Friday, 30 March 2012

Measurements of Jet Substructure

One of the hot topics in the particle theory community at the moment is the use of jet substructure.  Jets are one of the most common types of objects we actually see in colliders: streams of roughly collinear particles.  They arise due to the fact that the strong nuclear force is, well, strong.  The fundamental particles that interact through this force---quarks and gluons---can not exist in isolation.  If you try and pull two quarks apart, for example, the energy in the interaction between them is so great it can spontaneously create more particles from the vacuum.  This means that when a particle collider like the LHC creates a quark or gluon in an interaction, that quark or gluon quickly acquires a number of followers which bind into (meta-)stable particles like pions, kaons, protons and neutrons.

These collections of particles are what show up in the actual detectors.  In older experiments, all we really cared about was the direction and energy of these things, which is roughly the same as the original quark or gluon.  You can do a lot of good physics just on that information alone.  But two things are different about the LHC.  First, the angular resolution of the experiments is much better, allowing us to truly resolve the individual particles within the jet.  Second, the large energy of the collisions leads to new types of events where heavy objects like Ws, Zs and tops can be produced with relativistic velocities.

As an example, let us consider the top quark.  The top is the most massive fundamental particle discovered, with a mass of 175 times that of a proton, or comparable to a tungsten atom.[1]  The top decays essentially all the time to a lighter bottom quark and a W boson; the W decays two-thirds of the time to two quarks.  This gives us a standard top signal of three jets (corresponding to the three quarks).[2]  If the top quark is produced at rest, then those three jets will be well separated:

However, what happens if the top quark is not at rest?  Conservation of momentum means that the decay products will tend to travel in the same direction as the top.  When the speed of the top becomes large, comparable to the speed of light, all its decay products are going in the same direction and it becomes difficult to resolve the three jets:

The idea behind jet substructure is to capture cases like this.  The idea is that a jet that starts from one quark will look different, on small scales, to one started by three (or two, or any other number).  In particular, this method offers hope to identify these objects as tops (or similar for other possibilities), when older analyses would miss them.

The problem is that all the analysis so far has been theoretical, using computational tools that approximate the production of jets from quarks.  These tools are necessarily somewhat ad hoc, as we cannot do fully correct calculations in reasonable time.  This is where a recent paper comes in.  The ATLAS collaboration at the LHC have, for the first time, measured a number of substructure variables and compared them to the predictions.  The data set was small, only the data taken in 2010, so we expect it all to be well described by the SM.  And it is!  This is very encouraging; not only does it show that our theoretical tools are working well, it offers genuine hope for extracting the maximum amount of information from the data we'll be taking.  The only thing to note is that this analysis took some time, and so it's clear that these searches are not going to lead to quick discoveries.

So, what variables did they measure?  Well, there where three.  First was the Jet Mass.  This is something intrinsically related to Special Relativity.  One of the most famous equations of all time is Einstein's relation between energy and mass:
Here, E is energy, m is mass and c is the speed of light.  However, this equation is only accurate for objects at rest.  For objects with a speed v this is modified to
$E = \sqrt{1 + \frac{v^2}{c^2}} m c^2$.
This can be further modified, using the momentum, p:
$E^2 = p^2 c^2 + m^2 c^4 $.
In particular, this last expression works when m is zero or very small, which is the case that applies here.

The jet mass is simply defined.  Measure the energies and momenta of all the particles that make up the jet, add them up, and then use the mass defined by the above equation rewritten as
$m^2 = \frac{E^2}{c^4} - \frac{p^2}{c^2}$.
The only problem is that of measuring the energy deposited with sufficient angular resolution that each particle is separately identified.  As noted above, this is one area the LHC wins in compared to previous experiments.  Here are some results:
These plots show the number of jets (on the vertical axis) as a function of the jet mass (on the horizontal axis).  The top plots show the absolute data, and two different predictions; the bottom plots show the differences between the data and the predictions.  The left/right plots correspond to different ways of defining the plots; in particular, for the right plot the jets have been "cleaned" and removed of low-energy stuff unrelated to the initial quark or gluon.  The main point here is that these quantities have been measured, and agree very well with theory.  Which is encouraging!

The two other variables that were measured are the N-subjettiness and the kt-splitting scales.  These are slightly more complex quantities that define how much the jet looks like it is made of smaller, sub-jets.  In the top-quark example I gave above, the N-subjettiness parameter would be much higher for N = 3 than for N = 2, as it has three components.  Defining it is a bit tricky, though, and this post has run on much longer than I initially planned.  Here are some results:
Again, the agreement between prediction and experiment is very pleasing, especially given the more complex nature of the variable.

Given that only a small fraction (about 1%) of the total data collected so far has been analysed this way, this offers the hope of further gains in searches to come.  Conveniently, many of the weaknesses in current limits on things like supersymmetry relate to the top quark, and this paper offers the promise that those gaps can be addressed.  I'm excited!

[1] Actually closer to ytterbium, but I chose tungsten as it's better known.

[2] It is usually easier to detect the case where the W decays to an electron and neutrino, or a muon and neutrino, as the backgrounds from other processes are smaller.  However, I'm skipping over a lot of details about identification as they simply detract from my point.

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