Duality Symmetry in High Energy Scattering

Duality Symmetry in High Energy Scattering Alex Prygarin Hamburg University 10 March 010

Outline Introduction BFKL Hamiltonian Duality symmetry of forward BFKL Duality symmetry of BK and non-forward BFKL (Levin and A.P.,Phys.Rev.C78:0650,008; A.P.,ariv:0908.386 [hep-ph],ariv:0911.579 [hep-ph]) Summary

Introduction What is Pomeron? σ pp 1.7s 0.08 + 51.6s 0.45 σ p p 1.7s 0.08 + 98.4s 0.45 mb mb Regge limit s >> t where A(s, t) s α P(t) σ tot s α(t) 1 Regge trajectory α(t) α 0 + α t Pomeron (phenomenological) is the Regge trajectory with α 0 1

Pomeron in QCD BFKL equation in Leading Log Approximation (LLA) (Kuraev, Lipatov and Fadin 77, Balitsky and Lipatov 78 ) A(s, t) s α(t) obtained by summing (g log s) n Effective Emission Vertex = + + + + Virtual Corrections (gluon Regge trajectory) = + + + +

Pomeron in QCD k k q Real Part of the BFKL Kernel K real (k, χ) = q k (χ q) χ (k q) (χ k) (χ k) Virtual Part of the BFKL Kernel ɛ( k ) = αsnc d k χ 4π χ (χ k) BFKL equation F log s = K O BFKL F with KBFKL = K real + K virtual

BFKL equation «ɛ( k ) ɛ( (k q) ) F(k, q) = αsnc d χ K real(k, χ) F(χ, q) χ (χ q) k k q Violates Froissart Bound σ tot Const log s σ BFKL tot sω P log s, ω P = 4 αsnc π log >> 0.08 pheno Bartels 80, Kwiecinski and Praszalowicz 80 (BKP) Balitsky 95, Kovchegov 99 (BK)

Multicolor BFKL (Lipatov 90 93) Schrödinger-like equation for colorless compound state of n-reggeized gluons f m, m ( ρ 1, ρ,..., ρ n ; ρ 0 ) E m, m f m, m = H f m, m, ω m, m = g N c 8π E m, m ωm, m j-plane singularity ω m, m = j 1 = σ tot s ρ k aρ k + b cρ k + d, Möbius group ρ k = x k + iy k, ρ k = x k + iy k Möbius group generators M 3 k = ρ k k, M k = k, M + k = ρ k k, Casimir operators M = ( n k=1 Ma k Mk f m, m = m(1 m)f m, m, Mk f m, m = m(1 m)f m, m Conformal weights m = 1/ + iν + n/, m = 1/ + iν n/ Anomalous dimension γ = 1/ + iν, Conformal spin n )

Holomorphic separability H = 1 (h + h ), [h, h ] = 0 holomorphic and anti-holomorphic Hamiltonians h = n h k,k+1, h = k=1 BFKL operator n k=1 h k,k+1 h k,k+1 = log(p k )+log(p k+1 )+ 1 p k log(ρ k,k+1 )p k + 1 p k+1 log(ρ k,k+1 )p k+1 +γ here ρ k,k+1 = ρ k ρ k+1, p k = i /( ρ k ), γ = ψ(1)-euler const f m, m ( ρ 1,..., ρ n ; ρ 0 ) = r,l wave function factorizes c r,l f r m(ρ 1,..., ρ n ; ρ 0 )f l m(ρ 1,..., ρ n; ρ 0)

Duality Symmetry Two different normalization conditions for w.f. n n f A = d ρ r ρ 1 r,r+1 f, f B = r=1 r=1 (Lipatov 98) n n d ρ r p r f r=1 r=1 Indeed, there are two similarity transformations to relate h t with h h t = n n p r h pr 1 = r=1 r=1 n r=1 ρ 1 r,r+1 h n r=1 ρ r,r+1 = [h, A] = 0, A = ρ 1 ρ 3...ρ n1 p 1 p...p n Moreover, there is a family of mutually commuting integrals of motion [q r, q s ] = 0, [q r, h] = 0 as q r = n i 1<i <...<i r ρ i1,i ρ i,i 3...ρ ir,i 1 p i1 p i...p in

Duality Symmetry Integrals of motion q r = n i 1<i <...<i r ρ i1,i ρ i,i 3...ρ ir,i 1 p i1 p i...p in and h are invariant under i i + 1 ( Bose symmetry at N c ). Also have duality symmetry h ρ = ρ i 1,i p i ρ i,i+1 obvious from representation h = h ρ + h ( p ) n h p = ln(p k ) + 1 ρ k,k+λ ln(p k )ρ 1 k,k+λ + γ k=1 λ=±1 ( ) n ln(ρ k,k+1 ) + 1 p 1 k+(1+λ)/ ln(ρ k,k+1)p k+(1+λ)/ + γ k=1 λ=±1 Duality symmetry is realized as unitary transformation only for n p = p r = 0 r=1 then one can parametrize gluon momenta p r = x r x r+1

Duality Symmetry as Integral Equation For p = 0 the w.f. of the composite state of n reggeized gluons is ψ m, m ( ρ 1, ρ 3,..., ρ n1 ) = d ρ 0 π f m, m( ρ 1, ρ,..., ρ n ; ρ 0 ) The conformal weights have property m = 1 m, m = 1 m, thus (ψ m,m ( ρ 1, ρ 3,..., ρ n1 )) ψ 1 m,1 m ( ρ 1, ρ 3,..., ρ n1 ) and the duality symmetry can be written as integral equation n 1 Y ψ m, m( ρ 1,..., ρ n1) = λ m n k=1 d ρ k 1,k π ny k=1 e i ρ k,k+1 ρ k ρ k,k+1 ψ m,m( ρ 1,..., ρ n1 ) normalization is from Aψ m, m = λ mψ m, m, A = ρ 1...ρ n1p 1...p n and µ = Duality symmetry is related to the integrability of multicolor QCD (Faddeev and Korchemsky, 94) and used to solve Odderon in QCD (Lipatov, 98 ).

Reggeized gluons vs color(less) dipoles q q For N c = SU(N c ) U(N c ) BFKL (BK ) in dipole model (Mueller 94, Balitsky 95, Kovchegov 99 ) N(x 1, x ) = αsnc d x1 x 13 x13 {N(x 1, x 3 ) + N(x 3, x ) N(x 1, x ) N(x 1, x 3 )N(x 3, x )} x 3 Restores unitarity N 1 at y However, BK is solved only numerically and ignores subleading corrections in 1/Nc (present in Balitsky formulation)

Duality symmetry: dipoles vs gluons (A.P.,ariv:0908.386 [hep-ph],ariv:0911.579 [hep-ph]) BFKL in dipole model for impact parameter b = (x 1 + x )/ = 0 F(k, q) N(x 1 ) = αsnc αsnc 4π = α sn c d x1 x 13 x13 x 3 {N(x 13 ) N(x 1 )} BFKL in original formulation d k χ χ (χ k) + (k q) (χ q) (χ k) q «χ (χ q) F(χ, q) d k αsnc χ χ F(k, q) (χ k) 4π d (k q) χ χ F(k, q) (χ k + q) F(k) = α sn c for transferred momentum q = 0 ( d k ) χ χ (χ k) F(χ) α sn c 4π k d χ χ (χ k) F(k) the same equation for N(x 1 ) and F(k)!

Commonly used Fourier transform is not suitable F(k) 1 d x 1 (π) x1 e ik x1 N(x 1 ) Conformal invariance of BFKL allows to use dimensionless coordinates ρ ij = x ij x ij, χ i = k i k, κ = k k and define N (κ) 1 (π) d ρ 1 e iκ ρ1 N(ρ 1 ) this Fourier transform preserves duality symmetry. Linear BFKL is easily transformed to itself. What about the non-linear BK term?

N(x 1, x ) = αsnc BK equation d x1 x 13 x13 {N(x 1, x 3 ) + N(x 3, x ) N(x 1, x ) N(x 1, x 3 )N(x 3, x )} x 3 Open up the Kernel x1 x13 x13 = x 3 x13 x «3 x3 = 1 x13 + x 13 x 3 x13 x3 + 1 x3 d 1 x 3 x13 N(x 13 )N(x 3 ) = d χ 1 d χ 1 χ χ χ 1 χ N (χ 1 + χ κ)n (κ) d x 13 x 3 x 3 x13 x3 N(x 13 )N(x 3 ) = d χ 1 d χ 1 χ χ χ 1 χ N (χ 1 κ)n (χ + κ) Dipole size conservation x 13 x 3 = x 1 is built-in in the dipole model. Apply constraint dual to dipole size conservation d x 3 x 13 x 13 x 3 x3 d 1 x 3 x13 N(x 13 )N(x 3 )= δ () (κ) N(x 13 )N(x 3 )= d χ 1 χ 1 κ χ 1 χ 1 (χ 1 κ) N (χ 1 κ)n (χ 1 ) The last terms is self-dual and the BK equation has duality symmetry (in the absence of b dependence)! Consistent with 4 vertex for reggeized gluons (Bartels and Wusthoff 94)

Non-forward BFKL q 0 BFKL for q = 0 F(k) = αsnc d χ k χ (χ k) «F(χ) αsnc 4π d k χ χ (χ k) F(k) BFKL for q 0 F(k, k q) =+ αsnc + αsnc αsnc d χ k αsnc χ F(χ, χ q) (χ k) 4π d χ (k q) αsnc (χ q) F(χ, χ q) (χ k) 4π d χ q χ F(χ, χ q) (χ q) d χ k χ F(k, k q) (χ k) d χ (k q) (χ q) F(k, k q) (χ k) k x x k q k k q q k q k q x x x = + + x x x x x x x Non forward Uncut Cut Uncut Forward Forward Forward Similar to UCU structure found in RFT by Ciafaloni, Marchesini and Veneziano 75

Non-forward BFKL q 0 Introduce dual coordinates (with dimensions of mass!) k = k 1 = x 1 x = x 1 q k = k = x x 3 = x 3 q = k 3 = x 3 x 1 = x 31 k i = 0 i k x x x x k q x x x x F(x 1, x 3 ) = + αsnc + αsnc αsnc d z x1 j z (z x 1 ) F(z, z + x 31 ) 1 ff F(x 1, x 3 ) d z x 3 z (z + x 3 ) d z x 31 z (z + x 31 ) F(z, z + x 31) j F(z x 31, z) 1 ff F(x 1, x 3 )

Non-diagonal dipole (Levin and A.P.,Phys.Rev.C78:0650,008) Color dipole with different sizes to the left and to the right of the 1 unitarity cut 1 Described by function M(1; 1 ), which has a meaning of the total cross section, since for = it becomes M(1; 1) = N(1) (σ tot = ImA Optical Theorem) M(1 1 ) = ᾱs π 8 d < ρ 3 : 1 ρ 13 ρ 13 ρ 3 ρ 3! 0 M(1 1 ) 1 @ ρ 13 ρ 13 1 ρ 3 A ρ M(1 1 ) 3 + ρ 13 ρ ρ! 0 3 @ ρ 13 13 ρ 3 ρ 13 1 ρ 3 A n ρ N(13) + M(3 3 o ) 3 0 1 0 1 @ ρ 3 ρ 3 ρ 3 A ρ 3 @ ρ 13 ρ 13 Solved by (also non-linear version) Also has UCU structure! ρ 13 ρ ρ! 0 3 @ ρ 3 13 ρ 3 ρ 3 0 ρ 3 A ρ M(1 13) 1 3 M(1 1 ) = N(1) + N(1 ) N( ) @ ρ 3 ρ 3 1 ρ 3 A ρ M(13 1 ) 3 1 ρ 9 3 A ρ M(1 1 = ) ; 3

Duality symmetry for q 0 (A.P.,ariv:0911.579 [hep-ph]) Using M(1 1 ) = N(1) + N(1 ) N( ) rewrite the non-diagonal dipole evolution as (N(1) + N(1 ) N( )) = ᾱs ρ 1 d ρ 3 π ρ {N(13) + N(3) N(1)} 13 ρ 3 + ᾱs ρ 1 d ρ 3 N(13) + N(3 π ρ ) N(1 ) 13 ρ 3 M(1 1 ) = ᾱs ρ 1 d ρ 3 π ρ 13 ρ 3 ρ 1 d ρ 3 + ᾱs π ρ 13 ρ 3 ᾱs ρ d ρ 3 N(3) + N(3 π ρ ) N( ) 3 ρ 3 j M(3 3 ) 1 M(1 1 )+M(3 ) 1 ff M(1 ) j M(3 3 ) 1 M(1 1 )+M(3 ) 1 ff M(1 ) ᾱ s π ρ d ρ 3 M(3 3 ) ρ 3 ρ 3 Extra terms compared to BFKL! They are removed imposing the BFKL condition

BFKL condition F(k, k q) =+ αsnc + αsnc αsnc d χ k αsnc χ F(χ, χ q) (χ k) 4π d χ (k q) αsnc (χ q) F(χ, χ q) (χ k) 4π d χ q χ F(χ, χ q) (χ q) d χ k χ F(k, k q) (χ k) Note F(k, k q) = F(k 1, k ) the arguments should satisfy k 1 k = q it is present in BFKL by construction. d χ (k q) (χ q) F(k, k q) (χ k) We can associate dual coordinates (with dimension of mass) with dipole coordinates ρ 1 = x 1 ; ρ 1 = x 3 ; ρ = x 31

BFKL equation in dual coordinates F(x 1, x 3 ) = + αsnc + αsnc αsnc Non-diagonal dipole equation M(ρ 1 ρ 1 ) ᾱ s = π + ᾱ s π ᾱ s π d z x 1 z (z x 1 ) d z x 3 z (z + x 3 ) d z x 31 z (z + x 31 ) F(z, z + x 31) ρ 1 d ρ 3 ρ 3 (ρ 3 ρ 1 ) ρ 1 d ρ 3 ρ 3 (ρ 3 ρ 1 ) j F(z, z + x 31 ) 1 ff F(x 1, x 3 ) j F(z x 31, z) 1 ff F(x 1, x 3 ) j M(ρ 3 ρ 3 + ρ ) 1 ff M(ρ 1 ρ 1 ) ρ d ρ 3 ρ 3 (ρ 3 + ρ ) M(ρ 3 ρ 3 + ρ ) In the associated coordinates ρ 1 = x 1 ; ρ 1 = x 3 ; ρ = x 31 j M(ρ 3 ρ ρ 3 ) 1 ff M(ρ 1 ρ 1 ) one obtains the same evolution equation! Extended form of the Duality Symmetry holds also for non-forward BFKL

Impact parameter and Initial Condition Duality symmetry can be viewed as a symmetry under rotation of the BFKL Kernel in the transverse plane from s-channel to t-channel. k k q K s channel t channel K 1 1 Non-trivial result: transferred momentum q is dual to difference in dipole sizes, and not to impact parameter. Impact parameter dependence is incompatible with the duality symmetry, since it brakes translational invariance of the dual coordinates. Impact parameter relates evolution to the initial conditions, whereas the duality symmetry deals only with evolution. Prescription: first, find a dual evolution description, then match initial condition according to the physical system.

Summary and Discussions Non-linear Balitsky-Kovchegov equation and non-forward BFKL also posses Duality Symmetry Uncut-Cut-Uncut structure allows to relate forward to non-forward BFKL solutions, the same can be done for BK UCU structure can be used for analysis of multiparticle production in QCD Duality symmetry can explain real-virtual mixing of BFKL Does duality symmetry hold beyond LLA?