Recursive equations for arbitrary scattering processes Costas G. - PowerPoint PPT Presentation

✬ ✩ � Let n external particles with momenta p µ i , i = 1 . . . , n , and define the momentum P µ � P µ = p µ I ⊂ { 1 , . . . , n } i , i ∈ I � the binary vector � m = ( m 1 , . . . , m n ), where its components take the values 0 or 1 : n � P µ = m i p µ i . i =1 � Moreover this binary vector can be uniquely represented by the integer n � 0 ≤ m ≤ 2 n − 1 2 i − 1 m i , m = i =1 � Replace b µ ( P ) → b µ ( m ) . ✫ ✪ HEP - NCSR Democritos

✬ ✩ ♣ Convenient ordering of integers in binary representation ⇒ level l , defined by n � l = m i . i =1 ✫ ✪ HEP - NCSR Democritos

✬ ✩ ♣ Convenient ordering of integers in binary representation ⇒ level l , defined by n � l = m i . i =1 ♣ external momenta are of level 1 ✫ ✪ HEP - NCSR Democritos

✬ ✩ ♣ Convenient ordering of integers in binary representation ⇒ level l , defined by n � l = m i . i =1 ♣ external momenta are of level 1 ♣ the total amplitude corresponds to the unique level n integer 2 n − 1 A = b (1) · b (2 n − 2) ✫ ✪ HEP - NCSR Democritos

✬ ✩ ♣ Convenient ordering of integers in binary representation ⇒ level l , defined by n � l = m i . i =1 ♣ external momenta are of level 1 ♣ the total amplitude corresponds to the unique level n integer 2 n − 1 A = b (1) · b (2 n − 2) This ordering dictates the natural path of the computation : start- ing with level-1 sub-amplitudes, we compute the level-2 ones using the Dyson-Schwinger equations and so on up to level n − 1 ✫ ✪ HEP - NCSR Democritos

✬ ✩ The solution e − (1) e + (2) → e − (4) ¯ ¯ ν e (8) u (16) d (32) 1 10 33 2 -2 8 1 1 12 33 4 -2 8 1 1 48 34 16 -3 32 4 2 26 -4 10 33 16 -3 . . . 2 62 -2 10 33 52 -1 2 62 -2 12 33 50 -1 2 62 -2 58 31 4 -2 2 62 -2 58 32 4 -2 2 62 -2 60 31 2 -2 2 62 -2 60 32 2 -2 ✫ ✪ HEP - NCSR Democritos

✬ ✩ + e 2 + W 10 _ ν 8 ✫ ✪ HEP - NCSR Democritos

✬ ✩ • Dirac algebra simplification: 2-dim vs 4-dim and chiral representation, including m f � = 0. • The sign factor: ǫ ( P 1 , P 2 ) → ǫ ( m 1 , m 2 ) we define ǫ ( m 1 , m 2 ) = ( − 1) χ ( m 1 ,m 2 )   i − 1 2 � �   χ ( m 1 , m 2 ) = m 1 i ˆ m 2 j ˆ i = n j =1 where hatted components are set to 0 if the corresponding external particle is a boson. • Full EWK theory, both Unitary and Feynman gauges. A. Denner, Fortsch. Phys. 41, 307 (1993). ✫ ✪ HEP - NCSR Democritos

✬ ✩ HELAC • Construction of the skeleton solution of the Dyson-Schwinger equations. At this stage only integer arithmetic is performed. This is part of the initialization phase. • Dressing-up the skeleton with momenta, provided by PHEGAS and wave functions, propagators, n -point functions in general. • Unitary and Feynman gauges implemented. Due to multi-precision arithmetic,tests of gauge invariance can be extended to arbitrary precision. • All fermions masses can be non-zero. • All Electroweak and QCD vertices are implemented, including Higgs and would-be Goldstone bosons. ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • Ordinary approach SU ( N )-type � A a 1 ...a n = Tr ( T a σ 1 . . . T a σn ) A ( σ 1 . . . σ n ) ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • Ordinary approach SU ( N )-type � A a 1 ...a n = Tr ( T a σ 1 . . . T a σn ) A ( σ 1 . . . σ n ) � a σ ′ a σ ′ Tr ( T a σ 1 . . . T a σn ) Tr ( T 1 . . . T n ) C ij = ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • Ordinary approach SU ( N )-type � A a 1 ...a n = Tr ( T a σ 1 . . . T a σn ) A ( σ 1 . . . σ n ) � a σ ′ a σ ′ Tr ( T a σ 1 . . . T a σn ) Tr ( T 1 . . . T n ) C ij = Quarks and gluons treated differently ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • New approach U ( N )-type Each color-configuration amplitude is proportional to D i = δ 1 ,σ i (1) δ 2 ,σ i (2) . . . δ n,σ i ( n ) where σ i represents the i -th permutation of the set 1 , 2 , . . . , n . ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • New approach U ( N )-type Each color-configuration amplitude is proportional to D i = δ 1 ,σ i (1) δ 2 ,σ i (2) . . . δ n,σ i ( n ) where σ i represents the i -th permutation of the set 1 , 2 , . . . , n . ⋆ quarks 1 . . . n ⋆ antiquarks σ i (1 . . . n ) and ⋆ gluons = q ¯ q ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • New approach U ( N )-type Each color-configuration amplitude is proportional to D i = δ 1 ,σ i (1) δ 2 ,σ i (2) . . . δ n,σ i ( n ) where σ i represents the i -th permutation of the set 1 , 2 , . . . , n . ⋆ quarks 1 . . . n ⋆ antiquarks σ i (1 . . . n ) and ⋆ gluons = q ¯ q � D i D j = N α C ij = α = � σ 1 , σ 2 � c , ✫ ✪ HEP - NCSR Democritos

✬ ✩ Colour Configuration - EWK ⊕ QCD • New approach U ( N )-type Each color-configuration amplitude is proportional to D i = δ 1 ,σ i (1) δ 2 ,σ i (2) . . . δ n,σ i ( n ) where σ i represents the i -th permutation of the set 1 , 2 , . . . , n . ⋆ quarks 1 . . . n ⋆ antiquarks σ i (1 . . . n ) and ⋆ gluons = q ¯ q � D i D j = N α C ij = α = � σ 1 , σ 2 � c , ♠ exact color treatment ⇒ low color charge Problem: number of colour connection configurations: ∼ n ! where n is the number of ✫ ✪ q pairs. ⇒ Monte-Carlo over continuous colour-space. gluons or q ¯ HEP - NCSR Democritos

✬ ✩ g g (i, σ i) σ i) δ σ (j, i j (j, σ j) g � EF = − i f abc t a AB t b CD t c 4( δ AD δ CF δ EB − δ AF δ CB δ ED ) δ 1 σ 2 δ 2 σ 3 δ 3 σ 1 ✫ ✪ HEP - NCSR Democritos

✬ ✩ _ q σ i) (0, g σ i) (i, (i,0) q � CD = 1 1 t a AB t b 2( δ AD δ CB − δ AB δ AC ) N c 1 √ 2 ✫ ✪ HEP - NCSR Democritos

✬ ✩ _ q (0, σ i) g δ i σ i (0,0) (i,0) q � CD = 1 1 t a AB t b 2( δ AD δ CB − δ AB δ AC ) N c 1 √ 2 N c ✫ ✪ HEP - NCSR Democritos

✬ ✩ (3,σ ) 3 (1,σ ) (2,σ ) 1 2 (4,σ ) 4 δ 1 σ 3 δ 3 σ 2 δ 2 σ 4 δ 4 σ 1 2 g 12 g 34 − g 13 g 24 − g 14 g 23 ✫ ✪ HEP - NCSR Democritos

✬ ✩ 1 The N c expansion � D i D j = N α C ij = c , α = � σ 1 , σ 2 � The leading term σ 1 = σ 2 , N n c The subleading terms: how many δ ’s survive after contraction, which in Combinatorial Analysis are known to be the Stirling numbers ( − ) n − m S ( m ) n where n is the number of ‘objects’ and m the number of ‘surviving’ δ ’s, called cycles. N c -term is related to n ( n − 1) 1 For instance the permutations ! 2 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color � Tr ( t a 1 . . . t a n ) A ( { p i } n M ( { p i } n 1 , { ε i } n 1 , { a i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color � Tr ( t a 1 . . . t a n ) A ( { p i } n M ( { p i } n 1 , { ε i } n 1 , { a i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � M ( { p i } n 1 , { ε i } n 1 , { I i , J i } n δ I 1 ,P ( J 1 ) . . . δ I n ,P ( J n ) A ( { p i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color � Tr ( t a 1 . . . t a n ) A ( { p i } n M ( { p i } n 1 , { ε i } n 1 , { a i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � M ( { p i } n 1 , { ε i } n 1 , { I i , J i } n δ I 1 ,P ( J 1 ) . . . δ I n ,P ( J n ) A ( { p i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � 1 ) | 2 = g 2 n − 4 � � A i C ij A ∗ |M ( { p i } n 1 , { ε i } n 1 , { a i } n j { a i } n 1 { ε i } n ε ij 1 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color � Tr ( t a 1 . . . t a n ) A ( { p i } n M ( { p i } n 1 , { ε i } n 1 , { a i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � M ( { p i } n 1 , { ε i } n 1 , { I i , J i } n δ I 1 ,P ( J 1 ) . . . δ I n ,P ( J n ) A ( { p i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � 1 ) | 2 = g 2 n − 4 � � A i C ij A ∗ |M ( { p i } n 1 , { ε i } n 1 , { a i } n j { a i } n 1 { ε i } n ε ij 1 � ∼ n ! P (2 ,...,n ) ✫ ✪ HEP - NCSR Democritos

✬ ✩ Summation/Integration over color � Tr ( t a 1 . . . t a n ) A ( { p i } n M ( { p i } n 1 , { ε i } n 1 , { a i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � M ( { p i } n 1 , { ε i } n 1 , { I i , J i } n δ I 1 ,P ( J 1 ) . . . δ I n ,P ( J n ) A ( { p i } n 1 , { ε i } n 1 ) ∼ 1 ) P (2 ,...,n ) � 1 ) | 2 = g 2 n − 4 � � A i C ij A ∗ |M ( { p i } n 1 , { ε i } n 1 , { a i } n j { a i } n 1 { ε i } n ε ij 1 � ∼ n ! P (2 ,...,n ) � ∼ 3 n × 3 n ✫ ✪ { I i ,J i } n 1 HEP - NCSR Democritos

✬ ✩ ✫ ✪ HEP - NCSR Democritos

✬ ✩ The Dyson-Schwinger recursion equation for gluon in a general way can be written as follows: n � [ A µ ( P ); ( A, B )] = [ δ ( P − p i ) A µ ( p i ); ( A, B ) i ]+ i =1 � [ ( ig ) Π µ ρ V ρνλ ( P, p 1 , p 2 ) A ν ( p 1 ) A λ ( p 2 ) σ ( p 1 , p 2 ); ( A, B ) = ( C, D ) 1 ⊗ ( E, F ) 2 ] � [ ( g 2 ) Π µ σ G σνλρ ( P, p 1 , p 2 , p 3 ) A ν ( p 1 ) A λ ( p 2 ) A ρ ( p 3 ) σ ( p 1 , p 2 + p 3 ); − ( A, B ) = ( C, D ) 1 ⊗ ( E, F ) 2 ⊗ ( G, H ) 3 ] � ψ ( p 1 ) γ ν ψ ( p 2 ) σ ( p 1 , p 2 ); ( A, B ) = (0 , D ) 1 ⊗ ( C, 0) 2 ] ¯ [ ( ig ) Π µ + ν P = p 1 + p 2 ✫ ✪ where A, B, C, D, E, F, G, H = 1 , 2 , 3. HEP - NCSR Democritos

✬ ✩ L = − 1 4 F a µν F µνa , F a µν = ∂ µ A a ν − ∂ ν A a µ + gf abc A b µ A c ν L = − 1 µν H µνa + 1 2 H a 4 H a µν F µνa . ✫ ✪ HEP - NCSR Democritos

✬ ✩ n � [ A µ ( P ); ( A, B )] = [ δ ( P − p i ) A µ ( p i ); ( A, B ) i ]+ i =1 [ ( ig ) Π µ ρ V ρνλ ( P, p 1 , p 2 ) A ν ( p 1 ) A λ ( p 2 ) σ ( p 1 , p 2 ); ( A, B ) = ( C, D ) 1 ⊗ ( E, F ) 2 ] + [ ( ig ) Π µ σ ( g σλ g νρ − g νλ g σρ ) A ν ( p 1 ) H λρ ( p 2 ) σ ( p 1 , p 2 ); ( A, B ) = ( C, D ) 1 ⊗ ( E, F ) 2 ] + ψ ( p 1 ) γ ν ψ ( p 2 ) σ ( p 1 , p 2 ); ( A, B ) = (0 , D ) 1 ⊗ ( C, 0) 2 ] ¯ [ ( ig ) Π µ ν and � [ H µν ( P ); ( A, B )] = [ ( ig ) ( g µλ g νρ − g νλ g µρ ) A λ ( p 1 ) A ρ ( p 2 ) σ ( p 1 , p 2 ); P = p 1 + p 2 ( A, B ) = ( C, D ) 1 ⊗ ( E, F ) 2 ] . ✫ ✪ HEP - NCSR Democritos

✬ ✩ ✫ ✪ HEP - NCSR Democritos

✬ ✩ ( A, B ) = ( C, 0) ⊗ (0 , D ) = ( C, D ) w =1 , if C � = D. ✫ ✪ HEP - NCSR Democritos

✬ ✩ ( A, B ) = ( C, 0) ⊗ (0 , D ) = ( C, D ) w =1 , if C � = D. ( A, B ) = ( C, 0) ⊗ (0 , D ) = (1 , 1) w 1 ⊕ (2 , 2) w 2 ⊕ (3 , 3) w 3 , if C = D. ✫ ✪ HEP - NCSR Democritos

✬ ✩ ( A, B ) = ( C, 0) ⊗ (0 , D ) = ( C, D ) w =1 , if C � = D. ( A, B ) = ( C, 0) ⊗ (0 , D ) = (1 , 1) w 1 ⊕ (2 , 2) w 2 ⊕ (3 , 3) w 3 , if C = D. (1 , 0) ⊗ (0 , 1) = (1 , 1) 2 / 3 ⊕ (2 , 2) − 1 / 3 ⊕ (3 , 3) − 1 / 3 ✫ ✪ HEP - NCSR Democritos

✬ ✩ � � 2 n q n q − 1 n q − A − B � � � n q ! N CC = δ ( n q = A + B + C ) A ! B ! C ! A =0 B =0 C =0 ✫ ✪ HEP - NCSR Democritos

✬ ✩ � � 2 n q n q − 1 n q − A − B � � � n q ! N CC = δ ( n q = A + B + C ) A ! B ! C ! A =0 B =0 C =0 NALL NF Process NCC CC (%) CC gg → 2 g 6561 639 59.1 gg → 3 g 59049 4653 68.4 gg → 4 g 531441 35169 77.4 gg → 5 g 4782969 272835 85.0 gg → 6 g 43046721 2157759 90.4 gg → 7 g 387420489 17319837 94.0 gg → 8 g 3486784401 140668065 96.4 ✫ ✪ HEP - NCSR Democritos

✬ ✩ NALL NF Process NCC CC (%) CC gg → u ¯ u 729 93 93.5 gg → gu ¯ u 6561 639 91.6 gg → 2 gu ¯ u 59049 4653 92.6 gg → 3 gu ¯ u 531441 35169 94.6 gg → 4 gu ¯ u 4782969 272835 96.4 gg → 5 gu ¯ u 43046721 2157759 97.8 gg → 6 gu ¯ u 387420489 17319837 98.6 gg → c ¯ cc ¯ c 6561 639 99.1 gg → gc ¯ cc ¯ c 59049 4653 98.8 gg → 2 gc ¯ cc ¯ c 531441 35169 99.0 gg → 3 gc ¯ cc ¯ c 4782969 272835 99.3 gg → 4 gc ¯ cc ¯ c 43046721 2157759 99.6 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Process σ MC ± ε (nb) ε (%) (0.53185 ± 0.01149) × 10 − 2 gg → 7 g 2.1 (0.33330 ± 0.00804) × 10 − 3 gg → 8 g 2.4 (0.17325 ± 0.00838) × 10 − 4 gg → 9 g 4.8 (0.38044 ± 0.01096) × 10 − 3 gg → 5 gu ¯ u 2.8 (0.95109 ± 0.02456) × 10 − 5 gg → 3 gc ¯ cc ¯ c 2.6 (0.81400 ± 0.02583) × 10 − 6 gg → 4 gc ¯ cc ¯ c 3.2 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Process σ MC ± ε (nb) ε (%) (0.18948 ± 0.00344) × 10 − 3 gg → Zu ¯ ugg 1.8 gg → W + ¯ (0.62704 ± 0.01458) × 10 − 3 udgg 2.3 (0.16217 ± 0.00420) × 10 − 6 gg → ZZu ¯ ugg 2.6 gg → W + W − u ¯ (0.27526 ± 0.00752) × 10 − 5 ugg 2.7 d ¯ (0.38811 ± 0.00569) × 10 − 5 d → Zu ¯ ugg 1.5 d → W + ¯ (0.18765 ± 0.00453) × 10 − 5 d ¯ csgg 2.4 d ¯ (0.99763 ± 0.02976) × 10 − 7 d → ZZgggg 2.9 d → W + W − gggg (0.52355 ± 0.01509) × 10 − 6 d ¯ 2.9 ✫ ✪ HEP - NCSR Democritos

✬ ✩ d σ /dM jj 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0 100 200 300 400 500 600 700 800 M jj Figure 1: Invariant mass distribution of 2 gluons in the gg → 5 g process. Solid line crosses denote SPHEL case whereas dashed, the ✫ ✪ Monte Carlo one. HEP - NCSR Democritos

✬ ✩ • SPHEL approximation based on MHV amplitudes � 1 ) | 2 = 2 g 2 n − 4 N n − 2 |M ( { p i } n 1 , { ε i } n 1 , { a i } n ( N 2 c − 1) × c a,ε 2 n − 2( n + 1) � � 1 ( p i · p j ) 4 ( p 1 · p 2 )( p 2 · p 3 ) . . . ( p n · p 1 ) , n ( n − 1) 1 ≤ i ≤ j ≤ n P (2 ,...,n ) ✫ ✪ HEP - NCSR Democritos

✬ ✩ Process t SPHEL t MC t MC /t SPHEL 0 . 372 × 10 − 3 0 . 519 × 10 − 1 gg → 2 g 139.52 0 . 776 × 10 − 3 0 . 135 × 10 0 gg → 3 g 173.97 0 . 252 × 10 − 2 0 . 364 × 10 0 gg → 4 g 144.44 0 . 122 × 10 − 1 0 . 143 × 10 1 gg → 5 g 117.21 0 . 806 × 10 − 1 0 . 497 × 10 1 gg → 6 g 61.66 0 . 639 × 10 0 0 . 133 × 10 2 gg → 7 g 20.81 0 . 569 × 10 1 0 . 334 × 10 2 gg → 8 g 5.87 0 . 567 × 10 2 0 . 923 × 10 2 gg → 9 g 1.63 0 . 620 × 10 3 0 . 267 × 10 3 gg → 10 g 0.43 ✫ ✪ HEP - NCSR Democritos

✬ ✩ The high-colour processes The idea is to replace colour summation with integration and then follow a MC approach � � 8 � √ iB − 1 G µ ǫ µ z iA z ∗ G a ( P i ) η a ( z ) = AB ( P i ) = 6 3 δ AB λ ( P i ) a =1 √ ψ A ( P i ) = 3 u ( P i ) z iA √ ¯ u ( P i ) z ∗ ψ A ( P i ) = 3 ¯ iA In such a representation the amplitude can be seen � M ( z 1 , z 2 , . . . ) = z 1 · z σ ( i ) 1 z 2 · z σ ( i ) 2 . . . A i ✫ ✪ HEP - NCSR Democritos

✬ ✩ and the MC is over � � � � 3 3 � � dz i dz ∗ z i z ∗ [ dz ] ≡ δ ( i − 1) i i =1 i =1 where � √ � � � � � √ B − 1 D − 1 z A z ∗ z C z ∗ [ dz ] G AB G CD = [ dz ] 6 3 δ AB 6 3 δ AB � � = 1 δ AD δ CB − 1 3 δ AB δ CD 2 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Multi-jet processes Beyond any colour treatment a summation over different flavours is also needed. Up to now the most straightforward way was to count the distinct processes and then multiply with a multiplicity factor, i.e. process Flavour gg → ggg 1 q ¯ q → ggg 8 qg → qgg 8 qg → qgg 8 gg → q ¯ qg 5 q → q ¯ q ¯ qg 8 q ¯ q → r ¯ rg 32 qq → qqg 8 q ¯ r → q ¯ rg 24 qr → qrg 24 qg → qq ¯ q 8 qg → qr ¯ r 32 gq → qq ¯ q 8 ✫ gq → qr ¯ ✪ r 32 HEP - NCSR Democritos

✬ ✩ initial-state type distinct processes multiplicity factor A ( gg ) C 1 ( n ) χ ( n 0 , n 1 , . . . , n f ; f ) B ( q ¯ q ) C 2 ( n ) χ ( n 0 , n 2 , . . . , n f ; f − 1) C 2 ( n − 1) χ ( n 0 , n 2 , . . . , n f ; f − 1) C ( gq and qg ) D ( qq ) C 2 ( n − 2) χ ( n 0 , n 2 , . . . , n f ; f − 1) ( qq ′ and q ¯ q ′ ) E C 3 ( n − 2) χ ( n 0 , n 3 , . . . , n f ; f − 2) In order to clarify what we mean we consider the example of the type A initial state. Each distinct process is defined by an array ( n 0 , n 1 , . . . , n f ). For instance, in the case of four-jet production we have (4,0,0,0,0,0) gg → gggg (2,1,0,0,0,0) gg → ggq ¯ q (0,2,0,0,0,0) gg → q ¯ qq ¯ q (0,1,1,0,0,0) gg → q ¯ qr ¯ r ✫ ✪ HEP - NCSR Democritos

✬ ✩ � Θ( n 1 ≥ n 2 ≥ . . . ≥ n f ) C 1 ( n ) = n 0+2 n 1+ ... +2 nf = n � Θ( n 2 ≥ n 3 ≥ . . . ≥ n f ) C 2 ( n ) = n 0+2 n 1+ ... +2 nf = n and � C 3 ( n ) = Θ( n 3 ≥ n 4 ≥ . . . ≥ n f ) n 0+2 n 1+ ... +2 nf = n A distinct process, given by the array ( n 0 , n 1 , . . . , n f ) has a multiplicity factor : χ ( n 0 , n 1 , . . . , n f ; f ) = n f ( n f − 1) ... ( n f − j + 1) /j ! f � j = f if n i � = 0 i =1 f − 1 � j = f − 1 if n i � = 0 i =1 . . . j = 1 if n 1 � = 0 j = 0 otherwise ✫ ✪ HEP - NCSR Democritos

✬ ✩ Now we can think of a flavour-MC, so the wave function is multiplied by an � � f = � N f ( f 1 , f 2 , ... ) such that N f -dimensional array representing flavour , � f i f j = δ ij with a weight proportional to the relevant pdf for initial state flavours In that case a process like gg → ggq ¯ qq ¯ q will actually represent a plethora of processes. The number of distinct processes is now given by 9 k + 3 if n = 2 k and 9 k + 7 if n = 2 k + 1 # of jets 2 3 4 5 6 7 8 9 10 # of D-processes 12 16 21 24 30 34 39 43 48 # of dist.processes 10 14 28 36 64 78 130 154 241 total # of processes 126 206 621 861 1862 2326 4342 5142 8641 ✫ ✪ HEP - NCSR Democritos

✬ ✩ Multi-jet rates θ ij > 30 o | η i | < 3 p T i > 60 GeV, # jets 3 4 5 6 7 8 2.97 × 10 − 2 2.21 × 10 − 3 2.12 × 10 − 4 σ ( nb ) 91.41 6.54 0.458 % Gluon 45.7 39.2 35.7 35.1 33.8 26.6 A new code ⇒ JetI • anybody to tell us how many Feynman graphs in gg → 8 g ? • or gg → 2 g 3 u 3¯ u ? ✫ ✪ HEP - NCSR Democritos

✬ ✩ • Feynman graphs in gg → 8 g 10,525,900 !! • or gg → 2 g 3 u 3¯ u 946,050! ✫ ✪ HEP - NCSR Democritos

✬ ✩ PHEGAS • Phase space n �� � δ 3 �� � � d 3 p i d Φ n = (2 π ) 4 − 3 n E i − w δ p i � 2 E i i =1 • RAMBO , VEGAS -based nice but completely inefficient! dσ n = FLUX × |M 2 → n | 2 d Φ n need appropriate mappings of peaking structures, plus optimization! • Efficiency ⇒ to a large number of generators, each one for a specific class of processes. ✫ ✪ HEP - NCSR Democritos

✬ ✩ Multichannel approach � f ( � � x ) I = f ( � x ) dµ ( � x ) = x ) p ( � x ) dµ ( � x ) p ( � M ch M ch � � p ( � x ) = α i p i ( � x ) α i = 1 i =1 i =1 �� f ( � � � f ( � � � 2 x ) x ) E 2 N → − I 2 I → p ( � x ) p ( � x ) ⋆ Optimize α i ⇒ Minimize E ⋆ R.Kleiss and R.Pittau, Comput. Phys. Commun. 83, 141 (1994). ✫ ✪ HEP - NCSR Democritos

✬ ✩ New Dyson-Schwinger equations: subamplitude is a combination of several peaking structures! problem unsolved? QCD antennas P.D.Draggiotis, A.van Hameren and R.Kleiss, hep-ph/0004047. ✫ ✪ HEP - NCSR Democritos

✬ ✩ New Dyson-Schwinger equations: subamplitude is a combination of several peaking structures! problem unsolved? QCD antennas P.D.Draggiotis, A.van Hameren and R.Kleiss, hep-ph/0004047. Old Feynman graphs: exhibit single peaking structure! problem solved ✫ ✪ HEP - NCSR Democritos