Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $\epsilon$-expansion series with numerical coefficients. The algorithm is based on using generalized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, $p^2=9 m^2$, in an expansion in $\epsilon$ up to $\epsilon^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $\epsilon$-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.Comment: Discussion extende

The Four-Loop Dressing Phase of N=4 SYM

We compute the dilatation generator in the su(2) sector of planar N=4 super Yang-Mills theory at four-loops. We use the known world-sheet scattering matrix to constrain the structure of the generator. The remaining few coefficients can be computed directly from Feynman diagrams. This allows us to confirm previous conjectures for the leading contribution to the dressing phase which is proportional to zeta(3).Comment: 19 pages, v2: referenced adde

Four-loop quark form factor with quartic fundamental colour factor

We analytically compute the four-loop QCD corrections for the colour structure $(d_F^{abcd})^2$ to the massless non-singlet quark form factor. The computation involves non-trivial non-planar integral families which have master integrals in the top sector. We compute the master integrals by introducing a second mass scale and solving differential equations with respect to the ratio of the two scales. We present details of our calculational procedure. Analytical results for the cusp and collinear anomalous dimensions, and the finite part of the form factor are presented. We also provide analytic results for all master integrals expanded up to weight eight.Comment: 16 pages, 2 figure

Three-loop massive form factors: complete light-fermion and large-NcN_c corrections for vector, axial-vector, scalar and pseudo-scalar currents

We compute the three-loop QCD corrections to the massive quark form factors with external vector, axial-vector, scalar and pseudo-scalar currents. All corrections with closed loops of massless fermions are included. The non-fermionic part is computed in the large-$N_c$ limit, where only planar Feynman diagrams contribute.Comment: 33 page

On orthogonal expansions of the space of vector functions which are square-summable over a given domain and the vector analysis operators

The Hilbert space L2(omega) of vector functions is studied. A breakdown of L2(omega) into orthogonal subspaces is discussed and the properties of the operators for projection onto these subspaces are investigated from the standpoint of preserving the differential properties of the vectors being projected. Finally, the properties of the operators are examined

Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in ϵ\epsilon

Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, $d$. The method was used to obtain analytical expressions for two missing constants in the $\epsilon$-expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the $\epsilon$-expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in $\epsilon$ of the three-loop non-planar massless propagator diagram. Only multiple $\zeta$ values at integer points are present in our result.Comment: Talk given at the International Workshop `Loops and Legs in Quantum Field Theory' (April 25--30, 2010, W\"orlitz, Germany)

On a general analytical formula for U_q(su(3))-Clebsch-Gordan coefficients

We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U_q(su(3)). We obtain a very compact general analytical formula for the U_q(su(3)) CGCs in terms of the U_q(su(2)) Wigner 3nj-symbols.Comment: 9 pages, LaTeX; to be published in Yad. Fiz. (Phys. Atomic Nuclei), (2001