New Dual Conformally Invariant Off-Shell Integrals

Evidence has recently emerged for a hidden symmetry of scattering amplitudes in N=4 super Yang-Mills theory called dual conformal symmetry. At weak coupling the presence of this symmetry has been observed through five loops, while at strong coupling the symmetry has been shown to have a natural interpretation in terms of a T-dualized AdS_5. In this paper we study dual conformally invariant off-shell four-point Feynman diagrams. We classify all such diagrams through four loops and evaluate 10 new off-shell integrals in terms of Mellin-Barnes representations, also finding explicit expressions for their infrared singularities.Comment: 21 pages, 9 figure

The two-loop electroweak bosonic corrections to sin⁡2θeffb\sin^2\theta_{\rm eff}^{\rm b}

The prediction of the effective electroweak mixing angle $\sin^2\theta_{\rm eff}^{\rm b}$ in the Standard Model at two-loop accuracy has now been completed by the first calculation of the bosonic two-loop corrections to the $Z{\bar b}b$ vertex. Numerical predictions are presented in the form of a fitting formula as function of $M_Z, M_W, M_H, m_t$ and $\Delta{\alpha}$, ${\alpha_{\rm s}}$. For central input values, we obtain a relative correction of $\Delta\kappa_{\rm b}^{(\alpha^2,\rm bos)} = -0.9855 \times 10^{-4}$, amounting to about a quarter of the fermionic corrections, and corresponding to $\sin^2\theta_{\rm eff}^{\rm b} = 0.232704$. The integration of the corresponding two-loop vertex Feynman integrals with up to three dimensionless parameters in Minkowskian kinematics has been performed with two approaches: (i) Sector decomposition, implemented in the packages FIESTA 3 and SecDec 3, and (ii) Mellin-Barnes representations, implemented in AMBRE 3/MB and the new package MBnumerics.Comment: 14 pp; v2: some explanations and Tab.2 added, version published in PL

On the Resolution of Singularities of Multiple Mellin-Barnes Integrals

One of the two existing strategies of resolving singularities of multifold Mellin-Barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modified form. The corresponding algorithm is implemented as a Mathematica code MBresolve.mComment: LaTeX, 10 page

Three-loop planar master integrals for heavy-to-light form factors

We calculate analytically the three-loop planar master integrals relevant for heavy-to-light form factors using the method of differential equations. After choosing a proper canonical basis, the boundary conditions are easy to be determined, and the solution of differential equations is greatly simplified. The results for seventy-one master integrals at general kinematics are all expressed in terms of harmonic polylogarithms.Comment: 18 pages, 2 figure

Hidden Beauty in Multiloop Amplitudes

Planar L-loop maximally helicity violating amplitudes in N = 4 supersymmetric Yang-Mills theory are believed to possess the remarkable property of satisfying iteration relations in L. We propose a simple new method for studying the iteration relations for four-particle amplitudes which involves the use of certain linear differential operators and eliminates the need to fully evaluate any loop integrals. We carry out this procedure in explicit detail for the two-loop amplitude and argue that this method can be used to prove the iteration relations to all loops up to polynomials in logarithms.Comment: 21 pages, harvmac; v2: minor change

Towards a four-loop form factor

The four-loop, two-point form factor contains the first non-planar correction to the lightlike cusp anomalous dimension. This anomalous dimension is a universal function which appears in many applications. Its planar part in N = 4 SYM is known, in principle, exactly from AdS/CFT and integrability while its non-planar part has been conjectured to vanish. The integrand of the form factor of the stress-tensor multiplet in N = 4 SYM including the non-planar part was obtained in previous work. We parametrise the difficulty of integrating this integrand. We have obtained a basis of master integrals for all integrals in the four-loop, two-point class in two ways. First, we computed an IBP reduction of the integrand of the N = 4 form factor using massive computer algebra (Reduze). Second, we computed a list of master integrals based on methods of the Mint package, suitably extended using Macaulay2 / Singular. The master integrals obtained in both ways are consistent with some minor exceptions. The second method indicates that the master integrals apply beyond N = 4 SYM, in particular to QCD. The numerical integration of several of the master integrals will be reported and remaining obstacles will be outlinedComment: 9 Pages, Radcor/Loopfest 2015 Proceeding

Computer algebra tools for Feynman integrals and related multi-sums

In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite nested integrals or sums. Furthermore, one seeks for solutions of coupled systems of linear differential equations, that can be represented in terms of indefinite nested sums (or integrals). In this article we elaborate the main tools and the corresponding packages, that we have developed and intensively used within the last 10 years in the course of our QCD-calculations