Phase-magnitude relation in gravitational lensing: Reformulation and applications of the Kramers-Kronig relation

It is known that the amplification factor, defined as the ratio of the lensed to the unlensed waveform in the frequency domain, satisfies the Kramers-Kronig (KK) relation, which connects the real and imaginary parts of the amplification factor for any lensing signal. In this work, we reformulate the KK relation in terms of the magnitude and phase of the amplification factor. Unlike the original formulation, the phase cannot be uniquely determined from the magnitude alone due to the possible presence of a Blaschke product. While this ambiguity does not arise in the case of a point-mass lens, it can appear in more complex lens models, such as those with an NFW (Navarro--Frenk--White) lens profile. As an application of our formulation, we demonstrate that the leading-order behavior of the phase in the low-frequency regime is completely determined by the leading-order behavior of the magnitude in the same regime. This reproduces known results from the literature, derived via low-frequency expansions for specific lens models. Importantly, our result does not rely on any particular lens model, highlighting a universal feature that the low-frequency behavior of the amplification factor is tightly constrained by the KK relation. As a further application, we present two examples in which the phase is constructed from a given analytic form of the magnitude using the newly derived KK relation. In particular, the second example allows for an analytic evaluation of the KK integral, yielding an explicit expression for the phase. This study offers a potentially powerful method for applying the KK relation in model-agnostic searches for lensing signals.