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

## TL;DR

This paper uses the Landau-Khalatnikov-Fradkin transformation to explain why even zeta-values appear in multi-loop Euclidean massless correlators, revealing a new basis that removes these values from perturbative series.

## Contribution

It introduces a non-perturbative identity linking massless propagators in different gauges and constructs a basis that eliminates even zeta-values in perturbative expansions.

## Key findings

- Elimination of even zeta-values in perturbative series.
- Exact relation between hatted and standard zeta-functions.
- Insight into the gauge dependence of massless correlators.

## Abstract

The Landau-Khalatnikov-Fradkin (LKF) transformation is a powerful and elegant transformation allowing to study the gauge dependence of the propagator of charged particles interacting with gauge fields. With the help of this transformation, we derive a non-perturbative identity between massless propagators in two different gauges. From this identity, we find that the corresponding perturbative series can be exactly expressed in terms of a hatted transcendental basis that eliminates all even Euler $\zeta$-functions. This explains the mystery of even $\zeta$-values observed in multi-loop calculations of Euclidean massless correlators for almost three decades now. Our construction further allows us to derive an exact formula relating hatted and standard $\zeta$-functions to all orders of perturbation theory.