Small-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math> evolution of the dipole amplitude in momentum space: Forward-off-forward correspondence

We have shown that the small-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>x</a:mi></a:math> evolution of the leading-log dipole scattering amplitudes, both pomeron and odderon, in the momentum space can be completely determined by the evolution of the respective amplitudes, with rescaled momenta. In position space, if there is translation symmetry (assumption of a large nucleus), the dipole cross section depends on the positions of quarks and antiquarks only through their separation. The present study is an equivalent proposition in the momentum space—where translation symmetry in momentum bifurcates the amplitudes into two translationally symmetric functions along the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi mathvariant="bold-italic">k</c:mi></c:math> line in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="bold-italic">k</f:mi><f:mo>−</f:mo><f:mi mathvariant="bold">Δ</f:mi></f:math> plane. It also shows that high energy evolutions of dipole generalized transverse momentum-dependent distributions can be achieved only by studying the evolution of dipole transverse momentum-dependent distributions at small-<j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>x</j:mi></j:math>.