Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

TL;DR

This paper computes the four-loop gluon anomalous dimension for arbitrary Lorentz spin, focusing on the transcendental part, and provides results that improve the precision of parton distribution function evolution in QCD.

Contribution

It introduces a novel analytic construction of the $eta$-independent part of the four-loop gluon anomalous dimension using advanced algorithms and supersymmetric theory insights.

Findings

Derived the $eta$-independent $oldsymbol{ ext{zeta}(3)}$ part of the four-loop anomalous dimension.

Established new self-tuning relations for the singlet anomalous dimension matrix.

Provided exact contributions to the rational part of the four-loop splitting function.

Abstract

We consider the anomalous dimension $\gamma_{gg}^{(3)}(N)$ of the twist-two gluon operator of arbitrary Lorentz spin $N$ in the quark flavor singlet sector of a general gauge theory at four loops and construct its contribution proportional to $\zeta(3)$ in analytic form by applying the Lenstra-Lenstra-Lov\'{a}sz algorithm to the available low-$N$ moments. We exploit generalized Gribov-Liptov reciprocity, establish new self-tuning relations for the anomalous dimension matrix of the singlet sector, and inject information from $\mathcal{N}=1,4$ supersymmetric Yang-Mills theories. We also present the contribution to the rational part of $\gamma_{gg}^{(3)}(N)$ with color factor $C_F^2n_f^2$. Exact contributions to the four-loop splitting function $P_{gg}^{(3)}(x)$ hence resulting via inverse Mellin transformation help us to reduce theoretical uncertainties in scaling violations of parton distribution functions in QCD.