Tree-level NLO corrections to inclusive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>ψ</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> production in high energy factorization

We consider inclusive <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>ψ</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mi>S</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> production in proton-proton collisions at collider energies in the framework of nonrelativistic QCD and high energy factorization beyond the low-order approximation. We utilize a matching scheme proposed earlier to merge the leading-order and tree-level next-to-leading-order (NLO) production amplitudes and now extend it to low transverse momenta and/or forward rapidities. With the improved scheme, we examine the possibility to simultaneously describe all unpolarized LHC data in the full kinematic range accessible to experimental measurements and try different transverse momentum-dependent gluon distributions in the proton. A global fit to the data is carried out to extract the color octet long-distance matrix elements for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>ψ</e:mi> <e:mo stretchy="false">(</e:mo> <e:mn>2</e:mn> <e:mi>S</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> mesons. We show that taking the NLO corrections into account leads to a better description of the data.