The Diagrammatic Coaction

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1 E H U N I V E R S I T T Y O H F R G E D I N B U The Dagrammatc Coacton Enan Gard (Hggs Centre, Ednburgh) In collaboraton wth: Samuel Abreu, Ruth Brtto, Claude Duhr and James Matthew References: Phys.Rev.Lett. 9 (7) no.5, 56, JHEP 76 (7) 4, JHEP 7 (7) 9 and work to appear soon

2 The Dagrammatc Coacton: Motvaton Many Feynman dagrams evaluate to Multple polylogarthms (MPLs). MPLs admt a coacton, whch maps complcated objects nto smpler ones. The coacton has many applcatons ncludng smplfyng expressons and takng dscontnutes. Can Feynman ntegrals themselves be endowed wth a coacton? The dea that there should be a Hopf Algebra on Feynman graphs / ntegrals that reproduces the one on MPLs was an nspraton for research over many years [Kremer, Brown]. At one loop we now have an explct constructon n dmensonal regularsaton wth a smple graphcal nterpretaton nvolvng contractons and cuts. Ths algebrac structure encodes the analytc propertes of the ntegrals. It facltates e.g. performng analytc contnuaton and dervng dfferental equatons. It also provdes a new perspectve on cuts and Master Integrals. Can t be extended to hgher loops?

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<latext sha_base64="mqpbotwmutowkq6annry6gabs=">aaab7ncbdlsgmxfibpek3vnpjlgevwmclosuhfwv6ghuompdogjpkhyrtkidw4irtz6po9/g9ijo6w+bj/+cq875orwy33/y9vy3nres3sffcpdo+osyentnkmmgdjslr7ygaffxhw3irsjqpdisipgd7n6a4za8eq9kmkoaqdxwpoqhvwqztglkftqnsv/y5ya8eqcgpeq9me3n7bmorjmugm6g/amkfacwwuxmblpkrnsahyeksjrhpl93s6dydxottlszd3xm5lcmoq6jbvdsqbmf/vopmnb8ocqzszqnjozgtxckdjvpc43mokdyjr3uxipjoy6xiquhdwtl6hrusoh64lteqyzgkca4caub3ean7qeodwawgd4gvcv969n+99brhlwfo4i+8j+nfy+l</latext> <latext sha_base64="mqpbotwmutowkq6annry6gabs=">aaab7ncbdlsgmxfibpek3vnpjlgevwmclosuhfwv6ghuompdogjpkhyrtkidw4irtz6po9/g9ijo6w+bj/+cq875orwy33/y9vy3nres3sffcpdo+osyentnkmmgdjslr7ygaffxhw3irsjqpdisipgd7n6a4za8eq9kmkoaqdxwpoqhvwqztglkftqnsv/y5ya8eqcgpeq9me3n7bmorjmugm6g/amkfacwwuxmblpkrnsahyeksjrhpl93s6dydxottlszd3xm5lcmoq6jbvdsqbmf/vopmnb8ocqzszqnjozgtxckdjvpc43mokdyjr3uxipjoy6xiquhdwtl6hrusoh64lteqyzgkca4caub3ean7qeodwawgd4gvcv969n+99brhlwfo4i+8j+nfy+l</latext> Coacton on Multple Polylogs Multple Polylogarthms (MPLs) are terated ntegrals of the form: dt G(a,a,...,a n ; z) = G(a {z },...,a n ; t) t a {z } weght n weght n They form an algebra wth the acton A A A : Defne a coacton: A A H z (G(~c; z)) = ~b ~c a 4 a 3 a a z left: modfy the ntegrand retanng only the poles n (keepng the orgnal contour). G(~a; z) G( ~ b; z) = G( ~ b; z) {z } weght ~ b ~ b<latext sha_base64="mqpbotwmutowkq6annry6gabs=">aaab7ncbdlsgmxfibpek3vnpjlgevwmclosuhfwv6ghuompdogjpkhyrtkidw4irtz6po9/g9ijo6w+bj/+cq875orwy33/y9vy3nres3sffcpdo+osyentnkmmgdjslr7ygaffxhw3irsjqpdisipgd7n6a4za8eq9kmkoaqdxwpoqhvwqztglkftqnsv/y5ya8eqcgpeq9me3n7bmorjmugm6g/amkfacwwuxmblpkrnsahyeksjrhpl93s6dydxottlszd3xm5lcmoq6jbvdsqbmf/vopmnb8ocqzszqnjozgtxckdjvpc43mokdyjr3uxipjoy6xiquhdwtl6hrusoh64lteqyzgkca4caub3ean7qeodwawgd4gvcv969n+99brhlwfo4i+8j+nfy+l</latext> ~c ~a qq ~ b G ~b (~c; z) {z } weght ~c ~ b a 4 a3 a G(~c; z), a z ~ b rght: modfy the contour to encrcle the poles n (keepng the orgnal ntegrand), ~ b<latext sha_base64="mqpbotwmutowkq6annry6gabs=">aaab7ncbdlsgmxfibpek3vnpjlgevwmclosuhfwv6ghuompdogjpkhyrtkidw4irtz6po9/g9ijo6w+bj/+cq875orwy33/y9vy3nres3sffcpdo+osyentnkmmgdjslr7ygaffxhw3irsjqpdisipgd7n6a4za8eq9kmkoaqdxwpoqhvwqztglkftqnsv/y5ya8eqcgpeq9me3n7bmorjmugm6g/amkfacwwuxmblpkrnsahyeksjrhpl93s6dydxottlszd3xm5lcmoq6jbvdsqbmf/vopmnb8ocqzszqnjozgtxckdjvpc43mokdyjr3uxipjoy6xiquhdwtl6hrusoh64lteqyzgkca4caub3ean7qeodwawgd4gvcv969n+99brhlwfo4i+8j+nfy+l</latext> Ths allows to derve functonal denttes by smple algebra. Goncharov, Brown, Goncharov-Spradln-Vergu-Volovch, Duhr,

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sha_base64="t3la5b5j6ctlpkedkaoe7jflbyg=">aaab/ncbdlssnafiynmu9rcwvm8eejqaboaallxsbpqjtnjo3qyctmnhikvoobf4q49tnc+toyy9yebj/+cwznzb4ngghznpvvfcxsfbd3dvf7ypdpo5trvmdxjw7ybojrhkdeagwdtrjesbykgedottx6jywdzbomb+rvuqhpwsmbwpvvsmgjrh59cjlwyemy3dytcuorvnjrwmbg4llkvetb+8kztemggmjdce/iwo4fswsdflnusihi+6xuxozxs9n5e3xmnb4oywebdxzf9kjnj6hawmmyiwiuqflfrnceovncypmenn8juyijxnavc44pregmdhcpubsvqbshybkbhuaufnkmtwka/j+olsr5neuak6rwkoktuq3eojhqiogw9ofjl9a79tfvbhymspr9bnd4erlhk=</latext> <latext sha_base64="t3la5b5j6ctlpkedkaoe7jflbyg=">aaab/ncbdlssnafiynmu9rcwvm8eejqaboaallxsbpqjtnjo3qyctmnhikvoobf4q49tnc+toyy9yebj/+cwznzb4ngghznpvvfcxsfbd3dvf7ypdpo5trvmdxjw7ybojrhkdeagwdtrjesbykgedottx6jywdzbomb+rvuqhpwsmbwpvvsmgjrh59cjlwyemy3dytcuorvnjrwmbg4llkvetb+8kztemggmjdce/iwo4fswsdflnusihi+6xuxozxs9n5e3xmnb4oywebdxzf9kjnj6hawmmyiwiuqflfrnceovncypmenn8juyijxnavc44pregmdhcpubsvqbshybkbhuaufnkmtwka/j+olsr5neuak6rwkoktuq3eojhqiogw9ofjl9a79tfvbhymspr9bnd4erlhk=</latext> <latext sha_base64="t3la5b5j6ctlpkedkaoe7jflbyg=">aaab/ncbdlssnafiynmu9rcwvm8eejqaboaallxsbpqjtnjo3qyctmnhikvoobf4q49tnc+toyy9yebj/+cwznzb4ngghznpvvfcxsfbd3dvf7ypdpo5trvmdxjw7ybojrhkdeagwdtrjesbykgedottx6jywdzbomb+rvuqhpwsmbwpvvsmgjrh59cjlwyemy3dytcuorvnjrwmbg4llkvetb+8kztemggmjdce/iwo4fswsdflnusihi+6xuxozxs9n5e3xmnb4oywebdxzf9kjnj6hawmmyiwiuqflfrnceovncypmenn8juyijxnavc44pregmdhcpubsvqbshybkbhuaufnkmtwka/j+olsr5neuak6rwkoktuq3eojhqiogw9ofjl9a79tfvbhymspr9bnd4erlhk=</latext> The Master Formula Inspred by the coacton on MPLs we defne a coacton on ntegrals:, We have nserted a complete set of master ntegrands and correspondng master contours these bass elements are pared accordng to: P ss j = j, P ss s a sem-smple projector: t retans sem-smple objects (x) =x ncludng and ratonal functons (and complete ellptc ntegrals) but elmnates logs and polylogs.

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sha_base64="sglyxrz5dnnssgnwqwyaksha=">aaab8hcbbns8naeiyn9avwr6phl8eecpjefry8okxgvqnptjdtmu3de3yqqn+ffw+kepneppfuglznyfh7emwfn3jdhtbvp+3kg5tbzvl3cre/shhuf4pkpjvbhajjgpvs9ettmttgy4bskio5lqbtm/zevejks+wbmcqejwlgefjrcfbgiaiasmqzwv73kronfqaktybvr8eojqmghcowvd9lzfbhsowwum8mkgtbmcuz7fukqonssfdcvbdoyi87sl9pghhomqtspeza3/6vuxpdbbmtswqojmupops7jnbz69ru5qyproardg7qsmqjaymeegr968jpgnf8vvrdko4jdgzdjfhwdu4gxagyca3fnc5l86787fsltnfzcn8kfp5azej/u=</latext> <latext sha_base64="sglyxrz5dnnssgnwqwyaksha=">aaab8hcbbns8naeiyn9avwr6phl8eecpjefry8okxgvqnptjdtmu3de3yqqn+ffw+kepneppfuglznyfh7emwfn3jdhtbvp+3kg5tbzvl3cre/shhuf4pkpjvbhajjgpvs9ettmttgy4bskio5lqbtm/zevejks+wbmcqejwlgefjrcfbgiaiasmqzwv73kronfqaktybvr8eojqmghcowvd9lzfbhsowwum8mkgtbmcuz7fukqonssfdcvbdoyi87sl9pghhomqtspeza3/6vuxpdbbmtswqojmupops7jnbz69ru5qyproardg7qsmqjaymeegr968jpgnf8vvrdko4jdgzdjfhwdu4gxagyca3fnc5l86787fsltnfzcn8kfp5azej/u=</latext> Propertes of the coacton Gven a bass of pared ntegrand and contour the coacton on ntegrals s defned by reproducng the propertes of the coacton on MPLs. Dscontnutes act on the frst entry Dsc apple = apple Dsc, ths s useful for takng dscontnutes. Dfferentaton acts on the second entry = useful for dervng dfferental equatons.,

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sha_base64="rkyygtze8pwnxudesdxkhjm=">aaab+hcbbns8naeiy3ftb6aphl4tf8fssiux4eu8vbafiayuzbptnjonfsepggfd/jf/jdsb9yehhnrlm9gskqy67reztr6xubvdcnu7uflmqhrytp5pdk8cypagbcqrmfsugkglgusggh4+tvfaohy3emkat9qyuggjovr9c6jkefdiellbad86fbfqzkvwcuhqni+uwvhjznakfdjjup6boj8xjyjlmb7qyge8tebqteyheyp5sfpqvngnpinbkarz9/dexjjjlfgoyogi7ncm5n/bopdq78tkgkrvb8swqsooxnavaq6gbo5xyyfwleyvli6yr5tvybgl95fvqqmf57qjsr+vxfmgjosnxcope5usimcscpesav5m5cl6cd+djbrm5dph5i+czx8lu5n</latext> <latext sha_base64="rkyygtze8pwnxudesdxkhjm=">aaab+hcbbns8naeiy3ftb6aphl4tf8fssiux4eu8vbafiayuzbptnjonfsepggfd/jf/jdsb9yehhnrlm9gskqy67reztr6xubvdcnu7uflmqhrytp5pdk8cypagbcqrmfsugkglgusggh4+tvfaohy3emkat9qyuggjovr9c6jkefdiellbad86fbfqzkvwcuhqni+uwvhjznakfdjjup6boj8xjyjlmb7qyge8tebqteyheyp5sfpqvngnpinbkarz9/dexjjjlfgoyogi7ncm5n/bopdq78tkgkrvb8swqsooxnavaq6gbo5xyyfwleyvli6yr5tvybgl95fvqqmf57qjsr+vxfmgjosnxcope5usimcscpesav5m5cl6cd+djbrm5dph5i+czx8lu5n</latext> Dagrammatc Coacton at one loop To translate the coacton to Feynman dagrams we need to fx a bass. At one loop, a convenent bass of unform weght Feynman ntegrals s: ej n =e E d D n k Y l n m D/ (k q j ), D = m j j= Ths fxes the bass for the frst entres n the coacton: for a gven ntegral these are the dagram tself and all ts possble contractons. ej n Next we need to dentfy the correspondng contours encrclng poles, defnng the second entres. These are cuts Frst guess: cut all propagators that aren t contracted n the frst entry.

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sha_base64="wwl4aa5eyjbfrakpjjwbsg=">aaab73cbbnswmxein6etwpojf8fr6dhghepfewhtevjprntadk6xqsv+efw+kepveppfml7nyag/vzjcnofn9b3v73cxubw9k5xt7s3f3b4vd4+armvaynpotsnygaffx3irsjnqpekksbnb+f9hnqw5v8sjmuw4qojy85o9nr6mhgsl++wk/uiusq5fcbi++as3ucxlufomqdhdwe9tokacwvuplblpkxnsiyesjmjc6wlfgblwzodesrsnlvm4vyemndfmkksum6ffrc/o/wjez8u45tlnleq/cjoblgkzi8na66rwtfxqjnmblfcrlrtlejrdcshryorrqcdx/vwlsvjkminmmlbhandbdbjsbgybneiu379f78d69jvrwctntugpvm8fsaoqea==</latext> <latext sha_base64="wwl4aa5eyjbfrakpjjwbsg=">aaab73cbbnswmxein6etwpojf8fr6dhghepfewhtevjprntadk6xqsv+efw+kepveppfml7nyag/vzjcnofn9b3v73cxubw9k5xt7s3f3b4vd4+armvaynpotsnygaffx3irsjnqpekksbnb+f9hnqw5v8sjmuw4qojy85o9nr6mhgsl++wk/uiusq5fcbi++as3ucxlufomqdhdwe9tokacwvuplblpkxnsiyesjmjc6wlfgblwzodesrsnlvm4vyemndfmkksum6ffrc/o/wjez8u45tlnleq/cjoblgkzi8na66rwtfxqjnmblfcrlrtlejrdcshryorrqcdx/vwlsvjkminmmlbhandbdbjsbgybneiu379f78d69jvrwctntugpvm8fsaoqea==</latext> Dagrammatc Coacton at one loop example: the three-mass trangle The dagrammatc coacton for a three-mass trangle wth massless propagators (to all orders n dmensonal regularsaton): Can be verfed, order-by-order n to reproduce the coacton on MPLs Defnng z z = p /p and ( z)( z) =p 3/p At leadng order n : T (z, z) = L (z) + L ( z) ln(z z)ln z z [T (z, z)] = T (z, z) + T (z, z)+ln p ln z z +ln p 3 ln z z +ln( p ) ln We see how th channel dscontnutes are captured by th bubbles. z( z) z( z)