NLO JIMWLK evolution unabridged
Alex Kovner
1
Michael Lublinsky
0
Yair Mulian
0
0
Physics Department, Ben-Gurion University of the Negev
, Beer Sheva 84105,
Israel
1
Physics Department, University of Connecticut
, 2152 Hillside road, Storrs,
CT 06269, U.S.A
In ref. [1] we presented the JIMWLK Hamiltonian for high energy evolution of QCD amplitudes at the next-to-leading order accuracy in s. In the present paper we provide details of our original derivation, which was not reported in [1], and provide the Hamiltonian in the form appropriate for action on color singlet as well as color nonsinglet states. The rapidity evolution of the quark dipole generated by this Hamiltonian is computed and compared with the corresponding result of Balitsky and Chirilli [2]. We then establish the equivalence between the NLO JIMWLK Hamiltonian and the NLO version of the Balitsky's hierarchy [3], which includes action on nonsinglet combinations of Wilson lines. Finally, we present complete evolution equation for three-quark Wilson loop operator, thus extending the results of Grabovsky [4].
Contents
1 Introduction and conclusions
2 JIMWLK Hamiltonian at leading and next to leading orders
NLO JIMWLK for color non-singlet operators. Comparison with [3]
3.1 Self-interaction
3.2 Pairwise interaction
3.3 Triple interaction
NLO evolution of three-quark Wilson loop
4.1 LO evolution
4.2 NLO evolution
4.3 NLO evolution of conformal operator in N = 4
A Action of the NLO JIMWLK Hamiltonian on a dipole
B Useful identities
Introduction and conclusions
The JIMWLK Hamiltonian [513] is the limit of the QCD Reggeon Field Theory (RFT),
applicable for computations of high energy scattering amplitudes of dilute (small parton
number) projectiles on dense (nuclei) targets. In general it predicts rapidity evolution of
any hadronic observable O via the functional equation of the form
In refs. [513], the JIMWLK Hamiltonian was derived in the leading order in s in pQCD.
It contains a wealth of information about high energy evolution equations. In the
dilutedilute limit it reproduces the linear BFKL equation [1416] and its BKP extension [17, 18].
Beyond the dilute limit, the Hamiltonian incorporates non-linear effects responsible for
unitarization of scattering amplitudes. For practical applications, the JIMWLK evolution
is usually replaced by the Balitsky-Kovchegov (BK) non-linear evolution equation [1922],
which at large Nc describes the growth of the gluon density with energy and the gluon
saturation phenomenon [23] as reflected in the evolution of the scattering amplitude of a
fundamental dipole s
s(x, y) = 1 tr[S(x) S(y)] .
The Wilson line S(x), in the high energy eikonal approximation represents the scattering
amplitude of a quark at the transverse coordinate x. There exist numerous
phenomenological applications of the BK equation to DIS, heavy ion collisions and proton-proton
collisions at the LHC [24]. Successful BK phenomenology mandates inclusion of next to
leading order corrections, since at leading order the evolution predicted by the BK equation
is too rapid to describe experimental data. Currently only the running coupling corrections
are included in applications, although it is clearly desirable to include all next to leading
corrections.
The complete set of such corrections to the evolution of a fundamental dipole was
calculated by Balitsky and Chirilli [2], following on the earlier works [25, 26]. This result
generalizes the NLO BFKL equation [27, 28] and reduces to it in the linearized
approximation. Grabovsky [4] computed certain, connected, parts of the NLO evolution equation
for three-quark Wilson loop operator in the SU(3) theory (which we will sometimes refer
to as baryon)
B ijk lmn Sil(u)Sjm(v)Skn(w) .
Projected on the charge conjugation odd sector, the operator B is related to the odderon,
which at NLO was independently studied in [29].
The NLO extension of the JIMWLK framework is imperative for calculation of more
general amplitudes, beyond the dipole, which determine interesting experimental
observables like single- and double inclusive particle production. Thanks to the above mentioned
major progress in the NLO computations, in [1] we have presented the NLO JIMWLK
Hamiltonian which reproduces these results by simple algebraic application to the relevant
amplitudes. Ref. [1] appeared simultaneously with [3], which directly calculated many
elements of the general Balitskys hierarchy at NLO. Our construction in [1] was based
upon two major pieces of input. First, the general form of the NLO JIMWLK Hamiltonian
was deduced from the hadronic wave-function computation in the light cone perturbation
theory [30]. This allowed us to parametrize the Hamiltonian in terms of only five kernels.
These kernels were then fully reconstructed by comparing the evolution generated by the
Hamiltonian with the detailed results of [2] and [4].
Using a similar strategy, in ref. [31] (see also ref. [32]) we have constructed the NLO
JIMWLK Hamiltonian for N = 4 SUSY (...truncated)