Evaluating single-scale and/or non-planar diagrams by differential equations

We apply a recently suggested new strategy to solve differential equations for Feynman integrals. We develop this method further by analyzing asymptotic expansions of the integrals. We argue that this allows the systematic application of the differential equations to single-scale Feynman integrals. Moreover, the information about singular limits significantly simplifies finding boundary constants for the differential equations. To illustrate these points we consider two families of three-loop integrals. The first are form-factor integrals with two external legs on the light cone. We introduce one more scale by taking one more leg off-shell, $p_2^2\neq 0$. We analytically solve the differential equations for the master integrals in a Laurent expansion in dimensional regularization with $\epsilon=(4-D)/2$. Then we show how to obtain analytic results for the corresponding one-scale integrals in an algebraic way. An essential ingredient of our method is to match solutions of the differential equations in the limit of small $p_2^2$ to our results at $p_2^2\neq 0$ and to identify various terms in these solutions according to expansion by regions. The second family consists of four-point non-planar integrals with all four legs on the light cone. We evaluate, by differential equations, all the master integrals for the so-called $K_4$ graph consisting of four external vertices which are connected with each other by six lines. We show how the boundary constants can be fixed with the help of the knowledge of the singular limits. We present results in terms of harmonic polylogarithms for the corresponding seven master integrals with six propagators in a Laurent expansion in $\epsilon$ up to weight six.Comment: 27 pages, 2 figure

Protecting forests or saving trees?: The EU's regulatory approach to global deforestation

Abstract Given the poor problem‐solving effectiveness of international environmental law and a decline in multilateralism, unilateral approaches to halting deforestation globally have acquired increasing significance. Within this context, the European Union (EU) has adopted the EU Timber Regulation and the first and second Renewable Energy Directive, all of which have extraterritorial implications. Given continuing high rates of deforestation, the European Union has also been assessing which mix of instruments might prove more effective in preventing global deforestation. This article contextualizes these regulatory endeavours, analyses the specific interactions and features of existing instruments under EU law, such as due diligence obligations, sustainability criteria, certification schemes and bilateral agreements, and discusses the challenges arising from World Trade Organization (WTO) law regarding potentially more effective mandatory instruments. It finds that while the existing framework contains promising pathways for future regulation, designing sustainability criteria that are technically meaningful and also feasible from the perspective of WTO law requires greater policy coherence

A planar four-loop form factor and cusp anomalous dimension in QCD

We compute the fermionic contribution to the photon-quark form factor to four-loop order in QCD in the planar limit in analytic form. From the divergent part of the latter the cusp and collinear anomalous dimensions are extracted. Results are also presented for the finite contribution. We briefly describe our method to compute all planar master integrals at four-loop order.Comment: 19 pages, 3 figures, v2: typo in (2.3) fixed and coefficients in (2.6) corrected; references added and correcte

Emergence of turbulence in an oscillating Bose-Einstein condensate

We report on the experimental observation of vortices tangle in an atomic BEC of Rb-87 atoms when an external oscillatory perturbation is introduced in the trap. The vortices tangle configuration is a signature of the presence of a turbulent regime in the cloud. We also show that this turbulent cloud has suppression of the aspect ratio inversion typically observed in quantum degenerate bosonic gases during free expansion. Instead, the cloud expands keeping the ratio between their axis constant. Turbulence in atomic superfluids may constitute an alternative system to investigate decay mechanisms as well as to test fundamental theoretical aspects in this field.Comment: accepted for Phys. Rev. Let