Fragmentation of $\omega$ and $\phi$ Mesons in $e^+ e^-$ and $p p$ Collisions at NLO

A combined analysis of both $e^+\,e^-$ (LEP, SLD) and $p\,p$ (RHIC-PHENIX and LHC-ALICE) hadroproduction processes are done for the first time for the vector meson nonet at the next-to-leading order (NLO) using a model with broken SU(3) symmetry. The transverse momentum ($p_T$) and rapidity ($y$) dependence of the differential cross section for $\omega$ and $\phi$ mesons of the $p\,p$ data are also discussed. The input universal quark (valence and singlet) fragmentation functions at a starting scale of $Q_0^2=1.5$ GeV$^2$, after evolution, have values that are consistent with the earlier analysis for $e^+\,e^-$ at NLO. However, the universal gluon fragmentation function is now well determined from this study with significantly smaller error bars, as the $p\,p$ hadroproduction cross section is particularly sensitive to the gluon fragmentation since it occurs at the same order as quark fragmentation, in contrast to the $e^+\,e^-$ hadroproduction process. Additional parameters involved in describing strangeness and sea suppression and octet-singlet mixing are found to be close to earlier analysis; in addition, a new relation between gluon and sea suppression in $K^*$ and $\phi$ hadroproduction has been observed.