Derivation Notes on In-Plane and Out-of-Plane Effect by Flow Decomposition

Derivation Notes on In-Plane Out-Of-plane Effect by Flow Decomposition

This note is based on Yogiro's suggestion of analytic calculation of one tube model on in-plane out-of-plane effect by using simplified flow decompositions. (Some of the results here can be found in arXiv:1207.6415 [hep-ph].)

The basic idea of the following calculations is that instead of doing hydrodynamic evolution, we assume the final one-particle distribution of any peripheral collision can be approximated by superposition of diverted flows due to the presence of tube on top of a background elliptic distribution. The former we took from 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.

Elliptic background with isotropic tube distribution
Assuming the following one particle azimuthal distribution in terms of background and flow components generated by one tube model


 * $$\begin{align}

&\frac{dN}{d\phi}(\phi,\phi_t) \equiv \frac{dN}{d\phi}(\phi)=\frac{dN_{background}}{d\phi}(\phi)+\frac{dN_{tube}}{d\phi}(\phi,\phi_t) \\ &\frac{dN_{tube}}{d\phi}(\phi,\phi_t) \equiv \frac{dN_{tube}}{d\phi}(\phi) \end{align}$$

where


 * $$\begin{align}

&\frac{dN_{background}}{d\phi}(\phi)=\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\phi-\Psi_b))) \\ & \frac{dN_{tube}}{d\phi}(\phi)=\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\phi-\phi_t-\Psi_b)) \end{align}$$

where $$\phi_t $$ is the azimuthal location of the tube, its value is to be understood in respect to the event plane $$ \Psi_b$$. We hope this is approximately valid for $$N_t \ll N_b$$  and  $$v_n^t \ll 1$$. We did not write the distributions explicitly as functions of $$\phi_t$$, bearing in mind that the angle  $$\phi_t$$  will always be integrated out when calculating observables (see below). We also note that it is not necessary to write it as


 * $$\begin{align}

\frac{dN_{tube}}{d\phi}(\phi)=\frac{N_t}{2\pi}(1+\sum_{n=2,3}2v_n^t\cos(n(\phi-\phi_t-\Psi_b))) \end{align}$$

due to the fact that the normalization term is isotropic therefore can always be absorbed into the background. Consequently, we would expect the particle distribution due to the tube $$\frac{dN_{tube}}{d\phi}(\phi)$$  does not contribute to the mixed event correlation (see below).

Two particle correlation at in-plane angle is


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi} \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \end{align}$$ where $$f(\phi_t)$$  should be normalized so that $$\int f(\phi_t) \frac{d\phi_t}{2\pi} =1$$ is the tube distribution function, in our first treatment, we take $$f(\phi_t)=1$$  for simplicity.

At out-of-plane angle one has


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi} \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \end{align}$$ where again $$ \int f(\phi_t) \frac{d\phi_t}{2\pi} =1 $$ With the parameterizations of our model determined, we set off to calculate first in-plane correlation of proper events


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi} \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t}{2\pi} \frac{dN}{d\phi}(\Psi_b) \frac{dN}{d\phi}(\Psi_b+\Delta\phi)  \\ &=\int \frac{d\phi_t}{2\pi}[\frac{N_b}{2\pi}(1+2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(-n(\phi_t))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(\phi_t))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t)) \\ &+\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b)\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))  +\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b\cos(2\Delta\phi))\sum_{n=2,3}2v_n^t\cos(n\phi_t)] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(\phi_t))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi)]\\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Then we calculate in-plane correlation of mixed events, the two events have to be aligned in by the event plane of $$v_2$$, here we adopt an assumption that it is determined solely by the background distribution, since the contribution of the tube is smaller comparing to the background.


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi} \frac{dN^1}{d\phi}(\Psi_b) \frac{dN^2}{d\phi}(\Psi_b+\Delta\phi)  \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[\frac{N_b}{2\pi}(1+2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(-n(\phi_t^1))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2))] \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(\phi_t^1))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2))] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi)) \end{align}$$

Here we see that the mixed events only take account of the background contribution, which was expected. Therefore


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} =(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

The summation of particle distribution due to the tube was only carried out for $$\sum_{n=2,3}$$, but it is straightforward to see, the result is also valid for a more general case where


 * $$\begin{align}

\frac{dN_{tube}}{d\phi}(\phi)=\frac{N_t}{2\pi}\sum_{n=1}^{+\infty}2v_n^t\cos(n(\phi+\phi_t-\Psi_b)) \end{align}$$

Now we calculate out-of-plane correlation of proper events


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t}{2\pi} \frac{dN}{d\phi}(\Psi_b+\frac{\pi}{2}) \frac{dN}{d\phi}(\Psi_b+\frac{\pi}{2}+\Delta\phi)  \\ &=\int \frac{d\phi_t}{2\pi}[\frac{N_b}{2\pi}(1-2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\frac{\pi}{2}))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\frac{\pi}{2})))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\frac{\pi}{2}))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\frac{\pi}{2}))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\frac{\pi}{2}))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi)] \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Then we calculate in-plane correlation of mixed events, the two events are aligned in by the background distribution which basically determines the event plane of $$v_2$$.


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi}\int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi} \frac{dN^1}{d\phi}(\Psi_b+\frac{\pi}{2}) \frac{dN^2}{d\phi}(\Psi_b+\Delta\phi+\frac{\pi}{2}) \\ &=\int  \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[\frac{N_b}{2\pi}(1-2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t^1+\frac{\pi}{2}))]  [\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2+\frac{\pi}{2}))] \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(-\phi_t^1+\frac{\pi}{2}))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2+\frac{\pi}{2}))] \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) \end{align}$$ Therefore


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed} =(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Comparing this expression with the in-plane result, we see that they are the same. Though the correlations of proper event have different forms, the cumulants are purely due the flow of the tube. This is also somehow expected.

As suggested by Yogiro, one may also evaluate the correlation for any fixed angle $$\phi^0$$  (with  $$\phi^0=0$$  in-plane,  $$\phi^0=\frac{1}{2}$$  out-of-plane). For proper event


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b-\phi^0)}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t}{2\pi} \frac{dN}{d\phi}(\Psi_b+\phi^0) \frac{dN}{d\phi}(\Psi_b+\phi^0+\Delta\phi)  \\ &=\int \frac{d\phi_t}{2\pi}[\frac{N_b}{2\pi}(1+2v_2^b\cos(2\phi^0))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\phi^0))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\phi^0))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\phi^0))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\phi^0))] \\ &=\int \frac{d\phi_t}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0))) +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi)] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

For mixed event


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi}\int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b-\phi^0)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi} \frac{dN^1}{d\phi}(\Psi_b+\phi^0) \frac{dN^2}{d\phi}(\Psi_b+\Delta\phi+\phi^0)  \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[\frac{N_b}{2\pi}(1+2v_2^b\cos(2\phi^0))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t^1+\phi^0))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2+\phi^0))] \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[(\frac{N_b}{2\pi})^2(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))  +(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(-\phi_t^1+\phi^0))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2+\phi^0))] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0))) \end{align}$$

Therefore


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{mixed} =(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Particle number fluctuations included
Now we like to show the non-trivial effect. When one includes particle number fluctuations, extra terms emerged. Then we show that with proper numerical parameterization, it may reproduce the main features of in-plane and out-of-plane correlations observed experimentally.

Firstly of all, $$N_b$$  measures the total yields. On the other hand, $$N_t$$  does not affect the overall yields, instead, it only modifies the shape of the one-particle distribution due to the presence of the tube. Here we only introduce event-by-event fluctuation of total yield $$N_b$$, and we will show it is already enough to describe the main feature.

We introduce an extra average over different events in the above calculation, and pay special attention to the particle number $$N_b$$


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}\frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv\int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))\\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) \\ & \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

both of which can been seen as special case of


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b-\phi^0)}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b-\phi^0)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0))) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{\phi^0-to-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b\cos(2\phi^0))(1+2v_2^b\cos(2(\Delta\phi+\phi^0)))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

In the enclosed pdf file I included a set of numerical result with a specific choice parameterization. It contains five plots. The first plot is two particle correlation due to the tube, which also equals to the resulting cumulant without number fluctuation. The second and third plots show the two particle correlations of the background for in-plane and out-of-plane cases, in our model, they are purely due to the elliptic flow. The fourth and fifth plots are the two particle correlations considering overall particle number fluctuations. One sees that it matches two important features observed experimentally. 1) Single peak on away side at in-plane angle and double peak at out-of-plane angle. 2) The near side ridge yield decrease from in-plane to out-of-plane. Both of the two features can be easily understood by looking at our resulting expressions. Near the focus is the importance of event-by-event fluctuations.

Comparing to Rone's numerical results of one-tube model, which is very similar, one might wonder if they are consistent with the results presented here. Since Rone did not explicitly introduce particle number fluctuation, and in fact, the background in his calculation is fixed. However, if one notices that here we have introduced fluctuation in the overall yields, in this context, due to the tube positioning, overall all yields were different from event to event, even though one uses the same background and the same tube parameterizations. Also, by looking at the parameters of the plots, one see that the fluctuations were actually not big.

Two comments. Firstly, one may also include fluctuations for tube distribution $$\frac{dN_{tube}}{d\phi}(\phi)$$, in this case one obtains essentially the same results


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}\frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv\int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Secondly, the resulting expression can be also generalize to include N tubes. Omitting 1) the interference between the tubes 2) fluctuation of tube number, the result is very similar.

Flow coefficient fluctuations included
Firstly, we note that fluctuations in flow coefficients might have very similar effect as particle number fluctuations.

Elliptic background with biased tube distribution
As a simplified model, we omit particle number fluctuation and introduce the following tube distribution
 * $$f(\phi_t)=1-c_2^b\cos(2\phi_t) $$

where $$\phi_t$$  is the azimuthal location of the tube, as defined above, its value is to be understood in respect to the event plane  $$\Psi_b$$. which satisfies


 * $$\begin{align}

\int f(\phi_t) \frac{d\phi_t}{2\pi} =1 \end{align}$$

At in-plane angle


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi} \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t}{2\pi} [1-c_2^b\cos(2\phi_t)]\frac{dN}{d\phi}(\Psi_b) \frac{dN}{d\phi}(\Psi_b+\Delta\phi)  \\ &=\int \frac{d\phi_t}{2\pi}[1-c_2^b\cos(2\phi_t)][\frac{N_b}{2\pi}(1+2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(-n(\phi_t))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)[\frac{N_b}{2\pi}(1+2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(-n(\phi_t))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)\{(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi)) \\ &+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(\phi_t))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))  +\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b)\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t)) \\ &+\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b\cos(2\Delta\phi))\sum_{n=2,3}2v_n^t\cos(n\phi_t)\} \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi)  \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)  \{(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^t\cos(n(\phi_t))\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t)) \\ &+\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b)\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))  +\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b\cos(2\Delta\phi))\sum_{n=2,3}2v_n^t\cos(n\phi_t)\} \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)  \left\{ \frac{N_tN_b}{(2\pi)^2}(1+2v_2^b)\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t))+\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b\cos(2\Delta\phi))\sum_{n=2,3}2v_n^t\cos(n\phi_t) \right\} \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)  \left\{ \frac{N_tN_b}{(2\pi)^2}(1+2v_2^b)2v_2^t\cos(2(\Delta\phi-\phi_t))+\frac{N_tN_b}{(2\pi)^2}(1+2v_2^b\cos(2\Delta\phi))2v_2^t\cos(2\phi_t) \right\} \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) -\frac{N_tN_b}{(2\pi)^2}c_2^b(1+2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{N_tN_b}{(2\pi)^2}c_2^b(1+2v_2^b\cos(2\Delta\phi))v_2^t \end{align}$$


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi} \frac{dN^1}{d\phi}(\Psi_b) \frac{dN^2}{d\phi}(\Psi_b+\Delta\phi)  \\ &=\int \frac{d\phi_t^1}{2\pi}[1-c_2^b\cos(2\phi_t^1)]\int  \frac{d\phi_t^2}{2\pi}[1-c_2^b\cos(2\phi_t^2)]  [\frac{N_b}{2\pi}(1+2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(-n(\phi_t^1))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2))] \\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi)) +\int \frac{d\phi_t^1}{2\pi}\int \frac{d\phi_t^2}{2\pi}(-c_2^b\cos(2\phi_t^1))\frac{N_t}{2\pi}2v_2^t\cos(2\phi_t^1)\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi))\\ &+\int \frac{d\phi_t^1}{2\pi}\int \frac{d\phi_t^2}{2\pi}(-c_2^b\cos(2\phi_t^2))\frac{N_b}{2\pi}(1+2v_2^b)\frac{N_t}{2\pi}2v_2^t\cos(2(\Delta\phi-\phi_t^2))\\ &=(\frac{N_b}{2\pi})^2(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi)) -\frac{N_tN_b}{(2\pi)^2}c_2^bv_2^t(1+2v_2^b\cos(2\Delta\phi)) -\frac{N_bN_t}{(2\pi)^2}c_2^b(1+2v_2^b)v_2^t\cos(2\Delta\phi) \end{align}$$


 * $$\begin{align}

\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} =(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

At out-of-plane angle


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t}{2\pi}[1-c_2^b\cos(2\phi_t)] \frac{dN}{d\phi}(\Psi_b+\frac{\pi}{2}) \frac{dN}{d\phi}(\Psi_b+\frac{\pi}{2}+\Delta\phi)  \\ &=\int \frac{d\phi_t}{2\pi}[1-c_2^b\cos(2\phi_t)][\frac{N_b}{2\pi}(1-2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\frac{\pi}{2}))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\frac{\pi}{2})))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\frac{\pi}{2}))] \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)[\frac{N_b}{2\pi}(1-2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t+\frac{\pi}{2}))]  [\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\frac{\pi}{2})))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t+\frac{\pi}{2}))] \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)\frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}2v_2^t\cos(2(\Delta\phi-\phi_t+\frac{\pi}{2}))  \\ &-\int \frac{d\phi_t}{2\pi}c_2^b\cos(2\phi_t)\frac{N_t}{2\pi}2v_2^t\cos(2(-\phi_t+\frac{\pi}{2}))\frac{N_b}{2\pi}(1+2v_2^b\cos(2(\Delta\phi+\frac{\pi}{2}))) \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) -c_2^b\frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}v_2^t\cos(2\Delta\phi+\pi)  -c_2^b\frac{N_t}{2\pi}2v_2^t\cos(\pi)\frac{N_b}{2\pi}(1+2v_2^b\cos(2\Delta\phi+\pi)) \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) +c_2^b\frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}v_2^t\cos(2\Delta\phi)  +c_2^b\frac{N_t}{2\pi}v_2^t\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi)) \end{align}$$


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi}\int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\int \frac{d\phi_t^1}{2\pi}[1-c_2^b\cos(2\phi_t^1)] \int  \frac{d\phi_t^2}{2\pi}[1-c_2^b\cos(2\phi_t^2)]  \frac{dN^1}{d\phi}(\Psi_b+\frac{\pi}{2}) \frac{dN^2}{d\phi}(\Psi_b+\Delta\phi+\frac{\pi}{2})  \\ &=\int \frac{d\phi_t^1}{2\pi}[1-c_2^b\cos(2\phi_t^1)] \int  \frac{d\phi_t^2}{2\pi}[1-c_2^b\cos(2\phi_t^2)]   [\frac{N_b}{2\pi}(1-2v_2^b)+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(-\phi_t^1+\frac{\pi}{2}))]  [\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi))+\frac{N_t}{2\pi}\sum_{n=2,3}2v_n^t\cos(n(\Delta\phi-\phi_t^2+\frac{\pi}{2}))] \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) +\int  \frac{d\phi_t^1}{2\pi}[-c_2^b\cos(2\phi_t^1)] \int  \frac{d\phi_t^2}{2\pi}\frac{N_t}{2\pi}2v_2^t\cos(2(-\phi_t^1+\frac{\pi}{2}))\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi))  \\ &+\int \frac{d\phi_t^1}{2\pi}\int  \frac{d\phi_t^2}{2\pi}[-c_2^b\cos(2\phi_t^2)] \frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}2v_2^t\cos(2(\Delta\phi-\phi_t^2+\frac{\pi}{2}))  \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) +[-c_2^b] \frac{N_t}{2\pi}v_2^t\cos(2(\frac{\pi}{2}))\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi))  +[-c_2^b] \frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}v_2^t\cos(2(\Delta\phi+\frac{\pi}{2})) \\ &=(\frac{N_b}{2\pi})^2(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi)) +c_2^b \frac{N_t}{2\pi}v_2^t\frac{N_b}{2\pi}(1-2v_2^b\cos(2\Delta\phi))  +c_2^b \frac{N_b}{2\pi}(1-2v_2^b)\frac{N_t}{2\pi}v_2^t\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}  =(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \end{align}$$

Without fluctuations, both results stay the same. Now we introduce fluctuations into N_b \ \ \ N_t, we note that the two factors in extra terms come from different events in the case of mixed event.


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}\frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\frac{\left\langle N_tN_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_tN_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b\cos(2\Delta\phi))v_2^t \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv\int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))  -\frac{\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b\cos(2\Delta\phi))v_2^t  \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\frac{\left\langle N_tN_b\right\rangle-\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_tN_b\right\rangle-\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}c_2^b(1+2v_2^b\cos(2\Delta\phi))v_2^t \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &+\frac{\left\langle N_bN_t\right\rangle}{(2\pi)^2}c_2^b(1-2v_2^b)v_2^t\cos(2\Delta\phi)+\frac{\left\langle N_bN_t\right\rangle}{(2\pi)^2}c_2^bv_2^t(1-2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))  +\frac{\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}c_2^b(1-2v_2^b)v_2^t\cos(2\Delta\phi)+\frac{\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}c_2^bv_2^t(1-2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) +\frac{\left\langle N_bN_t\right\rangle-\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}c_2^b(1-2v_2^b)v_2^t\cos(2\Delta\phi)\\ &+\frac{\left\langle N_bN_t\right\rangle-\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}c_2^bv_2^t(1-2v_2^b\cos(2\Delta\phi)) \end{align}$$

We note that all the fluctuation terms which contains the factor [1+v_2^b\cos(2\Delta\phi)]  help to reach the observed in-plane effect, while the factor  $$[1-2v_2^b\cos(2\Delta\phi)]$$  helps to establish the observed out-of plane feature. These terms may come from a lot of places, we showed they can come from the fluctuation of overall normalization, also they come from the tube shape fluctuation if one considers a biased tube distribution, also they may come from flow coefficient fluctuations.

Uneven distribution of the tube
In this case, we only need to introduce the following substitution $$N_t \rightarrow N_t[1+d_2^t\cos(2\phi_t)] $$ Mathematically, this term contribute the same way as $$f(\phi_t)$$   term, since both $$ c_2^b, d_2^t$$  are small. We take the following approximation


 * $$\begin{align}

&[1+d_2^t\cos(2\phi_t)][1-c_2^b\cos(2\phi_t)]\simeq [1+e_2\cos(2\phi_t)] \\ &e_2 \equiv d_2^t - c_2^b \end{align}$$

Therefore the final expressions are


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}\frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &+\frac{\left\langle N_tN_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b)v_2^t\cos(2\Delta\phi)+\frac{\left\langle N_tN_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b\cos(2\Delta\phi))v_2^t \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed}\equiv\int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi} \int \delta{(\phi-\Psi_b)}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))  +\frac{\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b)v_2^t\cos(2\Delta\phi)+\frac{\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b\cos(2\Delta\phi))v_2^t  \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &+\frac{\left\langle N_tN_b\right\rangle-\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b)v_2^t\cos(2\Delta\phi)+\frac{\left\langle N_tN_b\right\rangle-\left\langle N_t\right\rangle\left\langle N_b\right\rangle}{(2\pi)^2}e_2(1+2v_2^b\cos(2\Delta\phi))v_2^t \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}\equiv \int f(\phi_t) \frac{d\phi_t}{2\pi} \int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN}{d\phi}(\phi) \frac{dN}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b^2\right\rangle}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ &-\frac{\left\langle N_bN_t\right\rangle}{(2\pi)^2}e_2(1-2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_bN_t\right\rangle}{(2\pi)^2}e_2v_2^t(1-2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed}\equiv \int f(\phi_t^1) \frac{d\phi_t^1}{2\pi} \int f(\phi_t^2) \frac{d\phi_t^2}{2\pi}\int \delta{(\phi-\Psi_b-\frac{\pi}{2})}\frac{d\phi}{2\pi}  \frac{dN^1}{d\phi}(\phi) \frac{dN^2}{d\phi}(\phi+\Delta\phi) \\ &=\frac{\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))  -\frac{\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}e_2(1-2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}e_2v_2^t(1-2v_2^b\cos(2\Delta\phi)) \\ &\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{cumulant}\equiv \left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{proper}-\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{out-of-plane}_{mixed} \\ &=\frac{\left\langle N_b^2\right\rangle-\left\langle N_b\right\rangle^2}{(2\pi)^2}(1-2v_2^b)(1-2v_2^b\cos(2\Delta\phi))+\frac{\left\langle N_t^2\right\rangle}{(2\pi)^2}\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) -\frac{\left\langle N_bN_t\right\rangle-\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}e_2(1-2v_2^b)v_2^t\cos(2\Delta\phi)-\frac{\left\langle N_bN_t\right\rangle-\left\langle N_b\right\rangle\left\langle N_t\right\rangle}{(2\pi)^2}e_2v_2^t(1-2v_2^b\cos(2\Delta\phi)) \end{align}$$

ZYAM subtraction
This part of note is based on discussions with Yogiro and Fuqiang on analytic calculation of one tube model on in-plane out-of-plane effect by using simplified flow decompositions. The equation numbers below are always referred to the archive arXiv:1010.0690 (STAR measurements).

Eq.(12) of arXiv:1010.0690 (STAR data) and the corresponding discussions show that the $$v_2^t$$ and $$v_2^b$$ used in ZYAM are in fact determined by using two particle correlation method. As pointed by Fu-Qiang, since they are associated terms such as $$$$ and/or $$<(v_2^{tr}v_2^{as})^2>$$, it implies that these quantities have already included the effects of flow fluctuations. To be specific, one may imagine that, the correlation of proper event involves terms such as


 * $$\langle N^2 v_2^{tr}v_2^{as}\rangle^{in-plane} = \langle N_b^2 (1+2v_2^{b,tr})(1+2v_2^{b,as}\cos(2\Delta\phi)\rangle

+\langle N_t^2 v_2^{t,tr}v_2^{t,as}\rangle

$$

In the above expression, one makes a difference between elliptic flow of trigger particles and that of the associated particles by superscripts $tr$ and $as$. However, in the calculation, the elliptic flow of the trigger particle, namely, $$v_2^{tr} \equiv v_2^{t,R}$$, is determined by Eq.(2) (from the paper of Voloshin), where $v_2^t$ is determined by Eq.(12),


 * $$v_2^{t} = \frac{\langle v_2^{tr}v_2^{as}\rangle}{\sqrt{\langle v_2^{as}v_2^{as}\rangle}}$$

Even though $$v_2^t$$ contains fluctuations itself, one can be readily shown that in the in-plane direction (see the discussion of arXiv:1207.6415)


 * $$v_2^{tr} \equiv v_2^{t,R}= 1$$

which can be understood intuitively.

Now we want to point out that the above discussion does not "solve" the problem. We note that the resulting ZYAM formulae of mixed event contribution depends only on $$v_2^{as}$$ which according to Eq.(12)


 * $$v_2^{as} = \sqrt{\langle v_2^{as}v_2^{as}\rangle}$$

In our simple case, where trigger particles are taken to be from the same momentum interval as associated particles, the above expression together with Eq.(1) implies that the mixed event correlation has exactly the same form as the proper event correlations. To be specific, when there is no multiplicity fluctuation, one has


 * $$\begin{align}

&\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{in-plane}=\left\langle\frac{dN_{pair}}{d\Delta\phi}\right\rangle^{mixed} =\frac{N_b^2}{(2\pi)^2}(1+2v_2^b)(1+2v_2^b\cos(2\Delta\phi))+(\frac{N_t}{2\pi})^2\sum_{n=2,3}2v_n^{t2}\cos(n\Delta\phi) \\ \end{align}$$

So why the data from STAR shows the in-plane and out-of-plane effect which is reproduced by hydrodynamics.

Look closely at the analysis in arXiv:1207.6415, the point is that the conclusion can be achieved if there is ANY mechanism to underestimate the elliptic flow coefficients in the mixed event, namely


 * $$\langle v_2^bv_2^b\rangle^{proper} > \langle v_2^bv_2^b\rangle^{mixed}$$

There is one detail in the STAR analysis which was not considered up to this point. In order to remove the effect of jet, what STAR did was to use a pseudo-rapdity separation $$\eta_{gap}=0.7$$ when pairing particles. To use this method, what was implied was that event plane is not a function of pseudo-rapidity. If there exist any decorrelation of event plane, the above procedure has to be re-considered. First, if the event plane between two pseudo-rapidity region are decorrelated, namely,


 * $$\langle\cos2(\Psi_{\eta}-\Psi_{\eta}^{ref})\rangle < 1$$

The estimated $$v_2$$ is smaller than the global value, since by definition, $$v_2$$ obtains the biggest value when calculated with respect to event plane. As a result, this leads to a difference between proper event and mixed event, which gives effectively a (working) mechanism to reproduce the in-plane and out-of-plane effect. In the manuscript, the mechanism was either multiplicity fluctuation


 * $$\langle N^2\rangle^{proper} > {\langle N\rangle^2}^{mixed}$$

or flow fluctuation


 * $$\langle v_2^bv_2^b\rangle^{proper} > {\langle v_2^b\rangle^2}^{mixed}$$

Now, one attributes it to


 * $$\langle v_2^bv_2^b\rangle^{global} > \langle v_2^bv_2^b\rangle^{\eta \; gap}$$

Effectively, all these three causes work out the same here. The event plane decorrelation were discussed in this workshop by Victor Roy. I assume that we may obtain very similar results by using SPheRIO.