We construct the Lagrangian for an effective theory of highly energetic quarks with energy Q, interacting with collinear and soft gluons. This theory has two low energy scales, the transverse momentum of the collinear particles, p ⊥ , and the scale p 2 ⊥ /Q. The heavy to light currents are matched onto operators in the effective theory at one-loop and the renormalization group equations for the corresponding Wilson coefficients are solved. This running is used to sum Sudakov logarithms in inclusive B → X s γ and B → X u ℓν decays. We also show that the interactions with collinear gluons preserve the relations for the soft part of the form factors for heavy to light decays found by Charles et al., establishing these relations in the large energy limit of QCD.

The factorization of soft and ultrasoft gluons from collinear particles is shown at the level of operators in an effective field theory. Exclusive hadronic factorization and inclusive partonic factorization follow as special cases. The leading order Lagrangian is derived using power counting and gauge invariance in the effective theory. Several species of gluons are required, and softer gluons appear as background fields to gluons with harder momenta. Two examples are given: the factorization of soft gluons in B → Dπ, and the soft-collinear convolution for the B → X s γ spectrum.

In this paper we show how gauge symmetries in an effective theory can be used to simplify proofs of factorization formulas in highly energetic hadronic processes. We use the soft-collinear effective theory, generalized to deal with back-to-back jets of collinear particles. Our proofs do not depend on the choice of a particular gauge, and the formalism is applicable to both exclusive and inclusive factorization. As examples we treat the -␥ form factor (␥␥*→ 0 ), light meson form factors (␥*M →M ), as well as deep inelastic scattering (e Ϫ p→e Ϫ X), the Drell-Yan process (pp →Xl ϩ l Ϫ ), and deeply virtual Compton scattering

Using the soft-collinear effective theory we derive the factorization theorem for the decays B ! M 1 M 2 with M 1;2 ; K; ; K , at leading order in =E M and =m b . The results derived here apply even if s E M is not perturbative, and we prove that the physics sensitive to the E scale is the same in B ! M 1 M 2 and B ! M form factors. We argue that c c penguins could give long-distance effects at leading order. Decays to two transversely polarized vector mesons are discussed. Analyzing B ! we find predictions for B 0 ! 0 0 and jV ub jf B! 0 as a function of .Decays of B mesons to two light mesons are important for the study of CP violation in the standard model. In [1] it was suggested that since m b ; E M ; m M the amplitudes should factorize into simpler nonperturbative objects, and the proposed factorization theorem was checked at one-loop. This approach is often referred to as ''QCD factorization'' (QCDF). Factorization has also been considered in the ''perturbative QCD'' (pQCD) approach [3]. These approaches rely on a perturbative expansion in s E M . The results obtained from factorization are quite predictive and may allow us to answer fundamental questions about the standard model. At the current time several important issues remain to be answered. These include (i) the extent to which the results are model-independent consequences of QCD (since QCD is a predictive theory any model-independent limit must give the same answer in different approaches). A complete proof of a factorization theorem will answer this question. (ii) Unambiguous definitions of any nonperturbative hadronic parameters which appear are required. This allows the universality of parameters to be understood, as well as making clear the extent to which predictions rely on model dependent assumptions about parameter values. (iii) Does the power expansion converge? If power suppressed contributions really compete with leading order contributions as some studies [4,5] suggest then the expansion cannot be trusted. In this case the only hope is a systematic modification of the power counting to promote these effects to leading order, or an identification of certain observables that are free from this problem.The soft collinear effective theory (SCET) [6,7] provides the necessary tools to address these issues. A first study of SCET factorization for B ! has been made in [8]. In this paper we go beyond Refs. [1,3,8] in several ways. We first reduce the SCET operator basis to its minimal form and extend it to allow for all B ! M 1 M 2 decays (including two vectors). Our results show that all of the so-called ''hard spectator'' contributions are already present in the form factors, just with different hard Wilson coefficients. We also derive a form of the factorization theorem which does not rely on a perturbative expansion in s E M , and show that the nonperturbative parameters are still the same as those in the B ! M form factors. In our analysis long distance c c penguins [9,10] are investigated, but are left unfactorized. For the values of m b ...

In QCD the amplitude forB 0 → D ( * )+ π − factorizes in the large N c limit or in the large energy limit Q ≫ Λ QCD where Q = {m b , m c , m b − m c }. Data also suggests factorization in the processes B → D * π + π − π − π 0 and B → D * ωπ − , however by themselves neither large N c nor large Q can account for this. Noting that the condition for large energy release inB 0 → D + π − is enforced by the SV limit, m b ≫ m b − m c ≫ Λ, we propose that the combined large N c and SV limits justify factorization in B → D ( * ) X. This combined limit is tested with the B → D * X inclusive decay spectrum measured by CLEO. We also give exact large N c relations among isospin amplitudes forB → D ( * ) X andB → D ( * )D( * ) X, which can be used to test factorization through exclusive or inclusive measurements. Predictions for the modes B → D ( * ) ππ, B → D ( * ) KK and B → D