Screening masses of positive- and negative-parity hadron ground-states, including those with strangeness

Chen Chen; Fei Gao; Si-xue Qin

REVIEW 3 major objections 4 minor 1 cited by

Screening masses of positive- and negative-parity hadron ground-states, including those with strangeness

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that in a symmetry-preserving contact-interaction model of QCD, every ground-state hadron parity-partner pair becomes degenerate above the critical temperature, with positive-parity screening masses dipping near $T_c$…

desk verdict A technically careful SCI extension to strange and negative-parity hadron screening masses, but the headline diquark-survival prediction is substantially an input through an ad hoc g-factor, not a dynamical output. read the letter →

arxiv 2412.15045 v3 pith:KMGNNZQ5 submitted 2024-12-19 hep-ph hep-exhep-latnucl-th

Chen Chen , Fei Gao , Si-xue Qin This is my paper · ORCID

classification hep-phhep-exhep-latnucl-th

keywords screeningmassesparitypartnerschiralsymmetryrestorationdiquarkcorrelationscontactinteractionfinite-temperatureQCDstrangenessFaddeevequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper computes screening masses of the ground-state mesons and baryons of flavour SU(3) at finite temperature, using a symmetry-preserving contact-interaction model of quantum chromodynamics. It claims that every parity-partner pair—meson, baryon, and the diquark correlations that build baryons—becomes degenerate above the critical temperature, with positive-parity screening masses staying flat, dipping near $T_c$, and then rising to meet their negative-parity partners. It further claims that inside spin-1/2 baryons only the $J=0$ scalar and pseudoscalar diquark correlations survive at high temperature, their fractions approaching each other as the $J=1$ channels switch off. The significance is a qualitative guide to what lattice QCD should see and to which diquark degrees of freedom matter for baryon correlation functions after deconfinement.

What carries the argument

The engine of the calculation is the symmetry-preserving treatment of a vector$\times$vector contact interaction: the gluon propagator in the rainbow-ladder kernel is replaced by a momentum-independent constant $g^2D_{\mu\nu}(k)=\delta_{\mu\nu}\,4\pi\alpha_{\rm IR}/m_G^2$, which makes every equation algebraic. Temperature enters through the dressed-quark mass obtained from the gap equation, the temperature-dependent infrared regulator $\Lambda_{\rm ir}^m(T)$ (set to zero for $T>T_d$, implementing deconfinement), and three families of spin-orbit factors—$g^{q\bar q}_{0^+}$, $g^{q\bar q}_{1^+}$, $g^{qq}_{0^-}$, $g^{qq}_{1^-}$, and $g^{P_B P_d}_{DB}$—that multiply the kernel in channels where the rainbow-ladder truncation is known to fail; each interpolates linearly from a value fitted at $T=0$ to unity at $T\gtrsim T_c$. Baryons are built from the quark$+$diquark Faddeev equation, with all four diquark channels (scalar, pseudoscalar, axial-vector, vector) included, and diquark fractions are read off from the unit-normalised Faddeev amplitudes.

What would settle it

A lattice QCD determination of the transverse screening mass of the $a_1$ meson (or of the $\sigma$/$\kappa$ channels where accessible) between $T_c$ and $2T_c$ that shows no decrease near $T_c$ before turning upward would falsify the predicted dip for positive-parity mesons; likewise, a lattice baryon correlation function that projects onto scalar plus pseudoscalar versus axial-vector diquark quantum numbers at $T\sim 2T_c$ would test the claim that only $J=0$ diquark correlations survive.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that in the SCI-RL truncation, with the temperature-dependent infrared regulator $\Lambda_{\rm ir}^m(T)$ and the spin-orbit factors $g_{\rm SO}(T)$, the screening masses of all ground-state $J^P=0^\pm,1^\pm$ mesons, $J^P=1/2^\pm,3/2^\pm$ baryons, and the diquark correlations inside them behave in a common way: negative-parity masses rise monotonically, positive-parity masses stay nearly constant up to $T_c/2$, dip as $T$ approaches $T_c$, then turn around and merge with their partners above $T_c$. The quantitative degeneracy temperatures satisfy e.g. $T_c^{\delta m, S=0}_{\sigma\pi}=1.95\,T_c$ for the pion/$\sigma$ pair; for $J=1$ meson pairs and for baryons the degeneracy temperature grows with strangeness. Within $J=1/2$ baryons the scalar diquark fractions stay dominant, the pseudoscalar fractions grow strongly above $T_c$ and approach the scalar ones, while the axial-vector and vector fractions decay, so in the large-$T$ limit only $J=0$ scalar and pseudoscalar diquark correlations remain. The meson results reproduce the qualitative pattern seen in lattice QCD, including the ordering of screening masses after degeneracy.

Load-bearing premise

The central prediction relies on an ad hoc temperature interpolation of the spin-orbit strength factors; if that interpolation does not represent how spin-orbit effects actually weaken in QCD, the predicted dip and the quoted degeneracy temperatures would be artefacts of the model.

Editorial extensions

If this is right

Above $T_c$, screening masses of every parity-partner pair studied become degenerate, so the restoration of chiral symmetry in the screening spectrum is a model prediction that can be compared channel by channel with lattice data.
For $J=1$ meson pairs and for baryons, the temperature at which degeneracy sets in increases with strangeness, so strangeness delays parity restoration in the screening spectrum.
In positive-parity $J=1/2$ baryons, negative-parity (pseudoscalar and vector) diquark correlations, negligible at $T=0$, grow with temperature, so any realistic baryon calculation at $T>T_c$ should include them.
At very large temperature, the Faddeev equation for $J=1/2$ baryons can be truncated to scalar and pseudoscalar diquarks only, since the axial-vector and vector channels decouple.
Screening masses of mesons and baryons approach but do not exceed $2\pi T$ and $3\pi T$ in this model, so the model underestimates the approach to the free-gas limit compared to lattice results that show an overshoot.

Reading between the lines

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

The predicted dip of positive-parity screening masses near $T_c$ is likely a generic consequence of the spin-orbit factor interpolation rather than of QCD itself; a sharper test would be to compute the same quantities with the $T$-dependent factor replaced by a constant and compare how the dip and $T_c^{\delta m}$ change.
The diquark-fraction prediction (only $J=0$ scalar and pseudoscalar survive, with equal fractions) could be probed in lattice QCD via three-point correlation functions that measure the quark-quark content of the nucleon at finite temperature.
The same framework could be applied to heavy-quark hadrons; the monotonic increase of $T_c^{\delta m}$ with strangeness suggests that charm and bottom parity partners would degenerate at even higher temperatures relative to $T_c$.
The model's result that $J=1$ diquarks vanish while $J=0$ survive above $T_c$ echoes the pattern expected if diquark correlations transmute into scalar and pseudoscalar quark-quark condensates in the chirally symmetric phase, a connection the paper does not make.

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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

Technically careful extension of the SCI framework to finite-T screening masses with strangeness, all diquark channels, and negative-parity baryons. The catch: the most novel claim — only J=0 scalar and pseudoscalar diquarks survive at high T in spin-1/2 baryons — is largely an input to the model, put in through the temperature-dependent g-factor in Eq. (47), not a dynamical result of solving the equations.

What earns credit: the gap, Bethe-Salpeter and Faddeev equations are solved consistently for the specified model; the T=0 masses reproduce earlier SCI results; the meson screening masses show the qualitative lattice pattern (positive-parity masses dip near T_c then rise to meet their monotonic negative-parity partners); and strangeness dependence, negative-parity partners, and the T-evolution of diquark fractions are genuinely new outputs. The authors also flag limitations themselves: the J=3/2 negative-parity pattern they expect to change with a realistic kernel, and screening masses never crossing the free-theory bounds.

The soft spot is the g-factor machinery. Eqs. (23), (35) and (47) interpolate T=0 spin-orbit corrections — fitted to empirical mass splittings — up to about 1 as M_u(T)/M_u(0) falls. Eq. (47) switches the opposite-parity diquark suppression from 0.12 at T=0 to about 0.98 above T_c; the stress-test says 'exactly 1', for physical quark masses it is approximately 1, but the point stands. The pseudoscalar-diquark fractions in Fig. 7 rise because this suppression is switched off. There is some genuine dynamics in the endpoint — J=1 fractions fall partly because J=1 screening masses grow relative to J=0 — but the rate and the crossover shape are controlled by the interpolation. The abstract says 'we observe'; the text more honestly says 'we judge.' A sensitivity study, varying the rate of g(T) or holding g fixed while only the masses run, is needed to show what in Fig. 7 is robust. The same caution applies to the quoted parity-degeneracy temperatures.

Also inherited from Ref. [4] and honestly flagged: T=0 spinors and the representative momentum ratio in the Faddeev calculation. That affects precision, but it is not a new flaw.

Who this is for: the CSM/contact-interaction community, and lattice people who want cheap qualitative screening-mass trends to compare against. It deserves a serious referee. The referee should push for the sensitivity study and for the abstract to be set at the level of model-guided expectations. I would send it to review, and I would not bet on the diquark-fraction plots surviving a sensitivity test unchanged.

Referee Report

3 major / 4 minor

Summary. The paper computes screening masses of flavour-SU(3) ground-state mesons, diquarks, and baryons in a symmetry-preserving vector×vector contact interaction (SCI) at nonzero temperature. The authors solve the gap equation with a temperature-dependent infrared regulator, then solve the Bethe-Salpeter equations for J^P = 0±, 1± mesons and diquarks, and feed these into quark+diquark Faddeev equations for J^P = 1/2± and 3/2± baryons. The main findings are that every parity-partner pair degenerates above the chiral crossover temperature T_c = 0.197 GeV, and that in J^P = 1/2± baryons only J=0 scalar and pseudoscalar diquark correlations survive at very high temperature. The paper also reports critical degeneracy temperatures, e.g., T_c^{δm} = 1.95 T_c for the pion/sigma pair, and compares the meson screening-mass pattern qualitatively with lattice QCD.

Significance. If the results were robust, the paper would provide a useful systematic survey of screening masses across the flavour-SU(3) hadron spectrum in a single symmetry-preserving model, extending earlier SCI work to include all diquark channels and negative-parity baryons. The T=0 masses reproduce previous SCI results, and the consistent treatment of gap, Bethe-Salpeter, and Faddeev equations is a technical strength. However, the most novel claim—that only J=0 scalar and pseudoscalar diquarks survive in J^P=1/2± baryons at high T—is not an emergent dynamical prediction; it is largely controlled by an ad hoc temperature-dependent spin-orbit factor, Eq. (47). Similarly, the quantitative degeneracy temperatures and the dip of positive-parity screening masses near T_c are shaped by the prescribed T-dependence of the factors in Eq. (23). The qualitative parity-degeneracy pattern is consistent with chiral symmetry restoration, but the specific quantitative predictions should be treated as model-dependent expectations rather than robust QCD predictions. The paper is honest about several limitations, notably the acknowledged likely artefact in the J^P=3/2− behaviour near Eq. (55).

major comments (3)

[§V, Eq. (47), Fig. 7] The central claim that at high temperatures only J=0 scalar and pseudoscalar diquark correlations survive within J^P=1/2± baryons is not an emergent result of the Faddeev dynamics: every opposite-parity diquark amplitude in the Faddeev equation is multiplied by g_{P_B P_d}^{DB}(T), which is forced to 1 for T>T_c. Since the pseudoscalar-diquark fractions in positive-parity baryons and the scalar-diquark fractions in negative-parity baryons visibly track the rise of this factor in Fig. 7, the endpoint is effectively inserted by hand. The manuscript should either provide an independent dynamical justification for the T-dependence in Eq. (47), or demonstrate that the same endpoint is reached when the factor is held fixed at its T=0 value (or varied within a plausible range). Without such a sensitivity analysis, the headline prediction is not robust.
[§III, Eqs. (23), (27)–(28)] The quantitative parity-degeneracy temperatures, e.g., T_{[σπ]c}^{δm} = 1.95 T_c, and the non-monotonic dip of the positive-parity meson screening masses near T_c are dominated by the ad hoc interpolation g_{q\bar q,0+}^{SO}(T) and g_{q\bar q,1+}^{SO}(T). These factors are fitted at T=0 to mass splittings and then interpolated to unity via M_u(T)/M_u(0), which is not derived from the model's dynamics or from QCD. Because these factors directly enter the Bethe-Salpeter kernels in Eqs. (A.18) and (A.23), the predicted screening-mass curves and the quoted T_c^{δm} values are not parameter-free predictions. A sensitivity study using alternative T-dependences (e.g., a step function or a slower approach to unity) is needed to establish which qualitative features are robust.
[§V, after Eq. (48)] The Faddeev calculation relies on a 'representative value for the momentum ratio of a diquark in relation to that of the baryon', but the actual value and the sensitivity of the baryon screening masses to this choice are not reported. Since this is one of the parameters that controls the T=0 masses and hence the T>0 behaviour, the manuscript should state the chosen value and provide at least a brief demonstration that the results are stable with respect to it.

minor comments (4)

[§II.B, after Eq. (7)] The text says 'Pincar´e-invariant' where 'Poincaré-invariant' is intended.
[§III, before Eq. (18)] The word 'Matusbara' appears twice; it should be 'Matsubara'.
[Appendix A.4, first paragraph] The sentence 'Inserting Eq. (20) into (18) and using Eq.(47)' should refer to Eq. (23a) or the scalar-meson spin-orbit factor, not Eq. (47), which is defined later for the Faddeev baryon calculation.
[§V, after Eq. (48)] The phrase 'Bethe-Salpleter amplitudes' contains a typo; it should be 'Bethe-Salpeter amplitudes'.

Circularity Check

3 steps flagged · score 6.0 of 10

The headline parity-partner degeneracy and the growth of pseudoscalar-diquark fractions are built into the model through ad hoc, temperature-dependent spin-orbit factors (Eqs. 23, 35, 47) that are made to approach 1 above T_c.

fitted input called prediction [Section III, Eqs. (23a)-(23b) and text following Eq. (23); used in Eqs. (A.18), (A.23)]
"To calculate the screening masses, in this article we keep employing the method given in Ref. [4], viz., we multiply the SCI kernel (3) by the T -dependent factors g_{q\bar q,0+}^{SO}(T)=1-M_u(T)/M_u(0)(1-0.322), (23a) ... As T increases, g_{q\bar q,0+}^{SO}(T) and g_{q\bar q,1+}^{SO}(T) become smaller and trend to 1 for T > T_c; consequently, DCSB can be traced."

At T=0 these factors take values 0.322 and 0.252, fitted to reproduce empirical sigma-rho and a1-rho splittings; for T>T_c they are forced to 1. Because they multiply only the positive-parity scalar/axial-vector BSE kernels, the positive-parity screening masses are deliberately driven toward their negative-parity partners as T approaches T_c. The degeneracy temperatures quoted in Eqs. (27)-(28) are therefore consequences of this assumed interpolation, not independent QCD-level predictions.
fitted input called prediction [Section IV, Eqs. (35a)-(35b), with T=0 values in Eqs. (36) and text after Eq. (35)]
"It was found that for their diquark partners, i.e., the pseudoscalar and vector diquarks, the similar factors are also necessary [59, 61, 63]. We define g_{qq,0-}^{SO}(T)=1-M_u(T)/M_u(0)(1-0.582), (35a) ... For T \gtrsim T_c, g_{qq,0-}^{SO} and g_{qq,1-}^{SO} trend to 1."

The pseudoscalar and vector diquark channels are suppressed at T=0 by factors selected to keep baryon masses realistic, and are then set to 1 above T_c by the same M_u(T)/M_u(0) interpolation. This directly produces the diquark parity-partner degeneracy in Fig. 5 and the critical degenerate temperatures in Eqs. (39)-(40); the diquark-sector restoration is pre-installed in the kernel rather than emerging from the dynamics.

1 more flagged steps

fitted input called prediction [Section V, Eq. (47)-(48), with consequences shown in Fig. 7]
"Analogous to Eqs. (23) and (35), we define g_{P_B P_d}^{DB}(T) = ( 1 : if P_B = P_d; 1-M_u(T)/M_u(0)(1-0.12) : if P_B = -P_d ). (47) As a result, in Faddeev equation calculations, a diquark's Bethe-Salpeter amplitude \Gamma_{[fg]}^{JP_d}(Q_0;T) should be modified as g_{P_B P_d}^{DB}(T) \times \Gamma_{[fg]}^{JP_d}(Q_0;T). (48)"

This factor suppresses every opposite-parity diquark amplitude inside a spin-1/2 baryon by 0.12 at T=0 and is forced to 1 above T_c. The strong growth of pseudoscalar-diquark fractions in positive-parity baryons and of scalar-diquark fractions in negative-parity baryons, shown in Fig. 7, follows this factor. The abstract's headline that only J=0 scalar and pseudoscalar diquarks survive is therefore substantially installed by Eq. (47); the final scalar/pseudoscalar-symmetric state is an input, not an independent Faddeev prediction. (The separate decline of J=1 fractions is more dynamical.)

full rationale

The paper is transparent and internally consistent: screening masses are obtained by solving gap, Bethe-Salpeter, and Faddeev equations, and the meson results are explicitly benchmarked against contemporary lQCD. There is no hidden renaming or a uniqueness theorem imported by self-citation. However, the most prominent new results—parity-partner degeneracy above T_c for mesons, diquarks, and baryons, and the survival only of J=0 scalar/pseudoscalar diquarks in J^P=1/2 baryons—are substantially controlled by Eqs. (23), (35), and (47). Those equations multiply the parity-changing channels by T-dependent factors whose zero-temperature values are fitted to spectroscopy and whose high-temperature limit is simply set to 1. Thus the central predictions are partly baked into the model kernel by construction. The calculation is not a statistical fit to screening data, so the circularity is partial rather than total; but the novel endpoint claimed in the abstract reduces to an assumed interpolation, not to an emergent consequence of QCD dynamics. This warrants a score of 6.

Assumptions & free parameters 12 free parameters · 8 assumptions · 0 invented entities

The calculation rests on a substantial set of model inputs: interaction parameters (alpha_IR, m_G, Lambda_ir, Lambda_uv), current quark masses fitted to the pion and kaon, spin-orbit factors fitted to T=0 mass splittings, and an ad hoc temperature-dependent interpolation of those factors. It also relies on the rainbow-ladder truncation and the quark+diquark approximation. No new fundamental entities are introduced.

free parameters (12)

alpha_IR = 0.93*pi
Model parameter simulating zero-momentum strength of the running coupling; sets the interaction strength.
m_G = 0.8 GeV
Gluon mass scale in the SCI ansatz.
Lambda_ir = 0.24 GeV
Infrared regulator implementing confinement at T=0; becomes temperature-dependent via Eq. (17).
Lambda_uv = 0.905 GeV
Ultraviolet regulator; sets the scale of all dimensionful quantities in the non-renormalizable SCI.
m_u = 0.007 GeV
Current u/d quark mass, fitted to reproduce the pion mass.
m_s = 0.17 GeV
Current strange quark mass, fitted to reproduce the kaon mass.
g_qqbar_0+_SO(0) = 0.322
Spin-orbit suppression factor for scalar mesons, fixed by the empirical sigma-rho mass splitting.
g_qqbar_1+_SO(0) = 0.252
Spin-orbit suppression factor for axial-vector mesons, fixed by the empirical a1-rho mass splitting.
g_qq_0-_SO(0) = 0.582
Spin-orbit factor for pseudoscalar diquarks, selected so baryon masses are realistic.
g_qq_1-_SO(0) = 0.452
Spin-orbit factor for vector diquarks, selected so baryon masses are realistic.
g_PB_Pd_DB(0) = 0.12
Factor suppressing diquark amplitudes with parity opposite to the baryon, chosen to fix negative-parity baryon masses.
representative momentum ratio
A representative ratio used to remove relative-momentum dependence in the quark-diquark Faddeev equation, following Ref. [4].

assumptions (8)

domain assumption Rainbow-ladder truncation of the two-body scattering kernel.
Leading order of the symmetry-preserving truncation scheme; neglects non-RL diagrams.
domain assumption Contact interaction ansatz g^2 D_mu_nu(k) = delta_mu_nu * 4*pi*alpha_IR/m_G^2.
The gluon propagator is replaced by a momentum-independent contact interaction.
domain assumption Chiral symmetry restoration and deconfinement happen simultaneously at zero chemical potential.
Used to set T_d = T_c and to define the temperature-dependent infrared regulator.
ad hoc to paper Temperature-dependent infrared regulator given by Eq. (17).
The functional form is constructed, not derived from QCD, and is chosen to vanish above T_d.
ad hoc to paper Temperature-dependent spin-orbit factors g(T) = 1 - (M_u(T)/M_u(0))(1 - g(0)).
Interpolates between T=0 fitted values and unity above T_c; drives the parity-partner degeneracy pattern.
domain assumption Baryon spinors are frozen at their T=0 forms.
The paper uses T=0 Dirac and Rarita-Schwinger spinors at finite T for simplicity.
domain assumption Relative-momentum dependence in the Faddeev equation is removed by choosing a representative momentum ratio.
Collapses the relative-momentum dependence of the Faddeev amplitudes, following Ref. [4].
domain assumption Isospin symmetry for u and d quarks.
u and d quarks are treated as degenerate throughout.

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Pith. "Pith review of Screening masses of positive- and negative-parity hadron ground-states, including those with strangeness." pith.science (2026). https://pith.science/paper/KMGNNZQ5

@misc{pith2026241215045,
  author       = {Pith},
  title        = {Pith review of: Screening masses of positive- and negative-parity hadron ground-states, including those with strangeness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMGNNZQ5}},
  note         = {Machine review of arXiv:2412.15045}
}

abstract

Using a symmetry-preserving treatment of a vector $\times$ vector contact interaction (SCI) at nonzero temperature, we compute the screening masses of flavour-SU(3) ground-state $J^P=0^\pm$, $1^\pm$ mesons, and $J^P=1/2^\pm$, $3/2^\pm$ baryons. We find that all correlation channels allowed at $T=0$ persist when the temperature increases, even above the QCD phase transition. The results for mesons qualitatively agree with those obtained from the contemporary lattice-regularised quantum chromodynamics (lQCD) simulations. One of the most remarkable features is that each parity-partner-pair degenerates when $T>T_c$, with $T_c$ being the critical temperature. For each pair, the screening mass of the negative parity meson increases monotonously with temperature. In contrast, the screening mass of the meson with positive parity is almost invariant on the domain $T\lesssim T_c/2$; when $T$ gets close to $T_c$, it decreases but soon increases again and finally degenerates with its parity partner, which signals the restoration of chiral symmetry. We also find that the $T$-dependent behaviours of baryon screening masses are quite similar to those of the mesons. For baryons, the dynamical, nonpointlike diquark correlations play a crucial role in the screening mass evolution. We further calculate the evolution of the fraction of each kind of diquark within baryons respective to temperature. We observe that, at high temperatures, only $J=0$ scalar and pseudoscalar diquark correlations can survive within $J^P=1/2^\pm$ baryons.

Figures

Figures reproduced from arXiv: 2412.15045 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. (Left to right) Screening masses of mesons with strangeness [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Quark+diquark Faddeev equation, a linear inte [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. (Left to right) Screening masses of diquarks with strangeness [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. Screening masses of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Temperature evolutions of the fractions of various diquark components of the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Screening masses of [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

**Figure 9.** Figure 9: FIG. 9. Temperature evolutions of the fractions of various diquark components of the [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

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Works this paper leans on

113 extracted references · 70 canonical work pages · cited by 1 Pith paper

[4]

(21), it is straightforward to derive the Bethe-Salpeter equations for vector mesons

V ector ( 1−) mesons and axial-vector( 1+) diquarks Using Eq. (21), it is straightforward to derive the Bethe-Salpeter equations for vector mesons. For a vec- tor meson’s longitudinal component, the Bethe-Salpeter equation is 1 + K1−,∥ [f ¯g] (Q2 0 = −(m1− [f ¯g])2; T ) = 0 , (A.13) where K1−,∥ [f ¯g] = 2αIR 3πm2 G Z 1 0 dα Mf Mg − ˆαM 2 f − αM 2 g − 2α ˆ...
[1]

Symmetry For sensible calculations in hadron physics, accounting for relevant symmetries is crucial. In this work, we will utilise the axial-vector and vector Ward-Green-Takahashi identities (WGTIs) whenever necessary [105–107], and in 17 the SCI framework, they can be expressed as [4] Z 1 0 dα[Ciu(ςf g; T ) + Ciu 1 (ςf g; T ) + Riu(ςf g; T )] = 0 , (A.1)...
[2]

(9), and ϑ2(x) being the Jacobi theta function [89]

= ˆαM 2 f + αM 2 g + α ˆαQ2 0 , (A.2) with ˆα ≡ 1 − α; C iu n (ςf g; T ) = (−1)n n! dn dς n f g Ciu(ςf g; T ) , (A.3a) Ciu n (ςf g; T ) = ς n f gC iu n (ςf g; T ) , (A.3b) and Riu(ςf g; T ) = Z 1/Λ2 ir 1/Λ2uv dτ e−τ ςf g r π τ − d dτ − 1 2τ 2T ϑ2(e−τ 4π2T 2 ) , (A.4) with Ciu defined in Eqs. (9), and ϑ2(x) being the Jacobi theta function [89]. Eq. (A.1) i...
[3]

E0− [f ¯g] F 0− [f ¯g] # = 4αIR 3πm2 G

Pseudoscalar ( 0−) mesons and scalar ( 0+) diquarks Inserting Eq. (19) into (18) produces the Bethe- Salpeter equation for a pseudoscalar meson: " E0− [f ¯g] F 0− [f ¯g] # = 4αIR 3πm2 G " K0−,EE [f ¯g] K0−,EF [f ¯g] K0−,F E [f ¯g] K0−,F F [f ¯g] # " E0− [f ¯g] F 0− [f ¯g] # , (A.6) where K0−,EE [f ¯g] = Z 1 0 dα Ciu(ςf g(α, Q2 0); T ) + Mf Mg − α ˆαQ2 0 −...
[5]

Scalar ( 0+) mesons and pseudoscalar ( 1+) diquarks Inserting Eq. (20) into (18) and using Eq.(47), yields the Bethe-Salpeter equation for a scalar meson 1 + gq ¯q,0+ SO (T ) · K0+ [f ¯g](Q2 0 = −(m0+ [f ¯g])2; T ) = 0 , (A.18) where K0+ [f ¯g] = − 4αIR 3πm2 G Z 1 0 dα Ciu(ςf g(α, Q2 0); T ) − Ciu 1 (ςf g(α, Q2 0); T ) − (Mf Mg + α ˆαQ2 0)C iu 1 (ςf g(α, ...
[6]

1”, constituted from valence u- and d-quarks, and is antisymmetric; while t8={ds} indicates the diquark is labeled “8

Axial-vector ( 1+) mesons and vector ( 1−) diquarks At nonzero temperatures, the axial-vector mesons also separate into longitudinal and transverse compo- nents. Using Eqs. (22) and (23b), one obtains the Bethe- Salpeter equation for the longitudinal part of an axial- vector meson 1 + gq ¯q,1+ SO (T ) · K1+,∥ [f ¯g] (Q2 0 = −(m1+,∥ [f ¯g] )2; T ) = 0 , (A...
[7]

C. E. Detar and J. B. Kogut, Phys. Rev. Lett. 59, 399 (1987)

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[1] [4]

(21), it is straightforward to derive the Bethe-Salpeter equations for vector mesons

V ector ( 1−) mesons and axial-vector( 1+) diquarks Using Eq. (21), it is straightforward to derive the Bethe-Salpeter equations for vector mesons. For a vec- tor meson’s longitudinal component, the Bethe-Salpeter equation is 1 + K1−,∥ [f ¯g] (Q2 0 = −(m1− [f ¯g])2; T ) = 0 , (A.13) where K1−,∥ [f ¯g] = 2αIR 3πm2 G Z 1 0 dα Mf Mg − ˆαM 2 f − αM 2 g − 2α ˆ...

[2] [1]

Symmetry For sensible calculations in hadron physics, accounting for relevant symmetries is crucial. In this work, we will utilise the axial-vector and vector Ward-Green-Takahashi identities (WGTIs) whenever necessary [105–107], and in 17 the SCI framework, they can be expressed as [4] Z 1 0 dα[Ciu(ςf g; T ) + Ciu 1 (ςf g; T ) + Riu(ςf g; T )] = 0 , (A.1)...

[3] [2]

(9), and ϑ2(x) being the Jacobi theta function [89]

= ˆαM 2 f + αM 2 g + α ˆαQ2 0 , (A.2) with ˆα ≡ 1 − α; C iu n (ςf g; T ) = (−1)n n! dn dς n f g Ciu(ςf g; T ) , (A.3a) Ciu n (ςf g; T ) = ς n f gC iu n (ςf g; T ) , (A.3b) and Riu(ςf g; T ) = Z 1/Λ2 ir 1/Λ2uv dτ e−τ ςf g r π τ − d dτ − 1 2τ 2T ϑ2(e−τ 4π2T 2 ) , (A.4) with Ciu defined in Eqs. (9), and ϑ2(x) being the Jacobi theta function [89]. Eq. (A.1) i...

[4] [3]

E0− [f ¯g] F 0− [f ¯g] # = 4αIR 3πm2 G

Pseudoscalar ( 0−) mesons and scalar ( 0+) diquarks Inserting Eq. (19) into (18) produces the Bethe- Salpeter equation for a pseudoscalar meson: " E0− [f ¯g] F 0− [f ¯g] # = 4αIR 3πm2 G " K0−,EE [f ¯g] K0−,EF [f ¯g] K0−,F E [f ¯g] K0−,F F [f ¯g] # " E0− [f ¯g] F 0− [f ¯g] # , (A.6) where K0−,EE [f ¯g] = Z 1 0 dα Ciu(ςf g(α, Q2 0); T ) + Mf Mg − α ˆαQ2 0 −...

[5] [5]

Scalar ( 0+) mesons and pseudoscalar ( 1+) diquarks Inserting Eq. (20) into (18) and using Eq.(47), yields the Bethe-Salpeter equation for a scalar meson 1 + gq ¯q,0+ SO (T ) · K0+ [f ¯g](Q2 0 = −(m0+ [f ¯g])2; T ) = 0 , (A.18) where K0+ [f ¯g] = − 4αIR 3πm2 G Z 1 0 dα Ciu(ςf g(α, Q2 0); T ) − Ciu 1 (ςf g(α, Q2 0); T ) − (Mf Mg + α ˆαQ2 0)C iu 1 (ςf g(α, ...

[6] [6]

1”, constituted from valence u- and d-quarks, and is antisymmetric; while t8={ds} indicates the diquark is labeled “8

Axial-vector ( 1+) mesons and vector ( 1−) diquarks At nonzero temperatures, the axial-vector mesons also separate into longitudinal and transverse compo- nents. Using Eqs. (22) and (23b), one obtains the Bethe- Salpeter equation for the longitudinal part of an axial- vector meson 1 + gq ¯q,1+ SO (T ) · K1+,∥ [f ¯g] (Q2 0 = −(m1+,∥ [f ¯g] )2; T ) = 0 , (A...

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