Global $\chi^2$ RHIC and LHC Data Constraints on Soft-Hard Transport Properties of semi-Quark-Gluon-Monopole Plasmas with the CUJET3.1 Framework

We report results of a comprehensive new global $\chi^2$ analysis of nuclear collision data from RHIC (0.2ATeV), LHC1 (2.76ATeV), and recent LHC2 (5.02ATeV) energies. We use the updated CUJET3.1 framework to evaluate jet energy loss distributions in various models of the color structure of the QCD fluids produced in such reactions. The framework combines consistently viscous hydro predicted by VISHNU and the DGLV theory of jet elastic and inelastic energy loss generalized to sQGMP fluids with color structure including effective semi-QGP color electric $q+g$ densities as well as an effective color magnetic monopole density $m$ peaking near $T_c$. As in our previous CUJET3.0 analysis, the local q,g,m color density composition of the sQGMP is constrained by nonperturbative lattice QCD data on the QCD entropy density $s$, Polyakov loop $L$, light quark susceptibility $\chi_2^u$, and lattice data on the color electric and magnetic screening masses $\mu_E ,\mu_M$. We vary the two control parameters of the model: the maximum value of the running QCD coupling, $\alpha_c$, and the ratio, $c_m=\mu_M/\mu_E$, and predict the $p_T>10$GeV, the centrality, as well as the beam energy dependence of the nuclear modified jet fragment observables $R_{AA}$ and $v_2$. The global $\chi^2$ is found to be minimized to near unity for $\alpha_c=0.9\pm0.1$, and $c_m=0.25\pm0.03$. Thus, CUJET3.1 provides a nonperturbative solution to the long standing hard versus soft perfect fluidity puzzle. Most remarkably, estimates from this framework lead to a shear viscosity to entropy ratio ($\eta/s \sim T^3/\hat{q}\sim 0.1$) required for internal consistency of the sQGMP jet transport coefficient, $\hat{q}/T^3$, with the perfect fluidity property of QCD fluids near $T_c$. Predictions for future tests at LHC with 5.44ATeV XeXe and 5.02ATeV PbPb are also presented.


I. INTRODUCTION
High energy quark and gluon jets, produced initially in rare perturbative QCD processes lose energy and diffuse transversely along their paths due to interactions with the microscopic constituents in the hot quark-gluon plasma created in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC). Such hard (p T > 10 GeV) processes provide us an independent probe of the evolution history of the soft (p T < 2 GeV) QCD matter produced in such collisions. Recent high precision data from LHC Pb+Pb collisions on jet quenching and azimuthal asymmetry observables over wide kinematics and centrality ranges provide an opportunity to quantitatively constrain and differentiate competing models of jet-medium interactions as well as varied assumptions of the chromo electric and magnetic field structure of the bulk QCD "perfect fluids" produced in ultra relativistic nuclear collisions.
The simplest class of hard observables in a specific centrality class, C, is the p T and relative azimuthal angle dependence of the nuclear modification factor R f AA for final state hadrons (with flavor species denoted by f ), which is Fourier decomposed into harmonics as: where T AA (C) is the average number of binary nucleon-nucleon scattering per unit area in centrality class C. Typically C is expressed as a percentage interval of the inelastic cross  [31][32][33][34][35]. Similar CIBJET results for 30-40% centrality, with excellent agreement with experimental data, were shown in [36].
section, e.g. 10 -20% of the charged multiplicity per unit rapidity distribution. The p T and φ are transverse momentum and azimuthal angle of observed leading hadrons, relative to the bulk collective flow azimuthal harmonics. It may be noted that the experimental measurements of hard particle harmonics v f n are made with respect to the event-wise soft harmonics, and event-by-event fluctuations of the bulk initial condition may play an important role [37]. Within the CUJET3 framework, the influence of event-by-event fluctuations has been investigated with a generalized CIBJET(=ebeIC+VISHNU+DGLV) framework, with the results reported in Ref. [36]. The CIBJET results of R AA , v 2 and v 3 observables across a very wide range of p T for 30-40% centrality Pb+Pb collisions at 5.02 ATeV were shown in [36], with excellent agreement with experimental data. Here in Fig. 1, we further present the CIBJET results of R AA , v 2 and v 3 for a different centrality of 20-30%, which again show excellent agreement with experimental data and demonstrate the correct central-ity dependence of the CIBJET results. One conclusion found with CIBJET is that the p T and centrality dependence of the elliptic v f 2 (p T , C) azimuthal harmonics shows quantitative consistency at ∼ 10% level between calculations with averaged smooth bulk geometry and that with fluctuating initial conditions. This conclusion is true for varied centrality class and is in agreement with a similar consistency-check from the ebeIC+LBT+HT hard+soft framework in Ref. [13,38], while different from the ebeIC+vUSPhydro+BBMG framework in Ref. [37] which found much larger factor ∼ 2 sensitivity of the hard elliptic harmonic to event-by-event fluctuations. The finding from CIBJET justifies the use of averaged smooth geometry in the CUJET3 framework, as we shall adopt in the present paper.
The prime motivation of this work, is to conduct a comprehensive new global χ 2 analysis of nuclear collision data from RHIC(0.2ATeV), LHC1(2.76ATeV), and recent LHC2(5.02ATeV) energies for high p T light and heavy flavor hadrons. This analysis is performed with the updated CUJET3.1 framework to evaluate jet energy loss distributions in various models of the color structure of the QCD fluids produced in heavy ion collisions. The CUJET3.1 is based on our previous CUJET3.0 framework [28,29] and successfully addressed a few issues in CUJET3.0. We include a brief introduction about CUJET3.0 as well as a detailed discussion on the improvement in CUJET3.1 in the two appendices. We will show that the CUJET3.1 provides a non-perturbative solution to the long standing hard (R AA and v 2 ) versus soft "perfect fluidity" puzzle. We further examine the crucial issue of consistency between soft and hard transport properties of the QCD fluid in this framework. Predictions for future tests at LHC with 5.44 ATeV Xe+Xe and 5.02 ATeV Pb+Pb will also be presented.
The organization of this paper is as follows. We perform the model parameter optimization in Sec II, based on the quantitative χ 2 analysis with a comprehensive set of experimental data for light hadrons. In Sec. III we show the successful CUJET3.1 description of available experimental data for light hadrons as well as the successful independent test with heavy flavor hadrons. The temperature dependence of jet transport coefficient and the corresponding shear viscosity for the quark-gluon plasma, extracted from CUJET3.1, are presented in Sec. V. The CUJET3.1 predictions for on going experimental analysis are shown in Sec. IV.
Finally we summarize the paper in Sec. VI. A brief introduction of the CUJET3 framework as well as the improvements made in CUJET3.1 are also included in the two appendices.

II. GLOBAL χ 2 ANALYSIS WITH CUJET3
With the details discussed in Appendix A, the CUJET3 framework is a quantification model solving jet energy loss in a hydrodynamics background, implementing DGLV jet energy loss from both inelastic and elastic scattering, and interacts with both chromo electric and magnetic charges of the medium. There are two key parameters in the model. One parameter is α c in Eq.(A4), which is the value of QCD running coupling at the non-perturbative scale Q 2 = T 2 c and sensitively controls the overall opaqueness of the hot medium. The other is c m in Eq.(A7), which is the coefficient for magnetic screening mass in the medium and influences the contribution of the magnetic component to the jet energy loss.
To systematically constrain these two key parameters, a first step we take is to perform a quantitative χ 2 analysis and utilize central and semi-central high transverse momentum light hadron's R AA and v 2 for all available data. We compare the relative variance between theoretical expectation and experimental data, which is defined as the ratio of squared difference between experimental data point and corresponding CUJET3 expectation, to the quadratic sum of experimental statistic and systematic uncertainties for that data point: where i runs over all experimental data point in the momentum range 8 ≤ p T ≤ 50 GeV/c, and s denotes summing over all sources of uncertainties, e.g. systematic and statistic uncertainties. We compute χ 2 /d.o.f. for each of the following 12 data sets: • 200 GeV Au-Au Collisions, 0-10% Centrality Bin, R AA (π 0 ): PHENIX [39,40]; • 200 GeV Au-Au Collisions, 0-10% Centrality Bin, v 2 (π 0 ): PHENIX [40]; • 200 GeV Au-Au Collisions, 20-30% Centrality Bin, R AA (π 0 ): PHENIX [39,40]; and finally obtain the overall χ 2 /d.o.f. as the average over these data sets.  First of all, we perform the analysis in "slow" quark-libration scheme (χ L T -scheme) for a wide range of parameter space: 0.5 ≤ α c ≤ 1. In order to test the necessity of chromo-magnetic-monopole degree of freedom as well as to explore potential influence of the theoretical uncertainties of different quark liberation schemes, we perform the same χ 2 analysis with two other schemes: (a) the "fast" quark-libration scheme (χ u T -scheme); (b) the weakly coupling QGP (wQGP) scheme, being equivalent to CUJET2.0 mode, assuming no chromo-magnetic-monopole, i.e. taking f E = 1, f M = 0, and cec fraction χ T = 1, while the running coupling takes the Zakharov formula as in Eq. (A5). By using these 3 schemes with their corresponding most optimal parameter set: • (i) sQGMP χ L T -scheme: α c = 0.9, c m = 0.25, we show in Fig. 3 their comparison with above experimental data sets, including quantitative value of χ 2 /d.o.f. for each data set. While both sQGMP schemes (χ L T and χ u T ) give similar jet quenching variables, the QGP scheme gives similar R AA but less azimuthal anisotropy.
Especially, one can see clearly from the quantitative value of their χ 2 /d.o.f. that the theoretical expectations of both sQGMP schemes are in good consistency with data, and that of the QGP scheme, without cmm degree of freedom, differs significantly from the highly precise LHC v 2 measurements. The χ 2 analysis strongly supports the necessity of chromomagnetic-monopole degree of freedom, but remains robust on the specific quark liberation scheme.
While we will maintain the unification of CUJET3 model by using the same (global optimized) parameter set, it's worth mentioning that quantitative χ 2 analysis for different data set, e.g. different observable or different beam energy, flavors different parameter regime, as shown in Tab. I. When comparing to the R AA results, the azimuthal anisotropy measurement with more shrink uncertainties, yields higher χ 2 /d.o.f and hence On the other hand, in CUJET3 models, the RHIC results flavor stronger coupling (larger α c or c m ) than the LHC results; while the latter are more precise and give better distinction on different models. Especially with 5.02 TeV data, one can see explicitly that the sQGMP schemes are more phenomenologically flavored than the wQGP scheme.  With the high statistics of 5.02 TeV Pb-Pb data, we further expect that highly precise jet quenching observables for heavy flavored hadrons, e.g. D meson, could serve as an independent probe to discriminate sQGMP versus wQGP models. As shown in Fig. 4, we find sQGMP and wQGP models predicts similar R AA , while their significant different predictions of v 2 would need future experimental data with higher accuracy and for higher p T to provide a decisive distinction.  [45,46] are also shown. Corresponding R AA and v 2 data for light hadrons [32][33][34] are also shown with gray symbols.

III. COMPARISON WITH EXPERIMENTAL DATA
With the systematic χ 2 analysis, we obtained the optimal region of CUJET3 parameters constrained by only light hadron R AA and v 2 , for central and semi-central collisions. To provide a critical independent test of the model, we compute CUJET3 results for both light and heavy flavor hadrons, with all centrality ranges up to semi-peripheral collisions, and perform apple-to-apple comparisons with all available experimental data.
Starting from this section, in CUJET3 simulations we employed the χ L T -scheme assuming slow quark-libration, with keeping the theoretical uncertainties by taking the parameter region spanned by (α c = 0.8, c m = 0.22) and (α c = 1.0, c m = 0.28), which correspond to upper/lower bounds of R AA and lower/upper bounds of v 2 , respectively.

A. Light Hadrons
First of all, in Figures 5-10 [32] and CMS [33,34] data for 5.02 TeV Pb-Pb collisions. One can clearly see the excellent agreement for all centrality range at all these collision energies.
In particular, it is worth to emphasize again that after the aforementioned correction, the current CUJET3.1 simulation framework is able to correctly reproduce the p T and centrality dependence of both R AA and v 2 .  We note that such comprehensive data set covers a rich diversity of geometrical and thermal profiles of the QCD Plasma. In different centrality bins at various colliding energies, the bulk backgrounds are significantly distinctive in lifetime, size, ellipticity as well as temperature, and consequently, the path length of the jets, either direction averaging or depending, varies in a wide range. In Tab data [34].

B. Heavy Flavor Measurements
With successfully describing high-p T R AA and v 2 data for light hadrons, we now perform further independent test of the energy loss mechanism by using heavy flavor data [50].

IV. CUJET3 PREDICTIONS FOR OTHER EXPERIMENTAL OBSERVABLES
In above section we perform a successful test of the CUJET3 framework, which provides a united description for comprehensive sets of experimental data, from average suppression to azimuthal anisotropy, from light flavor to heavy flavor observables, with beam energy from preliminary data for charged hadron R AA from the ALICE [60], ATLAS [61], and CMS [62] collaborations (as shown in Fig. 20). See also Ref. [62] for a detailed data-model comparison.

V. JET TRANSPORT COEFFICIENT AND SHEAR VISCOSITY
As discussed above, the jet quenching observables of light hadrons provide stringent constraints on values of the jet energy loss parameters. In the meanwhile, the comparison between three different schemes, (i) sQGMP-χ L T , (ii) sQGMP-χ u T , and (iii) wQGP, shows the necessity of chromo-magnetic-monopole degree of freedom, while robustness on quark liberation rate. It is of great interests to further compare how the jet and bulk transport properties differ in these schemes. This will pave the way for clarifying the temperature dependence of jet quenching and shear viscous transport properties based on available high p T data in high-energy A+A collisions.
The jet transport coefficientq characterizes the averaged transverse momentum transfer squared per mean free path [66]. For a quark jet (in the fundamental representation F) with initial energy E, we calculate itsq in the same way as the previous CUJET3.0 computation in [28,29], viâ and similarly for a gluon/cmm jet: The quasi-parton density fractions of quark (q) or gluon (g), denoted as f q,g , are defined as respectively for sQGMP χ L T and χ u T scheme. The magnetically charged quasi-particle density fraction is hence f m = 1 − χ T = 1 − f q − f g . The color factors are given by One can find that while switching to the wQGP scheme, by taking f q = c q , f g = c g , f E = 1, f M = 0, turning off the cmm channel, and employing the running coupling α s (Q 2 ) defined in Eq.(A5), the jet transport coefficientq for a quark/gluon jet defined in Eq.(3/4) returns to that of the CUJET2.0 framework [67].  [17] with coupling λ = 4π · 3 · 0.31, are plotted for comparisons. The green blobs in inset (b) shows the JET collaboration [66] model average ofq F /T 3 while the boxes represent the uncertainties. (Right) The shear viscosity to entropy density ratio η/s estimated with scheme (i) (red solid), (ii) (red dashed), and (iii) (green dotdashed). The inset shows quasi-particle number density fraction of q, g, m in the liberation scheme χ L T (solid) and χ u T (dashed).
Once the jet transport coefficientq has been computed, one can extrapolateq(T, E) down to thermal energy scales E ∼ 3T /2 and estimate the shear viscosity to entropy density ratio η/s, based on kinetic theory in a weakly-coupled quasi-particle picture [68][69][70]. An estimate of η/s can be derived as The ρ a (T ) ≡ f a ρ(T ) is the quasi-parton density of type a = q, g, m. The mean thermal Mandelstam variable S ab ∼ 18T 2 . Clearly the η/s of the system is dominated by the ingredient which has the largest ρ a /q a .
In the left panel of Fig. 25, we show the temperature dependence of the dimensionless jet transport coefficientq F /T 3 for a light quark jet with initial energy E = 30GeV / 3GeV with all three schemes. Corresponding results from JET collaboration [66] model average and AdS/CFT limit [17] are also plotted for comparisons. As discussed in previous CUJET3.0 papers [28,29], the near-T c enhancement of dimensionless jet transport coefficient can be observed with robust dependence on quark liberation schemes.
In the right panel of Fig. 25, we show the shear viscosity to entropy density ratio η/s estimated in the kinetic theory using theq extrapolation Eq. (8) with scheme (i) (red solid) (ii) (red dashed) (iii) (green dotdashed). The inset shows quasi-particle number density fraction of q, g, m in the liberation scheme χ L T (solid) and χ u T (dashed). Note that in the near T c regime, in the χ u T scheme, the total η/s is dominated by q, while in the χ L T "slow" quark liberation scheme the total η/s is dominated by m. For each sQGMP scheme, there is a clear η/s minimum at T ∼ 210 MeV, which is comparable with the SYM limit (η/s) min = 1/4π.

VI. SUMMARY
In this paper we presented the CUJET3.1 framework and performed a global quantitative We end by emphasizing the important theoretical advantage of the CUJET3.1 framework.
It is not only χ 2 consistent with soft and hard observables data at RHIC and LHC, but also with nonperturbative lattice QCD data. Remarkably, estimates from this framework lead to a shear viscosity to entropy density ratio η s ∼ 0.1, which are not only consistent with extracted values from experimental soft+hard A+A phenomenology but also theoretically internally consistent with sQGMP kinetic theory link, η s ∼ T 3 q F (E→3T,T ) , between long distance collective fluid properties and short distance jet quenching physics especially near T c .
Initiative. The Indiana METACyt Initiative at IU was also supported in part by Lilly Endowment, Inc.
It might be worth noting that in the CUJET2 framework, assuming weakly-coupling QGP, the running coupling takes the form (with Λ QCD = 200 MeV) The Debye screening mass µ(z) is determined from solving the self-consistent equation as in [90]; χ 2 (z) = M 2 x 2 + +m 2 g (z)(1−x + ) regulates the soft collinear divergences in the color antennae and controls the Landau-Pomeranchuk-Migdal (LPM) phase, the gluon plasmon Since the sQGMP contains both chromo electrically charged quasi-particles (cec) and chromo magnetically charged quasi-particles (cmc), when jets propagate through the medium near T c , the total quasi-particle number density ρ is divided into EQPs with fraction χ T = ρ E /ρ and MQPs with fraction 1 − χ T = ρ M /ρ. The parameter f E and f M is defined via f E ≡ µ E /µ and f M ≡ µ M /µ, with µ E and µ M being the electric and magnetic screening mass respectively, following with the local electric "coupling" g(T (z)) = 4πα s (µ 2 (T (z))).
In current sQGMP modeling, the cec component fraction χ T remains a theoretical uncertainty related to the question of how fast the color degrees of freedom get liberated. To estimate χ T , one notices that: (1) when temperature is high, χ T should reach unity, i.e.
On the other hand, another useful measure of the non-perturbative suppression of the color electric DOF is provided by the quark number susceptibilities [97][98][99][100]. The diagonal susceptibility is proposed as part of the order parameter for chiral symmetry breaking/restoration in [97], and plays a similar role as properly renormalized L for quark DOFs.
In this scheme, we parametrize the lattice diagonal susceptibility of u quark number density, renormalized the susceptibility by its value at T → ∞, as (T in GeV) and define the cec component fraction in deconfinement scheme (χ u T -scheme) as: These two different schemes, for the rate of "quark liberation", with χ L T the "slow" and χ u B → D → channels, follows the same parameterization as in [102].
Appendix B: Improvements in CUJET3.1 In this Appendix, we discuss the improvements of CUJET3.1 framework with respect to the earlier CUJET3.0 framework. One important motivation for the CUJET3.1 upgrade work reported in the present paper was to uncover causes and correct the discrepancy of CUJET3.0 predictions for LHC 5.02 ATeV Pb+Pb collusions, reported by CMS in Ref. [33] with the nuclear modification factor R AA (see Fig. 26), as well as in Ref. [34] with their observed p T and especially centrality dependence of the hard elliptic asymmetry v 2 (see Fig. 27).
After a systematic examination, we found and corrected two issues that existed in previous CUJET3.0 simulations for 5.02 ATeV Pb+Pb collisions. (i) Firstly, the initial parton spectra were not consistently read in: the flavor factor of 3, was missed for light quark spectra. As the result, a higher fraction of the final hadrons were fragmented from gluon jets, which are more quenched relative to quark jets and caused the over-quenched R AA . (ii) Secondly, the probability distribution of initial jet production was incorrectly oriented (with x-and y-axis switched), hence wrong centrality dependence of v 2 was predicted. With correcting these two issues, the CUJET3.1 simulation correctly reproduces the p T and centrality dependence The curves represent calculations made with the CUJET 3.0 [29] and the SHEE [37] models.