Extracting the jet transport coefficient from hadron suppressions by confronting current NLO parton fragmentation functions

Abstract

Nuclear modification factors of single hadrons and dihadrons at large transverse momentum ($p_{\mathrm{T}}$) in high-energy heavy-ion collisions are studied in a next-to-leading-order (NLO) perturbative QCD parton model. Parton fragmentation functions (FFs) in $A+A$ collisions are modified due to jet energy loss which is proportional to the jet transport coefficient $\hat{q}$ characterizing the interaction between the parton jet and the produced medium. By confronting 6 current sets of NLO parton FFs for large $p_{\mathrm{T}}$ hadron productions, we extract $\hat{q}$ quantitatively via a global fit to data for both single hadron and dihadron suppressions and obtain $\hat{q}/T^3 = 4.74 - 6.72$ at $T = 370$ MeV in central $Au+Au$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV, and $\hat{q}/T^3 = 3.07 - 3.98$ at $T = 480$ MeV in central $Pb+Pb$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV. The numerical results show that the uncertainties for $\hat{q}$ extraction are brought by the different contributions of gluon-to-hadron in the six sets of FFs due to gluon energy loss being 9/4 times of quark energy loss.

Appendix

1.1 A: The $\hat{q}_0$ extractions with different scales for different sets of FFs

To check the nuclear or medium effects in $A+A$ collisions, phenomenologically, one need a suitable baseline of $p+p$ collisions by adjusting the scale $\mu$ to fit data well in $p+p$ collisions. Shown in the upper panel of Fig. 14 are the numerical results for $\pi ^0$ hadron spectra at large transverse momentum $p_{\mathrm{T}}$ with the five sets of FFs in $p+p$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV. Here, the suitable scale $\mu$ is chosen in the theoretical model with each set of FFs for fitting to the experimental data [1], respectively. The lower panel of Fig. 14 is for the ratios of the experimental data over theoretical calculations. One can see that with the appropriate scales $\mu$ in the model with different sets of FFs, the numerical results fit the data better relative to those in Fig. 1.

Fig. 14

$\pi ^0$ spectra given by the NLO pQCD parton model with the befitted scale $\mu$ for each set of FFs in $p+p$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV (upper panel), and spectrum ratios of data over the theoretical results (lower panel). The data are from Ref. [1]

Similarly, in $p+p$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV the charged hadron spectra are also calculated with different scales in the NLO pQCD parton model for the six sets of FFs, as shown in the upper panel of Fig. 15. The ratios of the experimental data over theoretical calculations are shown in the lower panel. With the appropriate scales, the theoretical results fit data very well.

Using the above hadron spectra in $p+p$ collisions as baselines, we extract the jet quenching parameter from single hadron and dihadron suppressions with suitable scale in each set of FFs. Figure 16 shows the $\chi ^2/d.o.f$ fits to nuclear modification factors in central $Au+Au$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV: panel (a) the fits to only single hadron $R_{AA}(p_{\mathrm{T}})$, panel (b) the fits to only dihadron $I_{AA}(z_{\mathrm{T}})$, and panel (c) the global fits to $R_{AA}(p_{\mathrm{T}})$ + $I_{AA}(z_{\mathrm{T}})$. From the panel (c), we can read the best-fitting values of jet transport coefficient: $\hat{q}_0=1.4$ GeV$^2$/fm with KRE FFs at $\mu =0.6p_{\mathrm{T}}$, $\hat{q}_0=1.4$ GeV$^2$/fm with KKP FFs at $\mu =1.2p_{\mathrm{T}}$, $\hat{q}_0=1.4$ GeV$^2$/fm with HKNS FFs at $\mu =0.6p_{\mathrm{T}}$, $\hat{q}_0=1.2$ GeV$^2$/fm with AKK08 FFs at $\mu =1.0p_{\mathrm{T}}$, and $\hat{q}_0=1.2$ GeV$^2$/fm with DSS FFs at $\mu =1.2p_{\mathrm{T}}$. The difference between $\hat{q}_0=1.2 \sim 1.4$ GeV$^2/$fm is narrow relative to the same scale case.

Fig. 15

Charged hadron spectra given by the NLO pQCD parton model with the befitted scale $\mu$ in $p+p$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV (upper panel), and spectrum ratios of data over the theoretical results (lower panel). The data are from Refs. [4, 5]

Figure 17 shows the $\chi ^2/d.o.f$ fits to nuclear modification factors in central $Pb+Pb$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV: (a) the fits to only single hadron $R_{AA}(p_{\mathrm{T}})$, (b) the fits to only dihadron $I_{AA}(p_{\mathrm{T}}^{\mathrm{assoc}})$, and (c) the global fits to $R_{AA}(p_{\mathrm{T}})$ + $I_{AA}(p_{\mathrm{T}}^{\mathrm{assoc}})$. We use six sets of fragmentation function parameterizations with different scales in our calculations, respectively. We get the best fitting values of the jet transport parameter as: $\hat{q}_0=2.2$ GeV$^2/$fm with KRE FFs at $\mu =1.5p_{\mathrm{T}}$, $\hat{q}_0=2.4$ GeV$^2/$fm with KKP FFs at $\mu =3.5p_{\mathrm{T}}$, $\hat{q}_0=2.3$ GeV$^2/$fm with BFGW FFs at $\mu =5.0p_{\mathrm{T}}$, $\hat{q}_0=2.4$ GeV$^2/$fm with HKNS FFs at $\mu =6.5p_{\mathrm{T}}$, $\hat{q}_0=2.5$ GeV$^2/$fm with AKK08 FFs at $\mu =7.5p_{\mathrm{T}}$, $\hat{q}_0=2.4$ GeV$^2/$fm with DSS FFs at $\mu =5.5p_{\mathrm{T}}$. The difference between $\hat{q}_0$ drops to $2.2\sim 2.5$ GeV$^2/$fm.

The above transport coefficients $\hat{q}/T^3$ extracted in the model with different scales in different sets of FFs from hadron suppressions at RHIC and the LHC are summarized in the right panel of Fig. 13.

Fig. 16

The $\chi ^2/d.o.f$ results from fitting to nuclear modification factors in central $Au+Au$ collisions at $\sqrt{s_{\mathrm{NN}}}=200$ GeV, a the fits to only single hadron $R_{AA}(p_{\mathrm{T}})$, b the fits to only dihadron $I_{AA}(z_{\mathrm{T}})$, and c the global fits to $R_{AA}(p_{\mathrm{T}})+I_{AA}(z_{\mathrm{T}})$. Five sets of fragmentation function parameterizations are used in the model with different scales, respectively

Fig. 17

The $\chi ^2/d.o.f$ results from fitting to nuclear modification factors in central $Pb+Pb$ collisions at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, a the fits to only single hadron $R_{AA}(p_{\mathrm{T}})$ [4, 5], b the fits to only dihadron $I_{AA}(p_{\mathrm{T}}^{\mathrm{assoc}})$ [75, 76], and c the global fits to $R_{AA}(p_{\mathrm{T}})+I_{AA}(p_{\mathrm{T}}^{\mathrm{assoc}})$. Six sets of fragmentation function parameterizations are used in the model with different scales, respectively

1.2 B: Characteristics of fragmentation function contributions for parton to hadron with different scales for different sets of FFs

Figure 18 shows the quark and gluon FFs of $\pi ^0$ hadrons at $p_{\mathrm{T}}=10$ GeV and 50 GeV for all the available FFs with different scales, respectively. Figure 19 shows the similar plot for charged-hadron FFs of quark and gluon with different scales in each FF. The difference between gluon FFs is a bit reduced with befitted scales compared with the same scale case.

Fig. 18

Comparisons of parton-to-$\pi ^0$ fragmentation functions between five sets of FFs: left panels are for quark FFs and right panels for gluon FFs. Upper panels are for the hadrons with $p_{\mathrm{T}}=10$ GeV, and lower panels with $p_{\mathrm{T}}=50$ GeV. The scale $\mu$ is different for each FF

Fig. 19

Comparisons of parton-to-$h^{\pm }$ FFs between six sets of FFs: left panels are for quark FFs and right panels for gluon FFs. Upper panels are for the hadrons with $p_{\mathrm{T}}=10$ GeV, and lower panels with $p_{\mathrm{T}}=50$ GeV. The scale $\mu$ is different for each FF

Fig. 20

The contribution fractions of quark (solid lines) and gluon (dashed lines) fragmentations to the inclusive $\pi ^0$ cross sections given by the model with only AKK08 FFs with $\mu =1.0p_{\mathrm{T}}$ and $\mu =1.2p_{\mathrm{T}}$ at $\sqrt{s_{\mathrm{NN}}}=200$ GeV, respectively

Fig. 21

The contribution fractions of quark (solid lines) and gluon (dashed lines) fragmentations to the inclusive charged-hadron cross sections given by the model with only AKK08 FFs with $\mu =1.5p_{\mathrm{T}}$ and $\mu =7.5p_{\mathrm{T}}$ at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, respectively

Taking the AKK08 FFs as examples, we show the relative contributions of quark and gluon to hadrons with the changed scales in $p+p$ collisions at RHIC and the LHC energies, as shown in Figs. 20 and 21. One can see that both at RHIC and the LHC, with $\mu$ increasing, the gluon contribution to final state hadrons will decrease; thus, we need a larger jet quenching parameter to compensate for the total jet energy loss.

For a given set of FFs in a NLO pQCD parton model, the different scales lead to the different fractions of gluon (quark) contributions to hadrons in $p+p$ collisions. Although the scale change can also affect the parton distribution functions and the hard cross sections, the fraction change is mainly contributed by parton fragmentation functions, as shown in Figs. 20 and 21. In detail, when the scale $\mu$ decreases from 1.2$p_{\mathrm{T}}$ to 1.0$p_{\mathrm{T}}$ at $\sqrt{s_{\mathrm{NN}}}=200$ GeV, the contribution of gluon-to-hadron becomes larger, as illustrated in Fig. 20, so a relatively smaller $\hat{q}_0$ is needed for the case of $\mu = 1.0p_{\mathrm{T}}$ shown in Fig. 13. Meanwhile, with the $\mu$ increasing from 1.5$p_{\mathrm{T}}$ to 7.5$p_{\mathrm{T}}$ at $\sqrt{s_{\mathrm{NN}}}=2.76$ TeV, the contribution of gluon-to-hadron reduces, as presented in Fig. 21; thus, a relatively larger $\hat{q}_0$ is needed for the case of $\mu = 7.5p_{\mathrm{T}}$ shown in Fig. 13. In a word, the different fraction of gluon (quark) contribution to hadrons will give different energy loss parameters due to gluon energy loss being 9/4 times of quark energy loss.