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The cross section for back-to-back hadron pair production in $e^+e^-$annihilation provides access to the dihadron fragmentation functions (DiFF)needed to extract nucleon parton distribution functions from the semi-inclusivedeep inelastic scattering (SIDIS) experiments with two detected final statehadrons. Particular attention is given to the so-called interference DiFF(IFF), which makes it possible to extract the transversity parton distributionof the nucleon in the collinear framework. However, previously unnoticeddiscrepancies were recently highlighted between the definitions of the IFFsappearing in the collinear kinematics when reconstructed from DiFFs enteringthe unintegrated fully differential cross sections of SIDIS and $e^+e^-$annihilation processes. In this work, to clarify this problem we rederive thefully differential cross section for $e^+e^-$ annihilation at the leading-twistapproximation. We find a mistake in the definition of the kinematics in theoriginal expression that systematically affects a subset of terms and thatleads to two significant consequences. First, the discrepancy between the IFFdefinitions in the cross sections for SIDIS and $e^+e^-$ annihilation isresolved. Second, the previously derived azimuthal asymmetry for accessing thehelicity dependent DiFF $G_1^\perp$ in $e^+e^-$ annihilation vanishes, whichexplains the nonobservation of this asymmetry in the recent experimentalsearches by the ${\tt BELLE}$ Collaboration. We discuss the recently proposedalternative option to extract $G_1^\perp$.

Semi-inclusive production of two back-to-back hadron pairs in $e^+e^-$ annihilation revisited