Abstract:We elaborate on the method of differential equations for evaluating Feynman integrals. We focus on systems of equations for master integrals having a linear dependence on the dimensional parameter. For these systems we identify the criteria to bring them in a canonical form, recently identified by Henn, where the dependence of the dimensional parameter is disentangled from the kinematics. The determination of the transformation and the computation of the solution are obtained by using Magnus and Dyson series expansion. We apply the method to planar and non-planar two-loop QED vertex diagrams for massive fermions, and to non-planar two-loop integrals contributing to 2 → 2 scattering of massless particles. The extension to systems which are polynomial in the dimensional parameter is discussed as well.

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one-and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D=4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.

The energy of three crystallographic faces of the PbSe lattice, with Miller indices (100), (110), and (111), is computed at the density functional theory level with a double-ζ polarized basis set. The addition energy of different organic ligands on the three faces is computed at the same level both in periodic infinite slabs and in finite clusters to explore the relative affinities and the possible modifications of the stability order of the faces. Neutral ligands are found to have the greatest affinity for (110), while propionate anion binds most strongly to (111) face: in the last case the stability order of the pure surfaces can be reversed by the presence of ligands. The prevalence of different faces is related to the shape of PbSe nanoclusters, and a model is proposed to explain the excess of Pb atoms found in nonstoichiometric clusters by some experiments.

We consider the two-loop self-mass sunrise amplitude with two equal masses M and the external invariant equal to the square of the third mass m in the usual d-continuous dimensional regularization. We write a second order differential equation for the amplitude in x = m/M and show as solve it in close analytic form. As a result, all the coefficients of the Laurent expansion in (d − 4) of the amplitude are expressed in terms of harmonic polylogarithms of argument x and increasing weight. As a by product, we give the explicit analytic expressions of the value of the amplitude at x = 1, corresponding to the on-mass-shell sunrise amplitude in the equal mass case, up to the (d − 4) 5 term included.