arXiv:1011.4264v2 [hep-th] 4 Dec 2010 CERN-PH-TH/2010-275 HU-EP-10/81 Yangian symmetry of light-like Wilson loops J. M. Drummond, PH-TH Division, CERN, Geneva, Switzerland L. Ferro, Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany E. Ragoucy, LAPTH, Universit´e de Savoie, CNRS B.P. 110, F-74941 Annecy-le-Vieux Cedex, France Abstract We show that a certain class of light-like Wilson loops exhibits a Yangian symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops we discuss are equivalent to one- loop MHV amplitudes inN= 4 super Yang-Mills theory in a certain kinematical regime. The fact that we find a Yangian symmetry constraining their functional form can be thought of as the effect of the original conformal symmetry associated to the scattering amplitudes in theN= 4 theory.
1Introduction Scattering amplitudes in gauge theories exhibit many surprising features hinting at an extraordi- nary simplicity that is not apparent in direct Feynman graph calculations. This is demonstrated at tree-level by the remarkable simplicity of the Parke-Taylor formula for maximally-helicity- violating amplitudes [1].Such simplicity continues to all tree-level amplitudes if one employs the on-shell recursive BCFW relations [2, 3] to construct them from their known singularity structure. The level of simplification is even greater when considering the maximally supersymmetric theory,N= 4 super Yang-Mills. In this case the recursive tree-level relations simplify [4, 5] and admit a closed-form solution [6]. Furthermore theN= 4 theory exhibits a very large symmetry algebra. On the colour-ordered tree-level amplitudes the original superconformal symmetry of the Lagrangian combines with another copy of superconformal symmetry, called dual superconformal symmetry [7] to form the Yangian of the superconformal algebra [8].The individual BCFW terms are each invariants under the full Yangian symmetry. They can be thought of as particular contour choices in the Grassmannian integral of [9] (or equivalently its T-dual version [10, 11]) which collects together all Yangian invariant objects into a single simple formula [12, 13, 14]. At loop level it has recently been realised that the above statements all hold at the level of the (unregulated) planar integrand. The integrand at a given loop order can be constructed from its singularities via a generalisation of the BCFW recursion relations and, remarkably, each term is individually invariant under the full Yangian symmetry up to a total derivative [15]. At the level of the actual amplitudes the situation with the full symmetry is less clear, one issue being that the amplitudes are infrared divergent and thus require regularisation. A particularly useful regulator is the one obtained by introducing vacuum expectation values for the scalar fields [16, 17, 18]. This regulator preserves the dual conformal symmetry so that the resulting integrals are invariant. For work relating this picture to higher dimensions see [19, 20, 21]. It has been known for a while that amplitudes inN= 4 super Yang-Mills are connected with light-like Wilson loops both at strong coupling (via the AdS/CFT correspondence) [22] and in perturbation theory [23, 24]. While at strong coupling the dependence on the helicity configu- ration of the amplitude appears as a subleading effect, in perturbation theory the correspondence with Wilson loops was originally limited only to the MHV amplitudes (with direct evidence of the correspondence coming up to two loops and six points [25, 26]). However the amplitude/Wilson loop relation has recently been generalised to cover all helicity configurations [27, 28]. From the Wilson loop perspective the dual conformal symmetry of the scattering amplitudes is the natural conformal symmetry of the light-like Wilson loops. Its effects are taken into ac- count via an anomalous Ward identity [29, 30] which fixes the finite part of the Wilson loops (or 1
equivalently MHV amplitudes) up to a function of conformally invariant cross-ratios. The Ward identity therefore expresses the consequence of the dual conformal symmetry of the scattering am- plitudes. What is not clear is how the original conformal symmetry of the scattering amplitudes is realised beyond tree-level. The question of what happens to the original conformal symmetry at one loop has been addressed before in several papers. In particular the non-invariance of the one-loop amplitudes itself is not just due to the obvious breaking due to the presence of infrared divergences.A further effect can be traced to the holomorphic anomaly which gives a contact term variation even at tree level due to collinear singularities [31, 32, 33, 34]. Here, by appealing to Yangian structure of the underlying algebra we will be able to give a simple realisation of the symmetry on the one-loop amplitudes. There are two ways of looking at the full Yangian symmetry. The first is to treat the original superconformal symmetry of the scattering amplitudes as fundamental.The additional dual conformal symmetry then extends this symmetry algebra to its Yangian [8].The second way exchanges the roles of the original and dual copies of the superconformal symmetry [12].The equivalence of these two pictures should be thought of as the algebraic realisation of the T-duality which maps scattering amplitudes to Wilson loops [35, 36, 37]. The second way of thinking about the symmetry is more important in this paper. We will show that there is a natural (dual) conformally invariant finite quantity, described most naturally in terms of Wilson loops, which exhibits a Yangian symmetry. The finite quantity in question is the ratio of Wilson loops defined in [38, 39], corresponding to a choice of OPE channel when one considers expanding some subset of light-like edges around its totally collinear configuration. We will be working with Wilson loops with special light-like contours contained in a two- dimensional subspace of the full spacetime [40]. The one-loop form of the light-like Wilson loops has been known for some time [24] to be equivalent to the one-loop MHV amplitudes inN= 4 super Yang-Mills theory [41]. In the special two-dimensional kinematics they can be expressed purely in terms of logarithms [40, 42]. Recently also two-loop functions have become available for six points in general kinematics [43, 44] and for an arbitrary number of points in the two- dimensional setup [45, 42, 39]. In the two-dimensional kinematics the conformal symmetry of the Wilson loops is broken to ansl(2)⊕sl(2) subalgebra of the full conformal algebrasl(4). The extra symmetry we find then corresponds to two commuting copies of the YangianY(sl(2)) and is best called a Yangian symmetry of the light-like Wilson loop. It should be thought of as the remaining effects of the original conformal symmetry of the scattering amplitudes in the special kinematics. We begin by discussing representations of Yangians in section 2. We will construct multi- parameter representations based on the coproduct and the freedom to change basis at each stage in building up the representation on a tensor product space. Then in section 3 we will construct 2
100%
Scan to connect with one of our mobile apps
Coinbase Wallet app
Connect with your self-custody wallet
Coinbase app
Connect with your Coinbase account
Open Coinbase Wallet app
Tap Scan
Or try the Coinbase Wallet browser extension
Connect with dapps with just one click on your desktop browser
Add an additional layer of security by using a supported Ledger hardware wallet