Quantum Diaries

Updated: Monday, 2011 August 29, to clarify shape of angular distribution plots.

It’s the $10 billion question: If experimentalists do discover a bump at the Large Hadron Collider, does it have to be the infamous higgs boson? Not. One. Bit. Plainly and simply, if the ATLAS & CMS collaborations find something at the end of this year it will take a little more data to know we are definitely dealing with a higgs boson. Okay, I suppose I should back up a little an add some context. 🙂

The Standard Model of Particle Physics (or SM for short) is the name for the very well established theory that explains how almost everything in the Universe works, from a physics perspective at least. The fundamental particles that make up the SM, and hence our Universe, are shown in figure 1 and you can learn all about them by clicking on the hyperlink a sentence back. Additionally, this short Guardian article does a great job explaining fermions & bosons.

Fig 1. The Standard Model is composed of elementary particles, which are the fundamental building blocks of the Universe, and rules dictating how the particles interact. The fundamental building blocks are known as fermions and the particles which mediate interactions between fermions are called bosons. (Image: AAAS)

As great as the Standard Model is, it is not perfect. In fact, the best way to describe the theory is to say that it is incomplete. Three phenomena that are not fully explained, among many, are: (1) how do fermions (blue & green boxes in figure 1) obtain their mass; (2) why is there so little antimatter (or so much matter) in the Universe; and (3) how does gravity work at the nanoscopic scale? These are pretty big questions and over the years theorists have come up with some pretty good ideas.

The leading explanation for how fermions (blue & green boxes in figure 1) have mass is called the Higgs Mechanism and it predicts that there should be a new particle called the higgs boson (red box at bottom of figure 1). Physicist believe that the Higgs Mechanism may explain the fermion masses is because this same mechanism very accurately predicts the masses for the other bosons (red boxes in figure 1). It is worth nothing that when using the Higgs Mechanism to explain the masses of the bosons, no new particle is predicted.

Unfortunately, the leading explanations for the huge disparity between matter & antimatter, as well as a theory of gravity at the quantum level, have not been as successful. Interestingly, all three types of  theories (the Higgs Mechanism, matter/antimatter, and quantum gravity) generally predict the existence of a new boson, namely, the higgs boson, the Z’ boson (pronounced: zee prime), and the graviton. A key property that distinguishes each type of boson from the others is the intrinsic angular momentum they each carry. The higgs boson does not carry any, so we call it a “spin 0” boson; the Z’ boson carries a specific amount, so it is called a “spin 1” boson; and the graviton carries precisely twice as much angular momenta as the Z’ boson, so the graviton is called a “spin 2” boson. This will be really important in a few paragraphs but quickly let’s jump back to the higgs story.

Fig 2. Feynman Diagrams representing a higgs boson (left), Z’ boson (center), and graviton (right)
decaying into a b quark (b) & anti-b quark (b).

In July, at the European Physics Society conference, the CDF & DZero Experiments, associated with the Tevatron Collider in Illinois, USA, and the CMS & ATLAS Experiments, associated with the Large Hadron Collider, in Geneva, Switzerland, reported their latest results in the search for the higgs boson. The surprising news was that it might have been found but we will not know for sure until the end of 2011/beginning of 2012.

This brings us all the way back to our $10/€7 billion question: If the experiments have found something, how do we know that it is the higgs boson and not a Z’ boson or a graviton? Now I want to be clear: It is insanely unlikely that the new discovery is a Z’ or a graviton, if there is a new discovery at all. If something has been been discovered, chances are it is the higgs boson but how do we know?

Now, here is where awesome things happen.

The Solution.

In all three cases, the predicted boson can decay into a b quark (b) & anti-b quark (b) pair, which you can see in the Feynman diagrams in figure 2. Thanks to the Law of Conservation of Momentum, we can calculate the angle between each quark and the boson. Thanks to the well-constructed detectors at the Large Hadron Collider and the Tevatron, we can measure the angle between each quark and the boson. The point is that the angular distribution (the number of quarks observed per angle)  is different for spin 0 (higgs), spin 1 (Z’), and spin 2 (graviton) bosons!

To show this, I decided to use a computer program to simulate how we expect angular distributions for a higgs → bb, a Z’→ bb, and a graviton → bb to look. Below are three pairs of plots: the ones to the left show the percentage of b (or b) quarks we expect at a particular angle, with respect to the decaying boson; the ones on the right show the percentage of quarks we expect at the cosine (yes, the trigonometric cosine) of the particular angle.

Figure 3. The angular distribution (left) and cosine of the angular distribution (right) for the higgs (spin-0) boson, m_H = 140 GeV/c². 50K* events generated using PYTHIA MSUB(3).

Figure 4. The angular distribution (left) and cosine of the angular distribution (right) for a Z’ (spin-1) boson, m_Z’ = 140 GeV/c². 50K* events generated using PYTHIA MSUB(141).

Figure 5. The angular distribution (left) and cosine of the angular distribution (right) for a graviton (spin-2) boson, m_G = 140 GeV/c². 40K* events generated using PYTHIA MSUB(391), i.e., RS Graviton.

Thanks to the Law of Conservation of Angular Momentum, the intrinsic angular momenta held by the spin 0 (higgs), spin 1 (Z’), and spin 2 (graviton) force the quarks to decay preferentially at some angles and almost forbid other angles. Consequentially, the angular distribution for the higgs boson (spin 0) will give one giant hump around 90°; for the Z’ boson will have two humps at 60° and 120°; and the graviton (spin 2) will have three humps at 30°, 90°, and 150°. Similarly in the cosine distribution: the spin-0 higgs boson has no defining peak; the spin-1 Z’ boson has two peaks; and the spin-2 graviton has three peaks!

In other words, if it smells like a higgs, looks like a higgs, spins like a higgs, then my money is on the higgs.

A Few Words About The Plots

I have been asked by a reader if I could comment a bit on the shape and apparent symmetry in the angular distribution plots, both of which are extremely well understood. When writing the post, I admittedly glossed over these really important features because I was pressed to finish the post before traveling down to Chicago for a short summer school/conference, so I am really excited that I was asked about this.

At the Large Hadron Collider, we collide protons head-on. Since the protons are nicely aligned (thanks to the amazing people who actually operate the collider), we can consistently and uniformly label the direction through which the protons travel. In our case, let’s have a proton that come from the left be proton A and a proton that comes from the right be proton B. With this convention, proton A is traveling along what I call the “z-axis”; if proton A were to shoot vertically up toward the top of this page it would be traveling along the “x-axis”; and if it were to travel out of the computer screen toward you, the reader, the proton would be traveling in the “y direction” (or along the “y-axis”). The angle between the z-axis and the x-axis (or z-axis and the y-axis) is called θ (pronounced: theta). You can take a look at figure 6 for a nice little cartoon of the coordinate system I just described to you.

Figure 6: A coordinate system in which proton A (pA) is traveling along the z-axis and proton B (pB) in the negative z direction. The angle θ is measure as the angle between the z-axis and the x-axis, or equally, between the z-axis and the y-axis.

When the quarks (spin 1/2) inside a proton collide to become a higgs (spin 0), Z’ (spin 1), or graviton (spin 2), angular momentum must always be conserved. The simplest way for a quark in proton A and a quark in proton B to make a higgs boson is for the quarks to spin opposite directions, while still traveling along the z-axis, so that their spins cancel out, i.e., spin 1/2 – spin 1/2 = spin 0. This means that the higgs boson (spin 0) does not have any angular momentum constraints when decaying into two b-quarks and thus the cosine of the angle between the two b-quarks should be roughly flat and uniform. This is a little hard to see in figure 3 (right) because, as my colleague pointed out, the resolution in my plots are too small. (Thanks, Zhen!)

Turning to the Z’ boson (spin 1) case, protons A & B can generate a spin 1 particle most easily when their quarks, again while traveling along the z-axis, are spinning in the same direction, i.e., spin 1/2 + spin 1/2 = spin 1. Consequentially, the spin 1 Z’ boson and its decay products, unlike the higgs boson (spin 0), are required to conserve 1 unit of angular momentum. This happens most prominently when the two b-quarks (1) push against each other in opposite directions or (2) travel in the same direction. Therefore, the cosine of the angle made by the b-quarks is dominantly -1 or +1. If we allow for quantum mechanical fluctuations, caused by Heisenberg’s Uncertainty Principle, then we should also expect b-quarks to sometimes decay with a cosine greater than -1 and less than +1. See figure 4 (right).

The spin 2 graviton can similarly be explained but with a key difference. The spin 2 graviton is special because like the Z’ boson (spin 1) it can have 1 unit of angular momentum, but unlike Z’ boson (spin 1) it can also have 2 units of angular momenta. To produce a graviton with 2 units of angular momenta, rarer processes that involve the W & Z bosons (red W & Z in figure 1) must occur. This allows the final-state b-quarks to decay with a cosine of 0, which explains the slight enhancement in figure 5 (right).

It is worth noting that the reason why I have been discussing the cosine of the angle between the the quarks and not the angle itself is because the cosine is what we physicists calculate and measure. The cosine of an angle, or equally sine of an angle, amplify subtle differences between particle interactions and can at times be easier to calculate & measure.

The final thing I want to say about the angular distributions is probably the coolest thing ever, better than figuring out the spin of a particle. Back in the 1920s, when Quantum Mechanics was first proposed, people were unsure about a keystone of the theory, namely the simultaneous particle and wave nature of matter. We know bosons definitely behave like particles because they can collide and decay. That wavy/oscillatory behavior you see in the plots are exactly that: wavy/oscillatory behavior. No classical object will decay into particles with a continuous distribution; no classical has ever been found to do so nor do we expect to find one, at least according to our laws of classical physics. This wave/particle/warticle behavior is a purely quantum physics effect and would be an indicator that Quantum Mechanics is correct at the energy scale being probed by the Large Hadron Collider. 🙂

Happy Colliding.

– richard (@bravelittlemuon)

PS I apologize if some things are a little unclear or confusing. I traveling this weekend and have not had time to fully edit this post. If you have a question or want me to clarify something, please, feel free to write a comment.

PPS If you are going to be at the PreSUSY Summer School in Chicago next week, feel free to say hi!

*A note on the plots: I simulated several tens of thousands of events for clarity. According to my calculations, it would take four centuries to generate 40,000 gravitons, assuming the parameters I chose. In reality, the physicists can make the same determination as we did with fewer than four years worth of data.