The measurements of the anisotropic flow-planes (\Psi_{n}) variations with rapidity, commonly known as flow-plane decorrelation, provide important insights towards understanding of the initial conditions of the produced matter in heavy-ion collisions. In this paper, we report first measurements of a novel cumulant observable $T_{n}\{ba;dc\}=\langle\langle\sin[n(\Psi^{b}_{n}-\Psi^{a}_{n})]\sin[n(\Psi^{d}_{n}-\Psi^{c}_{n})]\rangle\rangle$ for the study of the elliptic and triangular flow planes (i.e.~$n=2$ and $3$)

decorrelations using the STAR experiment datasets of Au+Au and isobar (Ru+Ru, Zr+Zr) collisions at $\sqrt{s_{_{NN}}}$\xspace = 200 GeV. The correlations of four flow-plane angles from backward ($\eta_{a}$), mid-($\eta_{b}$, $\eta_{c}$), and forward ($\eta_{d}$) rapidity regions are measured with the aim to probe the systematic longitudinal flow-plane (de)correlations over the wide rapidity range. In the mid-central collisions range of $5-30\%$ centrality, measured $T_{n}\{ba;dc\}$ values found to be very ``close-to-zero'' (slightly negative within $1-2$ sigmas) independent of the collisions system, favoring a picture of a ``random-walk'' variation of the flow-plane angles, with rapidity correlation length small compared to the rapidity region under study.  These measurements provide new information for establishing the pertinent decorrelation patterns present in the system and a quantitative estimate of a possible systematic variation of the anisotropic flow angles, such as ``twist'', between forward and backward rapidity regions, opening new possibilities for understanding the three-dimensional structure of the quark-gluon plasma created in the heavy-ion collision.
 
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Four-plane correlator observable $T_{n}$ has been measured in addition to the factorization ratio, $r^{\Psi}_{n}(\eta)$, in isobar (Ru+Ru, Zr+Zr) and Au+Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV to check the longitudinal flow-plane decorrelation phenomena in heavy-ion collisions.

 

Negative but very close-to-zero decorrelation~i.e. $$T^{\Delta\Psi_{n}^{a\rightarrow b}*\Delta\Psi_{n}^{c\rightarrow d}}\sim\langle\Delta\Psi^{a\rightarrow b}_{n}*\Delta\Psi^{c\rightarrow d}_{n}\rangle$$ of the order of $10^{-2}$-$10^{-3}$ (for $n=2$, $3$), has been observed independent of the collisions system size in the mid-central collisions with $10-50\%$ centrality. The factorization ratio observable, on the other hand, yields comparatively higher decorrelation measurements that are close-to-unity~i.e.~$(r_{2}^{\Psi}-1)\sim4\%$ in the same $10-50\%$ centrality.