Reparametrization invariant collinear operators

Abstract

In constructing collinear operators, which describe the production of energetic jets or energetic hadrons, important constraints are provided by reparametrization invariance (RPI). RPI encodes Lorentz invariance in a power expansion about a collinear direction, and connects the Wilson coefficients of operators at different orders in this expansion to all orders in α s . We construct reparametrization invariant collinear objects. The expansion of operators built from these objects provides an efficient way of deriving RPI relations and finding a minimal basis of operators, particularly when one has an observable with multiple collinear directions and/or soft particles. Complete basis of operators are constructed for pure glue currents at twist-4, and for operators with multiple collinear directions, including those appearing in e + e − → 3 jets, and for pp → 2 jets initiated via gluon-fusion. * Electronic address: cmarcant@mit.edu † Electronic address: iains@mit.edu 1 In Ref. [18] it was shown that the construction of heavy-to-light operators can be simplified if only operators in a particular frame are required, by taking linear combinations of the RPI transformations that only act in this frame. In Ref. [15] this was described as the derivation of RPI conditions on a projected surface, and the complete set of such transformations was used for the O(λ 2 ) analysis done there. The formalism derived here makes a full analysis sufficiently simple that the consideration of projected surfaces becomes unnecessary.

A complete basis of helicity operators for subleading factorization

“…If the ∂ us(i)0 operator acts on an n i -collinear field, then it can be eliminated using the equations of motion [89], and therefore such a combination does not need to be included in our basis. In contrast with the collinear case, we cannot eliminate then i · ∂ us using the equations of motion without introducing inverse ultrasoft derivative (e.g.…”

“…All other field and derivative combinations can be reduced to this set by the use of equations of motion and operator relations [89]. Since these building blocks carry vector or spinor Lorentz indices they must be contracted to form scalar operators, which also involves the use of objects like {n µ i ,n µ i , γ µ , g µν , µνστ }.…”

“…In this case, our basis is quite different, as we have used the gluon equations of motion in the effective theory to eliminate the n · B n field, as is commonly done in the SCET literature [89]. The operators…”

An effective field theory for forward scattering and factorization violation

“…At any order in the power counting the most general building blocks for n-collinear components of SCET operators for QCD (other than the leading power kinetic term) contain three terms [86] χ n ,…”

Examination of higher-order twist contributions in parity-violating deep-inelastic electron-deuteron scattering

“…In SCET in the Breit frame, at leading order in the power counting in 2 QCD /Q 2 and at tree level, the four-quark operator in W du µν will be matched onto a SCET operator [45], dγ µ d(x)ūγ ν u(0)…”