Solution of the NLO BFKL Equation (and a strategy for solving the all-order BFKL equation) Yuri Kovchegov The Ohio State University based on arXiv: [hep-ph] with Giovanni Chirilli

Solutions of Evolution Equations DGLAP equation – derived: 1972 (QED), 1977 (QCD) – solved: 1972, 1974 LO BFKL equation – derived: 1977, 1978 – solved: 1978 NLO BFKL equation – derived: 1998 (Fadin&Lipatov, Camici&Ciafaloni) – solved: in this talk (2013)

The Problem We want to find the BFKL Green function. It satisfies the BFKL equation with the initial condition We are working in the azimuthally-symmetric case for simplicity. K(k,q) represents a BFKL kernel at an unspecified order in  s. We need to find the eigenfunctions and eigenvalues for the kernel.

BFKL Equation in N=4 SYM Theory The form of the BFKL equation’s solution is straightforward to determine in N=4 SYM theory: there the eigenfunctions are fixed by conformal symmetry and are simply E n,. In the angle-independent case at hand we write the BFKL Green function in N=4 SYM theory as Perturbative expansion takes place in the exponent (the eigenvalue).

Solving BFKL Equation in QCD QCD is not a conformal theory: we can not fix the all-order BFKL eigenfunctions by a symmetry argument. While simple powers are eigenfunctions for the LO kernel, they are not eigenfunctions for the NLO kernel due to running coupling effects: LO BFKL eigenvalue 1-loop running coupling Conformal NLO terms NLO terms

The Strategy Since the BFKL kernel is known perturbatively up to NLO it appears logical to construct the eigenfunctions order-by- order in the coupling as well. (Solving NLO BFKL equation exactly would exceed the precision of the approximation.) To find the eigenfunctions we thus write and (perturbatively) impose the eigenfunction condition where the eigenvalue  is also an unknown.

NLO BFKL: Eigenfunctions We look for the NLO correction to the LO eigenfunction in the following form It turns out that the NLO eigenfunctions are obtained if we truncate the series at the quadratic order and fix for any c 0 and c 1. The eigenvalue is

Completeness We have the LO+NLO BFKL eigenfunctions: However, they also need to satisfy the completeness condition Imposing this fixes with and real (all calculations are done to order-  ).

NLO BFKL Eigenvalues We now have the eigenfunctions The corresponding eigenvalues are real, as expected for a hermitean operator such as the BFKL kernel! Note that the imaginary term canceled out to give a real eigenvalue! This term could also be removed by acting with the NLO BFKL kernel on (Fadin & Lipatov, ’98), but these are not NLO eigenfunctions.

Orthogonality The functions also satisfy the orthogonality condition Note that the orthogonality condition does not fix c 0 and Im[c 1 ].

Phase convention and the choice of Remaining degrees of freedom can be eliminated by: – Phase convention – Redefinition of the -variable These degrees of freedom allow us to put c 0 =0 and Im[c 1 ]=0 in the eigenfunctions and eigenvalues.

NLO BFKL Solution The NLO BFKL eigenfunctions are Note that the correction to the LO eigenfunctions is proportional to  2 : as expected it is entirely due to the running of the coupling.

NLO BFKL Solution The NLO BFKL eigenfunctions are The corresponding eigenvalues are The LO+NLO BFKL solution can be written as (main result!):

NLO BFKL Solution Note that the perturbative expansion is present both in the exponent and in the eigenfunctions (the prefactor). We have constructed NLO BFKL solution by expanding the eigenfunctions around the conformal LO eigenfunctions. The procedure can be repeated at higher orders in  s.

All-Order BFKL Solution The procedure can be iterated to any order in  s. The general form of the solution would look like with the perturbatively constructed eigenfunctions

Properties of the NLO BFKL Solution Using the explicit form of the eigenfunctions in the NLO solution, after some algebra one arrives at This expression is  -independent to order-  s, as expected for a solution of a  -independent equation with the  - independent initial condition (cf. Ivanov and Papa, ‘06). Our power counting is standard,

Properties of the NLO BFKL Solution The NLO solution can also be rewritten as This is just like in a conformal theory (e.g. N=4 SYM), except the couplings are now the running QCD couplings! This form is also explicitly  -independent.

NNLO BFKL One may then be tempted to guess that solves the NNLO BFKL equation. To verify this ansatz we need to know the running coupling terms in the NNLO kernel. They can be obtained from the NLO kernel: The two-loop QCD beta-function  3 is defined by

NNLO BFKL Unfortunately the NNLO solution ansatz does not work… (checked by a substitution) Instead we augmented the above ansatz to obtain NNLO BFKL solution valid only for Gives the  5 Y 3 term found in YK, Mueller ‘98; Levin ‘98; Armesto, Bartels, Braun ‘98; Ciafaloni, Taiuti, Mueller ’01 when evaluated at =0 saddle point. Needs to be verified by constructing perturbative NNLO eigenfunctions! (in progress)

DGLAP Anomalous Dimension To cross check our result we can obtain DGLAP anomalous dimension from it. To do this write (cf. Fadin&Lipatov ‘98) Plugging this into the Green function we found before yields The intercept is shifted by the same amount as in Fadin&Lipatov ‘98, and hence gives the same NNLO DGLAP anomalous dimension.

Conclusions We have derived a systematic way of constructing BFKL eigenfunctions and eigenvalues using a perturbative expansion around the conformal (LO) eigenfunctions. We used it to construct a solution for the NLO BFKL equation. Our method can be applied to solving BFKL equation with an arbitrary higher-order kernel. We have also conjectured the form of NNLO BFKL solution (not using the method!) Further applications will include applying our method to the azimuthal-angle-dependent case, and to NLO BFKL in coordinate space (E n, functions).

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