Evolution of the transverse-momentum dependent gluon distribution at small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math>

Using the color dipole picture for photon-nucleus interactions at small <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>x</a:mi></a:math> together with the color glass condensate (CGC) effective theory, we demonstrate that the next-to-leading (NLO) order corrections to the cross section for the inclusive production of a pair of hard jets encode not only the Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov and Kovner (JIMWLK) evolution with decreasing <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>x</c:mi></c:math>, but also the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution of the gluon distribution function and the Collins-Soper-Sterman (CSS) evolution of the gluon transverse momentum dependent (TMD) distribution. The emergent CSS equation takes the form of a rate equation describing the evolution of the dijet distribution in the transverse momentum imbalance <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>K</e:mi><e:mo>⊥</e:mo></e:msub></e:math> when increasing the dijet relative momentum <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>P</g:mi><g:mo>⊥</g:mo></g:msub></g:math>. All three types of evolution become important when both <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>P</i:mi><i:mo>⊥</i:mo></i:msub></i:math> and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>K</k:mi><k:mo>⊥</k:mo></k:msub></k:math> are much larger than the nuclear saturation momentum <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi>Q</m:mi><m:mi>s</m:mi></m:msub><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:math> and we propose a framework which encompasses all of them. The solution to the JIMWLK equation provides the source term for the DGLAP evolution with increasing <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:msub><q:mi>K</q:mi><q:mo>⊥</q:mo></q:msub></q:math>, which in turn generates the initial condition for the CSS evolution with increasing <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>P</s:mi><s:mo>⊥</s:mo></s:msub></s:math>.