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the soft - collinear effective theory ( scet ) constructed in @xcite offers a systematic description of processes involving energetic particles . it has an expansion in a small parameter @xmath10 , where @xmath11 is a typical transverse momenta and @xmath12 the large energy scale . hard exclusive and inclusive processes in qcd are usually described using the powerful techniques of qcd factorization and light - cone expansions  @xcite . scet encompasses and extends these frameworks , and in particular allows a model independent description of effects caused by the interplay between energetic collinear particles and soft particles beyond leading order in the power expansion . these effects can be described in a rigorous way based solely on qcd , but are not included in purely collinear expansions . the study of operators that describe these mixed collinear - ultrasoft ( collinear - usoft ) effects is the purpose of this paper . for recent applications of scet in hard scattering processes and b - decays see refs . @xcite .    since our focus is on mixed collinear - usoft interactions , we consider collinear quark fields @xmath13 , collinear gluon fields @xmath14 , usoft heavy quark fields @xmath15 , usoft light quark fields @xmath16 , and usoft gluons @xmath17 . ( we follow the notation in refs . @xcite , but for simplicity will often suppress the momentum label @xmath18 on the collinear fields . ) these degrees of freedom can interact in a local manner in lagrangians and currents . this is in contrast with collinear - soft couplings whose interactions are mediated by offshell fluctuations  @xcite , and appear in external operators . we comment on collinear - soft interactions at the end of the paper . the derivation of the leading order collinear quark and gluon lagrangians @xmath19 and @xmath20 can be found in ref . @xcite , and a description of the gauge symmetries of scet can be found in refs . @xcite . for details on power counting we refer to ref . the heavy - to - light currents at large energy , @xmath21 , were derived to leading order in ref . @xcite , including one - loop matching for all the wilson coefficients . the running of these wilson coefficients was considered in refs . @xcite .    in the context of the scet , power suppressed corrections were first considered in ref . @xcite , and the @xmath1 suppressed currents @xmath21 and collinear quark lagrangians were derived . the authors showed that a reparameterization invariance ( rpi ) uniquely fixes the wilson coefficients of their subleading currents and lagrangian in terms of the leading order coefficients . was used to derive constraints on matrix elements @xmath22 . ] in ref . @xcite the rpi of scet was extended to the most general three classes ( i , ii , iii ) , and the multipole expansion of the collinear quark lagrangian was treated to higher orders in @xmath23 and were shown not to receive anomalous dimensions . in ref . @xcite the presence of additional @xmath1 heavy - to - light currents was pointed out that were missing in ref . @xcite . the study of power corrections in scet was continued in ref . @xcite and several important results were obtained for mixed usoft - collinear operators . in particular the mixed usoft - collinear quark lagrangian @xmath24 was first considered and was derived to @xmath25 working at tree level , but to all orders in attachments of @xmath26 gluon fields . in a similar fashion heavy - to - light currents were derived to @xmath0 , and linear combinations of currents that are invariant under the three types of rpi were identified . it was also shown that the operators in @xmath24 are not renormalized based on an analysis of arbitrary @xmath27-loop diagrams in the hard region of qcd . the mixed usoft - collinear quark lagrangian @xmath24 was extended to a gauge invariant form with covariant derivatives in ref . @xcite . the purpose of the present paper is to answer some open questions regarding our knowledge of the power suppressed usoft - collinear lagrangian and heavy - to - light currents . this includes the number of @xmath21 currents at @xmath1 , since even at tree level the full reduction of dirac structures has not yet been implemented . for both @xmath21 and @xmath24 we also construct a complete basis which is valid for matching at any order in @xmath2 , and therefore includes all operators that can be induced by radiative corrections or operator mixing . we work in the most general possible frame throughout ( eg .  allowing @xmath28 , @xmath29 ) , and consider all the restrictions from rpi including the transformation of wilson coefficients . finally , we include the mixed usoft - collinear pure glue lagrangian beyond lo ( which follows from an extension of work in refs . the above results are obtained by considering the full implications of rpi , and including all possible operators allowed from collinear gauge invariance , power counting , and the reduction of dirac structures from the effective theory fields . we restrict ourselves to @xmath1 which is one order less than the order to which the tree - level matching results are known from ref . we treat @xmath24 to @xmath0 , and give a detailed account of how the gauge invariant form in ref . @xcite was derived . in cases where our results are restricted to those in refs . @xcite we find agreement , as discussed in more detail in the body of the paper . the results derived here are sufficient for the proof of a factorization theorem for heavy - to - light form factors to all orders in @xmath2 and leading order in @xmath30  @xcite . ]    for the heavy - to - light currents at @xmath1 an important result we find is a new type of `` three - body '' currents , which have not been previously considered in the literature . in refs . @xcite the attention was restricted to scet operators of two - body type @xmath31 , where the two products in parenthesis are collinear gauge invariant , and the ellipses denote combinations of collinear derivatives . beyond tree level but at the same order in @xmath23 , we find that three - body structures can appear for some of the currents , having the form @xmath32 with three collinear gauge invariant factors . we show the rpi can be used to determine for which currents this happens . we also show that rpi greatly restricts the form of the three - body operators , so that they always involve a collinear gluon field strength . the two - body operators have hard wilson coefficients which are functions of a single parameter @xmath33 , while the new three - body operators have two parameter coefficients @xmath34 . analogous three - body structures could appear in the usoft - collinear lagrangian @xmath24 at higher orders in perturbation theory , however using constraints from symmetries of scet we prove that this does not occur . our results are relevant to the study of decay channels for @xmath35 mesons which involve energetic hadrons in the final state . for instance , the results derived in this paper are necessary ingredients in the factorization formula for heavy - to - light form factors proven in ref . @xcite ( for earlier work on factorization in heavy - to - light form factors see ref . @xcite , and for results from qcd sum rules see refs . the factorization theorem is valid to all orders in @xmath2 and leading order in @xmath30 , @xmath36 , and separates contributions from the scales @xmath37 , @xmath38 , and @xmath39 , where @xmath40 is a hadronic scale . it states that a generic form factor can be split into two types of contributions @xmath41 where  @xcite @xmath42 @xmath43 , and the two terms both scale as @xmath44 . this scaling is model independent and is in agreement with that derived from qcd sum rules  @xcite . in eq . ( [ ffintro ] ) @xmath45 and @xmath46 are standard nonperturbative light - cone distribution amplitudes , c.f .  @xcite . the hard coefficients @xmath47 and @xmath48 can be calculated in an expansion in @xmath49 and are simply related to the wilson coefficients of the @xmath50 current operators @xmath21 . the jet function @xmath51 is dominated by momenta @xmath52 . if we wish to expand in @xmath53 then using the techniques developed in ref . @xcite , @xmath51 is _ calculable _ in terms of time - ordered products of the @xmath54 operators @xmath21 and @xmath24 that we study here . at tree - level [ ie . @xmath55 one finds that @xmath51 contains a @xmath56 , and in ratios of form factors the results for @xmath57 then agree with terms computed in ref . the @xmath58 dependence first shows up at @xmath59 as does possible dependence on @xmath60 . however , as we show in section  [ bpisection ] it is possible to absorb the @xmath60 terms into a redefinition of the @xmath61 to all orders in perturbation theory . the factorization formula provides a clean separation of the `` soft '' non - factorizable ( nf ) contributions and `` hard '' factorizable ( f ) terms without double counting . it also gives us a procedure to systematically improve the predictions to any order in perturbation theory at leading order in @xmath30 . the value of @xmath48 and @xmath47 can depend on which heavy - to - light process we consider , whereas @xmath45 and @xmath62 are universal functions . the @xmath63 s are also universal since only a @xmath64 appears for decays to pseudoscalars @xmath65 , and a @xmath66 and @xmath67 appear if @xmath65 is a vector meson . the jet functions @xmath51 are common among certain classes of form factors and also do not depend on the precise state ( eg . @xmath68 or @xmath69 ) . the @xmath70 terms satisfy the so - called large energy form factor relations  @xcite , as expected from the prior loose definitions of these terms as `` soft '' contributions  @xcite . note that we have not bothered to separate @xmath71 and @xmath72 fluctuations in the @xmath61 functions , since it is not clearly beneficial phenomenologically . the factorization theorem does tell us that @xmath73 , however it does not distinguish between factors of @xmath74 and @xmath75 in this @xmath76 . it also does not numerically favor the @xmath57 or @xmath70 term , for instance it is possible that the leading @xmath53 in @xmath51 is compensated by an analogous factor in @xmath61 . we start in section ii by reviewing the general constraints imposed on scet operators following from collinear gauge invariance , spin structure reduction , and reparameterization invariance . in section iii we study the implications of these predictions for the subleading usoft - collinear lagrangian @xmath77 . in section iv we present detailed results for scet currents . using the example of the scalar current as the pedagogical example , we demonstrate the construction of the complete basis of @xmath78 operators contributing to the weak currents , which closes under rpi transformations . explicit results are then also derived for the pseudo - scalar , vector , axial - vector , and tensor heavy - to - light currents to @xmath1 . in section v we summarize the one - loop matching results for the currents , give explicit results for @xmath24 feynman rules , and discuss the basis of currents in the particular frame @xmath7 , @xmath79 .
we construct operators that describe power corrections in mixed collinear - ultrasoft processes in qcd . 
 we treat the ultrasoft - collinear lagrangian to @xmath0 , and heavy - to - light currents involving collinear quarks to @xmath1 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 @xmath2 . 
 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 @xmath3 operators in its gauge invariant form . for the @xmath4 heavy - to - light currents there are @xmath5 subleading ( scalar , pseudo - scalar , vector , axial - vector , tensor ) currents , where @xmath6 have coefficients that are not determined by rpi . 
 in a frame where @xmath7 and @xmath8 the total number of currents reduces to @xmath9 , 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 .