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 $C_F^3 T_F n_l$ and 5-loop $C_F^4 T_F n_l$ terms in the cusp anomalous dimension $\Gamma(\varphi)$ 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 $\gamma_h$ and to the small-$\varphi$ expansion of $\Gamma(\varphi)$ 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 $\varphi^6$.Comment: 9 pages; v2: minor typos fixed; v3: some explanations and 3 references added; v4: QED cusp anomalous dimension extended to $\varphi^6$, comparison with the quark-antiquark potential added, several typos fixed, 3 references adde

Effective field theories

A pedagogical introduction to low-energy effective field theories. In some of them, heavy particles are "integrated out" (a typical example - the Heisenberg-Euler EFT); in some heavy particles remain but some of their degrees of freedom are "integrated out" (Bloch-Nordsieck EFT). A large part of these lectures is, technically, in the framework of QED. QCD examples, namely, decoupling of heavy flavors and HQET, are discussed only briefly. However, effective field theories of QCD are very similar to the QED case, there are just some small technical complications: more diagrams, color factors, etc. The method of regions provides an alternative view at low-energy effective theories; it is also briefly introduced.Comment: Lectures at Baikal school on elementary particle physics and astrophysics 2019 and Helmholtz - DIAS school "quantum field theory at the limits" 2019; v2: ref. [1] added, minor text changes. arXiv admin note: text overlap with arXiv:0908.439

On the Casimir scaling violation in the cusp anomalous dimension at small angle

We compute the four-loop $n_f$ contribution proportional to the quartic Casimir of the QCD cusp anomalous dimension as an expansion for small cusp angle $\phi$. This piece is gauge invariant, violates Casimir scaling, and first appears at four loops. It requires the evaluation of genuine non-planar four-loop Feynman integrals. We present results up to ${\mathcal O}(\phi^4)$. One motivation for our calculation is to probe a recent conjecture on the all-order structure of the cusp anomalous dimension. As a byproduct we obtain the four-loop HQET wave function anomalous dimension for this color structure.Comment: 13 pages, 2 figures, 1 ancillary file; v2: journal versio