Elastic dijet production in electron scattering on a longitudinally polarized proton at small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math>: A portal to orbital angular momentum distributions

We calculate the elastic production of dijets from electron collisions with a longitudinally polarized proton target at small values of the Bjorken <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>x</a:mi></a:math> variable. Building on the pioneering proposals of Hatta [Gluon orbital angular momentum at small-<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>x</c:mi></c:math>, .] and Bhattacharya [Signature of the gluon orbital angular momentum, ., Probing the quark orbital angular momentum at electron-ion colliders using exclusive <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>π</e:mi><e:mn>0</e:mn></e:math> production, ., and Exploring orbital angular momentum and spin-orbit correlation for gluons at the Electron-Ion Collider, .] for measuring the quark and gluon orbital angular momentum (OAM) distributions, our focus is on both the longitudinal double spin asymmetry (DSA) and longitudinal single spin asymmetry (SSA). We compute the numerators of these asymmetries in the small-<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>x</g:mi></g:math> formalism of the light-cone operator treatment. Utilizing the small-<i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>x</i:mi></i:math> expressions for the OAM distributions derived earlier in Kovchegov and Manley [Orbital angular momentum at small x revisited, .], we demonstrate that the DSA provides a robust probe for both the quark and gluon OAM distributions within the proton. In contrast, we find that while the SSA is also sensitive to the OAM distributions, extraction of the latter from the SSA would require new developments in small-<k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>x</k:mi></k:math> theory and phenomenology, and is probably not feasible at this point in time. These findings highlight the potential of DSA measurements in elastic dijet production at the future Electron-Ion Collider to provide the first-ever direct access to the quark and gluon OAM distributions at small <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>x</m:mi></m:math>, paving the way for new insights into the proton spin puzzle.