The two-loop QCD matrix element for e+e−→3 jets

Abstract

We compute the O(α 3 s ) virtual QCD corrections to the γ * → qqg matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the MS scheme. The infrared pole structure of the matrix elements agrees with the prediction made by Catani using an infrared factorization formula. The analytic result for the finite terms of both matrix elements is expressed in terms of one-and two-dimensional harmonic polylogarithms.Among jet observables, the three-jet production rate in electron-positron annihilation plays an outstanding role. The initial experimental observation of three-jet events at PETRA [1], in agreement with the theoretical prediction [2], provided first evidence for the gluon, and thus strong support for the theory of Quantum Chromodynamics (QCD). Subsequently the three-jet rate and related event shape observables were used for the precise determination of the QCD coupling constant α s (see [3] for a review). Especially at LEP, three-jet observables were measured to a very high precision and the error on the extraction of α s from these data is dominated by the uncertainty inherent in the theoretical next-to-leading order (NLO) calculation [4][5][6][7][8] of the jet observables. The planned TESLA [9] linear e + e − collider will allow precision QCD studies at even higher energies than at LEP. Given the projected luminosity of TESLA, one again expects the experimental errors to be well below the uncertainty of the NLO calculation.Related to e + e − → 3 jets by crossing symmetry are (2+1)-jet production in deep inelastic ep scattering and vector-boson-plus-jet production at hadron colliders. The experimental data from HERA on ep → (2 + 1) jets and related event shape observables have already reached a level of precision demanding predictions beyond the present NLO accuracy; a further improvement on these data is expected soon from the HERA high luminosity programme. Similarly, vector-boson production at large transverse momentum is a classic test of QCD in hadron-hadron collisions and demands the theoretical prediction to be as precise as possible. In this case, it is also an important background in searches for new physics at the Tevatron and the LHC.Besides its phenomenological importance, the three-jet rate has also served as a theoretical testing ground for the development of new techniques for higher order calculations in QCD: both the subtraction [4] and the phase-space slicing [5] methods for the extraction of infrared singularities from NLO real radiation processes were developed in the context of the first three-jet calculations. The systematic formulation of phase-space slicing [7] as well as the dipole subtraction [8] method w...

Two-loop splitting amplitudes and the single-real contribution to inclusive Higgs production at N3LO

“…Multiple unresolved limits at zero and one loop order were understood already some time ago [18][19][20]. Two-loop corrections to simple soft and simple collinear limits were derived only to finite order in the regulator [21,22] by taking the appropriate limits of the two-loop three-parton decay matrix elements [23,24]. With the two-loop soft gluon current recently derived to all orders in the regulator [25,26], N 3 LO calculations at the two-loop single real level are now missing only the two-loop corrections to simple collinear limits, described by the two-loop splitting amplitudes.…”

Analytic continuation of massless two-loop four-point functions

“…In [8], the full set of two-loop four-point master integrals with one external leg off-shell was computed, for the kinematical situation of a 1 → 3 decay, by solving the differential equations in external invariants [4] fulfilled by these master integrals. These integrals were employed in the calculation of the two-loop QCD corrections to the e + e − → 3 jets matrix element and to the corresponding helicity amplitudes in [9], which can be expressed as a linear combination (with rational coefficients in the invariants and the space-time dimension d) of the corresponding master integrals. The 2 → 2 scattering processes related to e + e − → 3 jets by analytic continuation and crossing are both of high phenomenological importance: hadronic vector-boson-plus-jet production and deep inelastic two-plus-one-jet production.…”

“…Only in the special case of a time-like 1 → 3 decay (relevant to e + e − → 3 jets), which corresponds to the simultaneous continuation of all three external invariants from Euclidean to Minkowskian values, this continuation can be carried out by simply replacing an overall scaling factor, while preserving all 2dH-PLs (which depend only on dimensionless ratios of the invariants). These results were used in [9] for the calculation of the two-loop matrix elements for e + e − → 3 jets.…”

The leading power Regge asymptotic behaviour of dimensionally regularized massless on-shell planar triple box

“…p 2 i = 0, i = 1, 2, 3, 4, the problem of analytical evaluation of two-loop four-point diagrams in expansion in ǫ = (4 − d)/2, where d is the space-time dimension, has been completely solved in [2,3,4,5,6,7]. The corresponding analytical algorithms have been successfully applied to the evaluation of two-loop virtual corrections to various scattering processes [8] in the zero-mass approximation.In the case of massless two-loop four-point diagrams with one leg off-shell the problem of the evaluation has been solved in [9,10], with subsequent applications [11] to the process e + e − → 3jets. (See [12] for recent reviews of the present status of NNLO calculations.…”

“…In the case of massless two-loop four-point diagrams with one leg off-shell the problem of the evaluation has been solved in [9,10], with subsequent applications [11] to the process e + e − → 3jets. (See [12] for recent reviews of the present status of NNLO calculations.…”