Feynman Diagrams and 'Vertices'

Here we switch gears in order to introduce how certain particles can be produced at the LHC.

Another thing that the Standard Model of Particle Physics tells us is the rules of the game: which of the fundamental particles interact with each another, how do they interact, and by how much.  There was an image earlier which tried to capture that idea, but here we'll be a bit more explicit.

There are funny little diagrams, called Feynman diagrams, which you may have seen from time to time.  Fans of the Big Bang Theory TV series might be familiar with such diagrams since they sometimes pop up on whiteboards on the sets of the show.

These diagrams certainly look like the kind of thing you might expect a physicist to have on their board (we're an easy target for that sort of stuff).  But they're more than just pictures: the diagrams are useful for making calculations.  They depict the interactions of particles with other particles.  Whenever the lines of the diagrams meet up, we draw a vertex

Now, the Standard Model dictates the rules for how these vertices work and which particles one is allowed to connect to other particles. Some vertices are allowed by the Standard Model.  Others simply aren’t.

A few examples of allowed Standard Model vertices.
A few examples of forbidden vertices according to the Standard Model.

Why are some allowed and some forbidden?  Because nature said so, that's why!  Another answer: we don't know!  But the existence of such rules has allowed the universe to evolve from the big bang to its present day form, and allowed us to put together a coherent theory of how (at least part of) the universe works on the smallest scale.

With only a few simple vertex rules we can make up a number (really an infinite number) of complicated diagrams.  They can have as many vertices in them as we want.

But the point is that you can start snapping them together like Lego and making your own diagrams.  You just have to follow the rules of what's allowed and what's not allowed (based on the theory, but also our observations – a number of particle physicists actually look for evidence of what should be forbidden vertices!).

Consider an electron and a positron (remember: they are each other's own anti-particles) that are made to collide with one another at extremely high speeds (crucially, at extremely high energies).  I'm going to focus on collisions of electrons here, but we'll get to collisions of protons later.

Colliding electrons and positrons.

Ok, so we're focusing on an electron and a positron interacting with one another at high energies.  One thing that can result from such an interaction is the production of a pair of W bosons. There's the conventional way you would think about this as a sequence of images – the electron-positron pair getting closer and closer until, ultimately, something happens, the electron-positron pair vanishes, and the W bosons simply materialize (remember the marble and bowling ball analogy?).  We'll call this the sequential cartoon representation.  I made that name up.

There's also the way a particle physicist would depict this interaction using a Feynman diagram.

Here's what I mean:

Notice how there's a particle of light or photon there in the Feynman diagram representation which you don't see in the other representation?  It's the Lego piece that allows the interaction to happen (according to the vertex rules set out by the Standard Model).  Was it really there?  Well, kind of.  In a sense, yes, and in a sense, no, not the way you'd think of it in your non-quantum day-to-day life.

But without the photon or other 'mediating' particles like it, this process would never happen. 

In the Feynman diagram above, don't get confused about the directions of the arrows for the lines associated with the electron and positron.  It's partly convention but recognizably it is confusing as written.  I chose, after some hesitation, to draw it in the conventional way, but if it's easiesr for you think of it really as both of those particles heading into the vertex (as you probably would have with no arrows indicated at all) then just stick to that!

Often there is more than one way you can get from A to B... By that I mean there are a ton of diagrams (again, technically an infinite number) you could draw which 'satisfy the rules of the game' and which start on the left with the electron-positron pair and end on the right with a pair of W bosons.

Here are a few possible alternate diagrams just to give you a taste:

Some of these even have loops in them where particles seemingly appear and disappear, some have quarks and anti-quarks which pop out of nowhere.  What's going on here?

In the end how do we know which process actually happened?  Which was the correct diagram?  Doesn't the simplest diagram depict the series of events that most likely transpired?

Well according to the laws of relativistic quantum mechanics as long as it happens on a short enough time scale, any of these processes can occur.  And because they all can happen, they all do happen.  Or rather, all of the diagrams contribute.  This is true even if it's on a timescale such that we could never measure it.

So none of them happened.  And all of them happened.

It's recognizably a strange concept.

Of course some diagrams are more important than others (luckily these are also the least complicated ones).  We say that those represent the dominant contribution.  As a general rule of thumb, the more vertices there are in a diagram (and hence the more convoluted), the less it contributes.  But that's not always the case and you have to be careful!  What I mean by that is that even considering only diagrams with equal numbers of vertices, some types of processes occur much more often than others.  So the different vertices have different strengths, but it matters how we've snapped the pieces together.  For a theorist to tell you how often something is going to happen, they need the full diagram – actually the full set of diagrams – with all possible numbers of vertices. That's about as far as we take it without any math.

So crucially, to 'get it right' for a given process, and to be able to calculate how often something should happen, it requires the inclusion of all relevant diagrams.  Not just one diagram, but all of them. 

No problem, right?  Just add up the contributions from an infinite number of diagrams... 

Ok it's not as bad as it sounds.  In practice, when calculating how often certain processes should occur, theorists need to consider only as many diagrams as they need for a given level of precision, but the more diagrams included the better, not to mention the more computationally intensive.  At the end of the day someone has to pick up the pieces and count.

It seems strange.  But again, "come on", you could think, "only one of those process actually took place, right?"  But nature simply doesn't work like that.  If you've heard about the famous double-slit experiment, you'll know that for an individual interaction sometimes it's simply not possible to know which process truly occurred.  It's inherently hidden from us and even the best instruments won't reveal what truly happened.  We have to just accept that somehow nature does all of the accounting and at the end of the day allows us to know only the probabilities.  If we start by colliding electron-positron pairs at a certain energy, a certain fraction of the time we will get a pair of W bosons as the end result, and a certain fraction of the time we'll get something else. Said another way, there is only a probability that we will end up with a certain number of W-boson pairs after a number of collisions.  Full stop.  And it's the Standard Model that allows us to make the calculation. 

And so far it's in agreement with what we see, strange as it may seem.

So again, the big picture here is that we're using the energy of motion of these two electrons (really an electron and anti-electron) to produce something with a mass higher than their combined mass!  Since the two particles being created are heavier than the two initial particles, this just can't happen spontaneously, or we say 'at rest'.  Said another way, the process requires that energy of motion be added to the mix in order to make up the difference. 

Going back to Einstein's equation, basically it all comes down to simple accounting:

These values aren't made up – this really is how much energy would be required!  The masses are the ones that have been measured and what you see in the table.  They have funny units, but just know that we could also convert those numbers to kilograms if we wanted to, the same way you convert pounds to kilograms – it's simply a more convenient unit given that the mass of an electron in kilograms is about 0.00000000000000000000000000000091 kg (!)

Why is the energy required a minimum or a threshold as suggested in the figure?  Any extra energy will just go into the energy of motion of the particles we've made, so the W bosons will just have some residual energy of motion.  Right at the threshold production energy there's just enough energy that nature will let the process happen, so the W bosons will essentially be motionless or 'at rest' once they're produced.  Make sense?  Less energy than that (i.e. below the threshold production energy) and the electron-positron collision process will almost never produce a pair of W bosons.

Ok, here's another hypothetical scenario:

Suppose some theorist were to propose the existence of some exotic particle X.  No one has seen such a particle before, and its mass is much higher than that of any of the particles we know about (let's say it's roughly 80 times more massive than a W boson just to pick a number).  Assuming this particle can be produced in pairs based on certain vertex rules which allow us to make production diagrams like those above, when could we produce it?  How much energy do we need? 

The answer?  There's no trick, it's the same idea as before:

Here the value of 13 TeV (i.e. 13,000 GeV) is made up!  But actually, that happens to be the exact upper limit on the energy of the proton-proton collisions at the LHC during 2015.  So if that were the real number, we'd be close to being able to see it.  Why did I say upper limit?  I mean, why would we only be close to being able to see it? 

Well, that brings up another subtle point.  As you'll see shortly, there are a lot of possible constituent particles inside the protons in the collisions which can play a role in the interaction.  To truly make two X particles we would need a constituent particle with exactly 6.5 TeV out of each of the two protons to be involved in the collision (leaving nothing at all for the rest of the constituents in the proton).  So it's just within reach theoretically, but extremely unlikely.  If the X particle were heavier though such that we suddenly needed much more energy to produce it, say 20 TeV, we'd definitely be out of luck!  The laws of physics (as we know them) strictly forbid such an event to occur.  We could run the accelerator for years and we wouldn't see a single one of those X particles (though curiously we might see evidence of it in other, indirect ways, but we won’t get into that…).   If we were only to need, say, 1 or 2 TeV, then ok, we'd be in business!  ...in the sense that once in a while we'd be able to produce a pair of X's.  This might make more sense after we've talked about the structure of the proton.

Changing the topic for a moment, remember that the Higgs boson has a mass around 125 GeV (we'll leave the speed of light squared in the units out for now).  Actually, that's pretty low compared to the energy of 7,000 GeV when the LHC was running in 2011, or 8,000 GeV in 2012.  But we had to produce vast numbers of Higgs bosons in order to be able to be sure of what we saw.  So we needed to be way above the threshold.  The point: Higgs bosons would have been produced already in previous particle accelerator experiments, but we simply didn't have sufficient statistics to clearly see what we were looking for.  We discovered the Higgs boson at the LHC in 2012, but we would have produced them before that in particle accelerator experiments.


Electron-Positron vs. Proton-Proton Colliders

The stuff in the preceding section might have been confusing.  You might have been thinking, "Wait, aren't we colliding protons with other protons?  Why were you talking about colliding electrons with positrons?".

Well, you'd have been right: the LHC isn't an electron-positron collider, it's a proton-proton collider. But here's an interesting fact you might not have known: eviscerate the LHC and you're left with a giant tunnel, right?  Well that same tunnel used to house the predecessor to the LHC which was called ... you ready? ... the LEP Collider (where LEP stands for Large Electron-Positron).  So same tunnel, different accelerator.  My Ph.D. supervisor worked on OPAL, one of the LEP collider experiments, just as I work on ATLAS, one of the LHC experiments.  There are also other options: the second largest such collider (after the LHC) is the Tevatron near Chicago, situated at a CERN-like facility called Fermilab, where protons are collided not with other protons, but rather with anti-protons.  And yet another CERN-like facility, DESY, near Hamburg in Germany featured electron-proton or positron-proton collisions.  Different types of collisions will shed light on the universe in different ways, so there are advantages to doing each.

But the point: everything we've talked about in the past sections also applies to colliding other particles such as protons.  More or less.  Electron-positron colliders are just a bit less complicated because we're colliding fundamental particles together – collisions involving protons are a lot messier!

At any rate, you're now sufficiently armed with knowledge such that you're ready to tackle this one:

Try to explain in words what's happening in this diagram.  What do you start with?  What do you end up with?

...

You could probably guess that, similarly to the earlier process we mentioned, the diagram above shows the annihilation of two particles (which, referring back to the tables at the top of the page, you can see are called gluons).  It ultimately leads to the production of a pair of top quarks (really a top and an anti-top quark).

So get this: that diagram is one of the dominant processes for the production of top quarks at the LHC!  Top quarks are the heaviest fundamental particle we know of (heavier than the Higgs boson!), but they don't really exist naturally anywhere here on Earth; if we want to study them there's no way around it, we simply have to roll up our sleeves and create them!

But hold on, where do the gluons in that reaction come from?  Aren't we colliding protons at the LHC?  So shouldn't the process look rather like this?

Well, that’s right!  Except that you might have noticed that while electrons appear in the tables of fundamental Standard Model particles, protons are nowhere to be found.  Why?  Remember, they're not fundamental.  Protons are made up of smaller pieces called quarks as we saw above, in the same way that neutrons are.  In fact, it's even more bizarre than that.  If you look really closely at a proton, you'll see there's a probability of finding other stuff in there too.  And it gets stranger still: the composition of the proton even changes, in a sense, as its speed changes.

Let's take a closer look inside the proton:

All kinds of good stuff in there, right?  Other types of quarks (not just up and down), and even gluons.  When protons are travelling very very fast, it's actually the gluons inside a proton that take on a leading role.  Strange as it may seem, it's again what the theory tells us and what we've observed from experimental data!  The sizes here as shown don't mean that the up- or down-type quarks are larger than the other pieces (everything we're dealing with here are treated as point particles).  They're drawn this way to illustrate their relative contribution – if you 'reach into' a proton at rest you're most likely to pull out a quark, and twice as likely it'll be an up-type quark compared with a down-type quark. [Note: the relative sizes are illustrative only to say something about the probabilities but don't take the sizes at face value – they're not really drawn to scale!]

The charges even work out: the tables we showed way back at the beginning of the page list the up-type quark as having a charge of +2/3 the charge of an electron (really a positron since it's positive here).  A down quark's charge is -1/3.   Add those up and what do you get?  (Ok, so this is more math...): up + up + down = 2/3 + 2/3 - 1/3 = 3/3 = 1.  So a proton has a charge of positive one.  An electron has a charge of negative one.  They're opposites, so what we learned in elementary school science class was right!

Coming back to the production of a top and anti-top quark pair, a better way to represent it would then maybe be something like this:

Now it's abundantly clear: the gluons came from within the protons, and the energy of motion of the interaction was sufficiently high that a pair of top quarks was produced.

You'll learn more about the structure of the proton and why it's important in Part VII.

Recap time.

So the LHC (particle accelerator) provides a steady stream of protons, half going in one direction, and half going in the other direction around the ring.  They're actually in bunches.  And it runs, appropriately enough, like Swiss clockwork (bad joke, I know).  The internal pieces making up the protons (whether they're quarks or gluons) get close enough to each other and have high enough energies of motion that they interact and can make new particles.

So we call them 'beams' of protons, but again, they're actually in really tight little bunches at evenly spaced intervals as the figure above shows.  One way to think about these bunches is to pretend that they're just like enormous swarms of mosquitos.  The bunches themselves are tiny but we can cross the beams just right such that bunches pass through one another.  But that's still not enough to ensure collisions – most protons in the bunches will simply fly right past each other.  Remember that they're also all positively charged, and since 'like charges repel', they naturally don't want to get very close to one another.  But the bigger issue here is that there's a huge amount of empty space in there, so they seem close but on a small scale they're really still incredibly far away from one another.  We use powerful electromagnetic fields to squeeze the bunches close together to increase the likelihood that some protons will get close enough that something amazing happens.

In all of this, remember to keep that tennis ball analogy out of your head!

It's neat to think that in the merging of galaxies – on an entirely different scale – you get the same sort of thing: even when galaxies are 'colliding', most of their constituent stars fly right past the other stars, since in actual fact they're still at incredible distances from one another relative to their size.  So there as well, it's super rare that stars themselves will truly collide with one another.

Of course even though most protons fly by each other like ships crossing in the night, collisions do, from time to time, occur!  And since we have so many bunch crossings per second and so many individual protons in each bunch, it actually adds up to a huge number of collisions.  Here by 'collision', remember there's no membrane surrounding the protons.  So what do we mean?  Nothing is really 'colliding' here in the way you'd imagine things colliding in our macroscopic world.  All we need is for two constituent pieces – say, two gluons – to get close enough to one another, and we call it a collision, but in a sense, interaction would be a more appropriate term.

Depending on how much energy the initial protons have, there's a certain probability that we get the final state we showed earlier (the pair of top quarks).  Often that's not the case.   But it's just a matter of statistics – there's a probability the interaction will lead to outcome A, there's a probability the interaction will lead to outcome B, and so on.  Crucially not all processes are equally likely to occur.  Outcome A might happen 10,000 more often than outcome B.  What happens if what you're interested in really is outcome B?....Well, you take what you get – namely what you want and don't want, and then you try to sort through the whole mess afterwards.

So again, it's rare that you'll even get an interaction between the gluons, but there are about 40 million bunch crossings, as we call them, every single second, and as a result gluon interactions like that happen all the time.  Sometimes they produce the really interesting particles we care about (top quarks, Higgs bosons, W/Z bosons), but often it's just uninteresting garbage that comes out, though recognizably one person's garbage is another person's treature.  The job of the physicist is to search for the interesting events. The physicist has to try, as much as possible, to throw out the garbage (as well as cases where you have the less interesting outcome A) and really try to identify cases where they have outcome B.  This sorting isn't done by hand of course.  It relies on some clever algorithms that you'll hear about later in Part IX.

Coming back to the process we drew above, sometimes, if we're lucky, the outcome will not be a pair of top quarks but rather a Higgs boson, which can go something like this:

A standard production mechanism for a Standard Model Higgs Boson at the LHC via a top quark loop.

See those top quarks in there?  They're in a little triangular 'loop'.  We haven't broken the vertex rules here – these lines all allowed!  This is just one way to snap the Lego together and get a single Higgs boson on the right-hand side.  In fact, it’s the dominant contribution to the way that Higgs bosons are produced at the LHC!  So even if there aren't any real top quarks coming out on the right-hand side, the top quarks definitely played a role in the production of the Higgs!  It's all just part of the quantum dance that's taking place underneath the surface on timescales too short for us to even measure.  So even in the production of Higgs bosons, top quarks (what I studied for my PhD) play an important role.  We never see them, nor any traces of them, but they were crucial players in allowing Higgs bosons to be produced!

It's through diagrams like these that one starts to see how all of the fundamental particles making up our universe are intricately tied together in a great cosmic fabric.


Real and Virtual Particles

There's something strange you might have noticed in some of the diagrams above.  It seems as if the way we drew some of them there are lighter particles decaying to really heavy particles.  Shouldn't that be forbidden?

Well, sometimes it is, and sometimes it isn't, and it brings up the concept of virtual particles:

 

The Z boson in the left-hand case we treat as a real particle.  We've made it at the LHC and if it's just sitting around (though never for long), it simply cannot decay to a pair of W bosons – their combined mass (higher than the mass of the Z boson) together with the laws of relativity and the conservation of energy do indeed forbid it! 

But as shown in the right-hand picture when the Z boson is just a small piece in an otherwise more complicated interaction it's actually allowed to exist for a fleeting moment with whatever mass is required (!).  It's a borrow-energy-from-the-universe sort of a deal, and here's the catch: if it happens on a short enough timescale that no one would be able to measure it anyway, it's fine!  So here we call it a virtual particle.  What is its mass?  Maybe it's close to the real Z boson mass; maybe it's a hundred times larger; maybe it's zero!  It doesn't just apply to Z bosons of course (this is just an example): any particle can in principle be virtual.  Even photons (which normally have a mass of zero) can take on whatever mass suits them if they're the one tying vertices together in order to satisfy the necessary accounting.

Sounds absolutely ridiculous, but in fact if such virtual particles were forbidden, particle physics as we know it wouldn't work!  It's built into the mathematical framework of the Standard Model and, whether we like it or not, the predictions comes out to just what we measure!

But hold on, you could think, the Z boson in the upper-left picture could have some extra energy of motion, right?  And if it's high enough that would give it enough energy to decay to whatever it wants, ha! 

"Unfortunately, no", says Einstein.  It comes down to relativity.  Even if the stationary you were to measure what you thought was the speed (or momentum, or energy – they're all related) of the Z boson and it came out to be something enormous, there exists a reference frame in which the Z boson is still just sitting there motionless (take for instance someone travelling along beside the Z boson at the same speed – to a person in that reference frame the Z boson is not moving at all!).  So no matter how you cut it, under no circumstances can a real Z boson decay to two W bosons (or anything heavier than it for that matter).  The 'Lego' rules introduced earlier may tell you that the vertex is allowed, but the laws of physics simply forbid it.

Note that if this weren't true we'd have heavy particles being produced all over the place: high-energy electrons have far more the sufficient amount of energy they need to make Higgs bosons despite their comparatively tiny mass, so why wouldn't they just spontaneously 'radiate away' a Higgs boson like this?  After all, the Lego rules aren't violated, right?

A forbidden process in which a free energetic electron radiates a Higgs boson.

The answer: it can't happen for the same reason as above with the Z boson.   In some reference frame the electron is stationary, so there's absolutely no way it can affort to shed off something with a mass about 250,000 times larger than it has to begin with – it's simply bad accounting, would lead to all kinds of inconsistencies in the theory, and so, understandably, is forbidden by nature.

The take-home message: such seemingly counter-intuitive 'borrow-energy-from-the-universe' type reactions (with so-called virtual particles) can and do happen, but they only happen for interactions involving at least pairs of particles (not individual particles) where the total interaction energy is a fixed number regardless of who's moving and how fast!  That last point we'll touch on in the next section.


Next up we'll have a look at the basic principles behind the ATLAS detector and at how we use the detector to probe the interaction or interactions we're most interested in.