Spectral power for LHC physics

Jets are the bread and butter of LHC physics; detector objects (tracks and towers) are clustered into jets, and jet-parton duality allows us to map the jets onto the unobservable quarks and gluons from which they evolved. This clustering is frequently accomplished via a sequential algorithm (like antikt). Such an algorithm loops over the following steps:

1. Examine all $N^2$ two-particle correlations between clustered objects and select one "special" correlation.
2. Depending on the correlation, finalize one object or combine a pair of objects. Return to #1.

Notice that the evolution of such clustering is guided by only one correlation at every step. A more holistic approach is used by event shape variables (e.g. thrust and sphericity), which simultaneously use every two-particle correlation. Unfortunately, shape variables tend to be 1-dimensional, so an event shape curve must be built from thousands of events.

Ideally, we would like a shape curve that describes a single event, from which we could:

• Extract jet kinematics
• Tag interesting signatures
• Remove pileup
• Probe QCD with new tools

Taking a cue from cosmology, why not take an event's energy density $\rho(\theta, \phi)$ (projected onto the unit sphere) and expand it into the spherical harmonics?\begin{equation}\rho_l^m = \int \text{d}\Omega\, {Y_l^m}^*(\theta, \phi)\,\rho(\theta, \phi)\end{equation}Perhaps then we could identify broad shapes in the power spectrum with known physics.

Such a tool would be extremely useful if it can identify is pileup, which will be a serious problem for accurate reconstruction of jet with moderate energy at the high-luminosity LHC (e.g. the 40-ish GeV jets needed by precision electroweak measurements).

Limitations of finite sampling

A particle detector detects $N$ objects (assumed massless), from which one can derive a discrete energy density \begin{equation} \rho(\theta, \phi) = E_\text{tot}^{} \sum_{i=1}^N f_i\, \delta^2(\theta_i^{}, \phi_i^{})\,,\label{eq:rho}\end{equation} using energy fraction $f_i^{} \equiv |\vec{p}_i^{}|/E_\text{tot}^{}$. This gives the power spectrum introduced by Fox and Wolfram in 1978 \begin{equation} H_l = \sum_{i,\,j}^{} f_i^{}\, f_j^{}\; P_l^{}(\cos\theta_{ij}^{})\,.\label{eq:H_l}\end{equation}The power spectrum $H_l^{}$ is rotationally invariant because it depends only on the interior angle $\theta_{ij}^{}$ between pairs of particles. It is also relatively trivial to show that $H_l^{}$ is infrared and collinear safe for low-to-moderate values of $l$.

We will now examine the power spectrum of a few 3-jet final states: (A) very two-jet like and (B) very 3-jet like. First, notice the lack of broad shapes in the 3-parton $H_l^{}$; three delta functions are not mostly homogenous, which makes them extremely oscillatory compared to the CMB. Even after the parton shower of event $B$, the most striking difference in the showered power spectrum is that it asymptotically flattens to a much lower power as $l\to\infty$.

It turns out that the height of this asymptotic plateau is inversely proportional to $N$, the number of particles in the final state being analyzed\begin{equation}H_l^{} \sim \frac{1}{N}(1+\text{small fluctuations})\,.\end{equation}The power spectrum attempts to treat a discrete distribution as if it were continuous. Therefore, a very fine grained sampling (high particle multiplicity) is required to extract useful information from high $l$ information. In fact, the power plateau can be interpreted as white noise from discrete sampling, and therefore limits the maximum useable $l$.

Power jets, jet shape and pileup

QCD power spectra lack broad shapes because QCD jets are fairly collimated, making individual power spectra very oscillatory. This also ensures that power spectra look very different event-by-event. In order to extract useful physics, we must take a very different approach than CMB fitting.

A detector observes a set of $N$ physics objects (tracks, towers). Assuming that each of these objects is massless, a finite density (Eq. \ref{eq:rho}) can be constructed to obtain the observed power spectrum $H_l^\text{obs}$ (Eq. \ref{eq:H_l}). We now wish to map this observed power spectrum onto a simpler model. The "power jet" solution is to take a handful of "jets" — $n$ massless, spatial delta functions, with $n\ll N$ — and calculate their power spectrum $H_l^\text{reco}$. The $n$ PS jets are then fit to observations by minimizing the residuals between $H_l^\text{reco}$ and$H_l^\text{obs}$ (using a nonlinear least squares minimization algorithm).

This prototype power jet model is able to reconstruct the original kinematics with decent accuracy, but the fit is limited to the first few $H_l^{}$ moments (i.e. the maximum $l$ in the fit $l_\text{max}^{}<10$). This is because the $n$ jets in the model are massless spatial delta functions, just like the detector objects from which $H_l^\text{obs}$ is calculated. Unfortunately, while infinitely localized samples from a larger density makes sense for the many particles seen by a detector, there are only a handful of jets. Jets are extensive, they should have shape.

The simplest possible shape we can use for power jets is to assume that each jet is distributed isotropically in its center-of-momentum frame, acquiring its characteristic jet shape when it is boosted into the lab frame. In this model, jet shape only depends on each jet's boost $\gamma=E/m$. Even though this shape model is overly simplistic (e.g. it assumes jets are spin-0), it works quite well. We can see this when we fit a three-jet event with three PS jets (depicted below). With massless jets, the fit diverges after $l=6$. Adding 3 degrees of freedom by giving the jets mass pushes that breakdown to $l=46$.

An important property of power jets is that each jet's kinematics are calculated by fitting the whole power spectrum, simultaneously utilizing all $N^2$ correlations. As such, PS jets do not make a unique assignment between particles and jets, so that there are no boundaries between jets. Furthermore, there is no arbitrary radius R parameter to inject jets with a arbitrary scale. Not only does this picture better fit the expectations of theory, it suggests that the power jet definition will be quite useful for determining jet substructure. 

Another important property of power jets its ability to discriminate QCD signal from pileup. It is quite reasonable to anticipate that the shape of pileup is the same in every event (at a given collider and collision energy), with only the pileup intensity fluctuating between events. Hence, the universal shape of pileup can be measured in min-bias events, allowing the pileup contribution of each individual event to be fit to $H_l^\text{obs}$. This approach is clearly superior to local pileup subtraction because it leverages the information embedded in the many pileup correlations from across the entire detector.

For more information about spectral power and its application to particle physics, please refer to my doctoral thesis.