Observance of such a small close-to-zero %$T^{\Delta\Psi_{n}^{a\rightarrow b}*\Delta\Psi_{n}^{c\rightarrow d}}$

$T_{n}$ measurements with a comparatively larger close-to-unity factorization ratio measurements from the experimental data implies that the genuine longitudinal flow-plane decorrelation is happening on a much smaller scale favouring a picture of ``random-walk" variation of the flow-planes with $\eta$. In such a picture, one can think of large number of small rapidity steps ($\Delta\eta$), independent and uncorrelated, sandwiched together to constitute the overall $\eta-$range such that the azimuthal orientation of flow-planes at each individual step is randomly tilted by a small amount with respect to that at the previous step. Such a random-walk-like variation of $\Psi_{n}$'s among different rapidity steps eventually results in a zero expectation value for the overall tilt ($\langle\Delta\Psi_{n}\rangle$) with the variance (i.e.~$\Psi_{n}$ spread) linearly increasing with the step size ($\langle(\Delta\Psi_{n})^{2}\rangle\propto\Delta\eta$). 

 

Furthermore, in the ``random-walk" variation, for example if flow-plane randomly changes orientation say to the right or to the left in every step (say 0.2 or so) in rapidity, the factorization ratio observable would still be deviated from the unity but $T_{n}$ estimates a very close-to-zero value as there would be no consistent change occurring on the orientation of flow-planes along the rapidity. On the other hand, if there is a consistent change happening in the orientation of flow-planes (say there is some sort of physics driving such change) in going from backward to forward in rapidity then one would expect a non zero values for $T_{n}$ which would then correspond to the picture of ``monotonic" or ``C"-shape twist establishing the pertinent decorrelation patterns presented in the experimental data. 

 

Slight negative result observed for $T_{n}$ although it is very much consistent with zero is probably because the flow-planes at mid-rapidities (defined by TPCs) are slightly fluctuating with respect to the real reaction planes while that at forward and backward rapidities (defined either by EPDs or BBCs) are mostly fixed by the spectators. This could result in a (-)ve signal because both planes inside the TPC (i.e. at~$+\eta$ TPC and~$-\eta$ TPC) would be shifted in the same direction.