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We address the question of the mismatch between the zero momentum limits ofthe transverse and longitudinal dielectric functions for a fixed direction ofthe driving field observed in the cuprates. This question translates to whetheror not the order in which the longitudinal and transverse momentum transfersare taken to zero commute. While the two limits commute for both isotropic andanisotropic Drude metals, we argue that a scaleless vertex interaction thatdepends solely on the angle between scattered electron momenta is sufficient toachieve non-commutativity of the two limits even for a system that isinherently isotropic. We demonstrate this claim for a simple case of the Drudeconductivity modified by electron-boson interactions through appropriate vertexcorrections, and outline possible consequences of our result to optical andelectron energy loss spectroscopy (EELS) measurements close to zero momentumtransfer

Inequivalence of the zero-momentum Limits of Transverse and Longitudinal Dielectric Response in the Cuprates