Iain W. Stewart

We construct operators that describe power corrections in mixed collinear-ultrasoft processes in QCD. We treat the ultrasoft-collinear Lagrangian to O(lambda^2), and heavy-to-light currents involving collinear quarks to O(\lambda) including new three body currents. A complete gauge invariant basis is derived which has a full reduction in Dirac structures and is valid for matching at any order in alpha_s. The full set of reparameterization invariance (RPI) constraints are included, and are found to restrict the number of parameters appearing in Wilson coefficients and rule out some classes of operators. The QCD ultrasoft-collinear Lagrangian has two O(lambda^2) operators in its gauge invariant form. For the O(lambda) heavy-to-light currents there are (4,4,14,14,21) subleading (scalar, pseudo-scalar, vector, axial-vector, tensor) currents, where (1,1,4,4,7) have coefficients that are not determined by RPI. In a frame where v_perp=0 and n.v =1 the total number of currents reduces to (2,2,8,8,13), but the number of undetermined coefficients is the same. The role of these operators and universality of jet functions in the factorization theorem for heavy-to-light form factors is discussed.

For Non-Relativistic QCD the velocity renormalization group correlates the renormalization scales for ultrasoft, potential and soft degrees of freedom. Here we discuss the renormalization of operators by ultrasoft gluons. We show that renormalization of soft vertices can induce new operators, and also present a procedure for correctly subtracting divergences in mixed potential-ultrasoft graphs. Our results affect the running of the spin-independent potentials in QCD. The change for the NNLL t-tbar cross section near threshold is very small, being at the 1% level and essentially independent of the energy. We also discuss implications for analyzing situations where mv^2 ~ Lambda_QCD.

In QCD the amplitude for B0 -> D(*)+pi- factorizes in the large Nc limit or in the large energy limit Q >> Lambda_QCD where Q = {m_b, m_c, m_b-m_c}. Data also suggests factorization in exclusive processes B-> D* pi+ pi- pi- pi0 and B-> D* omega pi-, however by themselves neither large Nc nor large Q can account for this. Noting that the condition for large energy release in B0-> D+ pi- is enforced by the SV limit, m_b, m_c >> m_b-m_c >> Lambda, we propose that the combined large Nc and SV limits justify factorization in B -> D(*) X. This combined limit is tested with the inclusive decay spectrum measured by CLEO. We also give exact large Nc relations among isospin amplitudes for B -> D(*)X and B -> D(*) D-bar(*)X, which can be used to test factorization through exclusive or inclusive measurements. Predictions for the modes B-> D(*) pi pi, B-> D(*)K K-bar and B-> D(*) D-bar(*) K are discussed using available data.

We describe in some detail the derivation of a power counting formula for the soft-collinear effective theory (SCET). This formula constrains which operators are required to correctly describe the infrared at any order in the Lambda_QCD/Q expansion (lambda expansion). The result assigns a unique lambda-dimension to graphs in SCET solely from vertices, is gauge independent, and can be applied independent of the process. For processes with an OPE the lambda-dimension has a correspondence with dynamical twist.