Twistors and Unitarity

Transcription

1 Twistors and Unitarity NLO six-gluon Amplitude in QCD Pierpaolo Mastrolia Institute for Theoretical Physics, University of Zürich IFAE 2006, April 19, 2006 in collaboration with: Ruth Britto and Bo Feng hep-ph/ P. Mastrolia, UniZh - Twistors & Unitarity, 1

2 Outline Trees Spinor Formalism MHV Amplitudes CSW diagrams BCFW Recurrence Relation Loops Need for NLO Unitarity & Cut-Constructibility General Algorithm for Cuts NLO six-gluon amplitude in QCD P. Mastrolia, UniZh - Twistors & Unitarity, 2

3 Spinortrix Xu, Zhang, Chang on-shell massless Spinors Berends, Kleiss, De Causmaeker Gastmans, Wu i ± k ± i u ±(k i ) = v (k i ), i ± k ± i ū ±(k i ) = v (k i ), k 2 = 0 : k aȧ k µ σ µ aȧ = λk a λ k ȧ or k/ = k ± k ± Spinor Inner Products Gunion, Kunzst i j i j + = ɛ ab λ a i λb j = q s ij e iφ ij, [i j] i + j = ɛȧḃ λȧ i λḃ j = i j, with s ij = (k i + k j ) 2 = 2k i k j = i j [j i]. Polarization Vector ɛ + µ (k; q) = q γ µ k] 2 q k, ɛ µ (k; q) = [q γ µ k 2[k q], with ɛ 2 = 0, k µ ɛ ± µ (k; q) = 0, ɛ+ ɛ = 1. Changes in ref. mom. q are equivalent to gauge trasformations. P. Mastrolia, UniZh - Twistors & Unitarity, 3

4 MHV Amplitudes n-gluon Amplitudes A n (1 ±, 2 +,..., n + ) = 0 A MHV n (1, 2, 3 +,..., n + ) = n 1 Holomorphic in ij -product!!! guessed by Parke & Taylor; proven by Berends & Giele via Recurrence Relation A NMHV n (1, 2, 3, 4 +,..., n + ) =, does contain both ij and [ij] Kosower (1990) P. Mastrolia, UniZh - Twistors & Unitarity, 4

5 Gluons dance Twist Twistor Space Penrose (1967): (Z 1, Z 2, Z 3, Z 4 ) = (λ 1, λ 2, µ 1, µ 2 ), as a Fourier transform with respect to plus-helicity spinor. Witten, hep-th/ µȧ = i λȧ Witten s observation In Twistor Space, as a consequence of the holomorphy, Ã(λ i, µ i ) = Z Y d 2 λi i exp i X j µȧ j λ jȧ A(λ i, λ i ) external points lie on specific curves! MHV amplitudes for gluon scattering are associated with collinear arrangements of points Collinear Operator: F i,j,k = ij λ k + ki λ j + jk λ i P. Mastrolia, UniZh - Twistors & Unitarity, 5

6 CSW conjecture All the non-mhv n-gluon tree amplitudes are expressed as sum of tree graphs whose vertices are MHV amplitudes continued off-shell, and connected by scalar propagators Cachazo, Svrček, Witten; hep-th/ MHV rules for A n`q, (n q) + 1. Draw all the skeletons with q-1 nodes and q-2 links and assign helicity to the links, Keep only MHV-node configurations YES 2 3 NO 2 YES = P P P 2 P 3 4 P P P. Mastrolia, UniZh - Twistors & Unitarity, 6

7 CSW off-shell continuation i P i P/ η] where η is an arbitrary reference momentum (which MUST disappear from any physical quantity). Massless Projection Bena, Bern, Kosower P µ = P µ + P 2 2P η η µ i P i P (some) Applications: - fermion-gluon coupling Georgiou & Khoze - massive Higgs Dixon, Glover & Khoze - Vector Boson Currents Bern, Forde, Kosower & PM - massless QED Ozeren & Stirling P. Mastrolia, UniZh - Twistors & Unitarity, 7

8 Consider A(1, 2,..., n), and pick up any two special legs, say k and n. Analytic continuation, A A(z): BCFW Recurrence Relation k 2 1 n Britto, Cachazo, Feng; hep-th/ & Witten; hep-th/ p µ k p µ k (z) pµ k + z k γµ n] p µ n p µ n (z) pµ n z k γµ n] If n {i,..., j} && k / {i,..., j} P 2 ij (p i +... p j ) 2 P 2 ij (z) = P 2 ij z k P / ij n] the propagator develops a simple z = z ij P ij 2 k P/ ij n]. P. Mastrolia, UniZh - Twistors & Unitarity, 8

9 Cauchy Residue Theorem I 1 dz 2πi z A(z) = C = A(0) + X α Res z=z α A(z) z To get back the physical amplitude, A(0), use the residue theorem (the surface term does vanish C = 0) k k j 1 j holds in the massive case too Badger, Glover, Khoze & Svrček: hep-th/ A(0) = n = X i,j A L A ± R n Forde & Kosower: hep-th/ Ozeren & Stirling: hep-ph/ A(0) = X i,j,h A h L (z ij) 1 P 2 ij i + 1 i A h R (z ij) 1 Loops (!) unreal poles in complex momentum space from [a b] splitting functions: a b non-trivial factorization Bern, Dixon & Kosower; hep-ph/ Bern, Bjerrum-Bohr, Dunbar & Ita; hep-ph/ P. Mastrolia, UniZh - Twistors & Unitarity, 9

10 Need for NLO Less sensitivity to unphysical input scale ( renormalization & factorization scale) Precision measurments More accurate estimates of background for new physics More physics: initial state radiation, final state parton merging in jets, richer virtuality Matching with resummed calculation ( MC@NLO, NLOwPS) P. Mastrolia, UniZh - Twistors & Unitarity, 10

11 Need for NLO Less sensitivity to unphysical input scale ( renormalization & factorization scale) Precision measurments More accurate estimates of background for new physics More physics: initial state radiation, final state parton merging in jets, richer virtuality Matching with resummed calculation ( MC@NLO, NLOwPS) NLO Bricks Born level n-point Real contribution (n + 1)-point Virtual contribution n-point IR safety: R + V P. Mastrolia, UniZh - Twistors & Unitarity, 11

12 Need for NLO Less sensitivity to unphysical input scale ( renormalization & factorization scale) Precision measurments More accurate estimates of background for new physics More physics: initial state radiation, final state parton merging in jets, richer virtuality Matching with resummed calculation ( MC@NLO, NLOwPS) NLO Bricks Born level n-point Real contribution (n + 1)-point Virtual contribution n-point IR safety: R + V )( Virtual contribution: Tensor Reduction I µνρ... = Z d D l lµ l ν l ρ... D 1 D 2... P. Mastrolia, UniZh - Twistors & Unitarity, 12

13 NLO QCD 2 3 Bern, Dixon, Kosower (1993) Kunszt, Signer, Trocsanyi (1994) 1 4 Campbell, Glover, Miller (1997) 2 4 Bern, Dixon, Kosower (1997) relevant for status pp 3 jets pp V + 2 jets pp t t + jet w.i.p pp H + 2 jets w.i.p EW 2 4 Denner, Dittmaier, Roth, Wieders (2005) Boudjema et al. (2005) relevant for status e + e 4 fermions e + e HHν ν )( 6-point amplitude relevant for status pp W + W + 2 jets? pp t t + b b? P. Mastrolia, UniZh - Twistors & Unitarity, 13

14 NLO QCD 2 3 Bern, Dixon, Kosower (1993) Kunszt, Signer, Trocsanyi (1994) 1 4 Campbell, Glover, Miller (1997) 2 4 Bern, Dixon, Kosower (1997) EW 2 4 Denner, Dittmaier, Roth, Wieders (2005) Boudjema et al. (2005) )( 6-point amplitude (some) Algebraic-Semi-Numerical Approaches Ferroglia, Passera, Passarino, Uccirati Ellis, Giele, Glover, Zanderighi gg gggg [EGiZ,hep-ph/ ] Binoth, Guillet, Heinrich, Pilon, Schubert GRACE group Nagy, Soper Anastasiou, Daleo P. Mastrolia, UniZh - Twistors & Unitarity, 14

15 Analytic Approach P. Mastrolia, UniZh - Twistors & Unitarity, 15

16 One Loop Amplitudes Britto, Buchbinder, Cachazo & Feng; hep-ph/ Britto, Feng & PM; hep-ph/ PV Tensor Reduction A g = X i c 4,i + X j c 3,j + X k c 2,k + rational Since the D-regularised scalar functions associated to boxes (I (4m) 4, I (3m) 4, I (2m,e) 4, I (2m,h) 4, I (1m) 4 ), triangles (I (3m) 3, I (2m) 3, I (1m) 3 ) and bubbles (I 2 ) are analytically known Bern, Dixon & Kosower (1993) A g is known, once the coefficients c 4, c 3, c 2 and the rational term are known: they all are rational functions of spinor products i j, [i j] P. Mastrolia, UniZh - Twistors & Unitarity, 16

17 Supersymmetric decomposition Bern, Dixon, Dunbar & Kosower (1994) A g = (A g + 4A f + 3A s ) {z } 4 (A f + A s ) {z } + {z} A s N = 4 N = 1 N = 0 non-zero coefficients N Box Triangle Bubble rational Massless SuSy 1-Loop amplitudes constructible from 4-dim tree amplitudes, i.e. determined by their discontinuities (or absorptive parts) fully P. Mastrolia, UniZh - Twistors & Unitarity, 17

18 Unitarity & Cut-Constructibility Discontinuity accross the Cut Cut Integral in the P 2 ij -channel j l 2 j + 1 C i...j = (A 1-loop n ) = Z i + 1 l 1 i i 1 dµ A tree (l 1, i,..., j, l 2 )A tree ( l 2, j + 1,..., i 1, l 1 ) with dµ = d 4 l 1 d 4 l 2 δ (4) (l 1 + l 2 P ij ) δ (+) (l 2 1 ) δ(+) (l 2 2 ) C i...j = (A 1-loop n ) = X i c 4,i + X j c 3,j + X k c 2,k The Cut carries information about the coefficients. In 4-dim we lose any information about the rational term P. Mastrolia, UniZh - Twistors & Unitarity, 18

19 IR Divergence In Dim-reg, D = 4 2ɛ, the divergent behaviour of A g is: A 1 loop g singular = 1 ɛ Atree g Divergence of scalar functions Function I (4m) 4 I (3m,2m,1m) 4 I (3m) 3 I (2m,1m) 3 I 2 Divergence none ( P 2 ) ɛ ɛ 2 none ( P 2 ) ɛ ɛ 2 1 ɛ No need for 1m- & 2m-triangles Their coefficients are related to the Box-coefficients, as required by the mutual cancelation of the corresponding divergent pieces. P. Mastrolia, UniZh - Twistors & Unitarity, 19

20 Reduced Basis C i...j = (A 1-loop n ) = X i c 4,i + X j c 3,j + X k c 2,k with boxes (I (4m) 4, I (3m) 4, I (2m,e) 4, I (2m,h) 4, I (1m) 4 ), triangles (I (3m) 3 ) and bubbles (I 2 ) The global divergence of the amplitude is carried by the bubble coefficients: X k c 2,k = A tree g coefficients show up entangled in a given cut: how do we disentangle them? The polylogarithmic structure of boxes, 3m-triangles, and bubbles is different. Therefore their double cuts have specific signature which enable us to distinguish unequivocally among them. P. Mastrolia, UniZh - Twistors & Unitarity, 20

21 Quadruple Cuts Multiple Cuts Bern, Dixon, Dunbar, Kosower (1994) Boxes k k + 1 j + 1 A k Am m j A j A i m + 1 i + 1 i The discontinuity across the leading singularity, via quadruple cuts, is unique, and corresponds to the coefficient of the master box Britto, Cachazo, Feng (2004) c 4,i A tree j A tree k Atree m Atree i P. Mastrolia, UniZh - Twistors & Unitarity, 21

22 Double Cuts 3m-Triangles & Bubbles C i...j = (A 1-loop n ) = Z dµ A tree (l 1, i,..., j, l 2 )A tree ( l 2, j + 1,..., i 1, l 1 ) with dµ = d 4 l 1 d 4 l 2 δ (4) (l 1 + l 2 P ij ) δ (+) (l 2 1 ) δ(+) (l 2 2 ) Twistor-motivated Integration Measure Cacahazo, Svrček & Witten (2004) Use the δ (4) integral to reduce just to a single loop momentum variable Z d 4 l δ (+) (l 2 ) f( l, l]) = ˆl = l µ γ µ = l [l t λ [λ, t R Z 0 t dt Z λ dλ [λ dλ] t α f (α) ( λ, λ]) λ]= λ P. Mastrolia, UniZh - Twistors & Unitarity, 22

23 Cutting a Bubble I 2 = = Z d 4 l δ (+) (l 2 ) δ (+) ((l K) 2 ) = Z 0 t dt Z λ dλ [λ dλ] δ (+) ((l K) 2 ) Since δ (+) ((l K) 2 ) = δ (+) (K 2 2l K) = δ (+) (K 2 l K l]) = «1 t λ K λ] δ(+) K2 λ K λ] after the t-integration I 2 = Z K 2 λ dλ [λ dλ] λ K λ] 2 Derivative Form (via Integration-by-Parts) [λ dλ] λ K λ] 2 = [dλ λ]] [η λ] λ K η] λ K λ], η : η2 = 0 P. Mastrolia, UniZh - Twistors & Unitarity, 23

24 Holomorphic Anomaly Cachazo, Svrček, Witten (2004) The last integration is carried out along the contour λ] = λ, and that give rise to a delta-function like contribution, similar to z 1 z a = 2πδ(z a) In this case, Cauchy Residue Theorem would imply the contribution at the pole λ = K η] ( λ] = K η ) With the final result I 2 = lim λ K η] λ K η] K 2 [η λ] λ K η] λ K λ] «= 1. The discontinuity of a bubble is rational!!! P. Mastrolia, UniZh - Twistors & Unitarity, 24

25 Cutting a 3m-Triangle I 3 = K 2 K 1 K 3 = Z d 4 l δ (+) (l 2 ) δ(+) ((l K 1 ) 2 ) (l + K 3 ) 2 after the t-integration I 3 = Z 1 λ dλ [λ dλ] λ K 1 λ] λ Q λ] with ˆQ = (K 2 3 /K2 1 ) ˆK 1 + ˆK 3 Feynman parameter I 3 = Z 1 0 dz Z 1 λ dλ [λ dλ] λ (1 z)k 1 + zq λ] = 2 Z dz [(1 z)k 1 + zq] 2 P. Mastrolia, UniZh - Twistors & Unitarity, 25

26 Finally I 3 = 1 3m ln «2a + b 3m 2a + b + 3m with a = (Q K 1 ) 2, b = 2((K 1 Q) K 2 1 ) and 3m = (K 2 1 )2 2K 2 1 K2 2 2K2 1 K2 3 (K2 2 K2 3 ) The discontinuity a 3m-Triangle is a ln(irrational argument)!!! The double cut detect box-coefficient as well. One can show that the discontinuity of a 1m-,2m-,3m-box is a ln(rational argument). When it appears, it is flagged and forgotten (later used as crosscheck), since we prefer to compute it from 4-ple cut. P. Mastrolia, UniZh - Twistors & Unitarity, 26

27 Cut-Constructible A g A g = 2 1 n k = X i c 4,i + X j c 3,j + X k c 2,k c 2,i = " 2 1 n k # rational c 3,i = " 2 1 n k # ln(irr.) c 4,i = 2 1 n k = " 2 1 n k # ln(rat.) P. Mastrolia, UniZh - Twistors & Unitarity, 27

28 General Algorithm Sew Trees Remove l 1 or l 2 dµ CSW = tdt λ dλ [λ dλ] Canonical Decomposition G( λ, λ]) λ P a] α λ P λ] β G( λ, λ]) λ P Q λ α λ P λ] β F ( λ ) λ P a] α λ P λ][λ a] F ( λ ) λ P Q λ α λ P λ] λ Q λ] Derivative Form Feynman Param. Higher Poles Feynman Int. Holom. Anomaly Cauhchy Residue Th. Bubble coeff. 3m-Triangle coeff. P. Mastrolia, UniZh - Twistors & Unitarity, 28

29 6-gluon Amplitude in QCD Amplitude N = 4 N = 1 N = 0 CC N = 0 rat ( ) BDDK 94 BDDK 94 BDDK 94 BDK 05, KF 05 ( ) BDDK 94 BDDK 94 BBST 04 ( ) BDDK 94 BDDK 94 BBST 04 ( + ++) BDDK 94 BBDD 04 BBDI 05, BFM ( + ++) BDDK 94 BBCF 05, BBDP 05 BFM ( + + +) BDDK 94 BBCF 05, BBDP 05 BFM Quadruple Cuts Bidder, Bjerrum-Bohr, Dunbar & Perkins (2005) Double Cuts & P. Mastrolia, UniZh - Twistors & Unitarity, 29

30 Analytic Tools for One-Loop Amplitudes Bern, Dixon, Dunbar & Kosower (Jurassical) Brandhuber, Mc Namara, Spence, & Travaglini (2004,2005) Unitarity-based methods Quigley & Rozali (2004) Britto, Buchbinder, Cachazo, Svrček & Witten (2004, 2005) Britto, Feng & PM (2006) terms with discontinuities 4-dimension cuts rational terms D-dimension cuts Recurrence Relations Bern, Dixon & Kosower (2005) Bern, Bjerrum-Bohr, Dunbar & Ita (2005) Forde & Kosower (2005) = When possible, are more elegant! terms with discontinuities within the same class of polylogarithm rational terms Correction Terms driven by the polylog structure P. Mastrolia, UniZh - Twistors & Unitarity, 30

31 On-shell scattering amplitudes seem to have properties not evident in the expansion of off-shell Green-functions in terms of Feynman diagrams (later taken on-shell) Trees CSW as a formalism dealing with on-shell objects: MHV amplitudes as vertices, and scalar propagators BCFW recurrence relation for amplitudes, from general properties of complex functions, without any specific process-dependence Loops Unitarity & Cut-Constructibility allow to transfer knowledge from tree-amplitudes BBCF+BFM developed a systematic technique for finite cuts in any theory: CSW integration measure Canonical Decomposition: algebraic spinor reduction to Box, Bubble and 3m-Triangle functions One-Feynman Parameter for the residual integration (purely) Algebraic treatment of higher poles (characteristic of non-sym) gg gggg: numerically known (EGiZ), and its CC-part analytically known multileg NLO QCD: squeezing out of the bottleneck! P. Mastrolia, UniZh - Twistors & Unitarity, 31

32 P. Mastrolia, UniZh - Twistors & Unitarity, 32