A large size of the pion-like excitations in the stringy fluid above $T_{ch}$ is model independent and required by current algebra

L. Ya. Glozman

REVIEW 4 major objections 4 minor 35 references

A large size of the pion-like excitations in the stringy fluid above $T_{ch}$ is model independent and required by current algebra

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper argues that current algebra forces pion-like states to swell above Tch, making the effect independent of any particular confining model.

desk verdict The claim that pion swelling above Tch is required by current algebra does not hold up; a vanishing F_pi does not force a large radius. read the letter →

arxiv 2608.04563 v1 pith:ISOWRWP3 submitted 2026-08-05 hep-ph hep-exhep-lathep-thnucl-th

L. Ya. Glozman This is my paper

classification hep-phhep-exhep-lathep-thnucl-th

keywords hotQCDstringyfluidpion-likeexcitationschiralsymmetryrestorationcurrentalgebraGMORrelationBethe-Salpeteramplitudelattice

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the huge pion-like excitations seen in the confining, chirally symmetric stringy fluid above the chiral restoration temperature $T_{\rm ch}\simeq 155$ MeV are not an artifact of any particular model. The author's route is current algebra: in the hadron-gas phase the pion-mass relation of current algebra applies, so the pion decay constant $F_\pi$ also serves as the amplitude for the quark and antiquark to meet at one point. At $T_{\rm ch}$, $F_\pi$ must vanish; since lattice correlators above $T_{\rm ch}$ show clear pion-like peaks that are completely unlike a free quark loop, the remaining option is a correlated state so extended that the local amplitude is zero. The paper concludes that pion-like excitations above $T_{\rm ch}$ must be essentially larger than pions in the hadron gas, though it cannot quantify the factor. If correct, this converts a model prediction into a general consequence of QCD symmetries and gives a microscopic explanation for the highly collective behavior of the hot matter.

What carries the argument

The load-bearing identity is the GMOR relation, $m_\pi^2 F_\pi^2 = -(m_u+m_d)\langle\bar q q\rangle + O(m_q^2)$, combined with the physical reading of $F_\pi$ as the Bethe-Salpeter amplitude - the amplitude for the quark and antiquark inside the pion to sit at the same point. Because $F_\pi$ is an order parameter, it must go to zero at $T_{\rm ch}$; the paper then forces a choice between uncorrelated plane-wave quarks and a giant correlated state. Lattice correlators, far from the free quark loop and carrying $\pi$, $\pi'$ peaks, select the second branch. It is this dichotomy that converts a model result into what the paper calls a model-independent constraint.

What would settle it

A lattice calculation of the spatial size (root-mean-square radius) of the pion-like Bethe-Salpeter amplitude just above $T_{\rm ch}$ would settle the claim: if the radius remains within a factor of order one of the vacuum pion radius while $F_\pi$ drops sharply, the swelling is not realized. Existing correlator data cannot distinguish a genuinely huge bound state from a compact wavefunction with a node that also makes the central amplitude vanish.

Watch

Extended reading notes

Core claim

Section 4 states the claim directly: 'the appearance of the huge pion-like excitations above $T_{\rm ch}$ is a model-independent statement based on current algebra and lattice data.' The argument starts from the GMOR relation, valid below $T_{\rm ch}$, and notes that $F_\pi$ - the Bethe-Salpeter amplitude at zero separation - must vanish at chiral restoration. Two cases are possible: the quark and antiquark above $T_{\rm ch}$ are uncorrelated plane waves, or they form a correlated state whose size is so large that the amplitude at the origin is zero. The lattice pion correlators, which differ dramatically from free-quark-loop correlators and show $\pi$ and $\pi'$ peaks up to about $3T_{\rm ch}$, rule out the first case. The paper therefore asserts that the pion-like excitations in the stringy fluid are bound states or resonances with essentially larger size than in the hadron gas, and that this is required by current algebra, not by the confining model of Ref. [15].

Load-bearing premise

The argument assumes that a quark-antiquark pair with a vanishing decay constant must be either a pair of free, uncorrelated quarks or a single very large correlated state, and that a vanishing local amplitude cannot belong to a compact state.

Editorial extensions

If this is right

The swelling of pion-like states above $T_{\rm ch}$ follows from current algebra and lattice data, so it is not tied to the specific confining model in which it was first seen.
Pion-like peaks that persist until roughly $3T_{\rm ch}$ are compatible with, and indeed expected from, the presence of large correlated color-singlet states.
The result gives a microscopic reason for the small mean free path and high collectivity of the hot matter created in heavy-ion collisions: overlapping, large color-singlet systems.
The argument establishes that the size is 'essentially larger' but not by how much; quantifying the factor requires dedicated lattice studies above $T_{\rm ch}$.

Reading between the lines

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

If the logic extends beyond the pion channel, kaon-like and eta-like pseudoscalar excitations above $T_{\rm ch}$ should also swell, since they share the same Goldstone-structure relation to the order parameter.
A compact state with a nodal wavefunction, for instance a 'donut'-shaped quark-antiquark distribution, would also make the local amplitude vanish while keeping the radius normal; a direct size measurement, not correlator shapes, is what would distinguish these alternatives.
If the swelling is real, the stringy-fluid picture predicts transport imprints - near-zero mean free path and strong collectivity - that could be tested against measured elliptic-flow systematics across beam energies.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read. The paper's central claim—that the swelling of pion-like excitations above Tch is model-independent and required by current algebra—is not established. The load-bearing step is the two-option dichotomy in Section 4: F_pi vanishes either because quarks are uncorrelated plane waves (deconfinement) or because the pion-like state is huge with a vanishing Bethe-Salpeter amplitude at the origin. Lattice correlators are used to rule out (i), leaving (ii). The gap is that a vanishing local amplitude does not imply a large radius. A normalized state with a node at the origin, or with support displaced from r=0, can have F_pi=0 and finite root-mean-square size. Current algebra constrains the local matrix element, not the spatial distribution of the wavefunction. The model in Section 3 produces a divergent wavefunction at p=0, but that divergence is a dynamical consequence of the confining potential, not a consequence of current algebra. So the "model-independent and required" conclusion overreaches.

What the paper does well: it states the problem clearly, assembles the known ingredients (GMOR, F_pi as an order parameter, lattice pion peaks) in one place, and is honest about its limitations—it explicitly concedes it cannot quantify the swelling. The connection to collectivity is worth taking seriously, and if a large swelling were real, it would matter for the hot-QCD program. The heavy self-citation is defensible because the model prediction originates in the author's earlier work, and the lattice inputs are published results from other groups.

The weakest point is exactly where the stress-test lands: the dichotomy in Section 4 is not exhaustive, and the sub-claim that a correlated state with vanishing local amplitude must be huge is not proven. A donut-shaped amplitude or a node at the origin is a clear counterexample to the logic. The paper is not incoherent—it is a plausible physical scenario supported by a specific model—but the model independence claim fails as stated.

For a referee: yes, I would send this to peer review. The question is interesting, the argument is compact and checkable, and the flaw is isolatable and fixable. The right outcome is major revision—either weaken the claim to a model-based prediction with current-algebra support, or prove the dichotomy under stated assumptions. I would not cite the model-independence claim as established, but the paper is a useful contribution to the discussion.

Referee Report

4 major / 4 minor

Summary. The paper argues that the large spatial swelling of pion-like excitations above the chiral restoration temperature Tch is not a peculiarity of the confining model of Ref. [15] but a model-independent consequence of current algebra together with lattice evidence. The argument is: (1) the pion decay constant F_pi, defined through the axial-current matrix element in Eq. (4), vanishes at chiral restoration; (2) this can happen either because the quark and antiquark are uncorrelated plane waves or because the pion-like state is a correlated state with a huge size; (3) lattice correlators reviewed in Section 2 rule out uncorrelated plane waves; hence (4) the pion-like state above Tch must be huge. The paper explicitly concedes that the argument does not quantify the size, and refers to dedicated lattice studies for that purpose.

Significance. If the central claim were established, it would be significant: it would elevate a qualitative model prediction to a robust consequence of QCD symmetries, with direct implications for the interpretation of the hot QCD medium as a 'stringy fluid' and for the explanation of its small mean free path. The paper is also careful and transparent in presenting the lattice correlators, the model equations, and the model wavefunctions, and it contains no fitted parameters in the 'proof' part. However, the key logical step connecting F_pi=0 to a large spatial size is not justified by current algebra; the argument rests on a non-exhaustive dichotomy and therefore does not deliver the claimed model-independent result. The limitations are acknowledged by the author, but the central conceptual gap is load-bearing.

major comments (4)

[Section 4, paragraph beginning 'For us it is important'] The dichotomy between option (i) 'not correlated plane waves' and option (ii) 'a correlated state with a huge size so that the Bethe-Salpeter amplitude ... at the origin vanishes' is not exhaustive. The pion decay constant in Eq. (4) is the matrix element of the local axial current at x=0; it constrains the Bethe-Salpeter amplitude at vanishing quark-antiquark separation, but it does not determine the root-mean-square radius of the state. A normalized wavefunction with a node at the origin, or one whose probability density is displaced away from r=0, has a vanishing local amplitude at zero separation while having a finite and even modest root-mean-square radius. Current algebra and the GMOR relation contain no information about the spatial distribution of the state, so they do not force the swelling claimed in the conclusion; the inference requires an additional dynamical assumption that is not part of the 'model-independent' premise.
[Section 4, same paragraph; Section 2] The lattice evidence is used to rule out option (i), but it does not single out the large-size option. The correlators in Figures 1 and 3 differ strongly from the free quark loop, which demonstrates non-perturbative dynamics and the presence of pion-like spectral peaks, yet it does not measure the spatial size of those excitations. Strongly interacting but deconfined matter can also produce non-perturbative correlators and bound-state-like spectral structures without the specific divergence of the wavefunction at p=0 obtained in the model. Thus the combination of F_pi=0 and the lattice correlators is compatible with several dynamical scenarios, not uniquely with the 'huge size' scenario; the paper's inference from these data to the swelling is therefore not forced.
[Sections 3 and 4] The connection between a vanishing Bethe-Salpeter amplitude at the origin and a diverging root-mean-square radius is demonstrated in Section 3 only within the specific confining model of Eq. (1) solved in rainbow-ladder approximation. The wavefunctions in Figure 4 diverge at p=0 for a particular Hamiltonian with an instantaneous linear confining potential; this is a model result, not a current-algebra theorem. By importing as the only non-perturbative alternative a correlated state from that model, the Section 4 argument becomes circular in the sense that the only option considered is the one the model already produces. The paper would need an independent argument that a vanishing local amplitude implies an unbounded size in any confining theory; no such argument is provided.
[Section 4, Eq. (5)] The statement that F_pi 'must vanish at Tch' holds exactly only in the chiral limit. At physical quark masses the transition is a smooth crossover, and neither the quark condensate nor F_pi vanishes exactly. The GMOR formula (5) is an expansion in quark masses and is not valid above the chiral restoration crossover. Consequently, the model-independent argument, even if the earlier dichotomy were accepted, cannot by itself establish an 'essentially larger' size at physical quark masses; the factor of about 5 quoted in the Introduction and Section 3 is a model prediction, not a consequence of Eq. (5). The paper should state this limitation explicitly in the derivation, not only in the context of quantifying the size.

minor comments (4)

[Section 2, Figure 3 caption] The caption contains a duplicated word: 'with with the correlators' should read 'with the correlators'.
[Section 4, text after Eq. (4)] The phrase 'the Bethe-Salpeter amplitude' in Eq. (4) is potentially misleading: Eq. (4) is the matrix element of a local operator, while the Bethe-Salpeter wavefunction depends on the relative coordinate and enters in a more differential way. Rephrase to avoid implying that the local current matrix element alone fixes the full wavefunction.
[Section 4, last paragraph] There are small typographical errors: 'apriori' should be 'a priori', and 'K¨allen-Lehman' should be 'Källén-Lehmann'.
[References] Several references have 'at al.' instead of 'et al.' (for example Refs. [1], [2], [3]); these should be corrected.

Circularity Check

1 steps flagged · score 8.0 of 10

The claimed model-independent proof of pion swelling above Tch is circular: the 'huge size' conclusion is already written into the second disjunct of the dichotomy, and the identification of a vanishing origin amplitude with large radius is imported from the author's own confining model.

self definitional [Section 4, two-case dichotomy after the GMOR relation]
"This can happen in two cases: (i) The quark and antiquark in the chirally symmetric phase are not correlated plane waves. This is typical for the quark-gluon plasma where deconfined quarks and gluons satisfy perturbation theory. (ii) The quark-antiquark pion-like state is a correlated state (bound state or resonance) with a huge size so that the Bethe-Salpeter amplitude for the quark and antiquark to be at the origin vanishes. This is the case discussed within a model in the previous section."

Branch (ii) already asserts 'a huge size'; after lattice data rules out branch (i), the argument simply extracts that asserted size back out. Eq. (4) fixes only the local quark-antiquark amplitude at x=0, not the root-mean-square radius; a wavefunction with a node or with support away from the origin has vanishing local amplitude yet finite size. Current algebra gives no reason that F_pi=0 forces large radius. The identification is taken from the author's previous confining model ('This is the case discussed within a model in the previous section'), so the model-independent proof reduces to the model result it claims to go beyond.

full rationale

The paper's derivation chain is: GMOR/current algebra identifies F_pi as an order parameter that vanishes at Tch; the vanishing can happen either as free plane waves (deconfined) or as a correlated bound state with huge size whose Bethe-Salpeter amplitude at the origin vanishes; lattice correlators rule out the first alternative; hence the second holds. The circular step is in the dichotomy itself: the 'huge size' in the second disjunct is not derived from the vanishing of F_pi. The paper's Eq. (4) relates F_pi only to the local quark-antiquark amplitude at x=0, while 'size' refers to the root-mean-square radius; a normalized wavefunction with a node at the origin or with support displaced from the origin has vanishing local amplitude without being large. The paper supplies no current-algebra argument excluding such shapes. Instead, branch (ii) is explicitly labeled 'the case discussed within a model in the previous section', i.e., the author's confining-model solution from Ref. [15], and that model result is elevated to an exhaustive alternative. The conclusion is therefore already encoded in the premise, and the 'model-independent' claim reduces to re-asserting the model's conclusion via a self-citation. No fitting is involved and the lattice comparison is extrinsic evidence, but the central logical step is circular by construction. The paper is honest that it cannot quantify the size; that honesty does not cure the definitional character of the argument.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central argument of Section 4 uses no fitted numbers, which is a positive feature: it relies only on the GMOR relation, the interpretation of F_pi as the local quark-antiquark amplitude, and the lattice observation of pion-like peaks above Tch. The fragility is in the axioms, especially the asserted two-option dichotomy and the equivalence between a vanishing local amplitude and an infinite size, both imported from the model of Section 3 rather than derived from current algebra. No new entities are postulated; the stringy fluid and chromoelectric string concepts are carried over from the author's earlier work without new falsifiable handles.

assumptions (4)

domain assumption The GMOR relation (Eq. 5) holds in the hadron gas phase below Tch, with F_pi and the quark condensate as order parameters that vanish as T approaches Tch from below (a second-order transition in the chiral limit).
Section 4: 'In a dilute hadron gas below Tch the pion state can be defined as in vacuum. Hence the GMOR formula can be derived as in vacuum.' Near a crossover the gas is not dilute and the finite-temperature extension of GMOR is assumed rather than derived.
ad hoc to paper The only two ways for F_pi to vanish at Tch are uncorrelated plane-wave quarks (deconfined plasma) or a correlated state with a huge size.
Section 4: 'This can happen in two cases: (i)... (ii)...' No proof of exhaustiveness is given; the paper dismisses the possibility of a correlated state with a nodal wavefunction or a resonance whose axial coupling is suppressed by symmetry.
ad hoc to paper A correlated pion-like state with a vanishing Bethe-Salpeter amplitude at the origin must be infinitely large in the chiral limit and about a factor 5 larger at physical quark masses.
Section 4, option (ii): 'with a huge size so that the Bethe-Salpeter amplitude for the quark and antiquark to be at the origin vanishes.' This equivalence is the model result of Ref. [15]; current algebra alone gives only the vanishing of the local amplitude, not the divergence of the radius.
domain assumption Pion-like peaks persist as correlated states above Tch, disappearing around 3 Tch, and the QCD correlators differ from free-quark correlators so strongly that the perturbative deconfined option is excluded.
Sections 2 and 4, based on Refs [2,3,6]. The peaks come from spectral reconstruction [6] and the interpretation of the correlator degeneracies as a confining regime is the author's framework [12,13,28], which is not universally accepted.

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Pith. "Pith review of A large size of the pion-like excitations in the stringy fluid above $T_{ch}$ is model independent and required by current algebra." pith.science (2026). https://pith.science/paper/ISOWRWP3

@misc{pith2026260804563,
  author       = {Pith},
  title        = {Pith review of: A large size of the pion-like excitations in the stringy fluid above $T_ch$ is model independent and required by current algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ISOWRWP3}},
  note         = {Machine review of arXiv:2608.04563}
}

abstract

Multiple lattice evidences support the existence of a confining but chirally symmetric stringy fluid regime of QCD above the chiral symmetry restoration temperature at T_{ch} ~ 155 MeV. This regime is characterized by an approximate chiral spin symmetry and its extensions which means that the propagating excitations represent the chirally symmetric quarks connected into color singlets by the chromoelectric string. Clear \pi,\pi' peaks above T_{ch} were extracted on the lattice from the spatial and temporal correlators, which become broader with temperature and disappear roughly at $3T_{ch}$. The meson-like excitations above T_{ch} were studied within the manifestly confining and chirally symmetric model. It has been demonstrated that the chiral symmetry restoration in the confining regime happens because of Pauli blocking of the levels, required for the existence of the quark condensate, by the thermal quark excitation. The same Pauli blocking leads to a huge swelling of the low-spin mesons above T_{ch} which become infinitely large in the chiral limit. This property should be crucial for the explanation of the high collectivity and a very small mean-free path of the constituents above $T_{ch}$ seen experimentally. Here we demonstrate that the swelling of pions above T_{ch} is a model-independent effect required by current algebra.

Figures

Figures reproduced from arXiv: 2608.04563 by the authors.

**Figure 1.** Temporal correlation functions for 12 × 483 lattices. The l.h.s. shows correlators calculated with free noninteracting quarks with manifest U(1)A and SU(2)L × SU(2)R symmetries. The r.h.s. presents full QCD results at a temperature 220 MeV, which shows multiplets of all U(1)A, SU(2)L × SU(2)R, SU(2)CS and SU(4) groups. From Ref. [3] [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Left diagram - free quark loop correlator, right diagram -mesonic correlator with the dashed line [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Comparison of the full QCD spatial mesonic correlators (solid lines) with with the correlators [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The ground state (n = 0) wave functions ψ±(p) for J P C = 0−+ at different temperatures. For J P C = 0−+ and T < Tch, the corresponding pseudoscalar meson is a massless Goldstone boson with ψ+(p) = ψ−(p). At T > Tch ψ+(p) > ψ−(p), so for each temperature there are two …

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A large size of the pion-like excitations in the stringy fluid above $T_{ch}$ is model independent and required by current algebra

Introduction Lattice studies indicated the emergence of approximate chiral spin SU (2)CS and SU (4) symmetries in hot QCD matter above Tch thus suggesting that the matter should still be in the confining regime [1–4]. This regime of QCD was called a stringy fluid, where the propagating degrees of freedom are chirally symmetric quarks connected into color-...

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Comparison of the QCD meson correlators above Tch with the free quark loop correlators For pedagogical reasons and to establish a connection with the model-independent state- ment about the large size of the pion-like excitations above chiral restoration (that will be discussed in the subsequent sections) we begin with the overview of the lattice meson co...

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A giant swelling of the pion-like excitations above Tch in a confining and chirally symmetric model The model Hamiltonian is the QCD Hamiltonian in the Coulomb gauge where its gluonic part retains only the instantaneous confining term H = Z d3x ψ†(x, t) (−iα · ∇ + βm) ψ(x, t) + 1 2 Z d3x d3y ρa(x)Vconf (|x − y|)ρa(y), (1) which includes the interaction of...

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A model-independent derivation of a large swelling of the pion-like excitations above Tch Here we present a proof that the results of the previous section about a huge size of the pion-like excitation in the chirally symmetric confining phase, obtained within a model, are actually model-independent. The proof is based on the well-known results of current ...

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Conclusions In this paper we have presented a model-independent statement that the pion-like ex- citations above the chiral symmetry restoration transition are the correlated states (bound 7 states or resonances) with the essentially larger size than in the hadron gas phase. The ar- gument is based on the validity of the current algebra and of the Gell-Ma...

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The research is supported through the grant PAT3259224 of the Austrian Science Fund (FWF)

Acknowledgments The author thanks Tom Cohen and Aleksey Nefediev for a careful reading of the manuscript and consequent discussions. The research is supported through the grant PAT3259224 of the Austrian Science Fund (FWF)

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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L. Y. Glozman, Eur. Phys. J. C 85 (2025) no.11, 1358 doi:10.1140/epjc/s10052-025-15080-2 [arXiv:2508.05277 [hep-ph]]

arXiv 2025

[2] [1]

A large size of the pion-like excitations in the stringy fluid above $T_{ch}$ is model independent and required by current algebra

Introduction Lattice studies indicated the emergence of approximate chiral spin SU (2)CS and SU (4) symmetries in hot QCD matter above Tch thus suggesting that the matter should still be in the confining regime [1–4]. This regime of QCD was called a stringy fluid, where the propagating degrees of freedom are chirally symmetric quarks connected into color-...

work page Pith review arXiv 2026

[3] [2]

Comparison of the QCD meson correlators above Tch with the free quark loop correlators For pedagogical reasons and to establish a connection with the model-independent state- ment about the large size of the pion-like excitations above chiral restoration (that will be discussed in the subsequent sections) we begin with the overview of the lattice meson co...

work page

[4] [3]

The quark kinetic part is chirally symmetric while the confining part is invariant under larger symmetry groups: SU (2)CS , SU (2NF ), and SU (2NF ) × SU (2NF ) [12, 13]

A giant swelling of the pion-like excitations above Tch in a confining and chirally symmetric model The model Hamiltonian is the QCD Hamiltonian in the Coulomb gauge where its gluonic part retains only the instantaneous confining term H = Z d3x ψ†(x, t) (−iα · ∇ + βm) ψ(x, t) + 1 2 Z d3x d3y ρa(x)Vconf (|x − y|)ρa(y), (1) which includes the interaction of...

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[5] [4]

A model-independent derivation of a large swelling of the pion-like excitations above Tch Here we present a proof that the results of the previous section about a huge size of the pion-like excitation in the chirally symmetric confining phase, obtained within a model, are actually model-independent. The proof is based on the well-known results of current ...

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[6] [5]

Conclusions In this paper we have presented a model-independent statement that the pion-like ex- citations above the chiral symmetry restoration transition are the correlated states (bound 7 states or resonances) with the essentially larger size than in the hadron gas phase. The ar- gument is based on the validity of the current algebra and of the Gell-Ma...

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[7] [6]

The research is supported through the grant PAT3259224 of the Austrian Science Fund (FWF)

Acknowledgments The author thanks Tom Cohen and Aleksey Nefediev for a careful reading of the manuscript and consequent discussions. The research is supported through the grant PAT3259224 of the Austrian Science Fund (FWF)

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[8] [7]

Rohrhofer at al., Phys

C. Rohrhofer at al., Phys. Rev. D 96 (2017), 094501 [erratum: Phys. Rev. D 99 (2019), 039901]

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