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One-Loop Corrections to Two-Quark Three-Gluon Amplitudes

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One-Loop Corrections to Two-Quark Three-Gluon Amplitudes View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server hep-ph/9409393 SLAC{PUB{6663 Saclay/SPhT-T94/108 UCLA/94/TEP/33 Septemb er, 1994 T One-Lo op Corrections to Two-Quark Three-Gluon Amplitudes ] Zvi Bern Department of Physics University of California, Los Angeles Los Angeles, CA 90024 [email protected] ? Lance Dixon Stanford Linear Accelerator Center Stanford, CA 94309 [email protected] and David A. Kosower y Service de Physique Th eorique Centre d'Etudes de Saclay F-91191 Gif-sur-Yvette cedex, France [email protected] Abstract We present the one-lo op QCD amplitudes for two external massless quarks and three external gluons qqggg . This completes the set of one-lo op amplitudes needed for the next-to-leading-order corrections to three-jet pro duction at hadron colliders. We also discuss how to use group theory and sup ersymmetry to minimize the amount of calculation required for the more general case of one-lo op two-quark n-gluon amplitudes. We use collinear limits to provide a stringentcheck on the amplitudes. Submitted to Nuclear Physics B ] Research supp orted in part by the US Department of Energy under grant DE-FG03-91ER40662 and in part by the Alfred P. Sloan Foundation under grant BR-3222. ? Research supp orted by the US Department of Energy under grant DE-AC03-76SF00515. y Lab oratory of the Direction des Sciences de la Mati ere of the Commissariat a l'Energie Atomique of France. 1. Intro duction Jet physics at hadron colliders allows one to confront the theoretical predictions of QCD with exp erimental results and thereby prob e for new physics at the highest p ossible energies. Yet precise comparisons b etween theory and exp eriment are hamp ered by the lack of calculations b eyond the leading order of p erturbation theory, for all but the simplest pro cesses. In pure QCD, the next- to-leading-order corrections computed to date [1] have relied on the one-lo op amplitudes for four external partons, rst calculated by Ellis and Sexton [2]. More recently,wehave calculated the one-lo op amplitudes for ve external gluons ggggg [3], and Kunszt, Signer, and Tr ocs anyi KST have calculated the amplitudes for four quarks and a gluon qq qq g [4]. In this pap er we present the remaining one-lo op ve-parton amplitudes, for two massless quarks and three gluons qqggg . Combining these analytic results with the known six-parton tree amplitudes [5,6], one can now con- struct numerical programs for next-to-leading-order corrections to three-jet pro duction at hadron colliders, and examine the structure of jets, for example dep endence of cross-sections on the cone size, b eyond the leading non-trivial order prob ed in next-to-leading order two-jet programs [1,7]. Computation of the ratio of three-jet to two-jet events at hadron colliders at next-to-leading order in would also make p ossible the measurementof in purely hadronic pro cesses and at the s s largest energy scales available. Many metho ds develop ed in recentyears can b e used to simplify the computation of one- lo op multi-parton amplitudes, including spinor helicity metho ds [8], color decomp osition of am- plitudes [5,9,10], string-based techniques [11,12,13,14,3], sup ersymmetry Ward identities [15,16], sup ersymmetry-based decomp ositions [3,17,18], and p erturbative unitarity [19,20,21]; all of these techniques have b een used to obtain the amplitudes presented in this pap er. Wehave found it useful to organize the calculation in terms of gauge-invariant, color-ordered building blo cks, dubb ed primitive amplitudes.We show in the next section that all of the kine- matic co ecients partial amplitudes app earing in the color decomp osition of amplitudes with two quarks and n 2 gluons can b e expressed as sums over p ermutations of gauge invariant primitive amplitudes. The analytic structure of a primitive amplitude is generally simpler than that of a partial amplitude; a primitive amplitude receives contributions only from diagrams with a xed ordering of external legs, while the generic partial amplitude receives contributions from multiple orderings. Thus, fewer kinematic invariants app ear in each primitive amplitude. Although this organization was motivated in part by string theory, our discussion is entirely eld-theoretic. 2 We use sup ersymmetry to reduce the numb er of quantities to b e calculated. QCD ampli- tudes may b e decomp osed in terms of sup ersymmetric and non-sup ersymmetric parts. Through use of sup ersymmetry Ward identities, the sup ersymmetric parts of amplitudes with two external quarks and three external gluons may b e obtained directly from the previously calculated ve-gluon amplitudes [3]. Wehave also made use of the cut-reconstruction metho d describ ed in refs. [20,21]. If certain power-counting criteria are satis ed, amplitudes are entirely constructible from their cuts. Al- though QCD amplitudes are generally not cut-constructible, by taking linear combinations of QCD amplitudes with ones involving scalars and/or gluinos, the QCD amplitudes may b e separated into cut-constructible and non-cut-constructible parts. Wehave used this unitarity-based technique to obtain the cut-constructible comp onents of some of the primitive amplitudes forqqggg those that enter into the subleading-in-col or contributions to the virtual part of the cross-section. Here the cut-constructible comp onents are formed by adding to the desired diagrams a new set of diagrams, which di ers only in the replacement of virtual gluons in the lo op by scalars. For a sp eci c choice of the Yukawa coupling b etween the scalars and the quark line, the sum of gluon and scalar diagrams satis es the p ower-counting criteria see ref. [21]. We then calculate the scalar contributions di- rectly; they are not cut-constructible, but they are easier to calculate directly than the full gluon contributions. Finally we reassemble the desired gluon contributions. In order to ensure the correctness of the amplitudes, wehave p erformed a number of checks. As the momenta of two external legs b ecome collinear the amplitudes must factorize prop erly.Wehave veri ed this factorization for all amplitudes in all channels. This provides an extremely stringent constraint on the amplitudes. In fact, this constraint is suciently p owerful that it has b een used to construct ansatze for a numb er of amplitudes with xed helicities but an arbitrary number of external legs [22,23,20], whichwere then proven correct by either recursive [24] or unitarity [20,21] techniques. The recursive and unitarity techniques have also b een used to construct a varietyof other one-lo op amplitudes with an arbitrary numb er of external legs [24,20,21]. Wehave p erformed additional checks on certain helicity amplitudes by computing all diagrams that enter into a sup ersymmetry Ward identity [15], and explicitly verifying the identity. Not only do es this provide a check on amplitudes presented in this pap er, but also on the sup ersymmetric combinations of the ve-gluon amplitudes presented in ref. [3]. A similar sup ersymmetry check using the ve-gluon amplitudes has b een carried out [25] for theqq qqg amplitudes rep orted in ref. [4]. As a nal check, wehaveveri ed that the cuts in some amplitudes obtained by more direct diagrammatic means are consistent with unitarity. In section 2, we give the SU N color decomp osition for amplitudes involving two external c 3 quarks and n 2 external gluons, as a sum of color factors multiplied by partial amplitudes. We also give a formula for the sum over colors of the interference b etween tree and one-lo opqqggg amplitudes, in terms of partial amplitudes; this formula is required for the virtual part of the color-summed parton-level cross-section. The primitive amplitudes, which form the gauge-invariant building blo cks for the amplitudes, are describ ed in section 3. The precise relation of the primitive amplitudes to the partial amplitudes is given in section 4. In section 5 we give the main results of the pap er, the primitive amplitudes forqqggg . Section 6 contains our conclusions. Four app endices contain technical details related to color algebra and collinear checks. App endix I provides a derivation of the relation b etween primitive and partial amplitudes. App endix I I collects the one- lo op four-p oint amplitudes [16,11] that app ear in collinear limits ofqqggg amplitudes, namely gggg andqqgg . App endix I I I then illustrates the pro cedure for carrying out collinear checks, using these amplitudes and \splitting amplitudes" from ref. [20]. Finally, app endix IV shows how to use the two-quark n 2-gluon primitive amplitudes to construct amplitudes where some of the gluons are replaced by photons. 2. Color Decomp osition for Two-Quark n 2-Gluon Amplitudes In this section we describ e a color decomp osition of the one-lo op two-quark n 2-gluon amplitudeqqg : : : g, in terms of group-theoretic factors color structures multiplied by kinematic functions called partial amplitudes. In the following sections, we shall give formulae for all of the partial amplitudes in terms of color-ordered, gauge-invariant building blo cks called primitive amplitudes. A primitive amplitude is de ned as the sum of all one-lo op diagrams in which the n external legs have a xed order around the lo op the color order, with some additional restrictions to b e describ ed in the following section.
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