Cuts and discontinuities Ruth Britto Trinity College Dublin MHV@30, Fermilab 17 March 2016 Ruth Britto Cuts and discontinuities
Amplitudes as building blocks MHV amplitudes, for other helicity amplitudes ◮ Saw the Parke-Taylor formula in Witten’s 2003 paper. ◮ MHV diagrams build tree amplitudes on-shell [Cachazo, Svrcek, Witten] MHV MHV MHV Tree amplitudes, for loop amplitudes Cuts beyond one loop: what are we computing? What are cuts? Ruth Britto Cuts and discontinuities What are generalized cuts?
Amplitudes as building blocks MHV amplitudes, for other helicity amplitudes Tree amplitudes, for loop amplitudes ◮ Unitarity method: on-shell ”cut” techniques with master integarals [Bern, Dixon, Kosower; with Dunbar, Weinzierl, Del Duca, ...] ◮ Updated with MHV diagrams [Cachazo, Svrcek, Witten; Brandhuber, Spence, Travaglini] ◮ Twistor geometry led to differential operators for NMHV in N = 4 SYM; complex cuts for NNMHV [RB, Cachazo, Feng] ◮ Tree-level recursion observed as a byproduct [Roiban, Spradlin, Volovich; RB, Cachazo, Feng] ◮ Extended spinor integration and cuts for more realistic theories [collaborations with Anastasiou, Buchbinder, Cachazo, Feng, Kunszt, Mastrolia, Mirabella, Ochirov, Yang] l 2 . ... . . l 1 K Ruth Britto Cuts and discontinuities
Amplitudes as building blocks MHV amplitudes, for other helicity amplitudes Tree amplitudes, for loop amplitudes Cuts beyond one loop: what are we computing? What are cuts? What are generalized cuts? Ruth Britto Cuts and discontinuities
Cuts and Hopf algebra of Feynman integrals Cutkosky: Cut diagrams compute discontinuities across branch cuts. We conjecture: For integrals in the class of multiple polylogarithms (MPL), the discontinuities described by cuts are naturally found within the Hopf algebra of MPL. There is a diagrammatic Hopf algebra that ◮ involves cut diagrams, and ◮ corresponds to the Hopf algebra of MPL. [Based on: 1401.3546 with Abreu, Duhr, Gardi, and work in preparation ; 1504.00206 with Abreu and Gr¨ onqvist ] Ruth Britto Cuts and discontinuities
Cuts as discontinuities From the Largest Time Equation [Veltman] : � F + F ∗ = − Cut s F , s Hence: Disc s F = − Cut s F . Valid in a particular kinematic region: cut invariant s positive, others negative. Generalize to: Cut s 1 ,..., s k F = ( − 1) k Disc s 1 ,..., s k F . Ruth Britto Cuts and discontinuities
Multiple unitarity cuts Cut propagators: put them on shell. Choose propagators corresponding to a sequence of momentum channels. Cut s 1 ,..., s k F With real kinematics. Defined by: cut propagators + consistent energy flow + corresponding kinematic region Multiple cuts are taken simultaneously. + + + + Ruth Britto Cuts and discontinuities
Multiple polylogarithms (MPL) A large class of iterated integrals are described by multiple polylogarithms: � a n +1 dt I ( a 0 ; a 1 , . . . , a n ; a n +1 ) ≡ I ( a 0 ; a 1 , . . . , a n − 1 ; t ) t − a n a 0 Examples: � � 1 − z I (0; 0; z ) = log z , I (0; a ; z ) = log a � z a n ; z ) = 1 n ! log n � 1 − z � � I (0; � I (0; � , 0 n − 1 , a ; z ) = − Li n a a Harmonic polylog if all a i ∈ {− 1 , 0 , 1 } . n is the transcendental weight . Observation: most known Feynman integrals can be written in terms of classical and harmonic polylogs. Ruth Britto Cuts and discontinuities
Hopf algebra of MPL Goncharov’s coproduct formula for MPL (modulo π ): ∆ I ( a 0 ; a 1 , . . . , a n ; a n +1 ) k � � = I ( a 0 ; a i 1 , . . . , a i k ; a n +1 ) ⊗ I ( a i p ; a i p +1 , . . . , a i p +1 − 1 ; a i p +1 ) 0= i 0 < ··· < i k < i k +1 = n +1 p =0 Examples: ∆(log z ) = 1 ⊗ log z + log z ⊗ 1 ∆(log x log y ) = 1 ⊗ (log x log y ) + log x ⊗ log y + log y ⊗ log x + (log x log y ) ⊗ 1 n − 1 Li n − k ( z ) ⊗ log k z � ∆( Li n ( z )) = 1 ⊗ Li n ( z ) + Li n ( z ) ⊗ 1 + k ! k =1 Hopf algebra includes compatible counit and antipode. Graded by transcendental weight. Ruth Britto Cuts and discontinuities
Symbols of MPL The “symbol” S is essentially the maximal iteration. S ( F ) ≡ ∆ 1 ,..., 1 ( F ) ∈ H 1 ⊗ . . . ⊗ H 1 . � 1 � n ! log n z S = z ⊗ · · · ⊗ z � �� � n times S ( Li n ( z )) = − (1 − z ) ⊗ z ⊗ · · · ⊗ z � �� � ( n − 1) times Functions are all weight 1, i.e. log. (The symbol was introduced to us for remainder functions [Goncharov, Spradlin, Vergu, Volovich] and applied widely since then.) Ruth Britto Cuts and discontinuities
Coproducts of Feynman integrals Observation: without internal masses, coproduct can be written such that � ∆ 1 , n − 1 F = log( − s i ) ⊗ f s i i first entries are Mandelstam invariants, and each second entry f s i is the discontinuity of F in the channel s i . [Gaiotto, Maldacena, Sever, Viera] Thus: the coproduct captures standard unitarity cuts. What about generalized cuts? Ruth Britto Cuts and discontinuities
Coproduct entries If � ∆ 1 , 1 ,..., 1 , n − k F = log x 1 ⊗ · · · ⊗ log x k ⊗ g x 1 ,..., x k , � �� � { x 1 ,..., x k } k times then ∼ δ x 1 ,..., x k F g x 1 ,..., x k . = More precisely: match branch points. The “ ∼ =” means modulo π . Motivated by coproduct identity : ∆ Disc = (Disc ⊗ 1) ∆ [Duhr] and first-entry condition. Ruth Britto Cuts and discontinuities
Coproduct and discontinuities for Feynman integrals Disc s 1 F = ( − 2 π i ) δ s 1 F . ± (2 π i ) k δ x 1 ,..., x k F . � Disc s 1 ,..., s k F = x 1 ,..., x k Assume prior knowledge of alphabet (e.g. from cuts) Underlying claim: kinematics put us on the branch cuts, so that it is correct to use our definition of Disc. Ruth Britto Cuts and discontinuities
Basic example: triangle p 1 p 3 + k p 2 − k p 3 k p 2 2 � z ) + 1 � 1 − z �� − i T = Li 2 ( z ) − Li 2 (¯ 2 log( z ¯ z ) log p 2 z − ¯ z 1 − ¯ z 1 2 − i ≡ z P 2 p 2 z − ¯ 1 where z = p 2 z ) = p 2 2 3 z ¯ (1 − z )(1 − ¯ p 2 p 2 , 1 1 Ruth Britto Cuts and discontinuities
Coproduct of the triangle P 2 ⊗ 1 + 1 ⊗ P 2 + 1 z ) ⊗ log 1 − z z + 1 z )] ⊗ log ¯ z ∆ P 2 = 2 log( z ¯ 2 log[(1 − z )(1 − ¯ 1 − ¯ z P 2 ⊗ 1 + 1 ⊗ P 2 + 1 ⊗ log 1 − z z + 1 ⊗ log ¯ z � � � � − p 2 − p 2 = 2 log 2 log 2 3 1 − ¯ z + 1 1 ) ⊗ log 1 − 1 / ¯ z 2 log( − p 2 1 − 1 / z Alphabet: { z , ¯ z , 1 − z , 1 − ¯ z } . Ruth Britto Cuts and discontinuities
First cut of the triangle Cut in the p 2 2 channel. p 1 p 3 + k p 2 − k p 3 k p 2 Kinematic region: p 2 p 2 1 , p 2 2 > 0; 3 < 0. 2 π z ) log 1 − z Cut p 2 2 T = p 2 1 ( z − ¯ 1 − ¯ z = − Disc p 2 2 T 2 P 2 = 1 2 log 1 − z δ p 2 1 − ¯ z Disc p 2 2 T = ( − 2 π i ) δ p 2 2 T . Ruth Britto Cuts and discontinuities
Second cut of the triangle p 1 p 3 + k p 2 − k p 3 k p 2 4 π 2 i Cut p 2 2 T = 3 , p 2 p 2 1 ( z − ¯ z ) Kinematic region: p 2 3 , p 2 p 2 2 > 0; 1 < 0 Equivalently: ¯ z < 0 , z > 1. Now we have to match the alphabet with Mandelstam invariants: Disc p 2 3 T = Cut p 2 3 T . 2 , p 2 2 , p 2 3 T = 4 π 2 δ p 2 Disc p 2 2 , 1 − z T 2 , p 2 Ruth Britto Cuts and discontinuities
Reconstruction: from cut to symbol Integrability condition on symbols: for each k , � � � c i 1 ,..., i n d log a i k ∧ d log a i k +1 a i 1 ⊗ · · · ⊗ a i k − 1 ⊗ a i k +2 ⊗ · · · ⊗ a i n = 0 . i 1 ,..., i n Apparently related to exchanging order of cuts. Combine with first entry condition (=Mandelstam invariant) and known cut(s). Reconstruction of the symbol of the 2-loop 3-point ladder is unique from any of its single or double cuts. Various finite 1-loop examples also work. Ruth Britto Cuts and discontinuities
Reconstruction: from symbol to full function In general, integrating a symbol is an unsolved problem. But in many cases we have enough information to constrain the function uniquely & algebraically. In the same example, from the p 2 2 cut of triangle, given that: S ( T ) = 1 z ⊗ 1 − z z + 1 z ) ⊗ ¯ z 2 z ¯ 2 (1 − z )(1 − ¯ 1 − ¯ z and antisymmetry under z ↔ ¯ z , the solution T = P 2 ( z ) � 1 − z � �� z ) + 1 = Li 2 ( z ) − Li 2 (¯ 2 log( z ¯ z ) log 1 − ¯ z is unique. Ladder & massive triangles are easy too. At most, fix a single free constant by numerical evaluation at a point. Ruth Britto Cuts and discontinuities
Generalizations for internal masses [Abreu, RB, Gr¨ onqvist] First entries adjusted for thresholds � � m 2 � m 2 − p 2 � m 2 − p 2 p 2 + m 2 ⊗ m 2 − p 2 � = 1 � S + ⊗ ( m 2 − p 2 ) 2 + O ( ǫ ) . m 2 p 2 ǫ Include cuts of massive propagators Ruth Britto Cuts and discontinuities
Coproducts of diagrams [in preparation with Abreu, Duhr, Gardi] Second entries are discontinuities; first entries have discontinuities. Motivated by the identity ∆ Disc = (Disc ⊗ 1) ∆ . The companion relation ∆ ∂ = (1 ⊗ ∂ ) ∆ produces differential equations. Ruth Britto Cuts and discontinuities
Coproducts of diagrams [in preparation with Abreu, Duhr, Gardi] Second entries are discontinuities; first entries have discontinuities. Motivated by the identity ∆ Disc = (Disc ⊗ 1) ∆ . The companion relation ∆ ∂ = (1 ⊗ ∂ ) ∆ produces differential equations. Ruth Britto Cuts and discontinuities
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