Solution of the NLO BFKL Equation and γ γ cross-section at NLO

Solution of the NLO BFKL Equation and γ γ cross-section at NLO Giovanni Antonio Chirilli The Ohio State University Santa Fe - NM 13 May, 014 G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 1 / 8

Outline Leading-Log-Approximation at high-energy: general case. BFKL in N =4 SYM theory. Solution of the NLO BFKL equation in QCD. General Form of the Solution of Higher-Order BFKL equation. NLO photon impact factor and high-energy OPE γ γ cross section. Numerical results. Conclusions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 / 8

Linear Evolution equations in QCD DGLAP evolution equation Resum α s ln Q Λ QCD Eigenfunctions at any order: Powers of x B G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

Linear Evolution equations in QCD DGLAP evolution equation Resum α s ln Q Λ QCD Eigenfunctions at any order: Powers of x B BFKL evolution equation LO: resum (α s ln s) n. NLO: resum α s (α s ln s) n Eigenfunctions: LO: (k ) γ 1 NLO and higher order: Perturbative eigenfunctions (G.A.C. and Kovchegov). G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. Resum α s (α s Y) n NLO BFKL eq. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

Balitsky-Fadin-Kuraev-Lipatov equation Y G(k, k, Y) = d q K(k, q) G(q, k, Y) G(k, k, Y = 0) = 1 πk δ(k k ) k k and k k Y = ln s k k and s = (q 1 + q ) Resum (α s Y) n LO BFKL eq. Resum α s (α s Y) n NLO BFKL eq. The kernel is real and symmetric: K(k, k ) = K(k, k) K(k, k ) is Hermitian and the eigenvalues are real. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

LO BFKL equation Y G(k, k, Y) = d q K LO (k, q) G(q, k, Y) d q K LO (k, q) (q ) 1+γ = ᾱ µ χ 0 (γ)(k ) 1+γ ᾱ µ α µn c π (k ) 1+γ are eigenfunctions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 5 / 8

LO BFKL equation Y G(k, k, Y) = d q K LO (k, q) G(q, k, Y) d q K LO (k, q) (q ) 1+γ = ᾱ µ χ 0 (γ)(k ) 1+γ ᾱ µ α µn c π (k ) 1+γ are eigenfunctions. For γ = 1 + iν and ν real parameter (k ) 1+γ form a complete set. LO eigenvalues χ 0 (ν) = ψ(1) ψ( 1 + iν) ψ( 1 iν) are real and sym. ν ν LO BFKL is Conformal invariant. G(k, k, Y) = dν π k k ( k k ) iν eᾱµχ 0(ν) Y G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 5 / 8

BFKL equation in the N =4 SYM case In N = 4 SYM theory the coupling constant does not run. (k ) 1 +iν are eigenfunctions at any order. K(q, k) = α SYM K LO (q, k) + α SYM KNLO (q, k) +... G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 6 / 8

BFKL equation in the N =4 SYM case In N = 4 SYM theory the coupling constant does not run. (k ) 1 +iν are eigenfunctions at any order. K(q, k) = α SYM K LO (q, k) + α SYM KNLO (q, k) +... d q K(q, k)(q ) 1 +iν = [α SYM χ 0 (ν) + α SYM χ 1(ν)...](k ) 1 +iν G(k, k, Y) = ( dν π k k e[α SYMχ 0 (ν)+α SYM χ 1(ν)...] k ) iν k The eigenvalues ν ν. ᾱ µ χ 0 (ν) + ᾱ µ χ SYM 1 (ν) +... are real and symmetric for G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 6 / 8

γ γ scattering cross-section at LO Y A Y B <Y< Y A Y B G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 7 / 8

γ γ scattering cross-section at LO Y A Y B <Y< Y A Y B x y j α (x)j β (y)j ρ (x )j λ (y ) I αβ A (x, y; z 1, z ) I ρλ B (x, y ; z 1, z ) tr{u z1 U z } Y A tr{u z3 U z 4 } Y B G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 7 / 8

γ γ scattering cross-section at LO Y A e s 0 ( ) Y B <Y< Y A Y B e s 0 ( ) j α (x)j β (y)j ρ (x )j λ (y ) I αβ A (x, y; z 1, z ) I ρλ B (x, y ; z 1, z ) tr{u z1 U z } Y A A tr{u z3 U z 4 } Y B A G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 7 / 8

γ γ scattering cross-section at LO { U x = Pexp ig + dx + A (x + + x ) } α s A αβ ρλ (q 1, q ) i Q 1 Q ( dν I αβ Q ) iν LO (ν) Iρλ LO (ν) 1 χ 0(ν)(Y A Y B ) Q eᾱµ Y A = 1 ln s Q, Y B = 1 1 ln s Q, s = (q 1 + q ) α s A αβ ρλ (q 1, q ) i Q 1 Q ( dν I αβ Q ) iν LO (ν) Iρλ LO (ν) 1 χ 0(ν) ln Q eᾱµ s Q 1 Q G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 8 / 8

What about NLO? Can we repit the steps performed at LO also at NLO? Problems to be solved: Solve NLO BFKL equation G.A.C and Yu. Kovchegov Calculate NLO Impact Factor I. Balitsky and G.A.C. NLO Impact Factor has to be conformal invariant; Energy dependence of NLO Impact Factor needs to be eliminated; Composite Wilson line operators I. Balisky and G.A.C. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 9 / 8

BFKL equation at NLO in QCD K LO+NLO (k, q) ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) d q K LO+NLO (k, q) q γ = [ᾱ µ χ 0 (γ) ᾱ µ β χ 0 (γ) ln k µ + ᾱ µ ] δ(γ) k γ 4 ᾱ µ = α µn c π, β = 11 N c N f 1 N c G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 10 / 8

BFKL equation at NLO in QCD K LO+NLO (k, q) ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) d q K LO+NLO (k, q) q γ = [ᾱ µ χ 0 (γ) ᾱ µ β χ 0 (γ) ln k µ + ᾱ µ ] δ(γ) k γ 4 ᾱ µ = α µn c π, β = 11 N c N f 1 N c ᾱ µ β χ 0 (γ) ln k µ 1-loop running coupling. δ(γ) = β χ 0 (γ) + 4χ 1(γ) NLO Conformal terms Fadin-Lipatov (1998) χ 1 (γ) Real and symmetric in γ 1 γ γ = 1 + iν. d dγ χ 0(γ) χ 0 (γ) imaginary and not symmetric. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 10 / 8

NLO BFKL eigenfunctions G.A.C. and Yu. Kovchegov Perturbative eigenfunctions H γ (k) = k γ + ᾱ µ F γ (k) we have to determine F γ (k) so that d q K LO+NLO (k, q) H γ (q) = (γ) H γ (k) (γ) eigenvalues to be determined. F γ (k) has to satisfy: χ 0 (γ) F γ (k) + d q K LO (k, q)f γ (q) β χ 0 (γ) H γ (k) ln k µ = c(γ) H γ(k) For some function c(γ). G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 11 / 8

NLO eigenfunctions ansatz At this order we can write the condition for F γ (k) as: χ 0 (γ)f γ (k) + d q K LO (k, q)f γ (q) β χ 0 (γ) k γ ln k = c(γ) kγ µ Ansatz [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ + c 3(γ) ln 3 k µ +... G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 1 / 8

NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 13 / 8

NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ For c (γ) = β χ 0 (γ) χ 0 (γ) and for any c 0 (γ) and c 1 (γ) the eigenfunction is [ ( )] H γ (k) = k γ β χ 0 (γ) k 1 + ᾱ µ ln (γ) µ + c 1(γ) ln k µ + c 0(γ) and eigenvalues (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 0 ( 1 β χ 0 (γ) + χ 1(γ) + c 1 (γ)χ 0 (γ) + β ) χ 0 (γ) χ 0 (γ) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 13 / 8

NLO eigenfunctions Truncate the series at n = [ ] F γ (k) = k γ c 0 (γ) + c 1 (γ) ln k µ + c (γ) ln k µ For c (γ) = β χ 0 (γ) χ 0 (γ) and for any c 0 (γ) and c 1 (γ) the eigenfunction is [ ( )] H γ (k) = k γ β χ 0 (γ) k 1 + ᾱ µ ln (γ) µ + c 1(γ) ln k µ + c 0(γ) and eigenvalues (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 0 ( 1 β χ 0(γ) + χ 1 (γ) + c 1 (γ)χ 0(γ) + β ) χ 0 (γ) χ 0 (γ) Next step is to impose Completeness relation. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 13 / 8

Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 14 / 8

Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. Completeness relation at LO is satisfied with σ = 1 γ = 1 + iν with ν real parameter. we have to impose α µ -order to be 0 and Re[c 1 (ν)] = β χ 0 (ν) ν χ 0 (ν) ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 14 / 8

Completeness of the NLO eigenfunctions σ+i σ i dγ πi H γ(k) H γ (k ) = δ(k k ) Completeness relation has to be satisfied order-by-order in α µ. Completeness relation at LO is satisfied with σ = 1 γ = 1 + iν with ν real parameter. we have to impose α µ -order to be 0 and Re[c 1 (ν)] = β χ 0 (ν) ν χ 0 (ν) ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 If completeness relation is satisfied then orthogonality relation is also satisfied. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 14 / 8

Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 15 / 8

Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] and the eigenvalues are real (Im[c 1 (ν)]χ 0 (ν) is real function of ν) (ν) = ᾱ µ χ 0 (ν) + ᾱ µ [ ] χ 1 (ν) + Im[c 1 (ν)]χ 0(ν) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 15 / 8

Completeness of the NLO H-eigenfunctions provided that H 1 +iν(k) = k 1+iν ν Im[c 1(ν)] + Re[c 0 (ν)] = 0 [1 + ᾱ µ ( i β χ 0 (ν) k χ ln 0 (ν) µ + β ( χ 0 (ν) ν χ 0 (ν) the eigenfunctions are ) )] ln k µ + iim[c 1(ν)] ln k µ + Re[c 0(ν)] and the eigenvalues are real (Im[c 1 (ν)]χ 0 (ν) is real function of ν) (ν) = ᾱ µ χ 0 (ν) + ᾱ µ [ ] χ 1 (ν) + Im[c 1 (ν)]χ 0(ν) The imaginary term β ᾱ µχ 0 (γ) has canceled. We have freedom to chose Re[c 0 (ν)] and Im[c 1 (ν)] This freedom will not affect the solution. It is just an artifact of the phase and ν-reparametrization freedom of the H 1 +iν function. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 15 / 8

Phase freedom and ν-reparametrization Phase freedom of the H-functions: H 1 +iν(k) e iᾱµ(im[c 0(ν)] Im[c 1 (ν)] ln µ ) H 1 +iν(k) ν-reparametrization: ν = ν + ᾱ µ Im[c 1 (ν)] The Phase freedom and ν-reparametrization allow us to put c 0 (ν) = 0 and Im[c 1 (ν)] = 0. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 16 / 8

Solution of NLO BFKL equation NLO eigenfunctions: perturbation around the conformal LO eigenfunctions ( i ν H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ 0 (ν) k ln (ν) µ + 1 χ 0 ( ν ) )] χ 0 (ν) χ 0 (ν) ln k µ NLO eigenvalues (ν) = ᾱ µ χ 0 (ν) + ᾱ µ χ 1(ν) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 17 / 8

Solution of NLO BFKL equation NLO eigenfunctions: perturbation around the conformal LO eigenfunctions ( i ν H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ 0 (ν) k ln (ν) µ + 1 χ 0 ( ν ) )] χ 0 (ν) χ 0 (ν) ln k µ NLO eigenvalues (ν) = ᾱ µ χ 0 (ν) + ᾱ µ χ 1(ν) Solution of NLO BFKL equation G.A.C. and Yu. Kovchegov G(k, k, Y) = dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] Y H 1 +i ν(k) H 1 +i ν(k ) The perturbative expansion is in both the exponent and in the eigenfunctions (contrary to DGLAP case and N =4 BFKL). G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 17 / 8

µ-independence of the NLO solution The H γ (k) eigenfunctions diagonalize the LO+NLO BFKL kernel ᾱ µ K LO (k, q) + ᾱ µ KNLO (k, q) = 1 +i 1 i dγ π i (γ) H γ(k)h γ (q) LO+NLO BFKL kernel is µ-independent up to O(α 3 µ ) So is its diagonalization through H γ (k) eigenfunctions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 18 / 8

µ-independence of the NLO solution G(k, k, Y) = = dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1 (ν)] Y H γ (k)h γ (q) dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1(ν)] Y ( k G(k, k, Y) is µ-independent up to order O(α 3 µ). k ) i ν ) (1 ᾱ µ β χ 0 (ν) Y ln kk µ G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 19 / 8

µ-independence of the NLO solution G(k, k, Y) = = dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1 (ν)] Y H γ (k)h γ (q) dν π k k e[ᾱµ χ 0(ν)+ᾱ µ χ 1(ν)] Y ( k G(k, k, Y) is µ-independent up to order O(α 3 µ). At NLO we may write the solution as k ) i ν ) (1 ᾱ µ β χ 0 (ν) Y ln kk µ G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k ) χ 1 (ν)] Y ( k k ) i ν At this order the scale ᾱ λ s (k ) ᾱ λ s (k )ᾱs 1 λ (k k ) (for real λ) works as well. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 19 / 8

General Form of the Solution of All-Order BFKL equation Ansatz G(k, k, Y) = dν π e[ᾱs(k k ) χ 0(ν)+ᾱ s (k k ) χ 1(ν)+ᾱ 3 s (k k ) χ (ν)+...] Y χ (ν) and higher-order coefficients indicated by the ellipsis in the exponent are the scale-invariant (conformal) (ν ν)-even (real-valued) parts of the prefactor function generated by the action of the next-to-next-to-leading-order (NNLO) (and higher-order) kernels on the LO eigenfunctions. ( k k ) i ν G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 0 / 8

General Form of the Solution of All-Order BFKL equation Ansatz G(k, k, Y) = dν π e[ᾱs(k k ) χ 0(ν)+ᾱ s (k k ) χ 1(ν)+ᾱ 3 s (k k ) χ (ν)+...] Y To check the ansatz we have to plug it in the evolution eq. and we need the two-loop beta-function β 3 : ( k k ) i ν and µ dᾱ µ dµ = β ᾱ µ + β 3ᾱ 3 µ { d q K LO+NLO+NNLO (k, q) q 1+iν = ᾱ µ χ 0 (ν) [1 ᾱ µ β ln k +ᾱ µ [ i β χ 0 (ν) + χ 1(ν) ] [1 ᾱ µ β ln k µ k ln µ + ᾱ µ β 3 ln k µ } µ + ᾱ µ β ] + ᾱ 3 µ [χ (ν) + iδ (ν)] ] k 1+iν G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 0 / 8

General Form of the Solution of All-Order BFKL equation The ansatz does not work, but it allows us to recover the structure of the NNLO solution G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k )χ 1 (ν)+ᾱ 3 s (k k ) χ (ν)]y {1 + (ᾱ µ β ) [ 1 4 (ᾱ µy) 3 χ 0 (ν) χ 0(ν) + 1 4 (ᾱ µy) χ 0 (ν) provided that the imaginary part iδ (γ) is ( k k ( χ 0 (ν) ) i ν χ 0 (ν) χ 0(ν) ) + ᾱ µ Y χ 0 (ν) 4 ] } iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β This solution satisfies also the initial condition: the solution is unique so it is the right one. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 1 / 8

General Form of the Solution of All-Order BFKL equation The ansatz does not work, but it allows us to recover the structure of the NNLO solution G(k, k, Y) = dν π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s (k k )χ 1 (ν)+ᾱ 3 s (k k ) χ (ν)]y {1 + (ᾱ µ β ) [ 1 4 (ᾱ µy) 3 χ 0 (ν) χ 0(ν) + 1 4 (ᾱ µy) χ 0 (ν) provided that the imaginary part iδ (γ) is ( k k ( χ 0 (ν) ) i ν χ 0 (ν) χ 0(ν) ) + ᾱ µ Y χ 0 (ν) 4 ] } iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β This solution satisfies also the initial condition: the solution is unique so it is the right one. The imaginary part iδ (ν) has to be confirmed from the explicit calculation of the NNLO eigenfunction. So, let us calculate explicitly the NNLO eigenfunction. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 1 / 8

General Form of the Solution of All-Order BFKL equation The eigenfunction of the NNLO BFKL equation is ( i ν χ 0 (ν) k H 1 +i ν(k) = k 1+ [1 + ᾱ µ β i χ ln 0 (ν) µ + 1 ( ) ] k + ᾱ µ f µ,ν +... ( ν ) ) χ 0 (ν) χ 0 (ν) ln k µ The function f (k/µ,ν) denotes the NNLO corrections to the eigenfunctions. Ansatz for f (k/µ,ν): f (k/µ,ν) = c () 0 k (ν) + c() 1 (ν) ln µ + c() k (ν) ln µ + c() 3 (ν) ln3 k µ +... We have completely determined the c () 0 (ν), c() 1 (ν), c() (ν), c() 3 (ν) and confirmed the ansatz for the structure of the NNLO solution of the NNLO BFKL equation. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 / 8

γ γ scattering cross-section at NLO x y Y A Y B <Y< Y A x y Y B G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

γ γ scattering cross-section at NLO x y x y x y Y A Y B <Y< Y A + +... Y B x y y x x y Using high-energy Operator Product Expansion in composite Wilson line operator we get NLO Impact Factor that does not scale with energy. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

γ γ scattering cross-section at NLO x y x y x y Y A Y B <Y< Y A + +... Y B x y y x x y Composite Wilson line operators I. Balitsky and G.A.C. [tr{ûz η 1 Û z η } ] conf + α s 4π = tr{û η z 1 Û η z } d z 1 z 3 z [ 1 tr{û 13 z z η 3 N 1 Û z η 3 }tr{ûz η 3 Û z η } tr{ûz η 1 Û z η }] ln az 1 c z 13 z 3 + O(α s ) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

NLO Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. d 4 x e iq x p T{ĵ µ (x)ĵ ν (0)} p = s d k I µν (q, k )V a=xb (k ) k where the evolution of the dipole gluon distribution at NLO is a d da V a(k) = α sn c π ( 1 + α sb 4π [ Va (k ) (k k ) ln (k k ) + α sn c 4π d k {[ V a (k ) (k k ) (k, k )V a (k) ] k (k k ) [ ln µ k + N c(67 b 9 π 3 10n f ) ]) bα s 9N c 4π k V a (k) k k (k k ) ln (k k ) ] k [ ln (k /k ) (k k ) + F(k, k ) + Φ(k, k ) ] } V a (k ) + 3 α s N c π ζ(3)v a(k) V(k ) d z e i(k,z) V(z ) V(z ) = [1 1 N c Tr{U(z )U (0)}] G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

NLO Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. I µν (q, k ) = N c dν sinh πν 3 πν (1 + ν ) cosh πν ( 11 + 4 + 3ν)( 1 + α s ( k ) 1 iν{[( 9 Q ) ]} π + α sn c π F (ν) P µν 4 + ν)( 1 + α s π + α sn c ) π F 1(ν) P µν 1 P µν 1 = g µν q µq ν q P µν = 1 ( q q µ pµ q )( q ν pν q ) q p q p G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

NLO Photon Impact Factor for BFKL pomeron in momentum space k T -factorization form I. Balitsky and G.A.C. ( F 1() (ν) = Φ 1() (ν) + χ γ Ψ(ν), F 3 (ν) = F 6 (ν) + χ γ 1 ) Ψ(ν), γγ Ψ(ν) ψ( γ) + ψ( γ) ψ(4 γ) ψ( + γ), F 6 (γ) = F(γ) C γγ 1 γ γ 31 + χ γ 1 γ γ, + γγ Φ 1 (ν) = F(γ) + 3χ γ + γγ + 1 + 5 18( γ) + 1 γ 1 γ 7 18(1 + γ) + 10 3(1 + γ) Φ (ν) = F(γ) + 3χ γ + γγ + 1 + 1 γγ 7 ( + 3 γγ) + χ γ 1 + γ + χ γ(1 + 3γ) + 3 γγ, F(γ) = π 3 π sin πγ Cχ γ + χ γ γγ γ = 1 + iν G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 4 / 8

γ γ scattering cross-section at NLO G.A.C. and Yu. Kovchegov * I Q 1 I * I Q G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 5 / 8

γ γ scattering cross-section at NLO G.A.C. and Yu. Kovchegov * I Q 1 I * I Q σ γ γ LO+NLO (TT) = 1 4 λ 1,λ =±1 [ dν ( Q 1 Q 3π 6 ε λ1 ρ 1 (q 1 )ε λ1 σ 1 (q 1 )ε λ (q )ε λ σ (q ) N c 1 ρ N c α s (Q 1 )α s(q ) Q 1 Q ) iν ( )ᾱs(q s 1Q ) χ 0(ν)+ᾱ s (Q1Q)χ1(ν) Ĩ ρ1σ1 LO+NLO Q 1 Q (q 1,ν)Ĩ ρσ LO+NLO (q, ν) ] 1 + ᾱ s (Q 1 Q )Re[F(ν)] Re[F(ν)] is the NLO dipole-dipole scattering projected on the LO eigenfunctions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 5 / 8

Numerical results ΣTT, GeV ΣLL, GeV 0.0 0.10 0.15 0.05 0.10 0.05 4 6 8 10 Y 4 6 8 10 Y 0.05 0.05 0.10 NLO γ γ cross section with Q 1 = Q = 5 GeV (dashed line) and Q 1 = Q = 10 GeV (solid line) plotted as functions of rapidity Y. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 6 / 8

Numerical results Left panel depicts a Feynman diagram that contributes to the high-energy asymptotics of γ γ scattering. The right panel shows an example of the "box" diagram that are suppressed at high energies by a power of energy and, therefore, can be neglected. However, such types of diagrams become relevant, and therefore not anymore negligible at low rapidity where energy is not high enough to suppress such contributions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 7 / 8

Conclusions The NLO BFKL eigenfunctions have been constructed: they satisfy completeness and orthogonality condition NLO BFKL solution. NNLO Eigenfunctions has also been presented The structure of the NNLO solution has been found up to the still unknown conformal contribution χ (ν); but we predicted the imaginary part of the NNLO BFKL equation. Procedure to construct the solution of the BFKL equation to any order is now available. NLO γ γ cross-section using the solution of the NLO BFKL equation and the NLO impact factor calculated through the composite Wilson line operators. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 8 / 8

NNLO BFKL Solution: the NNLO eigenfunction This time we truncated the series up to c () 4 and proceeding similarly to the NLO case and we get ( Re[c () χ 1 ] = β 1 (ν)) χ 0 (ν) χ 1(ν)χ 0 (ν) χ 0 (ν) χ 1 (ν)χ 0 (ν) χ 0 (ν) + χ ) 0(ν)χ 1 (ν)χ 0 (ν) χ 3 0 ( (ν) 1 β 3 χ 0(ν)χ 0 (ν) ) χ 0 (ν) c () = β ( 5 χ 0 (ν)χ 0 (ν) 8 χ 4 0 (ν) χ0(ν)χ 0 (ν) 4χ 0 (ν) χ 0 (ν)χ 0 (ν) 4χ 3 0 (ν) 1 ) 8 ) χ 0 (ν) iβ 3 χ 0 (ν) ( χ1 (ν) +iβ χ 0 (ν) χ 0(ν)χ 1 (ν) χ 0 (ν) ( c () 3 = iβ 5 χ 0 (ν)χ 0 (ν) 1 χ 3 0 (ν) + χ ) 0(ν) 4χ 0 (ν) 4 = β χ 0 (ν) 8χ (ν) The Im[c () (ν)] is fixed to be 0 using again the ν-reparametrization. c () 1 G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 9 / 8

Structure of the NNLO BFKL Solution From explicit calculation of f (γ) we not only confirm the imaginary part iδ (ν) = i χ 0 (ν)β 3 + iχ 1 (ν)β but also confirm the structure of the NNLO BFKL solution obtained above in an indirect way ( ) G(k, k dν, Y) = π k k e[ᾱs(k k ) χ 0 (ν)+ᾱ s(k k ) χ 1 (ν)+ᾱ 3 s(k k ) χ (ν)] Y k i ν k [ {1 + (ᾱ µ β ) 1 4 (ᾱ µ Y) 3 χ 0 (ν) χ 0 (ν) + 1 ( χ 4 (ᾱ µ Y) χ 0 (ν) 0 (ν) ) χ 0 (ν) χ 0 (ν) + ᾱ µ Y χ 0 (ν) ] } 4 It looks like QCD is not just conformal part and running coupling contributions. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 30 / 8

BFKL at NNLO and higher orders VS. At NNLO there are Pomeron loop contributions which are inevitable in the symmetric case of γ -γ case a simple generalization of BFKL at NNLO (and higher) does not exist unless we consider large N c limit. In the asymmetric case of DIS, there are no pomeron loops. Linearization (with large N c limit) of the Balitsky-Kovchegov equation at any order provides a systematic procedure to consistently define a BFKL type of evolution equation at any order. BFKL to all order: linearization of all order BK equation in large N c limit. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 31 / 8

BFKL equation in DIS case K BFKL is k k symmetric Y sym = ln s k k G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

BFKL equation in DIS case VS. K BFKL is k k symmetric Y sym = ln s k k NLO K DIS BFKL Y DIS = ln s k ln 1 x B is not Symmetric G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

BFKL equation in DIS case VS. K BFKL is k k symmetric Y sym = ln s k k KNLO DIS = Ksym NLO 1 d D q K LO (q 1, q ) ln q q K LO(q, q ) NLO K DIS BFKL Y DIS = ln s k ln 1 x B is not Symmetric Fadin Lipatov(1998) KBFKL DIS is not symmetric eigenvalues not γ 1 γ symmetric. Eigenvalues get an extra term: DIS (γ) = sym (γ) 1 ᾱ µ χ 0 (γ)χ (γ) Reproduced lower order and predicted (and later confirmed) the 3-loop DGLAP anomalous dimension (Fadin-Lipatov (1998)) G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 3 / 8

DGLAP anomalous dimension from NLO BFKL solution G(k, k, Y) = dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] (Y DIS +ln k k ) H 1 +i ν(k) H 1 +i ν(k ) dν [ ] χ 0(ν)+ᾱ π e[ᾱµ µ χ 1(ν)] Y DIS H 1 +i ν(k) H 1 +i ν(k ) (1 + ᾱ µ χ 0 (ν) ln k ) k perform partial integration and exponentiate the Y DIS -dependent terms [ )] G(k, k, Y DIS dν [ ] ) = π e ᾱ µ χ 0 (ν)+ᾱ µ (χ 1 (ν)+ i χ 0 (ν)χ 0 (ν) Y DIS H 1 +i ν(k) H 1 +i ν(k ) ( 1 + i ) ᾱµ χ 0(ν) DIS (γ) = ᾱ µ χ 0 (γ) + ᾱ µ χ 1 (γ) 1 ᾱ µ χ 0 (γ)χ 0 (γ) Agrees with DGLAP 3-loop anomalous dimension. G. A. Chirilli (The Ohio State Uni.) NLO BFKL solu. & NLO γ γ Santa Fe - NM - 1-16 May, 014 33 / 8