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Posts Tagged ‘Feynman Diagrams’

Matter and energy have a very curious property. They interact with each other in predictable ways and the more energy an object has, the smaller length scales it can interact with. This leads to some very interesting and beautiful results, which are best illustrated with some simple quantum electrodynamics (QED).

QED is the framework for describing the interactions of charged leptons with photons, and for now let’s limit things to electrons, positrons and photons. An electron is a negatively charged fundamental particle, and a positron is the same particle, but with a positive charge. A photon is a neutral fundamental particle of light and it interacts with anything that has a charge.

That means that we can draw a diagram of an interaction like the one below:

An electron radiating a photon

An electron radiating a photon

In this diagram, time flows from left to right, and the paths of the particles in space are represented in the up-down direction (and two additional directions if you have a good enough imagination to think in four dimensions!) The straight line with the arrow to the right is an electron, and the wavy line is a photon. In this diagram an electron emits a photon, which is a very simple process.

Let’s make something more complicated:

An electron and positron exchange a photon

An electron and positron make friends by exchanging a photon

In this diagram the line with the arrow to the left is a positron, and the electron and positron exchange a photon.

Things become more interesting when we join up the electron and positron lines like this:

An electron and positron annihilate

An electron and positron get a little too close and annihilate

Here an electron and positron annihilate to form a photon.

Now it turns out in quantum mechanics that we can’t just consider a single process, we have to consider all possible processes and sum up their contributions. So far only the second diagram we’ve considered actually reflects a real process, because the other two violate conservation of energy. So let’s look at electron-positron scattering. We have an electron and a positron in the initial state (the left hand side of the diagram) and in the final state (the right hand side of the diagram):

What happens in the middle?  According to quantum mechanics, everything possible!

What happens in the middle? According to quantum mechanics, everything possible!

There are two easy ways to join up the lines in this diagram to get the following contributions:

Two possible diagrams for electron-positron scattering

Two possible diagrams for electron-positron scattering

There’s a multiplicative weight (on the order of a percent) associated with each photon interaction, so we can count up the photons and determine the contribution each process has. In this case, there are two photon interactions in each diagram, so each one contributes roughly equally. (You may ask why we bother calculating the contributions for a given pair of initial and final states. In fact what we find interesting is the ratio of contributions for two different pairs of initial and final states so that we can make predictions about rates of interactions.)

Let’s add a photon to the diagram, just for fun. We can connect any two parts of electron and positron lines to create a photon, like so:

Taking up the complexity a notch, by adding a photon

Taking up the complexity a notch, by adding a photon

A fun game to play in you’re bored in a lecture is to see how many unique ways you can add a photon to a diagram.

So how do we turn this into a fractal? Well we start off with an electron moving through space (now omitting the particle labels for a cleaner diagram):

A lonely electron :(

A lonely electron 🙁

Then we add a photon or two to the diagram:

An electron with a photon

An electron with a photon

An electron hanging out with two photons

An electron hanging out with two photons

An electron going on an adventure with two photons

An electron going on an adventure with two photons

Similarly let’s start with a photon:

A boring photon being boring

A boring photon being boring

And add an electron-positron pair:

Ah, that's a bit more interesting

Ah, that’s a bit more interesting

This is all we need to get started. Every time we see an electron or positron line, we can replace it with a line that emits and absorbs a photon. Every time we see a photon we can add an electron-positron pair. We can keep repeating this process as much as we like until we end up with arbitrarily complex diagrams, each new step adding more refinement to the overall contributions:

A very busy electron

A very busy electron

At each step the distance we consider is smaller than the one before it, and the energy needed to probe this distance is larger. When we talk about an electron we usually think of a simple line, but real electrons are actually made of a mess of virtual particles that swarm around the central electron. The more energy we put into probing the electron’s structure (or lack of structure) the more particles we liberate in the process. There are many diagrams we can draw and we can’t pick out a single one of these diagrams as the “real” electron, as they all contribute. We have to take everything to get a real feel of what something as simple as an electron is.

As usual, things are even more complicated in reality than this simple picture. To get a complete understanding we should add the other particles to the diagrams. After all, that’s how we can get a Higgs boson out of proton- in some sense the Higgs boson was “already there” inside the proton and we just liberated it by adding a huge amount of energy. If things are tricky for the electron, they are even more complicated for the proton. Hadrons are bound states of quarks and gluons, and while we can see an individual electron, it’s impossible to see an individual quark. Quarks are always found in groups, so have the take the huge fractal into account when we look inside a proton and try to simulate what happens. This is an intractable problem, so need a lot of help from the experimental data to get it right, such as the dedicated deep inelastic scattering experiments at the DESY laboratory.

The view inside a proton might look a little like this (where the arrows represent quarks):

The crazy inner life of the proton

The crazy inner life of the proton

Except those extra bits would go on forever to the left and right, as indicated by the dotted lines, and instead of happening in one spatial dimension it happens in three. To make matters worse, the valence quarks are not just straight lines as I’ve drawn them here, they meander to and fro, changing their characteristic properties as they exchange other particles with each other.

Each time we reach a new energy range in our experiments, we get to prober deeper into this fractal structure of matter, and as we go to higher energies we also liberate higher mass particles. The fractals for quarks interact strongly, so they are dense and have high discovery potential. The fractals for neutrinos are very sparse and their interactions can spread over huge distances. Since all particles can interact with each other directly or through intermediaries, all these fractals interact with each other too. Each proton inside your body contains three valence quarks, surrounded by a fractal mess of quarks and gluons, exactly the same as those in the protons that fly around the LHC. All we’ve done at the LHC is probe further into those fractals to look for something new. At the same time, since the protons are indistinguishable they are very weakly connected to each other via quantum mechanics. In effect the fractals that surround every valence particle join up to make one cosmological fractal, and the valence particles just excitations of that fractal that managed to break free from their (anti-)matter counterparts.

The astute reader will remember that the title of the post was the seemingly fractal nature of matter. Everything that has been described so far fulfils the requirements of any fractal- self similarity, increased complexity with depth and so on. What it is that makes matter unlike a fractal? We don’t exactly know the answer to that question, but we do know that eventually the levels of complexity have to stop. We can’t keep splitting space up into smaller and smaller chunks and finding more and more complex arrangements of the same particles over and over again. This is because eventually we would reach the Planck scale, which is where the quantum effects of gravity become important and it becomes very difficult to keep track of spatial distances.

Meanwhile, deep inside an electron, something weird happens at the Planck scale

Meanwhile, deep inside an electron’s fractal, causality breaks down and something weird happens at the Planck scale

Nobody knows what lies at the Planck scale, although there are several interesting hypotheses. Perhaps the world is made of superstrings, and the particles we see are merely excitations of those strings. Some models propose a unification of all known forces into a single force. We know that the Planck scale is about fifteen orders of magnitude higher in energy than the LHC, so we’ll never reach the energy and length scales needed to answer these questions completely. However we’ve scratched the surface with the formulation of the Standard Model, and so far it’s been a frustratingly good model to work with. The interactions we know of are simple, elegant, and very subtle. The most precise tests of the Standard Model come from adding up just a handful of these fractal-like diagrams (at the cost of a huge amount of labour, calculations and experimental time.)

I find it mind boggling how such simple ideas can result in so much beauty, and yet it’s still somehow flawed. Whatever the reality is, it must be even more beautiful than what I described here, and we’ll probably never know its true nature.

(As a footnote, to please the pedants: To get a positron from an electron you also need to invert the coordinate axes to flip the spin. There are three distinct diagrams that contribute to the electron positron scattering, but the crossed diagram is a small detail might confuse someone new to these ideas.)

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Theoretically, the Higgs boson solves a lot of problems. Theoretically, this Higgs boson is a problem.

Greetings from the good ol’ U.S. of A.

Now that Fall is here, classes are going, holidays are wrapping up, and research programs are in full steam. Unfortunately, all is not well in the Wonderful World of Physics. To refresh, back on 4th of July, the LHC experiments announced the outstanding and historical discovery of a new particle with properties consistent with the Standard Model Higgs boson. No doubt, this is a fantastic feat by the experiments, a triumph and culmination of a decades-long endeavor. However, there is deep concern about the existence of a 125 GeV Higgs boson. Being roughly 130 times the proton’s mass, this Higgs boson is too light. A full and formal calculation of the Higgs boson’s mass, according to the theory that predicts it, places the Higgs mass pretty close to infinity. Obviously, the Higgs boson’s mass is less than infinite. So let’s talk mass and why this is still a very good thing for particle physics.

For an introduction to the Higgs boson, click here, here, or here (This last one is pretty good).

The Standard Higgs

The Standard Model of Particle Physics (SM) is the theory that describes, well, everything with the exception of gravity (Yes, this is admittedly a pretty big exception).  It may sound pompous and arrogant, but the SM really does a good job at explaining how things work: things like the lights in your kitchen, or smoke detectors, or the sun.

Though if this “theory of almost-everything” can do all this, then when written out explicitly, it must be pretty big, right? Yes. The answer is yes. Undeniably, yes. When written out fully and explicitly, the “Lagrangian of the Standard Model” looks like this (click to enlarge):

Figure 1: The Standard Model Lagrangian in the Feynman Gauge. Credit: T.D. Gutierrez

This rather infamous and impressive piece of work is by Prof. Thomas Gutierrez of Cal Poly SLO. Today, however, we only care about two terms (look for the red circles):

Figure 2: The Standard Model Lagrangian in the Feynman Gauge with the Higgs boson tree-level mass and 4-Higgs vertex interactions terms circles. Original Credit: T.D. Gutierrez

The first term is pretty straightforward. It expresses the fact that the Higgs boson has a mass, and this can represented by the Feynman diagram in Fig 3. (below). As simple and uneventful as this line may appear, its existence has a profound impact on the properties of the Higgs boson. For example, because of its mass, the Higgs boson can never travel at the speed of light; this is the complete opposite for the massless photon, which can only travel at the speed of light. The existence of the diagram if Fig. 3 also tells us exactly how a Higgs boson (denoted by h) travels from one place in the Universe, let’s call is x, to another place in the Universe, let’s call it y. Armed with this information, and a few other details, we can calculate the probability that a Higgs boson will travel from point x to point y, or if it will decay at some point in between.

Figure 3: The tree-level Feynman diagram the represents a SM Higgs boson (h) propagating from a point x in the Universe to a point y somewhere else in the Universe. Credit: Mine

The second term is an interesting little fella. It expresses the way the Higgs boson can interact with other Higgs bosons, or even itself. The Feynman diagram associated with this second term is in Fig. 4. It implies that there is a probability a Higgs boson (at position w) and a second Higgs boson (at position x) can collide into each other at some point in the Universe, annihilate, and then produce two Higgs bosons (at point z and y). To recap: two Higgses go in, two Higgses go out.

Figure 4: The tree-level Feynman diagram the represents two SM Higgs bosons (h) at points w and x in the Universe annihilating and producing two new SM Higgs bosons at points z and y somewhere else in the Universe. Credit: Mine

This next step may seem a little out-of-the-blue and unmotivated, but let’s suppose that one of the incoming Higgs bosons was also one of the outgoing Higgs bosons. This is equivalent to supposing that w was equal to z. The Feynman diagram would look like Fig. 5 (below).

Figure 5: By making an incoming Higgs boson (h) the same as an outgoing Higgs boson in the 4-Higgs interaction term, we can transform the tree-level 4-Higgs interaction term into the 1-loop level correction to the Fig. 1, the diagram the represents the propagation of a Higgs boson in the Universe. Credit: Mine

In words, this “new” diagram states that as a Higgs boson (h) at position x travels to position y, it will emit and absorb a second Higgs boson somewhere in between x and y. Yes, the Higgs boson can and will emit and absorb a second Higgs boson.

If you look carefully, this new diagram has the same starting point and ending point at our first diagram in Fig. 3, the one that described the a Higgs boson traveling from position x to position y. According to the well-tested rules of quantum mechanics, if two diagrams have the same starting and ending conditions, then both diagrams contribute to all the same processes and both must be included in any calculation that has the same stating and ending points. In terms of Feynman diagrams, if we want to talk about a Higgs boson traveling from point x to point y, then we need to look no further than Fig. 6.

 

Figure 6: The tree-level (L) and 1-loop level (R) contributions to a Higgs boson (h) traveling from point x to point y. Credit: Mine

What Does This All Mean?

Now that I am done building things up, let me quickly get to the point. The second diagram can be considered a “correction” to the first diagram. The first diagram is present because the Higgs boson is allowed to have mass (mH). In a very real sense, the second diagram is a correction to the Higgs boson’s mass. In a single equation, the two diagrams in Fig. 6 imply

Equation 1: The theoretical prediction for the SM Higgs boson's observed mass, which includes the "tree-level" contribution ("free parameter"), and 1-loop level contribution ("cutoff"). Credit: Mine

In Eq. (1), term on the far left is the Higgs boson’s mass that has been experimentally measured, i.e., 125 GeV. Hence the label, “what we measure.” The term just right of that (the “free parameter”) is the mass of the Higgs boson associated with the first term in the SM Lagrangian (Fig. 2 and 3). When physicists talk about the Standard Model not predicting the mass of the Higgs boson, it is this term (the free parameter) that we talk about. The SM makes no mention as to what it should be. We have to get down, dirty, and actually conduct an experiment get the thing. The term on the far right can be ignored. The term “Λ” (the “cutoff scale“), on the other hand, terrifies and mystifies particle physicists.

Λ is called the “cutoff scale” of the SM. Physically, it represents the energy at which the SM stops working. I mean it: we stop calculating things when we get to energies equal to Λ. Experimentally, Λ is at least a few hundred times the mass of the proton. If Λ is very LARGE, like several times larger than the LHC’s energy range, then the observed Higgs mass gets an equally LARGE bump. For example, if the SM were 100% correct for all energies, then Λ would be infinity. If this were true, then

(the Higgs boson’s mass) = (something not infinity) + (something infinity) ,

which comes out inevitably to be infinity. In other words, if the Standard Model of Physics were 100% correct, then the Higgs boson’s mass is predicted to be infinity. The Higgs boson is not infinity, obviously, and therefore the Standard Model is not 100%. Therefore, the existence of the Higgs boson is proof that there must be new physics somewhere. “Where and at what energy?,” is a whole different question and rightfully deserves its own post.

 

Happy Colliding

– Richard (@bravelittlemuon)

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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.

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Paper vs. Protons (Pt. 2)

Tuesday, August 9th, 2011

Yup, it’s still summer conference season here in the Wonderful World of Physics. My fellow QD bloggers rocked at covering the European Physics Society meeting back in July, so check it out. Aside from the summer conferences, it is also summer school season for plenty of people (like me!). To clarify, I am not talking about repeating a class during the summer. Actually, it is quite the opposite: these are classes that are at most offered once a year and are taught in different countries, depending on the year.

To give you context, graduate students normally run out of courses to take in our second or third of our PhD program; and although the purpose of a PhD is to learn how to conduct research, there will always be an information gap between our courses and our research. There is nothing wrong with that, but sometimes that learning curve is pretty big. In order to alleviate this unavoidable issue, university professors often will teach a one-time-only “topics” course on their research to an audience of three or four students during the regular academic year. Obviously, this is not always sustainable for departments, large or small, because of fixed costs required to teach a course. The solution? Split the cost by inviting a hundred or so students from around the world to a university and cram an entire term’s worth of information into a 1- to 4-week lecture series, which, by the way, are taught by expert faculty from everywhere else in the world. 🙂

To be honest, it is like learning all about black holes & dark matter from the people who coined the names “black holes” & “dark matter.” So not only do graduate students get to learn about the latest & greatest from the people who discovered the latest & greatest, but we also get to hear all the anecdotal triumphs and setbacks that lead to the discoveries.

Fig. 1: Wisconsin’s state capitol in Madison, Wi., taken from one of the bike paths
that wrap around the city’s many lakes. (Photo: Mine)

This brings us to the point of my post. Back in July, I had the great opportunity to attend the 2011 CTEQ Summer School in Madison, Wi., where for 10 days we talked about this equation:

Now, this is not just any ordinary equation, it is arguably the most important equation for any physicist working at the Large Hadron Collider, the Tevatron, or any of the other half-dozen atom smashers on this planet. In fact, this equation is precisely what inspired the name Paper vs. Protons.

Since quantum physics is inherently statistical most calculations result in computing probabilities of things happening. The formula above allows you to compute the probability of what happens when you collide protons, something experimentalists can measure, by simply calculating the probability of something happening when you collide quarks, something undergraduates can do! Physicists love quarks very much because they are elementary particles and are not made of anything smaller, at least that is what we think. Protons are these messy balls of quarks, gluons, photons, virtual particles, elephant-anti-elephant pairs, and are just horrible. Those researchers studying the proton’s structure with something called “lattice QCD” have the eternal gratitude of physicists like me, who only deal with quarks and their kookiness.

Despite being so important the equation only has three parts, which are pretty straightforward. The first part, is that tail end of the second line:

which is just probability of this happening:

Fig. 2: Feynman diagram representing the qq-bar → γ → e+e- process.

If you read Paper vs. Protons (Pt. 1) you might recognize it. This Feynman diagram represents a quark (q) & an antiquark (q with a bar over it) combine to become a photon (that squiggly line in the center), which then decays into an electron (e-) & its antimatter partner, the positron (e+). Believe it or not, the probability of this “qq-bar → γ → e+e-” process happening (or cross section as we call it) is something that advanced college students and lower level graduate students learn to calculate in a standard particle physics course. Trust me when I say that every particle physicist has calculated it, or at the very least a slight variation that involves muons. By coincidence, I actually calculated it (for the nth time) yesterday.

Okay, time for the second part of the equation. To help explain it, I am using a great image (below) from the LHC experiment ALICE. So you & I know that all matter is made from atoms (left). Atoms, in turn, consist of a nucleus of protons & neutrons (center) that are being orbited by electrons (white dots, left). A proton (right) is made up of three quarks (three fuzzy, white dots, right) that bathe in a sea of gluons (red-blue-green fuzziness, right). About 45% of a proton’s energy at the LHC is shared by the three quarks; the remaining 55% of the proton’s energy is shared by the gluons.

Fig. 3: An atom (left), an atom’s nucleus (center), and a free proton (right). (Image: ALICE Expt)

How do we know those numbers? Easy, with something called a “parton distribution function”, or p.d.f. for short! A p.d.f. gives us back the probability of finding, for example, a quark in a proton with 15% of the proton’s energy. Since we want to know the probability of finding a quark (q) in the first proton (with momentum x1) and the probability of finding an anti-quark (q with a bar over its head) in the second proton (with momentum x2) we need to use our p.d.f. (which we will call “f”) twice. Additionally, since the quark and anti-quark can come from either of the two protons we need to use “f” a total of four times. Part 2 of our wonderful equation encapsulates the entire likelihood of finding the quarks we want to smash together:

Now the third (and final!) part is the simple to understand because all it tells us to do is to add: add together all the different ways a quark can share a proton’s energy. For example, a quark could have 5% or 55% of a proton’s energy, and even though either case might be unlikely to happen we still add together the probability of each situation happening. This the third part of our wonderful equation:

Putting everything together, we find that the probability of producing an electron (e-) and a positron (e+) when smashing together two protons is actually just the sum (part 3) of all the different ways (part 2) two quarks can produce an e+e- pair (part 1). Hopefully that made sense.

Though it gets better. When we plug our values into the formula, we get a number. This number is literally what we try to measure that the Large Hadron Collider; this is how we discover new physics! If theory “A” predicts a number and we measure a number that is way different, beyond any statistical uncertainty, it means that theory “A” is wrong. This is the infamous Battle of Paper vs Protons. To date, paper and protons agree with one another. However, at the end of this year, when the LHC shuts down for routine winter maintenance, we will have enough data to know definitively if the paper predictions for the higgs boson match what the protons say. Do you see why I think this equation is so important now? This is equation is how we determine whether or not we have discovered new physics. :p

Happy Colliding.

– richard (@bravelittlemuon)

PS. If you will be at the PreSUSY Summer School at the end of August, be sure to say hi.

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Paper vs. Protons (Pt. 1)

Tuesday, July 19th, 2011

It’s summer conference season! Well actually, it is summer school season for me…. but it is summer nonetheless. Last time, I briefly alluded to the fact that I am attending a 10-day summer school on how exactly physicists turn Feynman diagrams (Fig. 1) into numerical predictions, honest-to-goodness numbers that can be tested with an experiment (Fig. 9). Unfortunately, when I started writing my original post I of course decided to make a few pictures… let’s just say I got a little carried away and I am now dividing my summer school adventures into two parts. It’s also 3 am for me. 🙂

My goal for part 1 of “Paper vs. Protons” is to give an intuitive picture of how we generate electron (e-) & positron (e+) pairs when we physically collide two protons. Hopefully, the images are detailed enough so that you don’t have to read the text to understand what is happening. The words are there mostly for completeness.

Figure 1: A quark (q) & an anti-quark (q-bar) with equal and opposite charges combine and become a photon (γ).
The photon then decays into an electron (e-) & a positron (e+).

My colleague/fellow blogger Flip Tanedo has already done an awesome job describing Feynman diagrams, what they are, how they work, and why physicists love them so much. I do neither him nor Feynman justice when I say that the diagrams are simply ways for anyone (not just physicists!) to intuitively visualize how two or more pieces of matter can interact. The point I want to make with figure 1 (above) is that one way we can produce an electron and a positron pair at the Large Hadron Collider (LHC) is by having a quark from one proton and an anti-quark from another proton smash into each other and become a photon (γ). This photon then travels for a very short amount of time (and I mean very short) before it decays into an electron (e-) and a positron (e+). This process can also happen if we were to replace the photon (γ) with a related particle called the Z boson. You can forget about the Z boson for now, though we will need it for the very end of the post.

At the Large Hadron Collider (LHC), we are colliding protons (left black circle) with other protons (right black circle) in order to look for new physics.

Figure 2. Two protons (black circles) are moments from colliding.

We learned a while back ago that the proton is primarily composed of two up-type quarks and one down-type quark. The proton is also made up of something called “gluons,” they help mediate the Strong nuclear force. Gluons are emitted and absorbed from quarks at such a fantastic rate that the proton is ostensibly made of three quarks tied to one another with rigid rope. The three quarks are represented by the red/blue/green circles and the curly lines are the gluons.


Figure 3: The proton is actually made up quarks (red/green/blue circles), gluons (curly lines),
and virtual particles (black circles).

In the image right above you might have noticed that there are small little black circles, these are virtual particles. Quantum Mechanics and Special Relativity tell us that if we have enough energy, then matter can spontaneously form for a short amount of time. These could be muons (my personal favorite) or even other quarks. So long as matter and antimatter are produced in equal amounts all is well in the universe. Things get interesting when these virtual particles are produced right before two protons collide (below).


Figure 4: Two protons are about to collide right after an anti-quark (magenta circle) and its quark partner (not shown) were produced.

If, for instance, an anti-up quark (the magenta dot in the left circle, above… I did not come up with the color convention but I do like it.) were to form, it could then collide with a u-type quark from an oncoming proton (green circle in right circle, above) and become an photon. Jumping now to the image below, we can imagine the photon being that little black dot in center of the two incoming protons.

Figure 5: Two protons (gray circles) are about to collide resulting in an up quark (green circle, right) & an anti-up quark (magenta dot, left) becoming a photon (black dot, center), and decay into an electron & positron (two outgoing arrows).

If we now zoomed in on the collision (below), we would see the two protons physically overlap when they collide and it is at this moment the quark and its anti-partner combine to become a photon. I have removed the gluons just for clarity. Trust me, they are still there.

Figure 6: Two protons (gray circles) are about to collide resulting in an up quark (green circle, right) & an anti-up quark (magenta dot, left) becoming a photon (black dot, center), and decay into an electron & positron (two outgoing arrows).

Here is where things get messy. Imagine a firework exploding and fragmenting into a bunch of small pieces. Well, that is not too different from when two protons collide; they just kind of explode when they smash into each other while traveling at 99.99999% the speed of light. In the image below I left the q q-bar → e+ e- diagram in order to give you an idea how the protons, or what were formerly known as protons, fragment and decay. The dashed arrows should give you an idea of how they fan out.

Figure 7: Post collisions, the remnants of the two protons begin to fragment and decay.

Okay, let’s zoom all the way out because this is all happening in one of the LHC detectors!

Figure 8: How the q q-bar → e+ e- + fragmenting protons might look in a particle detector. The different colors represent the different layers in a collider detector. The beam travels horizontally through the center of the white region.

So one proton enters from the far left and the other proton comes from the far right. Again, the q q → e+ e- diagram has been left as a reference. After the two protons collide, an electron travels one way (long back arrow) and gets stopped pretty early. In a similar fashion, the positron heads out in the opposite direction from the electron in order to conserve momentum (the other long black arrow). The remaining proton fragments continue to decay and just start spewing out particles. The neatest thing about everything above is that we observe this stuff all the time at the LHC. Sadly, I could not find an event that matched our process perfectly. I did, however, find an real life event (below), seen with the ATLAS detector, where a quark and an anti-quark become a Z boson (Remember? Like a photon but heavier.) which then decays into an electron and positron (yellow lines). The remnants of the protons can be seen in teal.

Figure 9: A real q q-bar → Z → e+ e- from proton collisions at the LHC, seen with the ATLAS detector. Click on image for high-res version. The e- and e+ can be seen in yellow and proton fragments in teal.

I had a blast writing this post, even though I had a few WordPress issues. So what do you think? Cool right?

– richard (@bravelittlemuon)

PS Happy Colliding.

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