Event-by-Event Analysis and the Central Limit Theorem

Event-by-event analysis of heavy-ion collision events is an important tool for the study of the QCD phase boundary and formation of a quark-gluon plasma. A universal feature of phase boundaries is the appearance of increased fluctuations of conserved measures as manifested by excess measure variance compared to a reference. In this paper I consider a particular aspect of EbyE analysis emphasizing global-variables variance comparisons and the central limit theorem. I find that the central limit theorem is, in a broader interpretation, a statement about the scale invariance of total variance for a measure distribution, which in turn relates to the scale-dependent symmetry properties of the distribution.. I further generalize this concept to the relationship between the scale dependence of a covariance matrix for all conserved measures defined on a dynamical system and a matrix of covariance integrals defined on two-point measure spaces, which points the way to a detailed description of the symmetry dynamics of a complex measure system. Finally, I relate this generalized description to several recently proposed or completed event-by-event analyses.Comment: 40 pages, 10 figure

Comparing the same-side "ridge" in CMS p-p angular correlations to RHIC p-p data

The CMS collaboration has recently reported the appearance of a same-side "ridge" structure in two-particle angular correlations from 7 TeV p-p collisions. The ridge in p-p collisions at 7 TeV has been compared to a ridge structure in more-central Au-Au collisions at 0.2 TeV interpreted by some as evidence for a dense, flowing QCD medium. In this study we make a detailed comparison between 0.2 TeV p-p correlations and the CMS results. We find that 7 TeV minimum-bias jet correlations are remarkably similar to those at 0.2 TeV, even to the details of the same-side peak geometry. Extrapolation of azimuth quadrupole systematics from 0.2 TeV suggests that the same-side ridge at 7 TeV is a manifestation of the azimuth quadrupole with amplitude enhanced by applied cuts.Comment: 9 pages, 4 figures, response to referee comment

Extrapolating parton fragmentation to low Q2Q^2 in e+e^+-e−e^- collisions

We analyze the energy scale dependence of fragmentation functions from $e^+$-$e^-$ collisions using conventional momentum measures $x_p$ and $\xi_p$ and rapidity $y$. We find that replotting fragmentation functions on a normalized rapidity variable results in a compact form precisely represented by the beta distribution, its two parameters varying slowly and simply with parton energy scale $Q$. The resulting parameterization enables extrapolation of fragmentation functions to low $Q$ in order to describe fragment distributions at low transverse momentum $p_t$ in heavy ion collisions at RHIC.Comment: 19 pages, 20 figure

Correlation Analysis With Scale-local Entropy Measures

A novel method for correlation analysis using scale-dependent Renyi entropies is described. The method involves calculating the entropy of a data distribution as an explicit function of the scale of a d-dimensional partition of d-cubes, which is dithered to remove bias. Analytic expressions for dithered scale-local entropy and dimension for a uniform random point set are derived and compared to Monte Carlo results. Simulated nontrivial point-set correlations representing condensation and clustering are similarly analyzed.Comment: 12 pages, 6 figure

Parton fragment yields derived from minimum-bias jet angular correlations

Spectrum hard components from 200 GeV \auau collisions are accurately described by pQCD parton fragment distributions, indicating that a substantial population of parton fragments is present in hadron spectra at low $p_t$. Minimum-bias angular correlations contain jet-like correlation structure with most-probable hadron momentum 1 GeV/c. In this study we convert minimum-bias jet-like angular correlations to single-particle yields and compare them with spectrum hard components. We find that jet-like correlations in central 200 GeV \auau collisions correspond quantitatively to pQCD predictions, and the jet-correlated hadron yield comprises one third of the final state.Comment: 12 pages, 5 figure

Multiparticle correlations in Q-space

We introduce Q-space, the tensor product of an index space with a primary space, to achieve a more general mathematical description of correlations in terms of q-tuples. Topics discussed include the decomposition of Q-space into a sum-variable (location) subspace S plus an orthogonal difference-variable subspace D, and a systematisation of q-tuple size estimation in terms of p-norms. The "GHP sum" prescription for q-tuple size emerges naturally as the 2-norm of difference-space vectors. Maximum- and minimum-size prescriptions are found to be special cases of a continuum of p-sizes.Comment: 8 pages, 5 figure

Scale-local dimensions of strange attractors

We compare limit-based and scale-local dimensions of complex distributions, particularly for a strange attractor of the Henon map. Scale-local dimensions as distributions on scale are seen to exhibit a wealth of detail. Limit-based dimensions are shown to be averages of scale-local dimensions, in principle over a semi-infinite scale interval. We identify some critical questions of definition for practical dimension analysis of arbitrary distributions on bounded scale intervals.Comment: 12 pages, 5 figure

Autocorrelations from fluctuation scale dependence by inversion

Fluctuations in nuclear collisions can be measured as a function of momentum-space binning scale over a scale interval bounded by detector two-track resolution and acceptance. Fluctuation scale dependence is related to two-particle correlations by a Fredholm integral equation. That equation can be inverted by standard numerical methods to yield an autocorrelation distribution on difference variables as a projection of the full two-particle distribution which retains most of the correlation information in a more compact form. Autocorrelation distributions are typically more easily interpreted in terms of physical mechanisms than fluctuation measurements.Comment: 10 pages, 5 figure

Transverse-rapidity yt\bf y_t dependence of the nonjet azimuth quadrupole from 62 and 200 GeV Au-Au collisions

Previous measurements of a quadrupole component of azimuth correlations denoted by symbol $v_2$ have been interpreted to represent elliptic flow, a hydrodynamic phenomenon conjectured to play a major role in noncentral nucleus-nucleus collisions. $v_2$ measurements provide the main support for conclusions that a ``perfect liquid'' is formed in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC). However, conventional $v_2$ methods based on one-dimensional (1D) azimuth correlations give inconsistent results and may include a jet contribution. In some cases the data trends appear to be inconsistent with hydrodynamic interpretations. In this study we distinguish several components of 2D angular correlations and isolate a nonjet (NJ) azimuth quadrupole denoted by $v_2\{\text{2D}\}$. We establish systematic variations of the NJ quadrupole on $y_t$, centrality and collision energy. We adopt transverse rapidity $y_t$ as both a velocity measure and as a logarithmic alternative to transverse momentum $p_t$. Based on NJ quadrupole trends we derive a completely factorized universal parametrization of quantity $v_2\{\text{2D}\}(y_t,b,\sqrt{s_{NN}})$ which describes the centrality, $y_t$ and energy dependence. From $y_t$-differential $v_2(y_t)$ data we isolate a quadrupole spectrum and infer a quadrupole source boost having unexpected properties. NJ quadrupole $v_2$ trends obtained with 2D model fits are remarkably simple. The centrality trend appear to be uncorrelated with a sharp transition in jet-related structure that may indicate rapid change of Au-Au medium properties. The lack of correspondence suggests that the NJ quadrupole may be insensitive to such a medium. Several quadrupole trends have interesting implications for hydro interpretations.Comment: 19 pages, 14 figure

Challenging the utility of third-order azimuth harmonics in the description of ultra-relativistic heavy-ion collisions

In recent years it has become conventional practice to include higher-order cylindrical harmonics in the phenomenological description of two-particle angular correlations from ultra-relativistic heavy-ion collisions. These model elements, whose dependence on relative azimuth angle has the form $\cos[m(\phi_1-\phi_2)]$ where $m > 2$, were introduced to support a hydrodynamic flow interpretation of the same-side ($|\phi_1-\phi_2| < \pi/2$) 2D peak in the correlations. Previous studies have shown that the $m > 2$ harmonics are not required by the data, that they destabilize the fitting models, and that their net effect is to decompose the same-side peak into two components, one being dependent on and the other being independent of relative pseudorapidity. Thus we are lead to question whether descriptions of angular correlation data including higher-order harmonics inform our understanding of the same-side peak or heavy-ion collisions in general. Results from analysis of two-dimensional angular correlation data from the Relativistic Heavy-Ion Collider (RHIC) and the Large Hadron Collider (LHC) show that the RHIC data do not exclude a single-Gaussian hypothesis for the same-side peak. We find that the net effect of including the $m = 3$ harmonic or azimuth sextupole in the fitting model is the inclusion of small non-Gaussian dependence in the mathematical description of the same-side peak. Those non-Gaussian effects are systematically insignificant and can be accommodated by minor perturbations to the same-side 2D Gaussian peak model, which act locally at small relative azimuth. We also demonstrate that the 0-1% 2D angular correlation data for 2.76 TeV Pb+Pb collisions from ATLAS, which display an away-side double peak on azimuth, do not require a sextupole and exclude a positive same-side sextupole.Comment: 14 pages, 6 figures, 2 tables, submitted to Phys. Rev.