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The power spectrum - a way to accumulate texture information

In our work, a power spectrum analysis of event texture in pseudorapidity, $\eta$ and azimuthal angle $\zeta$ based on a Discrete Wavelet Transformation (DWT)[17], is performed on a number of large event ensembles sampled according to their multiplicity, thereby studying the impact parameter dependence of the observables. DWT quantifies contributions of different $\zeta$ and $\eta$ scales into the overall event's texture, thus testing for possible large scale enhancement.

DWT formalizes the images of the $PbPb$ collision events in pseudorapidity $\eta$ and azimuthal angle $\zeta$ by expanding them into a set of functions orthogonal with respect to scale and position in the ($\eta$, $\zeta$) space, and allows one to accumulate the texture information by averaging the power spectra of many events. While the DWT analyzes the object (an image, a sequence of data points, a data array) by transforming it, the full information content inherent in the object is preserved in the transformation. Mathematically, this is expressed by stating that the discrete wavelet family of functions constitutes a complete basis in the space of all measurable functions defined on the continuum of real numbers $L^2({\mathbb{R}})$. This statement is known as the multiresolution theorem and constitutes the theoretical ground for the multiresolution analysis.

The simplest DWT basis is the Haar wavelet, built upon the scaling function $\phi(x) = 1$ for $0\le x<1$ and 0 otherwise. If the interaction vertex lies on the detector's symmetry axis, every pad's acceptance is a rectangle in the $(\zeta,\eta)$ space. Then, the Haar basis is the natural choice, as its scaling function in two dimensions (2D) $\Phi(\zeta,\eta) = \phi(\zeta)\phi(\eta)$ is just a pad's acceptance (modulo units). We therefore set up a 2D wavelet basis:

$$\Psi^{\lambda}_{m,i,j}(\zeta,\eta) = 2^{m}\Psi^{\lambda}(2^{m}\zeta-i,2^{m}\eta-j).$$ (87)

$\Phi_{m,i,j}(\zeta,\eta)$ is constructed from $\Phi(\zeta,\eta)$ similarly. Here, $m$ is an integer scale fineness index; $i$ and $j$ index the discrete positions of pad centers in $\zeta$ and $\eta$ ( $1 \le m \le 4$ and $1\le i,j \le 16$ because we use $16=2^4$ rings and 16 sectors ). Different values of $\lambda$ (denoted as $\zeta$, $\eta$, and $\zeta \eta$) distinguish, respectively, functions with azimuthal, pseudorapidity, and diagonal texture sensitivity:

$$\Psi^\zeta=\psi(\zeta)\phi(\eta), \ \ \Psi^\eta=\phi(\zeta)\psi(\eta), \ \ \Psi^{\zeta\eta}=\psi(\zeta)\psi(\eta)$$ (88)

In the Haar basis, for any variable $x$

$$\psi(x) = \left\{ \begin{array}{r@{\quad:\quad}l} +1 & 0\le x<\frac{1}{2} \\ -1 & \frac{1}{2}\le x<1 \\ 0 & otherwise \end{array} \right.$$ (89)

is the wavelet function. Then, $\Psi^\lambda_{m,i,j}$ with integer $m$, $i$, and $j$ are known [17] to form an orthonormal basis in $L^2({\mathbb{R}}^2)$.

We adopt the existing [19] 1D DWT power spectrum analysis technique and expand it to 2D. The track density in an individual event is $\rho(\zeta,\eta)$ and its local fluctuation in a given event is $\sigma^2 \equiv \langle \rho - \bar{\rho},\rho - \bar{\rho}\rangle,$ where $\bar{\rho}$ is the average $\rho$ (over the acceptance) in the given event.

Using completeness of the basis, we expand

$$\rho - \bar{\rho} = \langle \rho,\Psi^\lambda_{m,i,j}\rangle \Psi... ...a_{m,i,j} - \langle \bar{\rho},\Psi^\lambda_{m,i,j}\rangle \Psi^\lambda_{m,i,j}$$ (90)

Notice that $\bar{\rho}$, being constant within detector's rectangular acceptance, is orthogonal to any $\Psi^\lambda_{m,i,j}$ with $m \ge 1$. Due to the orthonormality condition $\langle \Psi^\lambda_{m,i,j},\Psi^{\lambda'}_{m',i',j'}\rangle = \delta_{\lambda,\lambda'}\delta_{m,m'}\delta_{i,i'}\delta_{j,j'}$, the $\rho - \bar{\rho}$ components for different scales do not form cross-terms in the $\sigma^2$ sum, and the sum contains no cross-terms between $\rho$ and $\bar{\rho}$ for the four observable scales. Instead of a $\langle \rho, \Phi_{m=5,i,j} \rangle$ set, the Si detector energy amplitude array - its closest experimentally achievable approximation - is used as the DWT input. We used WAILI [81] software library to obtain the wavelet decompositions.

The Fourier power spectrum of a random white noise field is known to be independent of frequency [80]. We are looking for dynamical textures in the data, and therefore would like to treat the random white noise case as a ``trivial'' one to compare with. Therefore it is interesting to reformulate this property for wavelets, where scale plays the same role as frequency in Fourier analysis. To do that, we link scales with frequencies, or in other words, we must understand the frequency spectra of the wavelets. The Fourier images of 1D wavelet functions occupy a set of wave numbers whose characteristic broadness grows with scale fineness $m$ as $2^m$; $2^{2m}$ should be used in the 2D case. Discrete wavelets of higher orders have better frequency localization than the Haar wavelets. Despite this advantage, we use Haar because only Haar allows one to say that the act of data taking with the (binned !) detector constitutes the first stage of the wavelet transformation.

In 2D, we find it most informative to present the three modes of a power spectrum with different directions of sensitivity $P^{\zeta\eta}(m)$, $P^\zeta(m)$, $P^\eta(m)$ separately. We define the power spectrum as

$$P^\lambda(m) = \frac{1}{2^{2m}}\sum_{i,j}\langle \rho,\Psi^\lambda_{m,i,j}\rangle^2 ,$$ (91)

where the denominator gives the meaning of spectral density to the observable. So defined, the $P^\lambda(m)$ of a random white noise field is independent of $m$. In the first approximation, the white noise example provides a base-line case for comparisons in search for non-trivial effects.