A Complete Basis for Power Suppressed Collinear-Ultrasoft Operators
  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. 
Dan Pirjol, Iain W. Stewart