Bottomonia production and polarization in the NRQCD with k_T-factorization. II: Upsilon (2S) and chi _b(2P) mesons

N A Abdulov,A V Lipatov

doi:10.1140/epjc/s10052-020-8056-x

Abstract

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.

Highlights

S, orbital angular momentum L, total angular momentum J and color representation a from long-distance non-perturbative matrix elements (NMEs), which describe the transition of that intermediate Q Qstate into a physical quarkonium via soft gluon radiation
Our consideration was based on the off-shell production amplitudes for hard partonic subprocesses, non-relativistic QCD (NRQCD) formalism for the formation of bound states and Transverse Momentum Dependent (TMD) gluon densities in a proton
The latter were derived from the CCFM evolution equation, KMR scheme and recently proposed Parton Branching (PB) approach

Summary

Introduction

S, orbital angular momentum L, total angular momentum J and color representation a from long-distance non-perturbative matrix elements (NMEs), which describe the transition of that intermediate Q Qstate into a physical quarkonium via soft gluon radiation. For example, having the NMEs fixed from fitting the charmonia transverse momentum distributions, one disagrees with the polarization observables: if the dominant contribution comes from the gluon fragmentation into an octet Q Qpair, the outgoing meson must have strong transverse polarization The latter disagrees with the latest data [20,21,22,23,24], which show the unpolarized or even longitudinally polarized particles (so called “the polarization puzzle”). In this way the NMEs are represented in an explicit form inspired by the classical multipole radiation theory, that leads to unpolarized or only weakly polarized mesons either because of the cancellation between the 3 P1(8) and 3 P2(8) contributions or as a result of two successive color-electric E1 dipole transitions in the chain 3 S1(8) → 3 PJ(8) → 3 S1(1) This scenario was already successfully applied to describe the recent data on charmonia production and polarization [27,28].

Theoretical framework

Numerical results

Fit of color octet NMEs

Conclusion