David C. Dunbar

One-loop n-point gauge theory amplitudes, unitarity and collinear limits

Abstract We present a technique which utilizes unitarity and collinear limits to construct ansatze for one-loop amplitudes in gauge theory. As an example, we obtain the one-loop contribution to amplitudes for n -gluon scattering in N = 4 supersymmetric Yang-Mills theory with the helicity configuration of the Parke-Taylor tree amplitudes. We prove that our N = 4 ansatz is correct using general properties of the relevant one-loop n -point integrals. We also give the “splitting amplitudes” which govern the collinear behavior of one-loop helicity amplitudes in gauge theories.

Fusing gauge theory tree amplitudes into loop amplitudes

Abstract We identify a large class of one-loop amplitudes for massless particles that can be constructed via unitarity from tree amplitudes, without any ambiguities. One-loop amplitudes for massless supersymmetric gauge theories fall into this class; in addition, many non-supersymmetric amplitudes can be rearranged to take advantage of the result. As applications, we construct the one-loop amplitudes for n -gluon scattering in N = 1 supersymmetric theories with the helicity configuration of the Parke-Taylor tree amplitudes, and for six-gluon scattering in N = 4 super-Yang-Mills theory for all helicity configurations.

On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences

String theory implies that field theories containing gravity are in a certain sense ‘products’ of gauge theories. We make this product structure explicit up to two loops for the relatively simple case of N = 8 supergravity four-point amplitudes, demonstrating that they are ‘squares’ of N = 4 super-Yang-Mills amplitudes. This is accomplished by obtaining an explicit expression for the Ddimensional two-loop contribution to the four-particle S-matrix for N = 8 supergravity, which we compare to the corresponding N = 4 Yang-Mills result. From these expressions we also obtain the two-loop ultraviolet divergences in dimensions D = 7 through D = 11. The analysis relies on the unitarity cuts of the two theories, many of which can be recycled from a one-loop computation. The two-particle cuts, which may be iterated to all loop orders, suggest that squaring relations between the two theories exist at any loop order. The loop-momentum power-counting implied by our two-particle cut analysis indicates that in four dimensions the first four-point divergence in N = 8 supergravity should appear at five loops, contrary to the earlier expectation, based on superspace arguments, of a three-loop counterterm.

One-loop self-dual and N = 4 super Yang-Mills

Abstract We conjecture a simple relationship between the one-loop maximally helicity violating gluon amplitudes of ordinary QCD (all helicities identical) and those of N = 4 supersymmetric Yang-Mills (all but two helicities identical). Because the amplitudes in self-dual Yang Mills have been shown to be the same as the maximally helicity violating ones in QCD, this conjecture implies that they are also related to the maximally helicity violating ones of N = 4 supersymmetric Yang-Mills. We have an explicit proof of the relation up to the six-point amplitude; for amplitudes with more external legs, it remains a conjecture. A similar conjecture relates amplitudes in self-dual gravity to maximally helicity violating N = 8 supergravity amplitudes.

String based methods in perturbative gravity

Abstract String theory implies a relatively modest growth in computational complexity for perturbative gravity calculations as compared to gauge theory calculations, contrary to field theory expectations. An explicit string-based calculation, which would be extremely difficult using conventional techniques, is presented to illustrate this.

A mapping between Feynman and string motivated one-loop rules in gauge theories

Abstract Recently, computationally efficient rules for one-loop gauge theory amplitudes have been derived from string theory. We demonstrate the relationship of the compact string organization of the amplitude to Feynman diagrams. In particular, we explicitly show how large cancellations inherent in conventional Feynman diagram computations are avoided by the string motivated rules.

Calculation of graviton scattering amplitudes using string-based methods

Abstract Techniques based upon the string organisation of amplitudes may be used to simplify field theory calculations. We apply these techniques to perturbative gravity and calculate all one-loop amplitudes for four-graviton scattering with arbitrary internal particle content. Decomposing the amplitudes into contributions arising from supersymmetric multiplets greatly simplifies these calculations. We also discuss how unitarity may be used to constrain the amplitudes.

Recursive calculation of one-loop QCD integral coefficients

We present a new procedure, using on-shell recursion, to determine coefficients of integral functions appearing in the one-loop scattering amplitudes of gauge theories, including QCD. With this procedure, coefficients of integrals, including bubbles and triangles, can be determined without resorting to integration. We give criteria for avoiding spurious singularities and boundary terms that would invalidate the recursion. As an example where the criteria are satisfied, we obtain all cut-constructible contributions to the one-loop n-gluon scattering amplitude, Anone−loop(...−−−+++...), with split-helicity from an = 1 chiral multiplet and from a complex scalar. Using the supersymmetric decomposition, these are ingredients in the construction of QCD amplitudes with the same helicities. This method requires prior knowledge of amplitudes with sufficiently large numbers of legs as input. In many cases, these are already known in compact forms from the unitarity method.

MHV-Vertices for Gravity Amplitudes

We obtain a CSW-style formalism for calculating graviton scattering amplitudes and prove its validity through the use of a special type of BCFW-like parameter shift. The procedure is illustrated with explicit examples.

N=1 supersymmetric one-loop amplitudes and the holomorphic anomaly of unitarity cuts

Abstract Recently, it has been shown that the holomorphic anomaly of unitarity cuts can be used as a tool in determining the one-loop amplitudes in N = 4 super-Yang–Mills theory. It is interesting to examine whether this method can be applied to more general cases. We present results for a non-MHV N = 1 supersymmetric one-loop amplitude. We show that the holomorphic anomaly of each unitarity cut correctly reproduces the action on the amplitudes imaginary part of the differential operators corresponding to collinearity in twistor space. We find that the use of the holomorphic anomaly to evaluate the amplitude requires the solution of differential rather than algebraic equations.