MonteCarlo approach to particlefield interactions and the kinetics of the chiral phase transition
Abstract
The kinetics of the chiral phase transition is studied within a linear quarkmeson model, using a MonteCarlo approach to semiclassical particlefield dynamics. The meson fields are described on the meanfield level and quarks and antiquarks as ensembles of test particles. Collisions between quarks and antiquarks as well as the annihilation to mesons and the decay of mesons is treated, using the corresponding transitionmatrix elements from the underlying quantum field theory, obeying strictly the rule of detailed balance and energymomentum conservation. The approach allows to study fluctuations without making ad hoc assumptions concerning the statistical nature of the random process as necessary in LangevinFokkerPlanck frameworks.
1 Introduction
One of the motivations for the study of ultrarelativistic heavyion collisions is to gain a detailed understanding of the phase diagram of strongly interacting matter[1]. At the largest energies as achieved at the Large Hadron Collider (LHC) and the Relativistic Heavy Ion Collider (RHIC) a hot and dense fireball is formed which can be described to a surprising accuracy as a nearly perfect fluid of strongly coupled quarks and gluons (QGP) undergoing a transition to a hot hadronresonance gas. In these situations, where the netbaryon density or the baryonchemical potential are small, latticeQCD (lQCD) calculations indicate a crossover transition from confined to deconfined matter as well as from a phase where chiral symmetry is spontaneously broken to one where it is restored at a (common) transition temperature [2].
At lower collision energies, as studied in the RHIC beamenergy scan (BES) program, at the CERN SPS, and the future FAIR and NICA experiments, the produced medium starts at lower temperatures and larger netbaryon densities. Since in this situation the application of lQCD is challenging due to the “sign problem” at finite , one relies on effective chiral models, which predict the existence of a firstorder chiralphasetransition line ending in a critical point of a secondorder phase transition.
For theory the challenge is to provide possible observables for this phase structure in heavyion collisions, like the (“grandcanonical”) fluctuations of conserved charges like netbaryon number or electric charge. Not only the question, how to effectively model the phase transition (e.g., with the NambuJonaLasinio (NJL) or the (linear) model with extensions taking into account gluonic degrees of freedom implementing Polyakov loops) arises but also, which of the features of the phase structure predicted for such models applying thermal quantum field theory (describing a medium in thermal and chemical equilibrium) like (critical) fluctuations of conserved charges survive for a rapidly expanding and cooling fireball as created in heavyion collisions.
To address the latter question, one relies on transport simulations to describe the offequilibrium dynamics of the fireball. One approach is the use of ideal or viscous hydrodynamics to describe the bulk evolution of the fireball (assuming a state close to local thermal equilibrium), which successfully describes key phenomena of heavyion collisions, and adding the fluctuations by hand in a Langevin approach[3, 4, 5]. On the other hand this implies that the statistics of the random process has to be put in as an ad hoc assumption. Usually a Gaussian Markovian (“white noise”) is assumed, but the simulation of nonMarkovian (“colored noise”) processes is feasible in principle [6].
On the other hand, one would like to study the implication of different transition scenarios (crossover, 1 order, 2 order) as realized in the various quantumfield theoretical models on the nature of the hopefully observable fluctuations [7, 8, 9, 10, 11, 12, 13] on the statistics of the probably observable fluctuations.
In this work we present a novel MonteCarlo approach to address this challenging problem using the most simple quarkmeson linear model [14]. The meson fields are treated on the meanfield level and the quarks and antiquarks are realized in terms of a testparticle ensemble. Here the challenge is to implement “discrete” local interaction processes like elastic collisions and reactions like the annihilation to a meson and the decay of mesons to a pair admitting not only kinetic but also chemical equilibration starting from an offequilibrium situation, using the transitionprobability matrix elements of the underlying quantum field theory.
2 Linear quarkmeson model
To investigate the feasibility of a kinetic description of the offequilibrium dynamics of the chiral phase transition the most simple twoflavor chiral model, based on the chiral group is considered, using a flavor doublet of Dirac fields , describing and quarks and antiquarks and a fourdimensional realvalued set of scalar fields transforming under the SO(4) representation of the chiral group [15]. The Lagrangian reads
(1) 
with the meson potential
(2) 
where denotes the Yukawa coupling between quarks and mesons, the meson coupling constant (corresponding to a mass of ), the piondecay constant, and . The potential (2) contains the explicit breaking of the chiral symmetry due to the finite current quark masses resulting in a nonzero pion mass, , of the pseudoGoldstone modes . The constituent quark masses are given by .
The grandcanonical potential in meanfield approximation reads
(3) 
with
(4) 
where and the quarkdegeneracy factor . The mean fields have to be evaluated selfconsistently from the equilibrium condition
(5) 
In the following we restrict ourselves to vanishing pion mean fields. A nice feature of this model is that by varying the Yukawa coupling , one finds different kinds of phase transition as illustrated in Fig. 1.
3 Semiclassical particlefield dynamics
The challenge in applying the above model to an offequilibrium dynamical simulation of a system of particles (here quarks and antiquarks) and mean fields (representing the mesons) is that in order to reproduce the equilibriumphase structure as depicted in 1 as the stationary limit, one has to ensure that both kinetic and chemical equilibration is possible through the introduction of the appropriate elastic collision terms for and scattering as well as quarknumber changing processes such as . In a full kinetic approach this is achieved by a set of coupled BoltzmannVlasov equations, which read in our case schematically (again restricting ourselves to the case of vanishing pionmean fields)
(6)  
(7) 
Here, and denote collision integrals contributing to the mesonmeanfield and quarkphasespace distribution functions respectively.
In the following a novel scheme to MonteCarlo simulate such a system of kinetic equations is defined, where one describes the mesons solely with a mean field and the quarks and antiquarks in terms of test particles. While the elasticcollision term is realized in a straightforward way using the corresponding cross section from the underlying linear model, one has to find a way to realize the interactions in such a scheme, while still fulfilling energymomentum conservation and the principle of detailed balance, which are the fundamental principles constraining the offequilibrium dynamics and ensuring the proper (MaxwellBoltzmann) equilibrium limit.
In our recently developed model (Dynamical Simulation of a Linear Sigma Model, DSLAM) this challenge is solved as follows: In order to properly simulate the collision terms on the righthand sides of Eqs. (6) and (7) we define a spacetime grid. In each time step at each spatial cell the cross section for the annihilation process is used to stochastically determine an energymomentum transfer from the initial pair, located in the cell. This energymomentum change is transferred to the mean field in terms of an appropriate relativistic Gaussian wave packet (which simulates the gain term in ), and the pair is taken out of the testparticle ensemble (which simulates the corresponding loss term in ).
To simulate also the appropriate decay process, , we have to “particlize” the mean field locally in each spatial cell. This is done in the spirit of a coursegraining procedure: First, the total energymomentum content of the field within the cell in terms of the corresponding field energymomentum tensor is determined. Then one assumes a local thermal equilibrium phasespace distribution, equivalent to this energymomentum tensor. In order to fulfill detailed balance, the temperature has to be the same as that for the corresponding procedure for the quarks and antiquarks. The temperature is related to the meanfield value which depends on the scalar quarkantiquark density. In this way a temperature can be determined. It is important to note that it is defined in the local rest frame of the heat bath and thus the phasespace distribution is given by a MaxwellJüttner distribution , where is the fourvelocity, given by the total fieldfourmomentum in the spacial cell under consideration, . Now in each timestep within each spatial cell one can choose an ensemble of particles according to this local MaxwellJüttner distribution and using the corresponding decay rate to determine the gain term to . The loss term for the meanfield equation in the collision term is again achieved by taking the appropriate amount of energy and momentum out of the mean field in terms of a Gaussian wave packet.
In summary we have achieved a scheme which enables us to simulate the set of BoltzmannVlasov Eqs. (6) and (7) using test particles for the quarks and antiquarks and restricting the description of the mesons strictly to the meanfield level. The scheme by construction fulfills energymomentum conservation through the Gaussian wavepacket description for the exchange of energy and momentum between the mean field and the test particles. At the same time also the principle of detailed balance is fulfilled, using the coarsegraining approach to locally map the fieldenergymomentum distribution to a localequilibrium MaxwellJüttner distribution to reinterpret the mean field as a meson phasespace distribution and using the leadingorder transition rates (cross sections) of the underlying QFT linear model fulfilling the detailedbalance principle.
4 Proof of principle: “Box” calculations
As a plausibility check of the simulation method the stability of an equilibrium situation has been tested in a finite cubic box with periodic boundary conditions[14]. The stability of the energy conservation is demonstrated in Fig. 3. While the kinetic energy of the quarks and the energy of the mean field show anticorrelated thermal fluctuations the total energy stays stable within the numerical accuracy of the simulation.
The annihilation and creation processes of quarkantiquark pairs lead to thermal fluctuations in the total particle number, as shown in Fig. 3.
The energy distribution of the quarks shows the expected MaxwellBoltzmann distribution at the expected temperature, as demonstrated in Fig. 5. Also the spectral analysis of the mean field shows that the thermal equipartition theorem for the kinetic field energy is fulfilled at small wave numbers (long wavelengths), i.e., each mode contains an average energy of . On the other hand the “UV catastrophe” must be avoided due to the finite totalenergy content within a finite box. Indeed, the energymomentum transfer between particles and the field due to the pairannihilation and creation processes is not strictly “local” but occur within a finite volume whose scale is fixed by the finite width, , of the Gaussian wave packets used as field increments to keep care of the correct energymomentum transfer in each process. Thus the large wave numbers (short wave lengths) are effectively cut off at a scale , as shown by plotting the corresponding Gaussian on top of the distribution of the kinetic field energy (Fig. 5).
The dynamical behavior of the field in our thermalbox simulation is illustrated in Fig. 7, showing the field distribution within the plane of the simulation. Starting the simulation with a uniform meanfield equilibrium value, after a short time some local bloblike disturbances have developed (left panel) due to quarkpairannihilation processes, leading to the propagation of Gaussian wave packets on top of the still quite uniform average field value, as implemented by our concept of describing the exchange between field and particles in these processes. The right panel shows the fully developed equilibriumthermalfield fluctuations in the longtime limit of the simulation. In this limit the field values are Gaussian distributed around a mean value (cf. Fig. 7), which is expected due to many random energymomentum transfers between the field and particles. Thereby the average field value can slowly drift with time due to the thermal fluctuations of the field energy and momentum.
5 Conclusions and outlook
In this talk it was demonstrated that the dynamical description of the chiral phase transition in a simple quarkmeson model is feasible with a novel MonteCarlosimulation technique for a corresponding coupled Boltzmanntransport equation for the meson meanfield and the quark and antiquark phasespace distribution functions, implementing both elastic quark and antiquark scattering as well as quarkantiquarkpair creation and annihilation processes, enabling both kinetic and chemical equilibration between particles and the mean field (mesons).
The simulation is set up in a way that both energymomentum conservation and the principle of detailed balance are precisely realized (within the limits of achievable numerical accuracy). While the elastic quarkscattering processes are simulated with a straightforward testparticle realization, the particlefield kinetics has demanded the development of a novel scheme.
The annihilation process is evaluated by the MonteCarlo sampling according to the corresponding transition matrix elements from the underlying quantumfield theoretical interpretation of the linear model. The corresponding energy and momentum are precisely transferred to the mean field in terms of an appropriate disturbance in form of a relativistic Gaussian wave packet. To obey the principle of detailed balance, also the inverse decay has to be simulated. To that purpose a coarsegraining procedure based on the local energymomentum content of the field has been used for a “particlization” of the mean field in terms of a local BoltzmannJüttner equilibrium distribution, which in turn enables a MonteCarlo sampling of the meson decay according to the decay rate from the quantumfield theory picture of the model, which automatically implements detailed balance.
This scheme has now been applied to offequilibrium situations as in a “thermal quench” in a box where the particles and fields are initialized with different temperatures and it was demonstrated that the distribution can describe a possible phase transition from the initial state to a latetime (equilibrium) state.
Last but not least also the case of “expanding fireballs”, mimicking the situation of the medium created in heavyion collisions, is under study.
Acknowledgments
This work was partially supported by the Bundesministerium für Bildung und Forschung (BMBF Förderkennzeichen 05P12RFFTS) and by the Helmholtz International Center for FAIR (HIC for FAIR) within the framework of the LOEWE program (Landesoffensive zur Entwicklung WissenschaftlichÖkonomischer Exzellenz) launched by the State of Hesse. C. W. and A. M. acknowledge support by the Helmholtz Graduate School for Hadron and Ion Research (HGSHIRe), and the Helmholtz Research School for Quark Matter Studies in Heavy Ion Collisions (HQM). Numerical computations have been performed at the Center for Scientific Computing (CSC). H. v. H. has been supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number GR 1536/81. C.W. has been supported by BMBF under grant number 0512RFFTS.
References
References
 Friman B, Hohne C, Knoll J, Leupold S, Randrup J et al. 2011 Lect.Notes Phys. 814 pp. 980 URL http://dx.doi.org/10.1007/9783642132933
 Philipsen O 2013 Prog. Part. Nucl. Phys. 70 55–107 URL http://dx.doi.org/10.1016/j.ppnp.2012.09.003
 Nahrgang M, Leupold S, Herold C and Bleicher M 2011 Phys. Rev. C 84 024912 URL http://dx.doi.org/10.1103/PhysRevC.84.024912
 Nahrgang M, Leupold S and Bleicher M 2012 Phys. Lett. B 711 109–116 URL http://dx.doi.org/10.1016/j.physletb.2012.03.059

Herold C, Nahrgang M, Mishustin I and Bleicher M 2013 Phys. Rev. C 87 014907 URL http://dx.doi.org/10.1103/PhysRevC.87.014907
 Schmidt J, Meistrenko A, van Hees H, Xu Z and Greiner C 2015 Phys. Rev. E 91 032125 URL http://dx.doi.org/10.1103/PhysRevE.91.032125
 Stephanov M A, Rajagopal K and Shuryak E V 1999 Phys. Rev. D 60 114028 URL http://dx.doi.org/10.1103/PhysRevD.60.114028
 Schäfer B J, Pawlowski J M and Wambach J 2007 Phys. Rev. D 76 074023 URL http://dx.doi.org/10.1103/PhysRevD.76.074023
 Skokov V, Friman B and Redlich K 2011 Phys. Rev. C 83 054904 URL http://dx.doi.org/10.1103/PhysRevC.83.054904
 Skokov V, Stokic B, Friman B and Redlich K 2010 Phys. Rev. C 82 015206 URL http://dx.doi.org/10.1103/PhysRevC.82.015206
 BraunMunzinger P, Friman B, Karsch F, Redlich K and Skokov V 2012 Nucl. Phys. A 880 48–64 URL http://dx.doi.org/10.1016/j.nuclphysa.2012.02.010
 Schaefer B J 2012 Phys. Atom. Nucl. 75 741–743
 Morita K, Skokov V, Friman B and Redlich K 2014 Eur. Phys. J. C 74 2706 URL http://dx.doi.org/10.1140/epjc/s1005201327061
 Wesp C, van Hees H, Meistrenko A and Greiner C 2015 Phys. Rev. E 91 043302 URL http://dx.doi.org/10.1103/PhysRevE.91.043302
 GellMann M and Lévy M 1960 Nuovo Cim. 16 705 URL http://dx.doi.org/10.1007/BF02859738