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REVIEW 2 major objections 5 minor 119 references

Charged-current spectral functions of isospin-asymmetric dense matter match non-Abelian hydrodynamics up to the hard scale, and a product-formula extension capturing the infrared conformal sector is verified numerically.

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2026-08-01 08:47 UTC pith:HGHJLNA2

load-bearing objection Solid leading-order result for non-Abelian charged-current hydrodynamics at finite isospin density; the longitudinal extended-hydrodynamic claim rests on an admitted assumption and should be qualified in revision. the 2 major comments →

arxiv 2607.20991 v1 pith:HGHJLNA2 submitted 2026-07-23 hep-th cond-mat.stat-mechcond-mat.str-elgr-qcnucl-th

Flavour current correlators and the non-Abelian hydrodynamic approximation: the charged sector

Thomas Apostolidis , Matti J\"arvinen , Elias Kiritsis , Francesco Nitti , Andrea Olzi , Edwan Pr\'eau This is my paper
classification hep-th cond-mat.stat-mechcond-mat.str-elgr-qcnucl-th PACS 11.25.Tq
keywords non-Abelian hydrodynamicsholographic dense matterisospin chemical potentialcharged-current spectral functionsnear-extremal black holesextended hydrodynamic regimequasi-normal modesneutrino transport in neutron stars
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the charged-flavor-current correlators of a strongly coupled dense plasma with both a quark-number and an isospin chemical potential, in the holographic AdS5-Reissner-Nordström toy model, at temperature far below the total chemical potential. It establishes that, at leading order in ω, k, μ₃, T ≪ μ, the exact spectral functions collapse to the non-Abelian hydrodynamic forms, with the conductivity Σ = N_c(Mℓ)³w₀²/(8r_H) and diffusivity D = r_H/2 fixed purely by the horizon; the same formulas hold for ω, k ≪ T ≪ μ and for T ≪ ω, k ≪ μ, extending the validity of hydrodynamics up to the hard scale. It then proposes an extended hydrodynamic approximation — the hydrodynamic forms multiplied by the infrared power law (ω/μ)^{2Δ(k,μ₃)−1} from the near-horizon AdS₂ sector — and verifies numerically that it tracks the exact correlators better than plain hydrodynamics, especially near ω = 0, for μ_q/T = 65 and μ₃/μ_q = −0.1. A sympathetic reader would care because these correlators are the inputs to neutrino-transport rates in neutron-star-like matter, so the result turns a full microscopic computation into a two-constant fluid-dynamics problem up to the density scale.

Core claim

The central claim: at leading order in ω, k, T, μ₃ ≪ μ ≡ √(μ_q²+μ₃²), the charged-current spectral functions of strongly coupled isospin-asymmetric dense matter take the non-Abelian hydrodynamic forms ImΠ^±_⊥ = −Σω and ImΠ^±_∥ = Σω(k²−(ω±μ₃)²)/((ω±μ₃)²+(r_H/2)²k⁴), with Σ = N_c(Mℓ)³w₀²/(8r_H). The isospin background enters only through the shifted frequencies ω±μ₃ and through the horizon radius r_H(μ). The same leading forms hold for ω, k ≪ T ≪ μ and for T ≪ ω, k ≪ μ, so hydrodynamics extends to the hard scale. The paper further claims the extended approximation ImΠ^±_∥ ≈ −Σμ(ω/μ)^{2Δ−1}((ω±μ₃)²−k²)/((ω±μ₃)²+Dk⁴), with Δ = ½+½√(1+r_H²k²/3−r_H²μ₃²/9), which resums low-ω logarithms and capture

What carries the argument

The load-bearing apparatus is the near-extremal matching calculation: the bulk is split into an outer region where the fluctuation equations reduce to radial conservation laws and an inner region near the horizon where they become scalar equations in AdS₂-Schwarzschild, joined by matching the infalling AdS₂ solution to the outer solution. The matching produces the two constitutive constants Σ (conductivity) and D = r_H/2 (diffusivity) from horizon data alone. The second piece is the holographic product formula of reference [23] as applied in reference [11]: it rewrites the spectral function as the KMS-weighted product over poles of the two-sided correlator, which in the T→0 limit converts th

Load-bearing premise

The longitudinal 'extended hydrodynamic' improvement assumes the two-sided correlator has at most finitely many zeroes so the product formula applies — the paper states this assumption in §7 and concedes in §1.1 that a rigorous proof for the longitudinal sector is missing, since its Schrödinger potential is singular in the bulk (appendix J); the transverse channel, whose potential is regular, is not affected.

What would settle it

Compute the exact two-sided longitudinal correlator of the same holographic model at very low T/μ and count its zeroes for momenta up to where the hydro-like pole crosses the AdS₂ levels: an infinite set of zeroes — the known obstruction to the product formula — would invalidate the longitudinal extended approximation and require a modified representation. Independently, the leading-order claim has a sharp kinematic signature: the longitudinal spectral peak and pole must track ω = ∓μ₃ − i(r_H/2)k², shifting linearly with μ₃ at fixed k; a numerical calculation or an analogue measurement of char

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Charged-current spectral functions — the inputs to neutrino-emission and -absorption rates in dense matter — become two-constant predictions (Σ, D) in the near-extremal regime, needing no detailed microscopic input beyond the equation of state.
  • The validity of the hydrodynamic description extends from ω, k ≪ T ≪ μ to T ≪ ω, k ≪ μ for these correlators, so fluid dynamics remains predictive when frequencies and momenta exceed the temperature.
  • A nonzero isospin chemical potential shifts the diffusive pole to ω = ∓μ₃ − iDk² and lowers the infrared scaling dimension to 2Δ(k,μ₃)−1; both effects are confirmed in the quasi-normal-mode spectrum.
  • The extended hydrodynamic approximation improves on plain hydrodynamics mainly near ω = 0, with integrated relative differences at the few-percent level in the low-frequency corner and roughly 15–25% at ω/μ = k/μ = 1/2 for μ_q/T = 65, and it is the simplest among several IR-based approximations of comparable accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same matching logic applied to the neutral-current sector, where charged currents couple to stress-energy and baryon density, would predict coupled diffusive and sound-like modes; one can test whether the extended regime opens there too, which matters for a complete neutrino-opacity picture.
  • Because the longitudinal extended approximation rests on the finite-zeroes assumption, a direct numerical census of the zeroes of the two-sided longitudinal correlator as k increases — including where the hydro-like pole crosses the AdS₂ levels — would either certify or modify formula (7.15); this is the most direct stress test of the paper's advertised improvement.
  • The two-constant structure suggests a parameter-free cross-check: if the same μ, T scan were computed with a different bulk action (for instance with Chern–Simons terms or DBI flavor kinetics added), the value of Σ would change but the functional form of both approximations should persist — a signature of the near-extremal mechanism rather than of the specific model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper studies the two-point functions of the charged (U(1)_{I_3}-rotated) components of the SU(2) flavour currents in a strongly coupled dense holographic medium at finite quark and isospin chemical potentials μ_q and μ₃, described by an AdS_5–Reissner-Nordström black hole in an Einstein–Yang-Mills theory. The non-Abelian hydrodynamic form of the charged-current correlators is derived (Section 2), including the shifted Ward identity k±_μ ⟨J^{μ,±}J^{ν,∓}⟩ and the diffusive mode ω = ∓μ₃ − iDk². The holographic spectral functions are then computed from linearized Yang-Mills fluctuations (Section 4). The central technical result (Section 5) is a matched-asymptotic-expansion calculation in the near-extremal regime ω,k,T,μ₃ ≪ μ, which shows that the spectral functions reduce at leading order to the hydrodynamic forms ImΠ^±_⊥ = −Σω and ImΠ^±_∥ = Σω(k²−(ω±μ₃)²)/((ω±μ₃)²+D²k⁴), with Σ = N_c(Mℓ)³w₀²/(8r_H) and D = r_H/2 (Eqs. 5.20–5.21, 5.30–5.31). This generalizes earlier μ₃ = 0 results and supports the claim that the hydrodynamic regime extends to T ≪ ω,k ≪ μ. The paper further applies the holographic product formula (Section 7) to propose an extended hydrodynamic approximation (Eqs. 7.14–7.15) that resums the low-frequency AdS₂ scaling with a momentum-dependent IR dimension Δ(k,μ₃), and verifies both approximations against exact numerical correlators for μ_q/T ∈ {10⁴,65,5} and μ₃/μ_q ∈ {0,−0.1,−0.5} (Section 8 and Appendices L–O).

Significance. The paper's principal contribution is the parameter-free matching computation of Section 5, which derives the non-Abelian hydrodynamic spectral functions (1.2)–(1.3) with Σ and D given by (5.21) and (5.31) directly from the fluctuation equations: the hydrodynamic functional forms are derived independently in Section 2, so the agreement is not obtained by adjusting any parameter to the correlators. If correct, this establishes at leading order that the hydrodynamic description of charged currents — the input to neutrino transport rates in isospin-asymmetric dense matter — extends into the regime T ≪ ω,k,μ₃ ≪ μ. The numerical programme is extensive and well documented: exact spectral functions, quasi-normal-mode spectra, and integrated coarse- and fine-grained relative-difference tables over three densities and three isospin asymmetries (Appendices L–O). The paper is also transparent about the status of its assumptions, explicitly flagging in §1.1 and §7 that the product formula has not been proven for the longitudinal sector. The main caveat — the conditional status of the extended hydrodynamic approximation — is discussed in Major Comment 1; it does not affect the leading-order mat

major comments (2)
  1. [§7, Eq. (7.5); App. J, Eq. (J.14)] The longitudinal extended-hydrodynamic approximation (7.15) rests on the product formula (7.5), which assumes the longitudinal two-sided correlator has at most finitely many zeroes; the authors state this in §1.1 and §7. This is not a formality: V_∥ is singular where Ω±(r)=0 (Eq. J.14), and for real |ω|<|μ₃| such a point lies in the integration domain, whereas the product formula is proven for regular potentials. A singular potential can generate infinitely many zeroes, in which case (7.5) and (7.15) would fail. The §8 numerics (Tables 5–8) are supportive, but Appendix H shows other IR-based approximations achieve similar accuracy, so they do not select the product-formula mechanism. The §5 matching result is unaffected. Please state the conditional status of (7.15) in the abstract/conclusions, and either analyze the zero structure of the longitudinal two-sided correlator or label (7.15)
  2. [§7, Eqs. (7.11),(7.15); §6.3] The approximation (7.15) keeps only one hydro-like pole pair in its denominator, but the QNM analysis of §6.3 (Figs. 8–9) shows that a second hydro-like pole emerges at finite momentum in each charged correlator; the paper notes after (7.11) that 'there should be four factors in the denominator of (7.11) instead of two'. The missing poles become relevant at k ≃ √(μT), inside the near-extremal regime, so the abstract's wording that the approximation captures 'both hydrodynamic-like poles' is only accurate at leading order. Please qualify (7.15) as the leading-order truncation, with the second hydro-like pole entering at next-to-leading order.
minor comments (5)
  1. [Abstract; §1.1] The conditional status of the longitudinal extended-hydrodynamic result (the assumed product formula) is flagged in §1.1 and §7 but not in the abstract, which presents Eqs. (1.18)–(1.19) as obtained. Suggest adding a sentence in the abstract noting that the longitudinal application of the product formula relies on an unproven assumption.
  2. [Fig. 17 caption] In the caption of Figure 17, the right panel is referred to as '(left panel 17b)'; it should read '(right panel 17b)'.
  3. [§7, Eq. (7.6)] The KMS relation (7.6) uses the shifted time-translation generator K = H − μ₃Q₃. This convention is important for the definition of frequency in the presence of μ₃; the current discussion is adequate, but a one-sentence reminder of the convention where the spectral functions are plotted would help readers.
  4. [Abstract; §1.1] The abstract's statement that 'the traditional regime of validity of standard hydrodynamics extends...' could be read as a statement about fully non-linear hydrodynamics. Section 1 correctly emphasizes that only the linearized two-point functions are established; consider aligning the abstract's wording with this qualification.
  5. [Throughout] Minor typographical/grammatical slips should be corrected in a final pass: e.g., 'The right plots shows a subregion', 'The observables (l)ij is a 4×4 matrix', and the repeated 'we consider' phrasing around Eqs. (8.2)–(8.3).

Circularity Check

0 steps flagged

No significant circularity: the holographic-to-hydrodynamic matching is derived, not fitted; the longitudinal product-formula extension is an admitted unproven assumption, not a circular reduction.

full rationale

The central derivation is self-contained. Non-Abelian hydrodynamic correlators (2.93)-(2.94) are obtained from linearized covariant conservation and constitutive relations, with Σ and D left as undetermined microscopic coefficients. The holographic spectral functions are then computed from the bulk fluctuation equations (4.8) and (4.15), and the leading-order near-extremal results (5.20) and (5.30) are matched to the hydrodynamic forms, giving Σ = N_c(Mℓ)^3 w_0^2/(8 r_H) and D = r_H/2. These constants are derived analytically from the background/horizon data; they are not adjusted to match the numerical correlators. The numerical verification in section 8 uses exact solutions of the full fluctuation equations described in appendix K, so it is an external check of the approximations rather than a re-use of fitted inputs. The extended hydrodynamic approximation (7.14)-(7.15) is based on the product formula of [23] and the zero-temperature IR scaling; its parameters (Σ, D, Δ(k,μ3)) are again computed, not fitted. The only load-bearing caveat is the extension to the longitudinal sector, where the Schrödinger potential is singular at points where Ω±(r)=0 (J.14). The authors state the assumption explicitly in §7: 'we only assume that the longitudinal two-sided correlator has at most a finite number of zeroes', and in §1.1 they acknowledge that 'a rigorous proof that the results of [11] apply to the longitudinal sector in the non-Abelian case would require further investigation.' This is an admitted unproven assumption and a possible correctness gap, not circularity: no equation of the extended hydrodynamic approximation reduces by construction to the correlator it is meant to reproduce, and no parameter is fitted to the predicted quantity. The admitted gap should be weighed as a correctness risk, but it does not raise the circularity score.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The central derivation in §5 is parameter-free: the transport coefficients Σ and D and the IR conformal dimension Δ(k, μ₃) are computed from the holographic model, not fitted to the target correlators, and the numerical 'verification' in §8 compares the approximations against exactly computed correlators. The main ledger entries are (i) model parameters (Mℓ)³ and w₀ fixed by QCD matching in appendix E (prior-literature inputs, not fitted to the present claims); (ii) domain assumptions (RN-phase dominance per [62], no Chern-Simons terms, unbroken U(1)×SU(2)); (iii) one explicit ad hoc assumption — the product formula for the longitudinal charged correlator — flagged by the authors themselves. No invented entities are introduced.

free parameters (2)
  • Bulk couplings (Mℓ)³ and w₀ = (Mℓ)³ = 13/(6·45π²); w₀²(Mℓ)³ = 2/(3π²); numerics use N_c = 3, ℓ = 1, w₀ = 6√(5/13) ≈ 3.72
    Fixed in appendix E by matching the zero-density pressure to the ideal quark-gluon plasma and the baryon susceptibility to the ideal Fermi gas. These are prior-literature inputs setting the overall correlator normalization and r_H·μ; they are not fitted to the target correlators and do not affect the functional-form claims.
  • Scan points μ_q/T and μ₃/μ_q = μ_q/T ∈ {10⁴, 65, 5}; μ₃/μ_q ∈ {0, -0.1, -0.5}
    State parameters and chosen display points, not fit parameters. Included for transparency because the claimed accuracy of the approximations depends on the chosen scan: at μ₃/μ_q = -0.5 the approximations deteriorate even at small ω/μ and k/μ (appendix M.3), which bounds the regime where the abstract's 'agreement' holds quantitatively.
axioms (7)
  • domain assumption AdS/CFT correspondence with the standard holographic dictionary (Son-Starinets infalling prescription) for retarded two-point functions
    Invoked throughout §3-4 (eqs. 4.1-4.27); the entire computation is a holographic one and would be vacuous without the duality.
  • domain assumption Veneziano large-N limit (N_c, N_f → ∞ with N_f/N_c fixed) with classical bulk; toy model with conformal glue, no chiral symmetry breaking, no pions
    §3, appendix E; the paper states the model is a toy model, with V-QCD as the more realistic future direction.
  • domain assumption The U(1)_V × SU(2)_V symmetric AdS₅-RN solution with charges μ_q, μ₃ is the dominant saddle in the studied regime
    §1.1, §3.1; per [62] the ρ-meson (p+ip) superfluid condenses for larger μ₃; restricting to the RN phase bounds the domain of the results.
  • domain assumption No Chern-Simons terms: C_abc = C = β = γ = 0 (no chiral anomalies)
    §2.1, eq. (2.29); stated simplification, with anomalous transport deferred to future work.
  • domain assumption The canonical boundary time-translation is K = H - μ₃Q₃, which fixes the KMS relation (7.6) and the meaning of ω
    §7, near eq. (7.6); a convention, internally consistent with the hydrodynamic results, that makes spectral functions vanish at ω = 0.
  • ad hoc to paper The product formula of [23] applies to the charged two-sided correlators with at most finitely many zeroes in the longitudinal sector
    §7, eqs. (7.4)-(7.5); explicit assumption — the longitudinal Schrödinger potential is singular (appendix J, eq. J.14) and the authors concede a rigorous proof is outstanding (§1.1). Underlies the extended approximation (7.14)-(7.15).
  • standard math Matched-asymptotics structure: inner AdS₂-Schwarzschild and outer conservation-equation regions with an overlap window (requires k ≪ μ)
    §5, eqs. (5.5)-(5.12); standard boundary-layer matching; the validity windows are derived and checked in the text.

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read the original abstract

Flavor-current correlators are studied in strongly-coupled dense (holographic) matter, at finite quark chemical potential $\mu_q$ and finite isospin asymmetry. The non-Abelian hydrodynamic description of the charged currents is derived in the presence of an isospin chemical potential $\mu_3$. The two-point correlators of charged currents are then computed holographically at finite quark and isospin chemical potentials. In the near-extremal hydrodynamic regime, $\omega, k, T, \mu_3 \ll \mu \equiv \sqrt{\mu_q^2+\mu_3^2}$, relevant for cold strongly coupled matter, the IR properties of the correlators are studied. It is shown that in this regime, the correlators agree with the non-Abelian hydrodynamic predictions. Therefore, the traditional regime of validity of standard hydrodynamics extends beyond $\omega, k \ll T \ll \mu$ to the so-called extended hydrodynamic regime $T\ll \omega, k \ll \mu$. The holographic product formula is applied to the present non-Abelian system, and is used to propose an extended hydrodynamic approximation capturing both hydrodynamic-like poles and the leading effect of AdS$_2$ poles, by resumming the low-$\omega$ logarithms. The results are verified through a detailed numerical analysis of the exact correlators and quasi-normal mode spectrum.

Figures

Figures reproduced from arXiv: 2607.20991 by Andrea Olzi, Edwan Pr\'eau, Elias Kiritsis, Francesco Nitti, Matti J\"arvinen, Thomas Apostolidis.

Figure 1
Figure 1. Figure 1: Lowest lying transverse QNMs at fixed k = 0 and µ3/µq = 0.1, for four different values of µq/T. From top-left to bottom-right: µq/T = 0, 2, 10, 100. The frequency is measured in units of 2πT. – 42 – [PITH_FULL_IMAGE:figures/full_fig_p043_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Real part of the first five transverse poles as a function of the ratio of [PITH_FULL_IMAGE:figures/full_fig_p044_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative difference of the real part of the first transverse pole in units of [PITH_FULL_IMAGE:figures/full_fig_p045_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Imaginary parts of the nearly imaginary transverse poles (which become [PITH_FULL_IMAGE:figures/full_fig_p046_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Real parts of the first few poles of figure [PITH_FULL_IMAGE:figures/full_fig_p046_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: For µq/T = 5 and µ3/µq = 0.1, imaginary (left) and real (right) parts of the first few transverse poles as a function of momentum. -10 -5 0 5 10 -10 -8 -6 -4 -2 0 Re ω/2πT Im ω/ 2 πT -10 -5 0 5 10 -10 -8 -6 -4 -2 0 Re ω/2πT Im ω/ 2 πT -10 -5 0 5 10 -10 -8 -6 -4 -2 0 Re ω/2πT Im ω/ 2 πT -10 -5 0 5 10 -10 -8 -6 -4 -2 0 Re ω/2πT Im ω/ 2 πT [PITH_FULL_IMAGE:figures/full_fig_p047_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lowest lying longitudinal QNMs at fixed rHk 2 = 2πT and µ3/µq = 0.1, for four different values of µq/T. From top-left to bottom-right: µq/T = 0, 2, 10, 100. The frequency is measured in units of 2πT. The orange dashed lines show the location of −µ3/(2πT) on the real axis. The general pole structure in this case is shown in figure 7, for different values of µq/T and at fixed rHk 2 = 2πT 24 and µ3/µq = 0.1. … view at source ↗
Figure 8
Figure 8. Figure 8: Imaginary part of the first seven longitudinal QNMs as a function of [PITH_FULL_IMAGE:figures/full_fig_p048_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Real part of the first seven longitudinal QNMs as a function of momentum [PITH_FULL_IMAGE:figures/full_fig_p049_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Real part of the diffusive pole at µq/T = 300 and k = 0, as a function of µ3/µq. Blue points are numerical results, whereas the orange line corresponds to the linear hydrodynamic prediction. that increase with momentum. Even after the hydrodynamic-like poles leave the imaginary axis, they still displace the IR poles each time their (common) imaginary – 48 – [PITH_FULL_IMAGE:figures/full_fig_p049_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Real parts of the first six longitudinal QNMs with positive real parts, [PITH_FULL_IMAGE:figures/full_fig_p051_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Imaginary parts of the first few longitudinal QNMs as a function of [PITH_FULL_IMAGE:figures/full_fig_p052_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Real parts of the poles shown in figure 12. The orange dashed line shows the leading-order real part of the first hydrodynamic pole (6.7). For µq/T = 300 (right), the AdS2 poles have a finite positive real part of the order of (6.5), but it is too small to observe on the scale of the figure. We now focus on the hydrodynamic-like modes, and denote by n2 the AdS2 level that the second pole starts from. We t… view at source ↗
Figure 14
Figure 14. Figure 14: In the left panel 14a, we plot the imaginary part of the transverse charged current retarded polarization functions at zero isospin chemical potential for µq/T = 65 (the ± notation has been suppressed since we are working at µ3 = 0). The energy and momentum are expressed in units of µ, and the polarization function is normalized by −Σ/rH. The right panel 14b shows the plot of the relative difference 1 − I… view at source ↗
Figure 15
Figure 15. Figure 15: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 65 and µ3 = 0. The green line shows the locus ω = k. The right plots shows a subregion of the left plots. The dashed square in these right plots re… view at source ↗
Figure 16
Figure 16. Figure 16: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row) for µq/T = 65 and µ3/µq = −0.1. The green line describes the locus ω = k +µ3. The right plots shows a subregion of the left plots. The dashed square in these r… view at source ↗
Figure 17
Figure 17. Figure 17: Imaginary part of the longitudinal charged current retarded polarization [PITH_FULL_IMAGE:figures/full_fig_p067_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The percentage relative difference (8.1) of the imaginary part of the longi￾tudinal charged current polarization function with respect to the hydrodynamic ap￾proximation (7.13) (top row) and the extended hydrodynamic approximation (7.15) (bottom row) for µq/T = 65 and µ3 = 0. The green line describes the locus ω = k. The right plots shows a subregion of the left plots. The dashed square in these right plo… view at source ↗
Figure 19
Figure 19. Figure 19: The percentage difference (8.1) of the imaginary part of the longitudinal charged current polarization function with respect to the hydrodynamic approxima￾tion (7.13) (top row) and the extended hydrodynamic approximation (7.15) (bottom row) for µq/T = 65 and µ3/µq = −0.1. The green line describes the locus ω = k+µ3. The right plots shows a subregion of the left plots. The dashed square in these right plot… view at source ↗
Figure 20
Figure 20. Figure 20: The µ3 = 0 ratio ϱ0 (H.11) as a function of k/µ for µq/T = 104 (panel 20a), for µq/T = 65 (panel 20b), and for µq/T = 5 (panel 20c). row), for µq/T = 65 (middle row) and µq/T = 5 (bottom row). The plots in figure 23 show that raising µq/T the ratio ω 2∆−1/(ImG ±,norm. IR ) approaches 1 for smaller and smaller values of ω/µ, and k/µ. Now, we present several ways of approximating the imaginary part of the t… view at source ↗
Figure 21
Figure 21. Figure 21: The ratio ϱ + at k = 0 (H.13) as a function of µ3/µ, for µq/T = 104 (panel 21a), for µq/T = 65 (panel 21b), and for µq/T = 5 (panel 21c). ImΠ∥,± hydro = −Σ ω ω 2 − ⃗k 2 (ω ± µ3) 2 + D ⃗k 4 . (H.17) • Extended hydrodynamic approximation. ImΠ⊥,± ext-hydro = −Σ µ(ω/µ) 2∆−1 , (H.18) ImΠ∥,± ext-hydro = −Σ µ(ω/µ) 2∆−1 ω 2 − ⃗k 2 (ω ± µ3) 2 + D ⃗k 4 . (H.19) • Improved extended hydrodynamic approximation. ImΠ⊥,±… view at source ↗
Figure 22
Figure 22. Figure 22: The ratio ϱ + (H.10) at µ3/µq = 0.2 as a function of k/µ for µq/T = 104 (panel 22a), for µq/T = 65 (panel 22b), and for µq/T = 5 (panel 22c). • Approximation using the normalized IR-AdS2 correlator (H.14). ImΠ⊥,± norm. = −Σ µ 2∆−2 ImG ±,norm. IR , (H.22) ImΠ∥,± norm. = −Σ µ 2∆−2 ImG ±,norm. IR ω 2 − ⃗k 2 (ω ± µ3) 2 + D ⃗k 4 , (H.23) where the expression for G norm. IR is given in (H.14). • Approximation u… view at source ↗
Figure 23
Figure 23. Figure 23: The ratio ω 2∆0(k)−1/ImG norm. IR where ω and k are in units of µ. We plot this ratio as a function of ω. The top row shows the results for µq/T = 104 and µ3 = 0. The middle row shows the results for µq/T = 65 and µ3 = 0. The bottom row shows the results for µq/T = 5 and µ3 = 0. For each row, the left and right panels are associated with k/µ = 0.1 and k/µ = 1, respectively. The ± notation is suppressed si… view at source ↗
Figure 24
Figure 24. Figure 24: The imaginary part of the full-IR correlator ( [PITH_FULL_IMAGE:figures/full_fig_p092_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The relative difference (H.26). This is defined as 100 times the absolute value of the difference between the imaginary part of the full IR-AdS2 correlator and the hydrodynamic approximation (panel 25a) the extended hydrodynamic ap￾proximation (panel 25b) the improved extended hydrodynamic approximation (panel 25c) and the imaginary part of the normalized IR-AdS2 correlator (panel 25d). The background is … view at source ↗
Figure 26
Figure 26. Figure 26: The relative difference (H.26). This is defined as 100 times the absolute value of the difference between the imaginary part of the full IR-AdS2 correlator and the hydrodynamic approximation (panel 26a) the extended hydrodynamic ap￾proximation (panel 26b) the improved extended hydrodynamic approximation (panel 26c) the imaginary part of the normalized IR-AdS2 correlator (panel 26d). The back￾ground is cha… view at source ↗
Figure 27
Figure 27. Figure 27: The relative difference (H.26). This is defined as 100 times the absolute value of the difference between the imaginary part of the full IR-AdS2 correlator and the hydrodynamic approximation (panel 27a) the extended hydrodynamic ap￾proximation (panel 27b) the improved extended hydrodynamic approximation (panel 27c) the imaginary part of the normalized IR-AdS2 correlator (panel 27d). The back￾ground is cha… view at source ↗
Figure 28
Figure 28. Figure 28: The percentage relative difference (8.1) of the imaginary part of the transverse charged current polarization function with respect to the improved ex￾tended hydrodynamic approximation (H.20), for µq/T = 65 and µ3 = 0. The green line shows the locus ω = k. The right plot shows a subregion of the left plot. The dashed square in this right plot represents the standard hydrodynamic region ω/µ, k/µ ∈ [0, T/µ]… view at source ↗
Figure 29
Figure 29. Figure 29: As figure 28 but for the normalized correlator (H.22) instead of the improved extended hydrodynamic approximation [PITH_FULL_IMAGE:figures/full_fig_p109_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: As in figure 28 but for the full correlator (H.24) instead of the improved extended hydrodynamic approximation. (H.20), the approximation using the normalized IR-AdS2 correlator (H.22), and the approximation using the full IR-AdS2 correlator (H.24) respectively. We can confront these plots with the ones presented in section 8 regarding the hydrodynamic (top row of figure 15) and extended hydrodynamic (bot… view at source ↗
Figure 31
Figure 31. Figure 31: The percentage relative difference (8.1) of the imaginary part of the transverse charged current polarization function with respect to the improved ex￾tended hydrodynamic approximation (H.21), for µq/T = 65 and µ3 = 0. The green line shows the locus ω = k. The right plot shows a subregion of the left plot. The dashed square in this right plot represents the standard hydrodynamic region ω/µ, k/µ ∈ [0, T/µ]… view at source ↗
Figure 32
Figure 32. Figure 32: As figure 31 but for the normalized correlator (H.22) instead of the improved extended hydrodynamic approximation. correlator (H.25) respectively. We can confront these plots with the ones presented in section 8 regarding the hydrodynamic (top row of figure 18) and extended hydrodynamic (bottom row of figure 18) approximation. As for the transverse case, all the IR-based approximation presented in this se… view at source ↗
Figure 33
Figure 33. Figure 33: As figure 31 but for the full correlator (H.25) instead of the improved extended hydrodynamic approximation. (a) hydrodynamic 1/2 0.09 0.09 0.12 0.16 3/8 0.04 0.07 0.11 0.15 1/4 0.03 0.06 0.10 0.14 1/8 0.02 0.06 0.10 0.14 1/8 1/4 3/8 1/2 (b) extended-hydrodynamic 1/2 0.04 0.08 0.12 0.16 3/8 0.04 0.08 0.12 0.15 1/4 0.03 0.07 0.11 0.14 1/8 0.02 0.06 0.10 0.14 1/8 1/4 3/8 1/2 (c) improved 1/2 0.04 0.08 0.12 … view at source ↗
Figure 34
Figure 34. Figure 34: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p117_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 104 and zero µ3. The right plots shows a subregion of the left plots. The green line shows the locus ω = k. The black dot in bottom-right corner of… view at source ↗
Figure 36
Figure 36. Figure 36: As figure 35 but for the longitudinal polarization function instead of the transverse polarization function. The extended hydrodynamic approximation improves the low-frequency region: the region where the error is around 70% at ω ≃ 0 is now pushed to the top-left corner of the plot k/µ ≃ 1 (see the bottom-left plot). Also the 10% and 20% regions cover a much wider range, close to ω ≃ 0. At larger frequenc… view at source ↗
Figure 37
Figure 37. Figure 37: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p120_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 5 and zero µ3. The right plots shows a subregion of the left plots. The green line shows the locus ω = k. Longitudinal correlator In figure 39, we … view at source ↗
Figure 39
Figure 39. Figure 39: As figure 38 but for the longitudinal polarization function instead of the transverse polarization function. M.2 Numerical results at µ3/µq = −0.1 In this section, we present the results for the imaginary part of the charged cur￾rent polarization function at finite isospin chemical potential µ3/µq = −0.1 for µq/T ∈ {104 , 5}. First, we present the plots for the imaginary part of the polar￾ization function… view at source ↗
Figure 40
Figure 40. Figure 40: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p123_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 5 and µ3/µq = −0.1. The right plots shows a subregion of the left plots. The green line shows the locus ω = k + µ3. The longitudinal correlator In … view at source ↗
Figure 42
Figure 42. Figure 42: As figure 41 but for the longitudinal polarization function instead of the transverse polarization function. The effect of finite value of µ3 is also visible in the zoomed plots (right column): the contours shown are now at the 2%–4% level rather than at the sub-percent level, so the IR accuracy is already noticeably worse than at µ3 = 0 (see figure 36). In these plots, the extended hydrodynamic approxima… view at source ↗
Figure 43
Figure 43. Figure 43: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p126_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 5 and µ3/µq = −0.1. The right plots shows a subregion of the left plots. The green line shows the locus ω = k + µ3. In the zoomed plots (right colu… view at source ↗
Figure 45
Figure 45. Figure 45: As figure 44 but for the longitudinal polarization function instead of the transverse polarization function. M.3 Numerical results at µ3/µq = −0.5 In this section, we present the results for the imaginary part of the charged current polarization function at non-zero isospin chemical potential µ3/µq = −0.5 for µq/T ∈ {104 , 65, 5}. First, we present the plots for the imaginary part of the polarization func… view at source ↗
Figure 46
Figure 46. Figure 46: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p129_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 5 and µ3 = −0.5. The right plots shows a subregion of the left plots. The green line shows the locus ω = k + µ3. The extended approximation still i… view at source ↗
Figure 48
Figure 48. Figure 48: As figure 47 but for the longitudinal polarization function instead of the transverse polarization function. In figure 48, the longitudinal sector also shows a strong breakdown of the simple hydrodynamic picture. In the top-left plot, associated with the hydrodynamic ap￾proximation, the error is not monotonic in either variable. There is a low-error valley around ω/µ ≃ 0.6–0.8 and k/µ ≲ 0.3, but the error… view at source ↗
Figure 49
Figure 49. Figure 49: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p132_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 65 and µ3 = −0.5. The right plots shows a subregion of the left plots. The dashed square in the right column represents the hydrodynamic region ω/µ… view at source ↗
Figure 51
Figure 51. Figure 51: As figure 50 but for the longitudinal polarization function instead of the transverse polarization function. The longitudinal correlator [PITH_FULL_IMAGE:figures/full_fig_p134_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: Imaginary part of the transverse (left panel) and longitudinal (right panel) [PITH_FULL_IMAGE:figures/full_fig_p135_52.png] view at source ↗
Figure 53
Figure 53. Figure 53: The percentage relative difference (8.1) of the imaginary part of the trans￾verse charged current polarization function with respect to the hydrodynamic ap￾proximation (7.12) (top row) and the extended hydrodynamic approximation (7.14) (bottom row), for µq/T = 5 and µ3 = −0.5. The right plots shows a subregion of the left plots. The green line shows the locus ω = k + µ3. The longitudinal correlator In fig… view at source ↗
Figure 54
Figure 54. Figure 54: As figure 53 but for the longitudinal polarization function instead of the transverse polarization function. – 136 – [PITH_FULL_IMAGE:figures/full_fig_p137_54.png] view at source ↗

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