Longitudinal Dynamics from Hydrodynamics with an Order Parameter

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Shu Lin and Gezheng Zhou. Longitudinal Dynamics from Hydrodynamics with an Order Parameter[J]. Chinese Physics C.
Shu Lin and Gezheng Zhou. Longitudinal Dynamics from Hydrodynamics with an Order Parameter[J]. Chinese Physics C. shu
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Received: 2020-12-06
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Longitudinal Dynamics from Hydrodynamics with an Order Parameter

  • 1. School of Physics and Astronomy, Sun Yat-Sen University, Zhuhai, 519082, China

Abstract: We study coupled dynamics of hydrodynamic fields and order parameter in the presence of nontrivial longitudinal flow using the chiral fluid dynamics model. We find the longitudinal expansion provides an effective relaxation for the order parameter, which equilibrates in an oscillatory fashion. Similar oscillations are also visible in hydrodynamic degrees of freedom through coupled dynamics. The oscillations are reduced when dissipation is present. We also find the quark density initially peaked at the boundary of boost invariant region moves toward forward rapidity with the peak velocity correlated with velocity of longitudinal expansion. The peak gets broadened during the evolution. The corresponding chemical potential rises due to simultaneous dropping of density and temperature. We compare the cases with and without dissipation for the order parameter, and also the standard hydrodynamics without order parameter. We find the corresponding effect on temperature and chemical potential can be understood from the conservation laws and different speed of equilibration of order parameter in the three cases.

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    I.   INTRODUCTION
    • It is remarkable that relativistic hydrodynamics provides accurate description of bulk evolution of matter produced in heavy ion collisions, which consists of around a thousand particles [1, 2]. With the assumption of local equilibrium, the problem of complicated many particle dynamics is reduced to the conservation of energy, momentum and baryon number as equation of motion for relativistic hydrodynamics. Over the past decade, the framework of relativistic hydrodynamics has been furnished in many aspects for phenomenological application in heavy ion collisions: the inclusion of viscous correction leads to more accurate description of the bulk evolution [3, 4, 5]; the inclusion of noises allows for systematic treatment of fluctuations [6, 7, 8]; the inclusion of particle momentum anisotropy extends the regime of applicability to earlier time [9, 10] etc.

      A notable feature of quantum chromodynamics (QCD) is chiral phase transition. It is believed that the transition is a crossover at low baryon density based on lattice simulation and becomes first order at high baryon density based on high density perturbation theory. It is conjectured that the first order phase transition ends as second order phase transition point, which is commonly referred to as critical end point (CEP). Recently the beam energy scan (BES) program in relativistic heavy ion collider (RHIC) is devoted to pinning down possible existence of CEP in the QCD phase diagram [11]. In order to describe evolution of QCD matter across a phase transition, it is phenomenologically motivated to couple the dynamics of hydrodynamic degree of freedoms to the order parameter, known as chiral fluid dynamics model. Efforts along this line have been taken for studies of first order phase transition [12, 13, 14, 15] and extended to crossover as well [16, 17]. Close to the CEP, the inclusion of order parameter becomes indispensable due to the appearance of a new slow mode, which is a mixture of order parameter and baryon density [18, 19, 20]. The coupling of the critical mode and the hydrodynamic modes is found to alter the bulk evolution near the CEP [21, 22, 23], see also [24] for a recent review.

      In this paper, we use the chiral fluid dynamics model discussed above to study the effect of order parameter. In particular, we focus on the effect of a nontrivial longitudinal flow on the coupled dynamics of hydrodynamic fields and order parameter. The paper is organized as follows: In Section 2, we describe the hydrodynamics with order parameter based on linear sigma model. In Section 3, we present numerical solutions with nontrivial longitudinal expansion and discuss physical interpretation of the results. We conclude and discuss outlook in Section 4.

    II.   HYDRODYNAMICS WITH AN ORDER PARAMETER
    • We start with the Lagrangian of linear sigma model [25]:

      $ {\cal L} = {\bar q}(i{\not \partial}-g({\sigma}+i{\gamma}_5{\vec {\tau}}{\vec {\pi}}))q+\frac{1}{2}((\partial{\sigma})^2+(\partial{\vec {\pi}})^2)-U({\sigma},{\vec {\pi}}),$

      (1)

      with q, $ {\sigma} $ and $ {\vec {\pi}} $ being the quark, sigma and pions fields respectively. The condensation of $ {\sigma} $ gives mass to quarks $ M_q = g\left\langle {\sigma} \right\rangle $, breaking chiral symmetry. The symmetry is broken through the potential U given by

      $ U({\sigma},{\vec{\pi}}) = \frac{{\lambda}}{4}\left({\sigma}^2+{\vec{\pi}}^2-v^2\right)^2-c{\sigma}. $

      (2)

      Throughout the paper, we use mean-field approximation for $ {\sigma} $ and $ {\vec{\pi}} $. In the absence of isospin chemical potential, pions do not condense thus $ \left\langle {\vec{\pi}} \right\rangle = 0 $. $ \left\langle {\sigma} \right\rangle $ is the only order parameter in the model. This is to be included in hydrodynamics. The parameters in (1) and (2) are fixed as

      $ \begin{aligned}[b]&g = M_q/f_{\pi},\quad c = M_{\pi}^2f_{\pi},\quad \\ &{\lambda} = \frac{1}{2f_{\pi}^2}(M_{\sigma}^2-M_{\pi}^2),\quad v^2 = f_{\pi}^2-M_{\pi}^2/{\lambda}, \end{aligned} $

      (3)

      with the experimental input $ M_{\pi} = 138 \;{\rm{MeV}} $, $ M_{\sigma} = 600 \;{\rm{MeV}} $, $ f_{\pi} = 93 \;{\rm{MeV}} $. The condensate $ \left\langle {\sigma} \right\rangle $ is determined dynamically by minimizing the thermodynamic potential $ {\Omega} = U+{\Omega}_{q{\bar q}} $, with the quark contribution $ {\Omega}_{q{\bar q}} $ given by

      $\begin{aligned}[b] {\Omega}_{q{\bar q}}({\sigma},T,{\mu}) =& {\nu}_qT\int\frac{d^3k}{(2{\pi})^3}\big[\ln(1-n_q(T,{\mu},k))\\ &+\ln(1-n_{\bar q}(T,{\mu},k))\big]. \end{aligned} $

      (4)

      Here $ {\nu}_q = 2N_cN_f = 12 $ counts the spins, colors and flavors of the quark field. $ n_q $ and $ n_{\bar q} $ are Fermi-Dirac distributions for quark and anti-quark respectively:

      $n_q(T,{\mu},k) = \dfrac{1}{e^{(\sqrt{k^2+M_q^2}-{\mu})/T}+1},\quad n_{\bar q}(T,{\mu},k) = n_{q}(T,-{\mu},k).$

      (5)

      According to the chiral fluid dynamics model [14], the quark fields are treated as hydrodynamic degree of freedom, which is coupled to sigma mean field. The equations of motion (EOM) for $ {\sigma} $ and hydrodynamic fields are given by:

      $ D_{\mu} D^{\mu}{\sigma}+\frac{{\delta}{\Omega}}{{\delta}{\sigma}} = 0, $

      (6)

      $D_{\mu}\left(T_q^{{\mu}{\nu}}+T_{\sigma}^{{\mu}{\nu}}\right) = 0,$

      (7)

      $D_{\mu} J^{\mu} = 0. $

      (8)

      Note that we choose to write the EOM in curved spacetime with $ D_{\mu} $ denoting covariant derivative. This will be useful for adapting to Milner coordinates in the next section. The stress tensor component for quark field and sigma field and the quark current are given by respectively

      $ \begin{aligned}[b] &T_q^{{\mu}{\nu}} = ({\epsilon}+p)u^{\mu} u^{\nu}-pg^{{\mu}{\nu}}, \\ &T_{\sigma}^{{\mu}{\nu}} = \partial^{\mu}{\sigma}\partial^{\nu}{\sigma}-g^{{\mu}{\nu}}\left(\frac{1}{2}(\partial{\sigma})^2-U({\sigma})\right), \\ &J^{\mu} = nu^{\mu}, \end{aligned} $

      (9)

      with $ p = -{\Omega}_{q\bar{q}} $, $ n = \dfrac{\partial p}{\partial {\mu}} $ and $ {\epsilon} = T\dfrac{\partial p}{\partial T}+{\mu} n-p $. Using (6), we can express the divergence of $ T_{\sigma}^{{\mu}{\nu}} $ field as

      $ D_{\mu} T_{\sigma}^{{\mu}{\nu}} = -\frac{{\delta} {\Omega}_{q{\bar q}}}{{\delta}{\sigma}}\partial^{\nu}{\sigma}. $

      (10)

      We can thus rewrite conservation of stress energy tensor as

      $ D_{\mu} T_q^{{\mu}{\nu}} = \frac{{\delta} {\Omega}_{q{\bar q}}}{{\delta}{\sigma}}\partial^{\nu}{\sigma}. $

      (11)

      This has clear interpretation that the stress tensor from the quark field is conserved up to work and force by the sigma field. We further introduce a phenomenological dissipation term for the sigma field in (6):

      $ D_{\mu} D^{\mu}{\sigma}+\frac{u^{\mu} D_{\mu}{\sigma}}{{\tau}_{rel}}+\frac{{\delta}{\Omega}}{{\delta}{\sigma}} = 0. $

      (12)

      The limit $ {\tau}_{rel} = \infty $ corresponds to the case without dissipation. A finite $ {\tau}_{rel} $ characterizes the time scale in which the sigma field approaches equilibrium value. By varying the $ {\tau}_{rel} $, we can study the effect of the order parameter on the hydrodynamic degree of freedoms. We do not introduce a fluctuation for the sigma field for simplicity.

    III.   NUMERICAL SOLUTIONS
    • In this section, we solve (12) and (11) numerically. The linear sigma model involves possible first order phase transition. The state of matter is uniquely determined by $ {\sigma} $, T and $ {\mu} $. We will use these together with fluid velocity $ u^{\mu}(x) $ as our dynamical fields. We consider $ 1+1D $ longitudinal hydrodynamics. This corresponds to longitudinally expanding fluid only. While this ignores transverse expansion, which becomes important at late stage of matter evolution in heavy ion collisions, it can nevertheless provide some insights in the longitudinal dynamics of the fluid and the order parameter.

      It is convenient to express the EOM in terms of proper time $ {\tau} = \sqrt{t^2-z^2} $ and spacetime rapidity $ {\eta} = \tanh^{-1}\dfrac{z}{t} $ as

      $\partial_{\tau}^2{\sigma}\!+\!\frac{1}{{\tau}}\partial_{\tau}{\sigma}\!-\!\frac{1}{{\tau}^2}\partial_{\eta}^2{\sigma}\!+\!\frac{{\delta}{\Omega}}{{\delta}{\sigma}}\!+\!\frac{u^{\tau}\partial_{\tau}{\sigma}\!+\!u^{\eta}\partial_{\eta}{\sigma}}{{\tau}_{rel}} \!=\! 0, $

      (13)

      $ \partial_{\tau} T_q^{{\tau}{\tau}}+\frac{1}{{\tau}}T_q^{{\tau}{\tau}}+\partial_{\eta} T_q^{{\tau}{\eta}}+{\tau} T_q^{{\eta}{\eta}}-\frac{{\delta}{\Omega}_{q{\bar q}}}{{\delta}{\sigma}}\partial_{\tau}{\sigma} = 0,$

      (14)

      $ \partial_{\tau} T_q^{{\tau}{\eta}}\!\!+\!\!\frac{1}{{\tau}}\partial_{\eta} T_q^{{\tau}{\eta}}\!+\!\partial_{\eta} T_q^{{\eta}{\eta}}\!+\!\frac{2}{{\tau}}T_q^{{\tau}{\eta}}\!+\!\frac{1}{{\tau}^2}\!\frac{{\delta}{\Omega}_{q{\bar q}}}{{\delta}{\sigma}}\partial_{\eta}{\sigma} \!=\! 0,$

      (15)

      $\partial_{\tau} J^{\tau}+\frac{J^{\tau}}{{\tau}}+\partial_{\eta} J^{\eta} = 0. $

      (16)

      (13) will be solved with the following initial conditions at $ {\tau} = {\tau}_0 = 1 $fm

      $ \begin{aligned}[b] T({\tau} = {\tau}_0,{\eta})& = \frac{T_{\max}-T_{\min}}{2}\tanh\frac{{\eta}_E-{\eta}}{{\Delta}}+\frac{T_{\max}+T_{\min}}{2}, \\ {\mu}({\tau} = {\tau}_0,{\eta})& = {\mu}_{\max}e^{-({\eta}-{\eta}_E)^2/(2{\Delta}^2)}, \\ u^{\tau}({\tau} = {\tau}_0,{\eta})& = \frac{u_{\min}-u_{\max}}{2}\left[\tanh\frac{{\eta}_E-{\eta}}{{\Delta}}-\tanh\frac{{\eta}_E}{{\Delta}}\right]+u_{\min}, \\ {\sigma}({\tau} = {\tau}_0,{\eta})& = {\sigma}_{\rm{eq}}(T({\tau} = {\tau}_0,{\eta}),{\mu}({\tau} = {\tau}_0,{\eta})), \\ \partial_{\tau}{\sigma}({\tau} = {\tau}_0,{\eta})& = 0. \\[-15pt]\end{aligned} $

      (17)

      and boundary conditions

      $ \begin{aligned}[b] \partial_{\eta}{\sigma}({\tau},{\eta} = 0) =& \partial_{\eta} T({\tau},{\eta} = 0) = \partial_{\eta}{\mu}({\tau},{\eta} = 0) \\ =& \partial_{\eta} u^{\tau}({\tau},{\eta} = 0) = 0, \\ \partial_{\eta}{\sigma}({\tau},{\eta} = {\eta}_m) = &0. \end{aligned} $

      (18)

      The initial conditions (17) are motivated by heavy ion collisions. We use a Gaussian-type initial condition for $ {\mu} $ and a tanh-type initial condition for T. We use $ T_{\min} $ as a temperature cutoff at $ {\eta} = {\eta}_m = 10 $. This captures the main features of the fireball, which is hot in the nearly boost invariant mid-rapidity region and cold in the forward rapidity region. The baryons are localized on the edge of the boost invariant region [26, 27]. The initial condition of $ {\sigma} $ is taken to be the equilibrium value at given T and $ {\mu} $ in the initial profile. We also use a tanh-type profile for $ u^{\tau} $. We assume parity $ {\eta}\to-{\eta} $ is a symmetry of the system, which dictates $ u^{\tau}({\eta} = 0) = u_{\min} = 1 $. $ u_{\max} $ is a tunable parameter for the initial longitudinal flow, with $ u_{\max} = 1 $ corresponding to common choice of boost invariant flow. $ u_ {\max}>1 $ corresponds to a larger longitudinal flow in the forward rapidity region. While this is not the case for heavy ion collisions, keeping a tunable initial flow allows us to study its effect on evolution of the fireball. The boundary conditions (18) are chosen for the following reasons: the first line is dictated by parity symmetry $ {\eta}\to-{\eta} $; the second line of (18) is imposed at spatial boundary $ {\eta} = {\eta}_m $. Note that $ u^{\eta} $ is determined by $ u^{\tau} $ as

      $u^{\tau}{}^2-u^{\eta}{}^2{\tau}^2 = 1. $

      (19)

      However, at $ {\eta} = 0 $, $ \partial_{\eta} u^{\tau} = 0 $ and $ u^{\tau} = 1 $ leads to an undetermined $ \partial_{\eta} u^{\eta} $. A nonvanishing $ \partial_{\eta} u^{\eta} $ would mean non-boost invariant flow around $ {\eta} = 0 $. While this is in principle possible, we do not consider this possibility. Motivated by boost invariance at mid-rapidity, we impose $ \partial_{\eta} u^{\eta} = 0 $ at $ {\eta} = 0 $. (13) are solved by discretizing the rapidity space and integrating forward in proper time.

      Since our solution is not boost invariant, fluid cells starting with different initial rapidity trace out different trajectories in the phase diagram. The chiral phase transition corresponding each trajectory may be crossover, first order and second order, with the last case occurring when the trajectory passes through the CEP. Schematically, we have two possible scenarios: at high energy heavy ion collisions, the initial temperature is high and quark chemical potential is low, all the trajectories correspond to crossover transition; at low energy collisions, the initial temperature is low and quark chemical potential is high, some or even all trajectories correspond to first order transition. We expect the second scenario to be qualitatively different for the following reason: The first order transition is featured by potential barrier between two local minima of the sigma field. It follows that first order phase transition occurs through supercooling and bubble nucleation. At finite temperature, the nucleation process can be efficiently realized through thermal excitation, which requires the fluctuation of sigma field. This has been pursued in [14, 15, 17] by treating sigma field as a stochastic variable. In this paper, we focus on the first scenario for simplicity with an emphasis on the role of longitudinal dynamics and sigma field.

      Let us present numerical results without dissipation first. In Fig. 1, we show the evolution of T, $ n{\tau} $ and $ {\mu} $ with $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $. The value of $ T_ {\min} $ in our case is chosen to ensure stable numerical output. It is different from the case of actual heavy ion collisions with $ T_ {\min}\simeq0 $. We will refer to $ n{\tau} $ as quark number density. The reason for using $ n{\tau} $ is that this is the total charge per rapidity with $ {\tau} $ from the volume factor. It is conserved in boost invariant case. We see both the temperature and quark number density decrease with increasing $ {\tau} $ due to longitudinal expansion. The peak of $ n{\tau} $ broadens and moves towards forward rapidity. The behavior of $ {\mu} $ follows a similar trend as n, except that it rises at late time with increasing $ {\tau} $. The longitudinal flow build up gradually with increasing $ {\tau} $.

      Figure 1.  Top left: T as a function of $ {\tau} $ and $ {\eta} $. The temperature decreases with increasing $ {\tau} $ due to the longitudinal expansion. Top right: $ n{\tau} $ as a function of $ {\tau} $ and $ {\eta} $. The peak of $ n{\tau} $ broadens and moves towards forward rapidity. Bottom left: $ {\mu} $ as a function of $ {\tau} $ and $ {\eta} $. The peak of chemical potential rises further, broadens and moves towards forward rapidity, which can be understood from combined effect of $ n{\tau} $ and T. Bottom right: $ u^{\tau} $ as a function of $ {\tau} $ and $ {\eta} $. The longitudinal flow builds up gradually with increasing $ {\tau} $. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      The behavior of the peaks in $ n{\tau} $ and $ {\mu} $ can be understood as a combined effect of T and n: at low $ {\mu} $, we have $ {\mu} = \dfrac{n}{{\chi}} $ with $ {\chi}\sim T^2 $. The dropping of the denominator is faster than the numerator, leading to the increase of $ {\mu} $. The moving of the peak follows from the longitudinal flow we impose in the initial condition. To illustrate this, we show in Fig. 2 $ n{\tau} $ at three different times for $ u_ {\max} = 1.5 $ and $ u_ {\max} = 2.5 $ respectively. The peak corresponding to larger $ u_ {\max} $ moves faster towards forward rapidity. The broadening of the peak is correlated with the behavior in temperature. At $ {\tau} = 1{\rm{fm}} $ the width of variation of temperature and chemical potential are set by $ {\Delta} $. As the system cools during the longitudinal expansion, the width of the variation of temperature increases, which causes a similar increase in the width of peak of chemical potential through their coupling in hydrodynamic equations.

      Figure 2.  $ n{\tau} $ as a function of $ {\eta} $ at three different $ {\tau} $ for $ u_ {\max} = 1.5 $ (thin lines) and $ u_ {\max} = 2.5 $ (thick lines). The initial profile at $ {\tau} = 1{\rm{fm}} $ are the same. At later $ {\tau} $, the case with larger longitudinal flow (larger $ u_ {\max} $) moves faster towards forward rapidity. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $ and ${\mu}_ {\max} = 80 \;{\rm{MeV}}$.

      Now we investigate effect of dissipation. In Fig. 3, we show evolution of $ {\sigma} $ with and without dissipation. As the system cools down due to longitudinal expansion, the chiral symmetry is breaking through growth of sigma field. In the absence of dissipation, this occurs in an oscillatory fashion. Similar behavior is also observed in a boost invariant setting [28]. The effective relaxation can be understood as coming from longitudinal expansion. It would not exist in a static system, in which an off-equilibrium sigma field is expected to oscillate indefinitely. To see this more quantatively, we specialize to the boost invariant case, where the oscillatory behavior is known to exist [28]. We linearize the EOM of $ {\sigma} $ field in a background solution, in which the $ {\sigma} $ field is set by equilibrium value at local temperature and chemical potential. When the temperature drops significantly, the equilibrium value of sigma field corresponds to chiral symmetry breaking phase. The linearized EOM is given by

      Figure 3.  $ {\sigma} $ as a function of $ {\tau} $ and $ {\eta} $ without dissipation (left) and with dissipative relaxation time $ {\tau}_{rel} = 0.5{\rm{fm}} $ (right) starting from the same initial condition. In the absence of relaxation, $ {\sigma} $ rises in an oscillatory fashion. In the presence of sigma field dissipation, the oscillatory behavior is significantly reduced. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      $ \partial_{\tau}^2{\delta}{\sigma}+\frac{1}{{\tau}}\partial_{\tau}{\delta}{\sigma}+\frac{{\delta}^2{\Omega}}{{\delta}{\sigma}^2}{\delta}{\sigma}+\frac{{\delta}^2{\Omega}}{{\delta}{\sigma}{\delta} T}{\delta} T+\frac{{\delta}^2{\Omega}}{{\delta}{\sigma}{\delta} {\mu}}{\delta} {\mu} = 0. $

      (20)

      Since the $ {\sigma} $ is at local minimum, $\dfrac{{\delta}^2{\Omega}}{{\delta}{\sigma}{\delta} T} = \dfrac{{\delta}^2{\Omega}}{{\delta}{\sigma}{\delta} {\mu}} = 0$ and $\dfrac{{\delta}^2{\Omega}}{{\delta}{\sigma}^2} > 0$. The dynamics of $ {\delta}{\sigma} $ decouples from those of $ {\delta} T $ and $ {\delta}{\mu} $:

      $ \partial_{\tau}^2{\delta}{\sigma}+\frac{1}{{\tau}}\partial_{\tau}{\delta}{\sigma}+\frac{{\delta}^2{\Omega}}{{\delta}{\sigma}^2}{\delta}{\sigma} = 0. $

      (21)

      Clearly by ignoring the middle term and considering $ \dfrac{{\delta}^2{\Omega}}{{\delta}{\sigma}^2}\equiv k^2 $ being a slow-varying function of $ {\tau} $, we obtain an approximate oscillatory solution $ {\delta}{\sigma}\sim e^{ik{\tau}} $. The middle term is due to the longitudinal expansion. It provides an effective friction term for $ {\delta}{\sigma} $, with the effective relaxation time set by $ {\tau} $. Physically the dissipation arises from the loss of energy in fluid cells due to the longitudinal expansion.

      When dissipation is present, the oscillations of sigma field are significantly reduced. This is clearly visible in Fig. 4, which compares $ {\sigma} $, T and $ {\mu} $ as a function of $ {\eta} $ at different times with and without dissipation. Interestingly it is the non-dissipative case that corresponds to faster equilibration. Fig. 4 suggests that the dissipation slows down the initial rise of sigma toward the chiral symmetry breaking minimum. The behavior of the sigma field is in accord with counterpart of temperature and chemical potential. The peak of chemical potential is enhanced in the non-dissipative case. Analogous but milder enhancement also exists in temperature. These can be understood from conservation of charge and energy: in both cases the charge and energy are conserved. The non-dissipative case converges more quickly to equilbrium, i.e. larger value of $ {\sigma} $. Note that the thermodynamics of the quark sector is nothing but that of free quark with constituent mass set by $ M_q = g{\sigma} $. At the same T and $ {\mu} $, the non-dissipative case (with larger $ M_q $) gives smaller charge density n and energy density $ {\epsilon} $. In order to maintain the same n with the dissipative case, $ {\mu} $ has to rise further to compensate for the larger $ M_q $. Similar rise in T is needed to maintain the same $ {\epsilon} $ with the non-dissipative case.

      Figure 4.  $ {\sigma} $ (top), T (bottom left) and $ {\mu} $ (bottom right) as a function of $ {\eta} $ at different $ {\tau} $ without (thin lines) dissipation and with dissipative relaxation time $ {\tau}_{rel} = 0.5{\rm{fm}} $ (thick lines) starting with the same initial condition. The non-dissipative case corresponds faster equilibration of $ {\sigma} $ and enhanced temperature and chemical potential. The behavior of T and $ {\mu} $ can be understood from counterpart of $ {\sigma} $ and conservation of charge and energy, see text for explanations. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      It is also instructive to compare trajectories of fluid cells starting with the same initial rapidity for the cases with and without dissipation. We can trace the trajectories by solving the following equation:

      $ u^{\tau}({\tau},{\eta}({\tau})) = \frac{d{\tau}}{\sqrt{d{\tau}^2-{\tau}^2d{\eta}^2}} = \left(1-{\tau}^2\frac{d{\eta}({\tau})}{d{\tau}}\right)^{-1/2},$

      (22)

      with the initial condition $ {\eta}({\tau} = {\tau}_0) = {\eta}_0 $ for different $ {\eta}_0 $. The LHS is known from numerical solution of $ u^{\tau} $. With the solution of (22), we can obtain different trajectories in the phase diagram. We show in Fig 5 comparison of trajectories with and without dissipation. The two cases are clearly distinguishable by the zig-zag shape present only in the case without dissipation. This is reminiscent of the oscillatory behavior in the sigma field in the absence of dissipation.

      Figure 5.  The trajectories of fluid cells starting with different initial rapidities for the case without dissipation (left) and with dissipative relaxation time $ {\tau}_{rel} = 0.5{\rm{fm}} $ (right). The initial rapidities for different trajectories are $ {\eta} = 1 $, $ {\eta} = 2 $, $ {\eta} = 2.5 $ and $ {\eta} = 3 $ from left to right. The trajectories without dissipation are featured by zig-zag shape, which is reminiscent of the oscillatory behavior in the sigma field. The zig-zag shape is absent in the case with dissipation. The phase boundary is also shown in the plots with solid line and dashed line corresponding to first order and crossover transition respectively. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      To have a close look at the evolution of the sigma field, we show the sigma field in the fluid cell starting with $ {\eta} = 3 $ for the case with and without dissipation in Fig 6. We also use the equilibrium sigma field determined by local temperature and chemical potential as references for the corresponding cases. The cases with and without dissipation do not show significant difference for the equilibrium sigma field, but the difference in the actual sigma field is clearly visible in that the dissipative case shows reduced oscillation and delayed equilibration.

      Figure 6.  $ {\sigma} $ as a function of $ {\tau} $ in the fluid cell starting with $ {\eta} = 3 $ for the case with and without dissipation. Equilibrium $ {\sigma} $ determined by local temperature and chemical potential in the cell are also included for references. No significant effect of dissipation is seen in the equilibrium $ {\sigma} $. The effect of dissipation is clearly visible from reduced oscillations and slow convergence to equilirium in the actual $ {\sigma} $. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      To illustrate the role of sigma field in the dynamics, we also compare the dynamics with and without the sigma field. For the former, we choose the non-dissipative case as a reference. For the latter, we use the standard hydrodynamic equations below:

      $ \begin{aligned}[b] &D_{\mu} T^{{\mu}{\nu}} = 0, \\ &D_{\mu} J^{\mu} = 0. \end{aligned} $

      (23)

      with

      $ \begin{aligned}[b] &T^{{\mu}{\nu}} = ({\epsilon}+p)u^{\mu} u^{\nu}-pg^{{\mu}{\nu}}, \\ &J^{\mu} = nu^{\mu}. \end{aligned} $

      (24)

      Here $ p = -{\Omega}_{\rm{eq}} $ and ${\epsilon} = T\dfrac{\partial p}{\partial T}+{\mu} n-p$. The equilibrium free energy $ {\Omega}_{\rm{eq}} = {\Omega}_{q\bar{q}} $ is evaluated at the minimum of $ {\Omega}_{q\bar{q}}+U $ for given temperature and chemical potential. While this is thermodynamically consistent, it is not consistent with the Lagrangian (1) because the vacuum potential U does not contribute to the stress tensor (24), but only enters in fixing the equilibrium value of $ {\sigma} $. With the caveat in mind, we proceed to solve (23) numerically with the following initial and boundary conditions

      $ \begin{aligned}[b] T({\tau} = {\tau}_0,{\eta})& = \frac{T_{\max}-T_{\min}}{2}\tanh\frac{{\eta}_E-{\eta}}{{\Delta}}+\frac{T_{\max}+T_{\min}}{2}, \\ {\mu}({\tau} = {\tau}_0,{\eta})& = {\mu}_{\max}e^{-({\eta}-{\eta}_E)^2/(2{\Delta}^2)}, \\ u^{\tau}({\tau} = {\tau}_0,{\eta})& = \frac{u_{\min}-u_{\max}}{2}\left[\tanh\frac{{\eta}_E-{\eta}}{{\Delta}}-\tanh\frac{{\eta}_E}{{\Delta}}\right]+u_{\min}, \\ \partial_{\eta} T({\tau},{\eta} = 0)& = \partial_{\eta}{\mu}({\tau},{\eta} = 0) = \partial_{\eta} u^{\tau}({\tau},{\eta} = 0) = 0.\\[-15pt] \end{aligned} $

      (25)

      This choice ensures two cases having the same charge density initially. But the total energy densities are different because the case without sigma field does not include vacuum energy density. We solve (23) numerically and compare the hydrodynamic evolution with and without sigma field.

      In Fig. 7, we show the evolution of temperature and chemical potential for the two cases. The case without sigma field shows enhanced peak in $ {\mu} $ and a moderate enhancement in T. The enhancement can be understood from the same mechanism discussed above. The case without sigma field can be understood as instantaneous equilibration. In our case, it is equilibration toward chiral symmetry breaking minimum, giving rise to free quark with larger mass. Since the evolution conserves n, $ {\mu} $ has to rise further to compensate for the larger mass in order to maintain the same charge density as the case with sigma field. Similar mechanism works for T although there is a subtlety that the energy in the quark sector is not strictly conserved due to the work by the sigma field. In Fig. 8, we compare the trajectories of fluid cells starting from different rapidities for the case with and without sigma field. The trajectories for the case without sigma field bend toward region of larger $ {\mu} $ and T, which is consistent with the enhancement of $ {\mu} $ and T.

      Figure 7.  T (left) and $ {\mu} $ (right) as a function of $ {\eta} $ at different t with sigma field (thin lines) and without (thick lines). The case without sigma field shows an enhanced peak in $ {\mu} $. A moderate enhancement in T is also present. This can be understood from conservation of charge and energy density, see the text for explanations. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

      Figure 8.  The trajectories of fluid cells starting with different initial rapidities for the case with non-dissipative sigma field (left) and without (right). The initial rapidities for different trajectories are $ {\eta} = 1 $, $ {\eta} = 2 $, $ {\eta} = 2.5 $ and $ {\eta} = 3 $ from left to right. The trajectories for the case without sigma field bend toward larger $ {\mu} $ and T. The phase boundary is also shown in the plots with solid line and dashed line corresponding to first order and crossover transition respectively. Parameters used: $ T_{\max} = 360 \;{\rm{MeV}} $, $ T_ {\min} = 80 \;{\rm{MeV}} $, $ {\mu}_ {\max} = 80 \;{\rm{MeV}} $ and $ u_{\max} = 2 $.

    IV.   CONCLUSION AND OUTLOOK
    • We have studied the coupled evolution of hydrodynamic fields and sigma field as order parameter for chiral phase transition in the presence of nontrivial longitudinal flow using the chiral fluid dynamics model. We have chosen the initial condition with high temperature and low quark chemical potential, for which the phase transition in each fluid cell is a crossover. We have found that the presence of longitudinal expansion provides an effective relaxation for the sigma field, resulting in equilibration in an oscillatory fashion. The oscillation is reduced when dissipation for the sigma field is present. However, the dissipation also seem to slow down equilibration of sigma field itself.

      We have also found the quark density initially peaked at the boundary of the boost invariant region moves toward forward rapidity, with the peak velocity correlated with the velocity of the longitudinal expansion. In the mean time, the peak gets broadened during the evolution. As the system expands and cools, the width of temperature variation also gets broadened, which cause a similar broadening in the quark chemical potential. The peak of the chemical potential rises as a result of simultaneous dropping of density and temperature. The peak is enhanced in the non-dissipative case compared to the dissipative case. A mild enhancement is also present in temperature. These follow from charge and energy conservation and different speed of equilibration of sigma field.

      We have also compared the coupled evolution of hydrodynamic and sigma fields to the standard hydrodynamics without sigma field. It is found that the latter case leads to significant enhancement in the chemical potential and mild enhancement in the temperature. It can also be understood from conservation of charge and energy and instantaneous equilibration of sigma field in the standard hydrodynamics.

      While we restrict ourselves to case of crossover, it is more interesting to generalize the current study to the case of first order phase transition, in which fluctuation is expected to play an essential role. It would be interesting to understand how longitudinal expansion and dissipation of sigma field influence the first order phase transition. We leave it for future studies.

    ACKNOWLEDGMENTS
    • S.L. is grateful to Huichao Song for useful discussions.

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