Loop amplitudes for massless five particle scattering processes contain Feynman integrals depending on the external momentum invariants: pentagon functions. We perform a detailed study of the analyticity properties and cut structure of these functions up to two loops in the planar case, where we classify and identify the minimal set of basis functions. They are computed from the canonical form of their differential equations and expressed in terms of generalized polylogarithms, or alternatively as one-dimensional integrals. We present analytical expressions and numerical evaluation routines for these pentagon functions, in all kinematical configurations relevant to five-particle scattering processes. 1 1,2 3,4,5 s 12 , s 34 , s 45 > 0 s 35 > 0 s 23 , s 51 < 0 s 13 , s 15 , s 24 , s 25 < 0 2 5,1 2,3,4 s 51 , s 23 , s 34 > 0 s 24 > 0 s 12 , s 45 < 0 s 25 , s 35 , s 13 , s 14 < 0 3 4,5 1,2,3 s 45 , s 12 , s 23 > 0 s 13 > 0 s 51 , s 34 < 0 s 14 , s 24 , s 25 , s 35 < 0 4 3,4 5,1,2 s 34 , s 51 , s 12 > 0 s 25 > 0 s 45 , s 23 < 0 s 35 , s 13 , s 14 , s 24 < 0 5 2,3 4,5,1 s 23 , s 45 , s 51 > 0 s 14 > 0 s 34 , s 12 < 0 s 24 , s 25 , s 35 , s 13 < 0 6 3,5 1,2,4 s 12 > 0 s 35 , s 14 , s 24 > 0 s 23 , s 34 , s 45 , s 51 < 0 s 25 , s 13 < 0 7 1,4 2,3,5 s 23 > 0 s 14 , s 25 , s 35 > 0 s 34 , s 45 , s 51 , s 12 < 0 s 13 , s 24 < 0 8 2,5 3,4,1 s 34 > 0 s 25 , s 13 , s 14 > 0 s 45 , s 51 , s 12 , s 23 < 0 s 24 , s 35 < 0 9 1,3 4,5,2 s 45 > 0 s 13 , s 24 , s 25 > 0 s 51 , s 12 , s 23 , s 34 < 0 s 14 , s 35 < 0 10 2,4 5,1,3 s 51 > 0 s 24 , s 35 , s 13 > 0 s 12 , s 23 , s 34 , s 45 < 0 s 25 , s 14 < 0 Table 1. Kinematical channels in Minkowski region: the first five correspond to adjacent channels, while the remaining five are non-adjacent channels.

Virtual two-loop corrections to scattering amplitudes are a key ingredient to precision physics at collider experiments. We compute the full set of planar master integrals relevant to five-point functions in massless QCD, and use these to derive an analytical expression for the two-loop fivegluon all-plus-helicity amplitude. After subtracting terms that are related to the universal infrared and ultraviolet pole structure, we obtain a remarkably simple and compact finite remainder function, consisting only of dilogarithms.PACS numbers: 12.38BxThe precise theoretical description of scattering reactions of elementary particles relies on the perturbation theory expansion of the scattering amplitudes describing the process under consideration. In this expansion, higher perturbative orders correspond to more and more virtual particle loops. At present, one-loop corrections can be computed to scattering amplitudes of arbitrary multiplicity, while two-loop corrections are known only for selected two-to-one annihilation or two-to-two scattering processes.For many experimental observables at higher multiplicity, a substantial increase in statistical precision can be expected from the CERN LHC in the near future. Perturbative predictions beyond one loop will be in demand for many precision applications of these data, for example in improved extractions of standard model parameters or in search for indirect signatures of new high-scale physics in precision observables.Progress on multiloop corrections to high-multiplicity amplitudes requires significant advances in two directions. Feynman-diagrammatic approaches to the computation of these amplitudes yield enormously large expressions that contain many thousands of different Feynman integrals. These integrals are related among each other through Poincaré invariance and symmetries, such that only a limited set of independent so-called master integrals will remain in the final answer for a scattering amplitude. To express a generic two-loop multiparton amplitude in terms of the relevant master integrals (ideally circumventing the large algebraic complexity at intermediate stages that is generated by working in terms of Feynman diagrams) is a yet outstanding problem. A particular example where the reduction to a basis set of integrals was achieved [1, 2] is the two-loop five-gluon helicity amplitude with all helicities positive. In this case, the application of on-shell techniques led to a particularly compact integrand, which motivated a specific choice of basis integrals (which do not necessarily form a minimal set in the sense of being master integrals). In [1], these integrals were evaluated numerically for selected kinematical points. Although this specific helicity amplitude is not contributing to the second-order correction to the three-jet cross section (due to its vanishing at tree level), it provides an ideal testing laboratory for new calculational concepts and methods that will carry over to the general helicity case, as previously in the case for the four-point tw...

The contribution of quarks with masses m Λ QCD is the only part of the structure functions in deepinelastic scattering (DIS) which is not yet known at the next-to-next-to-leading order (NNLO) of perturbative QCD. We present improved partial NNLO results for the most important structure function F 2 (x, Q 2 ) near the partonic threshold, in the high-energy (small-x) limit and at high scales Q 2 m 2 ; and employ these results to construct approximations for the gluon and quark coefficient functions which cover the full kinematic plane. The approximation uncertainties are carefully investigated, and found to be large only at very small values, x 10 −3 , of the Bjorken variable.

In this paper, we analytically compute all master integrals for one of the two non-planar integral families for five-particle massless scattering at two loops. We first derive an integral basis of 73 integrals with constant leading singularities. We then construct the system of differential equations satisfied by them, and find that it is in canonical form. The solution space is in agreement with a recent conjecture for the nonplanar pentagon alphabet. We fix the boundary constants of the differential equations by exploiting constraints from the absence of unphysical singularities. The solution of the differential equations in the Euclidean region is expressed in terms of iterated integrals. We cross-check the latter against previously known results in the literature, as well as with independent Mellin-Barnes calculations.

This corrects the article DOI: 10.1103/PhysRevLett.116.062001.