Color screening and regeneration of bottomonia in high-energy heavy-ion collisions

The production of ground-state and excited bottomonia in ultrarelativistic heavy-ion collisions is investigated within a kinetic-rate equation approach including regeneration. We augment our previous calculations by an improved treatment of medium effects, with temperature-dependent binding energies and pertinent reaction rates, $B$-meson resonance states in the equilibrium limit near the hadronization temperature, and a lattice-QCD based equation of state for the bulk medium. In addition to the centrality dependence of the bottomonium yields, we compute their transverse-momentum ($p_T$) spectra and elliptic flow with momentum-dependent reaction rates and a regeneration component based on $b$-quark spectra from a nonperturbative transport model of heavy-quark diffusion. The latter has noticeable consequences for the shape of the bottomonium $p_T$ spectra. We quantify how uncertainties in the various modeling components affect the predictions for observables. Based on this we argue that the $\mathrm{{\rm Y}}\left(1S\right)$ suppression is a promising observable for mapping out the in-medium properties of the QCD force, while $\mathrm{{\rm Y}}\left(2S\right)$ production can help to quantify the role of regeneration from partially thermalized $b$ quarks.

Figure 1

Bottomonium binding energies for the SBS (vacuum $E_B$'s, dotted lines) [28] and $T$-matrix binding scenarios (TBS) with baseline value [29] $\eta =1.0$ (solid lines), and a 10% smaller ($\eta =0.9$, dashed lines) or larger ($\eta =1.1$, dash-dotted lines) reduction in $\mathrm{\Delta }{E}_B\left(T\right)$; cf. Eq. (3). The red, green, and blue lines are for $\mathrm{{\rm Y}}\left(1S\right)$, ${\chi }_b$, and $\mathrm{{\rm Y}}\left(2S\right)$ states, respectively.

Figure 2

Bottomonium dissociation rates in QGP for the SBS using gluo-dissociation with massless (upper panel) or massive gluons (dash-dotted lines in the middle and lower panel). The middle and lower panels also show the rates from inelastic “quasifree” scattering off massive quarks and gluons (dotted lines) without (middle panel) and with (lower panel) interference corrections, and their sum with massive gluo-dissociation rates (solid lines). All rates are evaluated at zero $Y$ 3-momentum with a strong coupling constant of $g=2.0$.

Figure 3

Bottomonium dissociation rates for the in-medium $T$-matrix binding scenario (TBS) in a massive thermal parton gas. Upper panel: baseline TBS (with $\eta =1.0$ in Fig. 1); middle (lower) panel: TBS with increased (decreased) binding energies $\eta =0.9$ ($\eta =1.1$); note that $\eta =0$ recovers the SBS. The dash-dotted and solid lines correspond to gluo-dissociation and inelastic parton scattering, respectively, while red, green, and blue colors represent $\mathrm{{\rm Y}}\left(1S\right)$, ${\chi }_b\left(1P\right)$, and $\mathrm{{\rm Y}}\left(2S\right)$ states, respectively. Dissociation rates are evaluated at bottomonium 3-momentum $p=0$. Interference corrections are included in the quasifree inelastic parton scattering.

Figure 4

Parton-induced quasifree dissociation rates (solid lines) for the baseline in-medium $T$-matrix binding scenario (TBS with $\eta =1.0$) in a massive thermal parton gas with (blue lines) or without (red lines) final-state Fermi blocking and Bose enhancement factors for quarks (dash–double-dotted lines) and gluons (dotted lines), respectively. The rates are evaluated at $p=0$ for $\mathrm{{\rm Y}}\left(1S\right)$. Interference corrections are included.

Figure 5

Three-momentum dependence of bottomonium dissociation rates in the SBS (vacuum binding). Upper panel: gluo-dissociation with massless partons at $T=180\mathrm{MeV}$ (dash-dotted lines) and $T=300\mathrm{MeV}$ (solid lines). Lower panel: gluo-dissociation (dash-dotted lines) and inelastic parton scattering (solid lines) for massive partons, at a temperature of $T=300\mathrm{MeV}$. In both panels red, green, and blue colors correspond to $\mathrm{{\rm Y}}\left(1S\right)$, ${\chi }_b\left(1P\right)$, and $\mathrm{{\rm Y}}\left(2S\right)$ states, respectively.

Figure 6

Three-momentum dependence of bottomonium dissociation rates from inelastic massive-parton scattering in the TBS for $T=180\mathrm{MeV}$ (upper panel) and $T=300\mathrm{MeV}$ (lower panel). The solid and dash-dotted lines correspond to the baseline TBS ($\eta =1.0$) and an increased (decreased) binding with $\eta =0.9$ ($\eta =1.1$), respectively. The red, green, and blue lines correspond to the $\mathrm{{\rm Y}}$, ${\chi }_b\left(1P\right)$, and $\mathrm{{\rm Y}}\left(2S\right)$ states, respectively.

Figure 7

Three-momentum dependence of bottomonium rates from gluo-dissociation in the TBS for the baseline scenario ($\eta =1.0$, upper panel) and increased (decreased) binding $\eta =0.9$, middle panel ($\eta =1.1$, lower panel) at $T=180\mathrm{MeV}$ (dash-dotted lines) and $T=300\mathrm{MeV}$ (solid lines). The red, green, and blue lines correspond to the $\mathrm{{\rm Y}}$, ${\chi }_b\left(1P\right)$, and $\mathrm{{\rm Y}}\left(2S\right)$ states, respectively.

Figure 8

Equilibrium limits of $Y$ states with (dashed lines) and without (dash-dotted lines) $B$-meson resonance degrees of freedom, and their smooth interpolation (solid lines) around a switching temperature of $T=220\mathrm{MeV}$. The red (blue) and yellow (light blue) curves are for $b$-quark only ($b$-quark plus resonance) degrees of freedom for $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)$, respectively. The default TBS parameter $\eta =1.0$ is used.

Figure 9

Temperature evolution of the expanding fire cylinder for central Pb-Pb (2.76 TeV) collisions using a lattice EoS with $T_{\mathrm{pc}}=170\mathrm{MeV}$ (red solid line), compared to a massless quasiparticle EoS with mixed phase at $T_{\mathrm{c}}=180\mathrm{MeV}$ (blue dashed line).

Figure 10

Centrality dependence of bottomonium yields in Au-Au (200 GeV) collisions using the baseline TBS ($\eta =1.0$) with updated feeddowns. The total (red band) and regenerated (blue lines) contributions are shown for inclusive $\mathrm{{\rm Y}}\left(1S\right)$ (upper panel), $\mathrm{{\rm Y}}(1S+2S+3S)$ (middle panel), and $\mathrm{{\rm Y}}(2S+3S)$ (lower panel) production at mid-rapidity ($\left|y\right|<0.5$) and compared to STAR [54] and PHENIX [21] data. The band width of the total yields is due to CNM effects with ${\sigma }_Y^{\mathrm{abs}}=0-3\mathrm{mb}$ [20].

Figure 11

Centrality dependence of bottomonium yields in U-U (193 GeV) collisions using the baseline TBS with updated feeddowns. The total (green band) and regenerated (light-blue line) contributions are shown for inclusive $\mathrm{{\rm Y}}\left(1S\right)$ (upper panel) and $\mathrm{{\rm Y}}(1S+2S+3S)$ (lower panel) at mid-rapidity ($\left|y\right|<1.0$) and compared to STAR data [22]. The band width of the total yields is due to CNM effects with ${\sigma }_Y^{\mathrm{abs}}=0-3\mathrm{mb}$ [20].

Figure 12

The $p_T$-dependent $R_{AA}$ for inclusive $\mathrm{{\rm Y}}\left(1S\right)$ in 0–60% Au-Au (200 GeV) collisions within the baseline TBS, compared to STAR data [54]. The red (blue) band is for the total (regeneration) yield, where the band width of the former is due to CNM effects with ${\sigma }_Y^{\mathrm{abs}}=0\text{–}3\mathrm{mb}$ [20].

Figure 13

Centrality dependence of bottomonium production in Pb-Pb (2.76 TeV) collisions within the SBS [28] with updated feeddowns. Upper panel: $\mathrm{{\rm Y}}\left(1S\right)$ (red band) and $\mathrm{{\rm Y}}\left(2S\right)$ (blue band) at mid-rapidity compared to CMS data [31]. Lower panel: $\mathrm{{\rm Y}}\left(1S\right)$ at forward rapidity compared to ALICE data [26], where we also show the regeneration contribution (dotted line). The band widths of the totals in both panels are due to a 0–25% variation in the shadowing suppression of the initial bottomonium yields.

Figure 14

Centrality dependence of $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)$ production in Pb-Pb (2.76 TeV) collisions within the TBS. Upper panel: inclusive $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)$ results at mid-rapidity for $\eta =0.9$, $\eta =1.0$, and $\eta =1.1$ scenarios compared to CMS data [31]. The red [blue] band is the total $\mathrm{{\rm Y}}\left(1S\right)\left[\mathrm{{\rm Y}}\right(2S\left)\right]R_{AA}$ for baseline $\eta =1.0$, the pink [light blue] band is the total $\mathrm{{\rm Y}}\left(1S\right)\left[\mathrm{{\rm Y}}\right(2S\left)\right]R_{AA}$ for $\eta =0.9$, the orange [green] band is the total $\mathrm{{\rm Y}}\left(1S\right)\left[\mathrm{{\rm Y}}\right(2S\left)\right]R_{AA}$ for $\eta =1.1$, and the yellow [cyan] band is the $\mathrm{{\rm Y}}\left(1S\right)\left[\mathrm{{\rm Y}}\right(2S\left)\right]$ regeneration contribution with $\eta =1.0$. The regeneration components for different scenarios have tiny differences. The bands reflect the uncertainty due to shadowing between 0–15%. Lower panel: inclusive $\mathrm{{\rm Y}}\left(1S\right)$ (red band) and regenerated component (blue band) at forward rapidity for $\eta =1.0$, compared to ALICE data [26]; the bands reflect the uncertainty due to a shadowing reduction between 0–30%.

Figure 15

Upper panel: comparison of $\mathrm{{\rm Y}}\left(1S\right)$, $\mathrm{{\rm Y}}\left(2S\right)$, and $\mathrm{{\rm Y}}\left(3S\right)R_{AA}$'s (solid lines: total; dashed lines: regeneration contribution) for the TBS with and without $B$-meson resonance states near $T_{\mathrm{pc}}$, assuming the default switching temperature of $T=220\mathrm{MeV}$. The red/blue curves are with/out resonances for $\mathrm{{\rm Y}}\left(1S\right)$, the cyan/yellow curve are with/out resonances for $\mathrm{{\rm Y}}\left(2S\right)$, and the magenta/green curves are with/out resonances for $\mathrm{{\rm Y}}\left(3S\right)$. Lower panel: sensitivity of the default scenario to the switching temperature.

Figure 16

Sensitivity of $\mathrm{{\rm Y}}\left(1S\right)$ production to the mean $b$-quark speed, $\langle v_b\rangle$, by which the correlation volume expands (red, black and blue lines are for $\langle v_b\rangle =0.6c$, $0.65c$, and $0.7c$, respectively). Solid (dashed) lines are for the total (regeneration component of the) $\mathrm{{\rm Y}}\left(1S\right)R_{AA}$ regeneration component.

Figure 17

Sensitivity of $\mathrm{{\rm Y}}\left(1S\right)$ production to a variation of the $Y$ formation times by $\pm 20\%$.

Figure 18

Sensitivity of $\mathrm{{\rm Y}}\left(1S\right)$ production to a variation of the QGP formation time over the range ${\tau }_0=0.1\text{–}0.3\mathrm{fm}/c$. The regeneration component (dashed lines) is virtually unaffected.

Figure 19

Sensitivity of $\mathrm{{\rm Y}}\left(1S\right)$ production to a variation of the fireball expansion acceleration over the range $a_{\mathrm{T}}=0.10\text{–}0.12/\mathrm{fm}$.

Figure 20

Sensitivity of $\mathrm{{\rm Y}}\left(1S\right)$ production to the $b$-quark thermalization time, ${\tau }_b$, over the range ${\tau }_b=8\text{–}12\mathrm{fm}$ at $2T_{\mathrm{pc}}$. Dashed lines are for the regeneration component.

Figure 21

Transverse-momentum dependent $R_{AA}$ for inclusive $\mathrm{{\rm Y}}\left(1S\right)$ (red band) and $\mathrm{{\rm Y}}\left(2S\right)$ (blue band) production and their regeneration component (pink and light blue bands, respectively) in minimum-bias Pb-Pb (2.76 TeV) collisions within the TBS for $\eta =1.0$ at mid- and forward rapidity (upper and lower panel, respectively), compared to CMS data [31]. The width of the total bands includes a 0–15% (0–30%) shadowing suppression for $\left|y\right|<2.4$ ($2.5<y<4.0$), the variation in the temperature window for the regeneration component (also shown separately) and the uncertainty in the $pp$ baseline spectra.

Figure 22

Same as upper panel in Fig. 21 but with the regeneration component evaluated by a thermal-blast-wave approximation for the respective $Y$ states.

Figure 23

The $p_T$ dependence of elliptic flow of $\mathrm{{\rm Y}}\left(1S\right)$ (upper panel) and $\mathrm{{\rm Y}}\left(2S\right)$ (lower panel) in semicentral Pb-Pb (2.76 TeV) at mid-rapidity within the TBS ($\eta =1.0$). In both panels the blue, green, and red curves are for the regeneration component, primordial component, and their weighted sum, respectively, where the band widths reflect uncertainties from varying the average regeneration temperatures.

Figure 24

Centrality dependence of $R_{AA}$ for $\mathrm{{\rm Y}}\left(1S\right)$ (upper panel) and $\mathrm{{\rm Y}}\left(2S\right)$ (lower panel) within the SBS in Pb-Pb (5.02 TeV) collisions at mid-rapidity. Red, green, and blue lines represent the total, primordial, and regeneration contributions, where the bands reflect a 0–25% shadowing effect.

Figure 25

Same as Fig. 24 but at forward rapidity.

Figure 26

Centrality dependence of bottomonium $R_{AA}$'s in Pb-Pb (5.02 TeV) collisions within the TBS ($\eta =1.0$). Upper panel: mid-rapidity $\mathrm{{\rm Y}}\left(1S\right)$, $\mathrm{{\rm Y}}\left(2S\right)$, and $\mathrm{{\rm Y}}\left(3S\right)$ (red, blue, and green bands, respectively) compared to CMS data [59, 60]; the bands are due to a 0–15% shadowing suppression. Lower panel: forward rapidity $\mathrm{{\rm Y}}\left(1S\right)$ compared to ALICE data [75, 76]; the bands are due to a 0–30% shadowing suppression.

Figure 27

Centrality dependence of the $\mathrm{{\rm Y}}\left(2S\right)/\mathrm{{\rm Y}}\left(1S\right)R_{AA}$ double ratio in 5.02 TeV Pb-Pb collisions at mid-rapidity within the TBS (with an uncertainty band from in-medium binding energies for $\eta =0.9-1.1$), compared to CMS data [77].

Figure 28

The $p_T$ dependence of $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)$ yields in MB Pb-Pb (5.02 TeV) collisions at mid- and forward rapidities within the TBS ($\eta =1.0$). Upper panel: mid-rapidity $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)R_{AA}$ for total (red and blue curves, respectively) and regeneration components (pink and light-blue curves, respectively), compared to CMS data [59, 60]; the bands reflect variations due to a 0–15% shadowing suppression and the average regeneration temperatures (${\overline{T}}_{\mathrm{reg}}$) of the two states. Middle panel: same as upper panel but at forward rapidity with a 0–30% shadowing range, compared to ALICE data [76]. Lower panel: $\mathrm{{\rm Y}}\left(2S\right)/\mathrm{{\rm Y}}\left(1S\right)R_{AA}$ double ratio compared to CMS data [77]; the band reflects variations in the ${\overline{T}}_{\mathrm{reg}}$'s.

Figure 29

The $p_T$ dependence of elliptic flow of $\mathrm{{\rm Y}}\left(1S\right)$ (upper panel) and $\mathrm{{\rm Y}}\left(2S\right)$ (lower panel) in semicentral Pb-Pb (5.02 TeV) at mid-rapidity within the TBS ($\eta =1.0$). In both panels the blue, green, and red curves are for the regeneration component, primordial component, and their weighted sum, respectively, where the band widths reflect uncertainties from varying the average regeneration temperatures.

Figure 30

Excitation function the MB $R_{AA}$ of $\mathrm{{\rm Y}}\left(1S\right)$ and $\mathrm{{\rm Y}}\left(2S\right)$ with TBS compared to STAR [54] and CMS [31, 59, 60] data at mid-rapidity.