Four-loop cusp anomalous dimension in QED

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 ${\gamma }_h$ are calculated analytically (the 4-loop one agrees with the recent numerical result [arXiv:1801.08292]). The 4-loop $C_F^3 T_F n_l$and5-loop$C_F^4 T_F n_l$termsinthecuspanomalousdimension$\Gamma(\varphi)$arecalculatedanalytically,exactlyin$\varphi$(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 ${\gamma }_h$ and to the small-$\phi$ expansion of $\mathrm{\Gamma }(\phi )$ up to ${\phi }^4$ [arXiv:1708.01221] (recently extended to ${\phi }^6$ [arXiv:1807.05145]) we now have the complete analytical 4-loop result for the Bloch–Nordsieck field anomalous dimension in QED, and the small-$\phi$ expansion of the 4-loop QED cusp anomalous dimension up to ${\phi }^6$.