NLO Productions of ω\omega and KS0K^0_{\rm S} with a Global Extraction of the Jet Transport Parameter in Heavy Ion collisions

In this work, we pave the way to calculate the productions of $\omega$ and $K^0_{\rm S}$ mesons at large $p_T$ in p+p and A+A collisions at the RHIC and the LHC. The $\omega$ meson fragmentation functions (FFs) in vacuum at next-to-leading order (NLO) are obtained by evolving NLO DGLAP evolution equations with rescaled $\omega$ FFs at initial scale $Q_0^2=1.5$ GeV$^2$ from a broken SU(3) model, and the $K^0_{\rm S}$ FFs in vacuum are taken from AKK08 parametrization directly. Within the framework of the NLO pQCD improved parton model, we make good descriptions of the experimental data on $\omega$ and $K^0_{\rm S}$ in p+p both at the RHIC and the LHC. With the higher-twist approach to take into account the jet quenching effect by medium modified FFs, the nuclear modification factors for $\omega$ meson and $K^0_{\rm S}$ meson at the RHIC and the LHC are presented with different sets of jet transport coefficient $\hat{q}_0$. Then we make a global extraction of $\hat{q}_0$ at the RHIC and the LHC by confronting our model calculations with all available data on 6 identified mesons: $\pi^0$, $\eta$, $\rho^0$, $\phi$, $\omega$, and $K^0_{\rm S}$. The minimum value of the total $\chi^2/d.o.f$ for productions of these mesons gives the best value of $\hat{q}_0=0.5\rm GeV^2/fm$ for Au+Au collisions with $\sqrt{s_{\rm NN}}=200$ GeV at the RHIC, and $\hat{q}_0=1.2\rm GeV^2/fm$ for Pb+Pb collisions with $\sqrt{s_{\rm NN}}=2.76$ TeV at the LHC respectively, with the QGP spacetime evolution given by an event-by-event viscous hydrodynamics model IEBE-VISHNU. With these global extracted values of $\hat{q}_0$, the nuclear modification factors of $\pi^0$, $\eta$, $\rho^0$, $\phi$, $\omega$, and $K^0_{\rm S}$ in A+A collisions are presented, and predictions of yield ratios such as $\omega/\pi^0$ and $K^0_{\rm S}/\pi^0$ at large $p_T$ in heavy-ion collisions at the RHIC and the LHC are provided.Comment: 9 pages, 13 figures, 1 tabl

Derivatives of tangent function and tangent numbers

In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.Comment: 17 page