adventures in loop integration
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Adventures in Loop Integration Jacob Bourjaily Niels Bohr - PowerPoint PPT Presentation

Amplitudes in the LHC Era Adventures in Loop Integration Jacob Bourjaily Niels Bohr International Academy based on work in collaboration with Caron-Huot, Herrmann, Trnka; Dixon, Dulat, Panzer; He, McLeod , Spradlin, von Hippel , Wilhelm ;


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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

  3. 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

  5. <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

  6. <latexit sha1_base64="itNnwRvq7L8eLiqHixXiXHgxgTw=">Agt3ic5Vlbj+NIFc4ut6WBZRoecmQmVXCpKPYTm9nV2ptaBhkAIsA7M7EKetsl1xivZt7HL6YplnXuEv8Kv4LbxwTpUdX9OXmV4hLd1Su+rUOafO5Tunym4zdFnMp9N/v/f+t79ne9+74PvH/zghz/68MePDn/yZRwkUVfWYEbRK9NElOX+fQVZ9ylr8OIEs906Vfm+S9x/astjWIW+H/iVyFdecTx2ZpZhAPJOHz0n6cf6S+Zs+EkioIL/bHOfA5/H9tnqb6+MIPLVJmGPEtTXWyWbiJqZ7NFlmXAtaVWqlPXzfR1RGB4mRJjmqUm/BFjBcdKplt2wGNBWSAFpJEXJZGFSPZyqlSnZn0qnmqpUswXcs9M15dHWshXB08/Os1dcZnHeGxMz2C25le3uqZVXSNuCGZPs7d8wjfxFaUvsrO0sWRgl4UpOdIegakxr7FTmIj3WN9Ju3IgL8Yok26bZI7NrEIbFoqgj2ilDdXGmFqtH08kxMBw15ovG/Fljrox2nl2jU58zZ6n/mngeGaJl14Yygj1kYGFtpYdRYBspO1XAWLE/cC30FWyfmKws/TaYCLkx3tCnlYcvDHo6v6gQ8qHa0ORZvXxmLkQLpPS6BKNjvWxzZmBaAhHYT4xW8ins7iMw0Ndwx1mC+NYknQdoQse2GI/A3jcUHT9YOnB7dWx3HLmfn9agPHKo7VB6iKy2IklFamZnParBbhzSI9ngvk3aF2o5/3E5iDkDk7aiRv1o2buWT82ck3uiHRkSQ0FYFy0LYH6C85NMstwb6+0gzW5COgB9baj6GLDaBfEOSAPscywLCGsAYAHc2A7bxmqG+NWzHWcKw9dGfXviaszu+L1TO1gVaRyKLBip72gNhVPn4Q8KagB0mp0PdQEO6o8yaE14aGKNb2NOruZ8PSrBDBOB5xk25GuJ6Le2V73FXrQUgvb17TFU4NCUZEPpJzDp5zN1OxoWCyWPWmEvSKN3NLAD8B1JWxszjMZs3wGr3ulwHXcer8eVXOLVIuQBKF+wFCuLM9emaQI2gus12lbSClOhQbiBM0xG4rEdwVZitNMwqk63mWD4C+UEJO95mpsusc6Lhg3N6OaGuGt8tbZXa3q1lpc3tnftZtJIbW87m1bamdJoYWqthYnGB+0Be9nqmxuqjkrtDlW7aota7N+jav/AnxnLIp7fdkLNbjdq6ObUflNjdmdQSk79Q0Besf3z3HLruo1BH9bt5AqQS0JIrXRv4QC+NE3k+2+FCF1TOZ1f/NcWA8GkwnU/HTbw+UfDo5T9fGIcf/ku3AyvxqM8tl8TxUkQ9JRFnlkuzAz2JKeTlnDg05YHlBms+jgJOPOdMbeKF7HYsCDwI3HxIvjK8+Ek6kqGW+C3DQ4Uu5bofRB5xRV70iLrk8kC/YHZwEVKfuPzq9Hg61S03MYu5AlaDKT5+DvIdvjlVPE/n9JvKFbP6XxywvwDPab5OuR8zU26DiIaUys+ZyEFgAjWKwg8TmN0B2bhnyTwaXsQNoVuDbj1GPXVC+eEfXphRUAHw7LaiQvG0OV1RblavYgCSIQSY+flQXQhJm6YdCyhwhAvZzibqQ4Iiwqlejup27egVy2a5ZaXMg9gmTXvaMAyAZHGT3BgAYmIyay0IEyNjN9k5A93AVhx9IhHifm/u12hJKp2BYv6eGq2qHgBU1Y4DKlsqazSv+k6AJ75kAie6ua6v/xdwOmn/Qyz0B/gHQaIqwz4sroW+kYo2GVUfvXSTeowvwzPQGkyUBAuo9fUGd9FaT6PQ5dx3AFVysl9NhOtOluqsB1WK17WzimQlgNltbRW6ZNK1xd4sxlsQq5ifuXSgdpYfZJ16l/s28CFDQbqHqmX+6SifVLMyZYaiESExciditQxHxqVTZ2IhBtmxcvYIi49HYDugdbS4bGsIj+dnIiTUxcyBSWrREU0l/ERHFL1D59t60L3bTQ/26O5XipwkIKqYEsj/CjfWt4Q3An6N8VRc1WcU/m6HNcNJ6aEo0t8x6VQDuAFjhpsEkXDgTIewB20EVUaObTq/pEyabgPBPQ+4MyjcTt6OCZh/jIReR2RBOGxbjIH2PQxKmho4FK+3BC3e5wr1h8/afH7DYHpZH6LCOzu0EhGq3RVnagtX0WqB0rbz5dC+iXsAsqWEOw+DlbA9SsKx39Efwv+/z7E0yKIfgH8NC6DgpNWeqU9S/7A/wuX9/YlhiSVzdV2LtzQZt8Ij3I753TduqWmIZV+c8E2XlE7R5hLuysBPJUBqRZH1Hw1xodM2XCDO8OayaYErCnJhJ2gvWVJDUQmSq8lG/Vjcmne9SxzDAngNatY8M0LZk+oWCo05Td1Z3ThvFkyKVv7bUqQNlgxsVcN24dAtxXOvKiRiSpumBbdxRe0DpcQfbepy0ljBW+oMnpVLdAFcY1uOG8OREtE8ZYFT/ifpKE+wmtWCKVC3R4vNtlKu1sM+/YhsjQoe0cOfZbBH7UTdqnJk/znXTIrNWLsnSBN/b6tQtUiEqVdoF0rSGCVcgf4XyNeIWiQ3xqwskanh7qiE1ibF3idciIxVvWwIPUNReXy2Ok8u4uQiHex1fuaFSUSfW4DP51nUuKLu9cf4W0jcPMuBT0j/XOr97AtHhA+3yvyuimy+Q3e3SBJLwQ6z9KBKiy06RrqXvY6b7P93E3g3IkcE27FE20MKRyLc7bOuH4JTbHgms2BZarO1Rk8lTk8G9xWwtUdt2AYl8/ZvM2t3F3xqwYPIbfyXHLWtHC42wjCrRM7XJaLcZhYnHzUB/O+w78UN+8JrdJ8iW0PvlQnCvj9h9ngs0/zF9wPej/r/bw37Cm9k95nvRe9L3qvetahefj3w38c/rP/Sd/or/sbyfr+e7nMT3u1n/6b/wKW61XE</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

  7. <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 ‣ numerically fast (and reliable) ‣ manifest “(transcendental) weight” ‣ undergrads (Euler/Abel/…) ‣ Goncharov/Brown/Bloch… ‣ minimal cancellation among terms ‣ Mathematica/GiNaC… ‣ manifest physical symmetries 
 ! 16 (non-redundantly)

  8. <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 [ 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

  10. <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

  11. 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

  12. 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

  13. <latexit 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<latexit 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<latexit 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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

  14. <latexit 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<latexit 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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

  15. 
 
 
 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

  16. <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 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 )

  17. <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 )

  18. <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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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

  19. <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: ∞ 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

  20. <latexit 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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

  22. 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

  23. <latexit 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<latexit 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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

  24. <latexit 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<latexit 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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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<latexit 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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

  26. <latexit 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<latexit 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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

  27. <latexit 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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) ]

  30. Loop Integral Zoology general complexity beyond polylogs 
 (& beyond elliptic polylogs)

  31. 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

  32. 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

  33. 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

  34. 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

  35. 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

  36. 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

  37. 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

  38. 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

  39. 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

  40. <latexit 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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

  48. 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

  49. 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

  50. 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

  51. 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

  52. 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

  53. A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] ! 32

  54. A Three-Loop Calabi-Yau 3-Fold ✦ Consider the simplest finite 3-loop wheel integral: [ JB , McLeod, von Hippel, Wilhelm ( in prep. ) ] ! 32

  55. 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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<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 ! 32

  58. <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 ! 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 ! 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

  61. <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 sha1_base64="MS2c+ocK/irW/W3bzoy9xBc5O/U=">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</latexit> 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

  62. <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 sha1_base64="NgdI/FBSMh4HcJCK2cvWsV4Xos=">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</latexit> 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

  63. <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 sha1_base64="NgdI/FBSMh4HcJCK2cvWsV4Xos=">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</latexit> 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

  64. <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 ) 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

  65. <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

  66. <latexit 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<latexit 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<latexit 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<latexit sha1_base64="y8fQf30hOkgELFhBSqpIVCo9o6E=">A5j3ic7Rtrc9vGUlfqfqy04/9QpW2SjaUQoCUJTujiN3nGRGTWKlzqOEhAHBI4kKBgApB7o9Q/1p/QX9N90dw+PwGgKEuZPsUxebe3r9vd2707wMO564Rt/uPt97+3vd/8MfvfPjzZ/89Gc/8WDh+9+GfqLwGavbd/1g6+HVshcx2OvIydy2dfzgFmzocu+Gp6/wPGvliwIHd/7Y3Q1Z6cza+I5Y8e2IgCZD3/zt8fbxokzmUZWEPgXxpbheBF8b43OYmN8MfQvY607j3gcGyQsngZsxPvHnHPAWjI7NpjrcmMcWNC8jC2zy+MhfFbw7bGDXvkRyFBjhEC1IiLlIhiCfS8q8ndYbFLv3rOkvrHQiY3jMFObx6dbj7ePkym4jozJwrN7hn0xtHVjVPryVOz3PnU4kYnmd7MiqahHcSv+Vl8vKPhLFLQSwS9ByBFbiqJBkzp+EIPTjgp03UyRgNrQD0moZzy2axRuiSWi39+ExPR3e6u3uAsKP0j5X+e0pfa2czu8ZJHTmTgfGRNZtZLdTs2tTaIEMYFsZOjXngj8zYOdRAWZIPWMfG3NV4Y2E6Z/G16ZDJ92pMHksTXGl0vd7o4PLW2NSEWo2xedyegLsP80AVaKPQ6IzQKxAaYoJgv/DbIr74VkPFZ+YeisZaocC0MsAnH9skv9aYSeFGcbm480bV8deaTIHt1sb2Naxrd/DqrhMW8RU6g7VrpaDbH8d4BRd4a6c8SdlJyYBiLgVq0hZPz1+1+VTUGf/ltGOCAtTw7BOUxaE+T7297lAuXWsl43UXxXpEOhjUzc6EKtVIV4R0hD2SxTCPcgClwbwrbg5Jimv7GcUvtHrZ7953Ze9RrB7cNlbPdCVayZFpgqWcdo+xqz25l+CNgQ+CYuJ3XyFcsc7VEB6bPYziXk2ivq5KztftPJihQ/Hcx4R8LSL6jfXVb9AXNQWjfXcyWhoUTQE2tcYCOo2kpy/brXQgx9El9BTUvqOCFQFf4bSx2Udr9OsKrL5Wce1Ultc9yZe4tZhHPjA/dmJcWZHjli8AB1h6gXYUsBSVSFBuP6ktWjTz7INoqiVcWjL3SUnhD+xyALKW1bzoWvZ52nChmS0OiFmia+Q9gpJr5DyksR212wmlOzVprOulM40JYXphRGiQ/SA+ay0/9eU1Ws1GpTlVdtuhYbt1i1/3sGXjsWaV+f58Ie7O719v+jcrXRKpeZaE7nkA7JcXkjQh+SvsQGaDnAHLutZn80EBnIXYoS/zRSe1+4td/UX4pGWMmBtZ4tgHyIdGuJiZ50hzFp/zZBh4sHG0i3soYFXnYaW8yrtIaJ3Hn3LzvCUjZVK5hPWyHovkt40A18AuKA96Oh5RBrMYAsITGEZ8BxULohXFVqlcDwvi30B+B6iYFfS/y6aIvbg35j0Vg2LgizGdGxC6j+MU3BRXK8BJMAXMBS7fbxfOMt2MwAoUjr+Ew/uxBNIDyE6QTrLCiI9QdHqiGgGOF/bcuMPuemdxceHGqzxIejCgtgYOeHcta7C6MplFChxho53GONiX2L2KSwjWJn9zcdeCzm7LXf12F4S2SGvjgEpQ0YSZDo5wnhkm5DNnG8GNKfdcXj5zEP4J/7PH4kn14erYZyfglrMjHOyXaWvkSCWMS2qVnitKbTpcT2JZS2enTAHYoUqtsJek9Fz7EB1UZsQLUojRlXBWIX3Kw1JDf9lVBOnFAtYSaN4rjNuCTuiFKeZaEUCKChVQ8e1rRQCpJuZvi/GeoqCeTphKA/NGiUP+Y/2TomuqP7Si+bVbWrtbNG63aMvuv7XptPu0lHYHw4BpKH9douaUV7F2XEAZXNqeZE/i6Ei8o9aRtzt4EfDb4N3ABPLXSMdIXg+AoVZjGjqiESTjKYj4vjmFlhqFYSaqoYkTGEp0SjcEmFpuaZrDkmv6hlWz65awRQq7kaENF2I68niqhI5LKoKhNWiXkixPQLsyprWSAFsN7aE4R7XLZMonKbqHK1+q0UA0pLu/p8TMdB3ui+0QE0ymFWnroT4smRf2J2D1iedq6hBK3Vbha+JDzmpEjaT4LKA9oLjViRBeAKj2b6cbrGLlDCU5rOKoTSzi+quqin28iWiCrQq3iU0da43kACGOD+sg3ajAkeqAi9uVAwyEBrod2owAtVATAJ5XEn/sYsjyTJDYfJC9L5mUA6l8/RACpNLjkmVg2j8SGUgR1JOUaEV1VKrKIVG6y9kdF7cmFQyFNsCNvESTVvVI2YVSq/yJRQV+h/lGN+hmt2CHmn7ojAHzgrPVndUXi5V7hAMrXQcU5JZhrZY4bskD6+LN7j0mgyhfVDirjHlXhHq3ALfmtGxiBIplWDOic16513dr/uVraQuOG/OQTtEpqGLR0rY5W2wlGr2aRstoejKN7CODlI4UTQpHpmQ4PZBVypF58rhwRbOdLrOV/PU3469n/OUZaupmrHZlVaQuDGXV8r9LhOdVol8TyTN7EwvLBJpVbxfeI6XikJBPF73XrPKtieVJeKL+jJU3d4XSJ+UzExGOCV01ZJj5JAWki/iIFpHmXCpvwsZRreWn07q6WIaSDfN9ye0fdyjevMgtD6xW/h8BdNRsPr1PpQRo9vR6bPRI4PREA0elFuxI4bgFZyrowFkJzkdt4wPMGeO0bJ8UjzPrTdfoJEz0lUxaKhupgmeVO4mdtoQqbjFBd7zCbOExTSVsFyjTE4vAukoEN2jyMHMdz32VAgFT8yER0c5rgldTCAlaGfT7t3jtOkxc43knEcj86HETVAIzdVSQdZ3mkO0IR81fDrGx0Vawgb6nTxXGW53Ai5a6+qnSKf1opMbmVWmkCKmyG2FIs2CiZtp/PcPMx3dOaDZne3S3+NckNLGs2N5O9z8+GDvxsj317MmBfZrhWGA3pIANUscmyX8U1jETJI/+fWhMWRb7v+OoEfmRFjfp0LVyOA6pEfm+G3asWRhezYa4zZcow6l/AYlghgxdFhmeH8wslxK3ETDXutw0LpyRfzFnuVGV4d73a5hu4th2tdAa1DFw9cXvUk0PdRmM7oPnjK86D482N13vE0jZMk4FIVxNGRjP2Ahs8NzZ86pFhK7S+8iAU4nRGbR1O82N0UevnuyInYzLlmRvobMI9d2D4Cop8CoWCu0yqIrKV6SQdEAQ24PSynjwtwKhU7diAl2aHub6cQ8cFBgRczIW0W9MrikWT/RLKe5F92Eao8VBcATYGl8ZRQ0sAJr6NhxCtjlZWRISVYNdgrIUCrIw8WwXlwGyJGqFYbxW/IpwGT2EAyd2zAkuID7T+q2FAwDseOCKCYjluD71I/aswdELjSY+cwfgKQc8XuTCviUGmUeT84vY9mfmaWoqAu5Hc+upPMN1mCZ92Dg7EUpAlqJzG2G0deUDHcTR0xvYM58zA2a2unAPs0vaXc0ujUtbNWbujL6iFfyP64T4IKApl5DdVJHFdRORM+6AFJYDkhYsfkOseDRDVik8CaTx07HIS25bLDJvBu9ko8Zg6X6Lu7+2LzTjQphEtWoeTS2cnvhKjl7WDiv4GnN+r4VxcKkMrAFb+kgX4EnlpeGqhJMjfDFvqKD1UTcZFu6i4NRTh6FrexGWwHGAW2MIVK+OJMGo1tU5Tb6tmZcGEyfPf0XaV+QMAp+9HzoyFZfNh25onjzKDWYUpgbiD/kAzegA4VDJOhzgShuK2FsbD0q4XsKQXf34AYS8YhRmCufqr6rl+ZKvm5q5XmeEPUJSAFmA7B2AxungPV7BvU/YH+A+X82x3LhB78FfBbmRsFOyb9Cn+Kr6E18kbwoeCSCSDzs0nfbAr9xFnJ85Nu2XMD9MJp/vK7mLJYvDvoixEsxoxeDGVxBYbRwOMLNwsnKqOgF0JYuHiLw/ZgkMa/AJLRWNeSO91ZOmKi+FmgLlapmYizRU1JA4JjA901opMceOIJX+Q0DqKBgyMTu1ykuFLRmWOpmIrMhU1fzRqArPL9WTOc6dXoFQRvDRgbCezOUad62onILt+pMQqkjAEpfhyxVSbiKHS1d5yIvirIbLyFlyqSv0LCNnaC1EqOB2jhj1GjU1RaU6Nomb1+JBXivG8tWCx1d4Xi8AlwjUFeAFAnsqA3sRYrah927wKQx8yJ+wDGcNGizlioRkpBUoBfxndl8EbCXNsTXS145tPBou/QFHB8l+cvQnxTyhbOElO6F9WSfK2STD/B7RY+WOKrO4qZOGIzaGdSuy02y6PHIXUCmCyRA2sru9DrigQ6WxiDg+wdvWBKt/AChd/UDvw692AL8Ktr2I9AybEDr5b/+gjK2tz3s6lBTGB4W7eyVtxd1whGkSXOTFcR2VEUnFPNRTYfw25FyvkFp0jinfxfZ6WfC+/X7jFJNlNrRZBMOjI41oFA/0GBqUilVcMvGdo0nVDOfGK25SsmGlp+cpkYZqTL23lzUR/X8ZNCtmzjBbPbNIbg0ReWpjiWrKsgCxT4VGvAol8pijEK3aBC7EBU/aLTe2DPHkJiPq/0NR0OHSKWwZy1l/UpzRrB4NAQF9umg+amnrZUW58qe9qEDCv+s3nz5KLkHc2frXx643Whraxv/F84+ONzdeb9itd1sHrQ9bR08fPt1/+runzwXq28lNL/cKPw9/eSfLRxsKw=</latexit> 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

  67. <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 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

  68. <latexit 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<latexit sha1_base64="y8fQf30hOkgELFhBSqpIVCo9o6E=">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</latexit> 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

  69. <latexit 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<latexit sha1_base64="y8fQf30hOkgELFhBSqpIVCo9o6E=">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</latexit> 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

  70. <latexit 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<latexit sha1_base64="y8fQf30hOkgELFhBSqpIVCo9o6E=">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</latexit> 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

  71. <latexit 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<latexit sha1_base64="JGRteqc+KDLiN9ALuKAB0z91v3g=">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</latexit> 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

  72. <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 := 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

  73. <latexit 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<latexit sha1_base64="y8fQf30hOkgELFhBSqpIVCo9o6E=">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</latexit> 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

  74. <latexit 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<latexit 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<latexit sha1_base64="JGRteqc+KDLiN9ALuKAB0z91v3g=">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</latexit> 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

  75. <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 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

  76. <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 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

  77. <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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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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