Yangian Symmetry for Fishnet Feynman Graphs
Deliang Zhong LPTENS
Paris
With Dmitry Chicherin, Vladimir Kazakov, Florian Loebbert, Dennis Müller arXiv: 1704.01967, 1708.00007 & work in progress
❖ Superconformal symmetry
❖ Planar integrability
• Origin: mysterious
❖ AdS/CFT correspondence
4D N = 4 Super Yang-Mills
� � twisted 4D N = 4 Super Yang-MillsBi-scalar limit [Gürdoğan, Kazakov ’15]
❖ Chiral, non-unitary theory
❖ 1-1 correspondence: correlator and fishnet Feynman diagram
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs
Dmitry Chicherin,1, ⇤ Vladimir Kazakov,2, † Florian Loebbert,3, ‡ Dennis Muller,3, § and De-liang Zhong2, ¶
1PRISMA Cluster of Excellence, Johannes Gutenberg University, 55099 Mainz, Germany
2Laboratoire de Physique Theorique, Departement de Physique de l’ENS,
Ecole Normale Superieure, PSL Research University, Sorbonne Universites,
UPMC Univ. Paris 06, CNRS, 75005 Paris, France3Institut fur Physik, Humboldt-Universiat zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany &
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
(Dated: July 31, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over theconformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensionsas well as the inclusion of fermions in four dimensions. The Yangian symmetry results in noveldi↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-ered fishnet graphs in three and four dimensions dominate the correlation functions and scatteringamplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJMtheory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrabilityof the planar AdS/CFT correspondence.
INTRODUCTION
Feynman diagrams represent the main tool for thestudy of complex physical phenomena—from fundamen-tal interactions of elementary particles to diverse solidstate systems. In spite of the great progress in comput-ing individual Feynman graphs with multiple loop inte-grations, examples of exact all-loop results for importantphysical quantities (such as amplitudes, correlators, etc.)are rare in dimensions greater than two. Remarkably,there exist certain types of planar graphs with a partic-ularly regular structure, which may be calculable at anyloop order. Examples are the regular tilings of the two-dimensional plane. These diagrams become accessibledue to their integrability properties, in close analogy tothe quantum integrable one-dimensional Heisenberg spinchains. Apart from providing new, powerful methods forthe computation of large classes of particular Feynmangraphs, these observations reveal the interplay betweenvarious physical systems and a rich variety of mathemat-ical aspects related to quantum integrability.
A prime example in the above class of Feynman graphsare scalar fishnets in four dimensions, built from four-point vertices connected by massless propagators (cf.Fig. 1). These represent one of the three regular tilingsof the Euclidean plane and, except for the simplest ex-ample, solving this class of Feynman integrals for genericexternal parameters is an open problem. On the otherhand, these square fishnet graphs are subject to out-standing properties: Firstly, they feature a (dual) confor-mal Lie algebra symmetry, which makes it natural to ex-press them using conformal cross ratios. They are finite,i.e. free of IR or UV divergencies, such that their confor-mal symmetry is unbroken for generic external kinemat-ics. Moreover, already in 1980 A. Zamolodchikov demon-strated that scalar fishnet graphs can be interpreted as
FIG. 1. Example of a conformal scalar fishnet Feynmangraph in four dimensions. Filled blobs denote loop integra-tions, white blobs represent external points xk.
integrable vertex models [1]. Furthermore, in the planarlimit fishnet graphs appear to dominate physical quanti-ties, such as scattering amplitudes and correlators, of thebi-scalar CFT recently found by O. Gurdogan and one ofthe authors [2] as a specific double-scaling limit of the in-tegrable �-twisted N = 4 SYM theory. This non-unitarybi-scalar CFT is defined by the Lagrangian
L� = NcTr�@µ�†
1@µ�1 + @µ�†2@µ�2 + ⇠2 �†
1�†2�1�2
�. (1)
Its basic physical quantities (anomalous dimensions, cor-relation functions etc.) are determined by a very limitednumber of Feynman graphs at each loop order and e�-ciently calculable via integrability [3, 4].In this letter we add a further remarkable property to
the above list of features of fishnet graphs. We demon-strate that their conformal symmetry extends to a non-local Yangian symmetry. This symmetry yields novel dif-ferential constraint equations for this class of Feynmanintegrals.A single scalar fishnet graph of the above type repre-
sents a single-trace correlator of the bi-scalar theory
K(x1, . . . , xn) = hTr[�1(x1) . . .�n(xn)]i. (2)
Here �k 2 {�1,�2,�†1,�
†2} and xi is the spacetime co-
ordinate of the field �i. Importantly, via the relation
K(x1, . . . , xn) = hTr[�1(x1) . . .�n(xn)]i.<latexit sha1_base64="ie6VLT5r35QrW3+QMtcDwLOzDTk=">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</latexit><latexit sha1_base64="ie6VLT5r35QrW3+QMtcDwLOzDTk=">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</latexit><latexit sha1_base64="ie6VLT5r35QrW3+QMtcDwLOzDTk=">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</latexit><latexit sha1_base64="ie6VLT5r35QrW3+QMtcDwLOzDTk=">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</latexit>
�k 2 {�1,�2,�†1,�
†2}
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❖ Conformal symmetry for specific choice of couplings
❖ Planar integrability
• From first principle
• Yangian symmetry for individual Feynman diagrams
❖ Is there an AdS dual?
� � twisted 4D N = 4 Super Yang-MillsBi-scalar limit [Gürdoğan, Kazakov ’15]
[Sieg-Wilhelm ’16] [Grabner, Gromov, Kazakov, Korchemsky ’17]
Scalar Fishnet Diagrams in 4D [Zamolodchikov ’80]
❖ Starting point: integrable lattice model
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
Fishnet
Fey
nman
Gra
phs
Feynman
graphs
madefro
msca
larfou
r-poin
t vertice
s in 4d(x
2 = xµ xµ
):
IVert
ex
:s d4 x
IProp
agator
j
k:
1x2
jk
©
1
(xj≠xk
)2
e.g.
Mostly uns
olved,
only cro
ss known [U
ssyukin
a ’93
Davydych
ev]:
=s
d4x0
x210
x220
x230
x240
Rem
arka
blepr
oper
ties
:
IConf
ormal sym
metry (un
broken
: nodiv
ergien
cies)
IRepr
esent
integra
blesta
tistica
l vertex
model [Zam
olodchiko
v
1980
]
IHere
: Newins
ights motiv
ated by
AdS/CFT inte
grabili
ty. . .
Florian
Loebbert: Yangia
n Symmetr
y forFish
netFeyn
manGrap
hs
1 / 13
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· · ·<latexit sha1_base64="nzBzl2EHHmDTqXdjvnR25IPCseI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiF48VjC20oWw223bpZjfsToQS+iO8eFDx6v/x5r9x0+ag1QcDj/dmmJkXpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkH3JK0EqdPo0Vmtqg3vCa3hzuX+KXpAEl2oP6Zz9WNEuYRCqIMT3fSzHMiUZOBZvV+plhKaETMmI9SyVJmAnz+bkz98QqsTtU2pZEd67+nMhJYsw0iWxnQnBslr1C/M/rZTi8DHMu0wyZpItFw0y4qNzidzfmmlEUU0sI1dze6tIx0YSiTagIwV9++S8JzppXTf/uvNG6LtOowhEcwyn4cAEtuIU2BEBhAk/wAq9O6jw7b877orXilDOH8AvOxzdRO48X</latexit><latexit sha1_base64="nzBzl2EHHmDTqXdjvnR25IPCseI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiF48VjC20oWw223bpZjfsToQS+iO8eFDx6v/x5r9x0+ag1QcDj/dmmJkXpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkH3JK0EqdPo0Vmtqg3vCa3hzuX+KXpAEl2oP6Zz9WNEuYRCqIMT3fSzHMiUZOBZvV+plhKaETMmI9SyVJmAnz+bkz98QqsTtU2pZEd67+nMhJYsw0iWxnQnBslr1C/M/rZTi8DHMu0wyZpItFw0y4qNzidzfmmlEUU0sI1dze6tIx0YSiTagIwV9++S8JzppXTf/uvNG6LtOowhEcwyn4cAEtuIU2BEBhAk/wAq9O6jw7b877orXilDOH8AvOxzdRO48X</latexit><latexit sha1_base64="nzBzl2EHHmDTqXdjvnR25IPCseI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiF48VjC20oWw223bpZjfsToQS+iO8eFDx6v/x5r9x0+ag1QcDj/dmmJkXpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkH3JK0EqdPo0Vmtqg3vCa3hzuX+KXpAEl2oP6Zz9WNEuYRCqIMT3fSzHMiUZOBZvV+plhKaETMmI9SyVJmAnz+bkz98QqsTtU2pZEd67+nMhJYsw0iWxnQnBslr1C/M/rZTi8DHMu0wyZpItFw0y4qNzidzfmmlEUU0sI1dze6tIx0YSiTagIwV9++S8JzppXTf/uvNG6LtOowhEcwyn4cAEtuIU2BEBhAk/wAq9O6jw7b877orXilDOH8AvOxzdRO48X</latexit><latexit sha1_base64="nzBzl2EHHmDTqXdjvnR25IPCseI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiF48VjC20oWw223bpZjfsToQS+iO8eFDx6v/x5r9x0+ag1QcDj/dmmJkXpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkH3JK0EqdPo0Vmtqg3vCa3hzuX+KXpAEl2oP6Zz9WNEuYRCqIMT3fSzHMiUZOBZvV+plhKaETMmI9SyVJmAnz+bkz98QqsTtU2pZEd67+nMhJYsw0iWxnQnBslr1C/M/rZTi8DHMu0wyZpItFw0y4qNzidzfmmlEUU0sI1dze6tIx0YSiTagIwV9++S8JzppXTf/uvNG6LtOowhEcwyn4cAEtuIU2BEBhAk/wAq9O6jw7b877orXilDOH8AvOxzdRO48X</latexit>
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❖ For : type scalar Feynman diagram ↵ =⇡
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[Baxter ’78]
Scalar Fishnet Diagrams in General
Di�erent Dimensions
I In d dimensions the scalar propagator iss
ddk eikx
k2 ≥ |x|2≠d.
I Intertwining relations generalize to d = 3, 4, 6:1
xd≠212
Ld2[”, • ]Ld
1[ ı , ” + d≠22 ] = Ld
2[” + d≠22 , • ]Ld
1[ ı , ”] 1xd≠2
12.
All Regular Tilings of the Plane give Yangian invariant integrals:
Dimension d = 3 d = 4 d = 6Propagator |xij |
≠1|xij |
≠2|xij |
≠4
ScalarFishnet
This is the complete set of fishnets introduced as statistical models byZamolodchikov in 1980!
Are there more?
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 11 / 13
❖ Feynman rules Vertex
Propagator
❖ UV/IR finite: unbroken conformal symmetry
Zddx ,
Fishnet Feynman Graphs
Feynman graphs made from scalar four-point vertices in 4d (x2 = xµxµ):
I Vertex :s
d4x
I Propagator j k : 1x2
jk©
1(xj≠xk)2
e.g.
Mostly unsolved, only cross known [Ussyukina ’93Davydychev ]: =
s d4x0x2
10x220x2
30x240
Remarkable properties:I Conformal symmetry (unbroken: no divergiencies)I Represent integrable statistical vertex model [Zamolodchikov
1980 ]I Here: New insights motivated by AdS/CFT integrability . . .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 1 / 13
, |xij |d�2
[Zamolodchikov ’80]
Yangian Symmetry for the Single Box
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
bPµI4 = 0, bPµ = � i2
4X
j<k=1
⇥(Lµ⌫
j +⌘µ⌫Dj)Pk,⌫�(j $ k)⇤�
4X
j=1
jPµj
I4 =
Zd4x0
1
x210x
220x
230x
240
======1
x213x
224
�(u, v) =
we see that the box integral is indeed Yangian-invariant. Parametrizing the box as I4 = 1
x213x
224�(u, v)
with the conformal cross ratios u = x212x
234
x213x
224
and v =x214x
223
x213x
224, this statement boils down to the following sec-
ond order di↵erential equation for the function �,
0 = �+ (3u� 1)@�
@u+ 3v
@�
@v+ (u� 1)u
@2�
@u2+ v2
@2�
@v2+ 2uv
@2�
@u@v
4 2
1
3
0
p8 p3
p7 p4
p1
p6
p2
p5
u =x212x
234
x213x
224
v =x214x
223
x213x
224
❖ We know everything about this building block!
• The integral can be expressed in terms of dilogarithms
• The integral is conformally invariant
[Ussyukina, Davydychev, ’93]
Ja 2 {D�=1,Lµ⌫ ,Pµ,Kµ}Dilatation Translation
Special Conf.Lorentz
JaI4 = 0, Ja =4X
j=1
Jaj
Yangian Symmetry for the Single Box
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
bPµI4 = 0, bPµ = � i2
4X
j<k=1
⇥(Lµ⌫
j +⌘µ⌫Dj)Pk,⌫�(j $ k)⇤�
4X
j=1
jPµj
I4 =
Zd4x0
1
x210x
220x
230x
240
======1
x213x
224
�(u, v) =
we see that the box integral is indeed Yangian-invariant. Parametrizing the box as I4 = 1
x213x
224�(u, v)
with the conformal cross ratios u = x212x
234
x213x
224
and v =x214x
223
x213x
224, this statement boils down to the following sec-
ond order di↵erential equation for the function �,
0 = �+ (3u� 1)@�
@u+ 3v
@�
@v+ (u� 1)u
@2�
@u2+ v2
@2�
@v2+ 2uv
@2�
@u@v
4 2
1
3
0
p8 p3
p7 p4
p1
p6
p2
p5
u =x212x
234
x213x
224
v =x214x
223
x213x
224
❖ We know everything about this building block!
• We also have non-local hidden symmetry
• Hidden symmetry gives two second order differential equation
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs
Dmitry Chicherin,1, ⇤ Vladimir Kazakov,2, † Florian Loebbert,3, ‡ Dennis Muller,3, § and De-liang Zhong2, ¶
1PRISMA Cluster of Excellence, Johannes Gutenberg University, 55099 Mainz, Germany
2Laboratoire de Physique Theorique, Departement de Physique de l’ENS,
Ecole Normale Superieure, PSL Research University, Sorbonne Universites,
UPMC Univ. Paris 06, CNRS, 75005 Paris, France3Institut fur Physik, Humboldt-Universiat zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany &
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
bPµI4 = 0, bPµ = � i2
4X
j<k=1
⇥(Lµ⌫
j +⌘µ⌫Dj)Pk,⌫�(j $ k)⇤�
4X
j=1
jPµj
Depend on the graphBilocal piece: universal
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs
Dmitry Chicherin,1, ⇤ Vladimir Kazakov,2, † Florian Loebbert,3, ‡ Dennis Muller,3, § and De-liang Zhong2, ¶
1PRISMA Cluster of Excellence, Johannes Gutenberg University, 55099 Mainz, Germany
2Laboratoire de Physique Theorique, Departement de Physique de l’ENS,
Ecole Normale Superieure, PSL Research University, Sorbonne Universites,
UPMC Univ. Paris 06, CNRS, 75005 Paris, France3Institut fur Physik, Humboldt-Universiat zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany &
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
bPµI4 = 0, bPµ = � i2
4X
j<k=1
⇥(Lµ⌫
j +⌘µ⌫Dj)Pk,⌫�(j $ k)⇤�
4X
j=1
jPµj
we see that the box integral is indeed Yangian-invariant. Parametrizing the box as I4 = 1
x213x
224�(u, v)
with the conformal cross ratios u = x212x
234
x213x
224
and v =x214x
223
x213x
224, this statement boils down to the following sec-
ond order di↵erential equation for the function �,
0 = �+ (3u� 1)@�
@u+ 3v
@�
@v+ (u� 1)u
@2�
@u2+ v2
@2�
@v2+ 2uv
@2�
@u@v
0 = �+ (u $ v)
Yangian Symmetry for Scalar Fishnets❖ The general fishnet can be obtained by connecting single boxes
= �(~u)
Generic Fishnets
I The above relations allow to pull the monodromy through an arbitraryn-point scalar fishnet, thus proving its Yangian invariance:
T (u)In = Ln[”+n , ”≠
n ]Ln≠1[”+n≠1, ”≠
n≠1] . . . L1[”+1 , ”≠
1 ]In = f(u)In
I The inhomogeneities ”±k are attributed via a simple rule, e.g.
[1,2] [1,2] [1,2]
[2,3][1,2] [1,2]
[2,3]
[2,3][4,5]
[3,4]
[2,3]
[3,4] [3,4] [3,4] [3,4]
[4,5]
[4,5]
[4,5]
I Eigenvalue:f(u) =
rjœout
(u + ”+j )(u + ”≠
j ) here= (u + 3)5(u + 4)5(u + 4)4(u + 5)4
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 10 / 13
❖ The hidden symmetries come from integrability !
❖ Fully encoded in the RTT relation
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v)
❖ What is R and T?
‣ R-matrix: Yang’s R-matrix
‣ T-matrix: product of Lax operator with inhomogeneities
‣ Lax operator: spinorial Lax operator
Yangian Symmetry for Scalar Fishnets
RTT Realization of the Yangian
Collect generators into monodromy:
T (u) ƒ 1+ 1u J + 1
u2‚J + . . . ,
Yangian algebra encoded in RTT-relationsR12(u≠v)T1(u)T2(v) = T2(v)T1(u)R12(u≠v) =with Yang’s R-matrix R12(u ≠ v) = 112 +uP12.
Monodromy is solution to RTT-relations:
T (u) =Ln(u+n , u≠
n )Ln≠1(u+n≠1, u≠
n≠1) . . . L1(u+1 , u≠
1 ).
with conformal Lax operator:inhomogeneities
Lk,–—(u+k , u≠
k ) = uk 1k,–— + Mab–—
[�a6d
,�b6d
]|proj.
di�erential conf. generators
J�kk,ab [Chicherin, Derkachov
Isaev 2013 ]
and variables: u+k := uk + �k≠4
2 , u≠k := uk ≠
�k2 .
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 7 / 13
T({u}) = Ln[�+n , �
�n ]Ln�1[�
+n�1, �
�n�1] · · ·L1[�
+1 , �
�1 ]T(~u)
[Chicherin, Derkachov Isaev ’13]
3
(a) (b)
FIG. 3. (a): Monodromy encircling a sample fishnet graphrepresenting the left hand side of (12). (b): Intermediate stepof the proof of Yangian symmetry.
and the algebra relations are rephrased via the Yang–Baxter equation with Yang’s R-matrix R(u) = 1+uP:
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)
We explicitly solve this RTT-relation by defining themonodromy as a product of conformal Lax operators [7]
Lk,↵�(u+k , u
�k ) = uk 1k,↵� +
12S
ab↵� J
�kk,ab, (11)
each of which obeys (10) with Tk ! Lk. Here we pack-age the inhomogeneities uk and the conformal dimensions�k into the symmetric variables u+
k := uk + �k�42 and
u�k := uk � �k
2 . The Jk,ab denote the di↵erential repre-sentation of the conformal algebra displayed in (4), andwe have Sab = i
4 [�a,�b]|upper block, with �a representing
six-dimensional gamma matrices for R2,4. The Yangiansymmetry of the box integral I4 and its n-point general-izations In now translates into the eigenvalue equation [8]
T(~u) In = �(~u) In 1, (12)
where T(~u) denotes the inhomogeneous monodromy
T(~u) = Ln(u+n , u
�n )Ln�1(u
+n�1, u
�n�1) . . .L1(u
+1 , u
�1 ).(13)
The choice of parameters u±k depends on the diagram
under consideration. It will be convenient to introducethe notation [�+k , �
�k ] := (u+�+k , u+��k ) and [�k] := u+�k.
By convention we choose the parameters on the boundarylegs at the top to be [1, 2]. Then the parameters on theright, bottom or left boundary legs have to be [2, 3], [3, 4]or [4, 5], respectively (see Fig. 3 for an example). TheLax operator defined in (11) acts on an auxiliary and on aquantum space. While the product in (13) is taken in theauxiliary space, each Lax operator acts on one externalleg of the considered Feynman graph which representsthe quantum space.
Proving the invariance statement (12) boils down toemploying the lasso method [9], i.e. to moving the mon-odromy through a given graph as displayed in Fig. 3.
(a) =
(b) = [� + 2]⇥
(c) =
FIG. 4. Rules employed to prove the Yangian invariance.(a): The intertwining relation (14). (b) and (c): Pulling themonodromy contour through an integration vertex, cf. (15).
The most important relation used in this process is thefollowing intertwining relation for the Lax operator andthe x-space propagator, cf. Fig. 4a:
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
. (14)
Moreover, we make use of the following relation, whichallows us to move a product of Lax operators through anintegration vertex, cf. Fig. 4b:
Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]
1
x201x
202x
203x
204
(15)
= [� + 2]
Zd4x0
1
x201x
202
L0[� + 1, � + 1]1
x203x
204
.
A third relation of this type is depicted in Fig. 4c. Finally,the Lax operator and its partially integrated version de-noted by LT act on a constant function as
L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)
These rules are su�cient to move the monodromy con-tour in Fig. 3a through the whole graph to end up withthe eigenvalue on the right hand side of (12). See Fig. 3bfor an intermediate step. The eigenvalue �(~u) in (12) iscomposed of the factors picked up in this process via therelations (15) and (16), cf. [9] for explicit expressions.
OFF- AND ON-SHELL LEGS
In the above construction, the external variables xµi
were in fact unconstrained. To interpret the xi as re-gion momenta for a scattering amplitude with masslesson-shell legs, we require that p2k = (xk � xk+1)2 = 0. No-tably, the delta-function imposing this constraint obeys
Inhomogeneities Differential conformal generator
Spin generator
u+k := uk +
�k � 4
2, u�
k := uk � �k
2[�+, ��] := (u+, u�) ⌘ (u+ �+, u+ ��), [�] := u+ �
[�+, ��] := (u+, u�) ⌘ (u+ �+, u+ ��), [�] := u+ �
Yangian Symmetry for Scalar Fishnets
❖ T includes all symmetry generators
Higher level generators
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs
Dmitry Chicherin,1, ⇤ Vladimir Kazakov,2, † Florian Loebbert,3, ‡ Dennis Muller,3, § and De-liang Zhong2, ¶
1PRISMA Cluster of Excellence, Johannes Gutenberg University, 55099 Mainz, Germany
2Laboratoire de Physique Theorique, Departement de Physique de l’ENS,
Ecole Normale Superieure, PSL Research University, Sorbonne Universites,
UPMC Univ. Paris 06, CNRS, 75005 Paris, France3Institut fur Physik, Humboldt-Universiat zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany &
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
Level 0 generators (Conformal symmetry)
Level 1 generators (Dual conformal symmetry)
(J
J<latexit sha1_base64="uMDlqDiuv75R9ucatjLE1qvuQzU=">AAACIHicbVBNSwMxEM3Wr7p+VT16CRbBU9kVwXorehFPFawtdEvJZqdtaJJdkqxQlv4VL/4VLx5U9Ka/xrRdUFsfBB7vzUxmXphwpo3nfTqFpeWV1bXiuruxubW9U9rdu9Nxqig0aMxj1QqJBs4kNAwzHFqJAiJCDs1weDnxm/egNIvlrRkl0BGkL1mPUWKs1C1VgxD6TGbUztBjNxDEDJTIrsc4CNxgQEz2I1kbZJSXdktlr+JNgReJn5MyylHvlj6CKKapAGkoJ1q3fS8xnYwowygHOzvVkBA6JH1oWyqJAN3JpheO8ZFVItyLlX3S4Kn6uyMjQuuRCG3lZF09703E/7x2anrVTsZkkhqQdPZRL+XYxHgSF46YAmr4yBJCFbO7YjogilBjQ3VtCP78yYukcVI5r/g3p+XaRZ5GER2gQ3SMfHSGaugK1VEDUfSAntALenUenWfnzXmflRacvGcf/YHz9Q1uAKSG</latexit><latexit sha1_base64="uMDlqDiuv75R9ucatjLE1qvuQzU=">AAACIHicbVBNSwMxEM3Wr7p+VT16CRbBU9kVwXorehFPFawtdEvJZqdtaJJdkqxQlv4VL/4VLx5U9Ka/xrRdUFsfBB7vzUxmXphwpo3nfTqFpeWV1bXiuruxubW9U9rdu9Nxqig0aMxj1QqJBs4kNAwzHFqJAiJCDs1weDnxm/egNIvlrRkl0BGkL1mPUWKs1C1VgxD6TGbUztBjNxDEDJTIrsc4CNxgQEz2I1kbZJSXdktlr+JNgReJn5MyylHvlj6CKKapAGkoJ1q3fS8xnYwowygHOzvVkBA6JH1oWyqJAN3JpheO8ZFVItyLlX3S4Kn6uyMjQuuRCG3lZF09703E/7x2anrVTsZkkhqQdPZRL+XYxHgSF46YAmr4yBJCFbO7YjogilBjQ3VtCP78yYukcVI5r/g3p+XaRZ5GER2gQ3SMfHSGaugK1VEDUfSAntALenUenWfnzXmflRacvGcf/YHz9Q1uAKSG</latexit><latexit sha1_base64="uMDlqDiuv75R9ucatjLE1qvuQzU=">AAACIHicbVBNSwMxEM3Wr7p+VT16CRbBU9kVwXorehFPFawtdEvJZqdtaJJdkqxQlv4VL/4VLx5U9Ka/xrRdUFsfBB7vzUxmXphwpo3nfTqFpeWV1bXiuruxubW9U9rdu9Nxqig0aMxj1QqJBs4kNAwzHFqJAiJCDs1weDnxm/egNIvlrRkl0BGkL1mPUWKs1C1VgxD6TGbUztBjNxDEDJTIrsc4CNxgQEz2I1kbZJSXdktlr+JNgReJn5MyylHvlj6CKKapAGkoJ1q3fS8xnYwowygHOzvVkBA6JH1oWyqJAN3JpheO8ZFVItyLlX3S4Kn6uyMjQuuRCG3lZF09703E/7x2anrVTsZkkhqQdPZRL+XYxHgSF46YAmr4yBJCFbO7YjogilBjQ3VtCP78yYukcVI5r/g3p+XaRZ5GER2gQ3SMfHSGaugK1VEDUfSAntALenUenWfnzXmflRacvGcf/YHz9Q1uAKSG</latexit><latexit sha1_base64="uMDlqDiuv75R9ucatjLE1qvuQzU=">AAACIHicbVBNSwMxEM3Wr7p+VT16CRbBU9kVwXorehFPFawtdEvJZqdtaJJdkqxQlv4VL/4VLx5U9Ka/xrRdUFsfBB7vzUxmXphwpo3nfTqFpeWV1bXiuruxubW9U9rdu9Nxqig0aMxj1QqJBs4kNAwzHFqJAiJCDs1weDnxm/egNIvlrRkl0BGkL1mPUWKs1C1VgxD6TGbUztBjNxDEDJTIrsc4CNxgQEz2I1kbZJSXdktlr+JNgReJn5MyylHvlj6CKKapAGkoJ1q3fS8xnYwowygHOzvVkBA6JH1oWyqJAN3JpheO8ZFVItyLlX3S4Kn6uyMjQuuRCG3lZF09703E/7x2anrVTsZkkhqQdPZRL+XYxHgSF46YAmr4yBJCFbO7YjogilBjQ3VtCP78yYukcVI5r/g3p+XaRZ5GER2gQ3SMfHSGaugK1VEDUfSAntALenUenWfnzXmflRacvGcf/YHz9Q1uAKSG</latexit>
❖ Yangian symmetry algebra [Drinfeld ’85]
[Ja, Jb] = f cabJc
<latexit sha1_base64="i51vcqjRrj2aoVhhfMmNk2MBBOQ=">AAACHnicbVDLSsNAFJ34rPUVdelmsAgupCQiPhZC0Y24qmBsIY1hMp20QyeTMDMRSsifuPFX3LhQEVzp3zhps6itBwbOnHMv994TJIxKZVk/xtz8wuLScmWlurq2vrFpbm3fyzgVmDg4ZrFoB0gSRjlxFFWMtBNBUBQw0goGV4XfeiRC0pjfqWFCvAj1OA0pRkpLvnnidiKk+iLKbnIfHcKJXwA9eAFDP0NB/oAnHVz1zZpVt0aAs8QuSQ2UaPrmV6cb4zQiXGGGpHRtK1FehoSimJG82kklSRAeoB5xNeUoItLLRvflcF8rXRjGQj+u4Eid7MhQJOUwCnRlsaSc9grxP89NVXjmZZQnqSIcjweFKYMqhkVYsEsFwYoNNUFYUL0rxH0kEFY60iIEe/rkWeIc1c/r9u1xrXFZplEBu2APHAAbnIIGuAZN4AAMnsALeAPvxrPxanwYn+PSOaPs2QF/YHz/AsEVol8=</latexit><latexit sha1_base64="i51vcqjRrj2aoVhhfMmNk2MBBOQ=">AAACHnicbVDLSsNAFJ34rPUVdelmsAgupCQiPhZC0Y24qmBsIY1hMp20QyeTMDMRSsifuPFX3LhQEVzp3zhps6itBwbOnHMv994TJIxKZVk/xtz8wuLScmWlurq2vrFpbm3fyzgVmDg4ZrFoB0gSRjlxFFWMtBNBUBQw0goGV4XfeiRC0pjfqWFCvAj1OA0pRkpLvnnidiKk+iLKbnIfHcKJXwA9eAFDP0NB/oAnHVz1zZpVt0aAs8QuSQ2UaPrmV6cb4zQiXGGGpHRtK1FehoSimJG82kklSRAeoB5xNeUoItLLRvflcF8rXRjGQj+u4Eid7MhQJOUwCnRlsaSc9grxP89NVXjmZZQnqSIcjweFKYMqhkVYsEsFwYoNNUFYUL0rxH0kEFY60iIEe/rkWeIc1c/r9u1xrXFZplEBu2APHAAbnIIGuAZN4AAMnsALeAPvxrPxanwYn+PSOaPs2QF/YHz/AsEVol8=</latexit><latexit sha1_base64="i51vcqjRrj2aoVhhfMmNk2MBBOQ=">AAACHnicbVDLSsNAFJ34rPUVdelmsAgupCQiPhZC0Y24qmBsIY1hMp20QyeTMDMRSsifuPFX3LhQEVzp3zhps6itBwbOnHMv994TJIxKZVk/xtz8wuLScmWlurq2vrFpbm3fyzgVmDg4ZrFoB0gSRjlxFFWMtBNBUBQw0goGV4XfeiRC0pjfqWFCvAj1OA0pRkpLvnnidiKk+iLKbnIfHcKJXwA9eAFDP0NB/oAnHVz1zZpVt0aAs8QuSQ2UaPrmV6cb4zQiXGGGpHRtK1FehoSimJG82kklSRAeoB5xNeUoItLLRvflcF8rXRjGQj+u4Eid7MhQJOUwCnRlsaSc9grxP89NVXjmZZQnqSIcjweFKYMqhkVYsEsFwYoNNUFYUL0rxH0kEFY60iIEe/rkWeIc1c/r9u1xrXFZplEBu2APHAAbnIIGuAZN4AAMnsALeAPvxrPxanwYn+PSOaPs2QF/YHz/AsEVol8=</latexit><latexit sha1_base64="i51vcqjRrj2aoVhhfMmNk2MBBOQ=">AAACHnicbVDLSsNAFJ34rPUVdelmsAgupCQiPhZC0Y24qmBsIY1hMp20QyeTMDMRSsifuPFX3LhQEVzp3zhps6itBwbOnHMv994TJIxKZVk/xtz8wuLScmWlurq2vrFpbm3fyzgVmDg4ZrFoB0gSRjlxFFWMtBNBUBQw0goGV4XfeiRC0pjfqWFCvAj1OA0pRkpLvnnidiKk+iLKbnIfHcKJXwA9eAFDP0NB/oAnHVz1zZpVt0aAs8QuSQ2UaPrmV6cb4zQiXGGGpHRtK1FehoSimJG82kklSRAeoB5xNeUoItLLRvflcF8rXRjGQj+u4Eid7MhQJOUwCnRlsaSc9grxP89NVXjmZZQnqSIcjweFKYMqhkVYsEsFwYoNNUFYUL0rxH0kEFY60iIEe/rkWeIc1c/r9u1xrXFZplEBu2APHAAbnIIGuAZN4AAMnsALeAPvxrPxanwYn+PSOaPs2QF/YHz/AsEVol8=</latexit>
[Ja, Jb] = f cabJc
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[Ja, [Jb, Jc]]� [Ja, [Jb, Jc]] = O(J3)<latexit sha1_base64="dUmR8+bxsD/sr9AaRwMSoYfMBXU=">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</latexit><latexit sha1_base64="dUmR8+bxsD/sr9AaRwMSoYfMBXU=">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</latexit><latexit sha1_base64="dUmR8+bxsD/sr9AaRwMSoYfMBXU=">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</latexit><latexit sha1_base64="dUmR8+bxsD/sr9AaRwMSoYfMBXU=">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</latexit>
Level 0
Level 1
Serre relation
Yangian Symmetry for Scalar Fishnets
MITP/17-049, HU-EP-17/20
Yangian Symmetry for Fishnet Feynman Graphs
Dmitry Chicherin,1, ⇤ Vladimir Kazakov,2, † Florian Loebbert,3, ‡ Dennis Muller,3, § and De-liang Zhong2, ¶
1PRISMA Cluster of Excellence, Johannes Gutenberg University, 55099 Mainz, Germany
2Laboratoire de Physique Theorique, Departement de Physique de l’ENS,
Ecole Normale Superieure, PSL Research University, Sorbonne Universites,
UPMC Univ. Paris 06, CNRS, 75005 Paris, France3Institut fur Physik, Humboldt-Universiat zu Berlin,
Zum Großen Windkanal 6, 12489 Berlin, Germany &
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
(Dated: December 13, 2017)
Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the
conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions
as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel
di↵erential equations for these families of largely unsolved Feynman integrals. Notably, the consid-
ered fishnet graphs in three and four dimensions dominate the correlation functions and scattering
amplitudes in specific double scaling limits of planar, �-twisted N = 4 super Yang–Mills or ABJM
theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability
of the planar AdS/CFT correspondence.
T(~u) In = �(~u) In 1, �(~u) =Y
j2out
�+j ��j = [3]5[4]9[5]4
(1)
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
(2)
T(u) = un 1+un�1 J + un�2 bJ + · · · (3)
[�+, ��] := (u+, u�) ⌘ (u+ �+, u+ ��), [�] := u+ �
❖ Yangian symmetry: eigenvalue equation for Feynman integral In<latexit sha1_base64="1AZ5p+Cgj3ENJ0TcLzjMZRpqGPU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL3qraGyhDWWz3bRLN5uwOxFK6E/w4kHFq//Im//GbZuDVh8MPN6bYWZemEph0HW/nNLS8srqWnm9srG5tb1T3d17MEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wjq6nfeuTaiETd4zjlQUwHSkSCUbTS3U1P9ao1t+7OQP4SryA1KNDsVT+7/YRlMVfIJDWm47kpBjnVKJjkk0o3MzylbEQHvGOpojE3QT47dUKOrNInUaJtKSQz9edETmNjxnFoO2OKQ7PoTcX/vE6G0XmQC5VmyBWbL4oySTAh079JX2jOUI4toUwLeythQ6opQ5tOxYbgLb78l/gn9Yu6d3taa1wWaZThAA7hGDw4gwZcQxN8YDCAJ3iBV0c6z86b8z5vLTnFzD78gvPxDY5BjYI=</latexit><latexit sha1_base64="1AZ5p+Cgj3ENJ0TcLzjMZRpqGPU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL3qraGyhDWWz3bRLN5uwOxFK6E/w4kHFq//Im//GbZuDVh8MPN6bYWZemEph0HW/nNLS8srqWnm9srG5tb1T3d17MEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wjq6nfeuTaiETd4zjlQUwHSkSCUbTS3U1P9ao1t+7OQP4SryA1KNDsVT+7/YRlMVfIJDWm47kpBjnVKJjkk0o3MzylbEQHvGOpojE3QT47dUKOrNInUaJtKSQz9edETmNjxnFoO2OKQ7PoTcX/vE6G0XmQC5VmyBWbL4oySTAh079JX2jOUI4toUwLeythQ6opQ5tOxYbgLb78l/gn9Yu6d3taa1wWaZThAA7hGDw4gwZcQxN8YDCAJ3iBV0c6z86b8z5vLTnFzD78gvPxDY5BjYI=</latexit><latexit sha1_base64="1AZ5p+Cgj3ENJ0TcLzjMZRpqGPU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL3qraGyhDWWz3bRLN5uwOxFK6E/w4kHFq//Im//GbZuDVh8MPN6bYWZemEph0HW/nNLS8srqWnm9srG5tb1T3d17MEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wjq6nfeuTaiETd4zjlQUwHSkSCUbTS3U1P9ao1t+7OQP4SryA1KNDsVT+7/YRlMVfIJDWm47kpBjnVKJjkk0o3MzylbEQHvGOpojE3QT47dUKOrNInUaJtKSQz9edETmNjxnFoO2OKQ7PoTcX/vE6G0XmQC5VmyBWbL4oySTAh079JX2jOUI4toUwLeythQ6opQ5tOxYbgLb78l/gn9Yu6d3taa1wWaZThAA7hGDw4gwZcQxN8YDCAJ3iBV0c6z86b8z5vLTnFzD78gvPxDY5BjYI=</latexit><latexit sha1_base64="1AZ5p+Cgj3ENJ0TcLzjMZRpqGPU=">AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FL3qraGyhDWWz3bRLN5uwOxFK6E/w4kHFq//Im//GbZuDVh8MPN6bYWZemEph0HW/nNLS8srqWnm9srG5tb1T3d17MEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wjq6nfeuTaiETd4zjlQUwHSkSCUbTS3U1P9ao1t+7OQP4SryA1KNDsVT+7/YRlMVfIJDWm47kpBjnVKJjkk0o3MzylbEQHvGOpojE3QT47dUKOrNInUaJtKSQz9edETmNjxnFoO2OKQ7PoTcX/vE6G0XmQC5VmyBWbL4oySTAh079JX2jOUI4toUwLeythQ6opQ5tOxYbgLb78l/gn9Yu6d3taa1wWaZThAA7hGDw4gwZcQxN8YDCAJ3iBV0c6z86b8z5vLTnFzD78gvPxDY5BjYI=</latexit>
= �(~u)
Generic Fishnets
I The above relations allow to pull the monodromy through an arbitraryn-point scalar fishnet, thus proving its Yangian invariance:
T (u)In = Ln[”+n , ”≠
n ]Ln≠1[”+n≠1, ”≠
n≠1] . . . L1[”+1 , ”≠
1 ]In = f(u)In
I The inhomogeneities ”±k are attributed via a simple rule, e.g.
[1,2] [1,2] [1,2]
[2,3][1,2] [1,2]
[2,3]
[2,3][4,5]
[3,4]
[2,3]
[3,4] [3,4] [3,4] [3,4]
[4,5]
[4,5]
[4,5]
I Eigenvalue:f(u) =
rjœout
(u + ”+j )(u + ”≠
j ) here= (u + 3)5(u + 4)5(u + 4)4(u + 5)4
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 10 / 13
Yangian Symmetry for Scalar Fishnets
❖ Proof of Yangian symmetry: intertwining relations
• Intertwiners
• Action on the interaction vertex
• Vacuum
3
(a) (b)
FIG. 3. (a): Monodromy encircling a sample fishnet graphrepresenting the left hand side of (12). (b): Intermediate stepof the proof of Yangian symmetry.
and the algebra relations are rephrased via the Yang–Baxter equation with Yang’s R-matrix R(u) = 1+uP:
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)
We explicitly solve this RTT-relation by defining themonodromy as a product of conformal Lax operators [7]
Lk,↵�(u+k , u
�k ) = uk 1k,↵� +
12S
ab↵� J
�kk,ab, (11)
each of which obeys (10) with Tk ! Lk. Here we pack-age the inhomogeneities uk and the conformal dimensions�k into the symmetric variables u+
k := uk + �k�42 and
u�k := uk � �k
2 . The Jk,ab denote the di↵erential repre-sentation of the conformal algebra displayed in (4), andwe have Sab = i
4 [�a,�b]|upper block, with �a representing
six-dimensional gamma matrices for R2,4. The Yangiansymmetry of the box integral I4 and its n-point general-izations In now translates into the eigenvalue equation [8]
T(~u) In = �(~u) In 1, (12)
where T(~u) denotes the inhomogeneous monodromy
T(~u) = Ln(u+n , u
�n )Ln�1(u
+n�1, u
�n�1) . . .L1(u
+1 , u
�1 ).(13)
The choice of parameters u±k depends on the diagram
under consideration. It will be convenient to introducethe notation [�+k , �
�k ] := (u+�+k , u+��k ) and [�k] := u+�k.
By convention we choose the parameters on the boundarylegs at the top to be [1, 2]. Then the parameters on theright, bottom or left boundary legs have to be [2, 3], [3, 4]or [4, 5], respectively (see Fig. 3 for an example). TheLax operator defined in (11) acts on an auxiliary and on aquantum space. While the product in (13) is taken in theauxiliary space, each Lax operator acts on one externalleg of the considered Feynman graph which representsthe quantum space.
Proving the invariance statement (12) boils down toemploying the lasso method [9], i.e. to moving the mon-odromy through a given graph as displayed in Fig. 3.
(a) =
(b)
[�, � + 1]
[� + 1, � + 2]
= [� + 2]⇥[� + 1, � + 1]
(c) =
FIG. 4. Rules employed to prove the Yangian invariance.(a): The intertwining relation (14). (b) and (c): Pulling themonodromy contour through an integration vertex, cf. (15).
The most important relation used in this process is thefollowing intertwining relation for the Lax operator andthe x-space propagator, cf. Fig. 4a:
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
. (14)
Moreover, we make use of the following relation, whichallows us to move a product of Lax operators through anintegration vertex, cf. Fig. 4b:
Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]
1
x201x
202x
203x
204
(15)
= [� + 2]
Zd4x0
1
x201x
202
L0[� + 1, � + 1]1
x203x
204
.
A third relation of this type is depicted in Fig. 4c. Finally,the Lax operator and its partially integrated version de-noted by LT act on a constant function as
L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)
These rules are su�cient to move the monodromy con-tour in Fig. 3a through the whole graph to end up withthe eigenvalue on the right hand side of (12). See Fig. 3bfor an intermediate step. The eigenvalue �(~u) in (12) iscomposed of the factors picked up in this process via therelations (15) and (16), cf. [9] for explicit expressions.
OFF- AND ON-SHELL LEGS
In the above construction, the external variables xµi
were in fact unconstrained. To interpret the xi as re-gion momenta for a scattering amplitude with masslesson-shell legs, we require that p2k = (xk � xk+1)2 = 0. No-tably, the delta-function imposing this constraint obeys
3
(a) (b)
FIG. 3. (a): Monodromy encircling a sample fishnet graphrepresenting the left hand side of (12). (b): Intermediate stepof the proof of Yangian symmetry.
and the algebra relations are rephrased via the Yang–Baxter equation with Yang’s R-matrix R(u) = 1+uP:
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)
We explicitly solve this RTT-relation by defining themonodromy as a product of conformal Lax operators [7]
Lk,↵�(u+k , u
�k ) = uk 1k,↵� +
12S
ab↵� J
�kk,ab, (11)
each of which obeys (10) with Tk ! Lk. Here we pack-age the inhomogeneities uk and the conformal dimensions�k into the symmetric variables u+
k := uk + �k�42 and
u�k := uk � �k
2 . The Jk,ab denote the di↵erential repre-sentation of the conformal algebra displayed in (4), andwe have Sab = i
4 [�a,�b]|upper block, with �a representing
six-dimensional gamma matrices for R2,4. The Yangiansymmetry of the box integral I4 and its n-point general-izations In now translates into the eigenvalue equation [8]
T(~u) In = �(~u) In 1, (12)
where T(~u) denotes the inhomogeneous monodromy
T(~u) = Ln(u+n , u
�n )Ln�1(u
+n�1, u
�n�1) . . .L1(u
+1 , u
�1 ).(13)
The choice of parameters u±k depends on the diagram
under consideration. It will be convenient to introducethe notation [�+k , �
�k ] := (u+�+k , u+��k ) and [�k] := u+�k.
By convention we choose the parameters on the boundarylegs at the top to be [1, 2]. Then the parameters on theright, bottom or left boundary legs have to be [2, 3], [3, 4]or [4, 5], respectively (see Fig. 3 for an example). TheLax operator defined in (11) acts on an auxiliary and on aquantum space. While the product in (13) is taken in theauxiliary space, each Lax operator acts on one externalleg of the considered Feynman graph which representsthe quantum space.
Proving the invariance statement (12) boils down toemploying the lasso method [9], i.e. to moving the mon-odromy through a given graph as displayed in Fig. 3.
(a) =
(b) = [� + 2]⇥
(c)
[�, � + 1]
=[� + 1, �]
FIG. 4. Rules employed to prove the Yangian invariance.(a): The intertwining relation (14). (b) and (c): Pulling themonodromy contour through an integration vertex, cf. (15).
The most important relation used in this process is thefollowing intertwining relation for the Lax operator andthe x-space propagator, cf. Fig. 4a:
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
. (14)
Moreover, we make use of the following relation, whichallows us to move a product of Lax operators through anintegration vertex, cf. Fig. 4b:
Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]
1
x201x
202x
203x
204
(15)
= [� + 2]
Zd4x0
1
x201x
202
L0[� + 1, � + 1]1
x203x
204
.
A third relation of this type is depicted in Fig. 4c. Finally,the Lax operator and its partially integrated version de-noted by LT act on a constant function as
L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)
These rules are su�cient to move the monodromy con-tour in Fig. 3a through the whole graph to end up withthe eigenvalue on the right hand side of (12). See Fig. 3bfor an intermediate step. The eigenvalue �(~u) in (12) iscomposed of the factors picked up in this process via therelations (15) and (16), cf. [9] for explicit expressions.
OFF- AND ON-SHELL LEGS
In the above construction, the external variables xµi
were in fact unconstrained. To interpret the xi as re-gion momenta for a scattering amplitude with masslesson-shell legs, we require that p2k = (xk � xk+1)2 = 0. No-tably, the delta-function imposing this constraint obeys
[ ? , �]
[� + 1, • ]
[ ? , � + 1]
[�, • ]
=<latexit sha1_base64="L9YnUhfbiHS23kj/yV0czku0Uz4=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUA9C0YvHFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3U795hMqzWN5b8YJBhEdSN7njBorNa675YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldpOnUYQjOIZT8OACanAHdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucP+1+MlQ==</latexit><latexit sha1_base64="L9YnUhfbiHS23kj/yV0czku0Uz4=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUA9C0YvHFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3U795hMqzWN5b8YJBhEdSN7njBorNa675YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldpOnUYQjOIZT8OACanAHdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucP+1+MlQ==</latexit><latexit sha1_base64="L9YnUhfbiHS23kj/yV0czku0Uz4=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUA9C0YvHFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3U795hMqzWN5b8YJBhEdSN7njBorNa675YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldpOnUYQjOIZT8OACanAHdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucP+1+MlQ==</latexit><latexit sha1_base64="L9YnUhfbiHS23kj/yV0czku0Uz4=">AAAB53icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUA9C0YvHFowttKFsttN27WYTdjdCCf0FXjyoePUvefPfuG1z0NYHA4/3ZpiZFyaCa+O6305hZXVtfaO4Wdra3tndK+8fPOg4VQx9FotYtUKqUXCJvuFGYCtRSKNQYDMc3U795hMqzWN5b8YJBhEdSN7njBorNa675YpbdWcgy8TLSQVy1Lvlr04vZmmE0jBBtW57bmKCjCrDmcBJqZNqTCgb0QG2LZU0Qh1ks0Mn5MQqPdKPlS1pyEz9PZHRSOtxFNrOiJqhXvSm4n9eOzX9yyDjMkkNSjZf1E8FMTGZfk16XCEzYmwJZYrbWwkbUkWZsdmUbAje4svLxD+rXlW9xnmldpOnUYQjOIZT8OACanAHdfCBAcIzvMKb8+i8OO/Ox7y14OQzh/AHzucP+1+MlQ==</latexit>
3
(a) (b)
FIG. 3. (a): Monodromy encircling a sample fishnet graphrepresenting the left hand side of (12). (b): Intermediate stepof the proof of Yangian symmetry.
and the algebra relations are rephrased via the Yang–Baxter equation with Yang’s R-matrix R(u) = 1+uP:
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)
We explicitly solve this RTT-relation by defining themonodromy as a product of conformal Lax operators [7]
Lk,↵�(u+k , u
�k ) = uk 1k,↵� +
12S
ab↵� J
�kk,ab, (11)
each of which obeys (10) with Tk ! Lk. Here we pack-age the inhomogeneities uk and the conformal dimensions�k into the symmetric variables u+
k := uk + �k�42 and
u�k := uk � �k
2 . The Jk,ab denote the di↵erential repre-sentation of the conformal algebra displayed in (4), andwe have Sab = i
4 [�a,�b]|upper block, with �a representing
six-dimensional gamma matrices for R2,4. The Yangiansymmetry of the box integral I4 and its n-point general-izations In now translates into the eigenvalue equation [8]
T(~u) In = �(~u) In 1, (12)
where T(~u) denotes the inhomogeneous monodromy
T(~u) = Ln(u+n , u
�n )Ln�1(u
+n�1, u
�n�1) . . .L1(u
+1 , u
�1 ).(13)
The choice of parameters u±k depends on the diagram
under consideration. It will be convenient to introducethe notation [�+k , �
�k ] := (u+�+k , u+��k ) and [�k] := u+�k.
By convention we choose the parameters on the boundarylegs at the top to be [1, 2]. Then the parameters on theright, bottom or left boundary legs have to be [2, 3], [3, 4]or [4, 5], respectively (see Fig. 3 for an example). TheLax operator defined in (11) acts on an auxiliary and on aquantum space. While the product in (13) is taken in theauxiliary space, each Lax operator acts on one externalleg of the considered Feynman graph which representsthe quantum space.
Proving the invariance statement (12) boils down toemploying the lasso method [9], i.e. to moving the mon-odromy through a given graph as displayed in Fig. 3.
(a) =
(b) = [� + 2]⇥
(c) =
FIG. 4. Rules employed to prove the Yangian invariance.(a): The intertwining relation (14). (b) and (c): Pulling themonodromy contour through an integration vertex, cf. (15).
The most important relation used in this process is thefollowing intertwining relation for the Lax operator andthe x-space propagator, cf. Fig. 4a:
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
. (14)
Moreover, we make use of the following relation, whichallows us to move a product of Lax operators through anintegration vertex, cf. Fig. 4b:
Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]
1
x201x
202x
203x
204
(15)
= [� + 2]
Zd4x0
1
x201x
202
L0[� + 1, � + 1]1
x203x
204
.
A third relation of this type is depicted in Fig. 4c. Finally,the Lax operator and its partially integrated version de-noted by LT act on a constant function as
L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)
These rules are su�cient to move the monodromy con-tour in Fig. 3a through the whole graph to end up withthe eigenvalue on the right hand side of (12). See Fig. 3bfor an intermediate step. The eigenvalue �(~u) in (12) iscomposed of the factors picked up in this process via therelations (15) and (16), cf. [9] for explicit expressions.
OFF- AND ON-SHELL LEGS
In the above construction, the external variables xµi
were in fact unconstrained. To interpret the xi as re-gion momenta for a scattering amplitude with masslesson-shell legs, we require that p2k = (xk � xk+1)2 = 0. No-tably, the delta-function imposing this constraint obeys
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3
(a) (b)
FIG. 3. (a): Monodromy encircling a sample fishnet graphrepresenting the left hand side of (12). (b): Intermediate stepof the proof of Yangian symmetry.
and the algebra relations are rephrased via the Yang–Baxter equation with Yang’s R-matrix R(u) = 1+uP:
R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)
We explicitly solve this RTT-relation by defining themonodromy as a product of conformal Lax operators [7]
Lk,↵�(u+k , u
�k ) = uk 1k,↵� +
12S
ab↵� J
�kk,ab, (11)
each of which obeys (10) with Tk ! Lk. Here we pack-age the inhomogeneities uk and the conformal dimensions�k into the symmetric variables u+
k := uk + �k�42 and
u�k := uk � �k
2 . The Jk,ab denote the di↵erential repre-sentation of the conformal algebra displayed in (4), andwe have Sab = i
4 [�a,�b]|upper block, with �a representing
six-dimensional gamma matrices for R2,4. The Yangiansymmetry of the box integral I4 and its n-point general-izations In now translates into the eigenvalue equation [8]
T(~u) In = �(~u) In 1, (12)
where T(~u) denotes the inhomogeneous monodromy
T(~u) = Ln(u+n , u
�n )Ln�1(u
+n�1, u
�n�1) . . .L1(u
+1 , u
�1 ).(13)
The choice of parameters u±k depends on the diagram
under consideration. It will be convenient to introducethe notation [�+k , �
�k ] := (u+�+k , u+��k ) and [�k] := u+�k.
By convention we choose the parameters on the boundarylegs at the top to be [1, 2]. Then the parameters on theright, bottom or left boundary legs have to be [2, 3], [3, 4]or [4, 5], respectively (see Fig. 3 for an example). TheLax operator defined in (11) acts on an auxiliary and on aquantum space. While the product in (13) is taken in theauxiliary space, each Lax operator acts on one externalleg of the considered Feynman graph which representsthe quantum space.
Proving the invariance statement (12) boils down toemploying the lasso method [9], i.e. to moving the mon-odromy through a given graph as displayed in Fig. 3.
(a) =
(b) = [� + 2]⇥
(c) =
FIG. 4. Rules employed to prove the Yangian invariance.(a): The intertwining relation (14). (b) and (c): Pulling themonodromy contour through an integration vertex, cf. (15).
The most important relation used in this process is thefollowing intertwining relation for the Lax operator andthe x-space propagator, cf. Fig. 4a:
1
x212
L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1
x212
. (14)
Moreover, we make use of the following relation, whichallows us to move a product of Lax operators through anintegration vertex, cf. Fig. 4b:
Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]
1
x201x
202x
203x
204
(15)
= [� + 2]
Zd4x0
1
x201x
202
L0[� + 1, � + 1]1
x203x
204
.
A third relation of this type is depicted in Fig. 4c. Finally,the Lax operator and its partially integrated version de-noted by LT act on a constant function as
L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)
These rules are su�cient to move the monodromy con-tour in Fig. 3a through the whole graph to end up withthe eigenvalue on the right hand side of (12). See Fig. 3bfor an intermediate step. The eigenvalue �(~u) in (12) iscomposed of the factors picked up in this process via therelations (15) and (16), cf. [9] for explicit expressions.
OFF- AND ON-SHELL LEGS
In the above construction, the external variables xµi
were in fact unconstrained. To interpret the xi as re-gion momenta for a scattering amplitude with masslesson-shell legs, we require that p2k = (xk � xk+1)2 = 0. No-tably, the delta-function imposing this constraint obeys
Integration by parts
Yangian Symmetry for Scalar Fishnets
❖ Proof of Yangian symmetry: example
[1,2] [1,2] [1,2]
[2,3]
[2,3]
[2,3][2,3]
[3,4]
[3,4]
[3,4]
[4,5]
[4,5]
[4,5]
[4,5]
[1,2] [1,2]
[2,2]
[2,3]
[2,3]
[3,4] [3,4]
[3,4]
[4,5]
[4,5]
[4,5]
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❖ The end of the story.
… wait a minute!
Scalar Fishnets in Momentum Space
❖ The dual Feynman diagram in momentum space
• Problem 1: different kind of interaction vertices
Split into two
Scalar Fishnets in Momentum Space
❖ The dual Feynman diagram in momentum space
• Solution: joining external points
Scalar Fishnets in Momentum Space
❖ The dual Feynman diagram in momentum space
• Problem 2: All external legs are off-shell
• Solution: adding delta-functions
p8 p3
p7 p4
p1
p6
p2
p5
x8 x4
x2
x6
p8 p3
p7 p4
p1
p6
p2
p5
x1 x3
x5x7
x8 x4
x2
x6
Off-shell external legs On-shell external legs
Scalar Fishnets in Momentum Space
❖ The dual Feynman diagram in momentum space
• Direct way of doing: Lax operator in momentum space
• We can obtain it by using Fourier transformation
• Similar intertwining relations
L(x)(u+, u�)
ZdDp eipxf(p) ⌘
ZdDp eipxL(p)(u+, u�)f(p)
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❖ New ingredients: Yukawa vertices & fermion line (dashed line)
❖ “Brick wall” diagrams
❖ Lax operator with spin
❖ Still have Yangian symmetry!
Fishnets with Fermions in 4D
Including Fermions in 4d
Obtain further integrable theories from limits of “-deformed N = 4 SYMtheory, e.g. two-fermion-three-scalar model: [Kazakov, Gurdogan
Caetano 2016 ]
Lint„Â = Nc Tr
!›2
1„†3„†
1„3„1 + ›22„†
2„†1„2„1 +
›1›2(Â1„1Â4 ≠ Â1„†
1Â4)".
I Now we have 4pt scalar and 3pt Yukawa vertices.I Use Lax operator for non-scalar representations [Chicherin, Derkachov
Isaev 2013 ].
Yangian-invariant “Brick Wall” Feynman Graphs:
Solid Unsolid
Florian Loebbert: Yangian Symmetry for Fishnet Feynman Graphs 12 / 13
4
5
3
1
20
00 0
[Chicherin, Derkachov Isaev ’13]
Summary
❖ Yangian symmetries (PDEs) for Feynman diagrams
❖ Integrability for non-vanishing dual Coxeter number
❖ New kind of fishnet diagrams: brick wall
Fishnet diagrams
N = 4 SYM<latexit sha1_base64="Y4ROKgcv2dBanylnNdwMbeEUvO8=">AAACAnicbVBNS8NAEN34WetX1ZteFovgqSRSUA9C0YsXpaKxlaaUzXbTLt1swu5ELCHgxb/ixYOKV3+FN/+N24+Dtj4YeLw3w8w8PxZcg21/WzOzc/MLi7ml/PLK6tp6YWPzVkeJosylkYhU3SeaCS6ZCxwEq8eKkdAXrOb3zgZ+7Z4pzSN5A/2YNUPSkTzglICRWoVtLyTQpUSkl9lJGXvYA/YA6fXdRdYqFO2SPQSeJs6YFNEY1Vbhy2tHNAmZBCqI1g3HjqGZEgWcCpblvUSzmNAe6bCGoZKETDfT4Q8Z3jNKGweRMiUBD9XfEykJte6HvukcXKwnvYH4n9dIIDhqplzGCTBJR4uCRGCI8CAQ3OaKURB9QwhV3NyKaZcoQsHEljchOJMvTxP3oHRccq7KxcrpOI0c2kG7aB856BBV0DmqIhdR9Iie0St6s56sF+vd+hi1zljjmS30B9bnD7OZlx4=</latexit><latexit sha1_base64="Y4ROKgcv2dBanylnNdwMbeEUvO8=">AAACAnicbVBNS8NAEN34WetX1ZteFovgqSRSUA9C0YsXpaKxlaaUzXbTLt1swu5ELCHgxb/ixYOKV3+FN/+N24+Dtj4YeLw3w8w8PxZcg21/WzOzc/MLi7ml/PLK6tp6YWPzVkeJosylkYhU3SeaCS6ZCxwEq8eKkdAXrOb3zgZ+7Z4pzSN5A/2YNUPSkTzglICRWoVtLyTQpUSkl9lJGXvYA/YA6fXdRdYqFO2SPQSeJs6YFNEY1Vbhy2tHNAmZBCqI1g3HjqGZEgWcCpblvUSzmNAe6bCGoZKETDfT4Q8Z3jNKGweRMiUBD9XfEykJte6HvukcXKwnvYH4n9dIIDhqplzGCTBJR4uCRGCI8CAQ3OaKURB9QwhV3NyKaZcoQsHEljchOJMvTxP3oHRccq7KxcrpOI0c2kG7aB856BBV0DmqIhdR9Iie0St6s56sF+vd+hi1zljjmS30B9bnD7OZlx4=</latexit><latexit sha1_base64="Y4ROKgcv2dBanylnNdwMbeEUvO8=">AAACAnicbVBNS8NAEN34WetX1ZteFovgqSRSUA9C0YsXpaKxlaaUzXbTLt1swu5ELCHgxb/ixYOKV3+FN/+N24+Dtj4YeLw3w8w8PxZcg21/WzOzc/MLi7ml/PLK6tp6YWPzVkeJosylkYhU3SeaCS6ZCxwEq8eKkdAXrOb3zgZ+7Z4pzSN5A/2YNUPSkTzglICRWoVtLyTQpUSkl9lJGXvYA/YA6fXdRdYqFO2SPQSeJs6YFNEY1Vbhy2tHNAmZBCqI1g3HjqGZEgWcCpblvUSzmNAe6bCGoZKETDfT4Q8Z3jNKGweRMiUBD9XfEykJte6HvukcXKwnvYH4n9dIIDhqplzGCTBJR4uCRGCI8CAQ3OaKURB9QwhV3NyKaZcoQsHEljchOJMvTxP3oHRccq7KxcrpOI0c2kG7aB856BBV0DmqIhdR9Iie0St6s56sF+vd+hi1zljjmS30B9bnD7OZlx4=</latexit><latexit sha1_base64="Y4ROKgcv2dBanylnNdwMbeEUvO8=">AAACAnicbVBNS8NAEN34WetX1ZteFovgqSRSUA9C0YsXpaKxlaaUzXbTLt1swu5ELCHgxb/ixYOKV3+FN/+N24+Dtj4YeLw3w8w8PxZcg21/WzOzc/MLi7ml/PLK6tp6YWPzVkeJosylkYhU3SeaCS6ZCxwEq8eKkdAXrOb3zgZ+7Z4pzSN5A/2YNUPSkTzglICRWoVtLyTQpUSkl9lJGXvYA/YA6fXdRdYqFO2SPQSeJs6YFNEY1Vbhy2tHNAmZBCqI1g3HjqGZEgWcCpblvUSzmNAe6bCGoZKETDfT4Q8Z3jNKGweRMiUBD9XfEykJte6HvukcXKwnvYH4n9dIIDhqplzGCTBJR4uCRGCI8CAQ3OaKURB9QwhV3NyKaZcoQsHEljchOJMvTxP3oHRccq7KxcrpOI0c2kG7aB856BBV0DmqIhdR9Iie0St6s56sF+vd+hi1zljjmS30B9bnD7OZlx4=</latexit>
Bi-scalar limit
� � twisted N = 4 SYM<latexit sha1_base64="8K/4ZFgD7m2fLBVnxWjDBkZGYPk=">AAACGXicbVDLSitBEO3xdTW+cnXppjEIbgwzIqiLC6IbN4qi8UEmhJpOJTZ2zwzdNVfDMN/hxl9x40LFpa78GzsxC18HGk6fU0VVnShV0pLvv3lDwyOjY3/GJ0qTU9Mzs+W/cyc2yYzAmkhUYs4isKhkjDWSpPAsNQg6UngaXe70/NP/aKxM4mPqptjQ0IllWwogJzXLQdgBrWElJLymnK7cRGwVIQ810IUAle8X/9a4+/f9o/O9olmu+FW/D/6TBANSYQMcNMsvYSsRmcaYhAJr64GfUiMHQ1IoLEphZjEFcQkdrDsag0bbyPunFXzJKS3eTox7MfG++rkjB21tV0eusrex/e71xN+8ekbtjUYu4zQjjMXHoHamOCW8lxNvSYOCVNcREEa6Xbm4AAOCXJolF0Lw/eSfpLZa3awGh2uVre1BGuNsgS2yZRawdbbFdtkBqzHBbtgde2CP3q137z15zx+lQ96gZ559gff6DjP8oSk=</latexit><latexit sha1_base64="8K/4ZFgD7m2fLBVnxWjDBkZGYPk=">AAACGXicbVDLSitBEO3xdTW+cnXppjEIbgwzIqiLC6IbN4qi8UEmhJpOJTZ2zwzdNVfDMN/hxl9x40LFpa78GzsxC18HGk6fU0VVnShV0pLvv3lDwyOjY3/GJ0qTU9Mzs+W/cyc2yYzAmkhUYs4isKhkjDWSpPAsNQg6UngaXe70/NP/aKxM4mPqptjQ0IllWwogJzXLQdgBrWElJLymnK7cRGwVIQ810IUAle8X/9a4+/f9o/O9olmu+FW/D/6TBANSYQMcNMsvYSsRmcaYhAJr64GfUiMHQ1IoLEphZjEFcQkdrDsag0bbyPunFXzJKS3eTox7MfG++rkjB21tV0eusrex/e71xN+8ekbtjUYu4zQjjMXHoHamOCW8lxNvSYOCVNcREEa6Xbm4AAOCXJolF0Lw/eSfpLZa3awGh2uVre1BGuNsgS2yZRawdbbFdtkBqzHBbtgde2CP3q137z15zx+lQ96gZ559gff6DjP8oSk=</latexit><latexit sha1_base64="8K/4ZFgD7m2fLBVnxWjDBkZGYPk=">AAACGXicbVDLSitBEO3xdTW+cnXppjEIbgwzIqiLC6IbN4qi8UEmhJpOJTZ2zwzdNVfDMN/hxl9x40LFpa78GzsxC18HGk6fU0VVnShV0pLvv3lDwyOjY3/GJ0qTU9Mzs+W/cyc2yYzAmkhUYs4isKhkjDWSpPAsNQg6UngaXe70/NP/aKxM4mPqptjQ0IllWwogJzXLQdgBrWElJLymnK7cRGwVIQ810IUAle8X/9a4+/f9o/O9olmu+FW/D/6TBANSYQMcNMsvYSsRmcaYhAJr64GfUiMHQ1IoLEphZjEFcQkdrDsag0bbyPunFXzJKS3eTox7MfG++rkjB21tV0eusrex/e71xN+8ekbtjUYu4zQjjMXHoHamOCW8lxNvSYOCVNcREEa6Xbm4AAOCXJolF0Lw/eSfpLZa3awGh2uVre1BGuNsgS2yZRawdbbFdtkBqzHBbtgde2CP3q137z15zx+lQ96gZ559gff6DjP8oSk=</latexit><latexit sha1_base64="8K/4ZFgD7m2fLBVnxWjDBkZGYPk=">AAACGXicbVDLSitBEO3xdTW+cnXppjEIbgwzIqiLC6IbN4qi8UEmhJpOJTZ2zwzdNVfDMN/hxl9x40LFpa78GzsxC18HGk6fU0VVnShV0pLvv3lDwyOjY3/GJ0qTU9Mzs+W/cyc2yYzAmkhUYs4isKhkjDWSpPAsNQg6UngaXe70/NP/aKxM4mPqptjQ0IllWwogJzXLQdgBrWElJLymnK7cRGwVIQ810IUAle8X/9a4+/f9o/O9olmu+FW/D/6TBANSYQMcNMsvYSsRmcaYhAJr64GfUiMHQ1IoLEphZjEFcQkdrDsag0bbyPunFXzJKS3eTox7MfG++rkjB21tV0eusrex/e71xN+8ekbtjUYu4zQjjMXHoHamOCW8lxNvSYOCVNcREEa6Xbm4AAOCXJolF0Lw/eSfpLZa3awGh2uVre1BGuNsgS2yZRawdbbFdtkBqzHBbtgde2CP3q137z15zx+lQ96gZ559gff6DjP8oSk=</latexit>
Outlook
• Yangian symmetry for the full γ-deformed theory (3 bosons and 3 fermions) ?
• Yangian symmetry in 3d and 6d for models containing both bosons and fermions?
• For on-shell massless scattering processes, how to understand Yangian symmetry if we have anomaly-like behaviour arising from collinear particles?
• Using Yangian PDEs to compute the Feynman integrals?
• Deriving Yangian symmetry of correlators and amplitudes directly from the Lagrangian ?
[Chicherin, Sokatchev ’17]
❖ For special cases of Fishnet (4pt case), integrability gives integral representations for fishnets.
• How to find their Yangian symmetry?
• For m, n large but their ratio fixed, one can solve the saddle point equation exactly.
• Continuum limit hints at AdS dual?
(z, z)
0
1
1
�†m1
�m1
�n2 �†n
2
Outlook
[Basso, Dixon ’17]
[Basso, Dixon, Kosower, D-l.Z ’In progress]
Thank you!