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When it’s been Integrated Obviously, “loop integrands should be integrated ” but what this really means depends on who’s talking (& why) This is so even when the integral is “just” a number ! 15
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When it’s been Integrated Obviously, “loop integrands should be integrated ” but what this really means depends on who’s talking (& why) This is so even when the integral is “just” a number Are these “ numbers” MZVs? YES! [ O. Schnetz ( private corr .) ] [ JB , Heslop, Tran (2015) ] implications for BES … ! 15
When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria X A L =2 , MHV = n a<b<c<d<a ( ` 1 ,N 1 )( ` 2 ,N 2 ) ≡ ≡ ( ` 1 ,a )( ` 1 ,a + 1)( ` 1 ,b )( ` 1 ,b + 1)( ` 1 , ` 2 )( ` 2 ,c )( ` 2 ,c + 1)( ` 2 ,d )( ` 2 ,d + 1) ! 16
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<latexit 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<latexit 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<latexit 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<latexit 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When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria X A L =2 , MHV = n a<b<c<d<a Z d 4 ` 1 d 4 ` 2 ( ` 1 ,N 1 )( ` 2 ,N 2 ) ≡ ≡ ( ` 1 ,a )( ` 1 ,a + 1)( ` 1 ,b )( ` 1 ,b + 1)( ` 1 , ` 2 )( ` 2 ,c )( ` 2 ,c + 1)( ` 2 ,d )( ` 2 ,d + 1) ! 16
<latexit sha1_base64="itNnwRvq7L8eLiqHixXiXHgxgTw=">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</latexit> When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria [ Symanzik (1972) ] [ Hodges (1977) ] ! 16
<latexit sha1_base64="itNnwRvq7L8eLiqHixXiXHgxgTw=">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</latexit> When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria [ Symanzik (1972) ] [ Hodges (1977) ] ✦ built of functions known to ‣ undergrads (Euler/Abel/…) ‣ Goncharov/Brown/Bloch… ‣ Mathematica/GiNaC… ! 16
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<latexit 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<latexit 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<latexit 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When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria [ JB , McLeod, Spradlin, von Hippel, Wilhelm (2017) ] Z ( a 0 , b 0 )( a 1 , b 1 )( a 2 , b 2 ) d 8 ~ ` ⇒ ( ` 1 , a 0 )( ` 1 , a 1 )( ` 1 , b 1 )( ` 1 , ` 2 )( ` 2 , a 2 )( ` 2 , b 2 )( ` 2 , b 0 ) i ∞ ∞ Z U Z h ih i - u z 1 1 · · · u z 7 d 6 ~ d 7 ~ Γ ( − z 1 ) 2 · · · = = ↵ z 7 F 3 0 − i ∞ ∞ 1 Z Z ds d 4 ~ = ↵ = H 3 ( s ) p f 1 f 2 g 2 4 s 3 − g 2 s − g 3 0 ✦ Certifiability ‣ against some reference (symbology, fibration bases, …) ‣ by checking physical limits/branch cuts/… ! 16
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When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] Z ( a 0 , b 0 )( a 1 , b 1 )( a 2 , b 2 )( a 3 , b 3 ) d 12 ~ ` ⇒ ( ` 1 , a 0 )( ` 1 , a 1 )( ` 1 , b 1 )( ` 1 , ` 2 ) · · · ( ` 3 , b 0 ) i ∞ ∞ ↵ U 2 Z Z h ih i - u z 1 1 · · · u z 16 d 8 ~ d 16 ~ Γ ( − z 1 ) 2 · · · = F 4 = z 16 0 − i ∞ ∞ 1 Z Z ds dz d 6 ~ = = H 4 ( s, z ) ↵ p f 1 f 2 f 3 g 3 4 s 3 − g 2 ( z ) s − g 3 ( z ) 0 ✦ Certifiability ‣ against some reference (symbology, fibration bases, …) ‣ by checking physical limits/branch cuts/… ! 16
<latexit sha1_base64="+Aa05goT4oRLk7W3qJj3LQVN3M=">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</latexit> When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] ✦ Certifiability ‣ against some reference (symbology, fibration bases, …) ‣ by checking physical limits/branch cuts/… ! 16
When’s it been Integrated? When the result is a function , this is more subtle— depending on various (often valid) criteria ✦ Certifiability ‣ against some reference (symbology, fibration bases, …) ‣ by checking physical limits/branch cuts/… ! 16
Rationalizing Loop Integration A surprisingly large class of planar UV finite multi- loop integrals can be directly integrated provided the right kind of naïveté (and mild cleverness): ✦ Feynman parameterize in 4d , one loop at a time ✦ Maintain manifest dual conformal invariance: ‣ regulate IR divergences with `DCI masses’ [ JB , Caron-Huot, Trnka (2013) ] ‣ rescale Feynman parameters to trivialize DCI [ JB , Dixon, Dulat, Panzer ( to appear ) ] ✦ Parameterize kinematic variables using: momentum twistors [ JB , McLeod, von Hippel, Wilhelm (2018) ] chosen non-redundantly ✦ Partial fraction to death (e.g. use HyperInt ) [ Panzer (2014) ] ! 17
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Planarity & Dual-Conformality ✦ We may parameterize momenta of planar loop (Feynman) integrals by their dual-graphs = Z d 4 ` 1 d 4 ` 2 ( a, c )( b, e )( d, f ) p a ≡ ( x a +1 − x a ) ≡ ( ` 1 , a )( ` 1 , b )( ` 1 , c )( ` 1 , ` 2 )( ` 2 , d )( ` 2 , e )( ` 2 , f ) ( a, b )=( b, a ) ≡ ( x b − x a ) 2 =( p a + . . . + p b − 1 ) 2 ≡ s a ··· b − 1 ( ` , a ) ≡ ( x ` − x a ) 2 and ✦ Dual-Conformal Invariance: conformality in x ’s ⌘ 2 ⌘ ( ab ; cd ) ⌘ ( a, b )( c, d ) [ Drummond, Henn, Smirnov, Sokatchev; ( a, c )( b, d ) Drummond, Korchemsky, Henn; … ] ! 18
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The Dual-Conformal Regulator ✦ The basic idea of the dual-conformal regulator is to give legs masses, but controlled by a parameter ‘ δ ’ that is dimensionless & has no conformal weight [ JB , Caron-Huot, Trnka (2013) ] a ⌘ x a + δ ( x a +1 � x a ) ( a � 2 , a ) a + δ ( p a − 1 + p a ) 2 ( p a + p a +1 ) 2 p 2 a 7! p 2 x a 7! x b ( a � 2 , a + 1) ( p a − 1 + p a + p a +1 ) 2 a + 1) = ( a, a + 1) + δ ( a � 1 , a + 1)( a, a + 2) a, d ( a, a + 1) 7! ( b ( a � 1 , a + 2) " Y !# Z Z Y L Y L ( ` i , a ) d 4 ` i I 7! I δ ⌘ d 4 ` i I = I ( ` i , b a ) ! 19 i =1 i =1 a R 3 , 1 R 3 , 1
Persevering Dual-Conformality ✦ Using the dual-conformal regularization scheme, all(?) UV-finite planar loop integrals take the form: [ JB , Dixon, Dulat, Panzer ( to appear ) ] ✦ Coefficients of each divergence can be obtained as strictly finite (Feynman-) parametric integrals— which can always be rendered manifestly DCI ! 20
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<latexit 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<latexit 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Restoring Conformality ✦ Feynman parameterization is naïvely at odds with maintaining (dual) conformal invariance ∞ ⇤ 1 1 Z Z d 4 ` d 3 ~ ⇥ ↵ = ⇒ F 2 ( ` , 1)( ` , 2)( ` , 3)( ` , 4) 0 F ≡ α 1 α 2 (1 , 2) + α 2 α 3 (2 , 3) + α 1 α 3 (1 , 3) + α 1 α 4 (1 , 4) + α 2 α 4 (2 , 4) + α 3 α 4 (3 , 4) At least when integrating one loop (at a time), conformality is always( ? ) restorable by rescaling Feynman parameters: (1 , 2)(2 , 3) α 1 7! α 1 (2 , 3) α 2 7! α 2 (1 , 3) α 3 7! α 3 (1 , 2) α 4 7! α 4 (2 , 4) ⇣ ⌘ F 7! (1 , 2)(2 , 3)(1 , 3) α 1 α 2 + α 2 α 3 + α 1 α 3 + α 4 ( α 1 v + α 2 + α 3 u ) | {z } ! 21 ( f 1 + α 4 f 2 )
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Restoring Conformality ✦ Feynman parameterization is naïvely at odds with maintaining (dual) conformal invariance ∞ ⇤ 1 1 Z Z d 4 ` d 3 ~ ⇥ ↵ = ⇒ F 2 ( ` , 1)( ` , 2)( ` , 3)( ` , 4) 0 ∞ ∞ ∞ ∞ ⇤ 1 1 1 Z Z Z Z d 3 ~ d 2 ~ d 2 ~ ⇥ ⇥ ⇤ ⇥ ⇤ ( f 1 + ↵ 4 f 2 ) 2 = ↵ ↵ d ↵ 4 ↵ F 2 ∝ f 1 f 2 0 0 0 0 At least when integrating one loop (at a time), conformality is always( ? ) restorable by rescaling Feynman parameters: (1 , 2)(2 , 3) α 1 7! α 1 (2 , 3) α 2 7! α 2 (1 , 3) α 3 7! α 3 (1 , 2) α 4 7! α 4 (2 , 4) ⇣ ⌘ F 7! (1 , 2)(2 , 3)(1 , 3) α 1 α 2 + α 2 α 3 + α 1 α 3 + α 4 ( α 1 v + α 2 + α 3 u ) | {z } ! 21 ( f 1 + α 4 f 2 )
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Symanzik Polynomial Obstructions ✦ It is easy to see that the trick just used breaks down at higher loops if one uses the Symanzik (graph) polynomial formalism. For example, consider: ∞ ∞ ⇤ U ↵ ] U n − 2( L +1) Z Z I ell d 6 ~ ⇥ [ d ~ ↵ db ∝ F n − 2 L F 3 0 0 � � F = ( a, c ) α a α c β d + β e + β f + γ � � + ( d, f ) β d β f α a + α b + α c + γ + . . . ✦ Nevertheless, it appears that this obstruction is always avoidable simply by parameterizing one loop at a time (true through at least three loops) ! 22
<latexit 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Symanzik Polynomial Obstructions ✦ It is easy to see that the trick just used breaks down at higher loops if one uses the Symanzik (graph) polynomial formalism. For example, consider: ∞ Z 1 I ell d 4 ~ ⇥ ⇤ ↵ db ∝ f 1 f 2 f 3 0 f 1 ⌘ ↵ (1 + � 1 ) + � 1 , f 2 ⌘ 1 + u 1 ↵ + v 1 � 1 + u 2 � 2 + v 2 � 3 , f 3 ⌘ (1 + u 3 ↵ ) � 2 + (1 + u 4 � 1 ) � 3 + � 2 � 3 + u 3 u 4 u 5 f 1 , u 1 ⌘ ( ab ; ce ) , u 2 ⌘ ( bd ; ef ) , u 3 ⌘ ( ab ; cf ) , v 1 ⌘ ( ea ; bc ) , v 2 ⌘ ( fb ; de ) , u 4 ⌘ ( bc ; da ) , u 5 ⌘ ( ac ; d f ) . [ JB , McLeod, Spradlin, von Hippel, Wilhelm (2017) ] ✦ Nevertheless, it appears that this obstruction is always avoidable simply by parameterizing one loop at a time (true through at least three loops) ! 22
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Dual-Conformal Sufficiency ✦ We may now ( regulate &) represent all of the following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals [ JB , Dixon, Dulat, Panzer ( to appear ) ] ( ∞ ) N ( ~ ↵ ) Z I k ∈ span d ~ ↵ F ( ~ u ) ↵ , ~ 0 ⌘ 2 ⌘ ( ab ; cd ) ⌘ ( a, b )( c, d ) where u’s are parity-even cross-ratios: ( a, c )( b, d ) ! 23
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Dual-Conformal Sufficiency ✦ We may now ( regulate &) represent all of the following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals [ JB , McLeod, von Hippel, Wilhelm (2018) ] ( ∞ ) N ( ~ ↵ ) Z I k ∈ span d ~ ↵ F ( ~ u ) ↵ , ~ 0 ⌘ 2 ⌘ ( ab ; cd ) ⌘ ( a, b )( c, d ) where u’s are parity-even cross-ratios: ( a, c )( b, d ) ! 23
Dual-Conformal Sufficiency ✦ We may now ( regulate &) represent all of the following integrals in the space of finite, manifestly conformal (Feynman-)parametric integrals [ JB , McLeod, von Hippel, Wilhelm (2018) ] ∞ 2 ( � 1 n 1 1 + � 2 n 1 + � 1 n 2 1 + � 2 n 2 ✓ ↵ L u 1 2 ) ◆ Z I ( L ) 2 d 2 L ~ ↵ [ d � ] � 1 8 , B ⌘ ( f 1 · · · f L − 1 ) g 1 g 2 g 3 g 1 g 2 k 0 X 1 ↵ j ↵ 1 ↵ 1 � 1 + . . . + ↵ k � � 2 u 2 + � 2 + . . . + ↵ k � ↵ i � 1 u 3 + � 1 � 2 u 2 u 3 u 4 + f k ⌘ 2 ; 1 2 ↵ 1 2 + . . . + ↵ L − 1 ↵ L 1 + ↵ L + ↵ L 1 � 2 u 2 + ↵ L g 1 ⌘ f L − 1 + � �� � � � i,j =1 ; � 1 + � 2 2 2 2 ↵ 1 ↵ 1 1 + . . . + ↵ L + ↵ L 1 + . . . + ↵ L + ↵ L � � � � g 2 ⌘ 2 u 5 + � 1 + � 2 u 1 ; g 3 ⌘ 2 + � 1 u 3 , ⌘ 2 ⌘ 1 1 ( ab ; cd ) ⌘ ( a, b )( c, d ) where u’s are parity-even cross-ratios: ( a, c )( b, d ) ! 23
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Conformal Complications ✦ Although a good start, we haven’t yet eliminated all conformal redundancies—just the rescalings —w hich is to say that parity-even cross-ratios are: • too great in number • the “wrong” variables… # rescaling- independent cross ratios: n ( n − 5) / 2 # actually independent cross ratios: ! 24 3 n − 15
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Conformal Complications ✦ Although a good start, we haven’t yet eliminated all conformal redundancies—just the rescalings —w hich is to say that parity-even cross-ratios are: • too great in number • the “wrong” variables… # rescaling- independent cross ratios: n ( n − 5) / 2 # actually independent cross ratios: ! 24 3 n − 15
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Conformal Complications ✦ Although a good start, we haven’t yet eliminated all conformal redundancies—just the rescalings —w hich is to say that parity-even cross-ratios are: • too great in number • the “wrong” variables… # rescaling- independent cross ratios: n ( n − 5) / 2 # actually independent cross ratios: ! 24 3 n − 15
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Conformal Complications ✦ Although a good start, we haven’t yet eliminated all conformal redundancies—just the rescalings —w hich is to say that parity-even cross-ratios are: • too great in number • the “wrong” variables… # rescaling- independent cross ratios: n ( n − 5) / 2 # actually independent cross ratios: ! 24 3 n − 15
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Conformal Complications ✦ Although a good start, we haven’t yet eliminated all conformal redundancies—just the rescalings —w hich is to say that parity-even cross-ratios are: • too great in number • the “wrong” variables… ‣ over-count the degrees of freedom ‣ insensitive to the rank of the Gramian ‣ do not rationalize Gramian dets ‣ satisfy (complex) algebraic relations p (1 − u − v − w ) 2 − 4 uvw # rescaling- independent cross ratios: n ( n − 5) / 2 # actually independent cross ratios: ! 24 3 n − 15
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Momentum-Twistor Magic [ Hodges (2009) ] ✦ Unsurprisingly (to most of us), momentum twistors are (closer to) the right kind of conformal variables [ Golden, Paulos, Spradlin,Volovich; Harrington; McLeod, … ] ‣ manifest the rank of the Gramian ‣ no constrained extra degrees of freedom ‣ rationalize all 6x6 Gram determinants ‣ positive domain Euclidean domain ⊂ ‣ positive domain is a cluster variety ‣ cluster coordinates given by plabic graphs ! 25
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Momentum-Twistor Magic [ Hodges (2009) ] ✦ Unsurprisingly (to most of us), momentum twistors are (closer to) the right kind of conformal variables [ Golden, Paulos, Spradlin,Volovich; Harrington; McLeod, … ] ‣ manifest the rank of the Gramian ‣ no constrained extra degrees of freedom ‣ rationalize all 6x6 Gram determinants ‣ positive domain Euclidean domain ⊂ ‣ positive domain is a cluster variety ‣ cluster coordinates given by plabic graphs ‣ easy to expose/probe kinematic boundaries ‣ easy to eliminate redundant parameters <b<c ! 25 <d<a [ JB , McLeod, von Hippel, Wilhelm (2018) ]
Loop Integral Zoology general complexity beyond polylogs (& beyond elliptic polylogs)
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? X A L =2 = f L n L [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? X A L =2 = f L n L L 8 9 > > < = , , ∈ > > : ; 8 9 > > > > > > > > < = f L ∈ , , > > > > > > > > : ; [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? X A L =2 = f L n L [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? #d.o.f. # cross # Kinematic Square Roots # elliptic ratios 4x4(+6x6)+cuts/coeffs curves [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? #d.o.f. # cross # Kinematic Square Roots # elliptic ratios 4x4(+6x6)+cuts/coeffs curves 17 20 70(+56)+10 16 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? #d.o.f. # cross # Kinematic Square Roots # elliptic ratios 4x4(+6x6)+cuts/coeffs curves 17 20 70(+56)+10 16 13 14 35(+7)+1 4 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? #d.o.f. # cross # Kinematic Square Roots # elliptic ratios 4x4(+6x6)+cuts/coeffs curves 17 20 70(+56)+10 16 13 14 35(+7)+1 4 9 9 15(+1)+0 1 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 27
The Two-Loop ‘Master’ Integrals ✦ How hard are the general “masters” at two loops? #d.o.f. # cross # Kinematic Square Roots # elliptic ratios 4x4(+6x6)+cuts/coeffs+curves curves 17 20 70(+56)+10+16 16 13 14 35(+7)+1+4 4 9 9 15(+1)+0+1 1 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 28
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! 29 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ✦ How hard are the general “masters” at two loops? # elliptic curves 16 4 1 Elliptic Curve s at Two Loops 4x4(+6x6)+cuts/coeffs+curves # Kinematic Square Roots 70(+56)+10+16 35(+7)+1+4 15(+1)+0+1 ratios # cross 20 14 9 #d.o.f. 17 13 9 ⊃ ⊃
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! 30 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ✦ How do the parents see their elliptic duaghters? Elliptic Curve s at Two Loops ⊃ ⊃
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! 30 [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ✦ How do the parents see their elliptic duaghters? Elliptic Curve s at Two Loops ⊃ ⊃
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Elliptic Curve s at Two Loops ✦ How do the parents see their elliptic duaghters? ( ) ⊃ ⊃ ⊃ [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 30
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Elliptic Curve s at Two Loops ✦ How do the parents see their elliptic duaghters? ( ) ⊃ ⊃ ⊃ [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 30
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Elliptic Curve s at Two Loops ✦ How do the parents see their elliptic duaghters? ( ) ⊃ ⊃ ( ) ⊃ ⊃ [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 30
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Elliptic Curve s at Two Loops ✦ How do the parents see their elliptic duaghters? ( ) ⊃ ⊃ ( ) ⊃ ⊃ [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 30
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Elliptic Curve s at Two Loops ✦ How do the parents see their elliptic duaghters? ( ) ⊃ ⊃ ( ) ⊃ ⊃ ( ) ⊃ [ JB , Duhr, Dulat, McLeod, Penante, von Hippel, Wilhelm, ( in progress ) ] ! 30
Traintracks Past Polylogarithms ✦ Despite their ubiquity at low multiplicity and low loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] [ Bloch, Kerr, Vanhove; Broadhurst;… ] T ( L ) ≡ ! 31
Traintracks Past Polylogarithms ✦ Despite their ubiquity at low multiplicity and low loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] [ Bloch, Kerr, Vanhove; Broadhurst;… ] T ( L ) ≡ pression: 1 1 Z T ( L ) = d L ~ d L ~ ⇥ ⇤ ↵ � , � � f 1 · · · f L g L 0 f k ⌘ ( a 0 a k � 1 ; a k b k � 1 )( a k � 1 b k ; b k � 1 a 0 )( a k b k ; a k � 1 b k � 1 ) f k � 1 k � 1 h X + ↵ 0 ( ↵ k + � k ) + ↵ k � k + ↵ j ↵ k ( b j a 0 ; a j a k ) j =1 i + ↵ j � k ( b j a 0 ; a j b k ) + ↵ k � j ( a 0 a j ; a k b j ) + � j � k ( a 0 a j ; b k b j ) , L h i X ↵ j ( b j a 0 ; a j b 0 ) + � j ( a 0 a j ; b 0 b j ) g L ⌘ ↵ 0 + . ! 31 j =1
Traintracks Past Polylogarithms ✦ Despite their ubiquity at low multiplicity and low loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] [ Bloch, Kerr, Vanhove; Broadhurst;… ] T ( L ) ≡ pression: 1 dx d L � 2 ~ 1 Z Z z T ( L ) = ) = d L ~ d L ~ ⇥ ⇤ G 0 ( x, ~ ↵ � , z ) , � � p f 1 · · · f L 4 x 3 � g 2 ( ~ g L z ) x � g 3 ( ~ z ) 0 f k ⌘ ( a 0 a k � 1 ; a k b k � 1 )( a k � 1 b k ; b k � 1 a 0 )( a k b k ; a k � 1 b k � 1 ) f k � 1 k � 1 h X + ↵ 0 ( ↵ k + � k ) + ↵ k � k + ↵ j ↵ k ( b j a 0 ; a j a k ) j =1 i + ↵ j � k ( b j a 0 ; a j b k ) + ↵ k � j ( a 0 a j ; a k b j ) + � j � k ( a 0 a j ; b k b j ) , L h i X ↵ j ( b j a 0 ; a j b 0 ) + � j ( a 0 a j ; b 0 b j ) g L ⌘ ↵ 0 + . ! 31 j =1
Traintracks Past Polylogarithms ✦ Despite their ubiquity at low multiplicity and low loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT � [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] L +1 L +1 � [ Bloch, Kerr, Vanhove; Broadhurst;… ] z }| { z }| { A ' 12 , . . ., ' 12 , ' 13 , ' 13 , ' 34 , . . ., ' 34 , ' 24 , ' 24 T ( L ) ≡ pression: 1 dx d L � 2 ~ 1 Z Z z T ( L ) = ) = d L ~ d L ~ ⇥ ⇤ G 0 ( x, ~ ↵ � , z ) , � � p f 1 · · · f L 4 x 3 � g 2 ( ~ g L z ) x � g 3 ( ~ z ) 0 f k ⌘ ( a 0 a k � 1 ; a k b k � 1 )( a k � 1 b k ; b k � 1 a 0 )( a k b k ; a k � 1 b k � 1 ) f k � 1 k � 1 h X + ↵ 0 ( ↵ k + � k ) + ↵ k � k + ↵ j ↵ k ( b j a 0 ; a j a k ) j =1 i + ↵ j � k ( b j a 0 ; a j b k ) + ↵ k � j ( a 0 a j ; a k b j ) + � j � k ( a 0 a j ; b k b j ) , L h i X ↵ j ( b j a 0 ; a j b 0 ) + � j ( a 0 a j ; b 0 b j ) g L ⌘ ↵ 0 + . ! 31 j =1
Traintracks Past Polylogarithms ✦ Despite their ubiquity at low multiplicity and low loop orders, iterated polylogarithms are far from the only class of integrals that are needed in QFT � [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] L +1 L +1 � [ Bloch, Kerr, Vanhove; Broadhurst;… ] z }| { z }| { A ' 12 , . . ., ' 12 , ' 13 , ' 13 , ' 34 , . . ., ' 34 , ' 24 , ' 24 T ( L ) ≡ pression: 1 dx d L � 2 ~ 1 Z Z z T ( L ) = ) = d L ~ d L ~ ⇥ ⇤ G 0 ( x, ~ ↵ � , z ) , � � p f 1 · · · f L 4 x 3 � g 2 ( ~ g L z ) x � g 3 ( ~ z ) 0 f k ⌘ ( a 0 a k � 1 ; a k b k � 1 )( a k � 1 b k ; b k � 1 a 0 )( a k b k ; a k � 1 b k � 1 ) f k � 1 k � 1 h X + ↵ 0 ( ↵ k + � k ) + ↵ k � k + ↵ j ↵ k ( b j a 0 ; a j a k ) j =1 i + ↵ j � k ( b j a 0 ; a j b k ) + ↵ k � j ( a 0 a j ; a k b j ) + � j � k ( a 0 a j ; b k b j ) , L h i X ↵ j ( b j a 0 ; a j b 0 ) + � j ( a 0 a j ; b 0 b j ) g L ⌘ ↵ 0 + . ! 31 j =1
A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] ! 32
A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] ! 32
A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] ! 32
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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ! 32
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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 ! 32
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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ! 32
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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ∞ d 4 x C Z Z ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) h i h i d 2 ~ d 2 ~ ↵ � = ( Q A ,Q A )( Q B ,Q B )( Q B ,C )( C,c 1 )( C,c 2 ) 0 ! 32
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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ∞ d 4 x C Z Z ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) h i h i d 2 ~ d 2 ~ ↵ � = ( Q A ,Q A )( Q B ,Q B )( Q B ,C )( C,c 1 )( C,c 2 ) 0 α 1 7! α 1 ( C,a 2 ) α 2 7! α 2 ( C,a 1 ) α 3 7! ( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) β 3 7! 1 β 1 7! β 1 β 2 7! β 2 ( a 1 ,b 1 ) ( a 1 ,b 2 ) ! 32
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ∞ d 4 x C Z Z ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) h i h i d 2 ~ d 2 ~ ↵ � = ( Q A ,Q A )( Q B ,Q B )( Q B ,C )( C,c 1 )( C,c 2 ) 0 α 1 7! α 1 ( C,a 2 ) α 2 7! α 2 ( C,a 1 ) α 3 7! ( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) β 3 7! 1 β 1 7! β 1 β 2 7! β 2 ( a 1 ,b 1 ) ( a 1 ,b 2 ) Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ Z ↵ d 2 ~ d 2 ~ � = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) 0 ! 32
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ∞ d 4 x C Z Z ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) h i h i d 2 ~ d 2 ~ ↵ � = ( Q A ,Q A )( Q B ,Q B )( Q B ,C )( C,c 1 )( C,c 2 ) 0 α 1 7! α 1 ( C,a 2 ) α 2 7! α 2 ( C,a 1 ) α 3 7! ( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) β 3 7! 1 β 1 7! β 1 β 2 7! β 2 ( a 1 ,b 1 ) ( a 1 ,b 2 ) Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ Z ↵ d 2 ~ d 2 ~ � = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) 0 ! 32 | Y C ) := | c 1 ) γ 1 + | R ) γ 2 + | S ) γ 3 + | c 2 ) η 3 =: | Q ) + | c 2 ) η 3
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) | Y A ) := | a 1 ) α 1 + | a 2 ) α 2 + | C ) α 3 + | B ) η 1 =: | Q A ) + | B ) η 1 | Y B ) := | b 1 ) β 1 + | b 2 ) β 2 + | Q A ) β 3 + | C ) η 2 =: | Q B ) + | C ) η 2 ∞ d 4 x C Z Z ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) h i h i d 2 ~ d 2 ~ ↵ � = ( Q A ,Q A )( Q B ,Q B )( Q B ,C )( C,c 1 )( C,c 2 ) 0 α 1 7! α 1 ( C,a 2 ) α 2 7! α 2 ( C,a 1 ) α 3 7! ( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) ( C,a 1 )( a 1 ,a 2 ) β 3 7! 1 β 1 7! β 1 β 2 7! β 2 ( a 1 ,b 1 ) ( a 1 ,b 2 ) Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ ∞ � ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) Z Z ↵ d 2 ~ ↵ d 2 ~ d 2 ~ d 2 ~ � d 2 � ~ = = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( Q,c 1 )( Q,Q ) 0 0 ! 32 | Y C ) := | c 1 ) γ 1 + | R ) γ 2 + | S ) γ 3 + | c 2 ) η 3 =: | Q ) + | c 2 ) η 3
<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ ∞ � ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) Z Z ↵ d 2 ~ ↵ d 2 ~ d 2 ~ d 2 ~ � d 2 � ~ = = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( Q,c 1 )( Q,Q ) 0 0 ! 33 | Y C ) := | c 1 ) γ 1 + | R ) γ 2 + | S ) γ 3 + | c 2 ) η 3 =: | Q ) + | c 2 ) η 3
<latexit 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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 0 Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ ∞ � ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) Z Z ↵ d 2 ~ ↵ d 2 ~ d 2 ~ d 2 ~ � d 2 � ~ = = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( Q,c 1 )( Q,Q ) 0 0 ! 33 | Y C ) := | c 1 ) γ 1 + | R ) γ 2 + | S ) γ 3 + | c 2 ) η 3 =: | Q ) + | c 2 ) η 3
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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := ( b 1 c 1 ; c 2 a 2 ) 0 y 1 := ( a 1 a 2 ; b 1 c 2 ) y 2 := ( b 1 b 2 ; c 1 a 2 ) y 3 := ( c 1 c 2 ; a 1 b 2 ) z 1 := ( b 2 c 1 ; c 2 b 1 ) z 2 := ( c 2 a 1 ; a 2 c 1 ) z 3 := ( a 2 b 1 ; b 2 a 1 ) Z d 4 x C ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) ∞ ∞ � ( a 1 ,a 2 ) 2 ( b 1 ,b 2 )( c 1 ,c 2 ) / ( a 1 ,b 1 ) Z Z ↵ d 2 ~ ↵ d 2 ~ d 2 ~ d 2 ~ � d 2 � ~ = = ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( C,R )( C,S )( C,c 1 )( C,c 2 ) ( ↵ 1 + ↵ 2 + ↵ 1 ↵ 2 )( Q,c 1 )( Q,Q ) 0 0 ! 33 | Y C ) := | c 1 ) γ 1 + | R ) γ 2 + | S ) γ 3 + | c 2 ) η 3 =: | Q ) + | c 2 ) η 3
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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := ( b 1 c 1 ; c 2 a 2 ) 0 y 1 := ( a 1 a 2 ; b 1 c 2 ) y 2 := ( b 1 b 2 ; c 1 a 2 ) y 3 := ( c 1 c 2 ; a 1 b 2 ) n 0 := y 1 ( x 1 x 2 x 3 y 1 y 2 y 3 ) z 1 := ( b 2 c 1 ; c 2 b 1 ) z 2 := ( c 2 a 1 ; a 2 c 1 ) z 3 := ( a 2 b 1 ; b 2 a 1 ) f 1 := α 1 + α 2 + α 1 α 2 ; f 2 := α 1 (1 + α 2 + β 1 + β 2 + γ 2 ) + α 2 (1 + x 1 z 2 ( z 3 β 1 + β 2 ) + γ 2 ) + β 1 y 1 (1 + x 1 x 3 y 2 z 2 β 2 + γ 2 ) + x 2 y 1 ( x 1 y 3 γ 1 + β 2 (1 + γ 2 )) ; h f 3 := α 1 (1 + α 2 + β 1 + β 2 + γ 2 ) γ 1 + β 2 (1 + α 2 + x 3 y 1 y 2 β 1 + γ 2 ) i h + z 3 β 1 (1 + α 2 + γ 2 ) α 2 (1 + x 1 ( z 3 β 1 + β 2 ) + γ 2 ) + γ 1 i + x 3 y 1 ( x 2 z 1 β 2 (1 + γ 2 ) + β 1 (1 + x 1 y 2 β 2 + γ 2 ) + (1 + γ 2 )(1 + α 2 + β 1 + β 2 + γ 2 )( z 3 α 2 β 1 + ( α 2 + x 3 y 1 y 2 β 1 ) β 2 ) ! 33
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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := ( b 1 c 1 ; c 2 a 2 ) 0 d 3 ~ Z q y 1 := ( a 1 a 2 ; b 1 c 2 ) y 2 := ( b 1 b 2 ; c 1 a 2 ) y 3 := ( c 1 c 2 ; a 1 b 2 ) = H 3 ( ~ q ) z 1 := ( b 2 c 1 ; c 2 b 1 ) z 2 := ( c 2 a 1 ; a 2 c 1 ) z 3 := ( a 2 b 1 ; b 2 a 1 ) p Q ( ~ q ) ! 33
<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := ( b 1 c 1 ; c 2 a 2 ) 0 d 3 ~ Z q y 1 := ( a 1 a 2 ; b 1 c 2 ) y 2 := ( b 1 b 2 ; c 1 a 2 ) y 3 := ( c 1 c 2 ; a 1 b 2 ) = H 3 ( ~ q ) z 1 := 0 z 2 := 0 z 3 := 0 p Q ( ~ q ) ! 33
<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := 1 0 d 3 ~ Z q y 1 := 1 y 2 := 1 y 3 := ( c 1 c 2 ; a 1 b 2 ) = H 3 ( ~ q ) z 1 := 0 z 2 := 0 z 3 := 0 p Q ( ~ q ) ! 33
<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := ( a 1 b 1 ; b 2 c 2 ) x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := ( c 1 c 2 ; a 1 b 2 ) = q ) H 3 ( ~ q ) z 1 := 0 z 2 := 0 z 3 := 0 G ( ~ ! 33
<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := 1 x 2 := 1 x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := 1 = q ) H 3 ( ~ q ) G ( ~ z 1 := 0 z 2 := 0 z 3 := 0 ! 33
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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := 1 x 2 := 1 x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := 1 = q ) H 3 ( ~ q ) → 20 ζ 5 G ( ~ z 1 := 0 z 2 := 0 z 3 := 0 ! 33
<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := 1 x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := 1 = q ) H 3 ( ~ q ) G ( ~ z 1 := 0 z 2 := 0 z 3 := 0 ! 33
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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := 1 x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := 1 = q ) H 3 ( ~ q ) G ( ~ z 1 := 0 z 2 := 0 z 3 := 0 " x 1 G ( { 0 , 0 , 0 , 1 , 0 , 0 } , x 1 ) − G ( { 0 , 0 , 1 , 0 , 0 , 0 } , x 1 ) + G ( { 0 , 1 , 1 , 0 , 0 , 0 } , x 1 ) − G ( { 0 , 0 , 0 , 1 , 1 , 0 } , x 1 ) = 1 − x 1 + G ( { 0 , 0 , 1 , 0 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 0 , 1 , 0 , 0 } , x 1 ) + G ( { 0 , 1 , 0 , 1 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 1 , 0 , 1 , 0 } , x 1 ) h i + ζ 2 G ( { 0 , 0 , 0 , 1 } , x 1 ) − G ( { 0 , 0 , 1 , 0 } , x 1 ) + G ( { 0 , 1 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 0 , 1 } , x 1 ) h i + 2 ζ 3 G ( { 0 , 1 , 0 } , x 1 ) − G ( { 0 , 0 , 1 } , x 1 ) + G ( { 0 , 1 , 1 } , x 1 ) − G ( { 0 , 0 , 0 } , x 1 ) # h i − 2(5 ζ 5 + ζ 2 ζ 3 ) G ( { 0 } , x 1 ) + 4( ζ 4 2 − ζ 2 + 6 ζ 4 G ( { 0 , 0 } , x 1 ) − G ( { 0 , 1 } , x 1 ) 3 ) + 3 ζ 6 ! 33
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<latexit 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<latexit 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A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] Z d 4 x A d 4 x B d 4 x C ( a 1 ,a 2 )( b 1 ,b 2 )( c 1 ,c 2 ) ( A,a 1 )( A,a 2 )( A,B )( B,b 1 )( B,b 2 )( B,C )( C,c 1 )( C,c 2 )( C,A ) ∞ Z n 0 ↵ d 2 ~ d 2 ~ � d 2 ~ � = f 1 f 2 f 3 x 1 := ( c 1 a 1 ; a 2 b 2 ) x 2 := 1 x 3 := 1 Z d 3 ~ 0 q y 1 := 1 y 2 := 1 y 3 := 1 = q ) H 3 ( ~ q ) G ( ~ z 1 := 0 z 2 := 0 z 3 := 0 " x 1 G ( { 0 , 0 , 0 , 1 , 0 , 0 } , x 1 ) − G ( { 0 , 0 , 1 , 0 , 0 , 0 } , x 1 ) + G ( { 0 , 1 , 1 , 0 , 0 , 0 } , x 1 ) − G ( { 0 , 0 , 0 , 1 , 1 , 0 } , x 1 ) = 1 − x 1 + G ( { 0 , 0 , 1 , 0 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 0 , 1 , 0 , 0 } , x 1 ) + G ( { 0 , 1 , 0 , 1 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 1 , 0 , 1 , 0 } , x 1 ) h i + ζ 2 G ( { 0 , 0 , 0 , 1 } , x 1 ) − G ( { 0 , 0 , 1 , 0 } , x 1 ) + G ( { 0 , 1 , 1 , 0 } , x 1 ) − G ( { 0 , 1 , 0 , 1 } , x 1 ) h i + 2 ζ 3 G ( { 0 , 1 , 0 } , x 1 ) − G ( { 0 , 0 , 1 } , x 1 ) + G ( { 0 , 1 , 1 } , x 1 ) − G ( { 0 , 0 , 0 } , x 1 ) # h i − 2(5 ζ 5 + ζ 2 ζ 3 ) G ( { 0 } , x 1 ) + 4( ζ 4 2 − ζ 2 + 6 ζ 4 G ( { 0 , 0 } , x 1 ) − G ( { 0 , 1 } , x 1 ) 3 ) + 3 ζ 6 → 20 ζ 5 ! 33
<latexit 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<latexit 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Bestiary of Loop Integral Geometry ✦ The bad news is that even elliptic polylogarithms are far from sufficient for loop integrals in QFT [ JB , McLeod, Spradlin, von Hippel, Wilhelm (2018) ] [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] CY 1 CY 2 (?) CY 3 CY L − 1 (?) CY L − 1 [ Bloch, Kerr, Vanhove; Broadhurst;… ] ! 34
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Bestiary of Loop Integral Geometry ✦ The bad news is that even elliptic polylogarithms are far from sufficient for loop integrals in QFT [ JB , McLeod, Spradlin, von Hippel, Wilhelm (2018) ] [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] CY 1 CY 3 CY 2 (?) CY 3 CY L − 1 (?) CY L − 1 [ Bloch, Kerr, Vanhove; Broadhurst;… ] ! 34
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<latexit sha1_base64="njM3wD/m7oxSYBzLqYyrAr080k=">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</latexit> Bestiary of Loop Integral Geometry ✦ The bad news is that even elliptic polylogarithms are far from sufficient for loop integrals in QFT [ JB , McLeod, Spradlin, von Hippel, Wilhelm (2018) ] [ JB , He, McLeod, von Hippel, Wilhelm (2018) ] CY 1 CY 3 CY 2 (?) CY 4 (¿?) CY 3 CY L − 1 (?) CY L − 1 [ Bloch, Kerr, Vanhove; Broadhurst;… ] ! 34
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