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REVIEW 3 major objections 5 minor 187 references

In physical QCD, the Chiral Separation Effect conductivity is suppressed in the hadronic phase and jumps sharply through the QCD crossover to the free-fermion value, while the Chiral Magnetic Effect conductivity vanishes in thermal equilibr

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2026-08-04 23:11 UTC pith:JS3NL3TJ

load-bearing objection First non-perturbative CSE conductivity at the physical point plus a convincing conserved-current explanation for the vanishing equilibrium CME; the rooting caveat is real but honestly disclosed. the 3 major comments →

arxiv 2509.06712 v1 pith:JS3NL3TJ submitted 2025-09-08 hep-lat

Lattice QCD Study of Anomalous Transport Phenomena in Strongly Interacting Matter

Eduardo Garnacho-Velasco This is my paper
classification hep-lat MSC 81V0581T2581T80 PACS 12.38.Gc11.30.Rd
keywords Chiral Magnetic EffectChiral Separation Effectlattice QCDanomalous transportQCD crossoverBloch's theoremconserved vector currentbackground magnetic fields
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 thesis uses first-principles lattice QCD with 2+1 flavors of staggered quarks at physical masses to settle what the Chiral Separation Effect (CSE) and Chiral Magnetic Effect (CME) do in thermal equilibrium. It reports the first non-perturbative calculation of the CSE conductivity in physical QCD: the conductivity is strongly suppressed at low temperature, rises sharply through the QCD crossover, and reaches the free massless-fermion value at high temperature. For the CME it shows that the conductivity vanishes in global thermal equilibrium, in agreement with Bloch's theorem, provided the electric current is the conserved vector current; earlier non-zero lattice results are traced to the use of a non-conserved local current. It also finds a localized equilibrium CME in inhomogeneous magnetic fields whose volume integral is zero, so it does not contradict Bloch's theorem. If correct, the results give a concrete transport signature of the QCD crossover and sharpen the distinction between real and artifact CME signals.

Core claim

Central claim: in physical QCD with 2+1 flavors, the CSE conductivity (the axial-current response to a baryon chemical potential at vanishing chemical potential) is severely suppressed at low temperature, rises sharply through the QCD crossover, and reaches the massless free-fermion value 1/(2π²) at high temperature — this is the first non-perturbative anomalous-transport conductivity in physical QCD. For the CME, the claim is that the conductivity vanishes in global thermal equilibrium, both for free fermions and in QCD, whenever the current is the conserved vector current, and that previous non-vanishing equilibrium CME results are artifacts of non-conserved local currents. The paper furth

What carries the argument

Bloch's theorem (no globally conserved current flows in the equilibrium ground state in the thermodynamic limit) is the organizing principle: it forbids a global equilibrium CME but not the CSE, whose axial current is not conserved. The technical carrier is the lattice definition of the currents: the conserved, point-split vector current and the anomalous axial current, contrasted with the non-conserved local current responsible for spurious CME signals. CSE and CME coefficients are defined as Taylor coefficients at zero chemical potential, avoiding the sign problem. For staggered fermions, the exponential chemical-potential insertion produces a tadpole term that must be kept for gauge invar

Load-bearing premise

The physical-QCD results rest on the fourth-root trick for staggered fermions — the assumption that the fourth root of the staggered determinant correctly describes one physical quark flavor in the continuum limit — which the author notes lacks a complete proof, and the Wilson-fermion crosschecks cover only free and quenched systems, not full physical QCD.

What would settle it

Simulate the CSE conductivity and the localized CME in 2+1 flavor QCD at physical masses with a discretization that has exact chiral symmetry, such as overlap or domain-wall fermions; if the sharp rise through the crossover or the localized signal changes beyond discretization errors, the rooted-staggered results are not the continuum answer, while agreement would confirm them.

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

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If this is right

  • The CSE conductivity becomes a transport-based thermometer: its inflection point at 136(1) MeV and its half-value temperature at 149(4) MeV offer new, anomaly-based definitions of the QCD crossover.
  • Equilibrium lattice studies of the CME must use the conserved vector current; results obtained with non-conserved local currents are not physical CME signals.
  • Because a localized equilibrium CME with zero total current exists in QCD for inhomogeneous magnetic fields, local measurements in a finite volume can see an equilibrium CME-like signal without contradicting Bloch's theorem.
  • The strong low-temperature CSE suppression implies the Chiral Magnetic Wave is weakened in the hadronic phase and strongest just above the crossover.
  • Previous non-vanishing equilibrium CME results from Wilson-fermion simulations are explained as artifacts of the non-conserved local current, consistent with the repeated calculation using the conserved current.

Where Pith is reading between the lines

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

  • A natural extension is to check whether the sharp CSE rise tracks deconfinement (the Polyakov loop) rather than chiral restoration; the quenched sector-dependence of the CSE hints that center-symmetry dynamics, not the chiral condensate, may be the controlling factor.
  • The conserved-versus-local current lesson likely transfers to condensed-matter lattice studies of Weyl semimetals, where the same ambiguity could generate spurious equilibrium CME conductivity.
  • The localized CME suggests that realistic heavy-ion magnetic-field profiles, which are inhomogeneous and short-lived, could produce local equilibrium currents easily mistaken for the dynamical CME signal; separating the two requires an out-of-equilibrium calculation.
  • Computing the CSE and the localized CME in full QCD with a rooting-independent discretization (overlap or domain-wall fermions) would directly test whether the fourth-root assumption changes the reported values.

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

3 major / 5 minor

Summary. This PhD thesis reports non-perturbative lattice QCD studies of the Chiral Separation Effect (CSE) and the Chiral Magnetic Effect (CME) in thermal equilibrium. Using 2+1 flavor rooted staggered fermions at the physical point, the CSE conductivity is computed across a wide temperature range, finding strong suppression at low T, a sharp rise through the QCD crossover, and approach to the massless free-fermion value at high T. The CME is shown to vanish in global equilibrium when the vector current is conserved, and the non-zero results of earlier Wilson-fermion studies are traced to the use of a non-conserved local current. The thesis also reports a localized equilibrium CME in inhomogeneous magnetic fields whose volume average is zero. The calculations are benchmarked against free-fermion analytic results and quenched Wilson-fermion crosschecks.

Significance. If the results hold, this is the first non-perturbative determination of an anomalous transport conductivity in physical QCD, and it provides a quantitative benchmark for model calculations of the CSE and Chiral Magnetic Wave. The CME result is a clean lattice confirmation of Bloch's theorem and clarifies a long-standing regularization ambiguity in the literature. The localized CME is a novel and falsifiable prediction. The thesis is strong in its use of analytic free-fermion benchmarks, in measuring rather than fitting the target observables, and in the transparency of its error analysis. The main limitation is that the full-QCD results rest on the unproven rooting trick for staggered fermions and on the omission of the multiplicative axial renormalization, so independent validation of the singlet axial current in the interacting theory remains an open load-bearing point.

major comments (3)
  1. [Sec. 2.6.2, Secs. 4.3.3 and 6.4] The central full-QCD results are obtained with rooted staggered fermions, while Sec. 2.6.2 states that rooting 'has a priori no reason to work' and that a complete proof is lacking. The promised Wilson crosschecks in Chs. 4 and 5 cover only free and quenched systems, not dynamical QCD. The singlet axial current is precisely where rooting is most fragile, because taste-breaking and the anomaly structure are entangled at finite lattice spacing. I recommend adding a dynamical Wilson or overlap crosscheck for at least one temperature, or a direct test of the rooted staggered singlet axial Ward identity, or a quantitative estimate of rooting systematics (e.g., comparing 4-flavor and rooted 1-flavor results at finite lattice spacing).
  2. [Sec. 4.3.3, footnote; Sec. 5.2] The singlet axial current requires multiplicative renormalization Z_A. The thesis argues that Z_A approaches unity in the continuum limit and therefore omits it. However, the continuum extrapolation is performed from finite-Nt ensembles (Nt=6,8,10,12), and if Z_A differs from unity by O(g^4) at these lattice spacings, the continuum value of C_CSE could be shifted, particularly the claim that the high-T value reaches the free-fermion result. Please report Z_A, or a perturbative estimate with a systematic uncertainty, and assess the effect on the CSE continuum band and on the localized CME signal.
  3. [Ch. 6 (Sec. 6.4)] The localized equilibrium CME is a novel result, but its physical interpretation depends on the choice of lattice axial-current operator. The thesis shows for free fermions that different staggered definitions (3-link vs 5-link, with/without U(1) links) have the same continuum limit, but this equivalence is not demonstrated in full QCD. I recommend repeating the localized measurement with the alternative staggered operator at least at one magnetic-field profile, or otherwise showing that the volume-averaged and local signals are insensitive to the operator definition in the interacting theory. This would considerably strengthen the claim that the localized effect is a real equilibrium QCD phenomenon and not a lattice-operator artifact.
minor comments (5)
  1. [Sec. 4.3.3] Two definitions of the crossover temperature are quoted: the inflection point Tc=136(1) MeV and the half-value Tc=149(4) MeV. Since these differ by more than the quoted errors, please clarify which definition is recommended for comparison with other observables or with model calculations.
  2. [Fig. 4.2] The 'local' Wilson data points show a divergent continuum limit. Please add an explicit note in the caption or legend that this refers to the local vector current of Eq. (4.22), not to a local axial current, to avoid confusion.
  3. [Eqs. (4.10)-(4.13)] The relation between the 3-link current Γ_ν5 and the 5-link current Γ_ν Γ_5 is introduced only later. It would be clearer to define both operators at the same point and state explicitly that they are not identical at finite lattice spacing but share the same continuum limit.
  4. [Eq. (4.16)] The notation for the disconnected terms uses the same angle brackets for ensemble averages and for the factorized products. This is understandable but worth a remark in the text, especially because the tadpole term is the least familiar contribution.
  5. [Ch. 6] For the localized CME, a figure showing the spatial profile of the current alongside the volume-averaged value would be helpful. If such a figure already exists, please make the sign convention for the inhomogeneous field explicit in the caption.

Circularity Check

0 steps flagged

No significant circularity: results are measured and cross-checked against external benchmarks; self-citations and rooting are validation issues, not circular inputs.

full rationale

The thesis's derivation chain is self-contained in the relevant sense. The central CSE result (Ch. 4) is obtained by measuring the axial-current derivative eq. (4.16) on lattice ensembles and extracting CCSE from the slope versus eB (Fig. 4.1); it is then compared to the independent free-fermion formula (3.72) and to Wilson-fermion crosschecks in free and quenched systems (Secs. 4.2.2, 4.3.1, 4.3.2). No parameter is fitted to the target value and no observable is defined in terms of the claimed outcome. The CME result (Ch. 5) likewise follows from expectation values of the conserved vector current, with the non-zero local-current signal in the literature explicitly traced to the use of a non-conserved local current—an explanation, not a definitional equivalence. Bloch's theorem is an external input used as motivation, and the lattice measurement confirms rather than assumes it. The localized CME (Ch. 6) is a spatial profile measured in an inhomogeneous background; its zero volume average is consistent with, but not imposed by, the conserved-current Ward identity. The self-citations to Refs. [152] and [138] indicate the source papers of the presented data/methods and are not load-bearing for the physical argument. The main limitation—rooted staggered fermions for the full-QCD and localized-CME results (Sec. 2.6.2), with Wilson crosschecks only free/quenched—is an acknowledged correctness/validation gap, not a circular step: the rooting trick is an assumption about the lattice discretization, not a re-labelling of the target observable. Therefore no circularity is present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central results depend on standard lattice QCD plus the unproven rooting trick, the assumed validity of Bloch's theorem in QCD, and the continuum extrapolation of the staggered axial current. No new particles or free parameters are introduced; the only fitted quantity is the measured conductivity itself.

axioms (4)
  • domain assumption Rooting trick for staggered fermions: the fourth root of the staggered fermion determinant describes one physical fermion flavor in the continuum limit.
    Used for all full-QCD results (Sec. 4.2.1, Eq. 4.8); no complete proof exists, author acknowledges this in Sec. 2.6.2.
  • domain assumption Bloch's theorem generalizes to interacting QCD: a global conserved current vanishes in the equilibrium ground state.
    Central to the CME zero claim in Ch. 5; cited to Yamamoto [134], assumed to hold for QCD.
  • domain assumption The 3-link staggered axial current (Eq. 4.11) has the correct continuum limit for anomalous transport coefficients.
    Verified for free fermions in Sec. 4.3.1, but assumed to extend to interacting QCD.
  • standard math Standard lattice QCD framework: path integral, Wilson gauge action, Symanzik improvement, continuum limit via asymptotic freedom.
    Background for all simulations, Sec. 2.

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Cite this review

Pith. "Pith review of Lattice QCD Study of Anomalous Transport Phenomena in Strongly Interacting Matter." pith.science (2026). https://pith.science/paper/JS3NL3TJ

@misc{pith2026250906712,
  author       = {Pith},
  title        = {Pith review of: Lattice QCD Study of Anomalous Transport Phenomena in Strongly Interacting Matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JS3NL3TJ}},
  note         = {Machine review of arXiv:2509.06712}
}
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In this thesis, we study the Chiral Magnetic Effect (CME) and the Chiral Separation Effect (CSE) using lattice QCD simulations. We completely characterize the CSE in QCD using $2+1$ simulations of staggered quarks tuned at the physical point. We find that the CSE conductivity is severely suppressed at low temperatures, and experiences a very sharp increase around the QCD crossover transition, reaching the value of massless non-interacting fermions at high temperatures. This is the first non-perturbative calculation of an anomalous transport conductivity in physical QCD. In the case of the CME, we show that this effect is not present in thermal equilibrium, in accordance with Bloch's theorem, both for free fermions and in QCD. We emphasize the crucial role of using a conserved vector current in the study of anomalous transport phenomena on the lattice, and how the use of non-conserved currents is behind non-vanishing results for the CME in equilibrium that can be found in the literature. Finally, we study the interplay between the CME and inhomogeneous magnetic fields, which leads us to find a novel localized CME signal in equilibrium. This does not contradict Bloch's theorem, since the total signal averages to zero when summed over the full volume. This effect is present in physical QCD, showing that a local equilibrium CME signal is possible in the presence of gluonic interactions.

Figures

Figures reproduced from arXiv: 2509.06712 by Eduardo Garnacho-Velasco.

Figure 2.1
Figure 2.1. Figure 2.1: Division of the Brillouin zone in two dimensions according to the areas defined in Eqs. (2.25), (2.26). and its dimensionless version πˆg = aπg. We can now consider the following regions in momentum space Ig =  k ; k = (k∅ + πg) mod 2π a , k∅ ∈ I∅  , (2.25) I∅ = n k ; − π 2a < kµ ≤ π 2a , ∀µ o , (2.26) such as the whole Brillouin zone can be decomposed as BZ = [ g∈G Ig . (2.27) Since this construction … view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Graphical representation of a plaquette Pµν, a 2 × 1 rectangle Rµν and a staple S 1 µν (see Sec. 2.6.2) on the lattice. for each individual spinor ψ(n). Since the fermionic action is a bilinear in the fields, we can perform the fermionic path integral analytically. Recalling that ψ(n), ψ¯(n) are Grassmann variables, the result of the integration yields the fermion determinant Z Dψ Dψ e ¯ −SF = det M . (2… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Illustration of the Chiral Magnetic Effect for a system of u and d quarks. The chiral imbalance is depicted by a chiral chemical potential µ5, which is taken to be positive so the number of right-handed particles is enhanced. When a magnetic field is present, particles with different charges get separated creating the CME. 3.3.1 CME with non-interacting fermions The CME can be expressed in terms of its c… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Feynman diagrams that can be used to calculate CCME in 1-loop pertur￾bation theory (free fermions). we refer to Ref. [138]. The calculation of the unregularized diagram yields the following integral CCME = 1 4π 3 Z ∞ −∞ dp3 dp4 m2 + p 2 4 − p 2 3 (m2 + p 2 4 + p 2 3 ) 2 . (3.54) It can be separated as CCME = (fm + fp), (3.55) with fm = 1 4π 3 Z ∞ −∞ dp3 dp4 m2 s (m2 s + p 2 4 + p 2 3 ) 2 , (3.56) and fp … view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: (Left): CME current at finite and zero µ5 as a function of the magnetic field. The slope of the linear fit of the finite µ5 points, rescalated by the appropriate factors, corresponds to CCME. The vector current used is the local version. (Right): Value of the CCME as a function of the lattice spacing. The continuum limit was estimated to be in the range [0.02, 0.03]. Figures taken from Ref. [117]. curren… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Analytic result (3.68) for the axial susceptibility χ5 for non-interacting fermions as a function of m/T. Figure from Ref. [138]. possible to question whether the non-conserved nature of the axial charge allows µ5 to control the axial imbalance. We can prove this, both analytically and on the lattice. We focus on the linear term of the Taylor expansion of the axial density N5 in the chiral chemical poten… view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Illustration of the Chiral Separation Effect, now considering a system of u and u¯ quarks. The quark chemical potential µ > 0 creates an imbalance of particles, which combine with the magnetic field to create the CSE. 3.4.1 CSE with free fermions The CSE conductivity can be defined as J35 = σCSE(µ) eB3 + O(B 3 ). (3.69) Analogously as for the CME, the conductivity can then be expanded in a Taylor series … view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Diagrams that can be used to obtain CCSE in 1-loop perturbation theory (free fermions). the CSE coefficient is well-defined even without regularization, and reads C free CSE = 1 2π 2 Z ∞ 0 dp h 1 + coshp p 2 + (m/T) 2 i−1 . (3.72) This is a well-known result in the literature [12, 13], which we plot in [PITH_FULL_IMAGE:figures/full_fig_p084_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Analytical behavior of CCSE for a system of interacting fermions from Eq. (3.72), as a function of m/T. Figure from Ref. [152] [PITH_FULL_IMAGE:figures/full_fig_p084_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Results for σCSE/µ as a function of temperature in two-color QCD, at fixed lattice spacing. The results suggest a suppression of the CSE at low temperatures. Figure taken from Ref. [151]. 3.4.2 CSE in QCD In the presence of strong interactions, less is known about CCSE. The main obstacle is that lattice simulations face the already mentioned sign problem at finite quark chemical potential. Two lattice wo… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Derivative of the axial current with respect to the chemical potential for staggered fermions, given by Eq. (4.16), as a function of the magnetic field for non￾interacting fermions in a 243 × 6 lattice (left) and in full QCD with 2 + 1 flavors and physical quark masses at T = 305 MeV (right). The dashed line shows the result of the linear fit, while the band is the estimation of the total error. Notice t… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: CSE conductivity coefficient for free fermions at different values of m/T using the staggered discretization. For m/T = 4, Wilson fermions results are also shown, as well as non-conserved versions of the vector currents. The dashed line corre￾sponds to the analytical value of CCSE for different values of m/T, given by Eq. (3.72). Figure from Ref. [152]. value of m/T. These results indicate that finite-si… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Continuum limit for the different operators in the staggered formulation. We show here the results for a particular m/T = 1, although the continuum limit agreement of all the possibilities also holds for other values. show the continuum limit of CCSE for these three definitions. We see that all of the combinations yield the correct continuum limit, but with different lattice artifacts. From these results… view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Value of CCSE as a function of m/T for different values of the imaginary chemical potential µI . Here averaged refers to take the average over µI = 0, ±2π/3. Figure from Ref. [152]. detailed discussion on the calculation). We see that choosing the value µI/T = ±2π/3 heavily modifies the functional form of CCSE, even reaching negative values. Therefore, we can expect that the imaginary sectors of the Poly… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Results for the Polyakov loop (left) and the derivative of the current with staggered fermions (right) in an ensemble of 100 configurations on a 323 × 8 lattice at T ≈ 400 MeV and at a magnetic field of eB = 0.74 GeV2 . The three sectors are clearly visible at this temperature, where the center symmetry is spontaneously broken, and the contribution of configurations with imaginary Polyakov loops is obser… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Results for CCSE with Wilson and staggered fermions in the quenched approximation, measured on configurations generated using the plaquette gauge ac￾tion (2.44), where the configurations were rotated to the real Polyakov loop sector. The pion mass is set to Mπ ≈ 415 MeV for staggered quarks and Mπ ≈ 710 MeV for Wilson quarks. The quenched critical temperature T q c is indicated by the dashed vertical lin… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: CCSE for a broad range of temperatures in full QCD with 2 + 1 flavors of staggered fermions. The continuum limit estimation is indicated by the orange band. The result of our low-energy model, involving p and Σ ±, Ξ − baryons, is shown by the continuous line at low T, while the parameterization for CCSE in the whole range of temperatures is displayed as a dashed-dotted line. Figure from Ref. [152]. contr… view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Continuum limits of CCSE for different flavor quantum numbers for the axial current and the chemical potential. We present six possible combinations, the phenomenologically relevant cases, involving an electrically charged axial current (left panel) and a baryon axial current (right panel). Figure from Ref. [152]. in quadrature. We also provided a parameterization of CCSE in the whole range of tem￾peratu… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Derivative of the vector current with respect to the chiral chemical po￾tential as a function of the magnetic field for a 243 × 6 lattice with staggered quarks in the free case with m/T = 1 (left) and in full QCD with 2 + 1 flavors and physical quark masses at T = 305 MeV (right). The continuous green line represents the result of the fit, and the green band represents the total error estimation. For com… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Continuum extrapolation of the CME conductivity for free fermions at different values of m/T using the staggered discretization, as well as Wilson fermions for m/T = 4. The red dot-dashed line corresponds to CCME = 1/(2π 2 ), while the black dashed line to CCME = 0. The quenched continuum limit estimation with Wilson fermions from Ref. [117] is depicted as an orange band for comparison. Figure from Ref. … view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Continuum limit for the different operator in the staggered formulation, As opposed to the case of CCSE (see [PITH_FULL_IMAGE:figures/full_fig_p112_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Axial susceptibility for free fermions with m/T = 3, using both the staggered and the Wilson discretizations. The black dashed line represents the value from the analytical expression (3.68) at this particular value of m/T. This agreement is also reproduced for other values of m/T. Figure from Ref. [138]. and 6.47, and again use both staggered and Wilson fermions for the observable. The cor￾relator is ca… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Results for the CME conductivity in the quenched theory, both with Wilson and staggered fermions. The conductivity is consistent with zero in all the cases (within 2σ) even without taking the continuum limit. Figure from Ref. [138] [PITH_FULL_IMAGE:figures/full_fig_p113_5_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Direct comparison of the study in Ref. [117] (open red circles), with our setup with a conserved vector current (filled blue squares) and a non-conserved one (filled black squares). The results with non-conserved currents agree with each other within errors and deviate from the correct value. The result with a conserved vector current is consistent with a vanishing CCME. Figure from Ref. [138]. that work… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Full dynamic QCD results with 2 + 1 flavors of staggered fermions at the physical point in a wide range of temperatures. The continuum limit for points at T > 100 MeV is shown as an orange band. Figure from Ref. [138]. temperature within errors. For completeness, we also perform the continuum limit for temperatures from 100 MeV to 400 MeV, considering the same spline-fit procedure as for CCSE. The obtain… view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Axial susceptibility for 2 + 1 flavors of staggered quarks in full QCD at the physical point. The results are normalized by the number of colors Nc = 3 to directly compare to the free fermions result (3.68). We observe χ5 to slowly grow above the crossover temperature, approaching the analytic result for massless non-interacting fermions (indicated by the black dashed line). Figure from Ref. [138]. we on… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Magnetic field profile (6.1) for different values of the parameter , that controls the width. In the limit  → ∞, the homogeneous magnetic field profile is recovered. For the implementation on the lattice, we consider a simplified functional form of the magnetic field. We choose a x1-dependent profile of the form6.1 B~ (x1) = B cosh−2 (x1/) ˆe3 , (6.1) motivated by its analytical properties [168]. The … view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Lattice data and continuum extrapolation of the CME correlator with eB/T2 = 14.14 and T = 1/3, normalized by the magnetic field, for free fermions. The analytical result is recovered in the continuum limit, confirming the validity of our setup. For comparison, the shaded area depicts the magnetic field profile (6.1) in an arbitrary normalization. Figure from Ref. [142]. Therefore, this result does not c… view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: Top panel: CME correlator for free fermions as a function of x1/L for different values of Nb. The results correspond to a 323 × 8 lattice with m/T = 1 and T = 1/3. The inset shows the magnetic field dependence of the central point. Bottom plot:  dependence of the CME correlator, calculated on a 323 ×8 lattice with m/T = 1 and Nb = 2. Notice that the current vanishes in the limit of homogeneous magnetic… view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: Fourier transform of the CME coefficient as a function of the momentum for different lattice sizes. The solid line represents the analytical result. In the infinite￾momentum limit, the result converges to −1/(2π 2 ), which arises purely from the Pauli￾Villars regulator fields in the analytic calculation, see Eq. (6.12). Figure from Ref. [142] [PITH_FULL_IMAGE:figures/full_fig_p124_6_4.png] view at source ↗
Figure 6.5
Figure 6.5. Figure 6.5: Lattice data for the CME correlator in QCD as a function of x1 at T = 113 MeV and Nb = 3 for different lattice spacings. The connecting lines serve to guide the eye. For comparison, the shaded area depicts the magnetic field profile from Eq. (6.1) in an arbitrary normalization. Figure from Ref. [142]. The error determined in this way is shown by the orange band in [PITH_FULL_IMAGE:figures/full_fig_p125_… view at source ↗
Figure 6.6
Figure 6.6. Figure 6.6: Continuum limit of the CME correlator in QCD as a function of x1 at T = 155 MeV. The bands show this correlator at three different magnetic field strengths: eB = 0.1, 0.2, 0.3 and 0.4 GeV2 . Figure from Ref. [142]. a non-trivial spatial structure, very similar to the behavior found for non-interacting fermions. In addition, using the results obtained on four different lattice spacings and different weak … view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: CME correlator in the continuum limit at x1 = 0 fm and x1 = 0.9 fm as a function of eB for temperatures below, at, and above the crossover. The weak-field behavior of the correlator at the center is indicated by the dot-dashed line. Figure from Ref. [142]. the QCD crossover temperature. Based on the slope of the x1 = 0 curve in [PITH_FULL_IMAGE:figures/full_fig_p127_6_7.png] view at source ↗
Figure 6.8
Figure 6.8. Figure 6.8: Left panel: heat plot of the correlator H(x1, x0 1 ) normalized by eB in the free case in the x1 − x 0 1 plane for m/T = 1 on a 403 × 10 lattice. The color scheme is shown on a logarithmic scale for the absolute value of the observable from 0.015 to 10−4 , and a linear scale from the latter to 0. Red (blue) colors indicate the sign of the correlator. The projections on the top and right axes correspond t… view at source ↗

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