Diffusion of conserved charges in relativistic heavy ion collisions

In order to characterize nuclear matter under extreme conditions, we calculate all diffusion transport coefficients related to baryon, electric and strangeness charge for both a hadron resonance gas and a simplified kinetic model of the quark-gluon plasma. We demonstrate that the diffusion currents do not depend only on gradients of their corresponding charge density. Instead, we show that there exists coupling between the different charge currents, in such a way that it is possible for density gradients of a given charge to generate dissipative currents of another charge. Within this scheme, the charge diffusion coefficient is best viewed as a matrix, in which the diagonal terms correspond to the usual charge diffusion coefficients, while the off-diagonal terms describe the coupling between the different currents. In this letter, we calculate for the first time the complete diffusion matrix including the three charges listed above. We find that the baryon diffusion current is strongly affected by baryon charge gradients, but also by its coupling to gradients in strangeness. The electric charge diffusion current is found to be strongly affected by electric and strangeness gradients, whereas strangeness currents depend mostly on strange and baryon gradients.

In order to characterize nuclear matter under extreme conditions, we calculate all diffusion transport coefficients related to baryon, electric and strangeness charge for both a hadron resonance gas and a simplified kinetic model of the quark-gluon plasma. We demonstrate that the diffusion currents do not depend only on gradients of their corresponding charge density. Instead, we show that there exists coupling between the different charge currents, in such a way that it is possible for density gradients of a given charge to generate dissipative currents of another charge. Within this scheme, the charge diffusion coefficient is best viewed as a matrix, in which the diagonal terms correspond to the usual charge diffusion coefficients, while the off-diagonal terms describe the coupling between the different currents. In this letter, we calculate for the first time the complete diffusion matrix including the three charges listed above. We find that the baryon diffusion current is strongly affected by baryon charge gradients, but also by its coupling to gradients in strangeness. The electric charge diffusion current is found to be strongly affected by electric and strangeness gradients, whereas strangeness currents depend mostly on strange and baryon gradients.

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Introduction. Ultrarelativistic hadronic collisions, performed in the largest particle accelerators in the world, create conditions required to study the properties of hot and dense hadronic and quark matter. In the last 10 years, these experiments have played a crucial role in uncovering novel transport properties of the quarkgluon plasma (QGP), a new state of nuclear matter in which quarks and gluons are no longer confined inside hadrons. In particular, several phenomenological studies [1][2][3][4][5][6][7][8] demonstrated that the QGP has one of the smallest shear viscosity to entropy density ratios in nature -a surprising result that is still not well understood from first principles. Additional theoretical and phenomenological studies [4,[9][10][11][12][13][14] have also improved our understanding of the bulk viscosity, showing that this coefficient can display novel behavior near the deconfinement transition of nuclear matter. Recently, much attention was paid to the electric conductivity; several studies on the lattice [15][16][17], in perturbative QCD (pQCD) [18][19][20] and effective theories [21][22][23] have been carried out.
On the other hand, at this stage very little is known about net-charge diffusion in hot and dense nuclear matter. This is due to the fact that in high energy heavy ion collisions, the net-charge density of the matter produced is extremely small in almost all space-time points and it becomes very difficult to observe any dissipative effects due to diffusion [24]. Recently, the Relativistic Heavy-Ion Collider (RHIC) started to perform hadronic collisions at lower energies within the beam energy scan (BES) program in order to investigate the phase diagram and transport properties of nuclear matter at finite netbaryon (and net-electric charge) density [25][26][27]. The central plateau in the rapidity distribution of baryon multiplicity dN B /dη is less pronounced at those low energies, such that strong gradients in the chemical potential of conserved charges are expected. At beam energies down to, e.g., √ s NN = 7.7 GeV in the RHIC BES, the baryon chemical potential can reach values up to µ B ∼ 400 MeV which is significant compared to the temperatures that are reached [28,29]. Therefore, one can expect that low energy collisions can be particularly useful to explore the properties of net-charge diffusion of nuclear matter, that were out of reach in higher energy collisions.
In the relativistic Navier-Stokes-Fourier theory, a netcharge (q) diffusion 4-current, j µ q , is determined by the following constitutive relation, where α q ≡ µ q /T is the thermal potential, with µ q being the charge chemical potential, T the temperature and κ q the corresponding net-charge diffusion coefficient. We further defined the transverse gradient ∇ µ ≡ ∆ µν ∂ ν , and the projection operator ∆ µν ≡ g µν − u µ u ν , where u µ is the local fluid velocity and g µν the space-time metric. We remark that this relativistic constitutive relation does not only describe the effects of charge diffusion but also includes the effects of heat flow. It is also important to emphasize that the constitutive relations satisfied by the diffusion 4-currents become different in the presence of more than one conserved charge. When discussing charge diffusion in the matter produced in heavy ion collisions, we must consider at least three conserved charges: baryon number (B), electric charge (Q), and strangeness (S). Since several hadrons (and quarks) carry more than one of these charges, the diffusion current of each charge will no longer be solely proportional to the gradient of the thermal potential (∇ µ α q ) of that specific charge. Instead, there will be a mixing be-tween the currents, with gradients of every single charge density being able to generate a diffusion current of any other charge. In general, one has leading to a diffusion coefficient that is a matrix instead of a number, κ qq ′ . Therefore, it is not sufficient to compute what is usually known as baryon, electric and strangeness diffusion coefficients, i.e., the diagonal terms in the matrix κ BB , κ QQ , κ SS , but also, one must calculate the off-diagonal terms or couplings terms κ QB , κ SB , κ SQ (it is sufficient to calculate only these three off-diagonal terms, since Onsager's theorem [30,31] guarantees that the diffusion matrix is symmetric). The dynamics of the thermal potentials α B , α S and α Q and their respective currents in heavy ion collisions is currently not well known. It is expect that the influence of diffusion currents on the hydrodynamical evolution of the net-charge currents can be very pronounced at lower collision energies, leading to significant effects on certain observables [24]. In this letter, we calculate for the first time the complete charge diffusion matrix, for the three charges listed above. We perform this task for a hadron resonance gas (HRG) and for a kinetic theory toy model of the QGP. We find that the coupling terms can be as large as the diagonal terms and, consequently, models simulating heavy ion collisions including only the diagonal contributions to net-charge diffusion may be missing crucial ingredients. Furthermore, it may not be a good approximation to perform simulations including only the dynamics of one charge since its gradients will necessarily give rise to diffusion currents of the remaining charges. We use natural units, = c = k B = 1 and Minkowski metric g µν = (1, −1, −1, −1). Greek indices run from 0 to 4.
First order Chapman-Enskog expansion. We consider a dilute gas consisting of N species particle species (either hadrons or quarks and gluons), with the i-th particle species having degeneracy g i , electric charge Q i , strangeness charge S i , baryonic charge B i and 4momentum k µ i . The state of the system is characterized by the single-particle momentum distribution function of each particle species, f i (x, k) ≡ f i k , with the time evolution of f i k being given by the relativistic Boltzmann equation. In contrast to previous work [22], we disregard any external field.
The single-particle distribution of each particle species is expanded in a Chapman-Enskog series, i.e., in a gradient expansion [32,33]. In this case, the Boltzmann equation is written as with C ij (x µ , k µ ) being the collision term and ǫ a bookkeeping parameter that will be set to one at the end of the calculation. The Chapman-Enskog expansion is just an expansion in powers of ǫ, jk is the j-th order solution of the expansion. The zeroth order solution of this series is the local equilibrium distribution function, leading to the equations of ideal fluid dynamics, while the first order solution contains terms that are of first order in gradients of velocity, temperature and chemical potential, leading to the equations of relativistic Navier-Stokes theory and the diffusion equation [32,33]. For the purposes of this letter, it is sufficient to calculate the first order contribution, which is the order that determines the diffusion coefficients. Without loss of generality, we only retain the terms of the expansion that contribute directly to the diffusion terms, omitting all others that contribute to shear and bulk viscosity.
This equation can be solved following the well known procedure outlined in [22,34]. Since the collision operator,Ĉ (1) ij , is linear, the solution for f 1k must be of the general form f i 1k = q a i q k µ i ∇ µ α q , where the coefficient a i q is a function of the energy in the local rest frame, where we defined In this work, the expansion in powers of energy is truncated at the lowest level possible, by setting M = 1. This assumption is mainly employed to simplify the numerical calculations we perform. Nevertheless, we have checked, in simpler examples solved using constant cross sections, that higher truncation values lead to only small corrections to the diffusion coefficients, as was also demonstrated in previous work [22,34] for other transport coefficients.
The q-th charge diffusion current is given as Substituting the expansion for f i 1k into Eq. (9), and comparing to Eq. (2), leads to the following expression for the diffusion coefficients Therefore, calculating κ qq ′ is reduced to evaluating the integrals in Eq. (8) and then solving the set of linear equations satisfied by a i q ′ ,m in Eq. (7). Both these tasks are performed numerically.
In order to perform these numerical calculations one has to first specify the differential cross sections for the particle interactions. In this letter, we restrict ourselves to elastic, isotropic (s-wave) scattering, employing all available √ s dependent cross sections from Ref. [35] shown in Fig. 1. Due to the lack of experimental data, we assume all missing hadronic cross sections to be constant, as done, e.g. in hadronic transport models [36][37][38]. The hyperon cross sections thus take constant values between 3 − 35 mb. We will also make an estimate of the diffusion coefficients of the QGP. For this purpose, we assume three flavors of massless quarks and gluons, and choose a unique total cross section σ tot in such a way that the shear viscosity to entropy density ratio is fixed to be η/s = 1/(4π), leading to σ tot ≈ 0.72/T 2 [39,40]. Further details on the choice of the cross sections will be presented in a forthcoming publication [41]. π +/0/-+π -/±/+ π 0 +p, π 0 +n π + +p π -+p π -+n p+p p+p n+p n+p K + +p K -+p K -+n K+π Results. We first remark that we checked that Onsager's theorem [30,31], which imposes that κ qq ′ = κ q ′ q , is fulfilled in all our calculations. We display our results for the diffusion coefficient matrix from Eq. (10) in Fig. 2 for µ B = 0, 300, 600 MeV. We fix µ Q and µ S such that we always retain an exact Isospin symmetry and vanishing net strangeness, since this is what approximatly occurs in heavy ion collisions [42,43]. For illustrative purposes, we show the HRG results below T = 160 MeV, and the QGP results above this temperature. We also compare here to the non-conformal holographic results from Ref. [21,23], since these results are the only ones in literature that contain all three diagonal coefficients. To the best of our knowledge, the off-diagonal coefficients have never been calculated before in any model.
First we note, that the HRG results are much richer in their T and µ B dependence, because of the multitude of scales involved here (masses and resonances). In contrast, the simple choice of a constant η/s in the QGP leads to the expected flat behavior for all coefficients [20] (it is known that a running strong coupling lets the coefficients increase for higher T in the QGP, see, e.g., Refs. [15,  18]). We note that only κ QB vanishes at µ B = 0 due to the symmetry of quark charges (and our simplified common cross section). At higher µ B , we see that these coefficients are found to be generally smaller in the QGP phase than they are in hadronic phase (κ SQ being the exception). This surprising behavior will be investigated in more detail in a forthcoming paper [41].
For the baryon diffusion current j µ B , we expect a strong dependence on both µ B and T , and indeed this can be seen from the functional behavior of the coefficient κ BB in Fig. 2. For µ B 300 MeV this coefficient rises rapidly with increasing temperature, as the system is less meson dominated at higher temperatures, and mesons act purely as a resistance for the diffusion of baryons. This effect is also visible in the off-diagonal coefficients −κ SB and κ QB . Comparing κ QB to κ BB , in Fig. 2, we infer that the electric charge gradients contribute to the baryon diffusion current about an order of magnitude less than the baryonic gradients. In contrast, gradients in strangeness can be as important as gradients in the baryon charge, as can be seen in the bottom right panel from the magnitude of the coefficient −κ SB , which is similar in magnitude to κ BB . We remark that this is due to the hyperons, which carry both B and S charge. The negative sign of κ SB indicates that gradients in strangeness act to reduce the baryon current.
We now discuss the coefficients κ QQ , κ SQ , κ QB , which characterize the diffusion of electric charges 1 . We see that κ QQ /T 2 decreases with temperature, and for increasing values of µ B . This happens because the particle density grows, but the ratio of charged to uncharged species stays the same. The small ratio κ QB /κ QQ indicates the little importance of baryon chemical potential gradients to the electric diffusion current, whereas κ SQ is (for T 100 MeV) of the same order of magnitude as κ QQ , indicating that strangeness gradients contribute significantly to the electric diffusion current.
Looking at the diffusion coefficients related to strangeness diffusion, we find that κ SS is larger than both −κ SB and κ SQ , being even larger in magnitude than the baryon diffusion coefficient (except for very small values of temperature). However, we find that baryonic gradients act to significantly reduce strangeness currents in both the QGP and HRG, since κ SB is negative and its magnitude is only about a factor two smaller than κ SS . Therefore, it is possible that cancellation effects due to coupling between the currents can lead to small strangeness diffusion currents. On the other hand, κ SQ is about an order of magnitude smaller than κ SS , indicating that electric gradients are less important for strangeness transport. We remark that the µ B dependence of κ SS , κ QQ and κ SQ is very weak, however their dependence on µ Q and µ S can behave differently. This dependence will be addressed in a future publication.
The holographic results from Ref. [21,23] match ours at high T (conformal limit). Their µ B dependence for the diagonal coefficients is as weak as for our QGP results, but slightly lower in magnitude. It is interesting how a simple kinetic calculation, that simply fixes η/s = 1/4π, is already capable of reproducing the basic trends of such holographic calculations. It would be interesting to see whether this holds for the off-diagonal coefficients.
Conclusion. We have calculated the complete diffusion coefficient matrix for the conserved baryon, electric and strange charges for a hot hadron gas and QGP, using the traditional Chapman-Enskog formalism. These 6 transport coefficients include the baryon diffusion coefficient κ BB , and the electric and strangeness diffusion coefficients κ QQ and κ SS , respectively. We present for the first time also the three off diagonal transport diffusion coefficients κ QB , κ SB and κ SQ , which describe the mixing between the different charge currents. In our semianalytic approach, we confine ourselves to classical statistics, elastic collisions and isotropic scattering. We include resonance and measured elastic hadron-hadron cross sections, when available, taking into account hadrons up to the Σ baryons. This constitutes the most extensive result of the charge diffusion matrix in the HRG to date. For calculations in the QGP phase, we fix η/s to be a constant. It is in fact very interesting that most of the diffusion coefficients in the QGP match the HRG results quite well nearby the conjectured phase transition region.
The diffusion coefficients can be readily used in, e.g., hydrodynamic simulations, or other model descriptions of high density heavy ion collisions, where diffusion processes are taken into account. Those models are and will be increasingly important for low energy and high density experiments like RHIC BES, NICA or FAIR. Our results emphasize that the mixing between different diffusion currents is in general important and should not be neglected when simulating low energy heavy ion collisions. For example, the contribution to the baryon diffusion current from gradients of baryon number density can be almost completely canceled by gradients in strangeness of comparable magnitude, whereas we found electric gradients to be almost negligible for baryon transport. Electric diffusion is mainly driven by electric and strangeness gradients. Strangeness diffusion is mostly affected by strangeness and baryon number gradients, with electric charge gradients being less important. The relevance of these effects for experimental observables remains to be investigated. We plan to extend our work to quantum statistics, a more realistic description of the QGP and possibly more particle species to achieve a fully comprehensive framework of diffusion properties. It would be desirable to compare our results to, e.g., lattice QCD results (which at present are only available for the electric conductivity). All coefficients should also be accessible from hadronic transport models, or other dynamical approaches.