Energy scale(s) and next-to-leading BFKL equation

Abstract

We complete the calculation of the next-to-leading kernel of the BFKL equation by disentangling its energy-scale dependent part from the impact factor corrections in largek dijet production. Using the irreducible part previously obtained, we derive the final form of the kernel eigenvalue and of the hard Pomeron shift for various scales. We also discuss the scale changes, the physical equivalence of a class of scales, and how to use the collinear safe ones. PACS 12.38.Cy 1 Work supported in part by EU Network contract FMRX-CT98-0194 and by M.U.R.S.T. (Italy) 1The theoretical effort devoted in the past few years [1-10] to the next-to-leading (NL) corrections to the BFKL equation [1], is now approaching its final steps [11,12].After the calculation of the relevant high-energy vertices [2][3][4][5][6][7] by Fadin, Lipatov and other authors, and of the qq part of the kernel [8,10] by the present authors, we computed the eigenvalue of the "irreducible" part of the gluonic kernel [9], by pointing out that the left-over "leading terms", after subtraction of a common scale, would lead to an additional contribution to the kernel, yet to be determined.The purpose of this note is to complete the calculation above with its missing part, and to discuss the energy-scale dependence of the kernel. Our results, which refer to large-k dijet production, are based on the separation of the NL kernel from the one-loop partonic impact factors, recently determined by one of us [11], and depending on some input scale s 0 . For the case s 0 = k 1 k 2 (where the k's are the transverse momenta of the jets), the full kernel derived here agrees with the one recently proposed by Fadin and Lipatov [12], who do not discuss the impact factors' contribution.

Rapidity evolution of gluon TMD from low to moderate x

“…The NLO corrections to the BFKL [44,45] and BK [17][18][19][20][46][47][48][49] equation are available but they suffer from the well-known problem that they lead to negative cross sections. This difficulty can be overcome by the "collinear resummation" of double-logarithmic contributions for the BFKL [50][51][52][53] and BK [54][55][56] equations and we hope that our eq.…”

“…Careful inspection of the expansions (A.19) and (A.30) reveals that there is no leading or next-to-leading terms with twist larger than four so we can recycle the final formulas (A. 45) …”

A proof of fulfillment of the strong bootstrap condition

“…This assumption was proved in the leading logarithmic approximation [14], but still remains a hypothesis beyond this approximation. Now a lot of results in the BFKL approach are obtained in the next-to-leading order for the kernel of the BFKL equation [2,3,4,5,6] and the impact factors [7,8,9,10,11,12,13]. These results are widely discussed in the literature.…”

The high-energy radiation pattern from BFKLex with double-log collinear contributions

“…One of the targets of the work here presented is to show how pre-asymptotic effects have sizable effects at present energies. This is the case for the simplest leading order (LO) [3][4][5][6][7][8] as well as for the more sophisticated next-to-leading order (NLO) [9,10] and higher-order calculations.…”