Collins-Soper-Sterman Hamiltonian: High energy evolution of rapidity dependent observables

We consider evolution of observables which depend on a small but fixed value of longitudinal momentum fraction <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>x</a:mi></a:math>, to high rapidity, such that <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>η</c:mi><c:mo>&gt;</c:mo><c:mi>ln</c:mi><c:mtext> </c:mtext><c:mn>1</c:mn><c:mo>/</c:mo><c:mi>x</c:mi></c:mrow></c:math>. We show that this evolution is not given by the Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) (or Balitsky-Kovchegov) equation. We derive the evolution Hamiltonian—<e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>H</e:mi><e:mrow><e:mi>CSS</e:mi><e:mo>−</e:mo><e:mi>JIMWLK</e:mi></e:mrow></e:msub></e:math> which generates this evolution in the cases of dilute and dense projectile wave function. The two limits yield identical results for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>H</g:mi><g:mrow><g:mi>CSS</g:mi><g:mo>−</g:mo><g:mi>JIMWLK</g:mi></g:mrow></g:msub></g:math>. We show that the resulting evolution for the gluon Transverse-Momentum-Dependent is identical to the (double logarithmic) perturbative Collins-Soper-Sterman evolution equation in the longitudinal resolution parameter at a fixed and very large transverse resolution.