Derivation Notes on ZYAM subtraction in azimuthal angle dependence of two particle correlation

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).

ZYAM subtraction
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 Fuqiang, since they are associated with 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) (originally from the paper of Voloshin when flow fluctuation was not a concern), 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 the value of $$v_2^t$$ contains fluctuations by 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 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 the multiplicity fluctuation


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

or the flow fluctuation


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

Now, one attributes it to event plane decorrelation


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

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

One may also consolidate the above arguments the other way around as follows. If $$v_2$$ is calculated from two particle correlation and therefore contains perfectly all the information on flow fluctuation (and therefore together with non-flow), then subtraction should also perfectly remove everything, so that nothing but white noise will be left. In order to exclude non-flow from the mixed event subtraction, STAR introduced a pseudo-rapidity gap in their analysis procedure, our argument is that this procedure causes the event plane decorrelation to come into play and as a consequence, this collective phenomenon has a bigger impact on the subtracted two particle correlation than that from non-flow.