The \(\Upsilon (2S)\) production and polarization at high energies is studied in the framework of \(k_T\)-factorization approach. Our consideration is based on the non-relativistic QCD formalism for bound states formation and off-shell production amplitudes for hard partonic subprocesses. The direct production mechanism, feed-down contributions from radiative \(\chi _b(3P)\) and \(\chi _b(2P)\) decays and contributions from \(\Upsilon (3S)\) decays are taken into account. The transverse momentum dependent gluon densities in a proton were derived from the Ciafaloni–Catani–Fiorani–Marchesini evolution equation, Kimber-Martin-Ryskin prescription and Parton Branching method. Treating the non-perturbative color octet transitions in terms of the mulitpole radiation theory, we extract the corresponding non-perturbative matrix elements for \(\Upsilon (2S)\) and \(\chi _b(2P)\) mesons from a combined fit to \(\Upsilon (2S)\) transverse momenta distributions measured by the CMS and ATLAS Collaborations at the LHC energies \(\sqrt{s} = 7\) and 13 TeV and from the relative production rate \(R^{\chi _b(2P)}_{\Upsilon (2S)}\) measured by the LHCb Collaboration at \(\sqrt{s} = 7\) and 8 TeV. Then we apply the extracted values to investigate the polarization parameters \(\lambda _\theta \), \(\lambda _\phi \) and \(\lambda _{\theta \phi }\), which determine the \(\Upsilon (2S)\) spin density matrix. Our predictions have a good agreement with the currently available data within the theoretical and experimental uncertainties.

中文翻译：

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NRQCD中的$ k_T $$ kT分解导致的足底分泌和极化。II：$$ \ Upsilon（2S）$$Υ（2S）和$$ \ chi _b（2P）$$χb（2P）介子

在\（k_T \）分解方法的框架下研究了高能量下的\（\ Upsilon（2S）\）的产生和极化。我们的考虑是基于非相对论性QCD形式主义的，用于结合态形成和硬部分子过程的脱壳产生幅度。直接生产机制，辐射\（\ chi _b（3P）\）和\（\ chi _b（2P）\）的馈电衰减和\（\ Upsilon（3S）\）的衰减衰减被考虑在内。质子中与横向动量有关的胶子密度取决于Ciafaloni-Catani-Fiorani-Marchesini演化方程，Kimber-Martin-Ryskin处方和Parton分枝法。根据多极辐射理论处理非扰动的彩色八位位组跃迁，我们从组合中提取\（\ Upsilon（2S）\）和\（\ chi _b（2P）\）介子的相应非扰动矩阵元素根据LHC能量\（\ sqrt {s} = 7 \）和13 TeV并根据相对生产率\（R ^ { ），拟合CMS和ATLAS合作测量的\（\ Upsilon（2S）\）横向动量分布。\ chi _b（2P）} _ {\ Upsilon（2S）} \）由LHCb协作在\（\ sqrt {s} = 7 \）和8 TeV下测量。然后，我们将提取的值应用于研究极化参数\（\ lambda _ \ theta \），\（\ lambda _ \ phi \）和\（\ lambda _ {\ theta \ phi} \）来确定\（ \ upsilon（2S）\）自旋密度矩阵。在理论和实验不确定性范围内，我们的预测与当前可获得的数据具有很好的一致性。