Z. Bern

An Automated Implementation of On-Shell Methods for One-Loop Amplitudes

We present the first results from BlackHat, an automated C++ program for calculating one-loop amplitudes. The program implements the unitarity method and on-shell recursion to construct amplitudes. As input to the calculation, it uses compact analytic formulae for tree amplitudes for four-dimensional helicity states. The program performs all related computations numerically. We make use of recently developed on-shell methods for evaluating coefficients of loop integrals, introducing a discrete Fourier projection as a means of improving efficiency and numerical stability. We illustrate the numerical stability of our approach by computing and analyzing six-, seven-, and eight-gluon amplitudes in QCD and comparing against previously obtained analytic results.

The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory

We give a representation of the parity-even part of the planar two-loop six-gluon MHV amplitude of N = 4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the ABDK/BDS all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky and Sokatchev in arXiv:0803.1466 [hep-th]. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides non-trivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.We give a representation of the parity-even part of the planar two-loop six-gluon maximally helicity violating (MHV) amplitude of N=4 super-Yang-Mills theory, in terms of loop-momentum integrals with simple dual conformal properties. We evaluate the integrals numerically in order to test directly the Anastasiou-Bern-Dixon-Kosower/Bern-Dixon-Smirnov all-loop ansatz for planar MHV amplitudes. We find that the ansatz requires an additive remainder function, in accord with previous indications from strong-coupling and Regge limits. The planar six-gluon amplitude can also be compared with the hexagonal Wilson loop computed by Drummond, Henn, Korchemsky, and Sokatchev in arXiv:0803.1466. After accounting for differing singularities and other constants independent of the kinematics, we find that the Wilson loop and MHV-amplitude remainders are identical, to within our numerical precision. This result provides nontrivial confirmation of a proposed n-point equivalence between Wilson loops and planar MHV amplitudes, and suggests that an additional mechanism besides dual conformal symmetry fixes their form at six points and beyond.

Three-Loop Superfiniteness of N=8 Supergravity

We construct the three-loop four-point amplitude of N=8 supergravity using the unitarity method. The amplitude is ultraviolet finite in four dimensions. Novel cancellations, not predicted by traditional superspace power-counting arguments, render its degree of divergence in D dimensions to be no worse than that of N=4 super-Yang-Mills theory -- a finite theory in four dimensions. Similar cancellations can be identified at all loop orders in certain unitarity cuts, suggesting that N=8 supergravity may be a perturbatively finite theory of quantum gravity.

Cancellations Beyond Finiteness inN=8Supergravity at Three Loops

We construct the three-loop four-point amplitude of N=8 supergravity using the unitarity method. The amplitude is ultraviolet finite in four dimensions. Novel cancellations, not predicted by traditional superspace power-counting arguments, render its degree of divergence in D dimensions to be no worse than that of N=4 super-Yang-Mills theory -- a finite theory in four dimensions. Similar cancellations can be identified at all loop orders in certain unitarity cuts, suggesting that N=8 supergravity may be a perturbatively finite theory of quantum gravity.

Simplifying multiloop integrands and ultraviolet divergences of gauge theory and gravity amplitudes

We use the duality between color and kinematics to simplify the construction of the complete four-loop four-point amplitude of N = 4 super-Yang-Mills theory, including the nonplanar contributions. The duality completely determines the amplitudes integrand in terms of just two planar graphs. The existence of a manifestly dual gauge-theory amplitude trivializes the construction of the corresponding N = 8 supergravity integrand, whose graph numerators are double copies (squares) of the N = 4 super-Yang-Mills numerators. The success of this procedure provides further nontrivial evidence that the duality and double-copy properties hold at loop level. The new form of the four-loop four-point supergravity amplitude makes manifest the same ultraviolet power counting as the corresponding N = 4 super-Yang-Mills amplitude. We determine the amplitudes ultraviolet pole in the critical dimension of D = 11/2, the same dimension as for N = 4 super-Yang-Mills theory. Strikingly, exactly the same combination of vacuum integrals (after simplification) describes the ultraviolet divergence of N = 8 supergravity as the subleading-in-1/N{sub c}{sup 2} single-trace divergence in N = 4 super-Yang-Mills theory.

Next-to-Leading Order W 5-Jet Production at the LHC

We present next-to-leading order QCD predictions for the total cross section and for a comprehensive set of transverse-momentum distributions in W + 5-jet production at the Large Hadron Collider. We neglect the small contributions from subleading-color virtual terms, top quarks and some terms containing four quark pairs. We also present ratios of total cross sections, and use them to obtain an extrapolation formula to an even larger number of jets. We include the decay of the

Progress on Ultraviolet Finiteness of Supergravity

W

Continuum Regularization of {QCD}

boson into leptons. This is the first such computation with six final-state vector bosons or jets. We use BlackHat together with SHERPA to carry out the computation.

Continuum Regularization of Quantum Field Theory

In this lecture we summarize recent calculations pointing to the possible ultraviolet finiteness of N = 8 supergravity in four dimensions. We outline the modern unitarity method, which enables multiloop calculations in this theory and allows us to exploit a remarkable relation between tree-level gravity and gauge-theory amplitudes. We also describe a link between observed cancellations at loop level and improved behavior of tree-level amplitudes under large complex deformations of momenta.

Type I superstrings in dimensions less than ten (II): Finiteness

Abstract We introduce a new stochastic regularization for continuum gauge theory. The scheme is a covariant derivative regularization of the Parisi-Wu Langevin equation, or equivalently, the Schwinger-Dyson equations. The regularized formulation is manifestly Lorentz invariant, gauge-invariant, ghost-free and finite to all orders. We verify a vanishing gluon mass at one loop.