Shear and bulk viscosities of strongly-interacting 'infinite' parton-hadron matter within the parton-hadron-string transport approach

Bottom line: this is a competent, useful extension of the PHSD/DQPM program — the first temperature scan of both eta/s and zeta/s across the PHSD crossover — and the mean-field mechanism behind the zeta/s peak is actually demonstrated rather than asserted. The comparison to lattice QCD, though, is partly built in and should be read that way. Neither point sinks the paper; the first is why it's worth reading, the second is why you shouldn't over-cite the lQCD agreement line (abstract:s6). What's new and done well: the box setup is clean (9³ fm³, 500 test particles), the Kubo extraction is checked properly — exponential decay of the correlator in Fig. 1, convergence in test-particle number in Fig. 2, volume independence in Fig. 3 — and the Kubo-versus-RTA agreement for eta (abstract:s4) is a real internal cross-check, not decoration. The best part is the ablation in Figs. 5 and 7: switching the scalar mean field on and off isolates the mechanism, and the claim in abstract:s8 that the zeta/eta maximum comes from the temperature dependence of the quasiparticle masses holds up on its own terms. Soft spots, in order of weight. First, the circularity: g²(T) is fitted to lattice data, s(T) comes from the lQCD equation of state, and the zeta/s peak is carried entirely by T² dm²/dT² — a derivative of that fitted mass function. So the peak's location and height track lambda, Ts/Tc, and the Nf=3 refit the paper explicitly does not display, deferring to Ref. [70]. That's the one thing I'd push on in review: show the Nf=3 g²(T) and do a sensitivity scan over the fit parameters. Second, zeta gets no Kubo cross-check while eta does, which matters because zeta is the headline. Third, the hadronic phase is at mu_q = 0; the authors flag this themselves, as they do the unresolved width decomposition in DQPM, so I won't count either against them. No code shipped, which is normal for PHSD but limits reproduction. Who should read it: heavy-ion transport and viscous-hydrodynamics people who need temperature-dependent transport coefficients from a model that already describes the bulk observables. Worth a serious referee — send it out, and ask specifically for the Nf=3 coupling and a robustness check on the zeta peak.

The headline (abstract:s6, s8) is that mean fields produce a rise in ζ/s near Tc that agrees with lQCD. In the RTA expression Eq. (25) the entire mean-field effect sits in the bracket [(1/3−v_s²)|p|² − v_s²(m² − T²dm²/dT²)]. The authors' own ablation (Fig. 5, Fig. 7: with vs. without mean field) shows that removing this term flattens ζ/s to an almost constant value, so the peak is carried entirely by T²dm²/dT². That derivative is a property of the DQPM ansatz: g²(T) in Eq. (1) is C/ln[λ²(T/Tc−Ts/Tc)²], whose logarithm passes through zero at T=0.56Tc, so g² and hence m² vary steeply just above Tc and dm²/dT² is large there. The peak's location and height are therefore set by λ, Ts/Tc and by the Nf=3 refit that the paper explicitly does not display ('we employ a slightly different analytical form ... For the details we refer the reader to Ref. [70]'). A second asymmetry: η is cross-checked by Kubo and RTA (abstract:s4), but ζ is RTA-only, so unlike η it has no independent dynamical check — it is a closed functional of the DQPM masses and widths (Eqs. 4-5, c=14.4). The mechanism claim in s8 is well supported by the ablation; the stronger 'in agreement with lQCD' in s6 is not, because the curve being compared is largely a derivative of a function fitted to the same lattice thermodynamics.

I largely agree with the reader. The reader's weakest_assumption correctly locates the load-bearing dependence on the DQPM mass parametrization; I would sharpen it in two ways. First, the specific mechanism is the derivative of a logarithmically steep fitted coupling, not merely 'the mass could be wrong' — the ablation in Figs. 5 and 7 already isolates T²dm²/dT² as the sole driver, so the question is whether that derivative is physical or an artifact of the fit ansatz. Second, ζ has no Kubo cross-check, so the internal validation that strengthens η (abstract:s4) does not extend to the headline ζ result. Credit where due: Eqs. (20) and (24) are mutually consistent — setting dm²/dT²=0 in Eq. (24) and multiplying the bracket by 3 reproduces (1−3v_s²)E²−m², i.e. Eq. (20) up to the 1/9 prefactor — which is a genuine internal check the paper does not spell out. The volume-independence check (Fig. 3) and the testparticle convergence (Fig. 2) are also real. The authors do acknowledge that the hadronic-phase results are at μq=0 and that the DQPM width decomposition is not addressed; those are stated limitations, not new findings. One further framing point: Fig. 5 compares 3-flavor PHSD ζ/s to pure-gauge lQCD without the caveat the authors give for Fig. 6, which slightly overstates the agreement. None of this warrants rejection; the reader's CONDITIONAL is the right level, and the lQCD comparison should be read as a consistency check.

The paper is a useful and technically careful contribution: it extracts transport coefficients from a transport approach that simultaneously describes the partonic and hadronic phases, without introducing viscosity as an external parameter, and it provides a genuine internal cross-check of η between two independent methods (Kubo dynamics versus RTA widths). The ablation of the mean-field term in Figs. 5 and 7 is a clean and informative diagnostic that isolates the mechanism behind the ζ/s peak. The results are relevant to the ongoing effort to constrain η/s and ζ/s near the crossover from heavy-ion phenomenology and lattice QCD. The main caveat is that the bulk-viscosity result rests on a single method and on a mass parametrization fitted to lattice thermodynamics, so the claimed agreement with lQCD is better described as a consistency check than as independent validation.

The paper asks a concrete question: if you take the transport model that already reproduces heavy-ion collision data and let it run in a periodic box until the matter equilibrates, what shear and bulk viscosities does the matter itself imply? The authors simulate partonic matter above the critical temperature $T_c$ and hadronic matter below it, then extract $/eta$ and $/zeta$ with two independent routes, the Green-Kubo correlation of the energy-momentum tensor and the relaxation-time approximation, and check that both give the same answer within statistics. The result they want to establish is that $/eta(T)/s(T)$ dips to roughly 0.1 near $T_c$, rises steeply in the hadronic phase as the entropy density falls, and approaches the perturbative QCD limit at high temperature, all in line with lattice QCD. The more distinctive claim concerns the bulk viscosity: $/zeta(T)/s(T)$ rises near $T_c$ only when the scalar mean field is included, and the authors trace that rise to the temperature dependence of the quasiparticle masses. If correct, this says the transport coefficients of strongly interacting matter are not extra dials but come out of the same in-medium masses and widths that fix the equation of state.

What carries the argument. The load-bearing object is the mean-field term in the relaxation-time expression for bulk viscosity, Eq. (25), where the bracket contains $(/tfrac{1}{3} - v_s^2)|p|^2 - v_s^2(m_i^2 - T^2 /, dm_i^2/dT^2)$. The derivative $T^2 dm^2/dT^2$ is fixed by the dynamical quasiparticle model: $m_q^2 = /tfrac{1}{3} g^2(T) T^2$ and $m_g^2 = /tfrac{3}{4} g^2(T) T^2$, with the resummed running coupling $g^2(T/T_c)$ fitted to lattice data. That derivative, not the scattering rate, is what generates the near-$T_c$ peak in $/zeta/s$ and in $/zeta/eta$. The shear channel is extracted independently from the exponential decay of the volume-averaged correlator $/langle /pi_{xy}(0)/pi_{xy}(t)/rangle$, with $/pi_{xy} = /frac{1}{V}/sum_i p_{xi}p_{yi}/E_i$, so that $/eta = (V/T)/langle/pi_{xy}(0)^2/rangle /tau$.

Core claim. The central claim is that a single off-shell transport approach, whose partonic sector is built on dynamical quasiparticle masses and widths fitted to lattice QCD, reproduces the temperature dependence of both shear and bulk viscosity of strongly interacting matter. The shear viscosity to entropy density ratio shows a minimum of about 0.1 close to $T_c = 158$ MeV, rises fast below $T_c$ because the entropy density of the hadron gas drops, and tends toward the perturbative QCD value at high $T$; the Kubo formalism and the relaxation-time approximation agree within error bars. For the bulk viscosity the authors report a significant rise of $/zeta(T)/s(T)$ in the vicinity of $T_c$, but only when the scalar mean field is consistently included; without potential effects the ratio is nearly flat and misses the lattice trend. The same mean-field term produces a pronounced maximum in $/zeta/eta$ near $T_c$, and the sound-channel combination $(/eta + 3/zeta/4)/s$ shows a nontrivial temperature dependence that broadly matches lattice gauge theory.

The premise it rests on. The peak in bulk viscosity is real only if the dynamical quasiparticle masses, built from a running coupling fitted to lattice thermodynamics, track the actual in-medium parton masses closely enough that the derivative $T^2 dm^2/dT^2$ is physical rather than an artifact of the fit.

Viscous hydrodynamic descriptions of heavy-ion collisions can take $/eta/s(T)$ and $/zeta/s(T)$ from this model rather than fitting them, since both follow from the same in-medium masses and widths that fix the equation of state.

The near-$T_c$ peak in $/zeta/s$ and the maximum in $/zeta/eta$ are predictions for the crossover region that can be checked against dynamical lattice QCD, not just against pure-gauge results.

Because the shear viscosity minimum sits close to the KSS value of $1/4/pi$, the matter simulated here behaves as a near-perfect fluid in the same temperature window where elliptic flow data demand small $/eta/s$.

The strong rise of $/eta/s$ below $T_c$ follows mostly from the entropy density falling exponentially, so the hadronic viscosity itself need not be large to produce a large ratio.

The approach extends to finite quark chemical potential, which the authors flag as the next step before the coefficients can be applied to lower-energy collisions.

The claim that mean fields, not scatterings, drive the bulk-viscosity peak is testable in a cleaner way than the paper attempts: hold the equation of state fixed and swap in a different in-medium mass parametrization, then check whether the peak survives. If it does not, the peak is a property of the fit rather than of the matter.

Since the peak is generated by $T^2 dm^2/dT^2$, its height should scale with how fast the quasiparticle masses drop through the crossover; a model with a sharper mass drop should show a sharper and taller $/zeta/s$ peak, which is a concrete cross-model prediction.

The agreement with pure-gauge lattice data is quoted as approximate, but the simulation is three-flavour; a matched comparison would need lattice results with dynamical quarks at the same $T_c$, which is the natural next check.

If the same mass derivative controls both the bulk viscosity and the trace anomaly, then the two should be correlated across temperatures, giving an independent handle on $/zeta/s$ from thermodynamic quantities alone.

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