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As a prerequisite of probing physics beyond the Standard Model (BSM) ofparticle physics, it is imperative to perform calculation of parity violatingelectric dipole ($E1_{PV}$) amplitudes within 0.5\% accuracy in atomic systems.Latest high precision calculations of $E1_{PV}$ of the $6s ~ ^2S \rightarrow 7s~ ^2S $ transition in $^{133}$Cs by three different groups claim achieving itsaccuracy below 0.5\%, but such claims contradict on the view that their finalvalues differ by 1\%. One of the major issues in these calculations is theopposite signs among the core correlation contribution from different worksleading to 200\% difference in its value. In a review [Rev. Mod. Phys. 90,025008 (2018)], a Letter [Phys. Rev. D 103, L111303 (2021)] and a Comment[Phys. Rev. D 105, 018301 (2022)], reliability of all these calculations arestrongly contended. We unearthed here the underlying reason for getting signdiscrepancies in various works by investigating how different electroncorrelation effects are encapsulated through the undertaken methods in theabove works to determine $E1_{PV}$. Detailed discussions presented in this workwould help in guiding theoretical studies to improve accuracy of $E1_{PV}$ inatomic systems to probe BSM physics.

Deciphering Core, Valence and Double-Core-Polarization Contributions to Parity Violating Amplitudes in $^{133}$Cs using Different Methods