1 Introduction
Recent measurements of charmed-meson ($D^{0}$, $D^{*}$, $D^{*+}$, and $D_{s}^{+}$) production in high energy hadronic collisional experiment reported by PHENIX Collaboration[1] and STAR collaboration[2-8] at Relativistic Heavy Ion Collider (RHIC), as well as ALICE Collaboration,[9-17] ATLAS Collaboration,[18] and LHCb Collaboration[19-20] at Large Hadron Collider (LHC) have shown that inclusive heavy-quark production provides us with an important tool for the test of the predictions of perturbative quantum chromodynamics (pQCD),[21-22] and the evolution of charm quark in hot partonic medium can be considered as the probe of the quark-gluon plasma (QGP) produced in ultrarelativistic heavy ion collisions.[23-38] Indeed, many phenomenological models such as the color glass condensate (CGC) effective theory,[39] general-mass variable-favor-number scheme,[40-42] $k_{T}$-factorization approach,[43-44] fragmentation approach,[45-47] Parton-Hadron-String Dynamics (PHSD) transport approach,[48] heavy quark recombination mechanism,[49-50] two-component HYDJET++ model,[51-52] gluon splitting with Langevin transport model,[53] POWLANG transport model,[54] double-parton scattering (DPS) mechanisms,[55] quark coalescence model,[56] and single- and central-diffractive mechanisms in the Ingelman-Schlein model[57] have been proposed for the calculation of charmed-meson production.
In this paper, we extend the hard photoproduction mechanism[58-64] in the quasi-real photon approximation, that plays an important role in the electron-proton deep inelastic scattering at the Hadron Electron Ring Accelerator (HERA), to the heavy quark fragmentation production of charmed-meson in Au-Au collisions at RHIC and Pb-Pb collisions at LHC. The photoproduction mechanism may be direct that the high-energy photons from the charged parton interact with the parton from the nucleus, as well as resolved that the hadron-like photons interact with the parton.
The paper is organized as follows. In Sec. 2 we present the hard photoproduction processes for charmed-meson at RHIC and LHC energies. The conclusion is given in Sec. 3.
2 General Formalism
The centrality dependence of differential cross section for inclusive charmed-meson hadroproduction from leading order (LO) can be expressed as
$\frac{dN^{\rm LO}}{dp_{T}^{2}dy}=\int d^{2}bdx_{a}dx_{b}T_{A}(\pmb{\it r})T_{B}(|\pmb{\it r}-\pmb{\it b}|) f_{a/A}(x_{a},Q^{2},\pmb{\it r}) \\ \times f_{b/B}(x_{b},Q^{2},|\pmb{\it r}-\pmb{\it b}|)\frac{x_{a}x_{b}}{x_{a}x_{b}-\tau}\frac{d\hat{\sigma}}{d\hat{t}}(ab \rightarrow cd) \\ \times \frac{D_{H/c}(z_{c})}{z_{c}},$
where the variables $x_{a}$ and $x_{b}=(x_{a}x_{2}-\tau)/(x_{a}-x_{1})$ are momentum fractions of the partons, $z_{c}=p_{H}/p_{c}$ is the momentum fraction of the final charmed-meson, $x_{1}=({1}/{2})(x_{T}^{2}+4\tau)^{1/2}\exp(y)$, $x_{2}=({1}/{2})(x_{T}^{2}+4\tau)^{1/2}\exp(-y)$, $x_{T}=2p_{T}/\sqrt{s}$, $\tau=(M/\sqrt{s})^{2}$, and $M$ is the mass of the charmed-meson; $({d\hat{\sigma}}/{d\hat{t}})(ab\rightarrow cd)$ is the differential cross section for the subprocess[65-66] such as $q\bar{q}\rightarrow Q\bar{Q}$, $qQ\rightarrow qQ$, $gQ\rightarrow gQ$, and $gg\rightarrow Q\bar{Q}$; $D_{H/c}(z_{c})$ is the Peterson heavy quark fragmentation function into charmed-meson,[67] as well as $T_{A}(\pmb{\it r})$ is the nuclear thickness function[68-69] normalized to $\int d^{2}r T_{A}(\pmb{\it r})=A$. We choose the transverse momentum scale as $Q^{2}=p_{T}^{2}$.
The parton distribution function (PDF) is given by[70]
$ f_{a/A}(x,Q^{2},\pmb{\it r})=R_{a/A}(x,Q^{2},\pmb{\it r}) \Bigl[\frac{Z}{A}f_{a/p}(x,Q^{2}) \\ +\Bigl(1-\frac{Z}{A}\Bigr)f_{a/n}(x,Q^{2})\Bigr],$
where $Z$ is the charge and $A$ the mass number of the nucleus. The PDF parametrization[71] will be used for nucleon parton distributions $f_{a/N}(x,Q^{2})$. The parton shadowing factor $R_{a/A}(x,Q^{2},\pmb{\it r})$ describes the nuclear modification of parton distributions per nucleon inside the nucleus and can be given by the parametrization.[72-75]
In the inelastic direct photoproduction processes (inel.dir.), the centrality dependence of differential cross section of inclusive large-$p_{T}$ charmed-meson production in the hadronic collisions can be expressed as
$ \frac{dN^{\rm inel.dir.}}{dp_{T}^{2}dy} = \int d^{2}bd^{2}rdx_{a} dx_{b}dz_{a}T_{A}(\pmb{\it r})T_{B}(|\pmb{\it r}-\pmb{\it b}|) \\ \times f_{a/A}(x_{a},Q^{2}, \pmb{\it r}) f_{\gamma/q}(z_{a}) f_{b/B}(x_{b},Q^{2},|\pmb{\it r}-\pmb{\it b}|) \\ \times\frac{x_{a}x_{b}z_{a}}{x_{a}x_{b}z_{a}-\tau} \frac{d\hat{\sigma}}{d\hat{t}}(\gamma b\rightarrow cd)\frac{D_{H/c}(z_{c})}{z_{c}}, $
$f_{\gamma/q}(x)=\frac{\alpha}{2\pi}e_{f}^{2}\Bigl\{\frac{1+(1-x)^{2}}{x}\Bigl[\ln\Bigl(\frac{E}{m}\Bigr)-\frac{1}{2}\Bigr] +\frac{x}{2}\Bigl[\ln\Bigl(\frac{2}{x}-2\Bigr) \\ +1\Bigr]+\frac{(2-x)^{2}}{2x}\ln\Bigl(\frac{2-2x}{2-x}\Bigr)\Bigr\},$
where $e_{f}$, $E$, and $m$ are the charge, energy, and mass of the parton, respectively.
In the inelastic resolved photoproduction processes (inel. res.), the centrality dependence of differential cross section for inclusive large-$p_{T}$ charmed-meson production in the hadronic collisions can be written as
$ \frac{dN^{\rm inel.res.}}{dp_{T}^{2}dy} = \int d^{2}bd^{2}rdx_{a} dx_{b}dz_{a}dz'_{a}T_{A}(\pmb{\it r})T_{B}(|\pmb{\it r}-\pmb{\it b}|) \\ \times f_{a/A}(x_{a},Q^{2},\pmb{\it r})f_{\gamma/q}(z_{a}) f_{\gamma}(z'_{a},Q^{2}) \\ \times f_{b/B}(x_{b},Q^{2},|\pmb{\it r}-\pmb{\it b}|)\frac{x_{a}x_{b}z_{a}z'_{a}}{x_{a}x_{b}z_{a}z'_{a}-\tau} \\ \times \frac{d\hat{\sigma}}{d\hat{t}} (a'b\rightarrow cd)\frac{D_{H/c}(z_{c})}{z_{c}},$
where $f_{\gamma}(z'_{a},Q^{2})$ is PDF for the hadron-like photon.[79]
The numerical results of our calculation for large-$p_{T}$ charmed-meson from the hard photoproduction processes are plotted in Figs. 1-4. Compared with the leading order (the dashed line) and the charmed-meson data of STAR Collaboration[3] and ALICE Collaboration,[9,12] we find that the large-$p_{T}$ charmed-meson produced by the hard photoproduction processes (the dashed-dotted line) cannot be negligible, that is because of the equivalent photon spectrum for the charged parton, $f_{\gamma/q}\propto\ln(E/m_{q})=\ln(\sqrt{s}/2m_{q})+\ln(x)$, becomes prominent since the collision energy $\sqrt{s}$ is very large at RHIC and LHC energies.
Fig. 1 (Color online) The centrality dependence of inclusive cross section of large transverse momentum $D^{0}$ meson production in the nucleus-nucleus collisions at RHIC and LHC energies. The dashed line (blue line) for the initial parton hard scattering processes (LO), the dashed-dotted line (red line) for the inelastic photoproduction processes, and the solid line (black) is for the sum of the above processes. The $D^{0}$ meson data points are from the STAR Collaboration[3] and ALICE Collaboration.[9] |
Fig. 2 (Color online) The centrality dependence of inclusive cross section of large transverse momentum $D^{*}$ meson production in the nucleus-nucleus collisions at RHIC and LHC energies. The dashed line (blue line) for the initial parton hard scattering processes (LO), the dashed-dotted line (red line) for the inelastic photoproduction processes, and the solid line (black) is for the sum of the above processes. The $D^{*}$ meson data points are from ALICE Collaboration.[9] |
Fig. 3 (Color online) The centrality dependence of inclusive cross section of large transverse momentum $D^{*+}$ meson production in the nucleus-nucleus collisions at RHIC and LHC energies. The dashed line (blue line) for the initial parton hard scattering processes (LO), the dashed-dotted line (red line) for the inelastic photoproduction processes, and the solid line (black) is for the sum of the above processes. The $D^{*+}$ meson data points are from ALICE Collaboration.[9] |
Fig. 4 (Color online) The centrality dependence of inclusive cross section of large transverse momentum $D_{s}^{+}$ meson production in the nucleus-nucleus collisions at RHIC and LHC energies. The dashed line (blue line) for the initial parton hard scattering processes (LO), the dashed-dotted line (red line) for the inelastic photoproduction processes, and the solid line (black) is for the sum of the above processes. The $D_{s}^{+}$ meson data points are from ALICE Collaboration.[12] |
3 Conclusion
We have investigated the heavy quark fragmentation production of large-$p_{T}$ charmed-meson from the direct and resolved hard photoproduction processes in Au-Au collisions at Relativistic Heavy Ion Collider (RHIC) and Pb-Pb collisions at Large Hadron Collider (LHC). At the early stage of relativistic heavy ion collisions, the charged parton of the incident nucleus can emit high photons (hadron-like photons) that can interact with the partons of the nucleus by the photon-gluon fusion (the quark-antiquark annihilation, quark-gluon Compton scattering, and gluon-gluon fusion interactions). The numerical results indicate that the contribution of charmed-meson produced by the hard photoproduction processes cannot be negligible, that is because of the very large photon spectrum in the quasi-real approximation, in the nucleus-nucleus collisions at RHIC and LHC energies.