Yangian symmetry for biscalar loop amplitudes
HJE
Yangian symmetry for biscalar 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
Deliang 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, HumboldtUniversiat 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 allloop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable biscalar 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 biscalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes.
Integrable Field Theories; AdSCFT 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 biscalar shnet graphs: regular boundary
5 Proof of Yangian symmetry: the lasso method
6 Yangian symmetry of biscalar 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], multipoint correlators [6], cusped Wilson loops, the
quarkantiquark 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 allloop 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 nonzero e ective coupling
3, this limit gives rise to a biscalar
quantum
eld theory with the fourdimensional 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 nonunitary,
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 doubletrace 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
doubletrace 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 nonunitary, 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 nonunitarity
is a price to pay for one of the remarkable features of the biscalar 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 biscalar
FT4 represents a
eldtheoretical 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 BDSsubtracted 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 singletrace 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 clockwise 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 biscalar 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 singletrace correlator of scalar
elds in
the planar approximation of the biscalar
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
onetoone 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 colorordered 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 \doublesheet" 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 doublecross 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 biscalar 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
LehmannSymanzik
Zimmermann (LSZ) procedure.
Notice that in the left gure 4 the dual graph with doubleline legs represents a
particular, doublebox 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 biscalar 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 biscalar 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 shnettype 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 lightlike
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 biscalar 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 levelone 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 socalled 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 YangBaxter equation
for the monodromy matrix T (u), the socalled RTTrelation:
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 Rmatrix
where P is the permutation matrix. It satis es the YangBaxter equation which is a
consistency relation for the structure constants of the Yangian algebra given by the RTTrelation.
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 noncompact
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 RTTrelation 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 ith 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 nondiagonal 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 noncompact spin chain with
JordanSchwinger representations of the algebra sl(N ) was studied, and the Roperator method
for the superconformal algebra gl(4j4) was investigated in [27{31] to construct treelevel
scattering amplitudes in N = 4 SYM, as well as in ABJM theory [32]. The Roperator
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 fourdimensional
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 nsite 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 nparticle 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 gammamatrices a for the sixdimensional space R2;4. These are
matrices in the eightdimensional 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 fourdimensional 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
RTTrelation (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 noncompact 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 nontrivial states of the noncompact spin chain.
The scalar propagator x122 is an intertwining operator permuting spectral parameters
of the twosite 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 uparameters 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 biscalar 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
multiloop multivariable Feynman integral. Fourdimensional 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 xspace 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 biscalar 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 xspace double cross integral is dual to the pspace doublebox
integral with six o shell legs, see
gure 4. To embrace the whole set of amplitudes of
the biscalar 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 ith 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 wellknown cross integral (oneloop 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
vepoint monodromy acting onto the integrand of (5.1). Thus, e ectively
we change the length of the noncompact 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 fourpoint 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
fourpoint monodromy. We have also explicitly veri ed that (5.2) is invariant under the Yangian levelone
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
vepoint
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
vepoint monodromy reduces to the
fourpoint 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 twopoint 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 doublecross 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] jdoublecrossi = [3]2[4]3[5] jdoublecrossi 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 eightpoint
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 sixpoint 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 Lloop 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 pseudovacuum 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 noncompact
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 biscalar 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 onetoone 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 spacetime point might cause divergences in loop diagrams. However
in some cases this junction results in nite loop integrals. Let us consider the product of
biscalar 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 righthand 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
spacetime 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 multisheet 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 momentumspace 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 momentumspace 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
(momentumspace) 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 nitedi 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
uexpansion 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 biscalar 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 lightlike 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 deltafunction
(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 biscalar 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 xspace
corresponds to the LSZ amputation of external propagators in the dual pspace 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 lightlike.
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 biscalar 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 lightlike momenta
ow into quartic vertices, and in the corners lightlike 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 complexvalued. So strictly speaking it does
not belong to the support of the deltafunction. 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 onefold
integral representation, which can be expressed in terms of elliptic integrals. The cut of
the doublecross 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 onshell generalization discussed in this section that
conformal symmetry of massless scattering processes typically comes with a subtle anomalylike
behavior arising from special (external or internal) momentum con gurations. This
subtlety usually originates from the conformal generators hitting a threepoint vertex with
two collinear particles, see e.g. [13, 14, 45{48]. Such an anomalylike 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 onshell 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 (nonlocal) 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 nitedimensional 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 levelone generators while the Lie algebra
generators, in the context of the Yangian, are called the levelzero generators. In addition
to the two Jacobiidentities
J A; J B; J C
+ cyclic = 0;
J A; J B; JbC
+ cyclic = 0;
(8.2)
the levelzero and levelone 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 CartanKilling 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 singlesite representation to a multisite 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 fourdimensional
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 levelone
generator
Here, f ABC are the inverse structure constants which follow from the ordinary structure
constants by lowering one Lie algebra index with the CartanKilling form. The socalled
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 levelone generators,
which annihilate the oneloop cross and the twoloop 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 singlesite 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 levelone generators which annihilate the cross integral.
Note that in the case of the conformal algebra it su ces to show invariance under one
levelone generator. The levelzero invariance together with the Yangian commutation
relations (8.1) guarantees that all the other levelone generators annihilate the cross integral
as well. In what follows, we will choose the simplest levelone generator which is the
levelone momentum generator Pb . Using the formula (8.5) we nd the following expression for
the bilocal 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 singlesite 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 levelone 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
jdoublecrossi =
Z
d4x0 d4x00
x210x220x320x400 x500 x600 x000
2 2 2
2 :
Applying the bilocal generator (8.9) to the double cross integral (8.13) yields
Pbbi jdoublecrossi = P
2 + 2P3 + 2P4 + 3P5 + 4P6 jdoublecrossi :
Again, we see that we can de ne an algebraically consistent levelone 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 levelzero 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 levelzero and levelone
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 bilocal piece of the levelone generator.
The second term is the product of two levelzero 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 levelzero generator (8.20) and can thus be dropped. Collecting the remaining
terms at order un 2 of (8.19), we thus nd the levelone 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 levelzero and levelone 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 oneloop box. At the same time one may
wonder whether an analogue of the above construction of the coordinatespace 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 levelone generator acting in onshell 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 biscalar 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 deltafunction (4)(x1
xn+1) reimposing the closure of the lightlike polygon.
This deltafunction corresponds to the momentum conserving deltafunction (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 onedimensional deltafunctions ensuring the lightlikeness of the edges
xi
xi+1. The reason for this is that the dual conformal generators as well as the
levelone onshell momentumspace generators manifestly respect the lightlikeness condition,
so that we can safely disregard these deltafunctions. 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 deltafunction 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 deltafunction 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 deltafunction 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 deltafunction present, which could balance out the inversion
weight of the bosonic deltafunction. 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 biscalar
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 deltafunction.
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 levelone momentum generator.
Yangian symmetry in momentum space. In this paragraph we will now demonstrate
that the generator K0 _
agrees with the conformal levelone 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 onshell 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 levelone momentum generator
as it follows from the formula (8.5) with the underlying levelzero 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 levelzero
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 nonvanishing evaluation parameters,
which is in complete analogy with the xspace levelone momentum generator that we
considered before.
10
Discussion and outlook
In this paper we established the conformal Yangian symmetry of singletrace correlators
and amplitudes in the planar approximation of the biscalar 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 multiloop 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 coordinatespace 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 levelzero Lie algebra
generators and the bilocal levelone 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 biscalar theory, we then demonstrated
that the dual conformal symmetry is equivalent to a Yangian levelone 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
(allloop) planar scattering matrix of the biscalar
FT4 is thus an exact statement. More
over, the breakdown of conformal symmetry by the doubletrace 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 bilocal (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 oneloop scalar box, has been solved. We
are optimistic that the discovered Yangian symmetry will open the door to computing the
respective higherloop integrals via the powerful toolbox of integrability, as it happened to
large classes of multiloop graphs of biscalar
FT4 theory, such as \wheel"graphs [16, 53]
and magnon correlator graphs [22] relevant for the computation of anomalous dimensions.
Taking the biscalar
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 Qbarequation and its levelone counterpart for the nite BDSsubtracted Smatrix 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
biscalar case considered here.
Recently, the Yangian symmetry of N = 4 super YangMills 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 biscalar 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 lightlike edges in the strongcoupling [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 biscalar
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 biscalar limit of a deformed smooth MaldacenaWilson loop [57, 58]
which, in addition to the gauge eld, also couples to the scalars of the theory.
Let us remind that every Yangianinvariant scattering amplitude in N = 4 SYM theory
can be written as an integral over a Grassmannian [15]. Identifying a similar geometric
structure for the biscalar amplitudes at hand would certainly be of great importance.
Note that the basic scalar correlators considered in this paper do not exhaust all
possible singletrace correlators of the biscalar 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 biscalar theory, which can be obtained by a similar double
scaling limit of the threedimensional deformed ABJM model (\triscalar" theory) [22],
where we deal with regular triangular shnet graphs. The same holds for a similar
sixdimensional triscalar 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 Dl.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" ERC2012AdG 320769 AdSCFTsolvable). 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.
B
Expansion of monodromy eigenvalue
Here we give the rst ve orders of the spectralparameter expansion of the monodromy
eigenvalue function:
(~u) = un + 1 un 1 X ^k + 1 un 2
2
8
+ 1 un 3
^i^j ^k
n
X
n
8 i;j=1
2 n
X ^i^j
1 Xn ^ j5
2 j=1
3
2 j;k
1 X Pjk ^ j ^k5
n
X
n
i=1
k = k+
k + 2 and we set Pjk = 1
2 jk.
Here we use the shorthand notations ^k =
k+ + k + 2 and ^
i =
i( i
Pjl lk
4Pjl jk
4 lk jl ^ j ^k ^l
(B.1)
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