Four-loop cusp anomalous dimension in QED
Andrey Grozin
TL;DR
The work addresses high-order corrections to the Bloch–Nordsieck field anomalous dimension $\gamma_h$ and the QCD/QED cusp anomalous dimension $\Gamma(\varphi)$ by exploiting Wilson-line exponentiation and single-web diagrams to obtain analytic, exact expressions for the $CF^{L-1} T_F n_l \alpha_s^L$ terms up to $L=5$. It provides the $L=4$ verification against numerical results and derives the $\varphi$-dependent form of $\Gamma(\varphi)$ at four loops, including the large-$\varphi$ asymptotics that match known $\Gamma_l$ behavior. In the QED limit, the results yield a complete analytic 4-loop expression for $\gamma_h$ and a detailed $\varphi$-dependent expansion for $\Gamma(\varphi)$ up to $\varphi^4$, with further terms constrained by known polylogarithmic constants. These analytic results enhance precision in infrared structure studies, enable stringent cross-checks with numerical evaluations, and strengthen connections between HQET, cusp physics, and the static quark potential.
Abstract
The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the HQET field anomalous dimension $γ_h$ are calculated analytically (the 4-loop one agrees with the recent numerical result [arXiv:1801.08292]). The 4-loop $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the cusp anomalous dimension $Γ(\varphi)$ are calculated analytically, exactly in $\varphi$ (the $\varphi\to\infty$ asymptotics of the 4-loop one agrees with the recent numerical result [arXiv:1707.08315]). Combining these results with the recent 4-loop $d_{FF} n_l$ contributions to $γ_h$ and to the small-$\varphi$ expansion of $Γ(\varphi)$ up to $\varphi^4$ [arXiv:1708.01221] (recently extended to $\varphi^6$ [arXiv:1807.05145]) we now have the complete analytical 4-loop result for the Bloch--Nordsieck field anomalous dimension in QED, and the small-$\varphi$ expansion of the 4-loop QED cusp anomalous dimension up to $\varphi^6$.