Diffusion coefficient matrix of the strongly interacting quark-gluon plasma
Jan A. Fotakis, Olga Soloveva, Carsten Greiner, Olaf Kaczmarek, Elena Bratkovskaya
DDiffusion coefficient matrix of the strongly interacting quark-gluon plasma
Jan A. Fotakis, ∗ Olga Soloveva, Carsten Greiner, Olaf Kaczmarek,
2, 3 and Elena Bratkovskaya
4, 1 Institut f¨ur Theoretische Physik, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Fakult¨at f¨ur Physik, Universit¨at Bielefeld, D-33615 Bielefeld, Germany Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics,Central China Normal University, Wuhan 430079, China GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstrasse 1, D-64291 Darmstadt, Germany (Dated: February 17, 2021)We study the diffusion properties of the strongly interacting quark-gluon plasma (sQGP) andevaluate the diffusion coefficient matrix for the baryon (B), strange (S) and electric (Q) charges - κ qq (cid:48) ( q, q (cid:48) = B , S , Q) and show their dependence on temperature T and baryon chemical potential µ B . The non-perturbative nature of the sQGP is evaluated within the Dynamical Quasi-ParticleModel (DQPM) which is matched to reproduce the equation of state of the partonic matter above thedeconfinement temperature T c from lattice QCD. The calculation of diffusion coefficients is based ontwo methods: i) the Chapman-Enskog method for the linearized Boltzmann equation, which allowsto explore non-equilibrium corrections for the phase-space distribution function in leading order ofthe Knudsen numbers as well as ii) the relaxation time approximation (RTA). In this work we explorethe differences between the two methods. We find a good agreement with the available lattice QCDdata in case of the electric charge diffusion coefficient (or electric conductivity) at vanishing baryonchemical potential as well as a qualitative agreement with the recent predictions from the holographicapproach for all diagonal components of the diffusion coefficient matrix. The knowledge of thediffusion coefficient matrix is also of special interest for more accurate hydrodynamic simulations. I. INTRODUCTION
An exploration of the properties of hot and dense mat-ter - created in heavy-ion collisions (HICs) at relativis-tic energies - is in the focus of extensive research. It isthe primary goal of experimental programs of the LHC(Large-Hadron-Collider) at CERN, the RHIC (Relativis-tic Heavy Ion Collider) at BNL, the future FAIR (Facilityfor Antiproton and Ion Research) at GSI, and the NICA(Nuclotron-based Ion Collider fAcility) facility at JINR,which reproduce in the laboratory the extreme conditionsof the early stages of our universe by ’tiny bangs’. In thecentral region of heavy-ion collisions the deconfined QCD(Quantum Chromo Dynamics) matter – a quark-gluonplasma (QGP) - is created which can achieve an approx-imate local equilibrium and exhibit hydrodynamic flow[1–3]. The hydrodynamic behaviour of the fluid can becharacterized by transport coefficients such as shear η ,bulk ζ viscosities, and diffusion coefficients κ , which de-scribe the fluid’s dissipative corrections at leading order.The interpretation of the experimental data, and espe-cially the elliptic flow v , in terms of the hydrodynamicmodels showed that the QGP behaves almost as a nearlyperfect fluid with a very low shear viscosity to entropydensity ( s ) ratio, η/s , which reflects that its propertiescorrespond to nonperturbative, strongly interacting mat-ter [4–6].By performing an experimental energy scan of HICsone can explore the different stages of the QCD phase di-agram. At ultra-relativistic heavy-ion collisions at LHC ∗ [email protected] and RHIC energies, the QGP is created at very largetemperatures T and almost zero or low baryon chemicalpotential µ B , where according to lattice QCD (lQCD) re-sults [7, 8] the transition from the QGP to the hadronicmatter is a crossover. By reducing the collision energyone can also explore the large µ B region where one mightexpect the existence of a critical point and a 1st orderphase transition. Such conditions are presently under in-vestigation within the RHIC BES (Beam Energy Scan)experiments and in future by the FAIR and NICA facil-ities.The theoretical description of the QCD matter at fi-nite µ B , and especially in the vicinity of the critical point,requires an appropriate description of transport of con-served charges – baryon B , strangeness S and electric Q charges. In order to study the phenomenon of baryon-stopping, the baryon diffusion was recently introducedto various fluid dynamic models [9–11]. Moreover, thebaryon diffusion coefficient has been studied in Refs. [12–17].In the recent past we have addressed the couplingof the conserved baryon number, strangeness and elec-tric charge; the diffusion coefficient matrix ( κ qq (cid:48) , where q, q (cid:48) = B , S , Q) was introduced and evaluated for ahadron gas and a simple model for quark-gluon plasma(QGP) [14, 16]. These investigations were followed by amore extended study in the hadronic phase from kinetictheory in the case of the electric cross-conductivities [18].Furthermore, a first study on the impact of the couplingof baryon number and strangeness was provided in Ref.[16]. For this the diffusion coefficient matrix of hot anddense nuclear matter has to be investigated thoroughlybeing important for accurate hydrodynamic simulations. a r X i v : . [ h e p - ph ] F e b It was further motivated that the off-diagonal coefficientsmay have implications on the chemical-composition ofthe hadronic phase [18].The study of the transport of conserved electric charge Q during heavy-ion collisions has been in the focus of in-tensive research. Due to its importance for the descrip-tion of soft photon spectra and rates [19–22] as well as forhydrodynamic approaches modelling the generation andevolution of electromagnetic fields [23–26], much atten-tion was paid to the electric conductivity within differ-ent theoretical approaches for the evaluation of the prop-erties of the partonic and hadronic matter [13, 14, 16–18, 27–42].The exploration of the QGP properties at finite ( T, µ B )are of special interest for an understanding of the phasetransition. The transport properties of the strongly in-teracting QGP has been studied using the DynamicalQuasi-Particle Model(DQPM) [43–47] that is matchedto reproduce the equation of state of the partonic sys-tem above the deconfinement temperature T c from lat-tice QCD. The DQPM is based on a propagator repre-sentation with complex self energies which describes thedegrees of freedom of the QGP in terms of strongly in-teracting dynamical quasiparticles which reflect the non-perturbative nature of the QCD in the vicinity of thephase transition where the QCD coupling grows rapidlywith decreasing temperature according to lQCD calcula-tions [48]. Moreover, the DQPM allows to explore theproperties of the QGP at finite ( T, µ B ), expressed interms of transport coefficients such as shear η , bulk ζ viscosities, baryon diffusion coefficients κ B and electricconductivity σ based on the RTA (relaxation time ap-proximation) [17, 47, 49].We note that an important advantage of a propaga-tor based approach is that one can formulate a consis-tent thermodynamics [50] and a causal theory for non-equilibrium dynamics on the basis of Kadanoff–Baymequations [51]. This allows to use the DQPM for thedescription of the partonic interactions and parton prop-erties in the microscopic Parton–Hadron–String Dynam-ics (PHSD) transport approach [46, 52–55] and to studythe QGP properties out-of equilibrium as created in HICsas well as in equilibrium by performing box calculationswith periodic boundary conditions [56]. Moreover, the( T, µ B ) dependence of partonic properties and interac-tion cross sections have been explored in a more recentstudy within PHSD 5.0 [17, 49, 57, 58].We note that the studies of transport coefficients( η/s, ζ/s, κ B , σ ) within the DQPM (and PHSD) hasbeen based on the relaxation-time approximation (RTA)as incorporated in Refs. [59–62] as well as on theKubo formalism [63–66] for η/s (cf. [49, 56]). In Refs.[14, 16, 39] the evaluation of the diffusion coefficient ma-trix has been done within the Chapman-Enskog method[67] which allows to explore non-equilibrium correctionsfor the phase-space distribution function in leading orderof the Knudsen numbers.In the present study we combine the developments of Refs. [14, 16, 17] and evaluate the diffusion coefficientmatrix of the strongly interacting non-perturbative QGPat finite ( T, µ B ), with properties described by the DQPMmodel, based on recently explored the Chapman-Enskogmethod [14, 16, 39]. This allows us to explore the in-fluence of traces of non-equilibrium effects by account-ing for the higher modes of the distribution function onthe transport properties and compare the results withthe often used kinetic RTA approximation. We providethe ( T, µ B ) dependence of the diffusion coefficients κ qq (cid:48) for q, q (cid:48) = B , S , Q charges for baryon chemical potentials µ B ≤ . II. FOUNDATIONS
Let x ≡ x µ be the four-coordinate and k ≡ k µ thefour-momentum. The single-particle distribution func-tion, f i, k ≡ f i ( x, k ), of a multi-component quasi-particlesystem obeys the effective Boltzmann equation [68] k µi ∂ µ f i, k + 12 ∂ µ (cid:0) m i (cid:1) ∂∂k i, µ f i, k = N species (cid:88) j = 1 C ij ( x, k ) , (1)where C ij is the collision term and the masses dependon temperature and chemical potentials, i.e. m i ≡ m i ( T, µ B , µ Q , µ S ). The (local) equilibrium state of thesystem is described by f (0) i, k = g i exp ( u µ k µi /T − µ i /T ) − a i , (2)where µ i = B i µ B + Q i µ Q + S i µ S is the chemical potential, g i is the degeneracy of the i -th species and a i ≡ +1 (Bosons) , − , . (3)Further, we define in short hand notation:˜ f (0) i, k ≡ a i f (0) i, k g i = 1 + a i exp ( u µ k µi /T − µ i /T ) − a i . (4)Furthermore, the isotropic local equilibrium pressure isdetermined by the temperature and chemical potentials, P ≡ P ( T, µ B , µ Q , µ S ). In this work, we adapt theisotropic pressure from lattice QCD [69, 70]. From theequation of state the energy density and the net chargedensities are defined: (cid:15) ≡ (cid:15) ( T, µ B , µ Q , µ S ) , n q ≡ n q ( T, µ B , µ Q , µ S ) ,q ∈ { B , Q , S } . (5)In kinetic theory the net charge densities are defined as: n q = N species (cid:88) i = 1 q i (cid:90) d K i E i, k f (0) i, k , q ∈ { B , Q , S } (6)where q is the type of the conserved quantum number, i.e.namely baryon number B, strangeness S or electric chargeQ, and q i is the quantum number (of type q ) of the i -thspecies. In this work we assume a partonic system withthree flavors and thus the following particle species: up-( u ), down- ( d ), strange-quark ( s ), the gluon ( g ), and thecorresponding anti-particles. Furthermore, the Landaumatching conditions were assumed [71]: N species (cid:88) i = 1 q i (cid:90) d K i E i, k (cid:16) f i, k − f (0) i, k (cid:17) = 0 , (7)using the notation d K i ≡ d k i (2 π ) E i, k , (8)with the on-shell energy E i, k = (cid:112) m i + k i .An (unpolarized) interaction is characterized bythe invariant matrix-element ¯ M i ...i n → j ...j m ≡ ¯ M ( k i . . . k i n → p j . . . p j m ), which is averaged overthe ingoing spin-states and is summed over the outgoingspin-states. The differential cross section for a binaryprocess of on-shell particles ( i + j → a + b ) in the center-of-momentum frame (CM), where the momenta of thecolliding particles obey k i + k (cid:48) j = p a + p (cid:48) b = P = 0 and k i + k (cid:48) j = √ s = p i + p (cid:48) j , is given byd σ ij → ab ( √ s, Ω) = 164 π s p out p in | ¯ M| dΩ , (9)where s in the Mandelstam variable and dΩ is the differ-ential solid angle corresponding to one of the final parti-cles. The momenta of the initial ( p in ) and final particles( p out ) in the CM frame are found to be: p i = (cid:112) ( s − ( m i + m (cid:48) i ) ) ( s − ( m i − m (cid:48) i ) )2 √ s , (10)where i = in / out, m i and m (cid:48) i being the masses of thecolliding partons. The total cross section is obtained via: σ ij → ab tot ( √ s ) ≡ πγ ij (cid:90) d cos( ϑ ) ddΩ σ ij → ab ( √ s, cos( ϑ ))= 132 πs p out p in γ ij (cid:90) − d cos( ϑ ) | ¯ M| , (11) where ϑ is the final polar angle of one of the final particlesin the CM frame, and γ ij = 1 − δ ij is the symmetryfactor.In this paper we use the short-hand “ { µ q } ” in-stead of “ µ B , µ Q , µ S ” in function arguments, and the(+ , − , − , − )-signature for the metric. Greek indices runfrom 0 to 3 and latin ones run from 1 to 3. Furthermore,we employ natural units, (cid:126) = c = k B = 1. A. First-order Chapman-Enskog approximation
If the perturbations from equilibrium are small, onecan expand the single-particle distribution function inorders of the Knudsen number (Kn): f i, k = f (0) i, k + (cid:15)f (1) i, k + O ( (cid:15) ) , (12)where (cid:15) is an assisting parameter for counting the or-ders of the gradients (or equivalently, the orders of theKnudsen number), which will be send to 1 afterwards.This approximation is known as the Chapman-Enskogexpansion to first order (CE) [67]. Neglecting second-order terms leads to the linearized effective Boltzmannequation: k µi ∂ µ f (0) i, k + 12 ∂ µ (cid:0) m i (cid:1) ∂∂k i, µ f (0) i, k = N species (cid:88) j = 1 C (1) ij [ f i, k ] , (13)with the linearized collision term N species (cid:88) j = 1 C (1) ij [ f i, k ] ≡ N species (cid:88) j,a,b = 1 (cid:90) R d P a (cid:90) R d P (cid:48) b (cid:90) R d K (cid:48) j × W ij → abkk (cid:48) → pp (cid:48) f (0) i, k f (0) j, k (cid:48) ˜ f (0) a, p ˜ f (0) b, p (cid:48) × (cid:32) f (1) a, p f (0) a, p ˜ f (0) a, p + f (1) b, p (cid:48) f (0) b, p (cid:48) ˜ f (0) b, p (cid:48) − f (1) i, k f (0) i, k ˜ f (0) i, k − f (1) j, k (cid:48) f (0) j, k (cid:48) ˜ f (0) j, k (cid:48) (cid:33) , (14)and W ij → abkk (cid:48) → pp (cid:48) ≡ (2 π ) δ (4) (cid:0) k i + k (cid:48) j − p a − p (cid:48) b (cid:1) (cid:12)(cid:12) ¯ M ij → ab (cid:12)(cid:12) (15)for the inelastic binary transition rate. To linear orderthe diffusion currents are given via: V µq = N species (cid:88) i = 1 q i (cid:90) R d K i k (cid:104) µ (cid:105) i f (1) i, k , (16)and therefore the explicit mass-term in Eq. (13) doesnot affect the currents due to the anti-symmetry of theintegrand [16]. Here q i is again the quantum number oftype q ∈ { B , S , Q } of the i -th particle species. For further evaluations with the CE method in thisstudy we consider a classical system of on-shell particles, a i = 0 ∀ i , and elastic binary collisions only, such that theon-shell transition rate for this case reads: W ij → abkk (cid:48) → pp (cid:48) = γ ij ( δ ia δ jb + δ ib δ ja )(2 π ) s × (cid:18) ddΩ σ ij → ab ( √ s, Ω) (cid:19) δ (4) (cid:0) k i + k (cid:48) j − p a − p (cid:48) b (cid:1) . (17)Additionally, we assume isotropic scattering processes.(The underlying assumptions will be discussed also inSection III.)Later we will incorporate total cross sections forelastic binary processes originating from the dynamicquasi-particle model (DQPM) [17] which depend ontemperature and baryon-chemical potential, σ ij → ab tot ≡ σ ij → ab tot ( √ s, T, µ B ) (Section II C). Following the steps taken in Refs. [14, 16, 39], we canexpress the diffusion coefficient matrix for a classical sys-tem under the assumption of elastic isotropic scatteringprocesses as κ qq (cid:48) = 13 N species (cid:88) i = 1 q i M (cid:88) m = 0 λ ( i ) m,q (cid:48) (cid:90) R d K i E mi, k (cid:0) m i − E i, k (cid:1) f (0) i, k , (18)where the scalars λ ( i ) m,q (cid:48) are solutions of the linearizedBoltzmann equation in the form [14, 16, 39]: M (cid:88) m = 0 N species (cid:88) j = 1 (cid:0) A inm δ ij + C ijnm (cid:1) λ ( j ) m,q = b iq,n , (19)with the abbreviations A inm ≡ N species (cid:88) (cid:96) = 1 γ i(cid:96) (cid:90) d K i d K (cid:48) (cid:96) d P i d P (cid:48) (cid:96) (2 π ) s (cid:18) ddΩ σ i(cid:96) → i(cid:96) (cid:19) δ (4) ( k i + k (cid:48) (cid:96) − p i − p (cid:48) (cid:96) ) f (0) i, k f (0) (cid:96), k (cid:48) E n − i, k k i, (cid:104) α (cid:105) (cid:16) E mi, p p (cid:104) α (cid:105) i − E mi, k k (cid:104) α (cid:105) i (cid:17) , C ijnm ≡ γ ij (cid:90) d K i d K (cid:48) j d P i d P (cid:48) j (2 π ) s (cid:18) ddΩ σ ij → ij (cid:19) δ (4) (cid:0) k i + k (cid:48) j − p i − p (cid:48) j (cid:1) f (0) i, k f (0) j, k (cid:48) E n − i, k k i, (cid:104) α (cid:105) (cid:16) E mj, p (cid:48) p (cid:48)(cid:104) α (cid:105) j − E mj, k (cid:48) k (cid:48)(cid:104) α (cid:105) j (cid:17) ,b iq,n ≡ (cid:90) R d K i E n − i, k (cid:0) m i − E i, k (cid:1) (cid:18) E i, k n q (cid:15) + P − q i (cid:19) f (0) i, k , (20)and where we further impose Landau’s definition of theframe [71], which leads to the additional constrain: W µ = N species (cid:88) i = 1 (cid:90) R d K i E i, k k (cid:104) µ (cid:105) i f (1) i, k ! = 0 ⇒ N species (cid:88) i = 1 M (cid:88) m = 0 λ ( i ) m,q (cid:90) R d K i E m +1 i, k (cid:0) m i − E i, k (cid:1) f (0) i, k ! = 0 . (21)Above we introduced the truncation order M ; for thesake of simplicity the order is fixed to M = 1 which cor-responds to the 14-moment approximation [72]. We fur-ther define the corresponding conductivities as, σ qq (cid:48) /T = κ qq (cid:48) /T and note that for q = Q or q (cid:48) = Q they areequivalent to the cross-electric conductivities introducedin Ref. [18]. Especially, κ QQ /T = σ el /T is the elec-tric conductivity, which was already evaluated in variousmodels [13, 14, 16–18, 27–42]. B. Relaxation time approximation
Anderson and Witting proposed an approximation tothe collision term by defining a governing relaxation time [73]. To first order, we write for each particle species i : N species (cid:88) j = 1 C (1) ij [ f i, k ] = − E i, k τ i (cid:16) f i, k − f (0) i, k (cid:17) = − E i, k τ i f (1) i, k + O (Kn ) . (22)The relaxation time τ i is related to the scattering rateΓ i ( k i , T, { µ q } ). For binary scattering we may write downthe momentum dependent on-shell relaxation time [60,74, 75]:1 τ i ( k i , T, { µ q } ) = Γ i ( k i , T, { µ q } ) = N species (cid:88) j = 1
12! 1 E i, k N species (cid:88) a,b = 1 (cid:90) R d K (cid:48) j d P a d P (cid:48) b f (0) j, k (cid:48) ˜ f (0) a, p ˜ f (0) b, p (cid:48) W ij → abkk (cid:48) → pp (cid:48) . (23)From this we can also define the momentum-averagedrelaxation time τ i, which may be used instead:1 τ i, ( T, { µ q } ) = Γ i, ( T, { µ q } ) ≡ n i (cid:90) R d K i E i, k Γ i ( k i , T, { µ q } ) f (0) i, k (24)with the on-shell particle density of species i : n i ( T, { µ q } ) ≡ (cid:90) R d K i E i, k f (0) i, k . (25)This is also known as the relaxation time approximation(RTA) (cf. [59–62]).In the classical limit and for the case of elastic, binaryprocesses with constant isotropic cross sections, σ ij → ij tot ≡ σ tot = const . , using Eq. (17) we can make the usualapproximation (see e.g. [16]):1 τ i, ( T, { µ q } ) = Γ i, ( T, { µ q } ) ≈ σ tot (cid:88) j n j = σ tot n tot . (26)Following Ref. [16], the diffusion coefficient matrix in theRTA can be expressed as: κ qq (cid:48) = 13 N species (cid:88) i = 1 q i (cid:90) R d K i τ i, ( T, µ B ) 1 E i, k (cid:0) m i − E i, k (cid:1) × (cid:18) E i, k n q (cid:48) (cid:15) + P − q (cid:48) i (cid:19) f (0) i, k ˜ f (0) i, k . (27) C. Dynamical quasi-particle model for thequark-gluon plasma
In the dynamical quasi-particle model (DQPM) [17,43–45, 53] the properties of the QGP are described interms of strongly interacting dynamical quasi-particles- quarks and gluons - with medium-adjusted properties.Their properties are constructed such that the equationof state (EoS) from lattice Quantum Chromo Dynamics(lQCD) is reproduced above the deconfinement tempera-ture T c . These quasi-particles are characterized by broadspectral functions ρ i ( i = q, ¯ q, g ), which are assumed tohave a Lorentzian form [44–46]. They depend on theparton masses m i and their associated widths γ i , ρ i ( ω, p ) = γ i ˜ E i, p (cid:32) ω − ˜ E i, p ) + γ i − ω + ˜ E i, p ) + γ i (cid:33) . (28)Here, we introduced the off-shell energy ˜ E i, p = (cid:112) p + M i − γ i . In the DQPM the effective (squared)coupling constant g is assumed to depend on tempera-ture T and baryon-chemical potential µ B [47, 76–78]. At µ B = 0 its temperature-dependence is parameterized viathe entropy density s ( T, µ B = 0) from lattice QCD fromRefs. [69, 70] in the following way: g ( T, µ B = 0) = d (cid:16) (cid:16) s ( T, /s QCDSB (cid:17) e − (cid:17) f , (29)with the Stefan-Boltzmann entropy density s QCDSB =19 / π T and the dimensionless parameters d = 169 . e = − . f = 1 . µ B , we use of the ’scaling hypothesis’ whichassumes that g is a function of the ratio of the effec-tive temperature T ∗ = (cid:112) T + µ B / (9 π ) and the µ B -dependent critical temperature T c ( µ B ) as [79]: g ( T /T c , µ B ) = g (cid:18) T ∗ T c ( µ B ) , µ B = 0 (cid:19) , (30)with T c ( µ B ) = T c ( µ B = 0) (cid:112) − αµ , T c ( µ B = 0) ≈ .
158 GeV and α = 0 .
974 GeV − . The ( T, µ B ) behaviourof the DQPM coupling g / (4 π ) is shown in Fig. 9 in Ap-pendix V A. At µ B = 0 one can see a good agreement be-tween the lQCD evaluation of the QCD running coupling α s = g / (4 π ) for N f = 2 [80] and the DQPM runningcoupling.With the coupling g fixed from lQCD, one can nowspecify the dynamical quasi-particle mass (for gluons andquarks) which is assumed to be given by the HTL thermalmass in the asymptotic high-momentum regime by [46,81] m g ( T, µ B ) = g ( T, µ B )6 (cid:32)(cid:18) N c + N f (cid:19) T + N c (cid:88) q µ q π (cid:33) ,m q (¯ q ) ( T, µ B ) = N c − N c g ( T, µ B ) (cid:32) T + µ q π (cid:33) , (31)where N c = 3 the number of colors, while N f = 3 denotesthe number of flavors. The strange quark has a largerbare mass which needs to be considered in its dynamicalmass. Empirically we find m s ( T, µ B ) = m u/d ( T, µ B ) +∆ m and ∆ m = 30 MeV. Furthermore, the quasi-particlesin the DQPM have finite widths, which are adopted inthe form [21, 79] γ i ( T, µ B ) = 13 C i g ( T, µ B ) T π ln (cid:18) c m g ( T, µ B ) + 1 (cid:19) , (32)where we use the QCD color factors for quarks, C q = C F = N c − N c = 4 /
3, and gluons, C g = C A = N c = 3. Fur-ther, we fixed the parameter c m = 14 .
4, which is relatedto the magnetic cut-off. We assume that the width ofthe strange quark is the same as that for the light ( u, d )quarks. The evaluated masses and widths in the DQPMare shown in Fig. 8 in Appendix V A.With the quasi-particle properties (or propagators)fixed as described above, one can evaluate the entropydensity s ( T, µ B ), the pressure P ( T, µ B ) and energy den-sity (cid:15) ( T, µ B ) in a straight forward manner by startingwith the entropy density s dqp and number density n dqp in the propagator representation from Baym [82, 83] andthen identifying, s = s dqp and n B = n dqp / P can then be obtained by using theMaxwell relation of a grand canonical ensemble: P ( T, µ B ) = P ( T ,
0) + T (cid:90) T s ( T (cid:48) , dT (cid:48) + µ B (cid:90) n B ( T, µ (cid:48) B ) dµ (cid:48) B , (33)where the lower bound is chosen between 0 . < T < . (cid:15) then follows from the Eulerrelation (cid:15) = T s − P + µ B n B . (34)In Ref. [84] we found a good agreement between the en-tropy density s ( T ), pressure P ( T ), energy density (cid:15) ( T )and interaction measure I ( T ) = (cid:15) − P resulting from theDQPM, and results from lQCD obtained by the BMWgroup [69, 70] at µ B = 0 and µ B = 400 MeV.From the above parametrizations of the masses, widthsand the couplings, cross sections for anisotropic, inelas-tic binary tree-level QCD interactions with the dressedpropagators and dressed couplings, have been evaluatedwhich depend on temperature and baryon-chemical po-tential [17, 49]. The corresponding total cross sectionsare shown in Fig. 9 in Appendix V A, and are used ofin the Chapman-Enskog evaluation described in SectionII A. Further, we provide new results for the complete dif-fusion coefficient matrix from the DQPM in the RTA byusing Eq. (27) and assuming relaxation times from Eq.(23). In the following the results from both approachesare presented. III. RESULTS
We provide first results for the diffusion coefficient ma-trix for the hot quark-gluon plasma at zero and finitebaryon chemical potential µ B by applying the Chapman-Enskog method, reviewed in Sec. II A and described indetail in Refs. [14, 16, 39], to a strongly interacting QGPsystem described by the DQPM (see Sec. II C and Ref.[17]). This is meant to be a significant and importantimprovement to the ’simplified’ model of a partonic sys-tem proposed in Refs. [14, 16]. These results - obtainedwithin the Chapman-Enskog method - are further com-pared to the results for the diffusion coefficient matrixcalculated within RTA approach based on the DQPM aswell as to various other models. The fact that the lin-earized Boltzmann equation is solved in the CE frame-work implies an improvement compared to approachesusing the RTA (also see Ref. [16]) in terms of accountingfor high moments of the distribution function. However,the proposed Chapman-Enskog method requires a fewapproximations for the QGP description, which are notin the spirit of the DQPM, in particular: 1. The system is assumed to obey classical (Maxwell-Boltzmann) statistics (i.e. a i = 0 for all particlespecies in Eq. (2).2. All particles are on-shell, therefore only the pole-masses from the DQPM, which depend on temper-ature and baryon-chemical potential, are assumedbut their widths are neglected (for general quasi-particle properties see Appendix V A, the Table Iand the Fig. 8).3. Inelastic scattering channels are neglected. Thatimplies that flavor-changing processes are not takeninto consideration, i.e. q ¯ q → q (cid:48) ¯ q (cid:48) are not allowed.4. All scattering processes are considered to beisotropic. We therefore feed total cross-sectionsinto the CE evaluation which are evaluated fromthe anisotropic differential cross section from theDQPM via Eq. (11). The dependence on √ s , tem-perature and baryon-chemical potential is takeninto account, σ ij → ij tot ≡ σ ij → ij tot ( √ s, T, µ B ) (see Ap-pendix V A, the Fig. 10 for an example at µ B = 0).We note that the CE method can in principle be im-proved such that approximations (1) and (3) become un-necessary. In Section III A we find indication that ap-proximation (3) might have a non-neglectable impact.Such improvements are left for future work. However,the nature of the method makes further improvement ofpoints (2) and (4) difficult and require further detailedstudy.The explicit manifestations of the isotropic pressure,the energy density and the net charge densities in the CEevaluation are fed with the same lattice data to which theDQPM model was fitted to (see Section II C) as far asthey are available. Since the net strangeness and net elec-tric densities are not available from the assumed lQCDresults, we compute their values from kinetic theory (seeEq. (6)).In the following the results from the RTA approachapplied to the original DQPM are denoted as “DQPMRTA” while the results from the Chapman-Enskogmethod applied to the DQPM under assumptions (1)-(4)(as described above) is denoted as “CE (DQPM)”. Weremind here that for constant cross sections the scaleddiffusion coefficients behave as κ qq (cid:48) /T ∼ /T , as foundin Ref. [16]. Such a decreasing behavior is indeed foundin Fig. 1. A. Model study: constant isotropic cross sections
In order to evaluate the systematic differences betweenthe DQPM RTA and the CE (DQPM) approaches weperform a here ’model study’ by assuming a (total) geo-metric cross section of σ tot = 10 mb for all interactions.For this comparison we consider the same assumptions asdescribed in the preface above, but assume all channelsto be characterized by the constant cross section. κ BB / T T [GeV]DQPM RTA:elasticinelastic CE (DQPM):RTAFull 0 0.002 0.004 0.006 0.008 0.2 0.25 0.3 0.35 0.4 κ QQ / T = σ e l / T T [GeV] σ T, isotr. = 10 mb µ B = 0 GeV Figure 1. The scaled baryon diffusion coefficient, κ BB /T (left), and the scaled electric conductivity, σ el /T (right), for a partonicsystem with geometric cross sections, σ tot = 10 mb at vanishing baryon-chemical potential as a function of temperature fromdifferent approaches. We compare the DQPM RTA results from Eq. (27) (for a system obeying quantum statistics, i.e. a i = ± a i = 0in Eq. (2)) or for “full” linearized collision term (orange solid line) from Eq. (18) ( a i = 0 in Eq. (2)). In Fig. 1 we show results for the scaled baryon coef-ficient, κ BB /T (left plot), and the scaled electric con-ductivity, σ el /T (right plot) at µ B = 0 in a temperaturerange from 160 MeV to 420 MeV. The DQPM RTA cal-culations are presented for two cases: firstly, where allbinary channels, including the inelastic ones, are consid-ered (blue dashed line), and for the case where only theelastic channels are accounted for (red dashed-double-dotted line). For the DQPM RTA results presented inthis paper we use Eq. (27) (for a system obeying quan-tum statistics, i.e. a i = ± a i = 0 in Eq. (2) ) under the assumption of the sim-plistic relaxation time, τ = 1 /n tot ¯ σ T (orange solid line),and for the second case we consider the full linearizedBoltzmann equation via Eq. (18) (for a system obey-ing classical statistics, i.e. a i = 0 in Eq. (2) ) (greendashed-dotted line).This ’model study’ shows the influence of the consid-eration of the linearized Boltzmann equation comparedto its relaxation time approximation, and the influenceof the inelastic channels compared to its neglection. Wefind that the consideration of the full linearized collisionterm effectively reduces the scattering rate of a specificparticle species, while in the RTA the scattering rate isoverestimated. This is because in the collision term notonly the scattering of particles from a specific momentumbin into all other disjoint momentum bins is considered,but also the rescattering into this particular momentum bin is accounted for (gain and loss term). As argued inRef. [16] such an overestimation of the scattering rateleads to a decrease of the diffusion coefficients from RTA(which are anti-proportional to the rate).Furthermore, we find that the inelastic channels lead toa further decrease of the diffusion coefficients due to therepeated effective increase of the scattering rate as shownin Fig. 1. Comparing the elastic version of the DQPMRTA evaluation with the CE (DQPM) calculation in theRTA limit (Eq. (27)), we find a good agreement of theresults at high temperatures. This is expected since theonly difference between both calculations – DQPM RTAand CE (DQPM) in the RTA limit – is the considera-tion of quantum corrections and the more sophisticated(momentum-dependent) relaxation time in DQPM RTA. B. Diffusion coefficient matrix of the quark-gluonplasma
In the following we show results for the scaled diffusioncoefficient matrix, κ qq (cid:48) /T , for the partonic phase fromthe DQPM (RTA) and CE (DQPM) evaluation underthe considerations described in the preface of Section III.Additionally we consider two cases: • We fix all chemical potentials to zero, µ q = 0 ( q =B , S , Q), and show the temperature dependence ofthe coefficients. • We fix the temperature to T = 2 T c ( µ B ), and showtheir dependence on the baryon chemical potential, (cid:6) (cid:5) (cid:6) (cid:6) (cid:5) (cid:7) (cid:6) (cid:5) (cid:8) (cid:6) (cid:5) (cid:9) (cid:6) (cid:5) (cid:10) (cid:6) (cid:5) (cid:11)(cid:6) (cid:5) (cid:6) (cid:6) (cid:6)(cid:6) (cid:5) (cid:6) (cid:6) (cid:11)(cid:6) (cid:5) (cid:6) (cid:7) (cid:6)(cid:6) (cid:5) (cid:6) (cid:7) (cid:11)(cid:6) (cid:5) (cid:6) (cid:8) (cid:6)(cid:6) (cid:5) (cid:6) (cid:8) (cid:11)(cid:6) (cid:5) (cid:6) (cid:9) (cid:6)(cid:6) (cid:5) (cid:6) (cid:9) (cid:11)(cid:6) (cid:5) (cid:6) (cid:10) (cid:6)(cid:6) (cid:5) (cid:11) (cid:7) (cid:5) (cid:6) (cid:7) (cid:5) (cid:11) (cid:8) (cid:5) (cid:6) (cid:8) (cid:5) (cid:11) (cid:9) (cid:5) (cid:6) (cid:9) (cid:5) (cid:11) (cid:10) (cid:5) (cid:6)(cid:6) (cid:5) (cid:6) (cid:6)(cid:6) (cid:5) (cid:6) (cid:7)(cid:6) (cid:5) (cid:6) (cid:8)(cid:6) (cid:5) (cid:6) (cid:9)(cid:6) (cid:5) (cid:6) (cid:10)(cid:6) (cid:5) (cid:6) (cid:11)(cid:6) (cid:5) (cid:6) (cid:12)(cid:6) (cid:5) (cid:6) (cid:13) k (cid:2)(cid:2) (cid:2) (cid:4) (cid:1) m (cid:1) (cid:5) (cid:2) (cid:8) (cid:4) (cid:6) (cid:18) (cid:20) (cid:2) (cid:19) (cid:27) (cid:26) (cid:24) (cid:3) (cid:1)(cid:19) (cid:27) (cid:26) (cid:24) (cid:1) (cid:28) (cid:30) (cid:16) (cid:1) (cid:1) (cid:8) (cid:1) (cid:6) (cid:1) (cid:5) (cid:1) (cid:8) (cid:2) (cid:2) m (cid:1) (cid:3) (cid:1) (cid:10) (cid:11) (cid:10) (cid:2) (cid:5) (cid:11) (cid:10) (cid:6) (cid:3) (cid:1) (cid:8) (cid:11) (cid:9) (cid:11) (cid:7) (cid:13) (cid:3) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:8) (cid:11) (cid:9) (cid:11) (cid:7) (cid:13) (cid:4) (cid:12) (cid:8) (cid:14) (cid:1) (cid:1) (cid:1) s (cid:3) (cid:4) (cid:2) (cid:4) (cid:1) (cid:3) (cid:1) k (cid:2)(cid:2) (cid:2) (cid:4) (cid:1) (cid:3) (cid:1) (cid:3) (cid:7) (cid:19) (cid:27) (cid:26) (cid:24) (cid:1) (cid:28) (cid:30) (cid:16) (cid:1) (cid:1)(cid:1) (cid:18) (cid:20) (cid:2) (cid:19) (cid:27) (cid:26) (cid:24) (cid:3) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:18) (cid:20) (cid:2) (cid:23) (cid:28) (cid:22) (cid:3) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:17) (cid:16) (cid:24) (cid:26) (cid:29) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:29) (cid:24) (cid:16) (cid:29) (cid:23) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:20) (cid:21) (cid:30) (cid:1) (cid:1) (cid:1) m (cid:3) (cid:1) (cid:6) (cid:1) (cid:4) (cid:1) (cid:7) (cid:10) (cid:9) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:15) (cid:8) (cid:4) (cid:7) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:25) (cid:1) (cid:15) (cid:8) (cid:1) (cid:1)% & ! $ (cid:31) " ! (cid:1) Figure 2. Left: Scaled electric conductivity, σ el /T = κ QQ /T , as a function of the scaled temperature T /T c at vanishingchemical potentials, µ q = 0, from various approaches. The results from the CE (DQPM) is shown by the red solid line, and fromDQPM RTA – by the black dashed line with crossed points. These are compared to results from the lattice QCD calculations(various shaped points with errorbars: quenched: orange circle-shaped points [38], light green rhombus-shaped points [85], N f = 2 : light cyan circle-shaped points [28], magenta rhombus-shaped points [37], and N f = 2 + 1 : dark cyan circle-shapedpoints [33] and blue stars [86]), the kinetic partonic cascade model BAMPS (dark-green solid line with triangular-shaped points)[34], and from conformal [13] (blue dotted line) and non-conformal [30] (violet dashed-dotted line) holographic models. Fortemperatures below T c = 0 .
158 GeV we show evaluations from hadronic models: SMASH [18, 42, 87] (grey short-dashed linewith squared points), effective field theory (EFT) [29] (blue dashed-dotted line), and CE tuned to a hadron gas [CE (HRG)]from Refs. [14, 16, 39] (dark-red dashed line). Right: Scaled electric conductivity of the QGP at fixed scaled temperature, T = 2 T c ( µ B ), and vanishing µ Q and µ S are shown for varying baryon chemical potential µ B from the DQPM RTA (blackdashed line with cross-shaped points) and the CE (DQPM) (red solid line with circle-shaped points) evaluation. µ B . Here we further set the other chemical poten-tials to zero, µ S = 0 and µ Q = 0.For most coefficients we find a rich µ B -dependence.This dependence originates from the fact that all quarkscarry baryon number and thus are sensitive to variationsin µ B . In Ref. [16] the temperature dependence of thesetransport coefficients was reviewed, and it was found thatthey roughly scale as κ qq (cid:48) /T ∼ / ( σ tot T ). In the caseof the DQPM at fixed chemical potential, the cross sec-tions depend on temperature as σ tot ∼ /T or ∼ /T (for the considered temperature range). This dependson the combinations of s − , t − , u − channels for differentparton-parton scatterings: for q − q , q − ¯ q and q − g scatterings σ tot ∼ /T , while for the g − g channel theterms 1 /T , 1 /T have equivalent contribution to thetotal cross-section σ tot ∼ c /T + c /T , where c , c depend on √ s, µ B (see e.g. Fig. 10 in Appendix V A).The temperature dependence of the cross-sections is inaccordance with the temperature scaling of the DQPMcoupling constant g ( T, µ B ) (see e.g. Fig. 9 in AppendixV A ). This leads to a roughly quadratic dependence intemperature, κ qq (cid:48) /T ∼ T , which is demonstrated inthe figures below.We remind that the diffusion coefficient matrix is sym- metric and therefore we may only show six instead of ninecoefficients [88, 89]. In the following we subdivide thepresentation of the conductivities in three sections: elec-tric conductivities ( κ QQ , κ QS and κ QB ), strange conduc-tivities ( κ SS and κ SB ), and finally the baryon conductivi-ties ( κ BB ). Since diffusion coefficients and conductivitiesare related to each other via temperature, κ qq (cid:48) = σ qq (cid:48) T ,we use their denomination interchangeably.
1. Electric conductivities
The electric conductivity, σ el /T , was evaluated in var-ious models (cf. Refs. [13, 14, 16–18, 27–42, 79]).In Fig. 2 we compare the results from DQPM RTAand CE (DQPM) to a variety of models for both, thepartonic [13, 28, 30, 31, 33, 34] and hadronic phase[14, 16, 18, 29, 39, 42], at µ q = 0 in a temperature rangebetween 0 and 3 T c , where here the deconfinement tem-perature is T c = 158 MeV. The Chapman-Enskog andRTA results for the dimensionless ratio of electric con-ductivity to temperature σ el /T (later referred to as scaledelectric conductivity) for µ q = 0 are presented in Fig.2(left) as solid red and dashed black lines. Results for both -3 ] κ Q B / T = σ Q B / T T [GeV]DQPM RTACE (DQPM)CE (HRG)SMASH 0.020.040.060.0 0.1 0.2 0.3 0.4[x 10 -3 ] κ Q S / T = σ Q S / T T [GeV] µ q = 0 GeV Figure 3. Scaled cross-electric conductivities, σ QB /T (left) and σ QS /T (right), from SMASH [18, 87] (grey short-dashed linewith square-shaped points), the DQPM RTA (black dashed line with cross-shaped points), and the CE (DQPM) (red solid linewith circle-shaped points) and CE (HRG) [14, 16] (dark-red dashed line) evaluation at vanishing chemical potentials, µ q = 0,for temperatures between 80 and 420 MeV. (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:4) (cid:2) (cid:1) (cid:5) (cid:2) (cid:1) (cid:6) (cid:2) (cid:1) (cid:7)(cid:2) (cid:1) (cid:2)(cid:2) (cid:1) (cid:7)(cid:3) (cid:1) (cid:2)(cid:3) (cid:1) (cid:7) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:4) (cid:2) (cid:1) (cid:5) (cid:2) (cid:1) (cid:6) (cid:2) (cid:1) (cid:7)(cid:2)(cid:3) (cid:2)(cid:4) (cid:2)(cid:5) (cid:2)(cid:6) (cid:2)(cid:7) (cid:2)(cid:8) (cid:2)(cid:9) (cid:2)(cid:10) (cid:12) (cid:3) (cid:2) (cid:1)(cid:2) (cid:11) k (cid:3) (cid:2) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3) (cid:2) (cid:1) (cid:3) m (cid:1) (cid:17) (cid:10) (cid:19) (cid:16) (cid:18)(cid:7) (cid:9) (cid:2) (cid:8) (cid:13) (cid:12) (cid:11) (cid:3) (cid:1)(cid:8) (cid:13) (cid:12) (cid:11) (cid:1) (cid:14) (cid:15) (cid:6) (cid:1) (cid:10) (cid:12) (cid:3) (cid:2) (cid:1)(cid:2) (cid:11) k (cid:3) (cid:4) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3) (cid:4) (cid:1) (cid:3) m (cid:1) (cid:17) (cid:10) (cid:19) (cid:16) (cid:18)(cid:1) (cid:15) (cid:1) (cid:5) (cid:1) (cid:4) (cid:1) (cid:15) (cid:2) (cid:2) m (cid:1) (cid:3) (cid:1) Figure 4. Scaled cross-electric conductivities, σ QB /T (left) and σ QS /T (right), from the DQPM RTA (black dashed line withcross-shaped points) and the CE (DQPM) evaluation at fixed scaled temperature, T = 2 T c ( µ B ), shown over baryon chemicalpotential µ B in range 0 to 0.5 GeV. Further, the other chemical potentials are fixed to zero, µ Q = 0 and µ S = 0. methods have a similar increase with temperature, whichis mainly a consequence of the temperature dependenceof the cross section (as discussed before) and also of theincreasing total electric charge density [16].We find that the results from DQPM RTA and CE(DQPM) are consistent with results from lattice QCD inthe vicinity of the crossover region, 1 ≤ T /T c ≤ .
5. We again point out the apparent quadratic dependence ontemperature which was shortly motivated in the prefaceof this section above. Due to our discussion from SectionIII A we suppose that a realistic result for the conductiv-ities may be between the evaluations from DQPM RTAand CE (DQPM).As follows from Fig. 2, the hadronic models presented0 (cid:4) (cid:3) (cid:9) (cid:5) (cid:3) (cid:4) (cid:5) (cid:3) (cid:9) (cid:6) (cid:3) (cid:4) (cid:6) (cid:3) (cid:9) (cid:7) (cid:3) (cid:4)(cid:4) (cid:3) (cid:4)(cid:4) (cid:3) (cid:5)(cid:4) (cid:3) (cid:6)(cid:4) (cid:3) (cid:7)(cid:4) (cid:3) (cid:8)(cid:4) (cid:3) (cid:9)(cid:4) (cid:3) (cid:10)(cid:4) (cid:3) (cid:11) (cid:4) (cid:3) (cid:9) (cid:5) (cid:3) (cid:4) (cid:5) (cid:3) (cid:9) (cid:6) (cid:3) (cid:4) (cid:6) (cid:3) (cid:9) (cid:7) (cid:3) (cid:4)(cid:2) (cid:4) (cid:3) (cid:6) (cid:4)(cid:2) (cid:4) (cid:3) (cid:5) (cid:10)(cid:2) (cid:4) (cid:3) (cid:5) (cid:6)(cid:2) (cid:4) (cid:3) (cid:4) (cid:12)(cid:2) (cid:4) (cid:3) (cid:4) (cid:8)(cid:4) (cid:3) (cid:4) (cid:4) k (cid:3)(cid:3) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3)(cid:3) (cid:1) (cid:3) (cid:2) (cid:1) (cid:2) (cid:1) (cid:6) (cid:12) (cid:11) (cid:10) (cid:1) (cid:13) (cid:14) (cid:4) (cid:1) (cid:1)(cid:1) (cid:5) (cid:7) (cid:2) (cid:6) (cid:12) (cid:11) (cid:10) (cid:3) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:5) (cid:7) (cid:2) (cid:9) (cid:13) (cid:8) (cid:3) (cid:1) (cid:1) (cid:1)(cid:1) (cid:17) (cid:19) (cid:18) (cid:19) (cid:16) (cid:21) (cid:15) (cid:20) (cid:17) (cid:22) (cid:1) (cid:1) (cid:1) m (cid:1) (cid:1) (cid:13) (cid:1) (cid:4) (cid:1) (cid:14) (cid:16) (cid:15) (cid:1) k (cid:3)(cid:2) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3)(cid:2) (cid:1) (cid:3) (cid:2) (cid:1) (cid:2) (cid:1) Figure 5. Scaled strange and strange-baryon diffusion coefficients, κ SS /T (left) and κ SB /T (right), as a function of scaledtemperature in the range from 0.5 to 3 T c at vanishing chemical potentials, µ q = 0. We compare results from CE (DQPM) (redsolid line with circles), DQPM RTA (black dashed-line with crossed-shaped points), the CE (HRG) [14, 16] (dark-red dashedline) and from conformal holography [13] (blue dotted line). (cid:3) (cid:2) (cid:3) (cid:3) (cid:2) (cid:4) (cid:3) (cid:2) (cid:5) (cid:3) (cid:2) (cid:6) (cid:3) (cid:2) (cid:7) (cid:3) (cid:2) (cid:8)(cid:3) (cid:2) (cid:3)(cid:3) (cid:2) (cid:4)(cid:3) (cid:2) (cid:5)(cid:3) (cid:2) (cid:6)(cid:3) (cid:2) (cid:7)(cid:3) (cid:2) (cid:8)(cid:3) (cid:2) (cid:9)(cid:3) (cid:2) (cid:10) (cid:3) (cid:2) (cid:3) (cid:3) (cid:2) (cid:4) (cid:3) (cid:2) (cid:5) (cid:3) (cid:2) (cid:6) (cid:3) (cid:2) (cid:7) (cid:3) (cid:2) (cid:8)(cid:1) (cid:3) (cid:2) (cid:4) (cid:9)(cid:1) (cid:3) (cid:2) (cid:4) (cid:5)(cid:1) (cid:3) (cid:2) (cid:3) (cid:11)(cid:1) (cid:3) (cid:2) (cid:3) (cid:7)(cid:3) (cid:2) (cid:3) (cid:3) k (cid:3)(cid:3) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3)(cid:3) (cid:1) (cid:3) m (cid:1) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) (cid:7) (cid:9) (cid:2) (cid:8) (cid:12) (cid:11) (cid:10) (cid:3) (cid:1)(cid:8) (cid:12) (cid:11) (cid:10) (cid:1) (cid:13) (cid:14) (cid:6) (cid:1) k (cid:3)(cid:2) (cid:1) (cid:3) (cid:1) (cid:2) s (cid:3)(cid:2) (cid:1) (cid:3) m (cid:1) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) (cid:1) (cid:14) (cid:1) (cid:5) (cid:1) (cid:4) (cid:1) (cid:14) (cid:2) (cid:2) m (cid:1) (cid:3) (cid:1) Figure 6. Scaled strange and strange-baryon diffusion coefficients, κ SS /T (left) and κ SB /T (right), from the DQPM RTA(black dashed line with cross-shaped points) and the CE (DQPM) evaluation at fixed scaled temperature, T = 2 T c ( µ B ), shownover baryon chemical potential µ B in range 0 to 0.5 GeV. Further, the other chemical potentials are fixed to zero, µ Q = 0 and µ S = 0. there – the hadronic transport model SMASH [18, 42, 87](grey short-dashed line with squared points), effectivefield theory (EFT) [29] (blue dashed-dotted line), and CEtuned to a hadron gas [CE (HRG)] from Refs. [14, 16, 39](dark-red dashed line) – substantially overestimate thelQCD data in the vicinity of T c as well as the results fromthe conformal [13] (blue dotted line) and non-conformal [30] (violet dashed-dotted line) holographic models. TheDQPM RTA results are in a good agreement in the vicin-ity of phase transition with the previous estimations forDQPM* from Ref. [79], where non-relativistic formulafor estimation the electric conductivity was used, whichresults in the linear dependence of the σ el /T on temper-ature while presented DQPM results show the quadratic1 (cid:3) (cid:2) (cid:8) (cid:4) (cid:2) (cid:3) (cid:4) (cid:2) (cid:8) (cid:5) (cid:2) (cid:3) (cid:5) (cid:2) (cid:8) (cid:6) (cid:2) (cid:3)(cid:3) (cid:2) (cid:3) (cid:3)(cid:3) (cid:2) (cid:3) (cid:8)(cid:3) (cid:2) (cid:4) (cid:3)(cid:3) (cid:2) (cid:4) (cid:8)(cid:3) (cid:2) (cid:5) (cid:3)(cid:3) (cid:2) (cid:5) (cid:8) (cid:3) (cid:2) (cid:3) (cid:3) (cid:2) (cid:4) (cid:3) (cid:2) (cid:5) (cid:3) (cid:2) (cid:6) (cid:3) (cid:2) (cid:7) (cid:3) (cid:2) (cid:8)(cid:3) (cid:2) (cid:3) (cid:3)(cid:3) (cid:2) (cid:3) (cid:8)(cid:3) (cid:2) (cid:4) (cid:3)(cid:3) (cid:2) (cid:4) (cid:8)(cid:3) (cid:2) (cid:5) (cid:3) k (cid:2)(cid:2) (cid:1) (cid:2) (cid:1) (cid:3) (cid:1) (cid:3) (cid:7) (cid:8) (cid:14) (cid:13) (cid:12) (cid:1) (cid:15) (cid:16) (cid:6) (cid:1) (cid:1)(cid:1) (cid:7) (cid:9) (cid:2) (cid:8) (cid:14) (cid:13) (cid:12) (cid:3) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:7) (cid:9) (cid:2) (cid:11) (cid:15) (cid:10) (cid:3) (cid:1) (cid:1) (cid:1)(cid:1) (cid:19) (cid:21) (cid:20) (cid:21) (cid:18) (cid:23) (cid:17) (cid:22) (cid:19) (cid:24) (cid:1) (cid:1) (cid:1) m (cid:1) (cid:1) (cid:9) (cid:1) (cid:3) (cid:1) (cid:10) (cid:12) (cid:11) (cid:1) k (cid:2)(cid:2) (cid:1) (cid:2) (cid:1) m (cid:1) (cid:5) (cid:2) (cid:8) (cid:4) (cid:6) (cid:1) (cid:7) (cid:9) (cid:2) (cid:8) (cid:14) (cid:13) (cid:12) (cid:3) (cid:1) (cid:1)(cid:8) (cid:14) (cid:13) (cid:12) (cid:1) (cid:15) (cid:16) (cid:6) (cid:1) (cid:1)(cid:1) (cid:16) (cid:1) (cid:5) (cid:1) (cid:4) (cid:1) (cid:16) (cid:2) (cid:2) m (cid:1) (cid:3) (cid:1) Figure 7. Scaled baryon diffusion coefficient, κ BB /T , (left) at vanishing chemical potentials, µ q = 0, as a function of the scaledtemperature from various approaches and (right) at fixed scaled temperature, T = 2 T c ( µ B ), as a function of baryon chemicalpotential µ B . The strange and electric potential are fixed to zero: µ S = 0 and µ Q = 0. We show results from the CE evaluationtuned to DQPM (red solid line), as described above, and tuned to a hadron gas from Refs. [14, 16] (dark-red dashed line).We again compare to the calculation from DQPM RTA [17] (black dashed line with crosses) and to conformal holography [13](blue dotted line) as done for the electric conductivity. dependence on temperature.Additionally to the electric conductivity, in Fig. 3 weshow the cross-electric conductivities, σ BQ and σ QS , fromthe CE (DQPM) and the DQPM RTA calculation to-gether with results achieved within SMASH [18] and theCE (HRG) evaluation from Refs. [14, 16] for the samethermal considerations for the hadronic phase. Com-paring the results in both phases, we find a significantdisagreement for σ QB around the crossover temperature.Further, we find such discrepancies to a smaller extendin the other electric conductivities and in the coefficientsto follow. Such disagreement may hint to a difference inthe chemical composition of the adjacent phases [18].Furthermore, as advertized in the preface, in Figs.2 (right) and 4 we present the sensitivity of the elec-tric conductivities on µ B at fixed scaled temperature, T = 2 T c ( µ B ). Compared to the coefficients directly con-nected to the baryonic sector, we find a rather weak de-pendence on µ B (see discussion of κ BB and κ SB ). Surpris-ingly also σ QB has such a weak dependence even thoughit also belongs to the baryonic sector. Further, σ QB isvery small - it has the smallest magnitude of all conduc-tivities in the diffusion matrix. One can discuss its plau-sibility with a symmetry argument: the σ QB coefficientrelates the generated electric current to the baryonic gra-dient which generates it (via the corresponding Navier-Stokes term). Assume a QGP with constant geometriccross section as discussed in Section III A. Further, as-sume that all quarks have the same mass. The down- andstrange-quark have the same baryon number and electric charge, B = +1 / − / e , while the up-quarkhas B = +1 / / e , i.e. the same baryonnumber but an electric charge which is twice the magni-tude but has the opposite sign (refer to Table I). Due tothe quarks carrying the same baryon number, a baryongradient generates a baryon current V µ B which is equallycomposed by a current of up-, down- and strange-quarks( V µi ): V µ B = (cid:80) i B i V µi , with V µi = V µ quark ∀ i . With thiswe can estimate the generated electric current: V µ Q = (cid:88) i Q i V µi = V µ quark (cid:18) − −
13 + 23 (cid:19) = 0 . (35)The same argument can be made additionally accountingfor the anti-quarks. The non-equal mass of the quarksand the varying cross sections lead to a non-vanishing σ QB . However, the above estimate illustrate the smallmagnitude of the respected coefficient.
2. Strange conductivities
We continue with the results for the strange sector: κ SS and κ SB . The coefficient κ SQ , or equivalently σ SQ = σ QS ,was already discussed above as part of the electric sec-tor. Fig. 5 shows the κ SS and κ SB as a function of tem-perature at vanishing chemical potentials. Further, weshow their µ B -dependence in Fig. 6 in the range µ B = 0to 0.5 GeV at fixed scaled temperature, T = 2 T c ( µ B )and for vanishing electric and strange chemical poten-tial. We compare results from the DQPM RTA and CE2(DQPM) computation to results from CE (HRG) in ourrecent publications [14, 16], and to results from conformalholography [13].We find that the baryon-strange diffusion coefficientis negative due to the definition of strangeness carriedby the s-quark as has been already advocated in Ref.[14, 16]. We obtain an almost quadratic dependence intemperature again, and a rather strong dependency on µ B . However, the results from DQPM RTA for κ SB inFig. 6 show a slightly different µ B -behavior than the re-sults from CE (DQPM) for µ B ≥ .
3. Baryon conductivities
In order to describe the deconfined QCD medium atthe non-zero baryon density one should first consider thebaryon diffusion coefficient κ BB . This diffusion coefficientwas already evaluated in various models [12–17]. Fig. 7(left) shows the temperature dependence of the baryondiffusion coefficient for the quark-gluon plasma estimatedwith the CE (DQPM) (red solid line) and DQPM RTAapproaches (black dashed line with crosses). We alsoshow the results from conformal holography [13] (bluedotted lines). For temperatures below T c we again re-fer to the CE (HRG) calculation from Refs. [14, 16](dark-red dashed line). The comparison is presented atzero chemical potentials µ q = 0. Around T c the resultsfrom DQPM RTA, CE (DQPM) and CE (HRG) seem tobe rather consistent with each other. Furthermore, weshow its dependence on µ B at fixed scaled temperature T = 2 T c ( µ B ) in Fig. 7 (right). The DQPM RTA showsa rather weak µ B dependence, while κ BB from the CE(DQPM) decreases with µ B . IV. CONCLUSION
In this study we have calculated the complete diffusioncoefficient matrix κ qq (cid:48) ( q, q (cid:48) = B , S , Q) of the stronglyinteracting quark-gluon plasma by using the Chapman-Enskog method as well as the relaxation time approxima-tion (RTA) from kinetic theory. We have explored the T and µ B dependencies of the diffusion coefficients byconsidering microscopical properties of quarks and glu-ons within the dynamical quasi-particle model (DQPM).The DQPM predictions of thermodynamic quantities forfinite µ B show a good agreement with the available lQCDEoS [84]. Moreover, for µ B = 0 the DQPM estimationsof the QGP electric conductivity ( σ el /T ) are in a goodagreement with the N f = 2 + 1 lQCD results and in case of the specific shear and bulk viscosities ( η/s, ζ/s ) theestimations are remarkably close to the predictions fromthe gluodynamic lQCD calculations [17].We find that the electric conductivities ( κ QQ , κ QS and κ QB ), strange conductivities ( κ SS and κ SB ), and finallythe baryon conductivity ( κ BB ) have a similar tempera-ture dependence in the vicinity of the phase transitionwhile the µ B dependence is rather different among theconsidered diffusion coefficients. In particular, the dif-fusion coefficients κ BB and κ QB decrease with µ B , whileother coefficients increase. A suppression of baryon dif-fusion in a sQGP with finite µ B has been seen also in theholographic calculations [13].One of the main endeavors of this paper is to de-liver reasonable estimates for the diffusion coefficientsof the strongly interacting quark-gluon plasma. Fur-thermore, we compare the RTA evaluations from re-cent DQPM publications [17, 57, 90] with the Chapman-Enskog method. We demonstrate that once the crosssections and the (thermal) properties (masses, equationof state, etc.) of a system are known, the CE frameworkat hand is able to deliver consistent results. We show agood agreement for both methods with the available pre-dictions from the literature for the partonic phase, in par-ticular results for the scaled electric conductivity are re-markably close to the lQCD estimates at µ B = 0, as wellas with the estimates for the hadronic phase. However, κ QB non-diagonal diffusion coefficient doesn’t coincidewell in the vicinity of the phase transition with the esti-mates for the hadronic phase, which can be interpretedas an indication of a difference in the chemical compo-sition of the adjacent phases. There are several modelcalculations of diagonal conductivities (mostly κ QQ ) inthe literature that are similar to the RTA approach butused numerous restrictive assumptions for the evaluationof the relaxation times or cross-sections. Studying thediffusion coefficients of the QGP should have future bene-fits when considering hydrodynamical description for thetime evolution of the deconfined QCD medium.3 V. APPENDIXA. Properties of partons in the DQPM
Name Spin Degeneracy Baryon Electric StrangenessNumber Charge g u / / / e u / − / − / e d / / − / e d / − / / e s / / − / e − s / − / / e +1Table I. Properties of the particle species considered in thequark-gluon plasma in this work. Here, e denotes the elemen-tary electric charge in natural units. ACKNOWLEDGMENTS
The authors acknowledge inspiring discussions with H.van Hees, T. Song and J. M. Torres-Rincon. Also the au-thors acknowledge support by the Deutsche Forschungs-gemeinschaft (DFG, German Research Foundation)through the CRC-TR 211 ’Strong-interaction matter un-der extreme conditions’– project number 315477589 –TRR 211. O.S. and J.A.F. acknowledge support fromthe Helmholtz Graduate School for Heavy Ion research.Furthermore, we acknowledge support by the DeutscheForschungsgemeinschaft by the European Union’s Hori-zon 2020 research and innovation program under grantagreement No 824093 (STRONG-2020) and by the COSTAction THOR, CA15213. The computational resourceshave been provided by the LOEWE-Center for ScientificComputing and the ”Green Cube” at GSI, Darmstadt. [1] J.-Y. Ollitrault, Phys. Rev. D , 229 (1992).[2] U. W. Heinz and P. F. Kolb, Nucl. Phys. A , 269(2002), arXiv:hep-ph/0111075.[3] E. Shuryak, Prog. Part. Nucl. Phys. , 48 (2009),arXiv:0807.3033 [hep-ph].[4] E. V. Shuryak, Nucl. Phys. A , 64 (2005), arXiv:hep-ph/0405066.[5] M. Gyulassy and L. McLerran, Nucl. Phys. A , 30(2005), arXiv:nucl-th/0405013.[6] P. Romatschke and U. Romatschke, Phys. Rev. Lett. ,172301 (2007), arXiv:0706.1522 [nucl-th].[7] M. Cheng et al. , Phys. Rev. D , 014511 (2008),arXiv:0710.0354 [hep-lat].[8] Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz,S. Krieg, and K. K. Szabo, JHEP , 088 (2009),arXiv:0903.4155 [hep-lat].[9] G. S. Denicol, C. Gale, S. Jeon, A. Monnai,B. Schenke, and C. Shen, Phys. Rev. C98 , 034916(2018), arXiv:1804.10557. [10] M. Li and C. Shen, Phys. Rev.
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The effective quark ( left ) and gluon ( right ) pole-masses M ( upper row ) and their widths γ ( lower row ) from theactual DQPM as a function of the temperature T and baryon chemical potential µ B [90]. The strange quark mass was assumedto be m s = m q + ∆ m , with ∆ m = 30 MeV, and the width is identical with the widths of the light quarks, γ s = γ q . In thecalculation of the diffusion matrix with the Chapman-Enskog method partons were assumed to be on-shell and thus the widthsvanish. (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:11) (cid:12) (cid:4) (cid:3)(cid:3) (cid:2) (cid:3)(cid:3) (cid:2) (cid:8)(cid:4) (cid:2) (cid:3)(cid:4) (cid:2) (cid:8)(cid:5) (cid:2) (cid:3)(cid:5) (cid:2) (cid:8)(cid:6) (cid:2) (cid:3)(cid:6) (cid:2) (cid:8) (cid:13) (cid:2) (cid:1) (cid:9) (cid:1) (cid:6) (cid:2) (cid:5)(cid:1) (cid:1) (cid:11) (cid:15) (cid:14) (cid:12) (cid:1) (cid:16) (cid:1) (cid:3) (cid:7) (cid:2)(cid:17) (cid:15) (cid:10) (cid:11) (cid:8) (cid:1) (cid:1) (cid:1) (cid:13) (cid:2) (cid:1) (cid:9) (cid:1) (cid:4)(cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:1) (cid:13) (cid:2) (cid:1) (cid:9) (cid:1) (cid:6) (cid:1) (cid:1) (cid:4) (cid:3) (cid:4) (cid:2) (cid:1) (cid:1) (cid:1) (cid:2) (cid:1) (cid:1) (cid:1) (cid:1) (cid:13) (cid:1) (cid:3) (cid:1) a) b)Figure 9. The running coupling α s = g / (4 π ) from the actual DQPM as a function of the scaled temperature T /T c at µ B = 0 a)and for moderate values of baryon chemical potential µ B < = 0 . N f = 0,(blue circles) are taken from Ref. [48] and for N f = 2 (black triangles) are taken from Ref. [80]. (cid:1) (cid:3) (cid:4) (cid:5) (cid:6)(cid:2) (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:3) (cid:1) (cid:1) (cid:2) (cid:5) (cid:4) (cid:3) (cid:15) (cid:16) (cid:3) (cid:9) (cid:15) (cid:16) (cid:6) (cid:2) (cid:1)(cid:3) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) (cid:1)(cid:18) (cid:18) (cid:3) (cid:9) (cid:18) (cid:18)(cid:1)(cid:18) (cid:18) (cid:3) (cid:9) (cid:18) (cid:18)(cid:1)(cid:18) (cid:13) (cid:3) (cid:9) (cid:18) (cid:13)(cid:1)(cid:18) (cid:13) (cid:3) (cid:9) (cid:18) (cid:13)(cid:1)(cid:17) (cid:17) (cid:3) (cid:9) (cid:17) (cid:17)(cid:1)(cid:17) (cid:17) (cid:3) (cid:9) (cid:17) (cid:17)(cid:1)(cid:1) (cid:1) (cid:1)(cid:8) (cid:1)(cid:5) (cid:2)(cid:1)(cid:11) (cid:1)(cid:8) (cid:1)(cid:5) (cid:4)(cid:6) (cid:7) (cid:1)(cid:10) (cid:14) (cid:12) (cid:1)(cid:15) (cid:15) (cid:3) (cid:9) (cid:15) (cid:15) a) (cid:2) (cid:4) (cid:5) (cid:6) (cid:7)(cid:3) (cid:2) (cid:1)(cid:3) (cid:3) (cid:2) (cid:2) (cid:3) (cid:2) (cid:3) (cid:3) (cid:2) (cid:4) (cid:1)(cid:1) (cid:1) (cid:1)(cid:3) (cid:1)(cid:2) (cid:1) (cid:1) (cid:1) (cid:2) (cid:5) (cid:4) (cid:3) (cid:6) (cid:2) (cid:1)(cid:3) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) (cid:16) (cid:1)(cid:13) (cid:1)(cid:4) (cid:3)(cid:6) (cid:5) (cid:9) (cid:1)(cid:15) (cid:18) (cid:17) (cid:12) (cid:1)(cid:1) (cid:1)(cid:19) (cid:19) (cid:2) (cid:14) (cid:19) (cid:19)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:20) (cid:20) (cid:2) (cid:14) (cid:20) (cid:20)(cid:16) (cid:1)(cid:13) (cid:1)(cid:4) (cid:3)(cid:7) (cid:8) (cid:10) (cid:1)(cid:15) (cid:18) (cid:17) (cid:12) (cid:1)(cid:1) (cid:1)(cid:19) (cid:19) (cid:2) (cid:14) (cid:19) (cid:19)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:20) (cid:20) (cid:2) (cid:14) (cid:20) (cid:20)(cid:16) (cid:13) (cid:4) (cid:3)(cid:5) (cid:11) (cid:1)(cid:15) (cid:18) (cid:17) (cid:12) (cid:1) (cid:1)(cid:19) (cid:19) (cid:2) (cid:14) (cid:19) (cid:19)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:20) (cid:20) (cid:2) (cid:14) (cid:20) (cid:20) b) (cid:1) (cid:3) (cid:4) (cid:5) (cid:6)(cid:2) (cid:1) (cid:1) (cid:2) (cid:1) (cid:2) (cid:2) (cid:1) (cid:3) (cid:1)(cid:13) (cid:1)(cid:10) (cid:1)(cid:4) (cid:3)(cid:5) (cid:8) (cid:4) (cid:1)(cid:12) (cid:15) (cid:14) (cid:1) (cid:1) (cid:1) (cid:2) (cid:5) (cid:4) (cid:3) (cid:6) (cid:2) (cid:1)(cid:3) (cid:3) (cid:1) (cid:5) (cid:2) (cid:4) (cid:17) (cid:17) (cid:2) (cid:11) (cid:17) (cid:17) (cid:9) (cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1) (cid:10) (cid:1)(cid:4) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4) (cid:3)(cid:6) (cid:1)(cid:12) (cid:15) (cid:14)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4) (cid:3)(cid:7) (cid:12) (cid:15) (cid:14) (cid:1)(cid:16) (cid:16) (cid:2) (cid:11) (cid:16) (cid:16) (cid:9) (cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1) (cid:10) (cid:1)(cid:4) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4) (cid:3)(cid:6) (cid:1)(cid:12) (cid:15) (cid:14)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:4) (cid:3)(cid:7) (cid:12) (cid:15) (cid:14) (cid:1) c)Figure 10. DQPM total cross sections between different partons for the on-shell case from Eq. (11) evaluated in the center ofmass of the collision system as a function of the collision energy √ s for a) µ B = 0 , T = 0 .
19 GeV for the elastic channels, b) for µ B = 0 , T = 0 . , . , .
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