Webs in multiparton scattering using the replica trick

Journal of High Energy Physics, Nov 2010

Soft gluon exponentiation in non-abelian gauge theories can be described in terms of webs. So far this description has been restricted to amplitudes with two hard partons, where webs were defined as the colour-connected subset of diagrams. Here we generalise the concept of webs to the multi-leg case, where the hard interaction involves non-trivial colour flow. Using the replica trick from statistical physics we solve the combinatorial problem of non-abelian exponentiation to all orders. In particular, we derive an algorithm for computing the colour factor associated with any given diagram in the exponent. The emerging result is exponentiation of a sum of webs, where each web is a linear combination of a subset of diagrams that are mutually related by permuting the eikonal gluon attachments to each hard parton. These linear combinations are responsible for partial cancellation of subdivergences, conforming with the renormalization of a multi-leg eikonal vertex. We also discuss the generalisation of exponentiation properties to beyond the eikonal approximation.

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Webs in multiparton scattering using the replica trick

Einan Gardi 5 Eric Laenen 1 3 4 Gerben Stavenga 2 Chris D. White 0 g 0 Department of Physics and Astronomy, University of Glasgow , Glasgow G12 8QQ, Scotland, U.K 1 Nikhef Theory Group, Science Park 105 , 1098 XG Amsterdam, The Netherlands 2 Fermi National Accelerator Laboratory , Batavia, IL 60510, U.S.A 3 ITFA, University of Amsterdam, Science Park 904 , 1090 GL Amsterdam, The Netherlands 4 ITF, Utrecht University , Leuvenlaan 4, 3584 CE Utrecht, The Netherlands 5 School of Physics and Astronomy, University of Edinburgh , Kings Buildings, Mayeld Road, Edinburgh EH9 3JZ, Scotland, U.K Soft gluon exponentiation in non-abelian gauge theories can be described in terms of webs. So far this description has been restricted to amplitudes with two hard partons, where webs were dened as the colour-connected subset of diagrams. Here we generalise the concept of webs to the multi-leg case, where the hard interaction involves non-trivial colour ow. Using the replica trick from statistical physics we solve the combinatorial problem of non-abelian exponentiation to all orders. In particular, we derive an algorithm for computing the colour factor associated with any given diagram in the exponent. The emerging result is exponentiation of a sum of webs, where each web is a linear combination of a subset of diagrams that are mutually related by permuting the eikonal gluon attachments to each hard parton. These linear combinations are responsible for partial cancellation of subdivergences, conforming with the renormalization of a multi-leg eikonal vertex. We also discuss the generalisation of exponentiation properties to beyond the eikonal approximation. 1 Introduction 1.1 Wilson lines and exponentiation 1.2 Motivation 1.3 Results and outline of the paper 2 3 4 Non-abelian exponentiation via evolution equations 2.1 The singularity structure in the massive case 2.2 The singularity structure in the massless case 2.3 Anomalous dimensions and webs Non-abelian exponentiation via webs using the replica trick 3.1 The replica trick formalism for non-abelian exponentiation 3.2 A replica-trick based algorithm for computing exponentiated colour factors General formula for exponentiated colour factors 4.1 Exponentiated colour factors in terms of conventional ones 4.2 Conventional colour factors in terms of exponentiated ones Exponentiated colour factors in special cases 5.1 Diagrams with distinct parton clusters 5.2 Webs connecting three lines at two loops 5.3 Webs in multi-parton scattering: three loops 5.4 Webs in multi-parton scattering: four loops and general discussion Cancellation of subdivergences 6.1 Renormalization of the multi-eikonal vertex 6.2 Identifying subdivergences 6.3 The mixing matrix as a projection operator 6.4 Cancellation of subdivergences: three-loop examples 6.5 Discussion Next-to-eikonal webs A Alternative derivation of ECF using eq. (4.3) B Example applications of the generalised Gatheral formula C Additional examples of three and four loop webs Introduction Wilson lines and exponentiation Wilson lines and their renormalisation properties are crucial ingredients in a variety of formal and phenomenological applications of quantum eld theory. It is well known that the vacuum expectation value of a product of Wilson line operators, which we may denote generically by S, renormalises multiplicatively to all orders in perturbation theory [1{4]. As a consequence, such quantities S obey linear evolution equations of the form S = 1 Z 2 d 2 2 S ( 2) ; 2 0 so correlators of Wilson lines generically have an exponential form. In an Abelian gauge theory, the anomalous dimension is simply a scalar function of the renormalized coupling and the kinematics and charges of the Wilson lines. In a nonAbelian gauge theory, things are more complicated due to the non-trivial colour structure. Specically, for a correlator involving four or more eikonal lines joining at a hard interaction vertex, or equivalently a self-intersecting Wilson loop, there are several dierent possible contractions of the colour indices, corresponding to dierent colour ows at the hard vertex, as illustrated in a simple case in gure 1. Because gluons carry colour charge, these colour ows mix under evolution, implying that ( 1.1) is a matrix equation. The exponential solution for S in (1.2) is then dened through its Taylor expansion and the matrices in each term are ordered in correspondence with the scale (this is indicated by the symbol P). A complementary picture by which exponentiation can be understood is that of webs. Considering the conguration of two eikonal lines meeting at a hard vertex (as exemplied by the diagrams of gure 2), webs in an abelian theory are simply the set of connected1 diagrams as depicted in the gure. Multiple photon exchange diagrams between the Wilson lines, such as the ladder or crossed diagrams of gure 3, are generated by exponentiation [5]: they are fully reproduced by higher-order terms in the expa (...truncated)


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Einan Gardi, Eric Laenen, Gerben Stavenga, Chris D. White. Webs in multiparton scattering using the replica trick, Journal of High Energy Physics, 2010, pp. 155, Volume 2010, Issue 11, DOI: 10.1007/JHEP11(2010)155