Structure constants of twist-2 light-ray operators in the triple Regge limit

Transcript

1 Structure constants of twist- light-ray operators in the triple Regge limit I. Balitsky JLAB & ODU XV Non-perturbative QCD 13 June 018 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

2 Outline 1 Setting the stage: correlator of 3 light-ray operators Three-point correlator of local operators: Light-ray operators as analytic continuation of local operators Three-point correlator of LR operators: what to expect CF of two light-ray operators in the BFKL limit Correlation function of two Wilson frames in the BFKL approximation. CF of two light-ray operators. 3 3-point CF in the BK limit 3-point CF of Wilson frames from the BK equation Structure constants at ω 1 = ω + ω point CF in the triple Regge limit BFKL kernel in a triple Regge limit Structure constants at the triple BFKL point 5 Conclusions and outlook I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

3 Supermultiplet of twist- operators SU 4 singlet operators. (Korchemsky et al) S l 1n(z) = F l n(z) + l 1 4 Λ l n(z) + S l n(z) = F l n(z) 1 4 Λ l n(z) S l 3n(z) = F l n(z) l + 1 Λ l n(z) + l(l 1) Φ l 4 n(z) l(l + 1) Φ l 7 n(z) (l + 1)(l + ) Φ l 4 n(z) F l n(z) i l tr F µn l n C 5 l ( n + n n ) F µ n + O(g ) Λ l n(z) i l 1 l 1 tr λ n C 3 ( n + n ) l 1 λ(z) + O(g ) n Φ l n(z) i l tr φ I l 1 n C 3 l ( n + n n ) φ I (z) + O(g ) C λ l (x) - Gegenbauer polynomials, n = 0, and F µ n F µν n ν etc. All operators have the same anomalous dimension γ S1 l (α s ) γ l (α s ) = α s π N c[ψ(l 1) + C] + O(αs ), γ S l = γ S1 l+1, γs3 l = γ S1 l+ I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

4 3-point CF of local operators Rychkov et al: CF of 3 operators with spin (n 1 = n = n 3 = 0) < O l 1 n1 (x)o l n (y)o l 3 n3 (z) >= The sum runs over m 1,m 13,m 3 0 λ m1,m 3,m l 1 l l 3 m 3 m 13 m 1 m 1 = l 1 m 1 m 13 0, m = l m 1 m 3 0, m 3 = l 3 m 13 m 3 0 where i is dimension and l i is Lorentz spin. 1 3 l 1 l l 3 = (V 1,3) l1 m1 m13 (V,31 ) l m1 m3 (V 3,1 ) l3 m13 m3 (H 1 ) m1 (H 13 ) m13 (H 3 ) m3 x y m 3 m 13 m 1+ 3 x z 1+ 3 y z V and H - some tensor structures I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 4

5 3s If we define forward operators Φ l n(x ) = du φ a AB l nφ ABa (un + x ), Λ l n(x ) = du i λ a A l 1 n σ n λ a A (un + x ) F l (x ) = du Fni a l n Fn ai (un + x ), the renorm-invariant operators are S l 1n = Fl n Λl n 1 Φl n, S l n = Fl n S l 3n = Fl n l + (l 1) Λl n (l + 1)(l + ) Φ l n l(l 1) 1 (l + 1) 4(l 1) Λl n + 6(l 1) Φl n and Rychkov s structures reduce to one (x n 1 = x n = x n 3 = 0) S l 1 n1 (x 1 )S l n (x )S k 3 n 3 (x 3 ) = = C(g, l i ) (n 1 n ) l1+l l3 1 (n 1 n 3 ) l 1 +l 3 l 1 (n n 3 ) l +l 3 l 1 1 x x x I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 5

6 3s If we define forward operators Φ l n(x ) = du φ a AB l nφ ABa (un + x ), Λ l n(x ) = du i λ a A l 1 n σ n λ a A (un + x ) F l (x ) = du Fni a l n Fn ai (un + x ), the renorm-invariant operators are S l 1n = Fl n Λl n 1 Φl n, S l n = Fl n S l 3n = Fl n l + (l 1) Λl n (l + 1)(l + ) Φ l n l(l 1) 1 (l + 1) 4(l 1) Λl n + 6(l 1) Φl n and Rychkov s structures reduce to one (x n 1 = x n = x n 3 = 0) S l 1 n1 (x 1 )S l n (x )S k 3 n 3 (x 3 ) = = C(g, l i ) (n 1 n ) l1+l l3 1 (n 1 n 3 ) l 1 +l 3 l 1 (n n 3 ) l +l 3 l 1 1 x x x Our aim is to find the structure constants C(g, l i ) in the BFKL limit l i 1 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 5

7 Light-ray operators Gluon light-ray (LR) operator of twist F a i(x + + x )[x +, x + ] ab F b i (x + + x ) Forward matrix element - gluon parton density 1 z µ z ν p Fµξ(z)[z, a 0] ab Fν bξ (0) p µ z =0 = (pz) dx B x B D g (x B, µ) cos(pz)x B Evolution equation (in gluodynamics) µ d dµ Fa i(x + + x )[x +, x + ] ab F b i (x + + x ) = x + x + dz + z + x + 0 dz + K(x +, x + ; z +, z + ; α s )F a i(z + + x )[z +, z + ] ab F b i (z + + x ) Forward LR operator F(L +, x ) = dx + F i(l a + + x + + x )[L + + x +, x + ] ab F (x bi + + x ) I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 6

8 Light-ray operators Conformal LR operator (j = 3 + iν) F µ j (x ) = 0 dl + L 1 j + F µ (L +, x ) Evolution equation for forward conformal light-ray operators µ d dµ F j(z ) = 1 γ j (α s ) is an analytical continuation of γ n (α s ) 0 du K gg (u, α s )u j F j (z ) I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 7

9 Supermultiplet of LR operators Gluino and scalar LR operators Λ(L +, x ) = i dx + [ λ a (L + + x + + x )[x + + x +, x + ] ab σ λ b (x + + x ) + c.c] Φ(L +, x ) = dx + φ a,i (L + + x + + x )[x + + x +, x + ] ab φ b,i (x + + x ) Λ j (x ) = 0 SU 4 singlet LR operators. dl + L j+1 + Λ(L +, x ), Φ j (x ) = 0 dl + L j+1 + Φ(L +, x ) S 1j (x ) = F j (x ) + j 1 8 Λ j(j 1) j(x ) Φ j (x ) 8 S j (x ) = F j (x ) 1 8 Λ j(j + 1) j(x ) + Φ j (x ) 4 S 3j (x ) = F j (x ) j + 4 Λ (j + 1)(j + ) j(x ) Φ j (x ) 8 All operators have the same anomalous dimension γ S 1 j (α s ) γ j (α s ) = α s π N c[ψ(j 1) + C] + O(αs ), γ S j = γ S 1 j+1, γs 3 j = γ S 1 j+ I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 8

10 Correlators of LR operators Since LR operators are analytic continuation of local operators, we expect (j 1 = 3 + iν 1, j = 3 + iν ) S j 1 n1 (x 1 )S j n (x ) = δ(ν 1 ν )f (α s, j) (n 1 n ) ω 1(µ ) γ(j 1,α s) x 1 (αs,j 1) for -point CF and similarly (j i 1 + ω i ) S j 1 n1 (x 1 ) S j n (x ) S j n3 (x 3 ) = (n 1 n ) ω1+ω ω3 (n 1 n 3 ) x F(α s, ω 1, ω, ω 3 ) (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) ω 1 +ω 3 ω ω +ω 3 ω 1 (n n 3 ) x x 3 µ γ(j 1) γ(j ) γ(j 3 ) for the 3-point CF ( = j + γ(j) = 1 + ω + γ ω - dimension). I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 9

11 Correlators of LR operators Since LR operators are analytic continuation of local operators, we expect (j 1 = 3 + iν 1, j = 3 + iν ) S j 1 n1 (x 1 )S j n (x ) = δ(ν 1 ν )f (α s, j) (n 1 n ) ω 1(µ ) γ(j 1,α s) x 1 (αs,j 1) for -point CF and similarly (j i 1 + ω i ) S j 1 n1 (x 1 ) S j n (x ) S j n3 (x 3 ) = (n 1 n ) ω1+ω ω3 (n 1 n 3 ) x F(α s, ω 1, ω, ω 3 ) (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) ω 1 +ω 3 ω ω +ω 3 ω 1 (n n 3 ) x x 3 µ γ(j 1) γ(j ) γ(j 3 ) for the 3-point CF ( = j + γ(j) = 1 + ω + γ ω - dimension). The goal is to calculate f (α s, j) and F(α s, j 1, j, j 3 ) at j i = 1 + ω i in the BFKL limit g 0, ω 0, and g ω = fixed I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 9

12 Warm-up exercise: LO Since LR operators are analytic continuation of local operators, we expect S j 1 n1 (x 1 ) S j n (x ) S j n3 (x 3 ) = (n 1 n ) ω1+ω ω3 (n 1 n 3 ) x = j + γ(j) - dimension Warm-up exercise: LO F(α s, ω 1, ω, ω 3 ) (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) ω 1 +ω 3 ω ω +ω 3 ω 1 (n n 3 ) x x F L 1 F x 1 n 1 F x n 3 F + perm. L n L3 x 3 F F I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

13 Warm-up exercise: LO S 1+ω 1 n 1 (x 1 )S 1+ω n (x )S 1+ω (N 3 c 1)F(α s, ω 1, ω, ω 3 ) n 3 (z 3 ) = 4π 6 (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) 1 (n 1 n ) ω 1 +ω ω 3 (n 1 n 3 ) ω 1 +ω 3 ω (n n 3 ) ω +ω 3 ω 1 x 1 x 13 x 3 x 1 x 13 x 3 F(α s, ω 1, ω, ω 3 ) = Γ ( 1 + ω 1 + ω ω 3 ) Γ ( 1 + ω + ω 3 ω 1 ) ( ω 1 + ω 3 ω ) Γ ) Γ(1 ω i )Γ ( ω 1 + ω ω 3 + ) Γ ( ω + ω 3 ω 1 + ) Γ ( ω 1 + ω 3 ω i { (e iπω 3 1 )[ e iπ(ω 1 ω ) + e iπ(ω ω 1 ) e iπω ] 3 + ( e iπω 1 1 )[ e iπ(ω ω 3 ) + e iπ(ω 3 ω ) e iπω ] 1 + ( e iπω 1 )[ e iπ(ω 3 ω 1 ) + e iπ(ω 1 ω 3 ) e iπω ] } +e iπ(ω 1+ω ω 3 ) + e iπ(ω +ω 3 ω 1 ) + e iπ(ω 1+ω 3 ω ) e iπ(ω 1+ω +ω 3 ) At small ω s F(ω 1, ω, ω 3 ) π (ω 1 + ω + ω 3) π (ω 1 + ω + ω 3 ω 1 ω ω 1 ω 3 ω ω 3 ) I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

14 In higher orders one should expect Sn 1+ω1 1 (x 1 )Sn 1+ω (x )Sn 1+ω3 3 (z 3 ) = = N c 1 1 4π 6 x1 x 13 x 3 (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) (n 1 n ) ω1+ω ω3 (x1 ) (n 1 n 3 ) ω 1 +ω 3 ω (x13 ) (n n 3 ) ω +ω 3 ω 1 (x3 ) ) ] n ω i F(ω 1, ω, ω 3 ; g ) F(ω 1, ω, ω 3 ) [1 + (g c n F(ω 1, ω, ω 3 ; g ), It could be obtained from the CF of three color dipoles with long sides collinear to n 1, n, and n 3 and transverse short sides. This means analyzing QCD (or N=4 SYM) in the triple Regge limit. Triple Regge limit: scattering of 3 particles moving with speed c in x, y, and z directions. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

15 In higher orders one should expect Sn 1+ω1 1 (x 1 )Sn 1+ω (x )Sn 1+ω3 3 (z 3 ) = = N c 1 1 4π 6 x1 x 13 x 3 (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) (n 1 n ) ω1+ω ω3 (x1 ) (n 1 n 3 ) ω 1 +ω 3 ω (x13 ) (n n 3 ) ω +ω 3 ω 1 (x3 ) ) ] n ω i F(ω 1, ω, ω 3 ; g ) F(ω 1, ω, ω 3 ) [1 + (g c n F(ω 1, ω, ω 3 ; g ), It could be obtained from the CF of three color dipoles with long sides collinear to n 1, n, and n 3 and transverse short sides. This means analyzing QCD (or N=4 SYM) in the triple Regge limit. Triple Regge limit: scattering of 3 particles moving with speed c in x, y, and z directions. What we can do in a meantime is to take n 3 n and consider the CF of a dipole in n 1 = n + direction and two dipoes in n = n 3 = n directions which can be obtained using the BK evolution. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

16 CF of two light-ray operators S j + (x ) = dx + dl + L 1 j + Fa i(l + + x + + x )[L + + x +, x + ] ab F (x bi + + x ) 0 + gluinos + scalars S j (y ) = dy dl L 1 j Fa i(l + y + y )[L + y, y ] ab F (y bi + y ) 0 + gluinos + scalars A general formula for the correlation function of two LR operators reads S + j= 3 +iν(x )S j = 3 +iν (y ) = δ(ν ν )a(j, α s )(µ ) γ(j,αs) ((x y) )j+1+γ(j,αs) δ(ν ν ) reflects boost invariance: x + λx +, x 1 λ x does not change the correlation functions which depend on x + y. We need to reproduce it and find γ(j, α s ) and a(j, α s ) as j 1. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

17 Correlator of two Wilson frames Wilson frame : light-ray operator with point-splitting in the transverse direction I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

18 Correlator of two color dipoles U σ (x 1, z )V σ+ (y 1, w ) = = 8g4 dν ν d ( z 0 (x 1 z) N ( ν ) (x 1 z 0 ) (z z 0) ( (y 1 w) ) 1 iν ( ) ℵ(ν) σ+ σ (y 1 z 0 ) (w z 0). σ +0 σ 0 where ℵ(ν) = αs π N c[ ψ(1) ψ ( 1 + iν 1) ψ ( 1 iν 1)]. From experience with 4-point CFs in the BFKL limit ) 1 +iν ( σ+ σ ) ℵ(ν) σ +0 σ 0 i (((x 1 y 3 ) ) ℵ(ν) ((x 3 y 1 ) ) ℵ(ν) sin πℵ(ν) (x13 ) ℵ(ν) (y 13 ) ℵ(ν) ((x 1 y 1 ) ) ℵ(ν) ((x 3 y 3 ) ) ℵ(ν) (x13 ) ℵ(ν) (y 13 ) ℵ(ν) (For small x13 and y 13 Wilson frames are approximately conformally invariant) XV Non-perturbative QCD 13 June I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit ).

19 Correlator of two Wilson frames V. Kazakov, E. Sobko, I.B. S +ω 1 S +ω gl+ gl = g 4 in 4π 3 ν δ(ω 1 ω ) ( ν ) ( x 13 y 13 ℵ(ν) )1+ where dν( )ℵ(ν) ω B( ω, ω ℵ(ν)) 1 eiπ(ℵ(ν) ω) sin πℵ(ν) ( ) ( x 13 ) 1 +iν ( y 13 ) 1 +iν ( x y G(ν) + (ν ν) )1+iν 4 1 iν π 3 (i ν) G(ν) = i Γ ( 3 iν)γ (1 + iν) sinh(πν). Now we can carry out the last integration over ν as the pole contribution at ω = ℵ(ν). We pick here the first pole Ψ-functions in pomeron intercept which corresponds to the operator with the lowest possible twist=. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

20 Correlator of two Wilson frames V. Kazakov, E. Sobko, I.B. + (x 1, x 3 )S 1+ω (y 1, y 3 ) δ(ω 1 ω )Υ( γ) (x 13 ) γ ω (y 13 ) γ ω x 13, y 13 0 ((x y) )+ γ, S 1+ω 1 Υ( γ) = N g 4 1 γ π γ Γ (1 γ )Γ ( 1 + γ ) sin(π γ)ˆℵ ( γ) γ = 1 + iν is the solution of ω = ˆℵ( γ) We use the point-splitting regularization in the orthogonal direction for our light-ray operators cutoffs are defined as Λ x = 1 x 13 and Λ y = 1 y 13 Rewrite + (x 1, x 3 )S 1+ω (y 1, y 3 ) δ(ω 1 ω )Υ(γ + ω) (x 13 ) γ (y 13 ) γ x 13, y 13 0 ((x y). )+ω+γ S 1+ω 1 The anomalous dimension γ = γ ω satisfies - Lipatov-Fadin formula ω = ˆℵ(γ + ω) = ˆℵ(γ) + ˆℵ (γ)ˆℵ(γ) + o(g 4 ). XV Non-perturbative QCD 13 June I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit

21 CF of three Wilson frames: one in + direction and two in - x 3 [x 1, x 3 ] Wilson frame (without F insertions) x1 x3 dx 1 dx 3 (x 1 x 3 ) ω [x 1, x 3 ] x 1 y 1 z 1 x 13 γj c(g YM, N c, ω) S +ω (x 1 ). x 13 0, ω 0 [x 1, x 3 ] U σ1 (x 1, x 3 ) with some cutoff σ 1 y 3 z 3 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

22 Using decomposition over Wilson lines we get: = D dx 1 dx 3 x ω1 31 x 1 S +ω 1 + (x 1, x 3 )S +ω (y 1, y 3 )S +ω 3 (z 1, z 3 ) = dy 1+ dy 3+ y ω 31+ y 1+ dz z 1+ dz 3+ z ω3 U σ 1 (x 1, x 3 )V σ + (y 1, y 3 )W σ 3+ (z 1, z 3 ), where D = N 3 c(ω 1 )c(ω )c(ω 3 ) ( x 1 x 3 )( y 1 y 3 )( z 1 z 3 ). I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 1

23 BK equation: where K BK in LO approximation: σ d dσ Uσ (z 1, z ) = K BK U σ (z 1, z ), = g π K LO BK U(z 1, z ) = d z 3 z 1 [U(z 1, z 3) + U(z 3, z ) U(z 1, z ) U(z 1, z 3)U(z 3, z )]. z 13 z 3 Schematically calculation of correlation function of 3 dipoles can be wrote as: ( ) dy 0(U Y 1 U Y 0 U Y 0 V Y ) (BK vertex at Y 0) U Y 0 W Y 3 where we introduced rapidity Y i = ln σ i I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

24 The structure of 3-point correlator in d - space Figure : The structure of 3-point correlator. Red circles correspond to BFKL propagators (the crossed one has extra multiplier ( ν 1 ) ). The blue blob corresponds to the 3-point functions of -dimensional BFKL CFT. The triple veritces correspond to E-functions. The αβγ-triangle in the first, planar, term and βγ-link in the second, nonplanar, term correspond to triple pomeron vertex. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

25 Result: Sn 1+ω1 1 (x 1, x 3 )Sn 1+ω (y 1, y 3 )Sn 1+ω3 (z 1, z 3 ) = = ig 10 δ(ω 1 ω ω 3 ) c(ω 1 )c(ω )c(ω 3 ) H Ψ(ν1, ν, ν 3 ) x 13 γ1 y 13 γ z 13 γ3 x y +γ1+γ γ3 x z +γ1+γ3 γ y z +γ+γ3 γ1 where ν i is a solution of BFKL equation for anomalous dimensions ω i = ℵ(ν i ) H = 10 (Nc 1) γ1( + γ 1) 4 ( + γ ) ( + γ 3) G(ν1 ) G(ν ) G(ν3 ) π Nc 5 ℵ (ν1 ) ℵ (ν ) ℵ (ν3 ), γ i = γ(j i) - anomalous dimension (j i = 1 + ω i) and ν πγ ( 1 G(ν) = + iν)γ( iν) ( 1 +, 4 ν ) Γ ( 1 iν)γ(1 + iν) Ψ(ν 1, ν, ν 3) = Ω(h 1, h, h 3) π Λ(h 1, h, h 3)Re(ψ(1) ψ(h 1) ψ(h ) ψ(h 3)), Nc where h i = 1 + iνi = 1 + γ i. Expression for Ω and Λ was obtained by G.Korchemsky in terms of higher hypergeometric and Meijer G-functions. XV Non-perturbative QCD 13 June 018 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit

26 n n 3 limit To identify the function Ψ(ν1, ν, ν 3 ) with structure constants of CF of three LR operators we need to consider limit n n 3 in the formula S j 1 n1 (x 1 ) S j n (x ) S j n3 (x 3 ) = (n 1 n ) ω1+ω ω3 x (n 1 n 3 ) ω 1 +ω 3 ω x C(α s, ω 1, ω, ω 3 ) (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) (n n 3 ) ω +ω 3 ω 1 x The limit n n 3 is tricky: in the limit n n 3 we get a zero mode coming from boost invariance at n = n 3 1 ( (n, n 3 ) ) ω +ω 3 ω 1 n n 3 dξe ξ(ω 1 ω ω 3 ) = δ(ω 1 ω ω 3 ) ω 1 ω ω 3 s I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

27 Limit n n 3 Rapidity integral at n = n 3 dy 1 dy dy 3 dy 0 θ(y 1 Y 0 )θ(y 0 + Y )θ(y 0 + Y 3 )e ω 1Y 1 ω Y ω 3 Y 3 e ℵ 1(Y 1 Y 0 )+ℵ (Y 0 +Y )+ℵ 3 (Y 0 +Y 3 ) = δ(ω 1 ω ω 3 ) (ω 1 ℵ 1 )(ω ℵ )(ω 3 ℵ 3 ) Let us take n n 3 but n 1 n n 1 n 3. We can use our formulas for n = n 3 case until longitudinal distances between frames and 3 are smaller than typical transverse separation, i.e. when (y 1 z 1 ) l l 3 n 3. In terms of rapidities Y = ln l n 1, Y 3 = ln l n 1 3 this restriction means Y + Y 3 r, r ln n 1 n n n 3. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

28 Limit n n 3 Rapidity integral with restriction Y + Y 3 r, r ln n 1 n n n 3. dy 1 dy dy 3 dy 0 θ(y 1 Y 0 )θ(y 0 + Y )θ(y 0 + Y 3 )θ ( Y + Y 3 < r ) e ω 1Y 1 ω Y ω 3 Y 3 +ℵ 1 (Y 1 Y 0 )+ℵ (Y 0 +Y )+ℵ 3 (Y 0 +Y 3 ) = e r (ω +ω 3 ω 1 ) (ω 1 ω ω 3 )(ω 1 ℵ 1 ) ( ω ℵ + ω 1 ω ω 3 ω +ω 3 ω 1 n 1 n ) ω +ω 3 ω 1 ( n n 3 (ω 1 ω ω 3 )(ω 1 ℵ 1 )(ω ℵ )(ω 3 ℵ 3 ) )( ω3 ℵ 3 + ω 1 ω ω 3 ) 1 ( (n, n 3 ) ) ω +ω 3 ω 1 n n 3 δ(ω1 ω ω 3 ) ω 1 ω ω 3 s I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

29 Structure constant in the BFKL limit V. Kazakov, E. Sobko, I.B. Finally for normalized structure constant C ω1,ω,ω 3 = C + ({ i },{1+ω i }) b1+ω1 b 1+ω b 1+ω3 we get: C ω1,ω,ω 3 = i 3/ g 4 N c 1 γ1( + γ 1) G(ν1 ) G(ν ) G(ν3 ) π 5 Nc ℵ (ν1 ) ℵ (ν ) ℵ (ν3 ) Ψ(ν 1, ν, ν3 ), where ω i = ℵ(ν i ) and ν πγ ( 1 G(ν) = + iν)γ( iν) ( 1 +, 4 ν ) Γ ( 1 iν)γ(1 + iν) Ψ(ν 1, ν, ν 3) = Ω(h 1, h, h 3) π Λ(h 1, h, h 3)Re(ψ(1) ψ(h 1) ψ(h ) ψ(h 3)) Nc with notation h i = 1 + iνi = 1 + γ i, ωi = ℵ(νi) The structure of the formula is C ω1,ω,ω 3 = g Nc 1 Nc f ( g ω 1, g ω, g ω 3 ) I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

30 Structure constant in the BFKL limit In the limit g ω i 0 we get the asymptotics: Ω(h 1, h, h 3) 16π3 γ1 [γ1(γ + γ 3 ) + γ(γ 1 + γ 3 )+ γ γ 3 +γ3(γ 1 + γ ) + γ 1 γ γ 3 )(1 + O(g /ω i )) Λ(h 1, h, h 3) 8π (γ 1 + γ + γ 3 ) γ 1 γ γ 3 (1 + O(g /ω i )) γ i = 8g ω i + o( g ω i ) C ω1,ω,ω 3 = ig N c 1 πn c 1 ω 5 1 ω 1 ω 1 3 (ω 1(ω + ω 3 )+ ω (ω 1 + ω 3 ) + ω 3(ω 1 + ω ) + ω 1 ω ω 3 )(1 + O(g )) I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

31 Wilson frames in the triple Regge limit n 1 Sn ω1 1 (z 1, z 1 ) = du 1 dl l ω1 Tr{F n1i(u 1 n 1 + ln 1 + z 1 )Fn i 1 (un 1 + z 1 )} 0 z 1 n 1 = n = n 3 = 0, z n i = 0, n 1 n n n 3 n 1 n 3 z 1 z 13 z 3 z 1 z 11 z z 33 z 3 n 3 z z n z 3 Triple Sudakov variables : k = αn 1 + βn + γn 3 + k I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018

32 Triple BFKL evolution n 1 z 1 z 1 k = k + αβs 1 + αγs 13 + βγs 3 s 1 n 1 n, s 13 n 1 n 3, s 3 n n 3 z 4 z 5 z 9 z 8 z 6 n 3 z 7 z 3 z z z 3 If α β, γ eikonal n α k + iɛ 1 β + γ + iɛα XV Non-perturbative QCD 13 June 018 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit

33 Check: BFKL kernel in the triple Regge regime n 1 k 1 k' 1 k k' n n 3 L µ (k, k 1 )L µ (k, k 1) = ( k 1 k 1) + k 1 k + k k 1 ( k 1 + k ) BFKL kernel with k = k + s 1s 3 s 13 β k, s ik = n i n k I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

34 BFKL evolution in triple Regge limit n 1 k 1 k' 1 k 1 k = α 1 n 1 + β 1 n + γ 1 n 3 + k 1 = α n 1 + β n + γ n 3 + k k = α n 1 + β n + γ n 3 + k k k' n n 3 The BFKL evolution is logarithmic until α 1 β s 1, α 1 γ s 13 m. In terms of rapidities η 1 ln α, η ln β, η 3 ln γ η 1 + η ln s 1, η 1 + η 3 ln s 13, I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

35 BFKL evolution of Wilson frames Evolution from α max l1 s z 11 to α min n 1 z 1 z 1 z 4 z 5 du 1 Tr{F n1i(u 1 n 1 + l 1 n 1 + z 1 )Fn i 1 (un 1 + z 1 ) z 11 0 = s N c dν Γ ( 3 16l 1 z 11 π 4 4iν iν) Γ(1 + iν) [ z Γ ( 11 z iν) Γ( iν) z 14 z 15 η 1 = ln l1 s z 11, y 1 = ln α min ] 1 +iν e ℵ(ν)(ln l1 y1) Tr{U z4 U z 5 } y1 At small z 11 we can close the contour over 1 + iν in the right half-plane and the leading contribution comes from the pole of ℵ(ν) at iν = 1 αsnc πω 1 XV Non-perturbative QCD 13 June I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit

36 Result = longitudinal integral transverse integral Longitudinal integral dη 1 dη dη 3 η1 η η3 dy 1 dy dy 3 e ω 1η 1 ω η ω 3 η 3 +ℵ 1 (η 1 y 1 )+ℵ (η y )+ℵ 3 (η 3 y 3 ) θ (y 1 + y ln ) θ (y + y 3 ln ) θ (y 1 + y 3 ln ) s 1 s 3 s 13 4 = (ω 1 ℵ 1 )(ω ℵ )(ω 3 ℵ 3 ) (s 1z 1 ) ω1+ω ω3 (s 13 z 13 ) ω 1 +ω 3 ω (s 3 z 3 ) ω +ω 3 ω 1 (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) η 1 ln l 1 s1 s 13 s 3 z 11, η ln l s1 s 3 s 13 z, η 3 ln l 3 s13 s 3 s 1 z 33 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

37 At the end of evolution As usual, the end of evolution means s leading-order tree diagrams m 1 but α s ln s m 1 n 1 z 4 z 5 z 9 z 8 n 3 z 6 z 7 n ln Z(1) 46 Z (1) 57 Z (1) 47 Z (1) 56 ln Z(3) 68 Z (3) 79 Z (3) 69 Z (3) 78 Z (kl) ij z ij (z ij n k )(z ij n l ) n k n l ln Z(3) 48 Z (13) 59 Z (3) 49 Z (13) 58 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

38 Transverse integral The transverse planes of the evolution of Wilson frames are different the resulting integral is two-dimensional but not translation invariant 1 a z z 1 1 a 1 1 a 1 z 5 z 4 1+a 1 ln z ln 56 6 z 9 1 a 1+a ln z 7 1+a 3 78 z 8 1 a a 3 z 3 = (z z ) + z = (z z ) +(z +z ) a ( 39) = (z 1 ) 1+a 1+a a 3 (z 13 ) 1+a 1+a 3 a (z 3 ) 1+a +a 3 a 1 Φ(a 1, a, a 3 ) a 1 = 1 iν 1 etc XV Non-perturbative QCD 13 June I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit

39 Result = longitudinal integral transverse integral Result = dν 1 dν dν 3 z iν z iν 1 z iν Longitudinal integral Transverse integral Longitudinal integral = 4 (ω 1 ℵ(ν 1 ))(ω ℵ(ν ))(ω 3 ℵ(ν 3 )) (s 1z 1 ) ω1+ω ω3 (s 13 z 13 ) ω 1 +ω 3 ω (s 3 z 3 ) ω +ω 3 ω 1 (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) Transverse integral = (z 1 ) 1 iν 1 iν +iν 3 (z 13 ) 1 iν 1 iν 3 +iν (z 3 ) 1 iν iν 3 +iν 1 Φ ( 1 iν 1, 1 iν, 1 iν ) 3 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

40 Three-point CF in the triple Regge limit Taking residues at ν 1 = ℵ 1 (ω 1 ), ν = ℵ 1 (ω ), ν 3 = ℵ 1 (ω 3 ) we get S j 1 n1 (z 1 ) S j n (z ) S j n3 (z 3 ) Φ( γ 1, γ, γ 3 ) = (ω 1 + ω ω 3 )(ω 1 + ω 3 ω )(ω + ω 3 ω 1 ) (s 1) ω1+ω ω3 (s 13 ) z ω 1 +ω 3 ω ω +ω 3 ω 1 (s 3 ) z z 3 µ γ(ω 1) γ(ω ) γ(ω 3 ) where i = 1 + ω i + γ i and γ i = 1 + iν i is a solution of ω i = ℵ(ν i ) To compare to previous results (for tree approximation and for ω 1 = ω + ω 3 ) we need Φ(a 1, a, a 3 ) at a 1, a, a 3 1. I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3

41 Conclusions and Outlook Conclusions Outlook QCD structure constants in the triple BFKL limit ω i 0 at g ω 1 are expressed in terms of the transverse integral Φ(γ 1, γ, γ 3 ). Calculate this transverse integral and compare to previous result in the limit n n 3 I. Balitsky (JLAB & ODU) Structure constants of twist- light-ray operators in the triple Regge limit XV Non-perturbative QCD 13 June 018 3