REVIEW 3 major objections 5 minor 111 references
$q\bar{q}$ scattering phase shift in the $\pi^0$ channel and ${\pi}^0$ meson spectral function under external magnetic field and finite meson momentum
T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read An external magnetic field turns the neutral pion's spectral function into a multi-peak structure with sharp phase-shift jumps.
desk verdict Solid NJL+RPA extension to finite momentum in a magnetic field, with a clean analytical core and honest caveats, but the numerical peak structure needs convergence tests before the field-induced claims can be trusted. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the one-loop quark-antiquark polarization function $\Pi_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k_3^2)$ evaluated in the random-phase approximation, with quark propagators built from Landau levels through the Ritus method and with Pauli-Villars regularization. Everything else is read off from its real and imaginary parts: the $S$-matrix element is $S=(1-2G\Pi(\omega-i\epsilon))/(1-2G\Pi(\omega+i\epsilon))=e^{2i\Phi_{\pi^0}}$, the phase shift is $\Phi_{\pi^0}=\arctan[2G\,\mathrm{Im}\,\Pi/(1-2G\,\mathrm{Re}\,\Pi)]$, and the spectral function is $\rho_{\pi^0}=2G\sin(2\Phi_{\pi^0})/(1-2G\,\mathrm{Re}\,\Pi)$. The singular thresholds of $\Pi$ do the causal work: unitary thresholds from the non-crossing terms $\omega=E_q+E_{q+k}$, where $\mathrm{Re}\,\Pi\to+\infty$ from below and $\mathrm{Im}\,\Pi\to+\infty$ from above; Landau thresholds from crossing terms $\omega=|E_{q+k}-E_q|$, whose divergence pattern depends on whether $\mathbf{k}_\perp$ or $k_3$ is nonzero; and Pauli-blocking thresholds set by the Fermi step when $T=0$ and $\mu>0$. The magnetic field enters through the Landau-level sums and the Laguerre-polynomial coefficients $B^\pm_{nlf}(\mathbf{k}_\perp^2)$, which is why the results depend on $\mathbf{k}_\perp^2$ and $k_3^2$ separately rather than on $\omega^2+\mathbf{k}^2$.
What would settle it
Recompute the polarization function at $eB=20m_\pi^2$ with the Pauli-Villars cutoff raised by about 20% and with more Landau levels kept: if the number or positions of the peaks, or the jump pattern of the phase shift, changes qualitatively, the multi-peak structure is a regulator artifact rather than a magnetic-field effect. An independent lattice QCD calculation of the $\pi^0$ spectral function at the same field strength, showing agreement with the predicted threshold zeroes, would confirm the claim.
Extended reading notes
Core claim
The paper's central claim is that, at field strength $eB=20m_\pi^2$, the pion spectral function $\rho_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k_3^2)$ is no longer a single delta-plus-continuum curve but a delta peak for the bound pion, several Breit-Wigner-type resonance peaks, and several non-Breit-Wigner peaks (peaks not tied to a pole of the propagator), with positions controlled by unitary thresholds $\omega_U(n,l)$ and Landau thresholds $\omega_L(n,l)$; at $T=0$ with finite quark chemical potential, Pauli-blocking thresholds additionally reshape the peak interiors. The companion claim is that the quark-antiquark scattering phase shift $\Phi_{\pi^0}$ is discontinuous: it jumps $0$ to $\pi$ at the bound-state pole, passes through $\pi/2$ at resonance poles, and jumps abruptly at unitary and Landau thresholds, exactly where the spectral function vanishes. Both features are caused by the external magnetic field, which quantizes quark momenta into Landau levels and breaks transverse-longitudinal symmetry; accordingly, finite transverse momentum $\mathbf{k}_\perp$ and finite longitudinal momentum $k_3$ modify the spectral function and phase shift differently, and only $k_3$ acts like a simple Lorentz shift at $T=\mu=0$.
Load-bearing premise
The central claim assumes that the multiple peaks and phase-shift jumps are real magnetic-field effects rather than artifacts of the ultraviolet cutoff or of truncating the quark energy-level sums, since the paper does not report convergence or cutoff-sensitivity tests.
Editorial extensions
If this is right
- In the magnetized NJL model, the $\pi^0$ spectral function vanishes exactly at every unitary and Landau threshold, so the multiple peaks are confined to intervals whose boundaries are fixed by the magnetic field and the quark mass.
- The $q\bar{q}$ phase shift jumps from $0$ to $\pi$ at the bound-state pole and from $\pi$ to a value below or above $\pi/2$ at unitary thresholds; with transverse momentum, Landau thresholds force a jump to $0$.
- Longitudinal pion momentum $k_3$ at zero temperature shifts the spectrum as $\omega\to\sqrt{\omega^2+k_3^2}$, while transverse momentum $\mathbf{k}_\perp$ does not, giving an explicit signal of magnetic-field-induced anisotropy.
- At zero temperature and finite quark chemical potential, Pauli-blocking thresholds, rather than unitary thresholds, set where the wide peaks begin and carve dips inside the resonance peaks.
- The computed spectral function and phase shift are the ingredients needed to extend Beth-Uhlenbeck-type thermodynamics of the quark-meson system to the magnetized case, which the paper identifies as a next step.
Reading between the lines
- Beyond the paper, the same threshold-and-jump mechanism should appear in other neutral channels of the same model, such as the sigma meson, because it follows from the Landau-level structure of the polarization function rather than from pion-specific dynamics.
- Beyond the paper, the predicted anisotropy between $\mathbf{k}_\perp$ and $k_3$ could be tested through angular distributions of dileptons or photons from pion decays in heavy-ion collisions with strong magnetic fields, although the paper does not draw that connection.
- Beyond the paper, the phase-shift jumps imply that thermodynamic quantities built from the phase shift via a Beth-Uhlenbeck formula acquire nonanalytic dependence on $eB$ at threshold energies, which could affect fluctuation observables near the chiral crossover.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript studies the q\bar q scattering phase shift in the π0 channel, Φπ0(ω²,k⊥²,k3²), and the π0 spectral function ρπ0(ω²,k⊥²,k3²) in the two-flavor Nambu-Jona-Lasinio model with a constant external magnetic field and finite meson momentum. The authors derive the RPA polarization function in a Landau-level basis, identify unitary, Landau, and Pauli-blocking thresholds, and present numerical results for the spectral function and phase shift in three regimes: chiral broken phase (T=μ=0), chiral restored phase at finite T, and chiral restored phase at finite μ. The central claims are that the magnetic field generates multiple Breit-Wigner and non-Breit-Wigner peaks, that the phase shift jumps abruptly at threshold points, and that the dependence on k⊥ versus k3 exhibits magnetic-field-induced anisotropy. An appendix contains the eB=0 limit and shows that the familiar NJL polarization function is recovered.
Significance. If the numerical results are robust, this paper would provide a systematic map of π0 spectral and phase-shift structure in a magnetized NJL model, including analytic threshold classifications that could be useful for further Beth-Uhlenbeck-type thermodynamics. The formal part has real strengths: the phase shift and spectral function are consistently defined from the same polarization function, the threshold tables are explicit and match the stated divergence structure, and the eB=0 limit in Appendix A is a meaningful consistency check. The main limitation is that the central quantitative claims—the multiple peaks and the abrupt phase-shift jumps—rest on numerical evaluations whose regularization and truncation details are not documented, so the physical interpretation as magnetic-field-induced structure is not yet fully supported.
major comments (3)
- [Section III, Eqs. (14)–(16)] The numerical implementation of the polarization function is not specified. The paper states that Pauli-Villars regularization is used, but does not give the PV subtraction masses, the order of subtraction, the cutoff for the Landau-level sums over n and l, or any convergence test. Since the peak structure between thresholds is explicitly said to 'rely on the numerical calculations' (Sec. III A.2, near the end of the T≠0, μ=0 subsection), and since the Landau-level sums and PV regulator directly control the threshold singularities, the multiple-peak and phase-shift-jump claims could be artifacts of the regulator or of a finite truncation. Please provide the implementation details and convergence checks, including variation of the PV parameters and of the Landau-level cutoff.
- [Tables I–III and Eqs. (14)–(16)] The analytic statements that ReΠ diverges to +∞ or −∞ and ImΠ diverges to +∞ at the unitary and Landau thresholds appear to be made for the unregularized principal-value integrals in (14)–(15). If Pauli-Villars regularization is applied before solving the gap equation and evaluating the polarization function, the regularized integrands differ from those shown, and the divergence pattern in Tables I–III may be modified. In particular, PV subtraction can render threshold singularities finite, which would soften the exact zeros of the spectral function and the abrupt phase-shift jumps. The paper needs to show the regularized threshold behavior and demonstrate that the jump pattern of Tables I–III survives regularization.
- [Section III A.3 and Fig. 2(c2)/3(c2)] The abstract and summary emphasize that in the large-ω region the phase shift jumps abruptly at the starting and end points of wide spectral peaks. However, for the T=0, μ≠0, k⊥≠0 case the authors restrict the plots to 0<ω<0.6 GeV because of the large number of thresholds (Sec. III A.3). The large-ω jump structure is therefore not actually documented for this central case. Either show the relevant ω region or state explicitly that the large-ω claim is an extrapolation based on the threshold structure; as written, the conclusion exceeds the presented numerical evidence.
minor comments (5)
- [Eq. (7)] The relation ρπ0 = 2G sin(2Φπ0)/(1−2G ReΠπ0) gives zero both below and above a bound-state pole, since sin(2π)=0; the delta peak is not represented by this formula. The text should state explicitly that the bound-state contribution is a separate delta-function term arising from the pole of the propagator, and that Eq. (7) describes only the continuum part.
- [Section II, Eq. (14)] The notation E_{q+k} is used before being defined; please define E_{q+k} = sqrt(2n|Q_f B| + (q3+k3)² + m_q²) immediately after Eq. (14) or before.
- [Figures 2 and 3] The threshold labels in the figures are written as e.g. 'U^d(0,0)' but the text refers to ω_U^d(n,l). The figures should use the same notation, e.g. ω_U^d(0,0), to avoid ambiguity.
- [Throughout] There are several typographical errors, including 'coefficient' in the introduction and 'difficult' in Sec. III A.3; a careful proofread is needed.
- [Introduction and Ref. [48]] Reference [48] (Mei et al., Phys. Rev. D 113, 074031 (2026)) appears to cover closely related ground on pion spectral functions in a magnetic field. The introduction should state explicitly what is new in this paper relative to [48], particularly the phase-shift analysis and the finite-momentum dependence.
Circularity Check
No significant circularity: phase shift and spectral function are derived from the same polarization function, but all thresholds and peaks follow from the model input rather than from fitted target data.
full rationale
The derivation chain is self-contained: the NJL Lagrangian, Eqs. (1)-(3), produces the polarization function Eqs. (14)-(16) via the Ritus method, and the scattering phase shift (Eq. (6)) and spectral function (Eq. (8)) are exact algebraic functions of that same polarization function. The paper does not present one as independent evidence of the other; rather, both are outputs of the same model calculation. The bound-state delta peak and the Breit-Wigner/non-Breit-Wigner peaks are read off from the pole equation 1 - 2G Re[Pi] = 0 and the threshold structure of Im[Pi], which are analytic consequences of the derived formulas, not fitted results. The NJL parameters G, Lambda, and m0 are fixed to vacuum observables (quark condensate, f_pi, m_pi) in Section III and taken from Ref. [49], and none of the eB-dependent thresholds, peak positions, or phase-shift jumps are adjusted to match the target spectral function or phase shift. Self-citations (e.g., [34], [48], [49]) provide methodological context and parameter values, but the load-bearing equations and threshold divergences are derived in the text itself, so no step reduces to an unverified self-citation. The eB=0 appendix acts as a consistency check rather than a circular reuse of the magnetized results. Concerns about Pauli-Villars regularization and Landau-level truncation are numerical robustness questions, not circularity, and therefore do not raise the circularity score.
Assumptions & free parameters
free parameters (3)
- NJL scalar/pseudoscalar coupling G =
3.44 GeV^-2
- Pauli-Villars cutoff Lambda =
1.127 GeV
- Current quark mass m0 =
0.005 GeV
assumptions (5)
- domain assumption The two-flavor NJL model with scalar and pseudoscalar four-fermion interactions is a valid effective theory for low-energy QCD in a magnetic field.
- domain assumption The RPA resummation of the quark bubble gives the full pi0 propagator; higher-order interactions are neglected.
- domain assumption The Leung-Ritus-Wang representation of the quark propagator (Landau-level sum) can be truncated in the numerical evaluation without changing the qualitative threshold structure.
- domain assumption Pauli-Villars regularization with Lambda=1.127 GeV removes UV divergences without artificially creating or destroying the peaks and jumps reported.
- standard math The gap equation at mean-field level determines the constituent quark mass mq consistently with the polarization function.
Cite this review
Pith. "Pith review of $q\bar{q}$ scattering phase shift in the $\pi^0$ channel and ${\pi}^0$ meson spectral function under external magnetic field and finite meson momentum." pith.science (2026). https://pith.science/paper/DOC6SBIC
@misc{pith2026260806788,
author = {Pith},
title = {Pith review of: $q\barq$ scattering phase shift in the $\pi^0$ channel and $\pi^0$ meson spectral function under external magnetic field and finite meson momentum},
year = {2026},
howpublished = {\url{https://pith.science/paper/DOC6SBIC}},
note = {Machine review of arXiv:2608.06788}
}
abstract
$q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k^2_3)$ and ${\pi}^0$ meson spectral function $\rho_{\pi^0}(\omega^2,\mathbf{k}_\perp^2,k^2_3)$ under external magnetic field $eB$ and finite meson momentum $\mathbf{k}_\perp^2,k^2_3$ are studied in the framework of a two-flavor Nambu-Jona-Lasinio (NJL) model. The $q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}$ is closely related to $\pi^0$ spectral function $\rho_{\pi^0}$. We consider three situations, chiral broken phase ($T=\mu=0$), chiral restoration phase ($T>T_{pc},\ \mu=0$) and chiral restoration phase ($T=0,\ \mu>\mu_{pc}$). For $T=\mu=0$ and $T>T_{pc},\ \mu=0$ cases, ${\pi}^0$ meson spectral function $\rho_{\pi^0}$ shows a delta peak, several Breit-Wigner peaks and several non-Breit-Wigner peaks. The delta peak indicates the bound state of $\pi^0$ meson, and the Breit-Wigner peak means the resonant state of $\pi^0$ meson. For $T=0,\ \mu>\mu_{pc}$ case, Pauli blocking effect plays a role, which changes the inner structure of these Breit-Wigner peaks and non-Breit-Wigner peaks. Such multiple peak structure is caused by the external magnetic field. The $q\bar{q}$ scattering phase shift in the $\pi^0$ channel $\Phi_{\pi^0}$ shows a jump from $0$ to $\pi$ when $\pi^0$ meson is in bound state. When $\pi^0$ meson is in resonant state, $\Phi_{\pi^0}$ has the value $\pi/2$ and changes continuously. In large $\omega$ region, at the starting and end points of wide peaks of spectral function, $\Phi_{\pi^0}$ jumps abruptly (from $\pi$ to finite value or from finite value to $0$), and such jumps are caused by the external magnetic field. Finite momentum $\mathbf{k}_\perp^2$ or $k^2_3$ modifies the spectral function $\rho_{\pi^0}$ and scattering phase shift $\Phi_{\pi^0}$, which demonstrates the anisotropy in the system induced by external magnetic field.
Figures
Figures from the paper (2 more)
Reference graph
Works this paper leans on
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[48]
and π0 meson spectral function ρπ0 (ω2, k2 ⊥,k 2
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[1]
2(a1) and Fig
T = µ = 0 ForT =µ = 0 andk2 3 = k2 ⊥ = 0 (see black dashed line in Fig. 2(a1) and Fig. 2(a2)), the π0 spectral function shows a delta peak and several peaks of Breit-Wigner type. The delta peak indicates the bound state of π0 meson, and the Breit-Wigner peak means the resonant state of π0 meson. The location of the delta peak can be identified as the pole...
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[2]
under external magnetic field eB and finite meson momentum k2 ⊥, k2 3 are studied in the framework of a two-flavor Nambu-Jona-Lasinio (NJL) model. The q ¯q scattering phase shift in the π0 channel Φπ0 is closely related to π0 spectral function ρπ0 . We consider three situations, chiral broken phase ( T = µ = 0 ), chiral restoration phase ( T > T pc, µ = 0...
work page Pith review arXiv 2026
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[3]
and π0 meson spectral function ρπ0 (ω2, k2 ⊥, k2
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[4]
(4) Being unimodular, it can be expressed in terms of the q ¯q scattering phase shift Φπ0 in the π0 channel Sπ0 (ω2, k2 ⊥,k 2
= 1 − 2GΠπ0 ((ω −iϵ)2, k2 ⊥,k 2 3) 1 − 2GΠπ0 ((ω +iϵ)2, k2 ⊥,k 2 3). (4) Being unimodular, it can be expressed in terms of the q ¯q scattering phase shift Φπ0 in the π0 channel Sπ0 (ω2, k2 ⊥,k 2
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[5]
(6) The q ¯q scattering phase shift in π0 channel Φπ0 (ω2, k2 ⊥,k 2
= e2iΦπ0 (ω2,k2 ⊥,k2 3), (5) and Φπ0 (ω2, k2 ⊥,k 2 3) = arctan[ 2G Im[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)] 1 − 2G Re[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)] ]. (6) The q ¯q scattering phase shift in π0 channel Φπ0 (ω2, k2 ⊥,k 2
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[6]
is closely related to the π0 spectral function ρπ0 (ω2, k2 ⊥,k 2 3), ρπ0 (ω2, k2 ⊥,k 2
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[7]
= 2G sin[2Φπ0 (ω2, k2 ⊥,k 2 3)] 1 − 2G Re[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)], (7) which is defined as ρπ0 (ω2, k2 ⊥,k 2
Show all 111 references
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[8]
= −2 Im[Dπ0 ((ω +iϵ)2, k2 ⊥,k 2 3)] = 8G2 Im[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)] (1 − 2G Re[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)])2 + (2G Im[Ππ0 ((ω +iϵ)2, k2 ⊥,k 2 3)])2, (8) and encodes the properties of bound and resonant states. Owing to the symmetry reduction from O(1, 3) toO(3) by the med...
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[9]
Accordingly, they are written as functions of (ω2, k2 ⊥,k 2
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[10]
Both the quark-antiquark scattering phase shift in π0 channel and theπ0 meson spectral function are controlled by the polarization function
throughout this manuscript. Both the quark-antiquark scattering phase shift in π0 channel and theπ0 meson spectral function are controlled by the polarization function. We now derive its analytical form, including the real and imaginary parts, and analyze the associated thresh...
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[11]
= Re[Π π0 ((ω +iϵ)2, k2 ⊥,k 2 3)] = −Nc ∑ f =u,d ∑ n,l=0 P.V. ∫ dq3 2π |QfB| 2π { B+ nlf (k2 ⊥) + B− nlf (k2 ⊥) 4 (F (Eq+k −µ) −F (−Eq+k −µ) Eq+k + F (Eq −µ) −F (−Eq −µ) Eq ) + 1 4Eq+kEq ( 2(n +l)|QfB| − (ω2 −k2 3) 2 [ B+ nlf (k2 ⊥) + B− nlf (k2 ⊥) ] − 2|QfB| √ nl [ B+ nlf (k2...
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[12]
= Im[Π π0 ((ω +iϵ)2, k2 ⊥,k 2 3)] = πNc ∑ f =u,d ∑ n,l=0 ∫ dq3 2π |QfB| 2π { 1 4Eq+kEq × ( 2(n +l)|QfB| − (ω2 −k2 3) 2 [ B+ nlf (k2 ⊥) + B− nlf (k2 ⊥) ] − 2|QfB| √ nl [ B+ nlf (k2 ⊥) − B− nlf (k2 ⊥) ]) × ( [F (Eq −µ) −F (Eq+k −µ)]δ(ω +Eq −Eq+k) + [F (−Eq −µ) −F (−Eq+k −µ)]δ(ω ...
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[13]
The imaginary part of polarization function ΠIm(ω2, k2 ⊥,k 2
diverges to +∞, and it remains finite for other values of ω. The imaginary part of polarization function ΠIm(ω2, k2 ⊥,k 2
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[14]
Therefore, at unitary thresholds, the spectral function is zero
diverges to +∞ asω →ωf U (n,l )+, and it remains finite for other values of ω. Therefore, at unitary thresholds, the spectral function is zero. With increasing ω, scattering phase shift Φπ0 jumps from value π to ϕU at unitary threshold. ϕU has a value greater (smaller) than π/...
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[15]
(18) arise from the cross terms 1 ω±Eq∓Eq+k
ρπ0 (ω2, k2 ⊥, k2 3) ω → ωf U (n, l)− +∞ finite π 0 ω → ωf U (n, l)+ finite +∞ ϕU 0 The Landau thresholds ωf L(n,l ) = max |Eq+k −Eq| = √ k2 3 + ( √ 2n|QfB| +m2q − √ 2l|QfB| +m2q)2, n,l = 0, 1, 2,... (18) arise from the cross terms 1 ω±Eq∓Eq+k . Note that with T = µ = 0 or van...
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[16]
The imaginary part ΠIm(ω2, k2 ⊥,k 2
diverges to −∞, and it remains finite for other values of ω. The imaginary part ΠIm(ω2, k2 ⊥,k 2
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[17]
The spectral function is zero at these Landau thresholds
diverges to +∞ asω →ωf L(n,l )−, and it remains finite for other values of ω. The spectral function is zero at these Landau thresholds. With increasing ω, scattering phase shift Φπ0 jumps from value ϕL1 to 0 at these Landau thresholds, and ϕL1 has a value greater (smaller) tha...
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[18]
In the case k2 3 ̸= 0, k2 ⊥ = 0 (see Table III), the coefficients defined in Eq.(16) become B± nlf (0) ∝ δnl, and this leads to the Landau thresholds ωf L(n,n )
is negative (positive). In the case k2 3 ̸= 0, k2 ⊥ = 0 (see Table III), the coefficients defined in Eq.(16) become B± nlf (0) ∝ δnl, and this leads to the Landau thresholds ωf L(n,n ). There is no divergence in the polarization function at ω = ωf L(n,n ), and its real part ΠRe...
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[19]
The spectral function is zero at these Landau thresholds
) has finite (zero) value. The spectral function is zero at these Landau thresholds. Scattering phase shift Φπ0 is a continuous function, and has the value ϕL2 = 0 (ϕL2 =π) at ω =ωf L(n,n ) when the sign of quantity 1 − 2GΠRe(ω2, k2 ⊥,k 2
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[20]
TABLE II
is positive (negative). TABLE II. Behavior near Landau thresholds with eB ̸= 0, k2 ⊥ ̸= 0, k 2 3 = 0. ΠRe(ω2, k2 ⊥, k2
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[21]
Behavior near Landau thresholds with eB ̸= 0, k2 ⊥ = 0, k 2 3 ̸= 0
ρπ0 (ω2, k2 ⊥, k2 3) ω → ωf L(n, l)− finite +∞ ϕL1 0 ω → ωf L(n, l)+ −∞ finite 0 0 TABLE III. Behavior near Landau thresholds with eB ̸= 0, k2 ⊥ = 0, k 2 3 ̸= 0. ΠRe(ω2, k2 ⊥, k2
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[22]
ρπ0 (ω2, k2 ⊥, k2 3) ω → ωf L(n, n)− finite 0 ϕL2 0 ω → ωf L(n, n)+ finite 0 ϕL2 0 Since |Qu| = 2|Qd| for quarks, we use the flavor index f =d when denoting unitary and Landau thresholds in the following context. III. NUMERICAL RESUL TS AND DISCUSSION Due to the non-renormaliz...
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[23]
With vanishing momentum k2 3 = k2 ⊥ = 0 (see black lines), the π0 spectral function shows a delta peak and several peaks of Breit-Wigner type
T ̸= 0, µ = 0 Figure 2(b1) and Figure 2(b2) show the π0 spectral function in chiral restoration phase with T = 0.2GeV, µ = 0 . With vanishing momentum k2 3 = k2 ⊥ = 0 (see black lines), the π0 spectral function shows a delta peak and several peaks of Breit-Wigner type. Apparen...
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[24]
GeV 2) a1) T μ 0 eB 20mπ 2 U d 0,0) U d 1,1)U d 0,0) U d 1,1) |k⟂|=k3=0 k3 = 0.5 GeV ★ ★ ★ ★ ★ ★ ★ ★ 0 0.5 1.0 1.5 0 5 10 15 20 25 30 ω GeV) π0 ω2,k⟂ 2 ,k3
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[25]
GeV 2) a2) T μ 0 eB 20mπ 2 U d 0,0) U d 1,1) U d 0,0) U d 0,1) U d 0,2)U d 1,1) U d 0,3) U d 1,2)U d 1,3) |k⟂|=k3=0 |k⟂| = 0.5 GeV ★ ★ ★ ★ ★ ★ ★ ★ 0 0.5 1.0 1.5 0 10 20 30 40 50 60 ω GeV) π0 ω2,k⟂ 2 ,k3
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[26]
GeV 2) b1) ★ 0.5 0.51 0.0 10 20 30 L d 0,0) U d 0,0) T 0.2 GeV μ 0 eB 20mπ 2 U d 0,0) U d 1,1) U d 2,2)U d 1,1) U d 2,2) |k⟂|=k3=0 k3 = 0.5 GeV ★ ★ ★ 0.08 0.1 0.12 0.14 0 10 20 30 40 50 60 (b2) U d (0,0) U d (0,0) ★ ★ ★ ★ 0.65 0.7 0.75 0.8 T = 0.2 GeV = 0 eB = 20m2 L d (2,0) U...
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[27]
(GeV-2) |k⟂|=k3=0 |k⟂| = 0.5 GeV ★ ★ ★ ★ ★ ★ ★ ★ 0 0.5 1.0 1.5 0 10 20 30 40 50 ω GeV) π0 ω2,k⟂ 2 ,k3
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[28]
GeV 2) c1) ★ 0.5 0 20 40 U d ( 0,0)PBL d,+ ( 0,0) T = 0 μ = 0.25 GeV eB = 20mπ 2 U d 1,1) U d 2,2)U d 1,1) U d 2,2) PBU d,± (0,0) PBU d,+(0,0)PBL d,-(0,0) |k⟂|=k3=0 k3 = 0.5 GeV ★ ★ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 ω GeV) π0 ω2,k⟂ 2 ,k3
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[29]
GeV 2) c2) ★ ★ ★ ★ 0.35 0.45 0 100 200 PBL d,± 1,0) L d 1,0) T = 0 μ = 0.25 GeV eB = 20mπ 2 PBL d, ± 1,0) U d 0,1) PBL d, ± 2,0) PBU d, ± 0,0) L d 1,0) PBU d, ± 0,0) k⟂ k3 0 k⟂ 0.5 GeV FIG. 2. Spectral function of π0 meson is shown as a function of ω for different T and µ at f...
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[30]
ω <ωd U (n + 1,n + 1), n = 0, 1, 2,
= 0 . ω <ωd U (n + 1,n + 1), n = 0, 1, 2,... . However, with k2 ⊥ ̸= 0 , k 2 3 = 0 , many Landau thresholds ωd L(n,l ),n,l = 0 , 1, 2,... and many unitary thresholdsωd U (n,l ), n,l = 0, 1, 2,... should be considered in the polarization function Eq.(14) and Eq.(15). This will ...
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[31]
This introduces additional Pauli blocking effect [31–33]
T = 0, µ ̸= 0 With zero temperature and finite quark chemical potential, the Fermi-Dirac distribution in Eq.(14) and in Eq.(15) reduces to a step function, F (Eq −µ) → Θ(µ −Eq). This introduces additional Pauli blocking effect [31–33]. We define the Pauli-blocking (PB) thresho...
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[32]
= 0 with vanishing imaginary part of 8 0 0.5 1.0 1.5 0 π 2 π ω GeV Φπ0 ω2,k⟂ 2 ,k3 2 a1 T μ 0 eB 20mπ 2 ★ ★ ★ ★ ★ ★ U d 0,0 U d 1,1U d 0,0 U d 1,1 |k⟂|=k3=0 k3 = 0.5 GeV 0 0.5 1.0 1.5 0 π 2 π ω GeV Φπ0 ω2,k⟂ 2 ,k3 2 a2 T μ 0 eB 20mπ 2 ★ ★ ★ ★ ★ ★ ★ ★ U d 0,0 U d 1,1 U d 0,0 U ...
-
[33]
9 polarization function ΠIm(ω2 δ, k2 ⊥,k 2
= π/2. 9 polarization function ΠIm(ω2 δ, k2 ⊥,k 2
-
[34]
= 0 , we obtain a step function Θ(x) Φπ0 (ω2, k2 ⊥,k 2
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[35]
At the pole of π0 propagator 1 − 2GΠRe(ω2 BW, k2 ⊥,k 2
= πΘ(ω2 −ω2 δ ), (21) with ω near ωδ. At the pole of π0 propagator 1 − 2GΠRe(ω2 BW, k2 ⊥,k 2
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[36]
= 0 with nonvanishing imaginary part of polarization function ΠIm(ω2 BW, k2 ⊥,k 2
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[37]
̸= 0 , we have Φπ0 (ω2, k2 ⊥,k 2
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[38]
Therefore, we mark a star at Φπ0 =π/2 in all panels of Figure 3
= π/2, (22) with ω =ωBW. Therefore, we mark a star at Φπ0 =π/2 in all panels of Figure 3. At unitary and Landau thresholds, jumps of scattering phase shift happen, as discussed in the previous section, see Table I, II, III
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[39]
With increasing ω, the scattering phase shift Φπ0 shows a jump from 0 to π at ω = ωδ
T = µ = 0 Figure 3(a1) and Figure 3(a2) black lines depict the q ¯q scattering phase shift in π0 channel with T =µ = 0 and k2 3 = k2 ⊥ = 0 . With increasing ω, the scattering phase shift Φπ0 shows a jump from 0 to π at ω = ωδ. With ωδ + 0 + < ω < ωd U (0, 0) − 0+, we have Φπ0 ...
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[40]
When ϕU > π/2, for instance at ω = ωd U (1, 1) + 0 + and ω = ωd U (0, 3) + 0 +, Φπ0 monotonically increases up to π until the next unitary threshold
is negative (positive). When ϕU > π/2, for instance at ω = ωd U (1, 1) + 0 + and ω = ωd U (0, 3) + 0 +, Φπ0 monotonically increases up to π until the next unitary threshold. Φπ0 in magenta lines can not recover the results in black lines with the replacementω → √ ω2 − k2 ⊥
-
[41]
Note that the values of ωδ, ωBW and ωd U (n,n ) are varied by temperature
T ̸= 0, µ = 0 Figure 3(b1) and Figure 3(b2) black lines depict the q ¯q scattering phase shift in π0 channel with T ̸= 0, µ = 0 and k2 3 = k2 ⊥ = 0 , which demonstrate similar behavior as in Figure 3(a1) and Figure 3(a2) black lines. Note that the values of ωδ, ωBW and ωd U (n...
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[42]
With increasing ω (0 < ω < 0.14GeV), Φπ0 increases continuously from zero to π at ω = ωd U (0, 0) − 0+, which crosses π/2 at ω = ωBW < ω d U (0, 0)
Since we meet many unitary and Landau thresholds, we plot two segments of the results, as in Figure 2(b2). With increasing ω (0 < ω < 0.14GeV), Φπ0 increases continuously from zero to π at ω = ωd U (0, 0) − 0+, which crosses π/2 at ω = ωBW < ω d U (0, 0). With ω = ωd U (0, 0) ...
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[43]
It should be mentioned that at PB thresholds, no jumps of scattering phase shift happen
T = 0, µ ̸= 0 With zero temperature and finite quark chemical potential, in addition to the unitary and Landau thresholds, we meet PB thresholds, due to the Pauli-blocking effect. It should be mentioned that at PB thresholds, no jumps of scattering phase shift happen. Figure 3...
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[44]
After that, Φπ0 keeps zero, until a jump from 0 to π at ω = ωδ
With 0 < ω < PBd,− L (0, 0), Φπ0 = π, and with PBd,− L (0, 0) < ω < PBd,+ L (0, 0), Φπ0 decreases from π to 0 continuously. After that, Φπ0 keeps zero, until a jump from 0 to π at ω = ωδ. Φπ0 keeps π with ωδ + 0 + < ω < ωd U (0, 0) − 0+, and jumps from π to ϕU < π/2 at ω = ωd ...
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[45]
At ω =ωd L(1, 0) + 0+, Φπ0 jumps to zero
Scattering phase shift Φπ0 shows a local maximum ( < π/2) around PB d,± L (1, 0), and a local maximum ( >π/ 2) withω <ωd L(1, 0) − 0+. At ω =ωd L(1, 0) + 0+, Φπ0 jumps to zero. After that, at ωδ, Φπ0 jumps from zero to π. It keeps π until ω = PBd,± L (2, 0), and starts to decr...
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[47]
The q ¯q scattering phase shift in the π0 channel Φπ0 is closely related to π0 spectral function ρπ0
under the external magnetic field eB and finite meson momentum k2 ⊥,k 2 3 are studied in the framework of a two-flavor Nambu-Jona-Lasinio (NJL) model. The q ¯q scattering phase shift in the π0 channel Φπ0 is closely related to π0 spectral function ρπ0 . We consider three situa...
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Usually, we start from a Lagrangian with current quarks and two-body interaction in scalar and pseudo-scalar channels
can provide input for understanding the thermodynamic properties in the magnetized quark-meson system. Usually, we start from a Lagrangian with current quarks and two-body interaction in scalar and pseudo-scalar channels. In mean-field approximation, the corresponding thermody...
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