Lance J. Dixon

Strings on orbifolds

String propagation on the quotient of a flat torus by a discrete group is considered. We obtain an exactly soluble and more or less realistic method of string compactification.

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.

The Conformal Field Theory of Orbifolds

A prescription for the calculation of any correlation function in orbifold conformal field theory is given. The method is applied to the scattering of four twisted string states, which allows the extraction of operator product coefficients of conformal twist fields. We derive Yukawa couplings in the effective field theory for fermionic strings on orbifolds.

Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

We compute the leading-color (planar) three-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4 - 2{epsilon} dimensions, as a Laurent expansion about {epsilon} = 0 including the finite terms. The amplitude was constructed previously via the unitarity method, in terms of two Feynman loop integrals, one of which has been evaluated already. Here we use the Mellin-Barnes integration technique to evaluate the Laurent expansion of the second integral. Strikingly, the amplitude is expressible, through the finite terms, in terms of the corresponding one- and two-loop amplitudes, which provides strong evidence for a previous conjecture that higher-loop planar N = 4 amplitudes have an iterative structure. The infrared singularities of the amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on resummation. Based on the four-point result and the exponentiation of infrared singularities, we give an exponentiated ansatz for the maximally helicity-violating n-point amplitudes to all loop orders. The 1/{epsilon}{sup 2} pole in the four-point amplitude determines the soft, or cusp, anomalous dimension at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and Vogt. Following similar logic, we are able to predict a term in the three-loop quark and gluon form factors in QCD.

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.

Moduli dependence of string loop corrections to gauge coupling constants

We consider one-loop corrections 4Δa to inverse gauge couplings ga−2 in supersymmetric vacua of the heterotic string. The form of these corrections plays an important role in scenarios for dynamical supersymmetry breaking in string theory. Specifically, we calculate the exact functional dependence of Δa(U) on any untwisted modulus field U of an orbifold vacuum; it has the universal form Δa(U, U)=Aa·log(|η(U)|4·lm U)+const., where Aa are easily computable rational constants. The dependence is nontrivial (Aa≠0) only if some sectors of the orbifold Hilbert space have precisely N=2 space-time supersymmetry. The expression for Δa has an expected invariance under modular transformations of U, since these are symmetries of the orbifold vacuum state. However, Δa is not the real part of a holomorphic function, in seeming contradiction with the existence of a supersymmetric effective lagrangian. The apparent paradox is an infrared problem, and can occur not just in string theory but in renormalizable supersymmetric field theories as well. We show how the paradox is resolved in the field theory case and argue that the same resolution applies also to the string theory case.

The Four-Loop Planar Amplitude and Cusp Anomalous Dimension in Maximally Supersymmetric Yang-Mills Theory

We present an expression for the leading-color (planar) four-loop four-point amplitude of N = 4 supersymmetric Yang-Mills theory in 4-2{epsilon} dimensions, in terms of eight separate integrals. The expression is based on consistency of unitarity cuts and infrared divergences. We expand the integrals around {epsilon} = 0, and obtain analytic expressions for the poles from 1/{epsilon}{sup 8} through 1/{epsilon}{sup 4}. We give numerical results for the coefficients of the 1/{epsilon}{sup 3} and 1/e{sup 2} poles. These results all match the known exponentiated structure of the infrared divergences, at four separate kinematic points. The value of the 1/{epsilon}{sup 2} coefficient allows us to test a conjecture of Eden and Staudacher for the four-loop cusp (soft) anomalous dimension. We find that the conjecture is incorrect, although our numerical results suggest that a simple modification of the expression, flipping the sign of the term containing {zeta}{sub 3}{sup 2}, may yield the correct answer. Our numerical value can be used, in a scheme proposed by Kotikov, Lipatov and Velizhanin, to estimate the two constants in the strong-coupling expansion of the cusp anomalous dimension that are known from string theory. The estimate works to 2.6% and 5% accuracy, providing non-trivial evidence in support of the AdS/CFT correspondence. We also use the known constants in the strong-coupling expansion as additional input to provide approximations to the cusp anomalous dimension which should be accurate to under one percent for all values of the coupling. When the evaluations of the integrals are completed through the finite terms, it will be possible to test the iterative, exponentiated structure of the finite terms in the four-loop four-point amplitude, which was uncovered earlier at two and three loops.

String calculation of fayet-iliopoulos D-terms in arbitrary supersymmetric compactifications☆

We calculate the Fayet-Iliopoulos D-terms generated at one loop for the heterotic string in any background which preserves four-dimensional space-time supersymmetry and maintains (2,0) world-sheet superconformal invariance. Although our calculation is performed in the full string theory, the result can be evaluated entirely in terms of properties of the massless spectrum. Furthermore it agrees with the result of a computation in the four-dimensional effective field theory using a stringy regularization. We also check our general result through a more explicit calculation in an orbifold background.

Dimensionally-regulated pentagon integrals☆

Abstract We present methods for evaluating the Feynman parameter integrals associated with the pentagon diagram in 4-2∈ dimensions, along with explicit results for the integrals with all masses vanishing or with one non-vanishing external mass. The scalar pentagon integral can be expressed as a linear combination of box integrals, up to O(∈) corrections, a result which is the dimensionally-regulated version of a D = 4 result of Melrose, and of van Neerven and Vermaseren. We obtain and solve differential equations for various dimensionally-regulated box integrals with massless internal lines, which appear in one-loop n -point calculations in QCD. We give a procedure for constructing the tensor pentagon integrals needed in gauge theory, again through O(∈ 0 ).

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.