We show that in the presence of massive particles, such as nucleons, the standard low-energy expansion, in powers of meson momenta and light-quark masses, in general converges in only part of the low-energy region. The expansion of the scalar form factor σ(t), for instance, breaks down in the vicinity of t = 4M 2 π . In the language of heavy baryon chiral perturbation theory, the proper behaviour in the threshold region results only if the multiple internal-line insertions generated by relativistic kinematics are summed up to all orders. We propose a method that yields a coherent representation throughout the low-energy region, while keeping Lorentz and chiral invariance explicit at all stages. The method is illustrated with a calculation of the nucleon mass and of the scalar form factor to order p 4 .

Using methods from effective field theory, an exact all-order expression for the Drell-Yan cross section at small transverse momentum is derived directly in q T space, in which all large logarithms are resummed. The anomalous dimensions and matching coefficients necessary for resummation at NNLL order are given explicitly. The precise relation between our result and the Collins-Soper-Sterman formula is discussed, and as a by-product the previously unknown three-loop coefficient A (3) is obtained. The naive factorization of the cross section at small transverse momentum is broken by a collinear anomaly, which prevents a process-independent definition of x T -dependent parton distribution functions. A factorization theorem is derived for the product of two such functions, in which the dependence on the hard momentum transfer is separated out. The remainder factors into a product of two functions of longitudinal momentum variables and x 2 T , whose renormalization-group evolution is derived and solved in closed form. The matching of these functions at small x T onto standard parton distributions is calculated at O(α s ), while their anomalous dimensions are known to three loops.

The infrared divergences of massless n-parton scattering amplitudes can be derived from the anomalous dimension of n-jet operators in soft-collinear effective theory. Up to three-loop order, the latter has been shown to have a very simple structure: it contains pairwise color-dipole interactions among the external par-tons, governed by the cusp anomalous dimension and a logarithm of the kinematic invariants s ij , plus a possible three-loop correlation involving four particles, which is described by a yet unknown function of conformal cross ratios of kinematic in-variants. This function is constrained by two-particle collinear limits and by the known behavior of amplitudes in the high-energy limit. We construct a class of relatively simple functions satisfying these constraints. We also extend the analysis to four-loop order, finding that three additional four-particle correlations and a single five-particle correlation appear, which again are governed by functions of conformal cross ratios. Our results suggest that the dipole conjecture, which states that only two-particle color-dipole correlations appear in the anomalous dimension , may need to be generalized. We present a weaker form of the conjecture, stating that to all orders in perturbation theory corrections to the dipole formula are governed by functions of conformal cross ratios, and are O(1/N 2 c) suppressed relative to the dipole term. If true, this conjecture implies that the cusp anomalous dimension obeys Casimir scaling to all orders in perturbation theory.

Methods from soft-collinear effective theory are used to perform the threshold resummation of Sudakov logarithms for the deep-inelastic structure function F2(x,Q2) in the end-point region x-->1 directly in momentum space. An explicit all-order formula is derived, which expresses the short-distance coefficient function C in the convolution F2 = C multiply sign in circle phi q in terms of Wilson coefficients and anomalous dimensions defined in the effective theory. Contributions associated with the physical scales Q2 and Q2(1-x) are separated from nonperturbative hadronic physics in a transparent way. A crucial ingredient to the momentum-space resummation is the exact solution to the integro-differential evolution equation for the jet function, which is derived. The methods developed in this Letter can be applied to many other hard QCD processes.

Partonic cross sections for the production of massive objects in hadronic collisions receive large corrections when the invariant mass of the initial-state partons is just above the production threshold. Since typically the center-of-mass energy of the hadronic collision is much higher than the mass of the heavy objects, it is not obvious that these contributions translate into large corrections to the hadronic cross section. Using a recent approach to threshold resummation based on effective field theory, we quantify to which extent the fall-off of the parton densities at high x leads to a dynamical enhancement of the partonic threshold region. With the example of Drell-Yan production, we study the emergence of an effective physical scale characterizing the soft emissions in the process. We derive compact analytical expressions for the resummed Drell-Yan cross section and rapidity distribution directly in momentum space. They are free of Landau-pole singularities and are trivially matched onto fixed-order perturbative calculations. Evaluating the resummed cross sections at NNNLL order and matching onto NNLO fixed-order calculations, we perform a detailed numerical analysis of the cross section and rapidity distribution in pp collisions.

Renormalization-group methods in soft-collinear effective theory are used to perform the resummation of large perturbative logarithms for deep-inelastic scattering in the threshold region x → 1. The factorization theorem for the structure function F 2 (x, Q 2 ) for x → 1 is rederived in the effective theory, whereby contributions from the hard scale Q 2 and the jet scale Q 2 (1 − x) are encoded in Wilson coefficients of effective-theory operators. Resummation is achieved by solving the evolution equations for these operators. Simple analytic results for the resummed expressions are obtained directly in momentum space, and are free of the Landau-pole singularities inherent to the traditional momentspace results. We show analytically that the two methods are nonetheless equivalent order by order in the perturbative expansion, and perform a numerical comparison up to next-to-next-to-leading order in renormalization-group improved perturbation theory.

Starting from a factorization theorem in Soft-Collinear Effective Theory, the thrust distribution in e + e − collisions is calculated including resummation of the next-to-next-tonext-to leading logarithms. This is a significant improvement over previous calculations which were only valid to next-to-leading logarithmic order. The fixed-order expansion of the resummed result approaches the exact fixed-order distribution towards the kinematic endpoint. This close agreement provides a verification of both the effective field theory expression and recently completed next-to-next-to-leading fixed-order event shapes. The resummed distribution is then matched to fixed order, resulting in a distribution valid over a large range of thrust. A fit to aleph and opal data from lep 1 and lep 2 produces α s (m Z ) = 0.1172 ± 0.0010 ± 0.0008 ± 0.0012 ± 0.0012, where the uncertainties are respectively statistical, systematic, hadronic, and perturbative. This is one of the world's most precise extractions of α s to date.

An exact formula is derived for the infrared singularities of dimensionally regularized scattering amplitudes in massless QCD with an arbitrary number of loops and legs. It is based on the conjecture that the anomalous-dimension matrix of n-jet operators in soft-collinear effective theory is fully determined by three functions of alpha{s}, which can be extracted from known perturbative results for the quark and gluon form factors. This allows us to predict the three-loop coefficients of all 1/;{k} poles for arbitrary n-parton scattering amplitudes, generalizing existing two-loop results.