, causing one to be deflected in the same direction as the ball was tossed. Similarly, catching the ball with negative momentum would push one in the direction that the ball came from.

Does it make sense classically? No! But that’s okay because they’re virtual particles.

That’s all for now, folks!
–Flip, on behalf of the US/LHC blog.

Tags: Feynman rules, muon, QED
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More Feynman Diagrams

Sunday, March 7th, 2010

Flip Tanedo

In a previous post we learned how to draw Feynman diagrams by drawing lines and connecting them. We started with a set of rules for how one could draw diagrams:

QEDrules

We could draw lines with arrows or wiggly lines and we were only permitted to join them using intersections (vertices) of the above form. These are the rules of the game. We then said that the arrowed lines are electrons (if the arrow goes from left to right) and positrons (if the arrow points in the opposite direction) while the wiggly lines are photons. The choice of rules is what we call a “model of particle interactions,” and in particular we developed what is called quantum electrodynamics, which is physics-talk for “the theory of electrons and photons.”

Where did it all come from?

One question you could ask now is: “Where did these rules come from? Why do they prohibit me from drawing diagrams with three wiggly lines intersecting?”

The short answer is that those are just the rules that we chose. Technically they came from a more mathematical formulation of the theory. It is not obvious at all, but the reason why we only allow that one particular vertex is that it is the only interaction that both respects the (1) spacetime (“Lorentz”) symmetry and (2) internal ‘gauge’ symmetry of the theory. This is an unsatisfying answer, but we’ll gradually build up more complicated theories that should help shed some light on this. Just for fun, here’s the mathematical expression that encodes the same information as the Feynman rules above: [caution: I know this is an equation, but do not be scared!]

Without going into details, the represents the electron (the bar turns it into a positron) while the A is the photon. The number e is the ‘electric coupling’ and determines the charge of the electron. Because equations can be intimidating, we won’t worry about them here. In fact our goal will be to go in the opposite direction: we will see that we can learn quite a lot by only looking at Feynman diagrams and never doing any complicated math. The important point is that our cute rules for how to connect lines really captures most of the physics encoded in these ugly equations.

Now a quick parenthetical note because I’m sure some of you are curious: In the equation above, the is a kind of derivative. Derivatives tell us about how things change, and in fact this term tells us about how the electron propagates through space. The e?A term tells us how the photon couples to the electron. The m term is the electron’s mass. We’ll have more to say about this down the road when we discuss the Higgs boson. Finally, the Fs are the “field strength” of the photon: it is the analog of the derivative term for the electron and tells us how the photon propagates through space. In fact, these F’s encode the electric and magnetic fields.

[Extra credit for advanced readers: notice that the electron mass term looks like the Feynman rule for a two-electron interaction with coupling strength m. You can see this by looking at the electron-electron-photon term and removing the photon.]

What we can learn from just looking at the rules

We learned that we could use our lines and intersections to draw diagrams that represent particle interactions. If you haven’t already, I encourage you to grab a piece of scratch paper and play with these Feynman rules. A good game to play is asking yourself whether a certain initial state can ever give you a certain final state. Here are a few exercises:

You start with one electron. Can you ever end up with a final state positron? [Answer: yes! Draw one such diagram.]
If you start with one electron, can you ever end up with more final state positrons than final state electrons? [Answer: no! Draw diagrams until you’re convinced it’s impossible.]
Draw a diagram where an electron and a photon interact to produce 3 electrons, 2 positrons, and 2 photons. Draw a few more to get a feel for how many different ways one can do this.
If you start with a photon, can you end up with a final state of only multiple photons? [This is actually a trick question; the answer is no but this is a rather subtle quantum mechanical effect that’s beyond our scope. You should be able to draw a diagram think that the answer is ‘yes.’]

So here’s what you should get out of this: Feynman rules are a nice way to learn what kinds of particle interactions can and cannot occur. (e.g. questions 1 and 2) In fact, the lesson you should have gleaned is that there is a conservation of electric charge in each diagram coming to the conservation of electric charge in each intersection. You can also see how complicated interactions can be reduced to simple interactions with “virtual particles” (intermediate particles that don’t appear in the initial state). We are able to do this simply by stating the Feynman rules of our theory and playing with drawings. No math or fancy technical background required.

Summing diagrams: an analogy to summing paths

There’s a lot more one could do with Feynman diagrams, such as calculating probabilities for interactions to occur. Actually doing this requires more formal math and physics background, but there’s still a lot that we can learn conceptually.

For example, there were two simple diagram that we could draw that represented the scattering of an electron and a positron off of one another:

We recall that we can describe these interactions in words by “reading” them from left to right:

The first diagram shows an electron and a positron annihilating into a photon, which then “pair produces” into another electron and positron.
The second diagram shows an electron and a positron interacting by sending a photon between them. This is definitely a different process since the electron and positron never actually touch, unlike the first diagram.

Remember that these diagrams are actually shorthand for complex numbers. The numbers represent the probability for each these processes to occur. In order to calculate the full probability that an electron and a positron will bounce off of one another, we have to add together these contributions as complex numbers.

What does this mean? This is just quantum mechanics at work! Recall another old post about the double slit experiment. We learned that quantum mechanics tells us that objects take all paths between an initial observed state to a final observed state. Thus if you see a particle at point A, the probability for it to show up at point B is given by the sum of the probability amplitudes for each intermediate path.

The sum of diagrams above is a generalization of the exact same idea. Our initial observed state is an electron and a positron. Each of these have some fixed [and observed] momentum. If you want to calculate the probability that these would interact and produce an electron and positron of some other momentum (e.g. they bounce off each other and head off in opposite directions), then one not only has to sum over the different intermediate paths, but also the different intermediate interactions.

Again, a pause for the big picture: we’re not actually going to calculate anything since for most people, this isn’t as fun as drawing diagrams. But even just describing what one would calculate, we can see how things reduce to our simple picture of quantum mechanics: the double slit experiment. (more…)

Tags: Feynman rule, QED
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Let’s draw Feynman diagrams!

Sunday, February 14th, 2010

Flip Tanedo

Greetings! This post turned into a multi-part ongoing series about the Feynman rules for the Standard Model and a few of its extensions. I’ll use this first post as an index for all of the parts of the series.

Let’s draw Feynman diagrams! (this post)
More Feynman diagrams.
Introducing the muon.
The Z boson and resonances.
Neutrinos.
The W boson, mixing things up.
Meet the quarks.
World of glue.
QCD and confinement.
Known knowns of the Standard Model. (summary)
When Feynman Diagrams Fail.
An idiosyncratic introduction to the Higgs.
A diagrammatic hint of masses from the Higgs
Higgs and the vacuum: Viva la “vev”
Helicity, Chirality, Mass, and the Higgs
The Birds and the Bs
The spin of gauge bosons
Who ate the Higgs?
Unitarization of vector boson scattering
Private lives of Standard Model particles (summary)

There are few things more iconic of particle physics than Feynman diagrams. These little figures of squiggly show up prominently on particle physicists’ chalkboards alongside scribbled equations. Here’s a ‘typical’ example from a previous post.

The simplicity of these diagrams has a certain aesthetic appeal, though as one might imagine there are many layers of meaning behind them. The good news is that’s it’s really easy to understand the first few layers and today you will learn how to draw your own Feynman diagrams and interpret their physical meaning.

You do not need to know any fancy-schmancy math or physics to do this!

That’s right. I know a lot of people are intimidated by physics: don’t be! Today there will be no equations, just non-threatening squiggly lines. Even school children can learn how to draw Feynman diagrams (and, I hope, some cool science). Particle physics: fun for the whole family. 🙂

For now, think of this as a game. You’ll need a piece of paper and a pen/pencil. The rules are as follows (read these carefully):

You can draw two kinds of lines, a straight line with an arrow or a wiggly line:

You can draw these pointing in any direction.
You may only connect these lines if you have two lines with arrows meeting a single wiggly line.

Note that the orientation of the arrows is important! You must have exactly one arrow going into the vertex and exactly one arrow coming out.
Your diagram should only contain connected pieces. That is every line must connect to at least one vertex. There shouldn’t be any disconnected part of the diagram.

In the image above the diagram on the left is allowed while the one on the right is not since the top and bottom parts don’t connect.
What’s really important are the endpoints of each line, so we can get rid of excess curves. You should treat each line as a shoelace and pull each line taut to make them nice and neat. They should be as straight as possible. (But the wiggly line stays wiggly!)

That’s it! Those are the rules of the game. Any diagram you can draw that passes these rules is a valid Feynman diagram. We will call this game QED. Take some time now to draw a few diagrams. Beware of a few common pitfalls of diagrams that do not work (can you see why?):

After a while, you might notice a few patterns emerging. For example, you could count the number of external lines (one free end) versus the number of internal lines (both ends attached to a vertex).

How are the number of external lines related to the number of internal lines and vertices?
If I tell you the number of external lines with arrows point inward, can you tell me the number of external lines with arrows pointing outward? Does a similar relation hole for the number of external wiggly lines?
If you keep following the arrowed lines, is it possible to end on some internal vertex?
Did you consider diagrams that contain closed loops? If not, do your answers to the above two questions change?

I won’t answer these questions for you, at least not in this post. Take some time to really play with these diagrams. There’s a lot of intuition you can develop with this “QED” game. After a while, you’ll have a pleasantly silly-looking piece of paper and you’ll be ready to move on to the next discussion:

What does it all mean?

(more…)

Tags: Feynman diagram, Feynman rule, particle physics, QED
Posted in Uncategorized | 36 Comments »

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