Double interference effects in Higgs precision tests in the $e^-e^+\to \nu\bar \nu h$ process

Higgs precision program at future lepton collider aims at (sub) percent level precision measurement of the Higgs properties, shedding light to new physics through the Higgs lamppost. Amongst many exclusive Higgs channels that can be measured precisely, the $WW$-fusion to Higgs with subsequent decays into $b\bar b$ final state are of particular importance. This channel provide leading constrains on Higgs total width in the $\kappa$-framework and greatly improves the constraints in the EFT framework as a distinct production mode other than the Higgsstrahlung process. We argue in this paper that, there are two interference effects both affects the physical information one can extract from the precision measurements. One takes place at quantum level from the interference between the two amplitudes that amounts to $-10\%$ of the $WW$-fusion signal strength, failing to take into account which will result in a $4$-$5\sigma$ discrepancy between theory and measurement. The other takes place at the classical level from the global fitting procedure where the $ZH$ process is the dominant background with its cross section around six times larger than the $WW$-fusion signal. Despite that $ZH$ process can be measured to great precision at future lepton colliders, failing take this interplay in the coupling extraction will result in a 100\% too aggressive constraints on $\kappa_W$, the Higgs coupling to $W$-boson pairs. This effect will also be important for lepton colliders running at slightly higher energies where the phase space overlap are still sizable between the two processes.