C. Anastasiou

Two-loop QCD corrections to massless quark–gluon scattering☆

Abstract We present the O (α 4 s ) virtual QCD corrections to the scattering process of massless quark q q →gg due to the interference of tree and two-loop amplitudes and to the self-interference of one-loop amplitudes. We work in conventional dimensional regularisation and our results are renormalised in the MS scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for the q q →gg and the qg→qg ( g q →g q ) scattering processes in terms of logarithms and polylogarithms that are real in the physical region. These results, together with those previously obtained for quark–quark scattering, are important ingredients in the next-to-next-to-leading order contribution to inclusive jet production at hadron colliders.

Two-loop QCD corrections to the scattering of massless distinct quarks☆

We present the two-loop virtual QCD corrections to the scattering of distinct massless quarks, qq→q′q′, in conventional dimensional regularisation. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for each of the s-, t- and u-channels in terms of polylogarithms. The results presented here form a vital part of the next-to-next-to-leading order contribution to inclusive jet production in hadron colliders and will play a crucial role in improving the theoretical prediction for jet cross sections in hadron–hadron collisions.

Two-loop QCD corrections to massless identical quark scattering

Abstract We present the two-loop virtual QCD corrections to the scattering of identical massless quarks, q q →q q , in conventional dimensional regularisation and using the MS scheme. The structure of the infrared divergences agrees with that predicted by Catani while expressions for the finite remainder are given for the q q →q q and the qq→qq ( q q → q q ) scattering processes in terms of polylogarithms. The results presented here form a vital part of the next-to-next-to-leading order contribution to inclusive jet production in hadron colliders and will play a crucial role in improving the theoretical prediction for jet cross sections in hadron–hadron collisions.

Two-loop QED and QCD corrections to massless fermion–boson scattering ☆

The authors present the NNLO QCD virtual corrections for q{bar q} {yields} g{gamma}, q{bar q} {yields} {gamma}{gamma} and the NNLO QED virtual corrections for e{sup +}e{sup -} {yields} {gamma}{gamma}, and all processes related by crossing symmetry. They perform an explicit evaluation of the two-loop diagrams in conventional dimensional regularization, and their results are renormalized in the {bar M}{bar S} scheme. The infrared pole structure of the amplitudes is in agreement with the prediction of Catanis general formalism for the singularities of two-loop amplitudes, while expressions for the finite remainder are given for all processes in terms of logarithms and polylogarithms that are real in the physical region.

The tensor reduction and master integrals of the two-loop massless crossed box with light-like legs

Abstract The class of the two-loop massless crossed boxes, with light-like external legs, is the final unresolved issue in the program of computing the scattering amplitudes of 2→2 massless particles at next-to-next-to-leading order. In this paper, we describe an algorithm for the tensor reduction of such diagrams. After connecting tensor integrals to scalar ones with arbitrary powers of propagators in higher dimensions, we derive recurrence relations from integration-by-parts and Lorentz-invariance identities, that allow us to write the scalar integrals as a combination of two master crossed boxes plus simpler-topology diagrams. We derive the system of differential equations that the two master integrals satisfy using two different methods, and we use one of these equations to express the second master integral as a function of the first one, already known in the literature. We then give the analytic expansion of the second master integral as a function of ϵ=(4−D)/2, where D is the space–time dimension, up to order O (ϵ 0 ) .

The two-loop scalar and tensor Pentabox graph with light-like legs

We study the scalar and tensor integrals associated with the pentabox topology: the class of two-loop box integrals with seven propagators — five in one loop and three in the other. We focus on the case where the external legs are light-like and use integration-by-parts identities to express the scalar integral in terms of two master-topology integrals and present an explicit analytic expression for the pentabox scalar integral as a series expansion in e=(4−D)/2. We also give an algorithm based on integration by parts for relating the generic tensor integrals to the same two master integrals and provide general formulae describing the master integrals in arbitrary dimension and with general powers of propagators.

Scalar one loop integrals using the negative dimension approach

Abstract We study massive one-loop integrals by analytically continuing the Feynman integral to negative dimensions as advocated by Halliday and Ricotta and developed by Suzuki and Schmidt. We consider n -point one-loop integrals with arbitrary powers of propagators in general dimension D . For integrals with m mass scales and q external momentum scales, we construct a template solution valid for all n which allows us to obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with m + q −1 variables. All solutions for all possible kinematic regions are given simultaneously, allowing the investigation of different ranges of variation of mass and momentum scales. As a first step, we develop the general framework and apply it to massive bubble and vertex integrals. Of course many of these integrals are well known and we show that the known results are recovered. To give a concrete new result, we present expressions for the general vertex integral with one off-shell leg and two internal masses in terms of hypergeometric functions of two variables that converge in the appropriate kinematic regions. The kinematic singularity structure of this graph is sufficiently complex to give insight into how the negative-dimension method operates and gives some hope that more complicated graphs can also be evaluated.

The On-shell massless planar double box diagram with an irreducible numerator

Abstract Using a Mellin-Barnes representation, we compute the on-shell massless planar double box Feynman diagram with an irreducible scalar product of loop momenta in the numerator. This diagram is needed in calculations of two loop corrections to scattering processes of massless particles, together with the double box without numerator calculated previously by Smirnov. We verify the poles in ϵ of our result by means of a system of differential equations relating the two diagrams, which we present in an explicit form. We verify the finite part with an independent numerical check.

Application of the negative-dimension approach to massless scalar box integrals

Abstract We study massless one-loop box integrals by treating the number of space-time dimensions D as a negative integer. We consider integrals with up to three kinematic scales ( s , t and either zero or one off-shell legs) and with arbitrary powers of propagators. For box integrals with q kinematic scales (where q =2 or 3) we immediately obtain a representation of the graph in terms of a finite sum of generalised hypergeometric functions with q −1 variables, valid for general D . Because the power each propagator is raised to is treated as a parameter, these general expressions are useful in evaluating certain types of two-loop box integrals which are one-loop insertions to one-loop box graphs. We present general expressions for this particular class of two-loop graphs with one off-shell leg, and give explicit representations in terms of polylogarithms in the on-shell case.

Erratum to “The two-loop scalar and tensor Pentabox graph with light-like legs” [Nucl. Phys. B 575 (2000) 416–436]

Immediately after acceptance of their paper the authors sent us an updated version ofSection 5 and the references for this paper. Unfortunately these updates have been ignoredandthe oldversionappearedin thepublicationofthe paper(Nucl.Phys.B 575(2000)416).Therefore the correct updated version of Section 5 and the references of this publication isgiven here in full.5. The master topologiesIn this section, we collect the explicit expressions in terms of hypergeometric functionsfor the general master-topology integrals