We construct an improved version of nonrelativistic QCD for use in lattice simulations of heavy quark physics, with the goal of reducing systematic errors from all sources to below 10%. We develop power counting rules to assess the importance of the various operators in the action and compute all leading order corrections required by relativity and finite lattice spacing. We discuss radiative corrections to tree level coupling constants, presenting a procedure that effectively resums the largest such corrections to all orders in perturbation theory. Finally, we comment on the size of nonperturbative contributions to the coupling constants.

We study the factorization of soft and collinear singularities in dimensionallyregularized fixed-angle scattering amplitudes in massless gauge theories. Our factorization is based on replacing the hard massless partons by light-like Wilson lines, and defining gauge-invariant jet and soft functions in dimensional regularization. In this scheme the factorized amplitude admits a powerful symmetry: it is invariant under rescaling of individual Wilson-line velocities. This symmetry is broken by cusp singularities in both the soft and the eikonal jet functions. We show that the cancellation of these cusp anomalies in any multi-leg amplitude imposes all-order constraints on the kinematic dependence of the corresponding soft anomalous dimension, relating it to the cusp anomalous dimension. For amplitudes with two or three hard partons the solution is unique: the constraints fully determine the kinematic dependence of the soft function. For amplitudes with four or more hard partons we present a minimal solution where the soft anomalous dimension is a sum over colour dipoles, multiplied by the cusp anomalous dimension. In this case additional contributions to the soft anomalous dimension at three loops or beyond are not excluded, but they are constrained to be functions of conformal cross ratios of kinematic variables.

We exhibit a solution to the evolution equation for the Sudakov form factor in QCD with massless quarks, which exponentiates infrared poles in dimensional continuation as well as logarithms of momentum transfer. We use this solution to construct an expression for the absolute value of the ratio of timelike to spacelike form factors, in which the infrared finiteness of the ratio is manifest. Finally, we compare this result to explicit calculations of the form factor available in the literature. Most of the large two-loop corrections to the absolute value of the ratio come from the exponentiation of one-loop corrections, including the effect of the running coupling.

We study the origin of subleading soft and collinear poles of form factors and amplitudes in dimensionally-regulated massless gauge theories. In the case of form factors of fundamental fields, these poles originate from a single function of the coupling, denoted G(α s ), depending on both the spin and gauge quantum numbers of the field. We relate G(α s ) to gauge-theory matrix elements involving the gluon field strength. We then show that G(α s ) is the sum of three terms: a universal eikonal anomalous dimension, a universal non-eikonal contribution, given by the coefficient B δ (α s ) of δ(1−z) in the collinear evolution kernel, and a process-dependent short-distance coefficient function, which does not contribute to infrared poles. Using general results on the factorization of soft and collinear singularities in fixed-angle massless gauge theory amplitudes, we conclude that all such singularities are captured by the eikonal approximation, supplemented only by the knowledge of B δ (α s ). We explore the consequences of our results for conformal gauge theories, where in particular we find a simple exact relation between the form factor and the cusp anomalous dimension.