Derivation Notes on Centrality Dependence of Two Particle Correlation

Derivation Notes on Centrality Dependence of Two Particle Correlation

This note is based on Yogiro's suggestion of extending our analytic study of one tube model on in-plane out-of-plane to that of centrality dependence of two particle correlation. The basic idea of the following calculations is that instead of doing hydrodynamic evolution, we assume the final one-particle distribution can be approximated by superposition of diverted flows due to the presence of tube on top of a background elliptic distribution. The former is essentially centrality independent while the later is centrality dependent. In particular, the flow due to the tube is taken from the one-particle distribution of one-tube model for central collisions.

By straightforward calculations of two particle correlations, one finds the two-cumulants have the same forms both for in-plane and out-of-plane angles. This is because the contributions of background elliptic distribution gets cancelled out when one integrates over the tube positions and subtracts mixed event correlations, event though the proper event correlations of in-plane angle is different from that of out-of-plane ones. However, the interesting feature of our model is that when one considers fluctuations of overall particle yields, one obtains extra terms whose resulting correlations are able to reproduce the main features of the experimental data with proper choice of parameters.

This calculation is aimed for a publication as a Phy. Rev. C rapid communication.

Similar to our previous study, we write down the one-particle distribution as a sum of two contributions: the distribution of the background and that of the tube.


 * $$\begin{align}

\frac{dN}{d\phi}(\phi,\phi_t) =\frac{dN_{bgd}}{d\phi}(\phi) +\frac{dN_{tube}}{d\phi}(\phi,\phi_t) \end{align}$$ where
 * $$\begin{align}

&\frac{dN_{bgd}}{d\phi}(\phi)=\frac{N_b}{2\pi}(1+2v_2^b\cos(2\phi))\\ &\frac{dN_{tube}}{d\phi}(\phi,\phi_t)=\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n[\phi-\phi_t]) \end{align}$$ For simplicity, we assume that the background is dominated by the elliptic flow, which is observed experimentally, especially in non-central collisions. In Eq.2(\ref{eq2}), the flow is parametrized in turn of the elliptic flow parameter $$v_2^b$$ and the overall multiplicity, denoted by $$N_b$$. As for the contributions from the tube, we take into account the minimal number possible of parameters to reproduce the two-particle correlation due to the sole existence of a peripheral tube in an isotropic background. Therefore, only two essential components $$v_2^t$$ and $$v_3^t$$ are retained in Eq.(\ref{eq3}). We note here that the overall triangular flow in our approach is generated only by the tube, i.e., $$v_3=v_3^t$$ and so its symmetry axis is correlated to the tube location $\phi_t$. The azimuthal angle $$\phi$$ of the emitted hadron and the position of the tube $$\phi_t$$ are measured with respect to the event plane $$\Psi_2$$ of the system. Since the flow components from the background are much bigger than those generated by the tube, as discussed below, $$\Psi_2$$ is essentially determined by the elliptic flow of the background $$v_2^b$$.

Following the methods used by the STAR experiment\cite{citation needed}, the subtracted di-hadron correlation is given by
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle  =\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{proper} -\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{mixed} , \end{align}$$ In one-tube model,
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{proper} =\int\frac{d\phi_s}{2\pi}\frac{d\phi_t}{2\pi}f(\phi_t) \frac{dN}{d\phi}(\phi_s,\phi_t)\frac{dN}{d\phi} (\phi_s+\Delta\phi,\phi_t), \end{align}$$ where $$f(\phi_t)$$ is the distribution function of the tube and $$\phi_s$$ is the azimuthal angle of the trigger particle. We will take $$f(\phi_t)=1$$, for simplicity.

The combinatorial background $$\left\langle{dN_{pair}}/{d\Delta\phi}\right\rangle^{mixed}$$ can be calculated by using either cumulant or ZYAM method \cite{zyam-1}. As shown below, both methods lend very similar conclusions in our model. Here, we first carry out the calculation using cumulant, which gives
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{mixed(cmlt)} =\int\frac{d\phi_s}{2\pi}\frac{d\phi_t}{2\pi}f(\phi_t) \int\frac{d\phi_t'}{2\pi}f(\phi_t')\frac{dN}{d\phi} (\phi_s,\phi_t)\frac{dN}{d\phi}(\phi_s+\Delta\phi,\phi_t'). \end{align}$$ Notice that, in the averaging procedure above, integrations both over $$\phi_t$$ and $$\phi_t'$$ are required in the mixed events, whereas only one integration over $$\phi_t$$ is enough for proper events. This will make an important difference between two terms in the subtraction of Eq.(\ref{eq4}).

Using our simplified parametrization, Eqs.(\ref{eq1}-\ref{eq3}) and, by averaging over events, one obtains
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{proper} =\frac{}{(2\pi)^2}(1+2(v_2^b)^2\cos(2\Delta\phi)) +(\frac{N_t}{2\pi})^2 \sum_{n=2,3}2(v_n^t)^2\cos(n\Delta\phi) \end{align}$$ and
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{mixed(cmlt)} =\frac{^2}{(2\pi)^2}(1+2(v_2^b)^2\cos(2\Delta\phi). \end{align}$$ Because of random distribution, contributions from peripheral tube are cancelled out upon averaging in the mixed-event correlation.  Observe the difference between the factors multiplying the background terms of the proper- and mixed-event correlation. By subtracting Eq.(\ref{mixedc}) from Eq.(\ref{proper}), the resultant correlation is
 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{(cmlt)} =\frac{-^2}{(2\pi)^2}(1+2(v_2^b)^2\cos(2\Delta\phi)) +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2({v_n^t})^2 \cos(n\Delta\phi). \end{align}$$ So, one sees that, as the multiplicity fluctuates, the background elliptic flow does contribute to the correlation. Now, experimental measurements showed that the elliptic flow coefficient increases when one goes to more peripheral collisions. It is therefore intuitive to expect, despite its simplicity, the above analytic model may reproduce the main characteristics of the observed data. If one uses a small $$v_2^b$$ is for central collisions and increase its value when one goes to peripheral collisions, the away-side double-peaks may disappear as a consequence of increasing of $$v_2^b$$.

The correlations in Fig.3 is plotted with the following parameters
 * $$\begin{align}

&=43.8 \rightarrow 43.8  \\ &v_2^t=v_3^t=0.1  \\ &-^2=2.15 \rightarrow   \\ &v_2^b=0.3 \rightarrow 0.25 \end{align}$$ We note that both the correlated yields and the flow harmonics are actually dependent on the specific choice of the $$p_T$$ interval of the observed hadrons as shown in Fig.1 and Fig.2. Since the transverse momentum dependence has not been explicitly taken into account in this simple model, the multiplicities in the above parameter set are only determined up to an overall normalization factor, and the flow coefficients are chosen to reproduce the qualitative behavior of the trigger-angle dependence of di-hadron correlation as shown by data.

Acknowledgments
We acknowledge funding from Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG) and Conselho Nacional  de Desenvolvimento Científico e Tecnológico (CNPq).