Regge limit and the soft anomalous dimension Simon Caron-Huot (McGill University) Based on: 1701.05241,1711.04850 & in progress with: Einan Gardi, Joscha Reichel & Leonardo Vernazza Galileo Galilei Institute: Amplitudes in the LHC era, Oct. 31 th 2018
Nonperturbative forward physics: total pp cross-section: 160 accelerator pp 104 103 140 accelerator p p 102 ARGO-YBJ 101 100 Akeno 99 120 98 Fly’s Eye 97 96 Auger 95 (mb) 100 94 TOTEM 4 10 tot σ 80 ~p Area 60 constrained 40 3 5 2 4 10 10 10 10 10 s (GeV) [fig: Menon& Silva `13] 2
• Why does it grow? • Relativity + quantum mechanics probability to decay in two partons p parton cascade wee parton target cross section Fig. 3-a • A cloud of virtual particles (pions, rho’s,..) builds around the proton! [cartoon: Gottsman, Level&Maor] 3
perturbative phenomenology of forward scattering: • Deep inelastic scattering/saturation (HERA,heavy ions) small x = Regge, large Q 2 ⇒ perturbative • Mueller-Navelet: pp->X+2jets, forward&backward ( ∆ η � 1 , ∆ φ ) for review of these: see 1611.05079
theoretical motivations: One of few limits where perturbation theory can be resumed Retain rich dynamics in 2D transverse plane: -toy model for full amplitude -nontrivial function spaces -predicts amplitudes and other observables in overlapping limits 5
The (multi-)Regge limit at higher points has been extensively studied, especially in planar N=4 SYM. It reveals an amazing integrable system (next talk?) Here we’ll focus on A 2 → 2, but in QCD at finite N c . It depends on: - energy: L ≡ log | s/t | − i π / 2 - IR regulator: 1/ 𝜁 ⇔ log(-t/ 𝜈 2 ) - color: C A , T 2 s , T 2 t , . . . 6
Nice variables 1. Coupling runs with transverse momenta, not CM energy ⇒ use α s ( − t ) 2. Crossing symmetry relates large-s & large-u limits ⇒ use symmetrical combination ✓ ◆ L ≡ 1 log − s − i 0 + log − u − i 0 2 − t − t � s � � � − i π → log � � t 2
Crossing symmetry: Project onto signature eigenstates: ⇣ ⌘ M ( ± ) ( s, t ) = 1 M ( s, t ) ± M ( � s � t, t ) 2 These simple definitions remove all i\pi’s. The following have nice& real coefficients: 1 M ( − ) ( L, ↵ s ( − t ) , ✏ ) , i ⇡ M (+) ( L, ↵ s ( − t ) , ✏ ) 8
2 → 2 kinematic limits exponentiation of IR div soft limit of BFKL 1 / ✏ = Regge limit of 𝜟 soft ∼ log − t soft µ 2 soft- Regge BFKL resummation Regge L ∼ log s − t 9
BFKL redux [Balitsky,Fadin,Kuraev,Lipatov ’76-78] A simple, and correct, approach to high-energy scattering: replace each fast parton by a null Wilson line R ∞ −∞ dx + A a + ( x + , 0 − ,x ⊥ ) T a U ( x ⊥ ) ≡ P e i
The subtlety: projectiles contain more than one parton cloud of ... radiated partons
Transverse distribution depends on energy resolution d = + � � d η � � � � z 1 (+perms.) � � � � z 2 Z d d η UU ∼ g 2 d 2 z 0 K ( z 0 , z 1 , z 2 ) ⇥ ⇤ U ( z 0 ) UU − UU • ‘shock’ = Lorentz-contracted t arget • 45 o lines = fast projectile partons • Each parton crossing the shock gets a Wilson line
The Balitsky-JIMWLK equation d 2 z 0 z 0 i · z 0 j � d d η ⌘ H = α s Z ⇣ �⌘ d 2 z i d 2 z j T a i,L T a j,L + T a i,R T a j,R � U ab T a i,L T b j,R + T a j,L T b � ad ( z 0 ) i,R z 2 0 i z 2 2 π 2 0 j • Well established and tested [Balitsky ’95, Mueller, Kovchegov, JIMWLK*] • Now well understood at NLL [Balitsky&Chirilli ’07&’13; Kovner,Lublinsky&Mulian ’13; SCH ’14] • Partial NNLL results [SCH&Herranen ’16; Henn& Mistlberger ’17; SCH,Gardi&Vernazza ’17] *Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov& Kovner 13
Main feature: index contractions preserve future two global symmetries: SU(N) past x SU(N) future past Spontaneously broken to diagonal in vacuum: h 0 | U ( x ⊥ ) | 0 i = 1 ‘Goldstone boson’ W = Reggeized gluon [Kovner& Lublinsky ’05] [SCH ’13] U ( x ⊥ ) = e igT a W a ( x ⊥ ) BFKL : expand in W’s and study linearized evolution
Multi-Regge exchanges are suppressed by coupling M (+) M ( − ) M ( − ) . . . + + … + . . . D j D i α g . . . LL = one-Reggeon NNLL NLL (two W’s) (one-W exchange) Z A LL ∝ s α g ( t ) d ν c ( ν ) s E ( ν ) A NLL ∝ t ‘Regge cut’ ‘Regge pole’
Perturbative structure of the BFKL Hamiltonian e igW a T a ∼ 1 + igWT + . . . n → n+k transitions: from LO B-JIMWLK ( W ) 1 g 2 g 4 g 6 ( W ) 1 0 0 · · · ( W ) 2 g 2 g 4 ( W ) 2 0 0 d ( W ) 3 g 4 g 2 ( W ) 3 0 0 = · · · · d η ( W ) 4 g 4 g 2 ( W ) 4 0 0 · · · · · · · · · Leading BFKL and required by symmetry of d/d η BKP kernels • Matrix is symmetrical: projectile/target symmetry • Growth/saturation: off-diagonal can’t be ignored • (‘Reggeon field theory’ which resums all, still elusive) 16
• At NNLL, something new happens: 1 and 3 Reggeon states mix • @ 2-loops: violation of Regge pole factorization [Del Duca, Falcioni, Magnea & Vernazza ’14] • @ 3-loops: first check of mixing matrix H 3 → 3 H 1 → 3 H 3 → 1
We don’t actually compute these diagrams: the LO B-JIMWLK Hamiltonian gives us simple 2d integrals all the work is to find the color factors that multiply them, starting from the Hamiltonian. H k → k +2 = ↵ 2 Z � 0 ( W i − W 0 ) z Tr [ dz i ][ dz 0 ] K ii ;0 ( W i − W 0 ) x W y s F x F y F z F a ⇤ ⇥ (3.11) � W a 3 ⇡ i + ↵ 2 Z [ dz i ][ dz j ][ dz 0 ] K ij ;0 ( F x F y F z F t ) ab h ( W i − W 0 ) x W y s 0 W z 0 ( W j − W 0 ) t 6 ⇡ � 2 i 0 ( W j − W 0 ) t − ( W i − W 0 ) x W y − W x i ( W i − W 0 ) y W z 0 ( W j − W 0 ) z W t . j � W a i � W b j
result for 2loops NNLL, in any gauge theory: ⇣ ⌘ 12 ( C A ) 2 ⌘� + ⇡ 2 R (2) ⇣ s − u ) 2 � 1 M ( − , 2) D (2) + D (2) + D (1) D (1) M (0) ˆ ˆ ( T 2 ij → ij = ij → ij , i j i j ✓ ◆ � 1 8 ✏ 2 + 3 4 ✏⇣ 3 + 9 R (2) = = ( r Γ ) 2 8 ✏ 2 ⇣ 4 + . . . (3. Color operator precisely corrects factorization violation! ✔ at 3-loops NNLL, we computed coefficients of 3 color structures: ij → ij = ⇡ 2 ⇣ C ( C A ) 3 ⌘ M ( − , 3 , 1) R (3) s − u ] + R (3) s − u + R (3) M (0) ˆ ˆ A T 2 s − u [ T 2 t , T 2 B [ T 2 t , T 2 s − u ] T 2 ij → ij ✓ 1 = 1 48 ✏ 3 + 37 ◆ R (3) 16 ( r Γ ) 3 ( I a − I c ) = ( r Γ ) 3 24 ⇣ 3 + . . . A ✓ ◆ ✔ Poles are consistent with IR exponentiation!
Removing the ‘hat’ requires the 3-loop gluon Regge trajectory: H 1 → 1 , which affects only the R C color structure. In N=4, we could fix it from [Henn& Mistlberger ’16] � � ✏ 3 + 49 ⇣ 4 1 1 ✏ + 107 − ⇣ 2 144 ⇣ 2 ⇣ 3 + ⇣ 5 H (3) = N 2 + N 0 4 + O ( ✏ ) 0 + O ( ✏ ) 1 → 1 = c c 144 192 Upshot: -new 3loop prediction in QCD, up to one constant -machine set up to compute NNLL M (-) to 4&higher loops, modulo same constant. In N=4, that constant is known. 1701.05241 20
Back to NLL: high-order Solution in the soft limit two-Reggeon exchange ( ∼ α ` s L ` − 1 ) . . . M (+) = leading order [1711.04850] evolution 21
Each rung = the BFKL Hamiltonian H 2 → 2 ˆ t ) ˆ t ) ˆ H = (2 C A − T 2 H i + ( C A − T 2 H m ‘integration’ part: d 2 � 2 ✏ k 0 Z ˆ r Γ (2 π ) 2 � 2 ✏ f ( p, k, k 0 ) [ Ψ ( p, k 0 ) − Ψ ( p, k )] H i Ψ ( p, k ) = ‘multiplication’ part: ◆ ✏ ◆ ✏ � ✓ p 2 p 2 ✓ H m Ψ ( p, k ) = 1 ˆ 2 − Ψ ( p, k ) − 2 ✏ k 2 ( p − k ) 2 Both increase transcendental weight by 1 22
Evolution equation: Ψ ( ` ) = ˆ Ψ (0) = 1 . H Ψ ( ` − 1) , Exact solution in adjoint channel: Ψ = 1 Cases where eigenfunctions are known: [Lipatov] - Color singlet dipoles (x-space conformal symmetry) - Color adjoint (p-space ‘dual’ conformal symmetry) Unfortunately, for d ≠ 4 / other color reps., eigenfunctions are not known ⇒ iterative solution 23
Outermost rungs are always easy (multiplication) 4-loop = single nontrivial integral NLL = − i ⇡ ( B 0 ) 4 p 2 ⇢ Z t ) 3 Ω mmm ( p, k ) M (+ , 4) ˆ ( C A − T 2 [D k ] k 2 ( p − k ) 2 3! � t ) 2 Ω mim ( p, k ) + (2 C A − T 2 t )( C A − T 2 T 2 s − u M (tree) = i ⇡ ( B 0 ) 4 ⇢ ✓ (2 ✏ ) 4 + 175 ⇣ 5 1 ◆ ( C A − T 2 t ) 3 ✏ + O ( ✏ 2 ) (2.32) 4! 2 ✓ ◆� 8 ✏ − 3 16 ⇣ 4 − 167 ⇣ 5 − ⇣ 3 + C A ( C A − T 2 t ) 2 ✏ + O ( ✏ 2 ) T 2 s − u M (tree) . 8 [SCH ’13] Note this has both leading& subleading IR divergences
How to predict the IR divergences at higher-loops? Facts: 1. Wavefunction is finite as ✏ → 0 Ψ ( ` ) ( p, k ) ⇒ poles can only appear from final integration Z Ψ ( ` ) ( p, k ) k → 0 2. Evolution closes in soft limit: k → 0 ψ ( ` ) ( p, k ) ∼ ˆ k → 0 ψ ( ` − 1) ( p, k ) lim H lim 25
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