Jet Quenching for Heavy Flavors in AA and pp Collisions

Abstract

We perform a global analysis of experimental data on jet quenching for heavy flavors for scenarios with and without quark-gluon plasma formation in pp collisions. We find that the theoretical predictions for the nuclear modification factor R_AA for heavy flavors at the LHC energies are very similar for these scenarios, and the results for R_AA and ${{{v}}_{2}}$ agree reasonably with the LHC data. The agreement with data at the RHIC top energy becomes somewhat better for the intermediate scenario, in which the quark-gluon plasma formation in pp collisions occurs only at the LHC energies. Our fits to heavy flavor R_AA show that description of jet quenching for heavy flavors requires somewhat bigger α_s than data on jet quenching for light hadrons.

APPENDIX

In this appendix we give, for the convenience of the reader, formulas for calculation of the gluon emission x-spectrum dP/dx. We use the representation of the induced gluon spectrum obtained in [31] with the prescription of [74] for incorporating the T-dependent running α_s. For a fast quark with momentum along the z-axis produced at z = 0 in the matter of thickness L, dP/dx has the form

$$\frac{{dP}}{{dx}} = \int\limits_0^L {dzn(z)\frac{{d\sigma _{{{\text{eff}}}}^{{BH}}(x,z)}}{{dx}},} $$

(11)

where n(z) is the medium number density, d$\sigma _{{{\text{eff}}}}^{{BH}}$/dx is an effective Bethe-Heitler cross section for q → gq process, given by

$$\begin{gathered} \frac{{d\sigma _{{{\text{eff}}}}^{{BH}}(x,z)}}{{dx}} = - \frac{{P_{q}^{g}(x)}}{{\pi M}} \\ \times \operatorname{Im} \int\limits_0^z {d\xi \sqrt {{{\alpha }_{s}}(Q(\xi ),T(z - \xi )){{\alpha }_{s}}(Q(\xi ),T(z + \xi ))} } \\ {{\left. { \times \exp \left( { - i\frac{\xi }{{{{L}_{f}}}}} \right)\frac{\partial }{{\partial \rho }}\left( {\frac{{F(\xi ,\rho )}}{{\sqrt \rho }}} \right)} \right|}_{{\rho = 0}}}. \\ \end{gathered} $$

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Here $P_{q}^{g}$(x) = (4/3)[1 + (1 – x)²]/x is the ordinary pQCD q → g splitting function, M = E_qx(1 – x), L_f = 2M/${{\epsilon }^{2}}$, ${{\epsilon }^{2}}$ = $m_{q}^{2}$x² + $m_{g}^{2}$(1 – x), Q²(ξ) = aM/ξ with a ≈ 1.85 [8], F is the solution to the radial Schrödinger equation

$$\begin{gathered} i\frac{{\partial F(\xi ,\rho )}}{{\partial \xi }} = \left[ { - \frac{1}{{2M}}{{{\left( {\frac{\partial }{{\partial \rho }}} \right)}}^{2}}} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}} + \,{v}(\rho ,x,z - \xi ) + \frac{{4{{m}^{2}} - 1}}{{8M{{\rho }^{2}}}}} \right]F(\xi ,\rho ) \\ \end{gathered} $$

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with the azimuthal quantum number m = 1, and the boundary condition F(ξ = 0, ρ) = $\sqrt \rho {{\sigma }_{{gq\bar {q}}}}$(ρ, x, z)$\epsilon$K₁($\epsilon$ρ) at ξ = 0 (K₁ is the Bessel function). The potential ${v}$ reads

$${v}(\rho ,x,z) = - i\frac{{n(z){{\sigma }_{{gq\bar {q}}}}(\rho ,x,z)}}{2},$$

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where ${{\sigma }_{{gq\bar {q}}}}$(ρ, x, z) is the three-body cross section of interaction of the gq$\bar {q}$ system with a medium constituent located at z (ρ is the transverse distance between g and the final quark q). In the transverse plane $\bar {q}$ is located at the center of mass of the gq pair. The ${{\sigma }_{{gq\bar {q}}}}$ can be written via the local dipole cross section ${{\sigma }_{{q\bar {q}}}}$(ρ, z) (for the color singlet $q\bar {q}$ pair)

$$\begin{gathered} {{\sigma }_{{gq\bar {q}}}}(\rho ,x,z){{{\text{|}}}_{{q \to gq}}} \\ = \frac{9}{8}[{{\sigma }_{{q\bar {q}}}}(\rho ,z) + {{\sigma }_{{q\bar {q}}}}((1 - x)\rho ,z)] - \frac{1}{8}{{\sigma }_{{q\bar {q}}}}(x\rho ,z). \\ \end{gathered} $$

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In the two-gluon approximation the dipole cross section reads

$${{\sigma }_{{q\bar {q}}}}(\rho ,z) = {{C}_{T}}{{C}_{F}}\int {d{\mathbf{q}}\alpha _{s}^{2}(q,T(z))\frac{{[1 - \exp (i{\mathbf{q}}\rho )]}}{{{{{[{{q}^{2}} + \mu _{D}^{2}(z)]}}^{2}}}}} ,$$

(16)

where C_{F, T} are the color Casimir for the quark and thermal parton (quark or gluon), and μ_D(z) is the local Debye mass.

For the QGP fireball in AA collisions the coordinate z coincides with the proper time τ, i.e. in terms of the real fireball number density, n_f(ρ, τ), we have n(z) = n_f(ρ_j(ρ_j0, τ), τ), where ρ_j0 is the jet production transverse coordinate, and ρ_j(ρ_j0, τ) = ρ_j0 + τp_T/|p_T| is the jet trajectory. We use the approximation of a uniform fireball. In this case, inside the fireball, the function n_f(ρ, τ) does not depend on the jet production point. This greatly reduces the computational cost, since one can tabulate the L-dependence of the induced gluon spectrum once, and then use it for calculations of the FFs for arbitrary jet geometry.