Soft-gluon expansions through NNNLO

i dxi φf/hi(xi, μF )] σ̂(s, t, u, μF , μR) , where σ̂ is the perturbatively calculable hardscattering cross section, and the parton distributions φ are determined from experiment. The renormalization and factorization scales are denoted by μR and μF respectively, and s, t, u are standard kinematical invariants formed from the momenta of the partons in the hard scattering. Near threshold for the production of a specified system, such as a top quark pair or a Higgs boson, there is restricted phase space for real gluon emission. The incomplete cancellation of infrared divergences between real and virtual graphs results in the appearance of large logarithms. If we define s4 = s+ t+u− ∑ m, with m the masses of the particles in the scattering, then s4 → 0 at threshold and these soft-gluon logarithmic corrections take the form of plus distributions, Dl(s4) ≡ [ln (s4/M )/s4]+, where M is a relevant hard scale, such as the mass of a heavy quark or the transverse momentum of a jet, and l ≤ 2n − 1 for the n-th order corrections. If we define moments of the cross section σ̂(N) = ∞ 0 ds4 e −Ns4/M σ̂(s4) then the soft corrections are transformed as [ln(s4/M )/s4]+ → [(−1) /(l + 1)] ln N + · · ·. We can formally resum these logarithms lnN to all orders in αs by factorizing the soft gluons from the hard scattering [1, 2]. Although the formal resummation in moment space is well defined, when inverting back to momentum space we encounter ambiguities due to the infrared singularity which require a prescription. Unfortunately different prescriptions can give different numerical results as well as have dubious theoretical underpinnings (see discussion in Ref. [3]). However, fixed-order expansions can provide us with solid, prescription-independent, theoretical and numerical results [3, 4]. At next-to-leading order (NLO) in αs, the cross section includes D1(s4) terms which are the leading logarithms (LL), and D0(s4) terms which are the next-to-leading logarithms (NLL). At next-to-next-to-leading order (NNLO), we have D3(s4) (LL), D2(s4) (NLL), D1(s4) (NNLL), and D0(s4) (NNNLL) terms. At next-to-next-to-next-to-leading order (NNNLO), we have D5(s4) (LL), D4(s4) (NLL), D3(s4) (NNLL), D2(s4) (NNNLL), D1(s4) (NNNNLL), and D0(s4) (NNNNNLL) terms.