Yangian symmetry for bi-scalar loop amplitudes

Journal of High Energy Physics, May 2018

Abstract We establish an all-loop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable bi-scalar field theory in four dimensions. This chiral theory is a particular double scaling limit of γ-twisted weakly coupled \( \mathcal{N}=4 \) SYM theory. Each amplitude with a certain order of scalar particles is given by a single fishnet Feynman graph of disc topology cut out of a regular square lattice. The Yangian can be realized by the action of a product of Lax operators with a specific sequence of inhomogeneity parameters on the boundary of the disc. Based on this observation, the Yangian generators of level one for generic bi-scalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes.

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Yangian symmetry for bi-scalar loop amplitudes

HJE Yangian symmetry for bi-scalar loop amplitudes Dmitry Chicherin 0 1 2 4 5 6 Vladimir Kazakov 0 1 2 5 6 Florian Loebbert 0 1 2 3 5 6 Dennis Muller 0 1 2 3 5 6 De-liang Zhong 0 1 2 5 6 0 Sorbonne Universites, UPMC Univ. Paris 06 , CNRS 1 Ecole Normale Superieure, PSL Research University 2 55099 Mainz , Germany 3 Institut fur Physik, Humboldt-Universiat zu Berlin 4 PRISMA Cluster of Excellence, Johannes Gutenberg University 5 Zum Gro en Windkanal 6 , 12489 Berlin , Germany 6 75005 Paris , France We establish an all-loop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable bi-scalar eld theory in four dimensions. This chiral theory is a particular double scaling limit of -twisted weakly coupled N = 4 SYM theory. Each amplitude with a certain order of scalar particles is given by a single shnet Feynman graph of disc topology cut out of a regular square lattice. The Yangian can be realized by the action of a product of Lax operators with a speci c sequence of inhomogeneity parameters on the boundary of the disc. Based on this observation, the Yangian generators of level one for generic bi-scalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes. Integrable Field Theories; AdS-CFT Correspondence; Conformal Field - This paper is dedicated to the memory of L.D. Faddeev. 1 Introduction 2 Yangian invariants and monodromy matrix 3 Conformal Lax operator 4 Yangian symmetry of bi-scalar shnet graphs: regular boundary 5 Proof of Yangian symmetry: the lasso method 6 Yangian symmetry of bi-scalar shnet graphs: irregular boundary 7 Scattering amplitudes and cuts of shnet graphs 8 Fishnet graphs and rst realization of the Yangian 8.1 8.2 8.3 First realization Pedagogic examples: cross and double cross Monodromy expansion 9 Dual conformal symmetry and the Yangian in momentum space 10 Discussion and outlook A Cyclicity B Expansion of monodromy eigenvalue operators | especially with the discovery of the quantum spectral curve (QSC) [2{4] | conformal structure constants [5], multi-point correlators [6], cusped Wilson loops, the quark-antiquark potential [7], as well as planar scattering amplitudes. Planar amplitudes | some of the most important physical quantities | have been regularized, destroy this symmetry [13, 14].1 However, as we will show in this paper, an all-loop Yangian symmetry can be constructed for the case of scalar amplitudes within the recently proposed, by one of the authors and O . Gurdogan in [16], double scaling limit of -deformed N = 4 SYM theory. This limit is obtained by taking the 't Hooft coupling g ! 0, the -deformation parameters e i j=2 ! 1, while keeping the new couplings j = ge i j=2(j = 1; 2; 3) xed. In the particular case of a single non-zero e ective coupling 3, this limit gives rise to a biscalar quantum eld theory with the four-dimensional Lagrangian L = Nc Tr 2 Here 1 and 2 are two complex scalar matrix elds in the adjoint representation of SU(Nc). The absence of the complex conjugate term Tr y2 y1 2 1 makes the model non-unitary, which implies speci c chiral properties with respect to its avor structure. In the 't Hooft limit Nc ! 1 the model behaves as a CFT for a majority of its correlation functions, except for those having the shortest initial or intermediate states of length L = 2, which correspond to the operators Tr( i j ) or Tr( i jy). These states induce singularities related to the double-trace couplings of the terms Tr( i j )Tr( iy jy) and Tr( iy j )Tr( i jy), which are generated in the action by the renormalization group. Even in the planar limit, the double-trace couplings are running with the renormalization scale [17{20]. However, the coupling does not run in the planar limit, which preserves the conformal behavior of the majority of physical quantities of the theory. The choice of imaginary parameters j in this theory makes it non-unitary, and the double scaling limit introduces a certain chiral structure (orientation of vertices) on the planar Feynman graphs, suggesting the name FT for such a QFT. The non-unitarity is a price to pay for one of the remarkable features of the bi-scalar theory | a great simpli cation of its Feynman graph content: for most of the physical quantities, each loop order of planar perturbation theory contains at most one Feynman graph. The bulk part of a su ciently large graph is always of \ shnet" type | it consists of a large chunk of regular square lattice. The vertical and horizontal lines of this lattice correspond to the lines of propagators of the elds 1 and 2, respectively. Hence, this bi-scalar FT4 represents a eld-theoretical realization of A. Zamolodchikov's integrable statistical mechanics model of shnet graphs [21]. The presence of two avors leads to a rich set of possibly integrable, i.e. potentially computable, individual 4D scalar Feynman graphs [16, 22]. 1See [14] for an approach towards Yangian symmetry of the nite BDS-subtracted scattering matrix of N = 4 SYM theory. In the Grassmannian formulation of scattering amplitudes, the integrand has manifest Yangian symmetry, cf. [15]. { 2 { HJEP05(218)3 graphs of shnet type with disc topology. The most general quantities which we will study here, and for which we will establish the Yangian symmetry, are the following single-trace correlators K(x1; x2; : : : x2M ) = hTr [ 1(x1) 2(x2); : : : 2M (x2M )]i ; (1.2) where i 2 f y1; y2; 1; 2g. In momentum space, this correlator can be interpreted as a scattering amplitude of very massive scalar \Higgs" particles which interact with each other only due to the exchange of massless scalars 1; 2, with momenta much smaller than the Higgs masses. Obviously, due to charge conservation, the number of elds 1 (which we denote by M1) should be equal to the number of elds y1, and the number M2 of elds 2 should be equal to the number of elds y2, such that M = M1 + M2. The interaction vertex conserves each of the two avors. Hence, in the corresponding Feynman graph of disc topology, the propagator lines of elds 1 continuously go from one external leg to another, and similarly for 2 . Lines of the same avor never cross and the intersection of two types of lines can only happen with one orientation: say the arrows on propagators of elds 1 and 2 around any vertex should follow in the clock-wise order, as is shown in gure 1. It is easy to convince oneself that the only possible planar graphs of this type correspond to a disc cut out of the regular square lattice along a sequence of 2M1 + 2M2 edges, as depicted by solid lines in gure 2.2 The two types of lines are made out of propagators of the eld 1 and the eld 2 respectively. Each such graph corresponds to a certain ordering of legs around the boundary, chosen from the set f y1; y2; 1 ; 2g. For a given ordering, there is a single possible planar graph. Although this statement might deserve an accurate global proof, it is a very natural statement: given a particular planar shnet graph, which xes a given ordering of 2This fact was mentioned in the conclusions of [22]. { 3 { a number of variables xi which are coordinates of external legs. Each solid line of the shnet graph represents the scalar propagator xij2. Integrations are over positions of vertices (denoted by blobs). The dual graph is drawn by dotted lines. The dual graph lives in the momentum representation with integrations over loop momenta. Its external momentum variables are de ned as pi = x i xi+1. The dual graph does not necessarily correspond to an amplitude in the bi-scalar theory, since it could have interaction vertices of valency di erent from four. The external legs of the dual graph are amputated. The in owing o -shell momenta are denoted by double lines. external elds, it seems impossible (at least locally) to change the structure of the graph without introducing new types of vertices (i.e. intersections of lines of the same color or of a wrong orientation). This means that each single-trace correlator of scalar elds in the planar approximation of the bi-scalar FT4 is described by a single graph, whose loop order equals the number of interaction vertices inside the disc. The number of intersections in turn, depends only on M1; M2 and on the ordering of the scalar elds under the trace.3 In the following, we will mostly drop the arrows on propagator lines since our symmetry considerations apply to generic Feynman graphs of shnet type | independently of their origin in a particular theory. Adding the arrows, however, is useful in order to illustrate that the FT4 furnishes a generating theory for all of these diagrams, with a conjectural one-to-one correspondence between correlators and Feynman graphs. We will show in the next section of this paper that the correlator (1.2), and hence its only Feynman diagram, obeys a Yangian symmetry. The correlator (1.2) is our master object from which we can obtain any amplitude. It is more general than just the color-ordered scattering amplitudes of massless bosons, but the latter can always be recovered from these correlators by passing to dual momentum 3Note that such a diagram of disc topology, although made out of the regular square lattice, can not always be drawn on the plane without overlaps, as can be seen in gure 3. The most general diagram can Riemann surface. We will call them singular square lattices. The irregularities should appear only at the boundary of this disc, the bulk being always a regular square lattice. { 4 { a simple sheet of regular square lattice. However, it can be cut out of a \double-sheet" regular square lattice with a branchpoint. More general graphs can be cut out of the lattices having various \conical" singularities, see footnote 3. variables pi = xi xi+1 and by putting all pi's on the light cone, as shown on the particular example of the double-cross diagram in the right of gure 4. This duality transformation can be represented graphically as a passage to the dual lattice, depicted by dotted lines on the right of gure 4. It represents a possible bi-scalar amplitude with quartic interaction vertices. The internal momenta to be integrated, correspond to square faces. To put the ith leg of this dual graph on shell, we simply cut the corresponding propagator by replacing it in coordinate space via xi i2+1 ! (xi2i+1) or in momentum space according to pi 2 ! (pi2). Hence, we amputate this external propagator as prescribed by the Lehmann-Symanzik Zimmermann (LSZ) procedure. Notice that in the left gure 4 the dual graph with double-line legs represents a particular, double-box amplitude, where the external vertices can have three neighbors (and the external legs can be on shell or not). A more general dual graph in gure 2 can also have ve neighboring propagators at some external vertices. Such graphs do not represent amplitudes in the bi-scalar theory de ned by (1.1) since the number of particles of each of the two avors cannot be conserved. However, we can recover from the graph in gure 2 a bi-scalar amplitude by doubling the external momentum legs at the convex corners and omitting them at concave corners of the dual graph, which corresponds to joining some external legs, as shown on the gure 5. Such a graph, with some external coordinates identi ed, will be called \irregular" in the following. N Importantly, it can be argued that all of the above shnet-type amplitudes are actually nite, both in the IR and UV regime. This means that, unlike the case of loop amplitudes in N = 4 SYM theory, the conformal and the full Yangian symmetry of these amplitudes, can be taken at face value. The rigorous establishment of this Yangian invariance, which is probably a close relative of the elusive (twisted) Yangian invariance of the full -deformed = 4 SYM theory, is the main result of this paper. First, we will demonstrate this invariance by application of a speci c monodromy matrix to the external legs around { 5 { p1 x6 p6 x1 p2 x2 x5 p5 x4 p3 x3 p4 p10 p9 x1 x10 x9 p1 momenta owing in loops) and its dual double cross topology in region momentum variables (integration over position of the vertices | lled blobs). (A): all in owing momenta (depicted by double lines) are o shell, pi2 = xi i+1 6= 0 at i = 1; : : : ; 6. (B): amplitude diagram for scattering of 2 massless particles, i.e. the in owing momenta (depicted by loosely dotted gray lines) are light-like pi2 = xi i+1 = 0 at i = 1; : : : ; 10. These constraints are imposed by means of delta functions (xi2i+1) 2 depicted by dashed black lines. dual graph depicted by dotted lines. It has doubled (w.r.t. the previous gure 2) external legs at the convex corners of the boundary, and no legs at the concave corners. The corresponding additional momentum variables at convex corners should satisfy the momentum conservation condition at each dual vertex. We obtain the admissible bi-scalar amplitude by identifying the points of the boundary of the original graph which end in the same square (i.e., the same site of the original lattice. Such points are surrounded by ellipses in the above gure. The masslessness is ensured by additional factors (pj2) multiplying the external legs of the dual graph. { 6 { the Feynman graphs under consideration (the \lasso" method). Then, expanding it with respect to the spectral parameter, we obtain the level-one generators of the conformal Yangian algebra and comment on its relation to dual conformal symmetry. 2 Yangian invariants and monodromy matrix In this section we brie y review the construction of Yangian invariants before specifying it to the case of the conformal algebra so(2; 4). Historically, the so-called RTT realization of the Yangian algebra rst appeared implicitly in the context of the quantum inverse scattering method [23, 24], much earlier than its general de nition by Drinfel'd. In this framework, the Yangian commutation relations are encoded into a Yang-Baxter equation for the monodromy matrix T (u), the so-called RTT-relation: R(u v) T (u) T (v) = T (v) T (u) R(u v) : The monodromy matrix T (u) is a formal series in the spectral parameter u. It encompasses the in nite set of Yangian generators J = 1; : : : ; 4 are matrix indices of the de ning representation of su(2; 2) so(2; 4), and the numerical matrix R(u) is Yang's R-matrix where P is the permutation matrix. It satis es the Yang-Baxter equation which is a consistency relation for the structure constants of the Yangian algebra given by the RTT-relation. The evaluation representation with J of (2.1), namely T (u) = (n) = 0 for n > 0 provides the simplest solution + u 1J (0). For the case at hand, the J (0) are linear combinations of the conformal algebra generators acting on a single site of a non-compact spin chain. Up to a conventional overall factor of u we call the above solution the Lax operator and denote it by L(u) u + J (0). The algebra given by the RTT-relation possesses a comultiplication structure. In particular, this means that the matrix product of several Lax operators (each acting on its own spin chain site) respects (2.1). We thus nd that the inhomogeneous monodromy T (~u) ' Ln(un) : : : L2(u2)L1(u1) speci ed by the set of parameters ~u = (un; : : : ; u1) furnishes a solution of eq. (2.1). Here Li(ui) acts on the i-th site of the spin chain. As was shown in [26, 27], the eigenvalue problem for an inhomogeneous monodromy constructed out of Lax operators, i.e. Ln(un) : : : L2(u2)L1(u1) j ; invi = (~u) j ; invi 1; (2.4) provides a natural way to obtain Yangian invariants j ; invi, which live on n sites of a noncompact spin chain. Both sides of eq. (2.4) are matrices and 1 denotes the identity matrix. { 7 { Here ui = u + i is the spectral parameter u shifted by the inhomogeneity i polynomial in u of degree n. Eq. (2.4) implies that the non-diagonal Yangian generators, obtained from the expansion (2.2), annihilate j ; invi and that the diagonal generators act covariantly on it, i.e. J (n)j ; invi = cn j ; invi with some coe cients cn = cn(~) speci ed by the expansion of the polynomial (~u) in powers of u. In [26] the eigenvalue problem (2.4) for a non-compact spin chain with JordanSchwinger representations of the algebra sl(N ) was studied, and the R-operator method for the superconformal algebra gl(4j4) was investigated in [27{31] to construct tree-level scattering amplitudes in N = 4 SYM, as well as in ABJM theory [32]. The R-operator method of [27] was also applied to construct form factors of composite operators [33, 34], the form factors of Wilson lines [35], the kernels of QCD parton evolutions [36], amplituhedron volume functions [37], and splitting amplitudes [38]. Here we apply the eigenvalue problem (2.4) to a di erent physical setup. We consider the principal series representation of the conformal algebra so(2; 4) of four-dimensional Minkowski space. We are going to show that any shnet graph G representing the correlator (1.2) solves the relation (2.4) for an appropriate choice of parameters ui = u + i, i.e. it is an eigenstate (which we denote from now on by jGi) of the n-site monodromy. The role of the spin chain sites is played by n external legs (or vertices) of a scalar amplitude, and the spin chain variables xi represent the 4D coordinates at each site. More precisely, the xi are region momenta for planar amplitudes. For an n-particle amplitude they are de ned as usual via pi = x i xi+1, where i = 1; : : : ; n and xn+1 x1. So they have the same dimension as the momenta. We can also think of the shnet graphs as of correlator diagrams. In this case xi are the usual position space coordinates. 3 Conformal Lax operator In order to build the Yangian generators, we need to specify the Lax operator for the conformal algebra. Considering scattering amplitudes with massless legs we have to stick to Minkowski signature, while for amplitudes with all external legs massive or for correlation functions we might also use Euclidean signature. To be speci c, let us choose Minkowski signature, so that the conformal algebra is so(2; 4). Let us denote the generators of so(2; 4) by Mab, a; b = 1; : : : ; 6. We will need the di erential representation of the conformal algebra generators on the space of scalar elds carrying conformal dimension . In this representation the generators ab (Mab) (3.1) take the familiar form. They are given by the following rst order di erential operators of translations (P ), dilatations (D), Lorentz rotations (L ), and conformal boosts (K ): D = P = i ; L K = 2 x (L ) 2i x : (3.2) { 8 { HJEP05(218)3 sab s(Mab) = Now we have all necessary ingredients to write down the main operator for all our considerations. We use the Lax operator for the conformal algebra in the form [39] which is a 4 4 matrix ( ; are matrix indices) with rst order di erential operator entries. It has a nice factorized form [39]: L(u+; u ) = This is a particular case of the principal series representation of weight ( ; 0; 0). The generators depend on one representation label which is the conformal dimension (or conformal spin). We will also need an irreducible spinor representation of so(2; 4). To construct these generators we use gamma-matrices a for the six-dimensional space R2;4. These are matrices in the eight-dimensional space V . Then we consider their commutators 4i [ a; b], which provide a spinor representation of so(2; 4). However, the latter take block diagonal form, i.e. they form a reducible representation with V = V+ V . Taking the Weyl projection onto V+ we obtain the irreducible four-dimensional representation HJEP05(218)3 where the block 2 2 matrices are x i x and p u+ u + u u 2 4 ; are linear combinations of the spectral parameter and the conformal dimension. Let us note that = u+ u + 2. In the following we will use both notations, L(u+; u ) and L(u; ), for the Lax operator (3.5). In Minkowski signature we have = (1; 1; 2; 3) and = (1; 1 ; 2 ; 3), where i, i = 1; 2; 3 are the standard Pauli matrices. The Lax operator (3.5) satis es the RTT-relation (2.1). For more details about the conformal Lax operator see [39]. Let us abbreviate xij xi xj , and x2 x x with Minkowski signature. We will need the following properties of the Lax operator (3.5): We denote by LT the transposed Lax operator in the non-compact physical space (not the auxiliary matrix space), i.e. we have x T = x equivalent to integration by parts. The inverse of the Lax operator coincides with its transposition (up to permutation and shift of parameters) L T (v 2; u 2)L (u; v) = uv : (3.7) { 9 { The Lax operator acts diagonally onto 1 at from this vacuum, we then construct non-trivial states of the non-compact spin chain. The scalar propagator x122 is an intertwining operator permuting spectral parameters of the two-site monodromy [39{42]: x122L1(u; v)L2(w; u + 1) = L1(u + 1; v)L2(w; u)x122: (3.9) Working in Minkowski signature, we can consider (x212), which is the unitary cut of the Feynman propagator 1=(x212 + i ). It satis es the same intertwining relation: (x122)L1(u; v)L2(w; u + 1) = L1(u + 1; v)L2(w; u) (x122): (3.10) The eigenvalue problems for monodromies with the same cyclic ordering of Lax operators are equivalent (cyclicity) [26]: Ln(un; n) : : : L1(u1; 1) jGi = jGi 1 Ln 1(un 1; n 1) : : : L1(u1; 1)Ln(un m 4; n) jGi = e jGi 1; (3.11) where by uk we mean di erent spectral u-parameters for each Lax operator.4 Here the eigenvalues and e related by un+un e = (un+ 2)(un 2) . For the bene t of the reader, in appendix A we adapt the proof from [26] to the conformal Lax operator (3.5). Implications of cyclicity for the rst realization of the Yangian generators are discussed in section 8. 4 Yangian symmetry of bi-scalar shnet graphs: regular boundary Now we consider shnet graphs for the correlators (1.2) and we show that they are invariants of the Yangian algebra. As it was established above, they have a disc topology with the square lattice, \ shnet" structure in the bulk. For simplicity we start with shnet graphs without junction of external legs, see gure 2. They are correlator graphs with all external points x1; : : : ; x2M in eq. (1.2) chosen to be di erent. We call such a boundary regular. These graphs are simpler to comprehend than the generic, irregular case discussed in the subsequent sections. The graphs in gure 1 have a regular boundary. We will demonstrate the Yangian symmetry and its proof for an arbitrary planar shnet graph with 2M disjoint external legs using a rather generic example of such a graph drawn by solid lines in gure 2. It lives in the coordinate representation. As we explained in the introduction, it can be cut out of a rectangular lattice by means of scissors, cutting 4Their relation to the below overall spectral parameter u of the monodromy (see e.g. the u in (4.1)) is given by uk = u + ak with some shift ak. We can also think of the variables xi as of region momenta. In this case they are kineHJEP05(218)3 a sequence of edges along a closed line forming the boundary. This boundary is simplyconnected (disc topology, no holes inside the graph) and it forms a polygon with all angles right. The coordinates of the external legs are arbitrary, so that the graph represents a multi-loop multi-variable Feynman integral. Four-dimensional integrations are assigned to quartic vertices (denoted by blobs in gure 2). The integral depends on variables xi , i = 1; : : : ; 2M , which are coordinates of external legs. These external points of the diagrams are depicted by small white blobs. We can think of the variables xi in two ways. We can consider them as true x-space coordinates, and in this case the graph is a correlator diagram corresponding to eq. (1.2). matic variables of the dual graph (drawn by dotted lines in gure 2) which is a momentumspace Feynman integral. Region momenta are attributed to external faces of the graph, as shown on the gure 4. Since they are related to the usual momenta pi by the duality transformation xi xi+1 = pi they automatically resolve the momentum conservation constraint, p1 + : : : + p2M = 0. Integrations in the dual graph are attributed to loops. Since all variables xi are independent, all momenta pi owing into the dual graph (denoted by solid double lines in gure 2) are o shell. As was noted before, the dual graph (of a shnet graph with regular boundary) does not necessarily correspond to an amplitude in the bi-scalar theory: all in owing momenta are o shell and vertices of valency di erent from four could appear. Nevertheless shnet graphs with regular boundary describe some amplitudes, for example the x-space double cross integral is dual to the p-space double-box integral with six o -shell legs, see gure 4. To embrace the whole set of amplitudes of the bi-scalar theory we will have to consider shnet graphs with more generic boundary set of external legs. notation (see section 6). do not appear. The loop integrals that we consider are nite. By power counting one can see that UV divergences are absent. Since all in owing momenta are o shell, IR or collinear divergences For the moment we consider a shnet graph with regular boundary. Then we draw the contour C, oriented clockwise along the boundary such that it crosses all 2M external legs, see gure 6. It denotes the monodromy matrix L2M : : : L1, and the blue segments depict conformal Lax operators (cf. eq. (3.5)) forming it. Red segments denote contractions of matrix (auxiliary space) indices. Slightly abusing notations we also denote by C the ordered We indicate the parameters of the i-th Lax operator of eq. (3.5) using the shorthand [ i+; i ] (ui+; ui ) (u + i+; u + i ) : (4.1) The rule to assign inhomogeneities along the monodromy is the following: At the rst leg we choose [ 1+; 1 ] = [1; 2] (of course the overall spectral parameter u is allowed to be shifted uniformly in all ui+ and ui along the contour). We do not change inhomogeneities of Lax operators when moving straight along a horizontal or vertical segment of the contour. [4,5] [4,5] [4,5] [3,4] L(u + i+; u + i ), cf. eq. (3.5). operators on the left hand side of eq. (4.2), on the shnet graph from gure 2. The monodromy is depicted by the oriented contour. Lax operators correspond to solid blue segments of the contour and dashed red lines denote summation over matrix indices. The contour is decorated by inhomogeneities [ i+; i ] of the monodromy, which indicate the shifts in the arguments of Lax operators We increase i+1 = i +1 at a convex corner i ! i +1, when turning by an angle =2. We decrease i+1 = i 1 at a concave corner i ! i + 1, when turning by an angle =2. Let us note that these rules are consistent with the cyclicity in (3.11). With the above prescription and according to eq. (3.6), representations of the conformal algebra carry the same conformal weight = 1 on all sites of the spin chain. In the following, also the spin chains carrying representations of di erent conformal weights will naturally arise. We claim that the shnet graph jGi is an eigenfunction of the monodromy ! ! Y Li[ i+; i ] jGi = Y [ i+][ i ] jGi 1 : i2C i2Cout While we have no formal proof for generic graphs, we provide a heuristic rule for how to read o the expression for the eigenvalue in the previous formula from the respective graph. Here we abbreviate [ k ] u + k : We split the set of 2M external legs into pairs of antipodes. For each pair we apply the following rule: a leg which we encounter rst when moving along the monodromy contour belongs to the set Cin, and a leg which we encounter last moving along the contour belongs to the set Cout. So we decompose the set of external legs according to (4.2) (4.3) C = Cin [ Cout. In the example in equals (u) = ([3] [4])5([4] gure 6, we have 2M = 18 and the eigenvalue in (4.2) Proof of Yangian symmetry: the lasso method Before presenting the general proof let us rst demonstrate how it works in several simple cases. We start with the well-known cross integral (one-loop box with four o -shell legs in momentum space): This integral can be evaluated in terms of dilogarithms [43], and the answer is eigenfunction of the monodromy z)(1 z) = xx221143xx222234 . We show that the cross integral is an L4[4; 5]L3[3; 4]L2[2; 3]L1[1; 2] jcrossi = [3][4]2[5] jcrossi 1 : The proof is given by the series of transformations of the monodromy contour in gure 7. Firstly, we extend the \spin chain" monodromy by one additional site corresponding to the integration point x0. More precisely, taking into account eq. (3.8), we multiply the fourpoint monodromy by the Lax operator L0T acting on 1 and integrate it by parts. In this way we obtain the ve-point monodromy acting onto the integrand of (5.1). Thus, e ectively we change the length of the non-compact spin chain and introduce representations with conformal dimension di erent from 1. Then we pull the monodromy through the scalar propagators, which form the integrand, using repeatedly eqs. (3.8) and (3.9).5 Finally, let us provide detailed algebraic expressions corresponding to di erent steps in the sequence of monodromy transformations in gure 7. Step 1. We act with four-point monodromy onto the cross integral (5.1), d4x0 L4[4; 5]L3[3; 4]L2[2; 3]L1[1; 2]x102x202x302x402 : Step 2. We multiply the monodromy by an identity matrix 1 = [2] 1 L0T [2; 0] 1 on the right hand side, eq. (3.7), and integrate the inserted Lax by parts, i.e. L0T ! L0, d4x0 L4[4; 5]L3[3; 4]L2[2; 3]L1[1; 2](L0T [2; 0] 1) x102x202x302x402 d4x0 L4[4; 5]L3[3; 4]L2[2; 3]L1[1; 2]L0[2; 0]x102x202x302x402 : (5.4) 5Using Mathematica, we have checked the invariance of the result (5.2) after integration under the four-point monodromy. We have also explicitly veri ed that (5.2) is invariant under the Yangian level-one generators of section 8. [3,4] [2] L3[3; 4]L0[2; 2]x302 = x302 L3[2; 4] L0[2; 3] : The Lax operators of the external part of the monodromy act on xed external coordinates. The Lax operator introduced around the integrated middle node acts on the coordinate of this node. Numerical factors appearing in the process are indicated above arrows. In the following steps we show that the integrand is an eigenfunction of the ve-point monodromy, L4[4; 5]L3[3; 4]L2[2; 3]L1[1; 2]L0[2; 0] x102x202x302x402 = [2][3][4]2[5] x102x202x302x402 : (5.5) Step 3. We use the intertwining relation6 (3.9) for adjacent Laxes L1 and L0, and then we act by L1 onto 1 according to eq. (3.8), L1[1; 2]L0[2; 0] x102 = x102 L1[0; 2] L0[2; 1] : So one Lax drops out of the monodromy and the ve-point monodromy reduces to the four-point monodromy. Step 4. We repeat analogous simpli cation for the adjacent Laxes L2 and L0 again reducing the length of the monodromy: L2[2; 3]L0[2; 1] x202 = x202 L2[1; 3] L0[2; 2] : Step 5. We implement the simpli cation for the adjacent Laxes L3 and L0: | [2{]z1 } | [3{]z1 } | [4{]z1 } Step 6. Finally, we obtain the two-point monodromy and nd that the propagator is its eigenfunction: L4[4; 5]L0[2; 3]x402 = x402 L3[3; 5]L0[2; 4] = x402 [4][5] 1 : Collecting all numerical factors which appeared in the previous steps we obtain eq. (5.5) and eq. (5.3). Let us consider the double-cross integral (double box with six o -shell legs in momentum L6[4; 5]L5[3; 4]L4[3; 4]L3[2; 3]L2[1; 2]L1[1; 2] jdouble-crossi = [3]2[4]3[5] jdouble-crossi 1 : An explicit expression for this integral in terms of familiar special functions is not yet known. It is believed to be given by a class of elliptic functions. The proof is given by the series of transformations shown in gure 8. Similarly to the cross integral, we use eq. (3.8) to extend the monodromy (the rst transformation in gure 8) by two additional Lax operators corresponding to the integration points x0 and x00 : L6[4; 5]L5[3; 4]L4[3; 4]L3[2; 3]L2[1; 2]L00 [2; 0]L1[1; 2]L0[2; 0] : (5.8) Then we show that the integrand of eq. (5.6) is an eigenfunction of the eight-point monodromy (5.8). We pull the monodromy through the scalar propagators and repeatedly use eqs. (3.8) and (3.9). Integrating the two auxiliary sites L0T [2; 0] 1 1 and L0T0 [2; 0] 1 1 by parts, cf. eq. (3.8), we come back to the six-point monodromy and we obtain the eigenvalue relation (5.7). 5.3 Multiloop integrals Now we want to consider a generic shnet graph, reproducing all possible features of boundary geometries. As a representative of such generic graphs, we can take the one depicted in gure 2. The monodromy around the boundary of this graph, the \lasso", is presented in gure 6. We interpret the shnet graph as an intertwiner for the monodromy matrix, since it is built from scalar propagators which are intertwiners according to eq. (3.9). So, according to our \lasso" method, we want to pull the 2M -point shnet graph through the 2M -site monodromy. We prefer to explain the proof of eq. (4.2) using pictures. Graphically, we pull the oriented contour C through the graph. To deform the monodromy contour we use the elementary transformations in gure 9. One can justify them using the same manipulations as in gure 7 for the single cross integral. [2,0] Numerical factors appearing in the process are indicated above the arrows. the graph which is not touched by the current transformation. Pushing the contour inside the graph involves integrations by parts. If the initial monodromy acts on an L-loop graph, then, after one local transformation, the new monodromy (transformed contour) acts on the (L 1)-loop integral. [4,3] [3,4] [4,5] [3,4] [3,3] [2,3] HJEP05(218)3 the monodromy. Blue segments denote Lax operators in the product. We see that we can always apply the transformations of the gure 9 to any part of this intermediate contour and thus shrink it further eventually reducing it to a point (i.e. the identity matrix). Now, using these transformations, we consecutively pull the monodromy contour in gure 6 through the graph using the transformations in gure 9. An intermediate step is shown in gure 10. We also use the intertwining relations (3.9) and the local pseudo-vacuum of eq. (3.8), to replace the Lax operator Lk[i; i+2] by the diagonal matrix [i+2] 1. In this way we shrink the monodromy contour and e ectively decrease the length of the non-compact spin chain represented by the monodromy. Finally the contour shrinks to a point, so that the monodromy is proportional to the identity matrix. Hence, relation (4.2) is proven. 6 Yangian symmetry of bi-scalar shnet graphs: irregular boundary Now we want to include into our considerations the coordinate space shnet graphs with more generic boundaries. These are the shnet graphs where some of the neighboring external legs are joined into one point. They correspond to correlator diagrams for K(x1; : : : ; x2M ) in eq. (1.2), where several neighboring external points x1; : : : ; x2M coincide. To be more speci c, let us consider a regular diagram decorated with avors, see gure 1 and gure 11. There is a one-to-one correspondence between a diagram and the ordering (up to cyclic shifts) of scalar elds j (xj ); j = 1; : : : ; 2M which carry one of the four avors. We want to identify coordinates of some of the adjacent elds. Putting several elds in a common space-time point might cause divergences in loop diagrams. However in some cases this junction results in nite loop integrals. Let us consider the product of bi-scalar elds of length 4L : : : y1 y2 1 2 y1 y2 1 2 : : : y1 y2 1 2 : : : ; | {z } of a regular boundary (2M = 22). This corresponds to identi cations of adjacent coordinates x3 = x4 = x5, x11 = x12 = x13 = x14, x19 = x20 and x21 = x22 in the correlator K, cf. eq. (1.2). In the picture on the right-hand side, the avors of elds are explicitly indicated. We are allowed to identify coordinates of only those adjacent elds in K which are ordered as in the interaction vertex y1 y2 1 2, see eq. (1.1), up to cyclic shifts. No further junctions are allowed in this picture. where the elds are ordered as in the interaction vertex of the Lagrangian (1.1). We cut a piece of arbitrary length out of this product (indicated by the curly bracket in the previous formula). We allow only this type of products of elds in K to be put at a common space-time point because they correspond to a vertex cut out of the original lattice. An example is given in gure 11, where we joined two, three and four neighboring points: 1 2 sits in x19 = x20; y2 1 sits in x21 = x22; y1 y2 1 sits in x3 = x4 = x5; 1 2 y1 y2 sits in x11 = x12 = x13 = x14. The external points xi in the left gure are decorated by small white blobs as opposed to the lled interior vertices (integration points). We can consider the junction of an arbitrary number of elds. The graphs with junction of ve or more elds live on a multi-sheet lattice, such as in gure 3. The number of external points in a graph with irregular boundary does not have to be even, so we denote the set of external points by x1; : : : ; xN . Let us stress that we do not allow for arbitrary junctions of neighboring legs, since the corresponding integrals are potentially IR divergent. Another, more complicated example of the graphs we consider is given in gure 12, along with its dual momentum-space graph. As we can see, it corresponds to a loop integral with all in owing momenta o shell. Note that dashed double lines in gure 12 correspond to trivial momentum-space propagator factors, involving only external xed momenta. However we will need them in the following. They correspond to coordinate space propagators stretched between two external vertices which are not involved into the integration. Note that the graph in gure 12 originates from a singular square lattice (see footnote 3 and gure 3): the plaquettes around the propagators meet in external points. Integration points are denoted by lled blobs. Propagators stretched between a pair of external vertices are inert with respect to loop integrations. The dual (momentum-space) graph is drawn by dotted lines. All incoming momenta are o shell and denoted by double lines. [2,3] [2,2] [3,2] [4,2] [5,2] two, (C) three, (D) four, (E) ve external legs. Rotating this picture by =2 we shift both inhomogeneities ! vertices with ve neighbors there cannot be cut out of a usual regular square lattice. Such a lattice should have a conical singularity with the angle of a cone equal to 52 . We remind that such conical singularities can appear only at the boundary of the disc. Let us now act with the monodromy on a correlator diagram with irregular boundary. The monodoromy contour C has to cross all external points. We need to specify how we depict a Lax operator which acts simultaneously onto a junction of several external legs. We draw the monodromy contour crossing one, two, three, etc. legs as in gure 13. The rule for the assignment of inhomogeneities to Lax operators is given there. The application of this rule to the correlator diagram from gure 12 is given in gure 14. The external vertices with junction of ve or more propagators appear in graphs living on a singular square lattice [5,5] [5,4] [4,4] [ 1,1 ] [2,1] [6,4] [5,3] [4,4] [4,3] [3,2] [3,3] [3,0] [5,2] i.e. it is an eigenfunction of the monodromy with the indicated inhomogeneities [ i+; i ]. The assignment of inhomogeneities follows the rule in gure 13. (with conical singularities). The junction of external legs changes the conformal weight (we have = k for the junction of k legs, and consequently + + 2 = k). Hence, the inhomogeneities of Lax operators, see eq. (3.6), di er from those for a regular boundary. So considering irregular boundaries, we deal with a spin chain carrying di erent principal series representations of so(2; 4) on di erent sites. A shnet graph jGi with a boundary consisting of the elements from gure 13 (and their rotations by a multiple of =2) is a Yangian invariant ! Y Li[ i+; i ] jGi = (~u) jGi 1 : i2C (6.1) The rule for assigning the inhomogeneities i is given in gure 13. An example graph is given in gure 14. One can establish this relation following the line of the proof for the regular boundary, section 5.3. So we pull the monodromy through the diagram using a sequence of elementary transformations given in gure 9. Note that we have already encountered irregular boundaries in section 5.3 at the intermediate steps of the shrinking contour, see gure 10, where one can see three neighbors at the same point at its intermediate boundary (like the vertex carrying the Lax operator with argument [4; 3]). Explicit expressions for can be worked out for each particular graph. We just need to keep track of numerical factors pulling the monodromy contour through the graph. There is, however, a quicker way to nd . Applying N times the cyclicity property of eq. (3.11), we obtain the initial eigenvalue problem with spectral parameter u shifted uniformly in all Lax operators. Comparing the eigenvalue of the initial eigenvalue problem and of the cyclically rotated one, we conclude that they satisfy the nite-di erence equation (u) (u 4) = P (u) P (u 2) ; (u) = Y(u + ai) ; P (u) = Y(u + i+)(u + i ): where (u) is a polynomial of degree N , and P (u) is a known (for each contour) polynomial of degree 2N , Eq. (6.2) is reminiscent of the Bethe Ansatz Equations (BAEs). In comparison with the usual BAEs, it is easy to solve equation (6.2), i.e. to nd roots of the polynomial (u). This and f i 2giN=1 have to contain at least N common elements. Then the parameters faigiN=1 determining the eigenvalue (u) are such that fai; i eq. (6.2) in the example in gure 14, we immediately nd (u) = [2][3]7[4]11[5]6[6]. 2giN=1 = fai 4; i giN=1. Solving Alternatively, we may perturbatively solve the relation (6.2) for the coe cients in the u-expansion of the monodromy eigenvalue . This will be helpful for making connection to the rst realization of the Yangian in section 8. For the rst two orders of the expansion we obtain (~u) = un + 2 In appendix B we display the rst ve orders of the expansion. Here we use the shorthand notations ^k = k+ + k + 2 and ^ i = i ( i Scattering amplitudes and cuts of shnet graphs Now we want to consider scattering amplitudes in the bi-scalar theory with massless external states. More generally, we want to put some of the in owing momenta in the momentum Feynman integrals considered above on shell, like in gure 12. We use region momenta xi as the amplitude variables. They are coordinates of the external legs of shnet graphs. All variables xi were independent and unconstrained in the previous considerations. However for a light-like momentum, pi2 = 0, the region momenta have to be constrained by xi2i+1 = 0. Fortunately, we can easily impose the constraints on region momenta in a way consistent with Yangian symmetry. We note that scalar propagators xij2 and distributions (xi2j ) have the same conformal weights and they satisfy identical intertwining relations | compare eqs. (3.9) and (3.10). So if we take a shnet graph jGi which respects the Yangian symmetry, eq. (6.1), and replace a scalar propagator xij2 by the delta-function (xi2j ), then we naively obtain another Yangian invariant jGicut which satis es the same eigenvalue relation (6.1) with the same inhomogeneities: T (~u) jGi = (~u) jGi 1 ) T (~u) jGicut = (~u) jGicut 1 : (7.1) in the bi-scalar theory. The shnet graph is formed by solid lines (which denote scalar propagators xij2) and dashed lines (which denote cut propagators (xi2j)). The dual graph is drawn by dotted lines which denote scalar propagators pi 2. All in owing momenta are on shell. They are denoted by thin dotted lines which cross cut propagators. This picture is obtained from gure 12 by cutting a number of (or all) propagators along the boundary in the correlator graph. This cutting in x-space corresponds to the LSZ amputation of external propagators in the dual p-space graph. In the literature, this substitution of a number of propagators 1=(p2 + i ) in a Feynman graph by their imaginary parts (p2) is known as the generalized cut. So, at rst sight, the cutting of Feynman graphs seems consistent with Yangian symmetry. Now, if we cut a propagator stretched between two adjacent external points xi and xi+1 of a shnet graph, we put the corresponding momentum pi of the dual graph on shell, i.e. we set pi2 = 0. For example, in the o -shell Feynman graph in gure 12 this cutting procedure leads to the Feynman integral in gure 15 with all in owing momenta light-like. So we describe the amplitudes using region momenta xi and consider them as distributions, jGicutn = Qn i=1 (xi2i+1) An, where An is a regular function. We can also consider mixed objects, where only a part of the in owing momenta are on shell. We just need to cut a smaller number of propagators in gure 12, which also results in distributions. One can see that for any scattering amplitude in the bi-scalar theory the corresponding dual shnet graph has the topology considered in sections 4 and 6. The loop amplitude diagrams for massless particles do not su er from IR or collinear divergences. Such divergences are typical for loop diagrams with emission of a massless particle through a cubic vertex. However, in our diagrams the external light-like momenta ow into quartic vertices, and in the corners light-like momenta enter in pairs (each of which is equivalent to one o -shell momentum).7 7We are grateful to G. Korchemsky and J. Henn for discussions on this point. According to eq. (7.1) we are allowed to cut internal propagators as well. In this way we localize some of the loop integrations reducing the complexity of the Feynman integral. Let us give several examples. Cutting four propagators of the cross integral (5.1) we localize all integrations and the result is an algebraic function 1=pdet g, where gij = xi2j , i; j = 1; : : : ; 4, [44]. It is a Yangian invariant, i.e. it satis es the same eigenvalue relation (5.3) as the cross integral. We checked this independently. Let us note that the solution x0 of the constraints x210 = x220 = x320 = x420 = 0 is complex-valued. So strictly speaking it does not belong to the support of the delta-function. In fact, we need to understand the cutting as deforming the integration contour and taking the residue in the pole speci ed by the cut propagator. Cutting all seven propagators of the double box with six external legs, we localize seven of eight Feynman integrations. The resulting cut integral has a one-fold integral representation, which can be expressed in terms of elliptic integrals. The cut of the double-cross integral, as well as the integral itself, satis es the eigenvalue relation (5.7), and so it is a Yangian invariant. Note with regard to the on-shell generalization discussed in this section that conformal symmetry of massless scattering processes typically comes with a subtle anomaly-like behavior arising from special (external or internal) momentum con gurations. This subtlety usually originates from the conformal generators hitting a three-point vertex with two collinear particles, see e.g. [13, 14, 45{48]. Such an anomaly-like behavior also plays an important role in the present context as can be seen in the recent discussion of conformal symmetry of loop integrals contained in [49]. We leave the detailed discussion of this subtlety in the context of the above on-shell formalism for future work. 8 Fishnet graphs and rst realization of the Yangian Above we have formulated the Yangian symmetry of shnet graphs in the language of the RTT realization. In order to study explicit constraints in form of (non-local) di erential equations in the future, it should be useful to make contact to Drinfel'd's rst realization of the Yangian algebra [50]. This rst realization is closer to the formulation of Lie algebras and given in terms of two types of generators, which can be obtained from the expansion of the monodromy T (u). 8.1 First realization In general, the Yangian is an in nite-dimensional algebra which can be understood as an extension of an underlying Lie algebra. In its rst realization, it is spanned by the Lie algebra generators J A and a second set of generators JbA transforming in the adjoint representation of the underlying Lie algebra, i.e. J A; J B = f AB C J C J A; JbB = f AB C JbC : (8.1) The generators JbA are typically referred to as the level-one generators while the Lie algebra generators, in the context of the Yangian, are called the level-zero generators. In addition to the two Jacobi-identities J A; J B; J C + cyclic = 0; J A; J B; JbC + cyclic = 0; (8.2) the level-zero and level-one generators should obey the Serre relations JbA; JbB; J C + cyclic = 4 1 f AGDf BH Ef CK F fGHK J fDJ EJ F g ; where the symbols fGHK follow from f GH K by lowering two indices with the inverse of the Cartan-Killing form of the underlying Lie algebra. The brackets f g denote total symmetrization of the enclosed indices. From an algebraic point of view the Yangian is a Hopf algebra and as such it comes equipped with a coproduct. The coproduct furnishes a prescription for how to extend a single-site representation to a multi-site representation and takes the following standard form Cross. The cross diagram corresponds to the following Feynman integral: Here, we again consider the four external coordinates x1; : : : ; x4 as independent, i.e. we interpret the cross graph as an o -shell correlator diagram. As discussed in the previous sections, the integral (8.6) is invariant under the generators of the four-dimensional conformal algebra so(2; 4) with generators as de ned in equation (3.2): Here J A is de ned as (J A) = J A (JbA) = JbA Based on this coproduct we can write down the following ansatz for a general level-one generator Here, f ABC are the inverse structure constants which follow from the ordinary structure constants by lowering one Lie algebra index with the Cartan-Killing form. The so-called evaluation parameters vk can be chosen arbitrarily without spoiling the above de ning relations of the Yangian. In many cases, like for example in the context of scattering amplitudes in N = 4 SYM theory [12], the local contribution (i.e. the single sum term) is absent and we have vk = 0 for all k. 8.2 Pedagogic examples: cross and double cross In order to illustrate the explicit form of the above Yangian generators on shnet graphs, we consider two simple examples. That is, we explicitly construct the level-one generators, which annihilate the one-loop cross and the two-loop double cross Feynman graph. J A jcrossi = 0 : J A = 4 X JiA ; i=1 (8.3) (8.4) (8.5) (8.6) (8.7) (8.8) with JiA 2 fPi; ; Li; ; Di; Ki; g being the single-site generators of the conformal algebra (3.2) with the conformal dimensions i xed uniformly to be equal to 1. We will now demonstrate how to construct level-one generators which annihilate the cross integral. Note that in the case of the conformal algebra it su ces to show invariance under one level-one generator. The level-zero invariance together with the Yangian commutation relations (8.1) guarantees that all the other level-one generators annihilate the cross integral as well. In what follows, we will choose the simplest level-one generator which is the levelone momentum generator Pb . Using the formula (8.5) we nd the following expression for the bi-local piece of Pb : Pbbi = (Lj + Dj )Pk; (j $ k) ; 2 X j<k=1 (8.9) HJEP05(218)3 where Lj , Dj and P k are the single-site conformal generators as introduced above. Applying this di erential operator to the box integral (8.6) and completing squares in the numerators yields Pbbi jcrossi = i Z d4x0 2x20 x210x420x320x420 + 4x30 x210x220x340x420 + 6x40 x210x220x320x440 (8.10) where we have dropped a total derivative term. The above expression can easily be seen to be equivalent to Pbbi jcrossi = P 2 + 2P3 + 3P4 jcrossi ; which makes it obvious that we can use the freedom to choose the vk's in equation (8.5) to construct a true symmetry generator. Explicitly, we de ne the full level-one momentum generator as Pbcr := Pbbi P 2 2P3 3P4 : Double Cross. As a second example, let us consider the double cross diagram. The corresponding Feynman integral reads jdouble-crossi = Z d4x0 d4x00 x210x220x320x400 x500 x600 x000 2 2 2 2 : Applying the bi-local generator (8.9) to the double cross integral (8.13) yields Pbbi jdouble-crossi = P 2 + 2P3 + 2P4 + 3P5 + 4P6 jdouble-crossi : Again, we see that we can de ne an algebraically consistent level-one momentum generator that annihilates the integral (8.13) by choosing the inhomogeneities as follows: Pbdcr := Pbbi P 2 2P3 2P4 3P5 4P6 : Given the discussion of shnet graphs within the RTT realization in the previous sections, it is in fact clear that the above procedure has to work for an arbitrary graph of shnet type. This is due to the fact that the monodromy matrix packages all Yangian generators in a very e cient way, cf. (2.2). For this reason, we will now derive in detail the generic relation between the monodromy matrix, the evaluation parameters vk and the explicit { 25 { (8.11) (8.12) (8.13) (8.14) (8.15) In order to obtain the explicit form of the Yangian generators, we expand the following monodromy matrix in the spectral parameter u: T (~u) = Ln(u + n+; u + n )Ln 1(u + n+ 1; u + n 1) : : : L1(u + 1+; u + 1 ) We employ the Lax operator given in (3.4) and (3.5), which yields T (~u) = un 1 + u ^k 1 + kabsab 1 n 1 X 2 n k=1 2 n k 1 + 8 u ^ 1 + jabsab j ^k 1 + ckdscd + : : : : at order un 1 as a symmetry operator which annihilates the considered graphs. At the next order, we rewrite 1 4 n X j<k=1 1 8 n X j<k=1 kab jcdsabscd = kab jcd[sab; scd] + kab jcdsabscd kab ckdsabscd: 1 8 n X j;k=1 1 Xn 8 k=1 T (~u) " (~u) 1 = 0 un 1 +un 1 1 Xn kabsab # 2 k=1 This allows to identify the level-zero generator +un 2 6 1 2 X 64 4 j<k=1 kab jcdsabscd + X ^j kabsab 1 Xn 4 k=1 j=1 n j6=k 8 1 Xn(4 k=1 3 5 k) k 177+: : : : Here we use the shorthand notation ^k = k+ + k + 2 and the abbreviations of section 3: sab = s(Mab) and k;ab = k (Mab). In order to identify the level-zero and level-one generators at order un 1 and un 2 of the above expansion, respectively, we also have to consider the function on the right hand side of the generic monodromy equation (6.1), whose expansion is given in (6.4) (cf. (4.2) for the case = 1): (~u) =un + 2 k=1 2 n 4 X i<j=1 ^i ^j 3 j=1 2 1 Xn ^ j 5 + O(un 3): Here ^ i = ( i k = k + 2. We can thus subtract the eigenvalue from the monodromy to nd the following operator which annihilates invariants under the Yangian algebra for arbitrary spectral parameter u: 4 1 n 2 u k J ab = 1 Xn 2 k=1 ab k (8.16) (8.17) (8.18) (8.19) (8.20) (8.21) The rst term on the right hand side reproduces the bi-local piece of the level-one generator. The second term is the product of two level-zero generators, which annihilates the diagrams under consideration and can thus be dropped. Noting that kab ckdsabscd = ( k 4) k 1 4 kabsab; we can rewrite the last term according to 1 Xn 8 k=1 kab ckdsabscd = 8 1 Xn (4 k=1 k) k 1 + kabsab: 1 Xn 2 k=1 Here the rst term cancels the piece proportional to the identity in (8.19), while the last term is the level-zero generator (8.20) and can thus be dropped. Collecting the remaining terms at order un 2 of (8.19), we thus nd the level-one generator to be given by b X j<k=1 ckd jef + 2 1 Xn vk kab: k=1 Here fcd;ef ab denotes the structure constants with [sab; scd] = fab;cdef sef and the evaluation parameters take the form vk = 2 1 Xn ^j : j=1 j6=k The level-zero and level-one generators in (8.20) and (8.24), respectively, agree with the expressions found on the single and double cross. 9 Dual conformal symmetry and the Yangian in momentum space In N = 4 SYM theory the Yangian invariance of scattering amplitudes is known to be equivalent to their superconformal and dual superconformal symmetry [11, 51]. It is thus natural to ask whether a similar statement can be made rigorous in the case at hand | namely, for planar scattering amplitudes in FT4 which enjoy ordinary conformal and dual conformal symmetry, cf. [52] for a discussion of the one-loop box. At the same time one may wonder whether an analogue of the above construction of the coordinate-space Yangian can also be performed in momentum space. In what follows, we will explicitly demonstrate that the generator of special conformal transformations in the dual (coordinate) space can be rewritten as a Yangian level-one generator acting in on-shell momentum space. We thus derive the Yangian symmetry in momentum space and establish its equivalence to dual conformal symmetry. Dual conformal symmetry. We start this section by brie y discussing the dual conformal symmetry of planar bi-scalar scattering amplitudes. To expose this symmetry it is convenient to pass to the dual variables which are de ned through the relation pi = xi xi+1 : (8.22) (8.23) (8.24) (8.25) (9.1) p3 FT4. All momenta are taken to be ingoing. In contrast to the situation in N = 4 SYM theory the dual conformal symmetry of the considered FT4 model is less universal in the sense that the dual conformal symmetry generators depend, at least in part, not only on the number of external legs but also on the structure of the amplitude itself. To make this statement more clear let us consider the three simple examples of amplitudes displayed in gure 16. Using Feynman rules we write down the following expressions for the amplitudes depicted above: At6 = 2 x14 x7) ; At8 = A81l = Z (4)(x1 d4x0 x210x230x520x70 x92) : (9.2) A few comments concerning these amplitudes are in order. In equation (9.2) we have already introduced the dual coordinates x1; x2; : : : ; xn which are related to the external momenta as stated in equation (9.1). Obviously, the denominators just represent the region momenta owing through the di erent propagators. Furthermore, note that in the above formulas we have relaxed the cyclicity condition on the x's at the cost of a fourdimensional delta-function (4)(x1 xn+1) reimposing the closure of the light-like polygon. This delta-function corresponds to the momentum conserving delta-function (4)(P), where P = Pn j=1 pj , and we have inserted it here for later convenience. On the contrary, we have not included the one-dimensional delta-functions ensuring the light-likeness of the edges xi xi+1. The reason for this is that the dual conformal generators as well as the levelone on-shell momentum-space generators manifestly respect the light-likeness condition, so that we can safely disregard these delta-functions. This being said, let us now take a closer look at the dual conformal properties of these amplitudes. The representation of the dual conformal algebra that we will use here was introduced in equation (3.2) and for pedagogical reasons we will start with a representation with conformal dimension i = 0 at each site. The amplitudes (9.2) are manifestly invariant under translations and rotations and thus they are annihilated by the corresponding generators. Acting with the dilatation generator on the amplitudes in equation (9.2) yields DAt6 = 6iAt6 ; DAt8 = 8iAt8 ; DA81l = 8iA81l ; (9.3) Note that due to the relaxed cyclicity condition, the dual conformal generators are summed up to n + 1 instead of n. From equation (9.3) we anticipate that acting with the conformal dilatation generator on an amplitude just yields the number of external legs times i and it is actually not too hard to convince oneself that this statement holds true for all the planar scattering amplitudes in FT4. In order to make the dilatation generator a true symmetry generator we will now use the freedom to choose the conformal weights for the terms on the right hand side of equation (9.3). The conformal weights i to compensate dilatation generator in the following way: There are actually many choices that lead to a modi ed generator D0 which annihilates the amplitudes (9.2) but the most natural one is: i X i=1 D0 = i : ~ At = (4 + 1; 0; 0; 1; 0; 0; 0) ; 6 (9.4) (9.5) (9.6) (9.7) (9.8) (9.9) where D = I[xi2j ] = 2 xij ; x2x2 i j Note that we can always choose n+1 = 0 as the delta-function allows us to eliminate the coordinate xn+1 in favour of x1. The factors of four in the above equations compensate for the weight that is introduced by the delta-function while all the other numbers are chosen such that the weight coming from the corresponding coordinate is cancelled out. Having discussed the dilatation symmetry, let us focus on special conformal transformations. Special conformal transformations are most conveniently studied by noting that the generator of special conformal transformations is related to the generator of translations by the following formula: Here, I is the inversion element which acts on the coordinates as I[x ] = x =x2. As a rst step towards establishing the action of the generator K on the amplitudes (9.2) let us study their inversion properties. Using that the region momenta invert as as well as the formula I[ (x1 xn+1)] = x81 (x1 xn+1),8 we nd I[At6] = x110x24At6 ; I[At8] = x110x23x46At8 ; I[A81l] = x110x23x25x27A18l : 8This formula can easily be derived by considering the de nition of the delta-function R d4x1 (x1 xn+1) = 1 and noting that under an inversion the measure transforms as I[d4x1] = d4x1=x18. In contrast to the situation in N = 4 SYM theory the amplitudes obviously do not transform in a completely covariant way. This is on the one hand due to the fact that there is no supermomentum conserving delta-function present, which could balance out the inversion weight of the bosonic delta-function. On the other hand, also the amplitude functions themselves do not transform in a completely homogeneous way, as some of the x's are simply not present while others come with a power higher than 2. Having studied the inversion properties of the amplitudes, we can now easily write down expressions for the action of the generator K on the three amplitudes (9.2). Using equation (9.7) we nd: K _ At6 = i 5x1_ + x4_ t A6 ; _ + x5 _ + x7_ A81l : D An = in An ; D0 = i ; n+1 i X i=1 with the i's as de ned in equation (9.6). Our examples and equation (9.6) make it clear that the generators D0 and K0 _ are no longer universal as the vector ~ does not only depend on the number of external legs but also on the amplitude itself. However, note that the generators D0 and K0 _ are perfectly consistent with the algebraic restrictions imposed on them by the conformal commutation relations. Hence, the generators P , L , D0 and K0 still furnish a representation of the conformal algebra so(2; 4). Finally, let us comment on the situation for a generic planar amplitude in bi-scalar FT4. As already mentioned above, the plain dilatation generator D acts on an amplitude as follows: For a given planar amplitude the modi ed dilatation generator annihilating the amplitude, can be constructed in the following way: rst, we determine which xi's will be absent in the amplitude by drawing the amplitudes' dual graph and where we employed the standard de nition: for later convenience we have written x and K as 2 2 matrices being de ned as As in the case of dilatation symmetry we can now de ne a modi ed generator K0 _ which annihilates the amplitudes by setting K _ _ x ; n+1 i X x i=1 i_ x _ K0 _ = K _ n i X i=1 i xi_ ; (9.10) (9.11) (9.12) (9.13) (9.14) (9.15) (9.16) (9.17) setting to zero all the corresponding i's. For the remaining xi's we set the corresponding i's equal to the number of lines which meet in the point xi. Finally, we set to zero n+1 and add a factor of 4 to 1 to compensate for the weight of the delta-function. The resulting generator D0 will then annihilate the considered amplitude. The generator of special conformal transformations annihilating this amplitude follows immediately from the algebra. Explicitly, it reads: i's satisfy the following relation: HJEP05(218)3 n X j<i=1 i X i=1 ~ _ x 1 n X j<i=1 ~ _ j j X j<i+1=1 i x1_ : and dropping the term that includes a derivative with respect to x, we nd K0 _ = i ~ _ i j j + i Note that the amplitudes can always be written as distributions depending exclusively on the spinor helicity variables f ig and f~ig. For this reason we could safely disregard We will use this equation in the next paragraph, when we establish the connection between the generator K0 _ and the level-one momentum generator. Yangian symmetry in momentum space. In this paragraph we will now demonstrate that the generator K0 _ agrees with the conformal level-one momentum generator up to terms which annihilate the amplitudes by themselves. The discussion follows closely the one presented in [51], where the statement was proved for the case of psu(2; 2j4). To rewrite K0 as an operator acting in on-shell spinor helicity space, we rst extend it such that it commutes with the constraint The result reads i K0 _ = K _ i xi_ : n X n i X i = n : (9.18) (9.19) (9.20) (9.21) (9.22) (9.23) K0 _ i X x i=1 i_ x x i i xi_ Using the inverse of equation (9.20) i=1 x i = x 1 { 31 { all terms containing a derivative with respect to the dual coordinates. Starting from equation (9.23) it is now however a straightforward exercise to rewrite K0 _ as K0 _ 1)Pj_ + P _ L + P _ L _ _ + P _ D P _ i X i=1 1_ L + x1 Li_ _ + x1_ Di ; (9.24) where we have introduced the conformal generators written in terms of spinor helicity variables HJEP05(218)3 Di = 1 12 ~i_ @~i _ + 1 : P j_ L + P j Li_ _ + Pj_ Di (i $ j) ; helicity, i.e. us with we see that most of the terms on the right hand side of equation (9.24) drop out, leaving 1 2 K0 _ = 2 Pb b_i + i X ( i j<i=1 1)Pj_ : (9.25) (9.26) (9.27) (9.28) The generator Pbb_i in equation (9.24) is the level-one momentum generator as it follows from the formula (8.5) with the underlying level-zero algebra being the conformal algebra spanned by the generators (9.25). Note that in order to bring K0 _ to the above mentioned form, we have also used the constraint equation (9.19), which allowed us to replace the i's by 1 in the term that contributes to x1_ Di. Finally, using the level-zero invariance of the amplitudes as well as the fact that all the external particles have zero 2 Pbb_i + i 2 i Xn Pi_ i=1 X ( i j<i=1 L Pi_ Pbb_i = X The local term would obviously vanish if all the i's were equal to 1 as they are for example in the case of N = 4 SYM theory. However, since this is not the case here, we arrive at a purely bosonic Yangian generator with non-vanishing evaluation parameters, which is in complete analogy with the x-space level-one momentum generator that we considered before. 10 Discussion and outlook In this paper we established the conformal Yangian symmetry of single-trace correlators and amplitudes in the planar approximation of the bi-scalar FT4 theory (1.1). This theory appeared in a speci c, double scaling limit of -twisted N = 4 SYM theory [16]. Each of the above observables is given by a single, generically multi-loop Feynman graph with the topology of a disc. In the bulk, the disc typically has the structure of Zamolodchikov's shnet Feynman graphs [21], i.e. it represents a piece of regular square lattice. The in nitedimensional Yangian over the conformal algebra so(2; 4) was explicitly constructed in its RTT realization. Here, the above coordinate-space Feynman graphs were shown to furnish eigenstates of an inhomogeneous monodromy matrix in the spirit of the work [26, 27] (see also [28]). The so(2; 4) Lax operators forming this monodromy have speci c inhomogeneity parameters depending on the shape of the boundary. Via expansion of this monodromy we obtained the respective level-zero Lie algebra generators and the bi-local level-one generators of the Yangian in its rst realization, which annihilate the expressions represented by shnet diagrams. For graphs with massless external legs, alias massless scattering amplitudes in the bi-scalar theory, we then demonstrated that the dual conformal symmetry is equivalent to a Yangian level-one symmetry in momentum space. Importantly, the above Feynman integrals can be argued to be free of divergencies, and hence there is no need for introducing a regulator which could break the conformal (Yangian) symmetry. As opposed to N = 4 SYM theory, the Yangian symmetry of the full (all-loop) planar scattering matrix of the bi-scalar FT4 is thus an exact statement. More over, the breakdown of conformal symmetry by the double-trace terms in the action [17{20] seems to be not an issue here since we simply do not have such anomalous amplitudes in the planar limit. The Yangian provides bi-local (with respect to the coordinates of external legs) di erential equations for all Feynman integrals of shnet type with the disc topology. Notably, at present only the simplest of these integrals, i.e. the one-loop scalar box, has been solved. We are optimistic that the discovered Yangian symmetry will open the door to computing the respective higher-loop integrals via the powerful toolbox of integrability, as it happened to large classes of multi-loop graphs of bi-scalar FT4 theory, such as \wheel"-graphs [16, 53] and magnon correlator graphs [22] relevant for the computation of anomalous dimensions. Taking the bi-scalar FT4 as a starting point, we may wonder what the above precise formulation of its integrability teaches us about N = 4 SYM theory. In particular, it would be interesting to look for a connection to the Yangian symmetry that lurks behind the Qbar-equation and its level-one counterpart for the nite BDS-subtracted S-matrix of N = 4 SYM theory [14]. As a starting point, one might try to understand the symmetries of scattering amplitudes in the -deformed N = 4 SYM theory as an expansion around the bi-scalar case considered here. Recently, the Yangian symmetry of N = 4 super Yang-Mills theory was understood on the level of its action [54]. It would be highly interesting to adapt the developed criterion for the integrability of planar gauge theories in four dimensions to the bi-scalar theory under investigation. Eventually, this might allow to derive our Yangian symmetry of correlators and amplitudes from the Lagrangian. As is well known, massless scattering amplitudes in N = 4 SYM theory are dual to polygonal Wilson loops with light-like edges in the strong-coupling [55] and weakcoupling [56] regimes. This duality serves as a natural explanation of the ordinary and dual conformal, alias Yangian symmetry of scattering amplitudes. Due to the absence of a gauge eld in the bi-scalar FT4, the de nition of a Wilson loop and hence a possible translation of this duality is not obvious. In order to better understand how to formulate a Wilson loop in this theory, it may be fruitful to rst forget about the polygonal contour and to consider the bi-scalar limit of a -deformed smooth Maldacena-Wilson loop [57, 58] which, in addition to the gauge eld, also couples to the scalars of the theory. Let us remind that every Yangian-invariant scattering amplitude in N = 4 SYM theory can be written as an integral over a Grassmannian [15]. Identifying a similar geometric structure for the bi-scalar amplitudes at hand would certainly be of great importance. Note that the basic scalar correlators considered in this paper do not exhaust all possible single-trace correlators of the bi-scalar theory, not even in the planar limit. For instance we can include composite elds (more general than those discussed in section 6). Using the OPE, such correlators can be obtained from the basic correlators studied in this paper. It would be interesting to understand whether the Yangian symmetry extends to such correlators. Finally, a similar Yangian symmetry of planar amplitudes exists in the threedimensional analogue of the bi-scalar theory, which can be obtained by a similar double scaling limit of the three-dimensional -deformed ABJM model (\tri-scalar" theory) [22], where we deal with regular triangular shnet graphs. The same holds for a similar sixdimensional tri-scalar theory with chiral cubic interactions, recently studied in [59], dominated by regular hexagonal shnet graphs (these models exhaust all three types of graphs whose integrability was noticed by A. Zamolodchikov).9 We will address these questions in future work. Acknowledgments We are thankful to B. Basso, J. Caetano, L. Dixon, J. Henn, G. Korchemsky and J. Plefka for discussions. The work of V.K. and D-l.Zh. was supported by the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme FP7/20072013/ under REA Grant Agreement No.317089 (GATIS), by the European Research Council (Programme \Ideas" ERC-2012-AdG 320769 AdS-CFT-solvable). V.K. is grateful to Humboldt University (Berlin) for the hospitality and nancial support of this work in the framework of the \Kosmos" programme. D.M. gratefully acknowledges the hospitality of the Mainz Institute for Theoretical Physics during the workshop \Amplitudes: Practical and Theoretical Developments". A Cyclicity In this appendix we prove the cyclicity property (3.11) of the eigenvalue problem adopting the arguments from [26] to the conformal Lax (3.5). 9Both 3D and 6D theories appear to be true CFT's in the planar limit [59]. We use shorthand notations L(u ) L(u+; u ) and u u+u . We need the inver L 1(u ) = u 1 L( u ) ; (Lt) 1(u ) = (u + 2) 1 Lt( u 4) (A.1) (A.2) To prove the cyclicity we apply several times inversions and matrix transpositions: Step 1. We start with the eigenvalue relation Step 2. We invert Ln, Ln(un ) : : : L1(u1 )jGi = jGi 1 Ln 1(un 1 ) : : : L1(u1 )jGi = Ln 1(un )jGi Lt1(u1 ) : : : Ltn 1(un 1 )jGi = Ln 1t(un )jGi Step 3. We act by matrix transposition on both sides of the previous eq., Step 4. We again invert Ln applying eqs. (A.1), (A.2), where e Ltn(un un 1 (un 2). 4)Lt1(u1 ) : : : Ltn 1(un 1 )jGi = ejGi 1 Step 5. We act by matrix transposition on both sides of the previous eq., Ln 1(un 1 ) : : : L1(u1 )Ln(un 4)jGi = e jGi 1 Eq. (3.11) is proven. 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Dmitry Chicherin, Vladimir Kazakov, Florian Loebbert, Dennis Müller, De-liang Zhong. Yangian symmetry for bi-scalar loop amplitudes, Journal of High Energy Physics, 2018, 3, DOI: 10.1007/JHEP05(2018)003