Viscous effects on the dynamical evolution of QCD matter during the first-order confinement phase transition in heavy-ion collisions

We investigate viscous effects on the dynamical evolution of QCD matter during the first-order phase transition, which may happen in heavy-ion collisions. We first obtain the first-order phase transition line in the QCD phase diagram under the Gibbs condition by using the MIT bag model and the hadron resonance gas model for the equation of state of partons and hadrons. The viscous pressure, which corresponds to the friction in the energy balance, is then derived from the energy and net baryon number conservation during the phase transition. We find that the viscous pressure relates to the thermodynamic change of the two-phase state and thus affects the timescale of the phase transition. Numerical results are presented for demonstrations.


I. INTRODUCTION
A phase diagram separates phases and determines conditions, at which different phases coexist at thermal equilibrium. The completion of the QCD phase diagram [1] is an ongoing task and essential for understanding the matter under strong interaction. Lattice QCD calculations [2] showed that the QCD phase transition at small baryon chemical potential is a crossover rather than a real phase transition. At high baryon chemical potential, theory predicts a first-order phase transition line [3] ending at a QCD critical point [4]. The phase transition of QCD matter can be investigated in experiments of heavy-ion collisions, where quark-gluon plasma (QGP) cools down due to expansion and hadronizes at certain temperature and baryon chemical potential. One major goal of the beam energy scan program at Relativistic Heavy Ion Collider (RHIC) [5][6][7][8] is to locate the critical point in the QCD phase diagram.
Usually, a phase transition is defined when two coexisting phases are in thermal equilibrium. The QCD matter produced in heavy-ion collisions possesses, however, a nonzero viscosity [9][10][11] and deviates from thermal equilibrium. If the system is not far away from thermal equilibrium, one can still define thermodynamic quantities such as temperature, pressure, and chemical potential, as done in viscous hydrodynamic calculations. With these thermodynamic quantities one can also identify the first-order phase transition for the expanding QGP, if the Gibbs condition holds. In this paper we consider nonzero viscosities of QCD matter and investigate viscous effects on the dynamical evolution of QCD matter during the first-order phase transition.
In Sec. II we first calculate the first-order phase transition line under the Gibbs condition for phase equilibrium * xuzhe@mail.tsinghua.edu.cn by using MIT bag model and hadron resonance gas model for the equation of state (EoS) of the parton and hadron phase. We then show in Sec. III how the shear and bulk viscosity affect the phase transition of the QCD matter produced in heavy-ion collisions. Two different expansion geometries are applied to the evolution of the QCD matter. In Sec. IV further discussions are given.

II. PHASE DIAGRAM
Since lattice QCD results of EoS at finite baryon chemical potential are not yet available, we use the EoS from model calculations. Based on these, we present in this section the first-order phase transition line in the temperature-baryon chemical potential diagram.
The parton phase is considered as a system of massless quarks and gluons, which interactions are described by perturbative QCD (pQCD) up to g 2 s terms [12][13][14]. The pressure and energy density are where B is the bag constant with B 1/4 = 200 MeV and a(T, µ q , g s ) µ i is the chemical potential of a quark flavor. n f is the number of quark flavors. We consider u, d, s quarks (n f = 3) and assume µ u = µ d ≡ µ q , µū = µd = −µ q , and µ s = µs = 0. The running coupling is given by [12][13][14].
with Λ QCD = 100 MeV. From the pressure we obtain the net baryon number density, which is one third of the net quark number density, (5) Here we neglect the logarithmic dependence of α s on µ q . The hadron phase is described by the hadron resonance gas model (HRG) [15][16][17]. Baryons, mesons, and their resonances having masses up to 2 GeV are included. The pressure and energy density of hadrons and the net baryon number density are given by where i, j, and m denote baryon, antibaryon, and meson, respectively. P 0 k , e 0 k , and n 0 k are the pressure, energy density, and number density of a hadron species k, when assuming a non-interacting hadron gas, where d k is the degeneracy factor and + sign is for baryons and − sign is for mesons. m k denotes the hadron mass and µ h k denotes the hadron chemical potential, which relates to the quark chemical potential as µ h k = ν k µ q , where ν k is the net quark number in hadron k. Thus, µ h k = 0 for mesons and µ h k = −µ h k when k denotes the antiparticle of k.
We take short range repulsive interactions among (anti)baryons into account [15,[18][19][20]. This is indicated in the denominators of the terms in Eqs. (6)- (8), where v i(j) = 4πr 3 i(j) /3 is the eigen volume of baryon i or antibaryon j. We assume a same hard sphere radius for all baryons.
During the first-order phase transition the Gibbs condition holds. The pressure, temperature and baryon chemical potential of both parton and hadron phase are equal. Equating the pressures from Eqs. (1) and (6) with the same temperature T and baryon chemical potential µ B = 3µ q , we obtain the phase boundary curve, which is shown in Fig. 1. The hard-core radius r i(j) of (anti)baryons is set to be 0.6 fm [15]. The calculated curve is similar as those given in [14,15,21].
For the later use the entropy density is given below, where the subscript i can be p and h denoting the parton and hadron phase, respectively. We calculate s p /n Bp and s h /n Bh along the first-order phase transition line. The results are shown in Fig. 2. We see that s p /n Bp is larger than s h /n Bh and both are decreasing with increasing µ B .

III. VISCOUS EFFECTS DURING THE FIRST-ORDER PHASE TRANSITION
In this section we show how the QCD matter in heavyion collisions crosses the first-order phase transition line.
For vanishing viscosity the total entropy is conserved. With the conservation of the net baryon number the ratio of the entropy density over the net baryon number density is conserved too. This means that s h /n Bh at the end of the phase transition should be equal to s p /n Bp at the beginning of the phase transition. From Fig. 2 we realize that s p /n Bp is always larger than s h /n Bh at any given µ B . Therefore, µ B (and T ) cannot keep constant during the first-order phase transition. Since in addition s h /n Bh increases with decreasing µ B , µ B of the two-phase state will change and move continuously to a smaller value along the first-order phase transition line (see the dashed line in Fig. 2), while T will accordingly move to a larger value [22]. The first-order phase transition will end up at a smaller µ B (or a larger T ). In the following we will study viscous effects on the dynamical evolution of QCD matter during the first-order phase transition. Obviously, µ B (or T ) along the first-order phase transition line will end up at even smaller (larger) value (see the dotted curve in Fig. 2), since more entropy will be produced due to nonzero viscosities.
We consider an expanding system of partons, which is undergoing the first-order confinement phase transition and hadronizing. There should be a clear spatial separation between the parton and hadron phase. Hadron bubbles will be formed. However, the description of the nucleation process is still a challenging issue [23,24]. We can imagine that due to the statistical nature, some of the bubbles are disappearing, while the others are growing and merging until the hadronization is complete. In this article we get around the fluctuating bubble picture in the nucleation process and describe the hadronization on an ensemble average. To this end we assume that each volume element, no matter how small it is, contains separated parton and hadron volumes. Suppose V is the volume of an expanding element in its local rest frame at proper time τ . We denote V p and V h as the volume of the parton and hadron phase, respectively. The fraction of the parton phase is then The time dependence of f p describes the hadronization on an ensemble average. The main conclusion of our study that we will present now shows that the effect of nonzero viscosity is to accelerate the decrease of µ B and to slow down the first-order phase transition.
The energy density and the net baryon number density of the two-phase system in the local rest frame of the considered expanding volume element are  14) gives (16) The left-hand side of the above equation can be obtained from the hydrodynamic equation for the energy density [25,26] where U µ is the fluid four-velocity, Π m = Π p f p + Π h (1 − f p ) is the total bulk pressure, and π µν m = π µν p f p + π µν h (1 − f p ) is the total shear tensor. Since we use the Landau's definition of the fluid four-velocity, there is no heat flow term in Eq. (17). Other symbols in this equation are defined as follows: We have then where Γ µ αβ ≡ 1 2 g µν (∂ β g αν − ∂ α g νβ − ∂ ν g αβ ) denotes the Christoffel symbol. By introducing the shear pressurẽ Eq. (17) changes to In the local rest frame, where U µ = (1, 0, 0, 0), we have De m = ∂e m /∂τ and By equating the right-hand side of both Eqs. (16) and (25) we obtain Analogously to the derivation from Eqs. (14) and (17) to Eq. (27), we can also derive df p /dτ from the net baryon number density (15) and its hydrodynamic evolution which indicates the net baryon number conservation. In the present study we have neglected the diffusion current induced by the heat conduction. We have then Equating Eq. (27) and Eq. (29) gives In the first-order theory of hydrodynamics, the bulk pressure and shear stress tensor are proportional to the bulk and shear viscosity [25,27,28], η m and ξ m are the shear and bulk viscosity of the twophase system and η m = η p f p + η h (1 − f p ) and ξ m = ξ p f p + ξ h (1 − f p ), where η p and η h (ξ p and ξ h ) are the shear (bulk) viscosity of the parton and hadron phase respectively. If all the viscosities and the fluid velocity are known, we can solve µ B (τ ) from Eq. (30) and then f p (τ ) from Eq. (27) or Eq. (29). As stated at the beginning of this section, µ B should decrease along the first-order phase transition line. However, from Eq. (30) it is not obvious that ∂µ B /∂τ is negative even for vanishing viscosities. We now look at the entropy of the two-phase system, which is according to Eqs. (14) and (15), and the definition (13). The time evolution of s m is obtained from the time evolution of e m and n Bm , namely Eqs. (17) and (28). We have for the first-order viscous hydrodynamics. Analogously to the derivation of Eq. (27) we get Equating Eq. (36) with Eq. (29) gives then where With the Gibbs-Duhem equation dP = sdT + ndµ one can prove that Eqs. (30) and (37) are identical.
Since both s p /n Bp and s h /n Bh decrease with increasing µ B (see Fig. 2), i.e., d(s p /n Bp )/dµ B < 0 and d(s h /n Bh )/dµ B < 0, one obtains easily ds p /dn Bp < s p /n Bp and ds h /dn Bh < s h /n Bh . We have then for s p /n Bp > s h /n Bh (see Fig. 2). From Eq. (37) we realize that ∂µ B /∂τ is always negative and its absolute value becomes larger for increasing viscosity. [Remember that Π m andπ m are negative for the first-order viscous hydrodynamics when the system is expanding and they are proportional to the bulk and shear viscosity according to Eqs. (24), (32), and (33).] The viscous effect leads to a stronger decrease of µ B during the first-order phase transition, compared to the ideal hydrodynamic expansion. Moreover, according to Eq. (29) the decrease of f p slows down in the viscous case. The larger the viscosity, the longer will the phase transition take. For demonstrating the viscous effects we now calculate explicitly the time evolution of µ B and f p during the first-order phase transition with given fluid velocity and viscosities. We compare the results with nonzero viscosities to those with zero viscosities.
Different from the Bjorken expansion, the Gubser expansion includes transverse expansion. The parameter a is set to be 4.5 fm. A similar value has been used to describe the hydrodynamic evolution of QGP in Au+Au collisions at RHIC with √ s N N = 200 GeV [30,33]. In addition we choose ρ = 0 in our calculations. The phase transition in volume elements with larger transverse radius ρ will occur earlier. For Bjorken expansion and for Gubser expansion at ρ = 0,τ is equal to the proper time in the local rest frame, τ .
Although there are calculations and model-to-data analyses on the shear and bulk viscosity of the parton and/or hadron phase in heavy-ion collisions [34][35][36][37][38][39][40][41], the shear and bulk viscosity of both phases during the firstorder phase transition are not yet fixed so far. We assume for simplicity that the shear and bulk viscosity in the parton phase are equal and the shear and bulk viscosity in the hadron phase are twice as much as those in the parton phase. Moreover, we set η p T /h p = ξ p T /h p to be constant during the phase transition. h p = e p + P c is the enthalpy density of partons. At µ B = 0, h p /T is equal to the entropy density. η p T /h p and ξ p T /h p are more relevant to characterize the viscous effect in slow expansion [42] as will happen in heavy-ion collisions with lower colliding energies.
In Fig. 3 the time evolution of µ B and f p are presented with three different values of viscosities, η p T /h p = ξ p T /h p = 0, 0.2, 0.4, and with two different expansion dynamics, Bjorken and Gubser expansion. As an example, the starting time of the phase transition τ c was set to be 1.5 fm/c and µ B at τ c has been chosen to be 0.5 GeV. With these settings we obtain n Bp = 0.1948/fm 3 and s p /n Bp = 20 at τ c . First we see that the results agree with the qualitative analyses done before. Compared with those with zero viscosities, the results with increasing viscosities show that the decrease of µ B becomes stronger, the phase transition takes longer, and thus µ B ends up with smaller final values, when the phase transition in the considered volume is complete. Second, the final value of µ B in ideal hydrodynamic expansion does not depend on the expansion geometry, because both the entropy and net baryon number are conserved. In viscous expansion the entropy production depends on the expansion geometry. Therefore, the final value of µ B in the Bjorken expansion is different from that in the Gubser expansion. Third, com-pared to the Bjorken expansion, the transverse expansion in the Gubser expansion leads to a stronger decrease of µ B and faster phase transition.
In addition, we show in the last panel of Fig. 3 the volume increase during the phase transition, V (τ )/V (τ c ). The points mark the volume increase at different end times of the phase transition. The rate of the volume increase is stronger in three-dimensional Gubser expansion than in one-dimensional Bjorken expansion. However, since the phase transition proceeds faster in Gubser expansion than in Bjorken expansion, the volume increase at the end of the phase transition is stronger in Bjorken expansion than in Gubser expansion.
The trajectory of s m /n Bm during the first-order phase transition is plotted in Fig. 2 for an ideal expansion (dashed straight line) and a viscous Bjorken expansion with η p T /h p = 0.2 (dotted curve).
Our results of the viscous effects on the dynamical evolution of QCD matter during the first-order phase transition could be refined, if more reliable EoS and transport coefficients of the parton and hadron phase were available and more realistic hydrodynamic expansion of QCD matter had been calculated at large baryon chemical potential. The present study is the basis for a further development of the dynamic transport simulation of the QCD phase transition in heavy-ion collisions [43].

IV. FURTHER DISCUSSIONS
In this section we discuss (or speculate) how the firstorder phase transition will proceed with much larger viscosities, with which calculations using the first-order hydrodynamics may be invalid.
Recalling Eq. (16) at the starting time of the phase transition τ c , where f p = 1, we have For an expanding system ∂e m /∂τ is always negative, while its absolute value depends on the form of the expansion (U µ ) and viscosity. The larger the viscosity, the smaller is |∂e m /∂τ |. If the second term on the right hand side of Eq. (44), (de p /dµ B )(∂µ B /∂τ ), is positive, df p /dτ should be negative, which indicates that the first-order phase transition will always proceed with any large viscosity, unless the viscous hydrodynamics breaks down.
On the other hand, if (de p /dµ B )(∂µ B /∂τ ) is negative, df p /dτ could be (mathematically) positive for sufficient large viscosity and slow expansion. A positive df p /dτ at τ c is not physical, which indicates that large viscosity may forbid the occurrence of first-order phase transition. Both the signs of de p /dµ B and ∂µ B /∂τ are determined by the EoS of the parton and hadron phase. For the EoS used in this work, both de p /dµ B and ∂µ B /∂τ are negative. Thus, df p /dτ at τ c is always negative.
We have to note that for large viscosity the friction heat will be so large, that ∂e m /∂τ becomes positive when using the first-order hydrodynamics [see Eqs. (32), (33), and (17)]. A positive ∂e m /∂τ can lead to a positive df p /dτ at τ c according to Eq. (44). However, positive ∂e m /∂τ is impossible for an expanding system. In this case the first-order hydrodynamics is invalid and higher order terms have to be included in the hydrodynamic description of the phase transition.
The phase transition for τ > τ c seems more complicated. It cannot be proven from Eq. (16) that df p /dτ is always negative with the used EoS, because de h /dµ B is positive. Thus, with sufficient large viscosity and slow expansion (slower than Bjorken and Gubser expansion), df p /dτ may become positive and a transition of the net baryon number from the hadron phase to the parton phase may happen. However, this will not lead to the disappearance of the hadron phase, since df p /dτ will be negative again at least at f p = 1, as proven before. Thus, we expect that for large viscosity and slow expansion, the time evolution of f p trends to decrease from 1 to 0, but maybe has some humps in between.
We note that at µ B being smaller than the value at the critical point, the QCD phase transition is a crossover. Our formalism developed in the previous section does not work in the crossover region. Although the exact position of the critical point connecting the crossover and first-order phase transition line is not known yet [3,4,7,8], various theoretical calculations suggest that its most probable location is in the µ B interval between 200 and 400 MeV [44,45]. For large viscosity, the moving (µ B , T ) point along the first-order phase transition line may pass the critical point. Then critical phenomena are expected to occur.
The whole derivations in the previous section from Eq. (14) to Eq. (39) are also valid for the first-order phase transition in a contracting medium, where dV /dτ is negative. In this case, µ B increases with time along the firstorder phase transition line. We then describe a transition from the hadron phase to the parton phase.
In heavy-ion collisions the produced QCD matter expands on the whole. But locally on a small spatial scale, expansion as well as contraction exist due to density fluctuations. Our study provides a potential hydrodynamic framework to describe the nucleation of partons.