Amplitudes in the LHC era Scattering Equations in Multi-Regge Kinematics Zhengwen Liu Center for Cosmology, Particle Physics and Phenomenology Institut de Recherche en Math´ ematique et Physique Base on 1811.xxxxx (with C. Duhr) and 1811.yyyyy Galileo Galilei Institute, Firenze November 15, 2018
Outline Introduction to scattering equations Multi-Regge kinematics (MRK) • Scattering equations in MRK • Gauge theory amplitudes in MRK • Gravity amplitudes in MRK Quasi Multi-Regge kinematics • Scattering equations in QMRK • Generalized Impact factors and Lipatov vertices Summary & Outlook Zhengwen Liu (UCLouvain) Scattering Equations in MRK 1/25
Scattering equations Let us start with a rational map from the moduli space M 0 ,n to the space of momenta for n massless particles scattering: 1 � σ 2 k µ dz ω µ ( z ) a = ω µ ( z ) 2 πi σ 4 σ 3 | z − σ a | = ǫ σ 5 n k µ P µ ( z ) � ω µ ( z ) = a = � n σ 1 z − σ a a =1 ( z − σ a ) a =1 Zhengwen Liu (UCLouvain) Scattering Equations in MRK 2/25
Scattering equations Let us start with a rational map from the moduli space M 0 ,n to the space of momenta for n massless particles scattering: 1 � σ 2 k µ dz ω µ ( z ) a = ω µ ( z ) 2 πi σ 4 σ 3 | z − σ a | = ǫ σ 5 n k µ P µ ( z ) � ω µ ( z ) = a = � n σ 1 z − σ a a =1 ( z − σ a ) a =1 ω µ ( z ) ω µ ( z ) = 0 ω µ ( z ) maps the M 0 ,n to the null cone of momenta 2 k a · k b 1 � dz ω ( z ) 2 = � 0 = , a = 1 , 2 , . . . , n 2 πi σ a − σ b b � = a | z − σ a | = ǫ which are named as the scattering equations . [Cachazo, He & Yuan, 1306.2962, 1306.6575] Zhengwen Liu (UCLouvain) Scattering Equations in MRK 2/25
Scattering equations The scattering equations: M 0 ,n → K n k a · k b � f a = = 0 , a = 1 , 2 , . . . , n σ a − σ b b � = a • This system has an SL (2 , C ) redundancy, only ( n − 3) out of n equations are independent • Equivalent to a system of homogeneous polynomial equations [Dolan & Goddard, 1402.7374] • The total number of independent solutions is ( n − 3)! Zhengwen Liu (UCLouvain) Scattering Equations in MRK 3/25
Scattering equations The scattering equations: M 0 ,n → K n k a · k b � f a = = 0 , a = 1 , 2 , . . . , n σ a − σ b b � = a • This system has an SL (2 , C ) redundancy, only ( n − 3) out of n equations are independent • Equivalent to a system of homogeneous polynomial equations [Dolan & Goddard, 1402.7374] • The total number of independent solutions is ( n − 3)! • The scattering equations have appeared before in different contexts, e.g., ◮ D. Fairlie and D. Roberts (1972): amplitudes in dual models ◮ D. Gross and P. Mende (1988): the high energy behavior of string scattering ◮ E. Witten (2004): twistor string • Cachazo, He and Yuan rediscovered them in the context of field theory amplitudes [CHY, 1306.2962, 1306.6575, 1307.2199, 1309.0885] Zhengwen Liu (UCLouvain) Scattering Equations in MRK 3/25
Scattering equations in 4d In 4 dimensions, the null map vector P µ ( z ) can be rewritten in spinor variables as follows: � n n � b ˜ λ α λ ˙ α � � = λ α ( z )˜ P α ˙ α ( z ) ≡ b λ ˙ α ( z ) ( z − σ a ) z − σ b a =1 b =1 deg λ ( z ) = d ∈ { 1 , . . . , n − 3 } , deg ˜ λ ( z ) = ˜ d , d + ˜ d = n − 2 . A simple construction is t i ˜ t I λ α λ ˙ α � � � � λ α ( z ) = I λ ˙ α ( z ) = i ( z − σ a ) , ( z − σ a ) z − σ I z − σ i a ∈ N I ∈ N a ∈ P i ∈ P We divide { 1 , . . . , n } into two subsets N and P , | N | = k = d +1 , | P | = n − k = ˜ d +1 . Zhengwen Liu (UCLouvain) Scattering Equations in MRK 4/25
Scattering equations in 4d In 4 dimensions, the null map vector P µ ( z ) can be rewritten in spinor variables as follows: � n n � b ˜ λ α λ ˙ α � � = λ α ( z )˜ P α ˙ α ( z ) ≡ b λ ˙ α ( z ) ( z − σ a ) z − σ b a =1 b =1 deg λ ( z ) = d ∈ { 1 , . . . , n − 3 } , deg ˜ λ ( z ) = ˜ d , d + ˜ d = n − 2 . A simple construction is t i ˜ t I λ α λ ˙ α � � � � λ α ( z ) = I λ ˙ α ( z ) = i ( z − σ a ) , ( z − σ a ) z − σ I z − σ i a ∈ N I ∈ N a ∈ P i ∈ P We divide { 1 , . . . , n } into two subsets N and P , | N | = k = d +1 , | P | = n − k = ˜ d +1 . Then the two spinor maps leads to t I t i t i t I ¯ I = ˜ � ˜ � E ˙ α λ ˙ α λ ˙ α E α i = λ α λ α I − i = 0 , I ∈ N ; i − I = 0 , i ∈ P σ I − σ i σ i − σ I i ∈ P I ∈ N Zhengwen Liu (UCLouvain) Scattering Equations in MRK 4/25
4D scattering equations Geyer-Lipstein-Mason (GLM) scattering equations: t I t i t i t I � � E ˙ ¯ I = ˜ α λ ˙ α λ ˙ ˜ α E α i = λ α λ α I − i = 0 , I ∈ N ; i − I = 0 , i ∈ P σ I − σ i σ i − σ I i ∈ P I ∈ N • These equations are originally derived from the four-dimensional ambitwistor string model, based on them tree superamplitudes in N =4 SYM and N =8 supergravity are obtained. [Geyer, Lipstein & Mason, 1404.6219] Zhengwen Liu (UCLouvain) Scattering Equations in MRK 5/25
Scattering equations in 4d Geyer-Lipstein-Mason (GLM) scattering equations: t I t i t i t I I = ˜ � ˜ � E ˙ ¯ α λ ˙ α λ ˙ α E α i = λ α λ α I − i = 0 , I ∈ N ; i − I = 0 , i ∈ P σ I − σ i σ i − σ I i ∈ P I ∈ N • These equations are originally derived from the four-dimensional ambitwistor string model, based on them tree superamplitudes in N =4 SYM and N =8 supergravity are obtained. [Geyer, Lipstein & Mason, 1404.6219] • Equivalent polynomial versions [Roiban, Spradlin & Volovich, hep-th/0403190; He, ZL & Wu, 1604.02834] n d = k − 1 � � t a σ m a ˜ λ ˙ α λ α ρ α m σ m a = 0 , m = 0 , 1 , . . . , d ; a − t a a = 0 a =1 m =0 • In 4d, the scattering eqs fall into “helicity sector” are characterized by k ∈ { 2 , . . . , n − 2 } � n − 3 � • In sector k , the number of independent solutions is k − 2 n − 2 � n − 3 � � = ( n − 3)! k − 2 k =2 Zhengwen Liu (UCLouvain) Scattering Equations in MRK 5/25
Multi-Regge Kinematics (MRK) Multi-Regge kinematics is defined as a 2 → n − 2 scattering where k 2 k 3 the final state particles are strongly ordered in rapidity while having comparable transverse momenta, q 4 y 3 ≫ y 4 ≫ · · · ≫ y n and | k 3 | ≃ | k 4 | ≃ . . . ≃ | k n | k 4 q 5 • In lightcone coordinates k a = ( k + a , k − a ; k ⊥ a ) with k ± a = k 0 a ± k z a k 5 and k ⊥ a = k x a + ik y a k + 3 ≫ k + 4 ≫ · · · ≫ k + n • We work in center-of-momentum frame: k n − 1 κ ≡ √ s q n k 1 = (0 , − κ ; 0) , k 2 = ( − κ, 0; 0) , k 1 k n • In this region, tree amplitudes in gauge and gravuty factorize 1 1 1 A n ∼ s spin C 2;3 V 4 · · · V n − 1 C 1; n t 4 t n − 1 t n Zhengwen Liu (UCLouvain) Scattering Equations in MRK 6/25
Multi-Regge Kinematics (MRK) Multi-Regge kinematics is defined as a 2 → n − 2 scattering where k 2 k 3 the final state particles are strongly ordered in rapidity while having comparable transverse momenta, q 4 k 4 y 3 ≫ y 4 ≫ · · · ≫ y n and | k 3 | ≃ | k 4 | ≃ . . . ≃ | k n | q 5 • In lightcone coordinates k a = ( k + a , k − a ; k ⊥ a ) with k ± a = k 0 a ± k z a k 5 and k ⊥ a = k x a + ik y a k + 3 ≫ k + 4 ≫ · · · ≫ k + n k n − 1 • We work in center-of-momentum frame: q n κ ≡ √ s k 1 = (0 , − κ ; 0) , k 2 = ( − κ, 0; 0) , k 1 k n • In this region, tree amplitudes in gauge and gravity factorize [Kuraev, Lipatov & Fadin, 1976; 1 1 1 A n ∼ s spin C 2;3 V 4 · · · V n − 1 C 1; n t 4 t n − 1 t n Del Duca, 1995; Lipatov, 1982] Zhengwen Liu (UCLouvain) Scattering Equations in MRK 6/25
When scattering equations meet MRK
Scattering equations in MRK • The simplest example: four points f 1 = − s 13 − s 14 σ 3 = s + t σ 1 = 0 , σ 2 → ∞ , = 0 = ⇒ σ 3 σ 4 σ 4 t In the Regge limit, s ≫ − t , we have � � σ 3 � s � � � � � ≃ � ≫ 1 ⇒ | σ 3 | ≫ | σ 4 | = � � � � σ 4 t � Zhengwen Liu (UCLouvain) Scattering Equations in MRK 7/25
Scattering equations in MRK • The simplest example: four points f 1 = − s 13 − s 14 σ 3 = s + t σ 1 = 0 , σ 2 → ∞ , = 0 = ⇒ σ 3 σ 4 σ 4 t In the Regge limit, s ≫ − t , we have | σ 3 /σ 4 | ≃ | s/t | ≫ 1 ⇒ | σ 3 | ≫ | σ 4 | = • The next-to-simplest: five points = k + = k + σ (1) a σ (2) a σ 1 = 0 , σ 2 → ∞ , , a = 3 , 4 , 5 a a k ⊥ k ⊥ ∗ a a In MRK, k + 3 ≫ k + 4 ≫ k + 5 , we have again | σ 3 | ≫ | σ 4 | ≫ | σ 5 | Zhengwen Liu (UCLouvain) Scattering Equations in MRK 7/25
Scattering equations in MRK • The simplest example: four points f 1 = − s 13 − s 14 σ 3 = s + t σ 2 → ∞ , ⇒ σ 1 = 0 , = 0 = σ 3 σ 4 σ 4 t In the Regge limit, s ≫ − t , we have | σ 3 /σ 4 | ≃ | s/t | ≫ 1 = ⇒ | σ 3 | ≫ | σ 4 | • The next-to-simplest: five points [Fairlie & Roberts, 1972] = k + = k + σ (1) a σ (2) a σ 1 = 0 , σ 2 → ∞ , , a = 3 , 4 , 5 a a k ⊥ k ⊥ ∗ a a In MRK, k + 3 ≫ k + 4 ≫ k + 5 , we have again | σ 3 | ≫ | σ 4 | ≫ | σ 5 | • Any n -point scattering eqs have a MHV (MHV) solution [Fairlie, 2008] = k + = k + σ (MHV) a a σ ( MHV ) σ 1 = 0 , σ 2 → ∞ , , a = 3 , . . . , n a a k ⊥ k ⊥ ∗ a a In MRK, k + 3 ≫ · · · ≫ k + n , we have | σ 3 | ≫ | · · · ≫ | σ n | Zhengwen Liu (UCLouvain) Scattering Equations in MRK 7/25
Recommend
More recommend
