Lattice NRQCD study of S- and P-wave bottomonium states in a thermal medium with $N_f=2+1$ light flavors

We investigate the properties of $S$- and $P$-wave bottomonium states in the vicinity of the deconfinement transition temperature. The light degrees of freedom are represented by dynamical lattice quantum chromodynamics (QCD) configurations of the HotQCD collaboration with $N_f=2+1$ flavors. Bottomonium correlators are obtained from bottom quark propagators, computed in nonrelativistic QCD under the background of these gauge field configurations. The spectral functions for the ${{}^{3}S}_1$ ($\mathrm{\Upsilon }$) and ${{}^{3}P}_1$ (${\chi }_{b1}$) channel are extracted from the Euclidean time correlators using a novel Bayesian approach in the temperature region $140\mathrm{MeV}\le T\le 249\mathrm{MeV}$ and the results are contrasted to those from the standard maximum entropy method. We find that the new Bayesian approach is far superior to the maximum entropy method. It enables us to study reliably the presence or absence of the lowest state signal in the spectral function of a certain channel, even under the limitations present in the finite temperature setup. We find that ${\chi }_{b1}$ survives up to $T=249\mathrm{MeV}$, the highest temperature considered in our study, and put stringent constraints on the size of the medium modification of $\mathrm{\Upsilon }$ and ${\chi }_{b1}$ states.

Figure 1

The lattice NRQCD correlators (top) and corresponding effective masses (bottom) $m_{\text{eff}}(\tau )=-\text{log}[D(\tau )/D(\tau +a_{\tau })]/a_{\tau }$ in the $S$-wave channel (left) and $P$-wave channel (right) at $T\simeq 0$ for $\beta =6.664,6.800,6.950$, and 7.280.

Figure 2

Bayesian reconstruction of the nonshifted spectra in the $S$-wave channel (left) and $P$-wave channel (right) at $T\simeq 0$ $\beta =6.664$ (dark red solid), 6.800 (dashed), 6.950 (solid) and 7.280 (light blue dashed). Note that at $\beta =7.280$ we only use the first 44 data points, i.e., where the signal is not yet lost to roundoff errors.

Figure 3

Energy shift constant $(C_{\text{shift}})$ determined from the position of the lowest lying peak in the $S$-wave channel spectral function for $\beta =6.664,6.800,6.950$, and 7.280.

Figure 4

Ratios of the Euclidean correlator at finite temperature and close to $T=0$ for the $\mathrm{\Upsilon }$ (top) and ${\chi }_{b1}$ (bottom) channel at the lattice spacings, where $T\simeq 0$ measurements were carried out.

Figure 5

Ratios of the Euclidean correlator at finite temperature and close to $T=0$ for the ${\eta }_b$ (top) and $h_b$ (bottom) channel at the lattice spacings, where $T\simeq 0$ measurements were carried out. Note the qualitative as well as quantitative similarity to the $\mathrm{\Upsilon }$ and ${\chi }_{b1}$ ratios shown in Fig. 4.

Figure 6

The mass-shift calibrated $S$-wave spectra from the new Bayesian approach (left column) and those from the MEM (right column) for fourteen different temperatures between 140 (dark violet) and 249 MeV (red). The middle row shows spectra for the three lowest temperatures (below $T_c$) and the three highest temperatures. The bottom figures contain spectra just above the deconfinement transition $T=155,\dots ,198\mathrm{MeV}$. Note that even though on a linear scale the ground state peak from the novel method does not appear to change (bottom left) its height decreases as can be seen in the top left panel.

Figure 7

Change in $S$-wave peak position and width when reconstructing the $T=0$ spectra at $\beta =6.664,\dots ,7.280$ (from the left to the right) from the full ${\tau }_{\max }=32/64$ data set, as well as from the subset with ${\tau }_{\max }=12$. Since for a fixed $\beta$ the high frequency regime remains unchanged when going from $T\simeq 0$ to $T>0$ the observed differences can serve as a measure of the limits to the reconstruction accuracy.

Figure 8

$S$-wave at 140 (left) and 249 MeV (right): Dependence of the reconstructed lowest peak position and peak width on choosing different subsets of data points along the $\tau$ axis. Note that the systematic effect of removing data points leads to a shift of the peak width and a broadening of the peak beyond the jackknife error bars.

Figure 9

$S$-wave spectral reconstructions at $T=140\mathrm{MeV}$ (top) and $T=249\mathrm{MeV}$ (bottom). To disentangle systematic and medium effects we compare the Bayesian $T>0$ spectra from all $N_{\tau }=12$ data points (colored solid) to the spectra obtained from the first ${\tau }_{\max }=12$ points of the $T\simeq 0$ correlator (dashed gray). To verify the presence of a bound state the spectra from noninteracting correlators are plotted (solid gray). $N_c$ denotes the number of correlators used in the analysis.

Figure 10

$S$-wave MEM spectral reconstructions at $T=140\mathrm{MeV}$ (top) and $T=249\mathrm{MeV}$ (bottom). Finite T spectrum (colored) as well as free spectrum with standard (solid gray) and extended $(N_{\text{ext}}=48)$ search space (dashed gray).

Figure 11

The mass-shift calibrated $P$-wave spectra from the new Bayesian approach (left column) and those from the MEM (right column) for fourteen different temperatures between 140 (dark violet) and 249 MeV (red). The middle row shows spectra for the three lowest temperatures (below $T_c$) and the three highest temperatures. The bottom figures contain spectra just above the deconfinement transition $T=155\mathrm{MeV},\dots ,198\mathrm{MeV}$.

Figure 12

Change in $P$-wave peak position and width when reconstructing the $T=0$ spectra from the full ${\tau }_{\max }=32/64$ data set, as well as from the subset with ${\tau }_{\max }=12$. Since for a fixed $\beta$ the high frequency regime remains unchanged when going from $T\simeq 0$ to $T>0$ the observed differences can serve as a measure of the limits to the reconstruction accuracy.

Figure 13

140 (left) and 249 MeV (right): Dependence of the reconstructed lowest peak position and peak width on choosing different subsets of data points along the $\tau$ axis. For the $P$-wave with worse signal-to-noise ratio, removing data points leads to an even stronger systematic shift of the peak width and a broadening of the peak than in the $S$-wave case.

Figure 14

$P$-wave spectral reconstructions at $T=140\mathrm{MeV}$ (left) and $T=249\mathrm{MeV}$ (right). To disentangle systematic and medium effects we compare the Bayesian $T>0$ spectra from all $N_{\tau }=12$ data points (colored solid) to the spectra obtained from the first ${\tau }_{\max }=12$ points of the $T\simeq 0$ correlator (blue dashed). To verify the presence of a bound state the spectra from noninteracting correlators are plotted (solid gray).

Figure 15

$P$-wave MEM spectral reconstructions at $T=140\mathrm{MeV}$ (left) and $T=249\mathrm{MeV}$ (right). Finite T spectrum (colored) as well as free spectrum with standard (solid gray) and extended $(N_{\text{ext}}=48)$ search space (dashed gray).

Figure 16

At 140 MeV: Dependence of the reconstructed lowest peak position and peak width on choosing different subsets of data points along the $\tau$ axis. In general the peak position is less susceptible than the peak width with the $\mathrm{\Upsilon }$ (top) showing consistently less dependence than the ${\chi }_{b1}$ (bottom).

Figure 17

At 249 MeV: Dependence of the reconstructed lowest peak position and peak width on choosing different subsets of data points along the $\tau$ axis. In general the peak position is less susceptible than the peak width with the $\mathrm{\Upsilon }$ (top) showing consistently less dependence than the ${\chi }_{b1}$ (bottom).

Figure 18

At 140 MeV: The raw $\mathrm{\Upsilon }$ reconstructed peak positions (top) and peak widths (bottom) for different choices of the number of jackknife bins and thus of the number of measurements contributing to the correlator average. Each plotted point corresponds to one spectral reconstruction for an individual jackknife bin. Their spread is a direct measure of the statistical uncertainty and reduces slightly with increasing number of jackknife bins. No systematic trend appears in reconstructed peak position, while the peak width seems to move to a smaller value until a plateau is reached at $N_{\text{conf}}=350$.

Figure 19

At 140 MeV: The raw ${\chi }_{b1}$ reconstructed peak positions (left) and peak widths (right) for different choices of the number of jackknife bins and thus of the number of measurements contributing to the correlator average. A slight trend to smaller values of the reconstructed peak position is visible, which however lies within the statistical uncertainty. No systematic trend appears for the peak width.

Figure 20

At 249 MeV: The raw $\mathrm{\Upsilon }$ reconstructed peak positions (left) and peak widths (right) for different choices of the number of jackknife bins and thus of the number of measurements contributing to the correlator average. No systematic trend appears in reconstructed peak position and width. In the reconstructed width the reduction of statistical uncertainty with increasing number of used measurements is clearly visible.

Figure 21

At 140 MeV: The raw ${\chi }_{b1}$ reconstructed peak positions (left) and peak widths (right) for different choices of the number of jackknife bins and thus of the number of measurements contributing to the correlator average. A slight trend to smaller values of the reconstructed peak position is visible, which however lies within the statistical uncertainty. No systematic trend appears for the peak width.

Figure 22

At 140 MeV: Dependence of the reconstructed peak position and width on different choices of the default model for the $\mathrm{\Upsilon }$ (top) and ${\chi }_{b1}$ (bottom) channel. Changing the functional form has a similar effect than moderately changing the overall normalization.

Figure 23

At 140 MeV: The actual spectral reconstructions for each of the choices of the prior model from which the values in Fig. 22 have been determined. The $S$-wave is shown on the left, the $P$-wave channel on the right.

Figure 24

At 248 MeV: Dependence of the reconstructed peak position and width on different choices of the default model for the $\mathrm{\Upsilon }$ (top) and ${\chi }_{b1}$ (bottom) channel. Changing the functional form has a similar effect to moderately changing the overall normalization.

Figure 25

At 248 MeV: The actual spectral reconstructions for each of the choices of the prior model from which the values in Fig. 24 have been determined. The $S$-wave is shown on the left, the $P$-wave channel on the right.

Figure 26

Free S-wave spectra (gray) for different choices of the default model $m(\omega )$, reconstructed from noninteracting NRQCD correlators using the effective mass values from the $T=140\mathrm{MeV}$ (left) and $T=249\mathrm{MeV}$ (right) lattices. In addition the interacting spectrum at the same temperature is shown (colored solid).

Figure 27

Free $P$-wave spectra (gray) for different choices of the default model $m(\omega )$, reconstructed from noninteracting NRQCD correlators using the effective mass values from the $T=140\mathrm{MeV}$ (left) and $T=249\mathrm{MeV}$ (right) lattices. In addition the interacting spectrum at the same temperature is shown (colored solid).

Figure 28

At 140 MeV: Dependence of the reconstructed peak position and peak width on different choices of the frequency interval for the $\mathrm{\Upsilon }$ (left) and ${\chi }_{b1}$ (right) channel. While a too narrow choice of $[{\omega }_{\min },{\omega }_{\max }]$ leads to significant changes in the reconstructed values, peak positions stabilize to a plateau at large enough interval lengths. The peak width is slightly less stable, since it depends more strongly on the default model normalization, which changes with each different choice of $[{\omega }_{\min },{\omega }_{\max }]$.

Figure 29

At 248 MeV: Dependence of the reconstructed peak position and peak width on different choices of the frequency interval for the $\mathrm{\Upsilon }$ (left) and ${\chi }_{b1}$ (right) channel. While a too narrow choice of $[{\omega }_{\min },{\omega }_{\max }]$ leads to significant changes in the reconstructed values, peak positions stabilize to a plateau at large enough interval lengths. The peak width is slightly less stable, since it depends more strongly on the default model normalization, which changes with each different choice of $[{\omega }_{\min },{\omega }_{\max }]$.

Figure 30

At 140 MeV: The dependence of the maximum entropy reconstructed peak width and position on different choices for the frequency interval in the $\mathrm{\Upsilon }$ (left) and ${\chi }_{b1}$ (right) channel. We find as expected from the arguments laid out in the subsection on the choice of the $\omega$ interval that moving ${\omega }_{\min }$ to lower and lower values degrades the quality of the reconstruction. Both the peak positions move to smaller values while the width grows. Neither in the $\mathrm{\Upsilon }$ nor in the ${\chi }_{b1}$ is a plateau reached within the inspected choices. Note that the error bars beyond ${\omega }_{\min }\le -1$ cannot be trusted as the minimizer remains at values of $L>100$. This is a clear indication for a lack of oscillatory degrees of freedom in the search space basis functions.

Figure 31

At 249 MeV: The dependence of the maximum entropy reconstructed peak width and position on different choices for the frequency interval in the $\mathrm{\Upsilon }$ (left) and ${\chi }_{b1}$ (right) channel.

Figure 32

The analytic free $S$-wave (left) and $P$-wave (right) spectral functions according to Eq. (3) for $n=2$ with the effective mass parameter chosen to agree with the values of the fully interacting lattices ($N_s=384$ used to obtain smooth curves). While lattice artifacts dominate the region above 8 GeV (i.e. above the first kink), the physics of the bound state resides at frequencies below 1 GeV.

Figure 33

The analytic free $S$-wave (left) and $P$-wave (right) spectral functions according to Eq. (b3) for $n=4$ with the effective mass parameter chosen to agree with the values of the fully interacting lattices ($N_s=384$ used to obtain smooth curves). Note that in comparison to Fig. 32 the high frequency range is much more limited, while the amplitude of the kink structure is increased. We also see that the region of relevant frequencies $w<1\mathrm{GeV}$ does not change appreciably.

Figure 34

The free spectra for $S$-waves (left) and $P$-waves (right) extracted from the free NRQCD correlators, based on the effective mass parameters $a_s(\beta =7.280)M_b=1.56$ and three different values of the parameter $n=2,3,4$. Consistent with expectations, the spectra are increasing in amplitude at higher frequencies $\omega >1\mathrm{GeV}$. In the $S$-wave channel the number of reconstructed wiggles remains stable, while there seems to be a slight increase in the amplitude of the ground state peak at $n=4$. In the $P$-wave channel for $n=4$, we obtain an additional wiggle but do not find indications that the strength of the low frequency structures changes.

Figure 35

The interacting spectra for $S$-waves (left) and $P$-waves (right) extracted from the NRQCD correlators measured on the $T>0$, $\beta =7.280$ lattices at three different values of the parameter $n=2,3,4$. Again consistent with expectations the spectra are increasing in amplitude at higher frequencies $\omega >1\mathrm{GeV}$. In both channels the ground state peak is virtually unaffected by the changes in the high frequency behavior of NRQCD.