Investigating effects of the electrical conductivity of QCD matter on charge-dependent directed flow

Chiho Nonaka; Hiroyuki R. Takahashi; Nicholas J. Benoit; Takahiro Miyoshi

REVIEW 2 major objections 4 minor 51 references

Investigating effects of the electrical conductivity of QCD matter on charge-dependent directed flow

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that the proton-antiproton directed-flow difference in symmetric heavy-ion collisions is a conductivity-sensitive observable, and that an initial positive charge can reverse its slope.

desk verdict A useful RRMHD extension to proton/antiproton Δv1 with a centrality scan, but the σ-scan anchoring the main claim is contaminated by an inconsistent EM-field initialization. read the letter →

arxiv 2502.04611 v2 pith:BQOTSRYN submitted 2025-02-07 nucl-th hep-exhep-phnucl-ex

Nicholas J. Benoit , Takahiro Miyoshi , Chiho Nonaka , Hiroyuki R. Takahashi This is my paper · ORCID

classification nucl-thhep-exhep-phnucl-ex

keywords relativisticresistivemagnetohydrodynamicscharge-dependentdirectedflowelectricconductivityquark-gluonplasmaheavy-ioncollisionselectromagneticfieldsAu+Aufreezeoutchemicalpotential

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the difference in directed flow between protons and antiprotons in symmetric gold-gold collisions at RHIC energies carries information about the electric conductivity of the quark-gluon plasma. Using a 3+1D relativistic resistive magnetohydrodynamic model, it shows that the slope of this charge-dependent flow difference changes systematically with conductivity across collision centralities. The model connects the conductivity to the lifetime of the magnetic field, which drives Faraday and Lorentz currents that leave an electric chemical potential imprinted on the freezeout surface. A positive initial electric charge in the plasma can flip the slope positive, offering an alternative explanation for the sign seen in data.

What carries the argument

The central object is the relativistic resistive magnetohydrodynamic (RRMHD) evolution coupled to Ohm's law $J^\mu = q u^\mu + \sigma F^{\mu\nu}u_\nu$. The conductivity $\sigma$ controls how the magnetic field decays like a non-linearly damped oscillator, and the resulting Lorentz and Faraday currents accumulate an electric chemical potential $\mu_Q$ on the freezeout hypersurface. The Cooper-Frye formula then converts $\mu_Q$ into a charge-dependent split of proton and antiproton directed flow, whose slope is the observable $\Delta v_1$.

What would settle it

A calculation of pre-equilibrium EM fields with a first-principles kinetic model (e.g., CGC/glasma) that gives fields an order of magnitude smaller than the analytic solution would also reduce the predicted $\Delta v_1$ slopes by roughly an order of magnitude. If that reduced slope falls below the sensitivity of STAR or ALICE data while conductivity values stay at lattice estimates, the paper's quantitative predictions fail; the qualitative conductivity dependence would still be testable in the centrality trend.

Watch

Extended reading notes

Core claim

The paper's central claim is that the slope of $\Delta v_1 = v_1^+ - v_1^-$ for protons and antiprotons in symmetric Au+Au collisions at $\sqrt{s_{NN}}=200$ GeV is sensitive to the electric conductivity $\sigma$ of the quark-gluon plasma. In the RRMHD model, a finite conductivity controls the decay of the magnetic field; larger $\sigma$ prolongs the field, strengthens the Lorentz and Faraday currents, and produces a larger negative slope of $\Delta v_1$ across centralities. The paper also shows that if the plasma initially carries a net positive charge, the slope can become positive, so the sign of the measured slope need not come solely from transported baryon charge.

Load-bearing premise

The initial electromagnetic fields are taken from an analytic in-medium solution that the paper itself notes overestimates the field strength, and reducing that input to 10% shrinks the predicted flow slope by an order of magnitude, so the quantitative size (and the sign-flip demonstration) rests on a known-overestimated input.

Editorial extensions

If this is right

The slope of proton-antiproton $\Delta v_1$ is a probe of QGP electric conductivity even in symmetric collisions, not only asymmetric ones.
Larger conductivity implies longer-lived magnetic fields and larger negative slopes of $\Delta v_1$, with Faraday current dominating Lorentz current.
An initial net positive electric charge in the plasma can reverse the slope to positive, providing an alternative to transported-baryon explanations of STAR data.
Conductivity dependence varies non-trivially with centrality, with lattice-like $\sigma$ showing the least change across centralities.
The result is qualitative; matching data quantitatively requires better initial conditions, a $\mu_Q$-dependent equation of state, and an afterburner.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the sign of the slope can be flipped by the initial charge density, measurements of $\Delta v_1$ slope alone cannot uniquely fix $\sigma$ without independent control of the initial charge distribution; joint fits to centrality trends might separate the two.
The same mechanism should generate charge-dependent flow for other identified hadrons (kaons, pions) with magnitudes set by their electric charge, which could be tested by comparing STAR and ALICE data across energies.
If the conductivity is shown to be tensor-valued in strong magnetic fields, the predicted $\Delta v_1$ slope could develop a dependence on the orientation of the field relative to the reaction plane, an effect this scalar-conductivity model cannot produce.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper's new pieces are genuine: proton/antiproton Δv1 in symmetric Au+Au, a centrality scan, and a first pass at an initial positive charge density. The model setup and freezeout are described carefully, and the authors are open about their simplifications. But the central claim—that Δv1 is sensitive to σ—rests on a comparison that is not cleanly defined. The initial EM fields come from the in-medium analytic solution of Eq. A1, which depends on σ, while the text says the earliest points in Fig. 1 are equal because the initial conditions are the same. Those two statements are in tension. At b=10 fm and τ0=0.4 fm/c, the analytic By changes by roughly seven orders of magnitude when σ goes from 0.0294 to 0.294 fm⁻¹. If the code uses a common initial field for all runs, the high-σ runs start with a field far above the physical value for that conductivity, which inflates the field lifetime and the resulting slope. If it uses σ-dependent fields, the earliest points would not be equal. The paper does not say which. Since Appendix C shows the slope scales roughly with the initial field strength, this ambiguity means Fig. 6 does not isolate the dynamical role of σ. That is a load-bearing problem, not a footnote.

The rest of the paper is more solid. The µQ-on-freezeout mechanism is laid out clearly, and the negative-slope result is consistent with Ref. [15]. The five-significant-figure slopes in Eqs. 25 and 26 are overpressed given the model's known sensitivity, and the 'first work' phrasing in Sec. VII is a bit strong next to Refs. [15,21]. Both are minor next to the initialization issue.

This is worth refereeing. The model extension to protons and the centrality pattern are potentially useful for the EM-field-in-heavy-ion community. But the referee should ask the authors to state explicitly what initial fields they used for each σ and, if they used a common field, to justify it physically or rerun with σ-consistent initial fields. Without that, the σ-sensitivity claim is not established. I'd bring it to a reading group as a case study in how a simulation comparison can be quietly contaminated by initial conditions.

Referee Report

2 major / 4 minor

Summary. The paper develops a 3+1D relativistic resistive magnetohydrodynamic (RRMHD) model for Au+Au collisions at sqrt(s_NN)=200 GeV. It initializes the QGP with an optical Glauber profile and initial electromagnetic fields from Tuchin's analytic in-medium solution, evolves the system with a constant scalar conductivity, and computes directed flow via Cooper-Frye freezeout with a finite electric chemical potential on the hypersurface. The central results are: (i) the time evolution of the magnetic field is sensitive to sigma, with larger sigma giving longer field lifetimes; (ii) the slope d(delta v1)/dy for protons and antiprotons is negative and grows in magnitude with sigma; (iii) the slope decreases toward central collisions; and (iv) adding an initial positive electric charge density (Zq=0.4) can flip the slope positive, as suggested in Ref. [15]. The authors explicitly list simplifying assumptions: constant scalar sigma, ideal equation of state, overestimated initial fields, no spectator currents during the evolution, and no afterburner.

Significance. The paper addresses an active experimental question and provides a concrete mechanism connecting the electric conductivity of QGP to a charge-dependent flow observable. Its strengths are the honest treatment of limitations, the consistency of the negative-slope sign with Ref. [15], and the explicit scaling test in Appendix C. If the conductivity-sensitivity claim were quantitatively robust, the model would yield falsifiable centrality trends that can be compared with STAR and ALICE data. However, because the slopes are roughly proportional to the initial field strength and the sigma-scan uses an ambiguous initialization convention, the present results establish a proof of mechanism rather than a quantitative extraction. The sign-flip demonstration with an initial charge density is illustrative, not a quantitative prediction.

major comments (2)

[Sec. V, Fig. 1, Appendix A] The controlled sigma-scan is undermined by an unspecified initialization convention for the electromagnetic fields. Equation (A1) makes the initial fields explicitly sigma-dependent through both a prefactor sigma and the exponential exp[-b^2 sigma/(4(t +/- vz))]. At a fixed initial time tau0=0.4 fm/c, the initial By values for different sigma are not equal; for a field point at transverse distance b=10 fm from a single charge, the exponential factor alone changes by about seven orders of magnitude when sigma is increased from 0.0294 to 0.294 fm^-1. The statement in Sec. V that 'we have assumed the initial conditions are the same, the point at the earliest time in Fig. 1 is equal' is therefore either inconsistent with Eq. (A1) or implies that a common reference conductivity was used to generate the initial fields for all runs. The manuscript never states which convention was adopted. If a common reference is used, the high-sigma runs start with a field that is unphysically large for their conductivity, inflating the field lifetime in Fig. 1 and the slopes in Fig. 6; if sigma-dependent initial fields are used, the earliest points in Fig. 1 cannot be equal. This ambiguity directly affects the abstract's claim of sensitivity across centralities, so it must be resolved and the sigma-scan rerun with a stated, physically motivated convention.
[Sec. VI and Appendix C] The quantitative slopes are not yet isolated from the known overestimation of the initial fields. Appendix C demonstrates that reducing the initial field strength to 10% reduces the delta v1 slope by an order of magnitude. Therefore, the statement in Sec. VI that 'an order increase in the electric conductivity sigma brought a similar increase in the slope' does not by itself establish a sigma-specific effect, because the same order-of-magnitude change can be generated by renormalizing the initial fields. The authors acknowledge this degeneracy in Appendix C, but the main-text presentation in Fig. 6 and the abstract should be conditioned on it. A two-dimensional scan in (sigma, initial field strength), or at least a normalization of the initial fields to a common physical reference, is needed before the centrality dependence in Fig. 6 can be attributed to sigma.

minor comments (4)

[Eqs. (11) and (15)] Both equations are written as □M(tau B2), but the surrounding text and Eq. (18) indicate that the third component B3 is intended.
[Fig. 6 caption] The caption spells the model as 'RRHMD'; it should be 'RRMHD'.
[Appendix B] The functions f±(eta), H(eta), and the normalization used in Eq. (B1) are not defined in the text; please provide explicit forms or precise references to Ref. [31].
[Sec. VI] The sentence 'an order increase in the electric conductivity sigma brought a similar increase in the slope' should specify the sigma range and centrality to which it applies, since Fig. 6 does not show a single power-law relation across all centralities.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the sigma-dependence of Delta-v1 is a controlled parameter study within the stated RRMHD equations, not a fitted or self-referential prediction.

full rationale

The paper's central claim is that Delta-v1 between protons and antiprotons responds to the electric conductivity sigma of the QGP. This is a parameter-sensitivity statement inside a specified RRMHD model, not a first-principles derivation of sigma from data. The derivation chain is explicit and open: Ohm's law Eq. (5) defines J^mu = q u^mu + sigma F^{mu nu} u_nu; the wave equations (13)-(18) determine how sigma changes the magnetic-field lifetime; the resulting comoving charge density q is related to the electric chemical potential mu_Q by Eq. (21); and mu_Q enters the Cooper-Frye distribution Eq. (20), producing Delta-v1. Every link is a computed model step; the sigma dependence is inherited from the stated current law, which is a modeling input, but this is not a circular reduction of the output to the input. No parameter is fitted to STAR Delta-v1 data and then renamed as a prediction; the comparison with STAR in Fig. 6 is qualitative. The initial EM fields in Appendix A (from Ref. [25]) do contain sigma and are acknowledged to overestimate the fields, but Sec. V explicitly states that the earliest-time points are equal across the sigma runs, so the Fig. 1 comparison is a controlled dynamical test rather than a normalization artifact. Whether that initialization convention is physically appropriate is a robustness and correctness concern, discussed in Appendix C, but it is not circularity. The self-citations to Refs. [21,22,23] refer to the authors' own code and earlier pion study, but the present proton results are computed in this paper rather than imported, and no uniqueness theorem or unexamined ansatz is smuggled in through self-citation. Therefore no specific reduction of a prediction to its own inputs by construction can be exhibited.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are postulated. The 'initial positive charge density' is a re-use of a standard baryon transport ansatz, not a new entity.

free parameters (3)

QGP electric conductivity σ = scanned: 0, 0.0294, 0.294, 2.94 fm^-1 (constant over spacetime)
Central knob of the study; taken as constant despite expected temperature dependence (Sec III, Sec VI). Not fitted to data, so the quoted Δv1 slopes are conditional on the chosen values.
Initial electric-to-baryon charge ratio Zq = 0.4
Chosen by hand in Appendix B to convert the phenomenological initial baryon density to a proton/charge density; this parameter drives the sign flip of Δ(dv1/dy) in Eq (25).
Freezeout energy density ϵf = 0.15 GeV/fm3
Sets Tf approximately 0.140 GeV and defines the Cooper-Frye hypersurface on which μQ is accumulated; changing it would change the slopes, and no scan is shown.

assumptions (6)

domain assumption QGP behaves as a relativistic ideal fluid described by ideal hydrodynamics plus a first-order Ohm's law with a constant scalar conductivity.
Sec II Eqs (1)-(6). No viscosity, no baryon current, no temperature dependence of σ; the paper acknowledges these limits in Sec VII.
domain assumption Initial EM fields are given by the analytic in-medium Maxwell solution of Ref [25], which overestimates late-time field strengths.
Sec III bullet 2 and Appendix A; overestimation is acknowledged and its effect estimated in Appendix C to be roughly a factor 10 in slope.
domain assumption During evolution, only plasma charges act as EM sources; spectator charges are excluded.
Sec III bullet 3; acknowledged to potentially underestimate late-time fields.
domain assumption Electric chemical potential on the freezeout hypersurface is linearly related to local charge density via μQ = q/(g_h T_f^2) (Eq 21), using a different EoS than during evolution.
Sec IV, Eq (21) and footnote 2; a linear approximation, and the inconsistency between EoSs is explicitly flagged by the authors.
ad hoc to paper Initial electric charge density is modeled by the wounded-nucleon ansatz of Appendix B with Zq = 0.4.
Appendix B; this is a rough phenomenological construction introduced specifically to demonstrate the positive-slope effect, with parameters not constrained by data.
domain assumption The Faraday and Lorentz current decomposition (γBy vs -γuzBy) captures the dominant contributions to Δv1.
Sec V, following Ref [15]; the paper verifies consistency with its own numerical results in Sec VI.

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Pith. "Pith review of Investigating effects of the electrical conductivity of QCD matter on charge-dependent directed flow." pith.science (2026). https://pith.science/paper/BQOTSRYN

@misc{pith2026250204611,
  author       = {Pith},
  title        = {Pith review of: Investigating effects of the electrical conductivity of QCD matter on charge-dependent directed flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BQOTSRYN}},
  note         = {Machine review of arXiv:2502.04611}
}

abstract

Charge dependent directed flow is an important observable of electromagnetic fields in relativistic heavy-ion collisions. We demonstrate how the difference in charge dependent directed flows between protons and antiprotons is sensitive to the resistivity, inverse of quark-gluon plasma's electric conductivity, over different collision centralities. Our model numerically solves the 3+1D relativistic resistive magneto-hydrodynamic (RRMHD) equations, assuming the electric conductivity to be a scalar. For this work, we focus on symmetric Au + Au collisions at the top RHIC energy of $\sqrt{s}=200$ GeV. We illustrate the time evolution of the electromagnetic fields in our model and connect that to the charge dependent directed flow results. Our results highlight the importance of modeling quark-gluon plasma's electric conductivity for charge dependent observables in relativistic heavy-ion collisions.

Figures

Figures reproduced from arXiv: 2502.04611 by the authors.

**Figure 1.** illustrates the impact of σ on the time evolution of By for our RRMHD model of QGP + EM fields. Following the setup in Sec. IV we prepare a peripheral Au-Au collision, with an impact parameter of b = 10 fm, at the collision energy of √ sNN = 200 GeV. The RRMHD simulation was run from τ0 = 0.4 fm/c until the energy density for all grid cells below the freezeout value ϵf = 0.15 GeV/fm3 . Focusing on the center of the… view at source ↗

**Figure 1.** We expect this dependence to be reflected in the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 3.** FIG. 3. Electric chemical potential [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG. 4. Longitudinal dependence of the electric chemical [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Charge dependent directed flow, calculated using our relativistic resistive magneto-hydrodynamic (RRMHD) model [ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The slope of the difference in charge dependent di [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG. 7. Charge dependent directed flow, calculated using [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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