The Diagrammatic Coaction Einan Gardi (Higgs Centre, Edinburgh) In - - PowerPoint PPT Presentation

the diagrammatic coaction
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The Diagrammatic Coaction Einan Gardi (Higgs Centre, Edinburgh) In - - PowerPoint PPT Presentation

I V N E U R S E I H T Y T O H F G R E U D B I N The Diagrammatic Coaction Einan Gardi (Higgs Centre, Edinburgh) In collaboration with: Samuel Abreu, Ruth Britto, Claude Duhr and James Matthew References: Phys.Rev.Lett.


slide-1
SLIDE 1

The Diagrammatic Coaction

Einan Gardi (Higgs Centre, Edinburgh)

In collaboration with: Samuel Abreu, Ruth Britto, Claude Duhr and James Matthew

T H E U N I V E R S I T Y O F E D I N B U R G H

References: 
 Phys.Rev.Lett. 119 (2017) no.5, 051601, 
 JHEP 1706 (2017) 114, 
 JHEP 1712 (2017) 090 
 and work to appear soon

slide-2
SLIDE 2

The Diagrammatic Coaction: Motivation

  • Many Feynman diagrams evaluate to Multiple polylogarithms (MPLs).


MPLs admit a coaction, which maps complicated objects into simpler ones. 
 The coaction has many applications including simplifying expressions and taking discontinuities.

  • Can Feynman integrals themselves be endowed with a coaction? 



 The idea that there should be a Hopf Algebra on Feynman graphs / integrals that reproduces the one on MPLs was an inspiration for research over many years [Kreimer, Brown].

  • At one loop we now have an explicit construction in dimensional regularisation with a

simple graphical interpretation involving contractions and cuts.

  • This algebraic structure encodes the analytic properties of the integrals. 


It facilitates e.g. performing analytic continuation and deriving differential equations. 
 It also provides a new perspective on cuts and Master Integrals.

  • Can it be extended to higher loops?
slide-3
SLIDE 3

Coaction on Multiple Polylogs

  • Multiple Polylogarithms (MPLs) are iterated integrals of the form:


  • They form an algebra with the action :
  • Define a coaction:



 
 
 
 
 
 
 


  • This allows to derive functional identities by simple algebra. 



 Goncharov, Brown, Goncharov-Spradlin-Vergu-Volovich, Duhr, …

G(a1, a2, . . . , an | {z }

weight n

; z) = Z z dt t − a1 G(a2, . . . , an | {z }

weight n−1

; t)

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G(~ a; z) G(~ b; z) = X

~ c 2~ a q q~ b

G(~ c; z) ,

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∆(G(~ c; z)) = X

~ b⊆~ c

G(~ b; z) | {z }

weight |~ b|

⊗ G~

b(~

c; z) | {z }

weight |~ c|−|~ b|

,

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A × A → A

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A → A ⊗ H

<latexit sha1_base64="4oxM985+U/U+z7ejcX3Bm24hC/I=">AB+HicdVDLSgMxFM3UV62Pjrp0EyCqzLTVkd3LW6rGAf0BlKJs20oZlkSDJCHfolblwo4tZPcefmD4EFT1w4XDOvdx7T5gwqrTjfFi5tfWNza38dmFnd2+/aB8cdpRIJSZtLJiQvRApwignbU01I71EhSHjHTDyfXc794Rqajgt3qakCBGI04jipE20sAuNnwtYMXmsZEwebALjlx3M95xwaUvU898KQWqV6VfWgW3YWKIEVWgP73R8KnMaEa8yQUn3XSXSQIakpZmRW8FNFEoQnaET6hnJk1gTZ4vAZPDXKEZCmuIaLtTvExmKlZrGoemMkR6r395c/Mvrpzq6DLKk1QTjpeLopRB8+s8BTikmDNpoYgLKm5FeIxkghrk1XBhPD1KfyfdCpl1/CbWqleWcWRB8fgBJwBF3igDpqgBdoAgxQ8gCfwbN1bj9aL9bpszVmrmSPwA9bJxtJkq8=</latexit><latexit sha1_base64="4oxM985+U/U+z7ejcX3Bm24hC/I=">AB+HicdVDLSgMxFM3UV62Pjrp0EyCqzLTVkd3LW6rGAf0BlKJs20oZlkSDJCHfolblwo4tZPcefmD4EFT1w4XDOvdx7T5gwqrTjfFi5tfWNza38dmFnd2+/aB8cdpRIJSZtLJiQvRApwignbU01I71EhSHjHTDyfXc794Rqajgt3qakCBGI04jipE20sAuNnwtYMXmsZEwebALjlx3M95xwaUvU898KQWqV6VfWgW3YWKIEVWgP73R8KnMaEa8yQUn3XSXSQIakpZmRW8FNFEoQnaET6hnJk1gTZ4vAZPDXKEZCmuIaLtTvExmKlZrGoemMkR6r395c/Mvrpzq6DLKk1QTjpeLopRB8+s8BTikmDNpoYgLKm5FeIxkghrk1XBhPD1KfyfdCpl1/CbWqleWcWRB8fgBJwBF3igDpqgBdoAgxQ8gCfwbN1bj9aL9bpszVmrmSPwA9bJxtJkq8=</latexit><latexit sha1_base64="4oxM985+U/U+z7ejcX3Bm24hC/I=">AB+HicdVDLSgMxFM3UV62Pjrp0EyCqzLTVkd3LW6rGAf0BlKJs20oZlkSDJCHfolblwo4tZPcefmD4EFT1w4XDOvdx7T5gwqrTjfFi5tfWNza38dmFnd2+/aB8cdpRIJSZtLJiQvRApwignbU01I71EhSHjHTDyfXc794Rqajgt3qakCBGI04jipE20sAuNnwtYMXmsZEwebALjlx3M95xwaUvU898KQWqV6VfWgW3YWKIEVWgP73R8KnMaEa8yQUn3XSXSQIakpZmRW8FNFEoQnaET6hnJk1gTZ4vAZPDXKEZCmuIaLtTvExmKlZrGoemMkR6r395c/Mvrpzq6DLKk1QTjpeLopRB8+s8BTikmDNpoYgLKm5FeIxkghrk1XBhPD1KfyfdCpl1/CbWqleWcWRB8fgBJwBF3igDpqgBdoAgxQ8gCfwbN1bj9aL9bpszVmrmSPwA9bJxtJkq8=</latexit><latexit sha1_base64="4oxM985+U/U+z7ejcX3Bm24hC/I=">AB+HicdVDLSgMxFM3UV62Pjrp0EyCqzLTVkd3LW6rGAf0BlKJs20oZlkSDJCHfolblwo4tZPcefmD4EFT1w4XDOvdx7T5gwqrTjfFi5tfWNza38dmFnd2+/aB8cdpRIJSZtLJiQvRApwignbU01I71EhSHjHTDyfXc794Rqajgt3qakCBGI04jipE20sAuNnwtYMXmsZEwebALjlx3M95xwaUvU898KQWqV6VfWgW3YWKIEVWgP73R8KnMaEa8yQUn3XSXSQIakpZmRW8FNFEoQnaET6hnJk1gTZ4vAZPDXKEZCmuIaLtTvExmKlZrGoemMkR6r395c/Mvrpzq6DLKk1QTjpeLopRB8+s8BTikmDNpoYgLKm5FeIxkghrk1XBhPD1KfyfdCpl1/CbWqleWcWRB8fgBJwBF3igDpqgBdoAgxQ8gCfwbN1bj9aL9bpszVmrmSPwA9bJxtJkq8=</latexit>

a1 a2 a3 a4

z

a1 a2 a3 a4

z

γ~

b

~ b

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left: modify the integrand 
 retaining only the poles in 
 (keeping the original contour).

~ b

<latexit sha1_base64="mqpBotw0MUToWKQ6AnNr0Y6Gabs=">AB7nicbZDLSgMxFIbPeK31VnXpJlgEV2WmCLosuHFZwV6gHUomPdOGJpkhyRTK0Idw40IRtz6PO9/G9IJo6w+Bj/+cQ875o1RwY3/y9vY3Nre2S3sFfcPDo+OSyenTZNkmGDJSLR7YgaFxhw3IrsJ1qpDIS2IpGd7N6a4za8EQ92kmKoaQDxWPOqHVWqztGlkfTXqnsV/y5yA8Eq1CGpeq90me3n7BMorJMUGM6gZ/aMKfaciZwWuxmBlPKRnSAHYeKSjRhPl93Si6d0ydxot1Tlszd3xM5lcZMZOQ6JbVDs1qbmf/VOpmNb8OcqzSzqNjiozgTxCZkdjvpc43MiokDyjR3uxI2pJoy6xIquhDWTl6HZrUSOH64LteqyzgKcA4XcAUB3EAN7qEODWAwgid4gVcv9Z69N+90brhLWfO4I+8j2+NFY+l</latexit><latexit sha1_base64="mqpBotw0MUToWKQ6AnNr0Y6Gabs=">AB7nicbZDLSgMxFIbPeK31VnXpJlgEV2WmCLosuHFZwV6gHUomPdOGJpkhyRTK0Idw40IRtz6PO9/G9IJo6w+Bj/+cQ875o1RwY3/y9vY3Nre2S3sFfcPDo+OSyenTZNkmGDJSLR7YgaFxhw3IrsJ1qpDIS2IpGd7N6a4za8EQ92kmKoaQDxWPOqHVWqztGlkfTXqnsV/y5yA8Eq1CGpeq90me3n7BMorJMUGM6gZ/aMKfaciZwWuxmBlPKRnSAHYeKSjRhPl93Si6d0ydxot1Tlszd3xM5lcZMZOQ6JbVDs1qbmf/VOpmNb8OcqzSzqNjiozgTxCZkdjvpc43MiokDyjR3uxI2pJoy6xIquhDWTl6HZrUSOH64LteqyzgKcA4XcAUB3EAN7qEODWAwgid4gVcv9Z69N+90brhLWfO4I+8j2+NFY+l</latexit><latexit sha1_base64="mqpBotw0MUToWKQ6AnNr0Y6Gabs=">AB7nicbZDLSgMxFIbPeK31VnXpJlgEV2WmCLosuHFZwV6gHUomPdOGJpkhyRTK0Idw40IRtz6PO9/G9IJo6w+Bj/+cQ875o1RwY3/y9vY3Nre2S3sFfcPDo+OSyenTZNkmGDJSLR7YgaFxhw3IrsJ1qpDIS2IpGd7N6a4za8EQ92kmKoaQDxWPOqHVWqztGlkfTXqnsV/y5yA8Eq1CGpeq90me3n7BMorJMUGM6gZ/aMKfaciZwWuxmBlPKRnSAHYeKSjRhPl93Si6d0ydxot1Tlszd3xM5lcZMZOQ6JbVDs1qbmf/VOpmNb8OcqzSzqNjiozgTxCZkdjvpc43MiokDyjR3uxI2pJoy6xIquhDWTl6HZrUSOH64LteqyzgKcA4XcAUB3EAN7qEODWAwgid4gVcv9Z69N+90brhLWfO4I+8j2+NFY+l</latexit><latexit sha1_base64="mqpBotw0MUToWKQ6AnNr0Y6Gabs=">AB7nicbZDLSgMxFIbPeK31VnXpJlgEV2WmCLosuHFZwV6gHUomPdOGJpkhyRTK0Idw40IRtz6PO9/G9IJo6w+Bj/+cQ875o1RwY3/y9vY3Nre2S3sFfcPDo+OSyenTZNkmGDJSLR7YgaFxhw3IrsJ1qpDIS2IpGd7N6a4za8EQ92kmKoaQDxWPOqHVWqztGlkfTXqnsV/y5yA8Eq1CGpeq90me3n7BMorJMUGM6gZ/aMKfaciZwWuxmBlPKRnSAHYeKSjRhPl93Si6d0ydxot1Tlszd3xM5lcZMZOQ6JbVDs1qbmf/VOpmNb8OcqzSzqNjiozgTxCZkdjvpc43MiokDyjR3uxI2pJoy6xIquhDWTl6HZrUSOH64LteqyzgKcA4XcAUB3EAN7qEODWAwgid4gVcv9Z69N+90brhLWfO4I+8j2+NFY+l</latexit>

right: modify the contour to 
 encircle the poles in 
 (keeping the original integrand)

slide-4
SLIDE 4

The Master Formula

  • Inspired by the coaction on MPLs we define a coaction on integrals:



 


  • We have inserted a complete set of master integrands and corresponding

master contours — these basis elements are paired according to:
 
 


ωi

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γi

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Pss ✓Z

γi

ωj ◆ = δij ,

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Pss

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  • is a semi-simple projector: it retains semi-simple objects 


including and rational functions (and complete elliptic integrals) but eliminates logs and polylogs.

∆(x) = x ⊗ 1

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π

<latexit sha1_base64="AaOvRKa3hf+wAefGSgag7W3DaDE=">AB63icbZBNSwMxEIZn/az1q+rRS7AInspuEfRY8OKxgv2AdinZdLYNTbJLkhVK6V/w4kERr/4hb/4bs+0etPWFwM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpyctk2SaYtlohEdyNqUHCFLcutwG6qkcpIYCea3OX1zhNqwxP1aKcphpKOFI85oza3+ikvDypVv+YvRNYhKAKhZqDyld/mLBMorJMUGN6gZ/acEa15UzgvNzPDKaUTegIew4VlWjC2WLXObl0zpDEiXZPWbJwf0/MqDRmKiPXKakdm9Vabv5X62U2vg1nXKWZRcWH8WZIDYh+eFkyDUyK6YOKNPc7UrYmGrKrIsnDyFYPXkd2vVa4PjhutqoF3GU4Bwu4AoCuIEG3EMTWsBgDM/wCm+e9F68d+9j2brhFTNn8Efe5w+ANI3R</latexit><latexit sha1_base64="AaOvRKa3hf+wAefGSgag7W3DaDE=">AB63icbZBNSwMxEIZn/az1q+rRS7AInspuEfRY8OKxgv2AdinZdLYNTbJLkhVK6V/w4kERr/4hb/4bs+0etPWFwM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpyctk2SaYtlohEdyNqUHCFLcutwG6qkcpIYCea3OX1zhNqwxP1aKcphpKOFI85oza3+ikvDypVv+YvRNYhKAKhZqDyld/mLBMorJMUGN6gZ/acEa15UzgvNzPDKaUTegIew4VlWjC2WLXObl0zpDEiXZPWbJwf0/MqDRmKiPXKakdm9Vabv5X62U2vg1nXKWZRcWH8WZIDYh+eFkyDUyK6YOKNPc7UrYmGrKrIsnDyFYPXkd2vVa4PjhutqoF3GU4Bwu4AoCuIEG3EMTWsBgDM/wCm+e9F68d+9j2brhFTNn8Efe5w+ANI3R</latexit><latexit sha1_base64="AaOvRKa3hf+wAefGSgag7W3DaDE=">AB63icbZBNSwMxEIZn/az1q+rRS7AInspuEfRY8OKxgv2AdinZdLYNTbJLkhVK6V/w4kERr/4hb/4bs+0etPWFwM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpyctk2SaYtlohEdyNqUHCFLcutwG6qkcpIYCea3OX1zhNqwxP1aKcphpKOFI85oza3+ikvDypVv+YvRNYhKAKhZqDyld/mLBMorJMUGN6gZ/acEa15UzgvNzPDKaUTegIew4VlWjC2WLXObl0zpDEiXZPWbJwf0/MqDRmKiPXKakdm9Vabv5X62U2vg1nXKWZRcWH8WZIDYh+eFkyDUyK6YOKNPc7UrYmGrKrIsnDyFYPXkd2vVa4PjhutqoF3GU4Bwu4AoCuIEG3EMTWsBgDM/wCm+e9F68d+9j2brhFTNn8Efe5w+ANI3R</latexit><latexit sha1_base64="AaOvRKa3hf+wAefGSgag7W3DaDE=">AB63icbZBNSwMxEIZn/az1q+rRS7AInspuEfRY8OKxgv2AdinZdLYNTbJLkhVK6V/w4kERr/4hb/4bs+0etPWFwM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpyctk2SaYtlohEdyNqUHCFLcutwG6qkcpIYCea3OX1zhNqwxP1aKcphpKOFI85oza3+ikvDypVv+YvRNYhKAKhZqDyld/mLBMorJMUGN6gZ/acEa15UzgvNzPDKaUTegIew4VlWjC2WLXObl0zpDEiXZPWbJwf0/MqDRmKiPXKakdm9Vabv5X62U2vg1nXKWZRcWH8WZIDYh+eFkyDUyK6YOKNPc7UrYmGrKrIsnDyFYPXkd2vVa4PjhutqoF3GU4Bwu4AoCuIEG3EMTWsBgDM/wCm+e9F68d+9j2brhFTNn8Efe5w+ANI3R</latexit>

∆ ✓Z

γ

ω ◆ ≡ X

i

Z

γ

ωi ⊗ Z

γi

ω ,

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slide-5
SLIDE 5

Properties of the coaction

∆ ✓Z

γ

ω ◆ ≡ X

i

Z

γ

ωi ⊗ Z

γi

ω

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ωi

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γi

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  • Discontinuities act on the first entry



 — this is useful for taking discontinuities.

  • Differentiation acts on the second entry



 
 
 
 — useful for deriving differential equations.

  • Given a basis of paired integrand and contour the coaction on

integrals is defined by 
 
 
 
 reproducing the properties of the coaction on MPLs. 


∆ ✓ ∂ Z

γ

ω ◆ = X

i

Z

γ

ωi ⊗ ∂ Z

γi

ω

  • ,
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∆ ✓ Disc Z

γ

ω ◆ = X

i

Disc Z

γ

ωi

Z

γi

ω ,

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slide-6
SLIDE 6

Diagrammatic Coaction at one loop

  • To translate the coaction 



 to Feynman diagrams we need to fix a basis.

  • At one loop, a convenient basis of uniform weight Feynman integrals is:

e Jn = eE✏ Z dDk i⇡D/2

n−1

Y

j=0

1 (k − qj)2 − m2

j

, D = 2 ln 2 m − 2✏

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  • Next we need to identify the corresponding contours encircling poles, 


defining the second entries. These are cuts!
 First guess: cut all propagators that aren’t contracted in the first entry.

  • This fixes the basis for the first entries in the coaction: for a given integral 


these are the diagram itself and all its possible contractions.

∆ ✓Z

γ

ω ◆ ≡ X

i

Z

γ

ωi ⊗ Z

γi

ω

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e Jn

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slide-7
SLIDE 7

Diagrammatic Coaction at one loop — example: the three-mass triangle

  • We see how the 3 channel discontinuities are captured by the 3 bubbles.
  • Can be verified, order-by-order in to reproduce the coaction on MPLs

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z ¯ z = p2

2/p2 1 and (1 − z) (1 − ¯

z) = p2

3/p2 1

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  • Defining 

  • The diagrammatic coaction for a three-mass triangle with massless

propagators (to all orders in dimensional regularisation):

At leading order in ✏ : T (z, ¯ z) = −2Li2(z) + 2Li2(¯ z) − ln(z¯ z) ln ✓1 − z 1 − ¯ z ◆

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∆ [T (z, ¯ z)] = T (z, ¯ z) ⊗ 1 + 1 ⊗ T (z, ¯ z) + ln

  • −p2

2

  • ⊗ ln 1 − ¯

z 1 − z + ln

  • −p2

3

  • ⊗ ln z

¯ z + ln(−p2

1) ⊗ ln ¯

z(1 − z) z(1 − ¯ z)

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slide-8
SLIDE 8

Diagrammatic Coaction at one loop — example: the three-mass triangle

  • The diagrammatic coaction for a three-mass triangle:
  • The trivial term with the full triangle on the second entry is recovered (to

all orders) though the “pole-cancellation identity”:
 
 
 
 
 
 which in general takes the form:

∆ [T (z, ¯ z)] = T (z, ¯ z) ⊗ 1 + 1 ⊗ T (z, ¯ z) + ln

  • −p2

2

  • ⊗ ln 1 − ¯

z 1 − z + ln

  • −p2

3

  • ⊗ ln z

¯ z + ln(−p2

1) ⊗ ln ¯

z(1 − z) z(1 − ¯ z)

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X

i∈[n]

Ci e Jn + X

i<j∈[n]

Ci,j e Jn = −✏ e Jn mod i⇡

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slide-9
SLIDE 9

The Diagrammatic Coaction at one loop: the general case

∆(JG) = X

;6=C✓EG

Z

γ

ωC ⊗ Z

γC

ωEG

∆ ✓Z

γ

ω ◆ ≡ X

i

Z

γ

ωi ⊗ Z

γi

ω

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  • Translating the coaction on integrals 



 
 
 to Feynman diagrams

JGC

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JGC

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  • The first entry in the diagrammatic coaction are : graphs whose edges

that are not in the subset C are contracted.

?

  • What is the corresponding contour defining the second entry?

γC

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γC = ⇢ ΓC∞ for |C| odd ΓC for |C| even

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  • The general solution:



 where encircles the poles in C while encircles in addition the branch point at infinite momentum (it’s a second type Landau singularity).

ΓC

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ΓC∞

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Pss ✓Z

γi

ωj ◆ = δij ,

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slide-10
SLIDE 10

Coaction for one loop Feynman diagrams

  • It turns out contours that encircle the branch point at infinite momentum at
  • ne loop can be recast in terms of ordinary cuts!

CC∞ e Jn = X

i∈[n]\C

CCi e Jn + X

i<j∈[n]\C

CCij e Jn mod iπ

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CC∞ e Jn = −2CC e Jn − X

i∈[n]\C

CCi e Jn mod iπ

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  • This is owing to the (homology) relations:

For even |C|: For odd |C|:

  • D. Fotiadi and F

. Pham (1966); Abreu et al. 1702.03163

∆(JG) = X

;6=C✓EG

JGC ⊗ @CCJG + aC X

e2EG\C

CCeJG 1 A

∆(JG) = X

;6=C✓EG

Z

γ

ωC ⊗ Z

γC

ωEG

  • The diagrammatic coaction at one loop:

aC =   

1 2

if |C| is odd if |C| is even

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with

slide-11
SLIDE 11

Coaction for one loop Feynman diagrams

∆(JG) = X

;6=C✓EG

JGC ⊗ @CCJG + aC X

e2EG\C

CCeJG 1 A

∆(JG) = X

;6=C✓EG

Z

γ

ωC ⊗ Z

γC

ωEG

  • The diagrammatic coaction at one loop, expressed in terms of cuts:

aC =   

1 2

if |C| is odd if |C| is even

<latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit>

with

∆ ✓Z

Γ∅

ω12 ◆ = Z

Γ∅

ω12 ⊗ Z

Γ12

ω12 + Z

Γ∅

ω1 ⊗ ✓Z

Γ1

+1 2 Z

Γ12

◆ ω12 + Z

Γ∅

ω2 ⊗ ✓Z

Γ2

+1 2 Z

Γ12

◆ ω12

<latexit sha1_base64="7D3ROsKtXVzX6OjZrd8jXNPEisg=">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</latexit><latexit sha1_base64="7D3ROsKtXVzX6OjZrd8jXNPEisg=">ADS3iclVJNaxRBEO2ZGI3rRzbm6GVwUSKBZWYR9CIEFeIxgpsEtpehprdmt0n39NBdIyzD/D8vXrz5J7x4SAg52PuBuFlRU9DwePWqXnVRWamkozj+FoQbtzZv39m627p3/8HD7fbOo2NnKiuwL4wy9jQDh0oW2CdJCk9Li6AzhSfZ2dtZ/uQTWidN8ZGmJQ41jAuZSwHkqXQnAP4OFQFXmNMelwWlNT8ErSHlqEuaOqSG41jSOuk13ArxN6/vrZf0gNSY1uRTjVzS8tf/PVr86rc+YNPs8tyC8qPbd1q0W497QsfcXx96NHdN2J+7G84jWQbIEHbaMo7T9lY+MqDQWJBQ4N0jikoY1WJCYdPilcMSxBmMceBhAX7SYT2/hSZ6plRlBvrX0HRnP29ogbt3FRnXqmBJu56bkb+KTeoKH81rGVRVoSFWBjlYrIRLPDikbSoiA19QCElX7WSEzAb4r8+bX8EpLrX14Hx71uEneTDy86B2+W69hij9kTtscS9pIdsPfsiPWZCD4H34Pz4CL8Ev4IL8OrhTQMljW7bCU2Nn8C4xsXUw=</latexit><latexit sha1_base64="7D3ROsKtXVzX6OjZrd8jXNPEisg=">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</latexit><latexit sha1_base64="7D3ROsKtXVzX6OjZrd8jXNPEisg=">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</latexit>

ωi γi ω12 Γ12 ω1 Γ1 + 1

2Γ12 = Γ∞1

ω2 Γ2 + 1

2Γ12 = Γ∞2

<latexit sha1_base64="v1+qz29P7QI1TudT3Y5DEy3b4Cc=">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</latexit><latexit sha1_base64="v1+qz29P7QI1TudT3Y5DEy3b4Cc=">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</latexit><latexit sha1_base64="v1+qz29P7QI1TudT3Y5DEy3b4Cc=">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</latexit><latexit sha1_base64="v1+qz29P7QI1TudT3Y5DEy3b4Cc=">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</latexit>

satisfy

Pss ✓Z

γi

ωj ◆ = δij

  • Example: bubble with two massive propagators
slide-12
SLIDE 12

Coaction for one loop Feynman diagrams: the two-mass hard box

∆(JG) = X

;6=C✓EG

JGC ⊗ @CCJG + aC X

e2EG\C

CCeJG 1 A

  • The diagrammatic coaction for the two-mass hard box:

aC =   

1 2

if |C| is odd if |C| is even

<latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">ACZXicdVFNaxsxENVuv9xt0zhp6aWHipqGnozWcerkUAj40mMKdRKwjNFqZ20RrXaRZkPNev9kbr3mkr9R+aOQlGZg4M2bmSfpKSm1csjY7yB8vTZ8xetl9Gr1ztvdt7+euqKyEkSx0YS8T4UArAyNUqOGytCDyRMNFcjVc9S+uwTpVmJ+4KGSi5lRmZICPTVtL8V0+I1ryJDXEU9gpkwtrBWLpta6iXhmhYx79IAj/MJaZbRZDpebgipHizRtOI8OfLJHh+AajJcCk26lI27VbI7dabvDuxr8OqQdHR/3BwIMBOzpn9C4y9bRIds4m7ZveFrIKgeDUgvnxjErceJVUkNXrdyUAp5JWYw9tCIHNykXrvU0M+eSWlWJ8G6Zq9v1GL3LlFnvjJXODc/dtbkf/rjSvMjie1MmWFYOTmoKzSFAu6spymyoJEvfBASKv8XamcC28s+o+JvAl/X0ofB+e9buzxj37ntLe1o0U+kE/kC4nJgJyS7+SMjIgkt0Er2Av2g7twJ3wXvt+MhsF25y15EOHP/32uDs=</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit><latexit sha1_base64="rNuqXouTbSaCquEchpgHuxfYQyw=">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</latexit>

with

∆ 

e4 e2 e3 e1

1 2

  • =

1 1

e2 e1

e4 e2 e3 e1

1 3 4 2

+

2 2

e3 e2

e4 e2 e3 e1

1 3 4 2

+

s s e3 e1

e4 e2 e3 e1

1 2

+

t t e4 e2

e4 e2 e3 e1

1 2

+

s

2 1

e3 e1 e2

e4 e2 e3 e1

1 2

+ (

e4 e2 e3 e1

1 2

+ 1 2

s

2 1

e3 e1 e2

+

e3 e1 e4 s

+

t

1

e4 e2 e1

+

t

2

e4 e2 e3

!) ⊗

e4 e2 e3 e1

1 2

slide-13
SLIDE 13

Differential equations from the coaction

  • Given that differentiation only acts on the second entries:



 
 
 
 
 we can derive differential equations from the diagrammatic coaction by focusing on the weight 1 component on the second entry.

  • For example, for a pentagon we get:

∆ ✓ ∂ Z

γ

ω ◆ = X

i

Z

γ

ωi ⊗ ∂ Z

γi

ω

  • ,
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slide-14
SLIDE 14

Coaction on hypergeometric functions

  • We have seen that the diagrammatic coaction holds order-by-order in ,

recovering the coaction on MPLs.

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  • Can it be established directly for the Hypergeometric functions? Yes!

ω1 = xa✏(1 − x)−1+b✏(1 − zx)c✏dx

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ω2 = xa✏(1 − x)b✏(1 − zx)−1+c✏dx

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γ1 = [0, 1]

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γ2 = [0, 1/z]

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Pss ✓Z

γi

ωj ◆ = δij

∆ Z

γ1

ω = Z

γ1

ω1 ⊗ Z

γ1

ω + Z

γ1

ω2 ⊗ Z

γ2

ω

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Example: Gauss Hypergeometric function 2F1

⇔ ⇔

The period matrix is readily diagonal, so suitable normalization yields: (Ruth Britto’s talk last week)

slide-15
SLIDE 15

Coaction on Appell functions

  • A similar construction can be applied to generalised Hypergeometrics. 


Let’s consider Appell F3, depending on two variables.

F3(α, α0, β, β0, γ; x, y) = Γ(γ) Γ(β)Γ(β0)Γ(γ − β − β0)× Z 1 Z 1v uβ1vβ01(1 − u − v)γββ01(1 − xu)α(1 − yv)α0 du dv

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ϕab = d log(u) ∧ d log(v) = du∧dv

uv

ϕbc = d log(1 − xu) ∧ d log(v) = − x du∧dv

(1−xu)v

ϕcd = d log(1 − xu) ∧ d log(1 − yv) =

xy du∧dv (1−xu)(1−yv)

ϕad = d log(u) ∧ d log(1 − yv) = − y du∧dv

u(1−yv)

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Φ =ua✏vb✏(1 − xu)c✏(1 − yv)d✏(1 − u − v)g✏

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Pss R

S ΦϕT

ab bc cd ad abg

1 ab✏2

bcg

1 bc✏2

cdg

1 cd✏2

adg

1 ad✏2

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a b d c g

1 x

1 Re(u)

1 y

1 Re(v)

Z

γ

ω with ω = Φϕ

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period matrix: basis of 2-forms:

∆ ✓Z

γ

ω ◆ ≡ X

i

Z

γ

ωi ⊗ Z

γi

ω

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  • The coaction can be obtained from
  • The result for the coaction can be fully expressed in terms of Appell F3.
slide-16
SLIDE 16

Coaction on hypergeometric functions — application to one loop diagrams

∆( 2F1(1, 1 + ✏, 2 − ✏, x)) = 2F1(1, ✏, 1 − ✏, x) ⊗ 2F1(1, 1 + ✏, 2 − ✏, x) + 2F1(1, 1 + ✏, 2 − ✏, x) ⊗ 2F1 ✓ 1, ✏, 1 − ✏, 1 x ◆

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q

a sum of two 2F1s

slide-17
SLIDE 17

Diagrammatic Coaction at two loops

  • The double triangle can be written in terms of two 3F2 and a 2F1
  • This system has 6 master integrands and contours, out of which only
  • ne is of the top topology.
  • This coaction was verified directly in terms of hypergeometric functions!
  • The correspondence between 1st and 2nd entries is just as at one loop. 


All 1st entries are two loop diagrams — in 2nd entries cuts affect both loops.

slide-18
SLIDE 18

Diagrammatic Coaction at two loops

  • In contrast to one loop, where the maximal cut is unique, many 


two-loop diagrams have multiple maximal cuts (multiple master integrals at the top topology).

  • We verified in several non-trivial examples that the second entries in

the coaction can all be expressed in terms of cuts of the original graph.

2F1

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F4

<latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit>

F1

<latexit sha1_base64="B6NDnNknojkqaJKSpdexYxC3ymQ=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSItRjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3A29QrhVdyGyDl4OFcjVHJS/+sOYpRFKwTVue5ifEzqgxnAmelfqoxoWxCR9izKGmE2s8Wq87IhXWGJIyVfdKQhft7IqOR1tMosJ0RNWO9Wpub/9V6qQmv/YzLJDUo2fKjMBXExGR+NxlyhcyIqQXKFLe7EjamijJj0ynZELzVk9ehXat6lu+vKo1aHkcRzuAcLsGDOjTgDprQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzmfP7qnjV4=</latexit><latexit sha1_base64="B6NDnNknojkqaJKSpdexYxC3ymQ=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSItRjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3A29QrhVdyGyDl4OFcjVHJS/+sOYpRFKwTVue5ifEzqgxnAmelfqoxoWxCR9izKGmE2s8Wq87IhXWGJIyVfdKQhft7IqOR1tMosJ0RNWO9Wpub/9V6qQmv/YzLJDUo2fKjMBXExGR+NxlyhcyIqQXKFLe7EjamijJj0ynZELzVk9ehXat6lu+vKo1aHkcRzuAcLsGDOjTgDprQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzmfP7qnjV4=</latexit><latexit sha1_base64="B6NDnNknojkqaJKSpdexYxC3ymQ=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSItRjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3A29QrhVdyGyDl4OFcjVHJS/+sOYpRFKwTVue5ifEzqgxnAmelfqoxoWxCR9izKGmE2s8Wq87IhXWGJIyVfdKQhft7IqOR1tMosJ0RNWO9Wpub/9V6qQmv/YzLJDUo2fKjMBXExGR+NxlyhcyIqQXKFLe7EjamijJj0ynZELzVk9ehXat6lu+vKo1aHkcRzuAcLsGDOjTgDprQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzmfP7qnjV4=</latexit><latexit sha1_base64="B6NDnNknojkqaJKSpdexYxC3ymQ=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSItRjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3A29QrhVdyGyDl4OFcjVHJS/+sOYpRFKwTVue5ifEzqgxnAmelfqoxoWxCR9izKGmE2s8Wq87IhXWGJIyVfdKQhft7IqOR1tMosJ0RNWO9Wpub/9V6qQmv/YzLJDUo2fKjMBXExGR+NxlyhcyIqQXKFLe7EjamijJj0ynZELzVk9ehXat6lu+vKo1aHkcRzuAcLsGDOjTgDprQAgYjeIZXeHOE8+K8Ox/L1oKTz5zCHzmfP7qnjV4=</latexit>

F4

<latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit><latexit sha1_base64="axu7X/QrTYDyq4NcZUd2WPaLsHo=">AB6nicbZBNS8NAEIYn9avWr6pHL4tF8FSUtBjQRCPFe0HtKFstpN26WYTdjdCf0JXjwo4tVf5M1/47bNQVtfWHh4Z4adeYNEcG1c9spbGxube8Ud0t7+weHR+Xjk7aOU8WwxWIRq25ANQousW4EdhNFNIoENgJjfzeucJleaxfDTBP2IjiQPOaPGWg+3g/qgXHGr7kJkHbwcKpCrOSh/9YcxSyOUhgmqdc9zE+NnVBnOBM5K/VRjQtmEjrBnUdItZ8tVp2RC+sMSRgr+6QhC/f3REYjradRYDsjasZ6tTY3/6v1UhNe+xmXSWpQsuVHYSqIicn8bjLkCpkRUwuUKW53JWxMFWXGplOyIXirJ69Du1b1LN/XK41aHkcRzuAcLsGDK2jAHTShBQxG8Ayv8OYI58V5dz6WrQUnzmFP3I+fwC/M41h</latexit>

2F1

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2F1

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(2 masters) (4 masters) (4 masters) (2 masters) (3 masters) (2 masters) Uncut integral Maximal cut

slide-19
SLIDE 19

Conclusions

  • We construct a coaction on integrals based on pairing between master integrands and master
  • contours. This reproduces the coaction on MPLs upon expansion.
  • It also extends naturally to hypergeometric functions including all Appell functions. In each

case, left entries span the space of functions related by integer shifts, while right entries span the space of solutions to the differential equation.

  • It also translates into a coaction of Feynman integrals for any one loop diagram, with any mass
  • configuration. Relations between cuts (homology) are essential to establish it.
  • The diagrammatic coaction leads to straightforward derivation of discontinuities and differential

equations.

  • Recently we used the coaction on hypergeometric functions to find the coaction of several

non-trivial two-loop diagrams. 
 The general diagrammatic coaction at two loops is not yet known.

  • New features at two loops include multiple Master Integrals at the top topology (several

maximal cuts) and elliptic polylogs. 
 Both these features are present in hypergeometric functions.