Small- x asymptotics of the leading-twist flavor-singlet quark TMDs

In this paper, we investigate the small- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>x</a:mi> </a:math> behavior of the flavor-singlet, leading-twist quark transverse-momentum-dependent parton distribution functions (TMDs) using the light-cone operator treatment. This formalism allows us to express TMD operators at small <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>x</c:mi> </c:math> in terms of polarized dipole amplitudes, enabling a systematic approach to their small- <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>x</e:mi> </e:math> evolution. We derive the evolution equations for these TMDs and solve them within the large- <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>N</g:mi> <g:mi>c</g:mi> </g:msub> </g:math> approximation under the linearized, double-logarithmic approximation, where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>N</i:mi> <i:mi>c</i:mi> </i:msub> </i:math> represents the number of quark colors. Expanding on previous work on unpolarized and helicity TMDs, we present the small- <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>x</k:mi> </k:math> asymptotics for a comprehensive set of TMDs, including the Sivers function, helicity worm-gear, transversity, pretzelosity, Boer-Mulders, and transversity worm-gear distributions. Our results provide a complete picture of the small- <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>x</m:mi> </m:math> asymptotic behavior for all leading-twist flavor-singlet quark TMDs. We also discuss the implications of our findings for phenomenological applications and outline potential avenues for further research, particularly in understanding nonlinear effects and extending beyond the double-logarithmic approximation and large- <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>N</o:mi> <o:mi>c</o:mi> </o:msub> </o:math> approximations.