Study of Self-similarity in Jet Events

Review of Fractal Dimensions :

Consider a structure distributed in three dimensions. If we put a cube around the structure so that the whole structure is contained, we can then divide the cube into sequentially smaller boxes - essentially rescaling the object, e.g., first the whole box (l=1), then 8 (l=1/2) boxes, then 64 (l=1/4) and so on. The fractal dimension of the structure (if it is a fractal) is determined by looking at the occupied cells as a function of the scale, l. If it is a fractal, then the object will appear with the same structure at each scale. It will be scale-invariant or self-similar. There are many different ways to measure fractal dimensions, all of them share the attirbute that one looks for a simple power law behavior as the strucutre is considered at different scales. Here are the three primary approaches:

• The Hausdorff, or box counting, dimension is given by

N(l) ~ l**-D

in the limit of small l. N(l) is the number of boxes in the space that include a data entry. D is the Hausdorff fractal dimension.

• The information-entropy dimension is given by

S(l) = -sum( from i=1 to N(l) ) p_i(l) * log( p_i(l) )

where p_i is the probability for an entry in the system to fall in the ith cell, and N(l) is the number of occupied cells for length scale l. For a fractal structure, in the limit of small l, S(l) --> const - sigma * log(l) and sigma is the information entropy dimension.

• The correlation integral dimension is given by

C(l) = 1/N**2 (Number of pairs of points whose separation distance is less than l)

where N is the number of entries in the data. For a fractal, C(l) obeys the power law:

C(l) ~ l**nu and nu is the correlation dimension.

The biggest problem with the first two is that finite multiplicities mean that at some scale each box contains a single particle and the distribution plateaus to a constant. This is the so-called convergence problem. It means that the region where log(N(l)) or log(S(l)) is linear may only consist of a few points, making the extracted fractal dimension ambiguous. The correlation ntegral approach mitigates this effect by including all the pairwise particle correlations in C(l), thus there are many points in the plot of log(C(l)) vs l and the linear region is huge. The disadvantage is that for a few hundred points the calculation can be CPU demanding.

Our motivation for studying the fractal structure of multiple jet events was originally to extract a topological event identifier, not necessarily to formulate a new technique to search for self-similarity in hadronic events. In the literature, two techniques for searching for self-similarity in hadronic event data are most commonly used: normalized factorial moments (NFM) - a generalization of the Hausdorf dimension -and bunching parameters (BP). In most published analyses the data is studied in the single dimension eta to mitigate algorithmic difficulties arising from finite multiplicities. In our work we demand that events maintain their integrity and insist on using as much of the available phase space as we can measure.

The next step in this work is to investigate the effects of imited particle spatial resolution. This issue does not seem prohibitive for the HEP-entropy algorithm, though correcting for variations in noise across the fiducial regions of the D0 detector presents a challenge.

We are embarking on a completely different approach to studying self-similarity in a hadron collider detector using sub-jets within jets reconstructed with the K_T algorithm. One of the advantages of the K_T algorithm over the cone algorithm is that it can be re-applied at finer levels within jets to reconstruct jets within jets at successively finer scales: sub-jets. The way that subjets are reconstructed at successively finer scales, however, does not lend itself to application of any standard form of fractal dimension calculating algorithm.

About the authors:

Suemee Shin - did her Masters of Science Thesis on this project and obtained many of the first results with code written in procedural c.

Andrew McDowell - a junior at Texas A&M University who worked on the project last summer at UTA. He developed a good deal of the code in OO c++. Subsequent students who have used his code (he wrote lots of code for us for many purposes) have referred to him as a programming "god."

Michael Strang - a 2nd year graduate student at UTA is confirming the earlier results and putting the finishing touches on the final results and producing the figures for the PRD draft.

Michael Vinson - is the Chair of the Physics Department at Shippensburg University and has supplied the nonlinear dynamics expertise for this work.

Ransom Stephens - made this web page.

We are also studying application of the Maximum Entropy Image Reconstruction techniques. We are attempting to fit underlying event distributions to simulated data in order to extract the hard scattering of two partons independent of the underlying event. One of our undergraduate students, Joe Sauder, has developed OO c++ software for this project and the earliest results promise that we will be able to measure the invariant mass of the two partons in a hard scattering with better resolution than can be attained with a simple E_T cut on jets in a final state. Be that as it may, these techniques promise accurate subtraction of minimum-bias interactions from mutliple interaction events. This will be important for QCD analyses at the LHC where we expect an average of 18 interactions per beam crossing at high luminosity.

reference - E.T. Jaynes, ``On The Rationale of Maximum-Entropy Methods,'' Proc. IEEE 70, 939 (1982); Phys. Rev. 106, 620 and 108, 171 (1957); J. Antolin, ``Maximum Entropy Formalism and Analytic Extrapolation,'' J. Math. Phys. 31, 791 (1990).