Heavy flavors in AA collisions: production, transport and final spectra

Abstract

A multi-step setup for heavy-flavor studies in high-energy nucleus-nucleus (AA) collisions—addressing within a comprehensive framework the initial $Q\overline{Q}$ production, the propagation in the hot medium until decoupling and the final hadronization and decays—is presented. The initial hard production of $Q\overline{Q}$ pairs is simulated using the POWHEG pQCD event generator, interfaced with the PYTHIA parton shower. Outcomes of the calculations are compared to experimental data in pp collisions and are used as a validated benchmark for the study of medium effects. In the AA case, the propagation of the heavy quarks in the medium is described in a framework provided by the relativistic Langevin equation. For the latter, different choices of transport coefficients are explored (either provided by a perturbative calculation or extracted from lattice-QCD simulations) and the corresponding numerical results are compared to experimental data from RHIC and the LHC. In particular, outcomes for the nuclear modification factor R _AA and for the elliptic flow v ₂ of D/B mesons, heavy-flavor electrons and non-prompt J/ψ’s are displayed.

Appendix: Lattice transport coefficients

For the sake of self-consistency, in this appendix, following the steps of Ref. [51], we display how the heavy quark momentum-diffusion coefficient κ can be given a general quantum field theory definition and how it can be expressed in terms of quantities accessible by lattice-QCD simulations.

One has to address the quite common situation of a “system” (the heavy quark) coupled to an “environment” (the thermal bath of gluons and light quarks).

We have shown in Eq. (10) how κ is related to the following force–force correlator:

$$ D^>(t)\equiv\frac{1}{3} \bigl\langle F^i(t)F^i(0) \bigr\rangle_\mathrm{HQ}. $$

(A.1)

Analogously, one defines D ^<(t)≡(1/3)〈F ⁱ(0)F ⁱ(t)〉_HQ. KMS boundary conditions entail for their Fourier transforms D ^<(ω)=e ^−βω D ^>(ω). One has then for the corresponding spectral function:

$$ \sigma(\omega)\equiv D^>(\omega)-D^<(\omega)= \bigl(1-e^{-\beta\omega }\bigr)D^>(\omega). $$

(A.2)

The momentum-diffusion coefficient reflects the ω→0 limit of the above spectral density. In fact:

$$ \kappa\equiv\int_{-\infty}^{+\infty} dt\, D^>(t)=D^>(\omega=0). $$

(A.3)

Hence one has

$$ \kappa=\lim_{\omega\to 0}\frac{\sigma(\omega)}{1-e^{-\beta\omega}} =\lim_{\omega\to 0} \frac{T}{\omega}\sigma(\omega), $$

(A.4)

σ(ω) being the quantity extracted from lattice-QCD simulations. For the latter one needs to consider the coupling of a (infinitely) heavy quark with the color field. The starting point is the M→∞ limit of the NRQCD Lagrangian

$$ \mathcal{L}=Q^\dagger(i\partial_0+gA_0)Q, $$

(A.5)

with the non-relativistic fields obeying the anticommutation relation

$$ \bigl\{Q_i(t,\mathbf{x}),Q_j^\dagger(t,\mathbf{y}) \bigr\}= \delta_{ij}\delta(\mathbf{x}-\mathbf{y}) $$

(A.6)

and the heavy-quark evolution being described the path-ordered exponential U(t,t ₀):

$$\begin{aligned} Q_i(t) =& \mathcal{P}\exp \biggl[ig \int_{t_0}^t A_0\bigl(t'\bigr)dt' \biggr]_{ij} Q_j(t_0) \\ =&U_{ij}(t,t_0)Q_j(t_0). \end{aligned}$$

(A.7)

One needs then to define the expectation value in Eq. (A.1)

$$ \bigl\langle F^i(t)F^i(0) \bigr\rangle_\mathrm{HQ} \equiv \frac{\sum_s\langle s|e^{-\beta H}F^i(t)F^i(0)|s\rangle}{\sum_s\langle s|e^{-\beta H}|s\rangle} , $$

(A.8)

which is taken over a thermal ensemble of states |s〉 of the environment plus one additional heavy quark, namely:

$$ \sum_s\langle s|\dots|s\rangle\equiv\sum _{s'}\int d\mathbf{x}\, \langle s'|Q_i(-T, \mathbf{x}) \dots Q_i^\dagger(-T,\mathbf{x})|s'\rangle. $$

(A.9)

Viewing the thermal weight e ^−βH as the imaginary-time translation operator, so that

(A.10)

and exploiting the anticommutation relation (A.6) one gets for the HQ partition function appearing in the denominator of Eq. (A.8)

$$\begin{aligned} Z_\mathrm{HQ} =&\sum_{s'}\int d\mathbf{x}\, \langle s'|Q_i(-T,\mathbf{x}) e^{-\beta H} Q_i^\dagger(-T, \mathbf{x})|s'\rangle \\ =&V_\mathrm{PS}\sum_{s'}\langle s'|e^{-\beta H}U_{ii}(-T-i\beta,-T)|s'\rangle \\ =&V_\mathrm{PS} \bigl\langle\mathrm{Tr}\,U(-T-i\beta,-T)\bigr\rangle Z_0, \end{aligned}$$

(A.11)

where now the thermal average is taken over the states of the environment only, with partition function Z ₀≡∑ _s′〈s′|e ^βH|s′〉, and the phase space volume arises from

$$ \int d\mathbf{x}\,\delta(\mathbf{x}-\mathbf{x})=\int d\mathbf{x}\int d\mathbf{p}/(2\pi)^3=V_\mathrm{PS}. $$

The numerator in Eq. (A.8) can be evaluated analogously starting from

(A.12)

One gets then

(A.13)

The above definition is the one used in Ref. [51], in which the AdS/CFT correspondence allows the derivation of real-time quantities in strongly coupled gauge theories ($\mathcal{N}=4$ SYM). However, in lattice-QCD one has to rely on lattice simulations carried on in Euclidean time. Equation (A.13) has to be accordingly generalized. In Refs. [33, 34] the authors evaluated then the following Euclidean electric-field correlator [52]:

$$ D_E(\tau)=-\frac{1}{3}\frac{\langle\mathrm{Tr}[U(\beta,\tau)gE^i (\tau)U(\tau)gE^i(0)]\rangle}{\langle \operatorname {Tr}U(\beta,0)\rangle} $$

(A.14)

and from the latter they extracted the spectral function σ(ω) entering in Eq. (A.4) according to

$$ D_E(\tau)=\int_0^\infty \frac{d\omega}{2\pi}\sigma(\omega)\frac {\cosh[(\beta/2-\tau)\omega]}{\sinh(\beta\omega/2)}. $$

(A.15)