Yangian Symmetry for Fishnet Feynman Graphstristan/HMI2018/slides/deliang.pdf · 4D N = 4 Super...

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


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=">AAACNHicbVBPS8MwHE3nvzn/TT16CQ5hgzFaEdSDMPQi6GHC5gZrKWmWbWFpWpJUNsq+lBe/hyc9eFDx6mcw7XrQzQeBx/u9X5L3vJBRqUzz1cgtLa+sruXXCxubW9s7xd29exlEApMWDlggOh6ShFFOWooqRjqhIMj3GGl7o6tk3n4gQtKAN9UkJI6PBpz2KUZKS27x9qY8dq2q3QuUrI7dmE8rFzZDfMCI7SM1FH7cFNOujYfUtRJrJbXCVOBa4BXHFqm/VnCLJbNmpoCLxMpICWRouMVnfRuOfMIVZkjKrmWGyomRUBQzMi3YkSQhwiM0IF1NOfKJdOI09RQeaaUH+4HQhyuYqr83YuRLOfE97UySyPlZIv4360aqf+bElIeRIhzPHupHDKoAJhXCHhUEKzbRBGFB9V8hHiKBsNJFJyVY85EXSeu4dl6z7k5K9cusjTw4AIegDCxwCurgGjRAC2DwCF7AO/gwnow349P4mllzRrazD/7A+P4BfjSq3A==</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



I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240






I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240




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

<latexit sha1_base64="D4X3B4p/6fD8iebLxmINCzh0RI8=">AAAB7HicbVBNS8NAEJ34WetX1aOXxSJ4KokI6q3oxWMF0xbaUCbbTbt2swm7G6GE/gcvHlS8+oO8+W/ctjlo64OBx3szzMwLU8G1cd1vZ2V1bX1js7RV3t7Z3duvHBw2dZIpynyaiES1Q9RMcMl8w41g7VQxjEPBWuHoduq3npjSPJEPZpyyIMaB5BGnaKzU7KJIh9irVN2aOwNZJl5BqlCg0at8dfsJzWImDRWodcdzUxPkqAyngk3K3UyzFOkIB6xjqcSY6SCfXTshp1bpkyhRtqQhM/X3RI6x1uM4tJ0xmqFe9Kbif14nM9FVkHOZZoZJOl8UZYKYhExfJ32uGDVibAlSxe2thA5RITU2oLINwVt8eZn457Xrmnd/Ua3fFGmU4BhO4Aw8uIQ63EEDfKDwCM/wCm9O4rw4787HvHXFKWaO4A+czx/36Y7s</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><latexit sha1_base64="nzBzl2EHHmDTqXdjvnR25IPCseI=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiF48VjC20oWw223bpZjfsToQS+iO8eFDx6v/x5r9x0+ag1QcDj/dmmJkXpYIb9Lwvp7Kyura+Ud2sbW3v7O7V9w8ejMo0ZQFVQuluRAwTXLIAOQrWTTUjSSRYJ5rcFH7nkWnDlbzHacrChIwkH3JK0EqdPo0Vmtqg3vCa3hzuX+KXpAEl2oP6Zz9WNEuYRCqIMT3fSzHMiUZOBZvV+plhKaETMmI9SyVJmAnz+bkz98QqsTtU2pZEd67+nMhJYsw0iWxnQnBslr1C/M/rZTi8DHMu0wyZpItFw0y4qNzidzfmmlEUU0sI1dze6tIx0YSiTagIwV9++S8JzppXTf/uvNG6LtOowhEcwyn4cAEtuIU2BEBhAk/wAq9O6jw7b877orXilDOH8AvOxzdRO48X</latexit>




❖ For : type scalar Feynman diagram ↵ =⇡

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�4<latexit sha1_base64="i52qBD7TfmvGCxT3mQQGES/jOcE=">AAAB7HicbVBNTwIxEJ3FL8Qv1KOXRmLiiewaEvRG9OIRExdIYCXd0oVKt920XROy4T948aDGqz/Im//GAntQ8CWTvLw3k5l5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEeoTyaXqhFhTzgT1DTOcdhJFcRxy2g7HNzO//USVZlLcm0lCgxgPBYsYwcZKrV4yYg+1frniVt050CrxclKBHM1++as3kCSNqTCEY627npuYIMPKMMLptNRLNU0wGeMh7VoqcEx1kM2vnaIzqwxQJJUtYdBc/T2R4VjrSRzazhibkV72ZuJ/Xjc10WWQMZGkhgqyWBSlHBmJZq+jAVOUGD6xBBPF7K2IjLDCxNiASjYEb/nlVeJfVK+q3l2t0rjO0yjCCZzCOXhQhwbcQhN8IPAIz/AKb450Xpx352PRWnDymWP4A+fzB6rojrk=</latexit><latexit sha1_base64="i52qBD7TfmvGCxT3mQQGES/jOcE=">AAAB7HicbVBNTwIxEJ3FL8Qv1KOXRmLiiewaEvRG9OIRExdIYCXd0oVKt920XROy4T948aDGqz/Im//GAntQ8CWTvLw3k5l5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEeoTyaXqhFhTzgT1DTOcdhJFcRxy2g7HNzO//USVZlLcm0lCgxgPBYsYwcZKrV4yYg+1frniVt050CrxclKBHM1++as3kCSNqTCEY627npuYIMPKMMLptNRLNU0wGeMh7VoqcEx1kM2vnaIzqwxQJJUtYdBc/T2R4VjrSRzazhibkV72ZuJ/Xjc10WWQMZGkhgqyWBSlHBmJZq+jAVOUGD6xBBPF7K2IjLDCxNiASjYEb/nlVeJfVK+q3l2t0rjO0yjCCZzCOXhQhwbcQhN8IPAIz/AKb450Xpx352PRWnDymWP4A+fzB6rojrk=</latexit><latexit sha1_base64="i52qBD7TfmvGCxT3mQQGES/jOcE=">AAAB7HicbVBNTwIxEJ3FL8Qv1KOXRmLiiewaEvRG9OIRExdIYCXd0oVKt920XROy4T948aDGqz/Im//GAntQ8CWTvLw3k5l5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEeoTyaXqhFhTzgT1DTOcdhJFcRxy2g7HNzO//USVZlLcm0lCgxgPBYsYwcZKrV4yYg+1frniVt050CrxclKBHM1++as3kCSNqTCEY627npuYIMPKMMLptNRLNU0wGeMh7VoqcEx1kM2vnaIzqwxQJJUtYdBc/T2R4VjrSRzazhibkV72ZuJ/Xjc10WWQMZGkhgqyWBSlHBmJZq+jAVOUGD6xBBPF7K2IjLDCxNiASjYEb/nlVeJfVK+q3l2t0rjO0yjCCZzCOXhQhwbcQhN8IPAIz/AKb450Xpx352PRWnDymWP4A+fzB6rojrk=</latexit><latexit sha1_base64="i52qBD7TfmvGCxT3mQQGES/jOcE=">AAAB7HicbVBNTwIxEJ3FL8Qv1KOXRmLiiewaEvRG9OIRExdIYCXd0oVKt920XROy4T948aDGqz/Im//GAntQ8CWTvLw3k5l5YcKZNq777RTW1jc2t4rbpZ3dvf2D8uFRS8tUEeoTyaXqhFhTzgT1DTOcdhJFcRxy2g7HNzO//USVZlLcm0lCgxgPBYsYwcZKrV4yYg+1frniVt050CrxclKBHM1++as3kCSNqTCEY627npuYIMPKMMLptNRLNU0wGeMh7VoqcEx1kM2vnaIzqwxQJJUtYdBc/T2R4VjrSRzazhibkV72ZuJ/Xjc10WWQMZGkhgqyWBSlHBmJZq+jAVOUGD6xBBPF7K2IjLDCxNiASjYEb/nlVeJfVK+q3l2t0rjO0yjCCZzCOXhQhwbcQhN8IPAIz/AKb450Xpx352PRWnDymWP4A+fzB6rojrk=</latexit>

[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?


❖ Feynman rules Vertex

Propagator

❖ UV/IR finite: unbroken conformal symmetry

Zddx ,



I Vertex :s

d4x


jk©

1(xj≠xk)2

e.g.


s d4x0x2

10x220x2

30x240




, |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)









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) =


x213x

224�(u, v)


234

x213x

224

and v =x214x

223

x213x



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









(Dated: December 13, 2017)









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

⇤[email protected]

†[email protected]

‡[email protected]

§[email protected]

¶[email protected]

Depend on the graphBilocal piece: universal

MITP/17-049, HU-EP-17/20


















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


x213x

224�(u, v)


234

x213x

224

and v =x214x

223

x213x



0 = �+ (3u� 1)@�

@u+ 3v

@�

@v+ (u� 1)u

@2�

@u2+ v2

@2�

@v2+ 2uv

@2�

@u@v




§[email protected]

¶[email protected]

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


❖ 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 .


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+ �


❖ T includes all symmetry generators

Higher level generators

MITP/17-049, HU-EP-17/20


















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)




§[email protected]

¶[email protected]

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

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[Ja, Jb] = f cabJc

<latexit sha1_base64="kUhZYNm6pQzIfj7sRLThkRdZ2ps=">AAACKnicbVDLSsNAFJ34rPVVdelmsAgupCQiqAuh1I24qmBtoYnhZjpph04ezEyEEvI/bvwVF7rQ4tYPcdJG0NYDA2fOuZd77/FizqQyzbGxsLi0vLJaWiuvb2xubVd2du9llAhCWyTikeh4IClnIW0ppjjtxIJC4HHa9oZXud9+pEKyKLxTo5g6AfRD5jMCSktupdG1A1ADEaQ3mQvH+OdnD0BpKXM97OBL7LspeNkDmfdJ2a1UzZo5AZ4nVkGqqEDTrbzavYgkAQ0V4SBl1zJj5aQgFCOcZmU7kTQGMoQ+7WoaQkClk05uzfChVnrYj4R+ocIT9XdHCoGUo8DTlfmqctbLxf+8bqL8cydlYZwoGpLpID/hWEU4Dw73mKBE8ZEmQATTu2IyAAFE6XjzEKzZk+dJ66R2UbNuT6v1RpFGCe2jA3SELHSG6ugaNVELEfSEXtA7+jCejTdjbHxOSxeMomcP/YHx9Q1ftqf5</latexit><latexit sha1_base64="kUhZYNm6pQzIfj7sRLThkRdZ2ps=">AAACKnicbVDLSsNAFJ34rPVVdelmsAgupCQiqAuh1I24qmBtoYnhZjpph04ezEyEEvI/bvwVF7rQ4tYPcdJG0NYDA2fOuZd77/FizqQyzbGxsLi0vLJaWiuvb2xubVd2du9llAhCWyTikeh4IClnIW0ppjjtxIJC4HHa9oZXud9+pEKyKLxTo5g6AfRD5jMCSktupdG1A1ADEaQ3mQvH+OdnD0BpKXM97OBL7LspeNkDmfdJ2a1UzZo5AZ4nVkGqqEDTrbzavYgkAQ0V4SBl1zJj5aQgFCOcZmU7kTQGMoQ+7WoaQkClk05uzfChVnrYj4R+ocIT9XdHCoGUo8DTlfmqctbLxf+8bqL8cydlYZwoGpLpID/hWEU4Dw73mKBE8ZEmQATTu2IyAAFE6XjzEKzZk+dJ66R2UbNuT6v1RpFGCe2jA3SELHSG6ugaNVELEfSEXtA7+jCejTdjbHxOSxeMomcP/YHx9Q1ftqf5</latexit><latexit sha1_base64="kUhZYNm6pQzIfj7sRLThkRdZ2ps=">AAACKnicbVDLSsNAFJ34rPVVdelmsAgupCQiqAuh1I24qmBtoYnhZjpph04ezEyEEvI/bvwVF7rQ4tYPcdJG0NYDA2fOuZd77/FizqQyzbGxsLi0vLJaWiuvb2xubVd2du9llAhCWyTikeh4IClnIW0ppjjtxIJC4HHa9oZXud9+pEKyKLxTo5g6AfRD5jMCSktupdG1A1ADEaQ3mQvH+OdnD0BpKXM97OBL7LspeNkDmfdJ2a1UzZo5AZ4nVkGqqEDTrbzavYgkAQ0V4SBl1zJj5aQgFCOcZmU7kTQGMoQ+7WoaQkClk05uzfChVnrYj4R+ocIT9XdHCoGUo8DTlfmqctbLxf+8bqL8cydlYZwoGpLpID/hWEU4Dw73mKBE8ZEmQATTu2IyAAFE6XjzEKzZk+dJ66R2UbNuT6v1RpFGCe2jA3SELHSG6ugaNVELEfSEXtA7+jCejTdjbHxOSxeMomcP/YHx9Q1ftqf5</latexit><latexit sha1_base64="kUhZYNm6pQzIfj7sRLThkRdZ2ps=">AAACKnicbVDLSsNAFJ34rPVVdelmsAgupCQiqAuh1I24qmBtoYnhZjpph04ezEyEEvI/bvwVF7rQ4tYPcdJG0NYDA2fOuZd77/FizqQyzbGxsLi0vLJaWiuvb2xubVd2du9llAhCWyTikeh4IClnIW0ppjjtxIJC4HHa9oZXud9+pEKyKLxTo5g6AfRD5jMCSktupdG1A1ADEaQ3mQvH+OdnD0BpKXM97OBL7LspeNkDmfdJ2a1UzZo5AZ4nVkGqqEDTrbzavYgkAQ0V4SBl1zJj5aQgFCOcZmU7kTQGMoQ+7WoaQkClk05uzfChVnrYj4R+ocIT9XdHCoGUo8DTlfmqctbLxf+8bqL8cydlYZwoGpLpID/hWEU4Dw73mKBE8ZEmQATTu2IyAAFE6XjzEKzZk+dJ66R2UbNuT6v1RpFGCe2jA3SELHSG6ugaNVELEfSEXtA7+jCejTdjbHxOSxeMomcP/YHx9Q1ftqf5</latexit>

[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=">AAACeHichZFLSwMxEMez66vWV9WjHqKl2oKWXRXUg1D0Ir2oYK3Qrstsmtpg9kGSFcqy38HP5s0v4sWL2bY+KzgQ+POb+SeTGS/iTCrLejHMicmp6ZncbH5ufmFxqbC8ciPDWBDaICEPxa0HknIW0IZiitPbSFDwPU6b3sNZlm8+UiFZGFyrfkQdH+4D1mUElEZu4anV9kH1hJ+0e6CSepq6sIPHobeDP1g9dYnj4N2vqvr/pk+WWfHJkBPgyUVa/rrmbr/iFopW1RoEHhf2SBTRKC7dwnO7E5LYp4EiHKRs2VaknASEYoTTNN+OJY2APMA9bWkZgE+lkwwml+KSJh3cDYU+gcID+t2RgC9l3/d0Zdak/J3L4F+5Vqy6R07CgihWNCDDh7oxxyrE2RpwhwlKFO9rAUQw3SsmPRBAlF5WXg/B/v3lcdHYqx5X7auDYu10NI0cWkObqIxsdIhq6BxdogYi6NVYN0rGlvFmbpjbZmVYahojzyr6EebeO6yowXo=</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


MITP/17-049, HU-EP-17/20


















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)




§[email protected]

¶[email protected]

[�+, ��] := (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



❖ Proof of Yangian symmetry: intertwining relations

• Intertwiners

• Action on the interaction vertex

• Vacuum

3

(a) (b)



R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)


Lk,↵�(u+k , u

�k ) = uk 1k,↵� +

12S

ab↵� J

�kk,ab, (11)


k := uk + �k�42 and

u�k := uk � �k




T(~u) In = �(~u) In 1, (12)


T(~u) = Ln(u+n , u

�n )Ln�1(u

+n�1, u

�n�1) . . .L1(u

+1 , u

�1 ).(13)



�k ] := (u+�+k , u+��k ) and [�k] := u+�k.



(a) =

(b)

[�, � + 1]

[� + 1, � + 2]

= [� + 2]⇥[� + 1, � + 1]

(c) =



1

x212

L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1

x212

. (14)


Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]

1

x201x

202x

203x

204

(15)

= [� + 2]

Zd4x0

1

x201x

202

L0[� + 1, � + 1]1

x203x

204

.


L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)





3

(a) (b)



R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)


Lk,↵�(u+k , u

�k ) = uk 1k,↵� +

12S

ab↵� J

�kk,ab, (11)


k := uk + �k�42 and

u�k := uk � �k




T(~u) In = �(~u) In 1, (12)


T(~u) = Ln(u+n , u

�n )Ln�1(u

+n�1, u

�n�1) . . .L1(u

+1 , u

�1 ).(13)



�k ] := (u+�+k , u+��k ) and [�k] := u+�k.



(a) =

(b) = [� + 2]⇥

(c)

[�, � + 1]

=[� + 1, �]



1

x212

L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1

x212

. (14)


Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]

1

x201x

202x

203x

204

(15)

= [� + 2]

Zd4x0

1

x201x

202

L0[� + 1, � + 1]1

x203x

204

.


L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)





[ ? , �]

[� + 1, • ]

[ ? , � + 1]

[�, • ]

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3

(a) (b)



R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)


Lk,↵�(u+k , u

�k ) = uk 1k,↵� +

12S

ab↵� J

�kk,ab, (11)


k := uk + �k�42 and

u�k := uk � �k




T(~u) In = �(~u) In 1, (12)


T(~u) = Ln(u+n , u

�n )Ln�1(u

+n�1, u

�n�1) . . .L1(u

+1 , u

�1 ).(13)



�k ] := (u+�+k , u+��k ) and [�k] := u+�k.



(a) =

(b) = [� + 2]⇥

(c) =



1

x212

L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1

x212

. (14)


Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]

1

x201x

202x

203x

204

(15)

= [� + 2]

Zd4x0

1

x201x

202

L0[� + 1, � + 1]1

x203x

204

.


L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)





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3

(a) (b)



R12(u� v)T1(u)T2(v) = T2(v)T1(u)R12(u� v). (10)


Lk,↵�(u+k , u

�k ) = uk 1k,↵� +

12S

ab↵� J

�kk,ab, (11)


k := uk + �k�42 and

u�k := uk � �k




T(~u) In = �(~u) In 1, (12)


T(~u) = Ln(u+n , u

�n )Ln�1(u

+n�1, u

�n�1) . . .L1(u

+1 , u

�1 ).(13)



�k ] := (u+�+k , u+��k ) and [�k] := u+�k.



(a) =

(b) = [� + 2]⇥

(c) =



1

x212

L2[�, • ]L1[ ? , � + 1] = L2[� + 1, • ]L1[ ? , �]1

x212

. (14)


Zd4x0 L2[� + 1, � + 2]L1[�, � + 1]

1

x201x

202x

203x

204

(15)

= [� + 2]

Zd4x0

1

x201x

202

L0[� + 1, � + 1]1

x203x

204

.


L↵� [�, � + 2] · 1 = LT↵� [� + 2, �] · 1 = [� + 2]�↵� . (16)





Integration by parts


❖ 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



• Solution: joining external points



• 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



• 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


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!

Yangian Symmetry for Fishnet Feynman Graphstristan/HMI2018/slides/deliang.pdf · 4D N = 4 Super...

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