Paper
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Yangian symmetry of light-like Wilson loops
Authors
J. M. Drummond,L. Ferro
E. Ragoucy,Hesham Osman,Mehdi Roohi,Chen Su-may Sheih,Shan-Ju L. Chang,Arcadi Santamaria,JIANXIN LU,Jean Turgeon,ivan lupandin,Alfredo Amigo,Pietro Giribone,Filiberto Hueyotl Zahuantitla,Ludovico BIAGI,Jose Vega,Kirk Jensen,Dian-Yong Chen,Sarvenaz Sarabipour,Christoph Gommel,Alejandro Cardenas-Avendano,Martin Goodchild,Trevor Glasbey,Alexandre Brea Rodríguez,Srinivasan Jagannathan,Kunal Vyas,Francisco Antonio Villaescusa-Navarro,Frédéric Dreyer,Saša Obradović,Peter Wolfsteiner,Ivana Dinić,Mirjana Knežević,Dragana Rejman Petrović,Milan Kostić,Nenad Tomić,Vladimir Ranković,Mikica Drenovak,Garrett Poe,Snežana Ljubisavljević,Katarina Borisavljević,Veljko Marinković,Ana Krstić,Vladimir Obradović,David Shih,Jesse Thaler,Alain Jaccard,Wade Ryberg,Carolyn Dzierba,Francesco Cilurzo,Mirjana Todorović,Biljana Jovković,Zoran Kalinić,Geofrey De Visscher,Stefan Hoeche,Andy Van Brocklin,Nemanja Karapavlović,Nemanja Lojanica,Luis Garcés-Erice,David Van Wie,Seok Ki Choi,Vishnu Ram OV,N/A ,Jonathan Wall,Marie Connett,Cemile Bardak,Milan Čupić,Thorsten Alexander Kern,Dilber Uzun Ozsahin,Martin Köhne,David Clarke,Frank Krauss,Jeff Forshaw,Andre Lukas,Ana Mª Roldán Gómez,Francisca Lopez-Granados,Ramon Gomez Jimenez,Valentin V Khoze,Michael Fry,CHIA-HO LIN,Cristian Armendariz Picon,Henrike Fleischhack,Maria Inês Tiraboschi Ferro,Sean Quinn,David Schofield,Sreenivasulu Gollapudi,Lee Fitzgerald,Santiago Folgueras,Roberto Rodriguez Madrera,Prateek Agrawal,Gustavo Alberto Burdman,ALFONSO OJEDA,Katarzyna Wójcik,Victor Terskikh,Shiraz Minwalla,Omar Diego Lopez,Rong-Xin Miao,William Linch,George Giannakopoulos,Neil Switz,David Straub,Ivonne Zavala,Sandipan Kundu,Shlomo Sergei Razamat,Livia Ferro,Kevin Goldstein,Arnthor Ævarsson,Ulf Gran,Carlos Alberto Vaquera-Araujo,metin bedir,Henning Samtleben,Vladimir Belavin,Kyohei Mukaida,Ezequiel Alvarez,Iosif Bena,Sai Putcha,Alina Sbruieva,Luis Caballero,Riccardo Scateni,Robert LoGalbo,Stephane Coulombe,Simon Thorrold,María de la Luz Casas-Perez,Kapil Dandekar,Per Gustavsson,Omar Tibolla,Aurel Schneider,MARIA ELISA ESTEVEZ LOPEZ,Guy Moore,Luca Silvestrini,Christian Pfeifer,Daisuke Yamauchi,Basudeb Dasgupta,Niayesh Afshordi,Dipankar Das,Ying Zhao,Nicholas Rodd,Riccardo Catena,Wolfgang Doerr,Constantinos Papageorgakis,Kazuya Yonekura,Gabriele Travaglini,Young-Kyun Kwon,Ana Núñez Carballosa,Neil Goulty,Mikael Chala,Vladimir Yuryev,Djamel BOUMEDIENE,Aleksandr N. 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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 in N = 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 the N = 4
theory.
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 in N = 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 the N = 4
theory.
1 Introduction
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 the N = 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 in N = 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
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 the N = 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 in N = 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 in N = 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 an sl(2) ⊕ sl(2) subalgebra of the full conformal algebra sl(4). The extra symmetry we find
then corresponds to two commuting copies of the Yangian Y (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
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 in N = 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 an sl(2) ⊕ sl(2) subalgebra of the full conformal algebra sl(4). The extra symmetry we find
then corresponds to two commuting copies of the Yangian Y (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
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