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The tune-out wavelength is usually viewed a zero in the frequency-dependent polarizability [1,2]. This view is appropriate for an atom in an optical lattice that is fixed in space. However, for an atom interacting with a traveling plane wave from a laser, it is more appropriate to view the tune-out wavelength as a zero in the Rayleigh scattering cross section for coherent scattering. In lowest order, the two approaches are equivalent, but not when higher-order retardation corrections are taken into account. This paper presents a development of the theory, starting from the relativistic scattering matrix of QED to obtain a formulation of the problem in the velocity gauge [3]. Gauge invariance is discussed, and an equivalent length form is obtained for the leading retardation correction for S-states. The $xp_z$ retardation correction to the tune-out wavelength of helium near 304 nm is calculated to be $0.000\,560\,0236$ nm.
[1] B. M. Henson et al., Phys. Rev. Lett. 115, 043004 (2015).
[2] Y.-H. Zhang et al., Phys. Rev. A 93, 052516 (2016).
[3] G. W. F. Drake, J. Manalo and P.-P. Zhang, Hyperfine Int. submitted (2019).
