pith. machine review for the scientific record. sign in

arxiv: 2604.27993 · v1 · submitted 2026-04-30 · ⚛️ nucl-th · hep-lat· hep-ph

Recognition: unknown

Hadron properties at finite temperature

Juan M. Torres-Rincon , Gl\`oria Monta\~na
Authors on Pith no claims yet

Pith reviewed 2026-05-07 05:16 UTC · model grok-4.3

classification ⚛️ nucl-th hep-lathep-ph
keywords finite-temperature QCDhadron in-medium propertieschiral perturbation theorylattice QCDheavy-ion collisionsspectral functionsthermal massesquarkonia
0
0 comments X

The pith

Thermal modifications to hadron masses, widths, and spectral functions emerge systematically from imaginary-time formalism, chiral perturbation theory, unitarized models, and lattice QCD.

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

This review provides a coherent picture of thermal effects on hadron properties by applying finite-temperature quantum field theory, specifically the imaginary-time formalism, to both light and heavy hadrons. Light hadrons are analyzed through chiral perturbation theory focusing on symmetry restoration, while heavy hadrons use unitarized and nonrelativistic effective field theories. These approaches are validated against Euclidean lattice QCD results for screening masses and spectral functions. The resulting thermal modifications are shown to affect key observables in relativistic heavy-ion collisions, including dilepton spectra and transport coefficients. Readers would care because this offers controlled theoretical tools to interpret experimental data from nuclear physics collisions.

Core claim

The authors argue that a systematic and controlled description of how finite temperature modifies hadron masses, decay widths, and spectral functions emerges from the combined use of the imaginary-time formalism in quantum field theory, chiral perturbation theory for light hadrons, unitarized approaches and nonrelativistic effective field theories for heavy hadrons, and lattice QCD in the Euclidean formulation. This framework allows for the extraction of in-medium properties that have direct phenomenological implications for relativistic heavy-ion collisions.

What carries the argument

The imaginary-time formalism of finite-temperature quantum field theory, applied inside effective field theories and lattice QCD to compute thermal masses, widths, and reconstructed spectral functions.

If this is right

  • Light hadron thermal masses arise from chiral symmetry restoration and are computable order by order in chiral perturbation theory.
  • Open heavy mesons and quarkonia acquire in-medium modifications that follow from self-consistent unitarized approaches and nonrelativistic effective field theories.
  • Euclidean lattice QCD supplies screening masses and reconstructed spectral functions that serve as independent benchmarks for the effective-theory results.
  • The thermal changes propagate directly into dilepton spectra, transport coefficients, and femtoscopy observables measured in relativistic heavy-ion collisions.

Where Pith is reading between the lines

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

  • The same set of frameworks could be pushed to higher temperatures to describe the approach to the quark-gluon plasma transition.
  • Precision measurements of thermal widths in dilepton channels at current or future colliders could test the quantitative accuracy of the unitarization procedures.
  • Analogous calculations at finite baryon density might connect to the properties of hadrons inside neutron stars.

Load-bearing premise

The assumption that imaginary-time formalism, chiral perturbation theory, unitarized approaches, nonrelativistic effective field theories, and lattice QCD together provide a sufficiently complete and controlled description of thermal modifications without major uncontrolled systematics in the hadronic phase.

What would settle it

A high-precision lattice QCD extraction of the rho-meson spectral function at a temperature below the critical temperature that deviates substantially from the prediction of chiral perturbation theory would challenge the claim of a coherent controlled picture.

Figures

Figures reproduced from arXiv: 2604.27993 by Gl\`oria Monta\~na, Juan M. Torres-Rincon.

Figure 1.1
Figure 1.1. Figure 1.1: Imaginary-time path applied along the temporal direction in the imaginary-time formalism of thermal quantum field theory. view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Pion screening mass (dashed line) and pion pole mass (solid line) according to the calculation of Ref. [ view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Pion thermal mass (left panel) and thermal decay width (right panel) calculated from ChPT in Ref. [ view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Left panel: Pion thermal mass computed from ChPT with the Inverse Amplitude Method and 2 flavors [ view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Left panel: ρ-meson in-medium spectral function from the low-energy effective Lagrangian of Refs. [61, 63]. Right panel: ρ-meson in-medium spectral function from different low-energy models (NLσM, hidden-local symmetry model, and LσM), from Ref. [64]. threshold. The result denoted as IAM SU(2)f ChPT is taken from Ref. [52] and uses two light flavors. Therefore, it is not coupled to the KK¯ channel and em… view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Thermal masses (left panels) and half-width (right panels) of the view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Thermal mass (left panel) and decay width (right panel) of the scalar view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Meson screening masses computed within the vector view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: Subtracted chiral condensate (2.17) as a function of the temperature, as computed in the lattice-QCD computation of Ref. [93]. transition. The chiral phase transition separates the Nambu-Goldstone (or broken) phase from the Wigner-Weyl (or symmetric) phase. An order parameter for the restoration of chiral transition can be defined through the quark condensate ⟨qq¯ ⟩, which takes a finite value in the Nam… view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Pion and σ thermal masses in the O(N) model for N = 4 as shown in Ref. [54]. We plot the case without an explicit symmetry￾breaking term ϵ = 0 (left panel), and the case with an explicit symmetry-breaking term ϵ ≠ 0 (right panel). 0 50 100 150 200 250 300 350 400 T (MeV) 0 200 400 600 800 Meson masses and condensate (MeV) v m¯ σ m¯ π 0 50 100 150 200 250 Temperature [MeV] 0 200 400 600 800 1000 1200 Mass… view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: Left panel: Pion and σ thermal masses from the LσM of Ref. [113] showing the parity partner degeneracy at high temperatures. Right panel: Kaon and κ thermal masses in the 3-flavor LσM of Ref. [118], also presenting a degeneracy above the chiral transition. The degeneracy between chiral partners can also be obtained in models where the hadrons are composed states, like in the NJL [38, 119, 120, 121, 39, … view at source ↗
Figure 2.11
Figure 2.11. Figure 2.11: Diagrammatic representation of the T-matrix equation (2.22),(2.23) in the meson sector of the (P)NJL model. Given that the kernel interaction is constant, one can obtain a factorization of the equation, and the corresponding T-matrix equation (suppressing flavor and Dirac structures) reduces to t ab = [ 2G 1 − 2GΠ ] ab , (2.23) 20 view at source ↗
Figure 2.12
Figure 2.12. Figure 2.12: Results for the pion and σ masses in the (P)NJL models at finite temperature in the calculation with two [124] and three flavors [43]. where t ab are the T-matrix elements. The polarization function Π ab at finite temperature is given by Π ab(iνm,p) = −⨋ k trγ [Ω¯ a ¯ji Si (iωn,k) Ω b i¯j S¯j (iωn − iνm,k − p)] , (2.24) where the trace is taken in Dirac space and the matrix Ω a i¯j = (Icolor ⊗ τ a i¯j ⊗… view at source ↗
Figure 2.13
Figure 2.13. Figure 2.13: Vector and axial-vector meson thermal masses as calculated in the 3-flavor PNJL model. We show the strangeness view at source ↗
Figure 2.14
Figure 2.14. Figure 2.14: Scalar (left panel) and pseudoscalar (right panel) screening meson masses from lattice QCD at finite temperature. Figures taken view at source ↗
Figure 2.15
Figure 2.15. Figure 2.15: Light (left panel), open-strangeness (middle panel), and hidden-strangeness (right panel) screening meson masses from lattice view at source ↗
Figure 2.16
Figure 2.16. Figure 2.16: Light-meson screening masses in different spin-parity channels from lattice QCD at finite temperature. Figures taken from the view at source ↗
Figure 2.17
Figure 2.17. Figure 2.17: Left panel: Damping coefficient of nucleons in a dilute gas of pions, from Ref. [ view at source ↗
Figure 2.18
Figure 2.18. Figure 2.18: Nucleon and ∆ baryon thermal masses from the QCD Dyson-Schwinger equations performed in Ref. [139]. 3π T N+ N￾0 0.5 1 1.5 2 0 1 2 3 T/Tc m [GeV ] 3π T Δ+ Δ￾0 0.5 1 1.5 2 0 1 2 3 T/Tc m [GeV ] view at source ↗
Figure 2.19
Figure 2.19. Figure 2.19: Nucleon (left panel) and ∆ baryon (right panel) screening masses from the vector × vector constaint interaction model of Ref. [75]. Both positive (solid lines) and negative (dashed lines) parity states have been included. where i runs over the three pion states (π − , π0 , π+ ), and M0 is the vacuum nucleon mass. In a more fundamental approach, the nucleon and ∆ baryon screening masses have been investi… view at source ↗
Figure 2.20
Figure 2.20. Figure 2.20: Nucleon screening mass from the calculation of Ref. [ view at source ↗
Figure 2.21
Figure 2.21. Figure 2.21: Diagrammatic representation of the T-matrix equation (2.35),(2.36) in the diquark sector of the (P)NJL model. quark-antiquark rescattering, with the key difference that the diquarks are not color singlets, but color antitriplets (the members of the color sextet representations cannot be part of baryons and are therefore discarded). Once the diquarks have been obtained, they are combined in appropriate s… view at source ↗
Figure 2.22
Figure 2.22. Figure 2.22: Diagrammatic representation of the quark-diquark rescattering in the (P)NJL model for the generation of baryon-like states. Left view at source ↗
Figure 2.23
Figure 2.23. Figure 2.23: Baryon thermal masses in the NJL model of Ref. [ view at source ↗
Figure 2.24
Figure 2.24. Figure 2.24: Baryon thermal masses in the PNJL model of Ref. [ view at source ↗
Figure 2.25
Figure 2.25. Figure 2.25: Temporal baryon masses calculated by the FASTSUM collaboration in Ref. [ view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Thermal properties of the D and D∗ mesons at rest in a pion gas obtained in Ref. [177]. Left: Collisional width Γ and mass shift Re Σ/2M as a function of the temperature. Right: Spectral functions at a temperature T = 200MeV, where dashed and solid curves correspond to the lowest-order and the first iteration of the self-consistent calculation. Figures taken from Ref. [177]. Following the work of Ref. [1… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: D-meson collisional width in a hot meson gas as a function of the temperature computed using the Boltzmann equation. The contributions to the total width from individual meson species are also displayed. Figure taken from Ref. [178]. The temperature dependence of charmed-meson masses has also been investigated in the context of chiral symmetry restoration using an extension of the linear sigma model for … view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Temperature dependence of the 0 ± nonstrange (left panel) and strange (middle panel) charm mesons obtained within the linear sigma model incorporating heavy quark symmetry in the mean field approximation in Ref [179]. The results for the nonstrange states are compared with non-unitarized chiral-EFT (right panel) calculations from the same reference. Figures taken from Ref. [179]. 3.3. Unitarized thermal … view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: (a) Coupled-channel Bethe-Salpeter equation. At finite temperature, the view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: shows the spectral functions of the D and D∗ mesons at various temperatures (top panels) and the corresponding temperature dependence of the thermal decay widths (bottom panels), as obtained in Ref. [174]. A clear thermal broadening is observed. In these calculations, the authors set the real parts of the corresponding self-energies to zero, thereby neglecting any temperature-induced mass shifts view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Thermal spectral functions of the ground-state open heavy-flavor mesons ( view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Temperature dependence of the masses (left) and widths (right) of the ground-state heavy-light mesons extracted from the spectral view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Imaginary part of the T-matrix diagonal elements showing the lineshapes of the dynamically generated excited heavy-light mesons in the charm and bottom sectors. Figure taken from Ref. [187]. 39 view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Spectral functions of the open-charm states reconstructed from lattice QCD Euclidean correlators at various temperatures. Figures view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Screening mass for sc¯ (left) and cc¯ states (right) as a function of the temperature extracted from lattice QCD spatial correlators. Figure obtained from Ref. [193]. More recently, Aarts et al. [188] introduced a double-ratio method to isolate the genuine temperature dependence of the ground-state masses in the hadronic phase from the temporal correlators, without full spectral reconstruction. The appr… view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Temperature dependence of the mass of the view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Temperature dependence of the mass of the view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Temperature dependence of the D-meson mass from QCD sum-rule analyses. Left panel: Scalar (0 +) channel from Ref. [195] (orange dotted curve), compared with the EFT results of Ref. [179] for both 0 + and 0 − states. Right: Pseudoscalar (0 −) channel from Ref. [173] (blue stars and red circles), alongside various heavy quark potentials and the EFT results of Ref. [110]. Figures obtained from Refs. [195, … view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Left panels: Heavy-baryon masses of opposed parity for different values of the anomalous coupling constant view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: Temperature dependence of the ground-state masses of the spin- view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Spectral function of J/ψ (left) and ηc (right) from quenched lattice QCD at finite temperature, reconstructed using MEM. Figure taken from [218]. More recent extractions of the charmonium spectral functions from quenched lattice QCD include those of Refs. [223, 224]. Both studies used MEM to reconstruct the spectral functions. Ref. [224] used anisotropic lattices following the strategy of earlier studies… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Temperature dependence of the pole mass of view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Pseudoscalar spectral functions from quenched lattice QCD reconstructed using MEM. Figures taken from [ view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Spectral functions of the charmonia states reconstructed from view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: S-wave (left) and P-wave (right) bottomonium spectral functions reconstructed from view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Spectral functions reconstructed for S-wave bottomonium obtained using the MEM (gray dashed), standard BR (colored solid), view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Temperature dependence of the in-medium mass shift for bottomonium (left) and charmonium (right) S-wave (top) and P-wave view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Thermal mass shift (left) and width (right of the view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Temperature dependence of the in-medium mass shift for view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Temperature dependence of the in-medium widths for view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Spectral functions from the lattice pNRQCD approach for bottomonium (left) and charmonium (right) S-wave (top) and P-wave view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Temperature dependence of the mass (top) and width (bottom) of the bottomonium (left) and charmonium (right) states from view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Charmonium (left) and bottomonium (right) spectral functions in the pseudoscalar (top) and vector (bottom) from MEM analysis view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: The S-wave bottomonium T-matrices in the complex-energy plane, evaluated at zero center-of-mass momentum for different temperatures. Figures obtained from Ref. [258]. panels show an increase of temperature from left to right, where different poles are clearly seen. These poles move in the complex energy plane with temperature reflecting the temperature dependence of their masses and widths. However, som… view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Comparison of various dissociation criteria for S-wave bottomonia: open squares denote the temperatures at which the width view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: Unitarized DD¯ ∗ scattering amplitude illustrating the thermal behavior of the X(3872) lineshape at several temperatures from two similar approaches that differ in their interaction kernels and implementation. Left: results Ref. [278]. Right: results from [279]. The finite-temperature analysis of exotic molecular states in Ref. [279] was extended to the spin partner of the X(3872), the X(4014), interpre… view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: Temperature dependence of the mass shift and half-width of the view at source ↗
Figure 4.19
Figure 4.19. Figure 4.19: Spectral functions of the T + cc(3875) obtained with two different interaction kernels, VA (top panels) and VB (bottom panels), at several temperatures and for two values of the molecular probability (left and right columns). Figures taken from [282].             view at source ↗
Figure 4.20
Figure 4.20. Figure 4.20: Finite-temperature properties of the X(3872) from the screened heavy-quark potential framework of Ref. [283]. Top left: Decay width scaled by the temperature as a function of the Debye mass. Top right: Binding energy and real part of the potential at large heavy-quark separations as functions of the Debye mass. Bottom left: Mean square radius, √ ⟨r 2⟩, and screening length, 1/mD, versus Debye mass. Bott… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: ρ meson spectral function at different temperatures and chemical potentials used by SMASH simulations in Ref. [308]. formalism of Ref. [309] where strangeness production in low-energy HICs is analyzed. 0 20 40 60 80 0.3 0.4 0.5 0.6 0 20 40 60 0.3 0.4 0.5 0.6 0.7 A(ω) [G e V -2 ] A(ω) [G e V -2 ] 2 ρ 0 2 ρ 0 2 ρ 0 2 ρ 0 1 . 5 ρ 0 1 . 5 ρ 0 1 . 5 ρ 0 1 . 5 ρ 0 ρ ρ 0 0 ρ 0 ρ 0 0. 5 ρ 0 0. 5 ρ 0 0. 5 ρ 0 T=5… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: K¯ meson spectral function at different temperatures and chemical potentials used by PHSD simulations in Ref. [309]. We note that both temperature and density effects are accounted for. While this review is focused on pure thermal effects, the effect of high-baryon density turns out to be more important for the spectral broadening of states. In the absence of net baryon density (µB = 0), the interaction … view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Left panel: Experimental dimuons spectra in semicentral In-In collisions at view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Left panel: Ratio of K1 over K∗ yields in Pb+Pb collision at √sNN = 5.02 TeV at different centralities. From Ref. [325]. Right panel: Temperature dependence of the integrated 3-body decay width of K+ 1 (1270) → π +π −K+. Figure adapted from Ref. [327]. 5.3. Determination of transport coefficients The transport coefficients of an interacting system—like the shear viscosity η, the bulk viscosity ζ, and the… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Left panel: Pion and σ meson masses in the LσM as functions of the temperature as calculated in Ref. [113]. Right panel: Corresponding value of the shear viscosity over entropy density calculated in the same model. 65 view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Drag force coefficient (left panel) and spatial diffusion coefficient (right panel) of view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Pion-proton (left panel) and pion-deuteron (right panel) femtoscopy correlation functions measured in high-multiplicity view at source ↗
read the original abstract

This review provides an overview of thermal effects on hadron properties, focusing on the theoretical frameworks used to describe in-medium modifications of masses, decay widths, and spectral functions. We examine the application of finite-temperature quantum field theory -- specifically the imaginary-time formalism (ITF) -- to analyze both light- and heavy-hadron sectors. For light hadrons, we discuss the role of chiral symmetry restoration and the different definitions of thermal masses in effective field theories, like chiral perturbation theory. In the heavy-flavor sector, we review recent progress in describing open-heavy mesons and quarkonia using self-consistent unitarized approaches and nonrelativistic effective field theories. All these results are complemented by analyses of recent lattice-QCD calculations using the Euclidean formulation of QCD at finite temperature, relevant to extract screening masses and reconstructed spectral functions. Finally, we discuss the phenomenological impact of the thermal modifications on experimental observables in relativistic heavy-ion collisions, including numerical simulations, dilepton spectra, transport coefficients, and hadron femtoscopy. By combining phenomenological considerations with robust theoretical tools, this review provides a coherent picture of how thermal effects emerge in the hadronic phase and how they can be systematically studied within controlled frameworks. Ultimately, the discussion serves as a bridge between experimental observations in relativistic heavy-ion collisions and fundamental developments in finite-temperature QCD and effective field theories for hadronic systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript is a review article surveying thermal modifications to hadron properties (masses, widths, spectral functions) in the hadronic phase. It covers the imaginary-time formalism applied to light hadrons via chiral perturbation theory (including chiral symmetry restoration and thermal-mass definitions), heavy-flavor systems via unitarized approaches and nonrelativistic EFTs, lattice QCD extractions of screening masses and reconstructed spectral functions, and the translation of these results into phenomenological observables (dilepton spectra, transport coefficients, femtoscopy) in relativistic heavy-ion collisions.

Significance. As a synthesis of established methods rather than a source of new derivations, the review's potential significance lies in organizing recent literature into a coherent narrative that connects finite-temperature EFTs, lattice results, and heavy-ion phenomenology. Credit is due for explicitly referencing standard controlled frameworks (ITF, chiral PT, unitarized NR EFTs, Euclidean lattice QCD) and for highlighting their phenomenological reach; however, the absence of new calculations or falsifiable predictions limits its novelty to the quality of the synthesis and the identification of open issues.

major comments (1)
  1. [Abstract] Abstract and introductory discussion of frameworks: the assertion that the chosen methods (ITF, chiral PT, unitarized approaches, NR EFTs, lattice QCD) 'together give a sufficiently complete and controlled description' is load-bearing for the central claim of a 'coherent picture within controlled frameworks.' The review must explicitly delineate the temperature and density ranges where each framework remains reliable and must flag major uncontrolled systematics (e.g., convergence of chiral PT near the crossover, model dependence in unitarization, or reconstruction ambiguities in lattice spectral functions) rather than leaving this implicit.
minor comments (3)
  1. Ensure consistent notation for thermal masses, screening masses, and spectral functions across sections; different definitions in chiral PT should be cross-referenced to the corresponding lattice observables.
  2. Add a short dedicated paragraph or table summarizing the temperature validity windows and dominant uncertainties of each framework to make the 'controlled' claim concrete for readers.
  3. Update and cross-check all citations to recent lattice-QCD results on quarkonia and open-heavy mesons to avoid inadvertent omission of key works published after the review's cutoff.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comment on the abstract and introduction. We agree that the central claim requires more explicit support regarding the applicability of the frameworks and will revise accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introductory discussion of frameworks: the assertion that the chosen methods (ITF, chiral PT, unitarized approaches, NR EFTs, lattice QCD) 'together give a sufficiently complete and controlled description' is load-bearing for the central claim of a 'coherent picture within controlled frameworks.' The review must explicitly delineate the temperature and density ranges where each framework remains reliable and must flag major uncontrolled systematics (e.g., convergence of chiral PT near the crossover, model dependence in unitarization, or reconstruction ambiguities in lattice spectral functions) rather than leaving this implicit.

    Authors: We agree that the abstract's phrasing and the introductory framing would be strengthened by making the validity ranges and systematics more explicit. In the revised version we will update the abstract to note the primary regimes of applicability (chiral perturbation theory for T ≲ 100–150 MeV where the expansion remains convergent, unitarized approaches extending to higher temperatures but carrying regularization dependence, and lattice QCD providing non-perturbative input subject to reconstruction uncertainties). We will also insert a concise paragraph early in the introduction that systematically lists the temperature and density domains for each method together with the principal uncontrolled systematics, including the degradation of chiral convergence near the crossover, cutoff and scheme dependence in unitarization, and the ill-posed character of spectral-function reconstruction from Euclidean correlators. These additions will directly support the claim of a coherent picture within controlled frameworks without changing the review’s overall scope or narrative. revision: yes

Circularity Check

0 steps flagged

No significant circularity in this review synthesis

full rationale

This is a review article that synthesizes existing literature on thermal modifications to hadron properties using established frameworks (imaginary-time formalism, chiral perturbation theory, unitarized approaches, nonrelativistic EFTs, and lattice QCD). No new derivations, predictions, or first-principles results are presented that could reduce by construction to the paper's own inputs. All technical content is referenced from external sources, and the assertion of providing a 'coherent picture' is a qualitative synthesis of prior work rather than a load-bearing mathematical claim or self-referential fit. Self-citations, if present for the authors' prior contributions, do not serve as the sole justification for uniqueness theorems or force the central conclusions. The paper is self-contained against external benchmarks and exhibits no self-definitional, fitted-input, or ansatz-smuggling patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a review the paper introduces no new free parameters, axioms, or invented entities; all technical content is drawn from the cited prior literature on finite-temperature QCD and effective field theories.

pith-pipeline@v0.9.0 · 5540 in / 1067 out tokens · 50649 ms · 2026-05-07T05:16:23.828282+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

298 extracted references · 282 canonical work pages · 3 internal anchors

  1. [1]

    E. V. Shuryak, Quark-Gluon Plasma and Hadronic Production of Leptons, Photons and Psions, Phys. Lett. B 78 (1978) 150.doi:10.1016/0370-2693(78)90370-2

    work page doi:10.1016/0370-2693(78)90370-2 1978

  2. [2]

    E. L. Feinberg, Direct Production of Photons and Dileptons in Thermodynamical Models of Multiple Hadron Production, Nuovo Cim. A 34 (1976) 391. 71

    1976

  3. [3]

    R. Rapp, J. Wambach, Chiral symmetry restoration and dileptons in relativistic heavy ion collisions, Adv. Nucl. Phys. 25 (2000) 1.arXiv:hep-ph/9909229,doi:10.1007/0-306-47101-9_1

    work page doi:10.1007/0-306-47101-9_1 2000

  4. [4]

    Y. Aoki, G. Endrodi, Z. Fodor, S. Katz, K. Szabo, The Order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675–678.arXiv:hep-lat/0611014,doi:10.1038/ nature05120

    work page Pith review arXiv 2006

  5. [5]

    Braun-Munzinger, K

    P. Braun-Munzinger, K. Redlich, J. Stachel, Particle production in heavy ion collisions, Quark–Gluon Plasma 3 (2003) 491–599arXiv:nucl-th/0304013,doi:10.1142/9789812795533_0008

    work page doi:10.1142/9789812795533_0008 2003

  6. [6]

    Decoding the phase structure of QCD via particle production at high energy,

    A. Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel, Decoding the phase structure of QCD via particle pro- duction at high energy, Nature 561 (7723) (2018) 321–330.arXiv:1710.09425,doi:10.1038/s41586-018-0491-6

    work page doi:10.1038/s41586-018-0491-6 2018

  7. [7]

    H. A. Weldon, Simple Rules for Discontinuities in Finite Temperature Field Theory, Phys. Rev. D28 (1983) 2007. doi:10.1103/PhysRevD.28.2007

    work page doi:10.1103/physrevd.28.2007 1983

  8. [8]

    A. K. Das, Finite Temperature Field Theory, World Scientific, New York, 1997

    1997

  9. [9]

    J. I. Kapusta, C. Gale, Finite-temperature field theory: Principles and applications, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2011.doi:10.1017/CBO9780511535130

    work page doi:10.1017/cbo9780511535130 2011

  10. [10]

    M. L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2011.doi:10.1017/CBO9780511721700

    work page doi:10.1017/cbo9780511721700 2011

  11. [11]

    Basics of Thermal Field Theor y,

    M. Laine, A. Vuorinen, Basics of Thermal Field Theory, Vol. 925, Springer, 2016.arXiv:1701.01554,doi: 10.1007/978-3-319-31933-9

    work page doi:10.1007/978-3-319-31933-9 2016

  12. [12]

    Petreczky, Lattice QCD at non-zero temperature, J

    P. Petreczky, Lattice QCD at non-zero temperature, J. Phys. G 39 (2012) 093002.arXiv:1203.5320,doi:10. 1088/0954-3899/39/9/093002

    work page arXiv 2012

  13. [13]

    and Fodor, Z

    S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, K. K. Szabo, Full result for the QCD equation of state with 2+1 flavors, Phys. Lett. B 730 (2014) 99–104.arXiv:1309.5258,doi:10.1016/j.physletb.2014.01.007

    work page doi:10.1016/j.physletb.2014.01.007 2014

  14. [14]

    and Bhattacharya, T

    A. Bazavov, et al., Equation of state in ( 2+1 )-flavor QCD, Phys. Rev. D 90 (2014) 094503.arXiv:1407.6387, doi:10.1103/PhysRevD.90.094503

    work page doi:10.1103/physrevd.90.094503 2014

  15. [15]

    Bazavov, et al., Meson screening masses in (2+1)-flavor QCD

    A. Bazavov, et al., Meson screening masses in (2+1)-flavor QCD, Phys. Rev. D 100 (9) (2019) 094510.arXiv: 1908.09552,doi:10.1103/PhysRevD.100.094510

    work page doi:10.1103/physrevd.100.094510 2019

  16. [16]

    Kubo, Statistical mechanical theory of irreversible processes

    R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in mag- netic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570–586.doi:10.1143/JPSJ.12.570

    work page doi:10.1143/jpsj.12.570 1957

  17. [17]

    P. C. Martin, J. S. Schwinger, Theory of many particle systems. 1., Phys. Rev. 115 (1959) 1342–1373.doi: 10.1103/PhysRev.115.1342

    work page doi:10.1103/physrev.115.1342 1959

  18. [18]

    Gasser, H

    J. Gasser, H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142.doi:10.1016/ 0003-4916(84)90242-2. 72

    1984

  19. [19]

    Gasser, H

    J. Gasser, H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465–516.doi:10.1016/0550-3213(85)90492-4

    work page doi:10.1016/0550-3213(85)90492-4 1985

  20. [20]

    Weinberg, Phenomenological Lagrangians, Physica A 96 (1-2) (1979) 327–340.doi:10.1016/0378-4371(79) 90223-1

    S. Weinberg, Phenomenological Lagrangians, Physica A 96 (1-2) (1979) 327–340.doi:10.1016/0378-4371(79) 90223-1

    work page doi:10.1016/0378-4371(79 1979

  21. [21]

    S. R. Coleman, J. Wess, B. Zumino, Structure of phenomenological Lagrangians. 1., Phys. Rev. 177 (1969) 2239– 2247.doi:10.1103/PhysRev.177.2239

    work page doi:10.1103/physrev.177.2239 1969

  22. [22]

    C. G. Callan, Jr., S. R. Coleman, J. Wess, B. Zumino, Structure of phenomenological Lagrangians. 2., Phys. Rev. 177 (1969) 2247–2250.doi:10.1103/PhysRev.177.2247

    work page doi:10.1103/physrev.177.2247 1969

  23. [24]

    Gell-Mann, M

    M. Gell-Mann, M. Levy, The axial vector current in beta decay, Nuovo Cim. 16 (1960) 705.doi:10.1007/ BF02859738

    1960

  24. [25]

    J. I. Kapusta, Quantum Chromodynamics at High Temperature, Nucl. Phys. B 148 (1979) 461–498.doi:10.1016/ 0550-3213(79)90146-9

    1979

  25. [26]

    R. D. Pisarski, F. Wilczek, Remarks on the Chiral Phase Transition in Chromodynamics, Phys. Rev. D 29 (1984) 338–341.doi:10.1103/PhysRevD.29.338

    work page doi:10.1103/physrevd.29.338 1984

  26. [27]

    Dolan, R

    L. Dolan, R. Jackiw, Symmetry Behavior at Finite Temperature, Phys. Rev. D 9 (1974) 3320–3341.doi:10.1103/ PhysRevD.9.3320

    1974

  27. [28]

    S. R. Coleman, R. Jackiw, H. D. Politzer, Spontaneous Symmetry Breaking in the O(N) Model for Large N*, Phys. Rev. D 10 (1974) 2491.doi:10.1103/PhysRevD.10.2491

    work page doi:10.1103/physrevd.10.2491 1974

  28. [29]

    Kaymakcalan, J

    O. Kaymakcalan, J. Schechter, Chiral Lagrangian of Pseudoscalars and Vectors, Phys. Rev. D 31 (1985) 1109. doi:10.1103/PhysRevD.31.1109

    work page doi:10.1103/physrevd.31.1109 1985

  29. [30]

    H. Gomm, O. Kaymakcalan, J. Schechter, Anomalous Spin 1 Meson Decays From the Gauged Wess-Zumino Term, Phys. Rev. D 30 (1984) 2345.doi:10.1103/PhysRevD.30.2345

    work page doi:10.1103/physrevd.30.2345 1984

  30. [31]

    U. G. Meissner, Low-Energy Hadron Physics from Effective Chiral Lagrangians with Vector Mesons, Phys. Rept. 161 (1988) 213.doi:10.1016/0370-1573(88)90090-7

    work page doi:10.1016/0370-1573(88)90090-7 1988

  31. [32]

    Bando, T

    M. Bando, T. Kugo, K. Yamawaki, Nonlinear Realization and Hidden Local Symmetries, Phys. Rept. 164 (1988) 217–314.doi:10.1016/0370-1573(88)90019-1

    work page doi:10.1016/0370-1573(88)90019-1 1988

  32. [33]

    Harada, K

    M. Harada, K. Yamawaki, Hidden local symmetry at loop: A New perspective of composite gauge boson and chiral phase transition, Phys. Rept. 381 (2003) 1–233.arXiv:hep-ph/0302103,doi:10.1016/S0370-1573(03)00139-X

    work page doi:10.1016/s0370-1573(03)00139-x 2003

  33. [34]

    Gasser, M

    J. Gasser, M. E. Sainio, A. Svarc, Nucleons with chiral loops, Nucl. Phys. B 307 (1988) 779–853.doi:10.1016/ 0550-3213(88)90108-3. 73

    1988

  34. [35]

    Bernard, N

    V. Bernard, N. Kaiser, U.-G. Meissner, Chiral dynamics in nucleons and nuclei, Int. J. Mod. Phys. E 4 (1995) 193–346.arXiv:hep-ph/9501384,doi:10.1142/S0218301395000092

    work page doi:10.1142/s0218301395000092 1995

  35. [36]

    D. U. Jungnickel, C. Wetterich, Effective action for the chiral quark-meson model, Phys. Rev. D 53 (1996) 5142– 5175.arXiv:hep-ph/9505267,doi:10.1103/PhysRevD.53.5142

    work page doi:10.1103/physrevd.53.5142 1996

  36. [37]

    Schaefer, J

    B.-J. Schaefer, J. M. Pawlowski, J. Wambach, The Phase Structure of the Polyakov–Quark-Meson Model, Phys. Rev. D 76 (2007) 074023.arXiv:0704.3234,doi:10.1103/PhysRevD.76.074023

    work page doi:10.1103/physrevd.76.074023 2007

  37. [38]

    Dynamical Model of Elementary Particles Based on an Analogy with Su- perconductivity. 1.,

    Y.Nambu, G.Jona-Lasinio, DynamicalModelofElementaryParticlesBasedonanAnalogywithSuperconductivity. 1., Phys. Rev. 122 (1961) 345–358.doi:10.1103/PhysRev.122.345

    work page doi:10.1103/physrev.122.345 1961

  38. [39]

    S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64 (1992) 649–708. doi:10.1103/RevModPhys.64.649

    work page doi:10.1103/revmodphys.64.649 1992

  39. [40]

    Fukushima, Chiral effective model with the Polyakov loop, Phys

    K. Fukushima, Chiral effective model with the Polyakov loop, Phys. Lett. B 591 (2004) 277–284.arXiv:hep-ph/ 0310121,doi:10.1016/j.physletb.2004.04.027

    work page doi:10.1016/j.physletb.2004.04.027 2004

  40. [41]

    Ratti, M

    C. Ratti, M. A. Thaler, W. Weise, Phases of QCD: Lattice thermodynamics and a field theoretical model, Phys. Rev. D 73 (2006) 014019.arXiv:hep-ph/0506234,doi:10.1103/PhysRevD.73.014019

    work page doi:10.1103/physrevd.73.014019 2006

  41. [42]

    Blanquier, Standard particles in the SU(3) Nambu-Jona-Lasinio model and the Polyakov-NJL model, J

    E. Blanquier, Standard particles in the SU(3) Nambu-Jona-Lasinio model and the Polyakov-NJL model, J. Phys. G 38 (2011) 105003.doi:10.1088/0954-3899/38/10/105003

    work page doi:10.1088/0954-3899/38/10/105003 2011

  42. [43]

    J. M. Torres-Rincon, B. Sintes, J. Aichelin, Flavor dependence of baryon melting temperature in effective models of QCD, Phys. Rev. C91 (6) (2015) 065206.arXiv:1502.03459,doi:10.1103/PhysRevC.91.065206

    work page doi:10.1103/physrevc.91.065206 2015

  43. [44]

    Song, Pions at finite temperature, Phys

    C. Song, Pions at finite temperature, Phys. Rev. D 49 (1994) 1556–1565.doi:10.1103/PhysRevD.49.1556

    work page doi:10.1103/physrevd.49.1556 1994

  44. [45]

    Gasser, H

    J. Gasser, H. Leutwyler, Light Quarks at Low Temperatures, Phys. Lett. B 184 (1987) 83–88.doi:10.1016/ 0370-2693(87)90492-8

    1987

  45. [46]

    J. L. Goity, H. Leutwyler, On the Mean Free Path of Pions in Hot Matter, Phys. Lett. B 228 (1989) 517–522. doi:10.1016/0370-2693(89)90985-4

    work page doi:10.1016/0370-2693(89)90985-4 1989

  46. [47]

    Schenk, Absorption and dispersion of pions at finite temperature, Nucl

    A. Schenk, Absorption and dispersion of pions at finite temperature, Nucl. Phys. B 363 (1991) 97–113.doi: 10.1016/0550-3213(91)90236-Q

    work page doi:10.1016/0550-3213(91)90236-q 1991

  47. [48]

    Toublan, Pion dynamics at finite temperature, Phys

    D. Toublan, Pion dynamics at finite temperature, Phys. Rev. D56 (1997) 5629–5645.arXiv:hep-ph/9706273, doi:10.1103/PhysRevD.56.5629

    work page doi:10.1103/physrevd.56.5629 1997

  48. [49]

    Quantum time and the time-dilation in- duced interaction transfer mechanism.Phys

    A. Schenk, Pion propagation at finite temperature, Phys. Rev. D47 (1993) 5138–5155.doi:10.1103/PhysRevD. 47.5138

    work page doi:10.1103/physrevd 1993

  49. [50]

    A. V. Smilga, Physics of thermal QCD, Phys. Rept. 291 (1997) 1–106.arXiv:hep-ph/9612347,doi:10.1016/ S0370-1573(97)00014-8

    work page arXiv 1997

  50. [51]

    J. M. Torres-Rincon, D. Teaney, Kinetics of hydrodynamic pions in chiral perturbation theory, Phys. Rev. D 106 (5) (2022) 056012.arXiv:2201.10495,doi:10.1103/PhysRevD.106.056012. 74

    work page doi:10.1103/physrevd.106.056012 2022

  51. [52]

    Fernandez-Fraile, A

    D. Fernandez-Fraile, A. Gomez Nicola, Chemical nonequilibrium for interacting bosons: Applications to the pion gas, Phys. Rev. D 80 (2009) 056003.arXiv:0903.0982,doi:10.1103/PhysRevD.80.056003

    work page doi:10.1103/physrevd.80.056003 2009

  52. [53]

    Oller, E

    J. Oller, E. Oset, J. Pelaez, Meson meson interaction in a nonperturbative chiral approach, Phys. Rev. D 59 (1999) 074001, [Erratum: Phys.Rev.D 60, 099906 (1999), Erratum: Phys.Rev.D 75, 099903 (2007)].arXiv:hep-ph/ 9804209,doi:10.1103/PhysRevD.59.074001

    work page doi:10.1103/physrevd.59.074001 1999

  53. [54]

    J. M. Torres-Rincon, Degeneracy Patterns of Chiral Companions at Finite Temperature, Symmetry 13 (8) (2021) 1400.arXiv:2106.15700,doi:10.3390/sym13081400

    work page doi:10.3390/sym13081400 2021

  54. [55]

    J. A. Oller, Unitarization Technics in Hadron Physics with Historical Remarks, Symmetry 12 (7) (2020) 1114. arXiv:2005.14417,doi:10.3390/sym12071114

    work page doi:10.3390/sym12071114 2020

  55. [56]

    Benedikt, et al., Future Circular Collider Feasibility Study Report: Volume 3, Civil Engineering, Implementation and Sustainability, Eur

    A. Salas-Bernárdez, Unitarity and Unitarization (3 2026).arXiv:2603.03256,doi:10.1140/epjs/ s11734-026-02146-9

    work page doi:10.1140/epjs/ 2026

  56. [57]

    Dobado, M

    A. Dobado, M. J. Herrero, T. N. Truong, Unitarized Chiral Perturbation Theory for Elastic Pion-Pion Scattering, Phys. Lett. B235 (1990) 134–140.doi:10.1016/0370-2693(90)90109-J

    work page doi:10.1016/0370-2693(90)90109-j 1990

  57. [58]

    Dobado, J

    A. Dobado, J. R. Pelaez, A Global fit of pi pi and pi K elastic scattering in ChPT with dispersion relations, Phys. Rev. D 47 (1993) 4883–4888.arXiv:hep-ph/9301276,doi:10.1103/PhysRevD.47.4883

    work page doi:10.1103/physrevd.47.4883 1993

  58. [59]

    Dobado, J

    A. Dobado, J. R. Pelaez, The Inverse amplitude method in chiral perturbation theory, Phys. Rev. D56 (1997) 3057–3073.arXiv:hep-ph/9604416,doi:10.1103/PhysRevD.56.3057

    work page doi:10.1103/physrevd.56.3057 1997

  59. [60]

    R. Rapp, J. Wambach, Low mass dileptons at the CERN SPS: Evidence for chiral restoration?, Eur. Phys. J. A 6 (1999) 415–420.arXiv:hep-ph/9907502,doi:10.1007/s100500050364

    work page doi:10.1007/s100500050364 1999

  60. [61]

    R.Rapp, SignaturesofthermaldileptonradiationatRHIC,Phys.Rev.C63(2001)054907.arXiv:hep-ph/0010101, doi:10.1103/PhysRevC.63.054907

    work page doi:10.1103/physrevc.63.054907 2001

  61. [62]

    C. Gale, J. I. Kapusta, Vector dominance model at finite temperature, Nucl. Phys. B 357 (1991) 65–89.doi: 10.1016/0550-3213(91)90459-B

    work page doi:10.1016/0550-3213(91)90459-b 1991

  62. [63]

    Rapp, The Vector probe in heavy-ion reactions, J

    R. Rapp, The Vector probe in heavy-ion reactions, J. Phys. G 31 (2005) S217–S230.arXiv:nucl-th/0409054, doi:10.1088/0954-3899/31/4/027

    work page doi:10.1088/0954-3899/31/4/027 2005

  63. [64]

    J. Alam, S. Sarkar, P. Roy, T. Hatsuda, B. Sinha, Thermal photons and lepton pairs from quark gluon plasma and hot hadronic matter, Annals Phys. 286 (2001) 159–248.arXiv:hep-ph/9909267,doi:10.1006/aphy.2000.6091

    work page doi:10.1006/aphy.2000.6091 2001

  64. [65]

    Ghosh, S

    S. Ghosh, S. Sarkar, S. Mallik, Analytic structure of rho meson propagator at finite temperature, Eur. Phys. J. C 70 (2010) 251–262.arXiv:0911.3504,doi:10.1140/epjc/s10052-010-1446-8

    work page doi:10.1140/epjc/s10052-010-1446-8 2010

  65. [66]

    Super-Schur polynomials for Affine Super Yangian Y(bgl1|1)

    A. Gómez Nicola, J. R. de Elvira, A. Vioque-Rodríguez, The pion-kaon scattering amplitude and theK∗ 0(700)and K∗(892) resonances at finite temperature, JHEP 08 (2023) 148.arXiv:2304.08786,doi:10.1007/JHEP08(2023) 148

    work page doi:10.1007/jhep08(2023 2023

  66. [67]

    Gao, Z.-H

    R. Gao, Z.-H. Guo, J.-Y. Pang, Thermal behaviors of light scalar resonances at low temperatures, Phys. Rev. D 100 (11) (2019) 114028.arXiv:1907.01787,doi:10.1103/PhysRevD.100.114028. 75

    work page doi:10.1103/physrevd.100.114028 2019

  67. [68]

    M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, QCD and Resonance Physics. Theoretical Foundations, Nucl. Phys. B 147 (1979) 385–447.doi:10.1016/0550-3213(79)90022-1

    work page doi:10.1016/0550-3213(79)90022-1 1979

  68. [69]

    M. A. Shifman, A. I. Vainshtein, V. I. Zakharov, QCD and Resonance Physics: Applications, Nucl. Phys. B 147 (1979) 448–518.doi:10.1016/0550-3213(79)90023-3

    work page doi:10.1016/0550-3213(79)90023-3 1979

  69. [70]

    A. I. Bochkarev, M. E. Shaposhnikov, Spectrum of the Hot Hadronic Matter and Finite Temperature QCD Sum Rules, Nucl. Phys. B 268 (1986) 220–252.doi:10.1016/0550-3213(86)90209-9

    work page doi:10.1016/0550-3213(86)90209-9 1986

  70. [71]

    Gubler, D

    P. Gubler, D. Satow, Recent Progress in QCD Condensate Evaluations and Sum Rules, Prog. Part. Nucl. Phys. 106 (2019) 1–67.arXiv:1812.00385,doi:10.1016/j.ppnp.2019.02.005

    work page doi:10.1016/j.ppnp.2019.02.005 2019

  71. [72]

    Ayala, C

    A. Ayala, C. A. Dominguez, M. Loewe, Finite Temperature QCD Sum Rules: a Review, Adv. High Energy Phys. 2017 (2017) 9291623.arXiv:1608.04284,doi:10.1155/2017/9291623

    work page doi:10.1155/2017/9291623 2017

  72. [73]

    Azizi, B

    K. Azizi, B. Barsbay, H. Sundu, Light scalarK∗ 0(700)meson in vacuum and a hot medium, Phys. Rev. D 100 (9) (2019) 094041.arXiv:1909.00716,doi:10.1103/PhysRevD.100.094041

    work page doi:10.1103/physrevd.100.094041 2019

  73. [74]

    M. A. Ramírez-Garrido, R. J. Hernández-Pinto, I. M. Higuera-Angulo, L. X. Gutiérrez-Guerrero, Screening masses for scalar and pseudoscalar mesons and their diquark partners: Insights from the contact interaction model, Phys. Rev. D 112 (3) (2025) 034024.arXiv:2508.01099,doi:10.1103/jfzy-2nt4

    work page doi:10.1103/jfzy-2nt4 2025

  74. [75]

    C. Chen, F. Gao, S.-x. Qin, Screening masses of positive- and negative-parity hadron ground states, including those with strangeness, Phys. Rev. D 112 (1) (2025) 014022.arXiv:2412.15045,doi:10.1103/527r-mtxb

    work page doi:10.1103/527r-mtxb 2025

  75. [76]

    Pagels, Departures from Chiral Symmetry: A Review, Phys

    H. Pagels, Departures from Chiral Symmetry: A Review, Phys. Rept. 16 (1975) 219.doi:10.1016/0370-1573(75) 90039-3

    work page doi:10.1016/0370-1573(75 1975

  76. [77]

    Coleman,Aspects of Symmetry: Selected Erice Lectures

    S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge, U.K., 1985. doi:10.1017/CBO9780511565045

    work page doi:10.1017/cbo9780511565045 1985

  77. [78]

    QCD phenomenology based on a chiral effective Lagrangian,

    T. Hatsuda, T. Kunihiro, QCD phenomenology based on a chiral effective Lagrangian, Phys. Rept. 247 (1994) 221–367.arXiv:hep-ph/9401310,doi:10.1016/0370-1573(94)90022-1

    work page doi:10.1016/0370-1573(94)90022-1 1994

  78. [79]

    Leutwyler, Principles of chiral perturbation theory, in: Hadrons 94 Workshop, 1994, pp

    H. Leutwyler, Principles of chiral perturbation theory, in: Hadrons 94 Workshop, 1994, pp. 1–46.arXiv:hep-ph/ 9406283

    1994

  79. [80]

    Alkofer, H

    R. Alkofer, H. Reinhardt, Chiral quark dynamics, Vol. 33, Springer Berlin, Heidelberg, 1995.doi:10.1007/ 978-3-540-49454-6

    1995

  80. [81]

    M. A. Nowak, M. Rho, I. Zahed, Chiral nuclear dynamics, World Scientific Publishing Company, 1996

    1996

Showing first 80 references.