Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the relevant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. Such observables include: transverse momentum distributions at p T much less then the high energy scattering scale, jet broadening, exclusive hadroproduction and decay, as well as the Sudakov form factor. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group". That is, a Collin-Soper like equation is realized as a renormalization group equation, but has a more universal applicability to observables beyond the traditional transverse momentum dependent parton distribution functions (TMDPDFs) and the Sudakov form factor. This formalism has the feature that it allows one to track the (non-standard) scheme dependence which is inherent in any scenario where one performs a resummation of rapidity divergences. We present a pedagogical introduction to the formalism by applying it to the well-known massive Sudakov form factor. The formalism is then used to study observables of current interest. A factorization theorem for the transverse momentum distribution of Higgs production is presented along with the result for the resummed cross section at NLL. Our formalism allows one to define gauge invariant TMDPDFs which are independent of both the hard scattering amplitude and the soft function, i.e. they are universal. We present details of the factorization and resummation of the jet broadening cross section including a renormalization in p ⊥ space. We furthermore show how to regulate and renormalize exclusive processes which are plagued by endpoint singularities in such a way as to allow for a consistent resummation.

We introduce a systematic approach for the resummation of perturbative series which involves large logarithms not only due to large invariant mass ratios but large rapidities as well. A series of this form can appear in a variety of gauge theory observables. The formalism is utilized to calculate the jet broadening event shape in a systematic fashion to next-to-leading logarithmic order. An operator definition of the factorized cross section as well as a closed form of the next-to-leading-log cross section are presented. The result agrees with the data to within errors.

Electroweak Sudakov corrections of the form alphanlogms/MW,Z2 are summed using renormalization group evolution in soft-collinear effective theory. Results are given for the scalar, vector, and tensor form factors for fermion and scalar particles. The formalism for including massive gauge bosons in soft-collinear effective theory is developed.

We study the Sudakov form factor for a spontaneously broken gauge theory using a (new) ∆regulator. To be well-defined, the effective theory requires zero-bin subtractions for the collinear sectors. The zero-bin subtractions depend on the gauge boson mass M and are not scaleless. They have both finite and 1/ǫ contributions, and are needed to give the correct anomalous dimension and low-scale matching contributions. We also demonstrate the necessity of zero-bin subtractions for soft-collinear factorization. We find that after zero-bin subtractions the form factor is the sum of the collinear contributions minus a soft mass-mode contribution, in agreement with a previous result of Idilbi and Mehen in QCD. This appears to conflict with the method-of-regions approach, where one gets the sum of contributions from different regions.

Previous work on electroweak radiative corrections to high-energy scattering using soft-collinear effective theory (SCET) has been extended to include external transverse and longitudinal gauge bosons and Higgs bosons. This allows one to compute radiative corrections to all parton-level hard scattering amplitudes in the standard model to next-to-leading-log order, including QCD and electroweak radiative corrections, mass effects, and Higgs exchange corrections, if the high-scale matching, which is suppressed by two orders in the log counting, and contains no large logs, is known. The factorization structure of the effective theory places strong constraints on the form of gauge theory amplitudes at high energy for massless and massive gauge theories, which are discussed in detail in the paper. The radiative corrections can be written as the sum of process-independent one-particle collinear functions, and a universal soft function. We give plots for the radiative corrections to q " q ! W T W T , Z T Z T , W L W L , and Z L H, and gg ! W T W T to illustrate our results. The purely electroweak corrections are large, ranging from 12% at 500 GeV to 37% at 2 TeV for transverse W pair production, and increasing rapidly with energy. The estimated theoretical uncertainty to the partonic (hard) cross section in most cases is below 1%, smaller than uncertainties in the parton distribution functions. We discuss the relation between SCET and other factorization methods, and derive the Magnea-Sterman equations for the Sudakov form factor using SCET, for massless and massive gauge theories, and for light and heavy external particles.