Four Loop Massless Propagators: an Algebraic Evaluation of All Master Integrals

TL;DR

This work introduces and exploits the glue-and-cut (GaC) symmetry of massless propagator-type Feynman integrals to derive explicit analytic expressions for all four-loop master integrals. By coupling GaC with reduction to master integrals, the authors express complex non-primitive masters in terms of a small set of primitive watermelon-like integrals, obtaining precise ε-expansions containing zeta-values up to weight 7. The method demonstrates three- and four-loop analytic solvability and argues for a feasible extension to five loops, with significant implications for ultraviolet counterterms, beta-functions, and anomalous dimensions in MS-scheme calculations. A key outcome is a structural explanation for the absence of even zetas in certain quantities (e.g., Adler function) and a framework for understanding higher-loop transcendental weight patterns in massless p-integrals. The results are complemented by extensive tests, cross-checks with sector decomposition, and discussions on generality and future extensions.

Abstract

The old "glue--and--cut" symmetry of massless propagators, first established in [1], leads --- after reduction to master integrals is performed --- to a host of non-trivial relations between the latter. The relations constrain the master integrals so tightly that they all can be analytically expressed in terms of only few, essentially trivial, watermelon-like integrals. As a consequence we arrive at explicit analytical results for all master integrals appearing in the process of reduction of massless propagators at three and four loops. The transcendental structure of the results suggests a clean explanation of the well-known mystery of the absence of even zetas (zeta_{2n}) in the Adler function and other similar functions essentially reducible to the massless propagators. Once a reduction of massless propagators at five loops is available, our approach should be also applicable for explicit performing the corresponding five-loop master integrals.

Figures (5)

• Figure 1: two- and three-loop master p-integrals. $\varepsilon^m$ after a master label stands for the maximal term in $\varepsilon$-expansion of the master integral which one needs to know for evaluation of the contribution of the integral to the final result.
• Figure 2: all master p-integrals for the four-loop Problem. In $M_{ij}$ the digit $i$ stands for the number of (internal) lines in the integral minus five and $j$ numerates different integrals with the same value of $i$. The integrals are ordered (if read from left to right and then from top to bottom) according to their complexity. $\varepsilon^{\scriptsize m}$ after $M_{ij}$ stands for the maximal term in $\varepsilon$-expansion of $M_{ij}$ which one needs to know for evaluation of the contribution of the integral to the final result for a four-loop p-integral after reduction is done. In other words, $m$ stands for the maximal power of a spurious pole $1/\varepsilon^m$ which could appear in front of $M_{ij}$ in the process of reduction to masters.
• Figure 3: the generalized two-loop p-integral; indexes besides lines show the powers of corresponding massless propagators. $n_i$ and $a_i$ are assumed to be integers.
• Figure 4: Four finite three-loop p-integrals displaying a remarkable feature of being equal at $\varepsilon =0$ and $q^2=1$.
• Figure 5: Two ways of cutting a line in the graph $E_4$ (a,b). The generic four-linear vertex (c) and three ways transforming it into a pair of the three-linear ones connected by an auxiliary propagator (d,e,f).