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The Landau-Khalatnikov-Fradkin (LKF) transformation is a powerful and eleganttransformation allowing to study the gauge dependence of the propagator ofcharged particles interacting with gauge fields. With the help of thistransformation, we derive a non-perturbative identity between masslesspropagators in two different gauges. From this identity, we find that thecorresponding perturbative series can be exactly expressed in terms of a hattedtranscendental basis that eliminates all even Euler $\zeta$-functions. Thisexplains the mystery of even $\zeta$-values observed in multi-loop calculationsof Euclidean massless correlators for almost three decades now. Ourconstruction further allows us to derive an exact formula relating hatted andstandard $\zeta$-functions to all orders of perturbation theory.

Landau-Khalatnikov-Fradkin transformation and the mystery of even $\zeta$-values in Euclidean massless correlators