Multiparticle Cuts of Scattering Amplitudes Pierpaolo Mastrolia - PowerPoint PPT Presentation

Multiparticle Cuts of Scattering Amplitudes Pierpaolo Mastrolia Institute of Theoretical Physics, University of Z¨ urich RAD COR 2007 Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 1

Outline All fundamental processes are reversible Feynman • Cutting Loops ⇔ Sewing Trees • Unitarity & Cut-Constructibility • General Algorithm for Multiple-Cuts in D -dim • Quadruple-Cut • Double-Cut • Triple-Cut • Applications Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 2

Spinor Formalism Xu, Zhang, Chang Berends, Kleiss, De Causmaeker • on-shell massless Spinors Gastmans, Wu Gunion, Kunzst | i � ≡ | k + [ i | ≡ � k + i � ≡ u + ( k i ) = v − ( k i ) , i | ≡ ¯ u + ( k i ) = ¯ v − ( k i ) , • k 2 = 0 : a ≡ k µ ! µ a ˜ a = � k � k k a ˙ or k = | k � [ k | + | k ] � k | / a ˙ a ˙ • Spinor Inner Products � j = −� i j � ∗ , ˙ | s i j | e i # ij , b ˜ i ˜ � i j � ≡ � i − | j + � = " ab � a i � b [ i j ] ≡ � i + | j − � = " ˙ � ˙ a b j = � a ˙ with s i j = ( k i + k j ) 2 = 2 k i · k j = � i j � [ ji ] . • Polarization Vector µ ( k ; q ) = � q | $ µ | k ] µ ( k ; q ) = [ q | $ µ | k � " + " − √ √ , , 2 � qk � 2 [ kq ] with " 2 = 0 , " + · " − = − 1 . k µ · " ± µ ( k ; q ) = 0 , Changes in ref. mom. q are equivalent to gauge trasformations. Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 3

One Loop Amplitudes P-V Tensor Reduction A = % + % + % c 4 , i c 3 , j c 2 , k +rational i j k Since the D -regularised scalar functions associated to boxes ( I ( 4 m ) , I ( 3 m ) , I ( 2 m , e) , I ( 2 m , h) , I ( 1 m ) , I ( 0 m ) ) , 4 4 4 4 4 4 triangles ( I ( 3 m ) , I ( 2 m ) , I ( 1 m ) ) and bubbles ( I 2 ) are analytically known 3 3 3 ’t Hooft & Veltman (1979) Bern, Dixon & Kosower (1993) Duplan ˇ cic & Ni ˇ zic (2002) • A is known, once the coef fi cients c 4 , c 3 , c 2 and the rational term are known: they all are rational functions of spinor products � i j � , [ i j ] Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 4

Unitarity & Cut-Constructibility • Discontinuity accross the Cut Cut Integral in the P 2 i j -channel j j + 1 � 2 i + 1 � 1 i i − 1 Z C i ... j = & ( A 1-loop d 4 # A tree ( � 1 , i ,..., j ,� 2 ) A tree ( − � 2 , j + 1 ,..., i − 1 , − � 1 ) ) = n with d 4 # = d 4 � 1 d 4 � 2 ' ( 4 ) ( � 1 + � 2 − P i j ) ' (+) ( � 2 1 ) ' (+) ( � 2 2 ) • loop-Reconstruction Bern, Dixon, Dunbar & Kosower Bern & Morgan; Anastasiou & Melnikov Bedford, Brandhuber, Mc Namara, Spence & Travaglini - channel-by-channel reconstruction of the loop-interal: ' (+) ( p 2 ) ↔ 1 / ( p 2 − i 0 ) - loop-tools integrations: PV-tensor reduction & integration-by-parts identitities • Unitarity-motivated loop-mometum decomposition Ossola, Papadopoulos & Pittau; Forde; Ellis, Giele & Kunszt → talks by Forde, Kunszt, Papadopoulos Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 5

Generalised Unitarity • coef fi cients show up entangled in a given cut: how do we disentangle them? The polylogarithmic structure of boxes, triangles, and bubbles is different. Therefore their multiple cuts have speci fi c signature which enable us to distinguish unequivocally among them. = c 4 + c 3 + c 2 = c 4 + c 3 = c 4 • Cuts in 4-dim carry informations about the coefficients • Cuts in 4-dim do not carry any informations about the rational term • Cuts in D -dim detect also rational term Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 6

Quadruple Cuts Boxes • Multiple Cuts Bern, Dixon, Dunbar, Kosower (1994) K 2 K 3 A 2 A 3 A 1 A 4 K 1 K 4 The discontinuity across the leading singularity, via quadruple cuts, is unique , and corresponds to the coef fi cient of the master box Britto, Cachazo, Feng (2004) A tree 1 A tree 2 A tree 3 A tree c 4 , i ( 4 with a frozen loop momentum: � µ = ) K µ 1 + * K µ 2 + $ K µ 3 + '" µ 1 K , +,! K + 2 K ! 3 Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 7

Double-Cut Phase Space Measure • 4-dim LIPS Cacahazo, Svr ˇ cek & Witten � 2 / 0 = 0 , � 0 = | � 0 � [ � 0 | ≡ t | � � [ � | Z � � d � � [ � d � ] K 2 � � Z Z Z d 4 # = d 4 � 0 ' (+) ( � 2 0 ) ' (+) (( � 0 − K ) 2 ) = t dt ' (+) ⇒ t − � � | K | � ] � � | K | � ] • D -dim LIPS Anastasiou, Britto, Feng, Kunszt, PM Z Z Z d 4 − 2 " # dµ − 2 " d 4 # µ , = - ( " ) � 1 − 4 µ 2 1 − K 2 with � 2 / L = � 0 + zK , 0 = 0 , � 0 ≡ t | � � [ � | z 0 = , 2 Z Z d 4 L ' (+) ( L 2 − µ 2 ) ' (+) (( L − K ) 2 − µ 2 ) d 4 # µ ⇒ = Z � � d � � [ � d � ] � � t − ( 1 − 2 z ) K 2 Z Z t dt ' (+) = dz ' ( z − z 0 ) � � | K | � ] � � | K | � ] Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 8

Double-Cut ⊕ Spinor-Integration Britto, Buchbinder, Cachazo & Feng (2005); Britto, Feng & PM (2006) Anastasiou, Britto, Feng, Kunszt & PM (2006) Britto & Feng (2006) A L A R Z Z dµ − 2 " & , d 4 # µ A tree ⊗ A tree M = - ( " ) & = L R • t -integration ⊕ Schouten identity � � t − ( 1 − 2 z ) K 2 A tree ( �, z , t ) A tree [ . � ] n ( �, z , t ) Z = % � � | P 1 | � ] n + 1 � � | P 2 | � ] ≡ % L R t dt ' G i ( | � � , z ) T i � � | K | � ] � � | K | � ] i i the 4D-discontinuity reads, Z Z & = % � � d � � [ � d � ] T i dz ' ( z − z 0 ) i 1. P 1 = P 2 = K (momentum across the cut) ⇒ 2-point function (cut-free term) 2. P 1 = K , P 2 � = K , or P 1 � = P 2 � = K ⇒ n -point functions with n ≥ 3 (Log-term) Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 9

Log-term of 4D-Double Cut • Feynman Parametrization: P 1 = K , P 2 � = K , or P 1 � = P 2 � = K [ . � ] n Z dx ( 1 − x ) n G i ( | � � , z ) T i = ( n + 1 ) � � | R | � ] n + 2 , R = x / / P 1 +( 1 − x ) / P 2 [ . � ] n [ . � ] n + 1 � � | P | � ] n + 2 = [ d � / | � ] ] • Integration-by-Parts in | � ] [ � d � ] � � | P | � ] n + 1 � � | P | . ] . ( n + 1 ) • Integration in | � � : Holomorphic ' -function (Cauchy-Pompeiu’s Formula ) Cachazo, Svrcek, Witten; Cachazo; Britto, Cachazo, Feng � � d � � [ d � / | � ] ] G i ( | � � , z ) [ . � ] n + 1 Z Z Z dx ( 1 − x ) n F i = � � d � � [ � d � ] T i = � � | R | � ] n + 1 � � | R | . ] � → � ij � �� i j � G i ( | � � , z ) [ . � ] n + 1 � G i ( / R | . ] , z ) � Z = F ( 1 ) + F ( 2 ) dx ( 1 − x ) n + % = lim i i ( R 2 ) n + 1 � � | R | � ] n + 1 � � | R | . ] j where | � i j � are the simple poles of G i , and R 2 = a ( x − x 1 )( x − x 2 ) • Double-Cut Z Z � � F ( 1 ) + F ( 2 ) dµ − 2 " M = - ( " ) dz ' ( z − z 0 ) % i i i Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 10

• I 2 Z � � d � � [ � d � ] Z d 4 � ' (+) ( � 2 ) ' (+) (( � − K ) 2 ) = K 2 = = 1 ; � � | K | � ] 2 The discontinuity of a bubble is rational !!! • I 3 m 3 K 2 Z � � d � � [ � d � ] Z 1 Z 1 d 4 � ' (+) ( � 2 ) ' (+) (( � − K 1 ) 2 ) � � d � � [ � d � ] 0 dx 1 Z Z = = � � | K 1 | � ] � � | Q | � ] = 0 dx = ( � + K 3 ) 2 � � | R | � ] 2 R 2 K 1 K 3 Q ⇒ R 2 quadratic in x Q = ( K 2 3 / K 2 1 ) / K 1 + / R = ( 1 − x ) / / K 3 , K 1 + x / / The discontinuity of a 3m-Triangle is a ln (irrational argument) !!! • I 4 The double cut detect box-coef fi cient as well. One can show that the discontinuity of a 1m-,2m-,3m- box is a ln (rational argument) – but boxes are known from 4-ple cuts. Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 11

� • I 2 1 − 4 µ 2 = K 2   � 1 − 4 µ 2 1 − • I 1 m = 1 K 2 3 K 2 ln   �   1 − 4 µ 2 1 + K 2   � 1 − 4 µ 2 ( s + t ) 1 − • I 0 m 2 s 4 = ln   � �   1 − 4 µ 2 ( s + t ) 1 − 4 µ 2 ( s + t ) st 1 + s s • µ -integration ≡ Dimension-Shift 2 ( 2 0 ) − 2 " ( µ 2 ) − 1 − " + r f ( µ 2 ) = ( 2 0 ) 2 r R d 1 − 1 − 2 " d − 2 " µ dµ 2 d 2 r − 2 " µ Z Z Z Z ( 2 0 ) − 2 " ( µ 2 ) r f ( µ 2 ) ( 2 0 ) 2 r − 2 " f ( µ 2 ) R d 1 2 r − 1 − 2 " = d 1 − 1 − 2 " d 2 r − 2 " µ Z − " ( 1 − " )( 2 − " ) ··· ( r − 1 − " )( 4 0 ) r ( 2 0 ) 2 r − 2 " f ( µ 2 ) = Pierpaolo Mastrolia - Multiparticle Cuts of Scattering Amplitudes , 12