Power counting and modes in SCET

Raymond Goerke,Michael Luke

doi:10.1007/jhep02(2018)147

Abstract

We present a formulation of soft-collinear effective theory (SCET) in the two-jet sector as a theory of decoupled sectors of QCD coupled to Wilson lines. The formulation is manifestly boost-invariant, does not require the introduction of ultrasoft modes at the hard matching scale Q, and has manifest power counting in inverse powers of Q. The spurious infrared divergences which arise in SCET when ultrasoft modes are not included in loops disappear when the overlap between the sectors is correctly subtracted, in a manner similar to the familiar zero-bin subtraction of SCET. We illustrate this approach by analyzing deep inelastic scattering in the endpoint region in SCET and comment on other applications.

Highlights

JHEP02(2018)147 the momentum components of collinear modes in the n direction are defined to scale as pn ∼ (p+n, p−n, pn⊥) ∼ Q(λ2, 1, λ), whereas the momenta of ultrasoft modes scale as pus ∼ Q(λ2, λ2, λ2)
While physical results should be independent of this choice — for example, it was shown in [10] that deep inelastic scattering (DIS) could be analyzed in soft-collinear effective theory (SCET) in either the target rest frame, where only ultrasoft and n-collinear modes were required, or the Breit frame, where n-collinear, n-collinear and ultrasoft modes were required — the degrees of freedom may differ in different reference frames, so the theory is not manifestly frame independent
In this paper we have shown that SCET may be written as a theory of decoupled sectors, where the invariant mass of particles in each sector is much less than the hard scale Q, while the invariant mass of any pair of sectors is of order Q

Summary

The effective theory for DIS at tree level

Qμ is the momentum transfer by the external current, and P is the momentum of the incoming proton. The amplitude (2.27) is reproduced by the one-gluon matrix element of O2, given by the diagrams, where the dashed line indicates an n-collinear gluon emitted from the nonlocal vertex of O2 This may be evaluated using the spinor-helicity formalism to give p2(n)±, k(n) ± |O2|p1(n)± p2(n)±, k(n) ∓ |O2|p1(n)±. Note that the QCD amplitude to produce a final state gluon has been reproduced by the amplitude in SCET to produce an n sector gluon: this is consistent with the power counting in the theory, where the invariant mass of the final state is much less than Q, and is described by the n sector. The leading amplitudes will be reproduced by the nsector gluon matrix elements of O2, while at O(1/Q) SCET will include contributions from operators with Bnfields, whose coefficients may be determined by considering incoming states with nsector gluons. The complete set of operators to O(1/Q2) relevant for dijet production in SCET is renormalized in [36]

One loop matching and overlap subtraction

Matching onto the PDF

Nc spins

Conclusions

A Matching onto helicity eigenstates