Matter Dependence of the Three-Loop Soft Anomalous Dimension Matrix

The resummation of soft gluon exchange for QCD hard scattering requires a matrix of anomalous dimensions, which has been computed through two loops. The two-loop matrix is proportional to the one-loop matrix. Recently there have been proposals that this proportionality extends to higher loops. One can test such proposals by computing the dependence of this matrix on the matter content in a generic gauge theory. It is shown that for the matter-dependent part the proportionality extends to three loops for arbitrary massless processes.Comment: 5 pages, 2 figures; v2 minor clarifications, references adde

Twistor String Theory and QCD

I review recent progress in using twistor-inspired methods to compute perturbative scattering amplitudes in gauge theory, for application to collider physics.Comment: 17 pages, 9 figures. Presented at the International Europhysics Conference on High Energy Physics (Lisbon, July, 2005

The Principle of Maximal Transcendentality and the Four-Loop Collinear Anomalous Dimension

We use the principle of maximal transcendentality and the universal nature of subleading infrared poles to extract the analytic value of the four-loop collinear anomalous dimension in planar ${\cal N}=4$ super-Yang-Mills theory from recent QCD results, obtaining $\hat{\cal G}_{0}^{(4)} = - 300 \zeta_7 - 256 \zeta_2 \zeta_5 - 384 \zeta_3 \zeta_4$. This value agrees with a previous numerical result to within 0.2 percent. It also provides the Regge trajectory, threshold soft anomalous dimension and rapidity anomalous dimension through four loops.Comment: 12 pages, no figures. v2, references adde

Ultraviolet Behavior of N=8 Supergravity

In these lectures I describe the remarkable ultraviolet behavior of N=8 supergravity, which through four loops is no worse than that of N=4 super-Yang-Mills theory (a finite theory). I also explain the computational tools that allow multi-loop amplitudes to be evaluated in this theory - the KLT relations and the unitarity method - and sketch how ultraviolet divergences are extracted from the amplitudes.Comment: 30 pages, 12 figures; lectures presented at International School of Subnuclear Physics, 47th Course, Erice Sicily, August 29-September 7, 200

Recent Developments in Perturbative QCD

I review recent progress in perturbative QCD on two fronts: extending next-to-next-to-leading order QCD corrections to a broader range of collider processes, and applying twistor-space methods (and related spinoffs) to computations of multi-parton scattering amplitudes.Comment: 13 pages, 7 figures. Talk presented at 13th International Workshop on Deep Inelastic Scattering (DIS 05), Madison, Wisconsin, April, 200

Gluing Ladders into Fishnets

We use integrability at weak coupling to compute fishnet diagrams for four-point correlation functions in planar $\phi^4$ theory. The results are always multi-linear combinations of ladder integrals, which are in turn built out of classical polylogarithms. The Steinmann relations provide a powerful constraint on such linear combinations, leading to a natural conjecture for any fishnet diagram as the determinant of a matrix of ladder integrals.Comment: 6 pages, 4 figures; v2, minor corrrections, e.g. in fig. 2 captio

Bootstrapping an NMHV amplitude through three loops

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function $U$, which is closely related to the parity-even part of the ratio function $V$, is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios $(u,v,w)$, as well as on a few two-dimensional planes. Finally, we present an empirical formula for $V$ in terms of elements of the coproduct of the six-gluon MHV remainder function $R_6$ at one higher loop, which works through three loops for $V$ (four loops for $R_6$).Comment: 69 pages, 12 figures, 1 table, 3 ancillary files; v2, minor typo's correcte