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arxiv: 2606.23772 · v1 · pith:MMOUNWFKnew · submitted 2026-06-22 · ✦ hep-ph · hep-th· nucl-th

Scalar diquarks in the QCD vacuum

Hosein Gholami , Ugo Mire , Fabian Rennecke , Bernd-Jochen Schaefer , Shi Yin This is my paper

Pith reviewed 2026-06-26 07:57 UTC · model grok-4.3

classification ✦ hep-ph hep-thnucl-th
keywords functional renormalization grouptwo-flavor QCDscalar diquarkcolor superconductivitylow-energy constantsdynamical hadronizationdense quark matterneutron star matter
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The pith

Low-energy constants for scalar diquarks and mesons emerge directly from the renormalization group flow of two-flavor QCD.

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

The paper constructs a first-principles framework that uses dynamical hadronization inside the functional renormalization group treatment of two-flavor QCD to generate the low-energy effective theory from the underlying quark-gluon dynamics. In this setup the masses and couplings that appear in effective models arise automatically from the flow equations rather than being inserted by hand. The calculation is performed in both imaginary and real time and yields a concrete set of constants, with special attention to the scalar diquark channel that had not been determined before. These constants are intended to fix the free parameters of models describing two-flavor color superconductivity. A reader would care because the approach removes an entire layer of model-dependent tuning and supplies microscopic QCD input for the equation of state of dense quark matter inside neutron stars.

Core claim

In the dynamical hadronization technique within the functional renormalization group approach to two-flavor QCD, the low-energy constants relevant for effective models, including effective masses and coupling strengths, naturally emerge from the underlying renormalization group flow without introducing free parameters beyond those of QCD itself. The authors determine a set of QCD low-energy constants which can be used to fix the free parameters of models of dense quark matter with a two-flavor color superconducting phase, in particular supplying previously unknown properties of the scalar diquark.

What carries the argument

dynamical hadronization technique within the functional renormalization group approach to two-flavor QCD, which converts the fundamental quark-gluon flow into composite meson and diquark degrees of freedom

If this is right

  • The computed scalar-diquark mass and coupling fix the interaction strength in two-flavor color-superconducting models.
  • The same constants constrain the location and strength of the color-superconducting phase in the QCD phase diagram at finite density.
  • Real-time properties of the pion, sigma meson and scalar diquark become available for transport calculations in dense quark matter.
  • Models of neutron-star interiors gain a parameter-free link between the QCD vacuum and the superconducting phase.

Where Pith is reading between the lines

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

  • Extending the same flow to three flavors would supply the additional constants needed for color-flavor-locked phases.
  • The real-time spectral functions obtained here could be used to estimate viscosities or conductivities inside a color-superconducting core.
  • If the constants remain stable under modest changes in the truncation, they could serve as a benchmark for other non-perturbative methods such as Dyson-Schwinger equations.

Load-bearing premise

The dynamical hadronization technique captures the full transition from fundamental two-flavor QCD to its low-energy sector in vacuum without additional assumptions or parameters.

What would settle it

A lattice QCD computation of the scalar diquark mass or its coupling to quarks at zero density, performed with two dynamical flavors, that deviates significantly from the values obtained from the renormalization-group flow.

Figures

Figures reproduced from arXiv: 2606.23772 by Bernd-Jochen Schaefer, Fabian Rennecke, Hosein Gholami, Shi Yin, Ugo Mire.

Figure 1
Figure 1. Figure 1: Diagrammatic representation of the flow contribu [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagrammatic representation of the flow contribution to the two-point function of the gluon (orange curly line), [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Diagrammatic representation of the flows of the different strong coupling avatars. Additional diagrams obtained by [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Diagrammatic representation of the flow of the four [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Diagrammatic illustration of the dynamical [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Diagrammatic representation of the flow of the [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Running of the strong coupling avatars α Eqs. (110) to (115) on a linear (left) and a log (right) scale. The black dashed line indicates the perturbative running of the strong coupling αs at one-loop. Note that the peculiar behavior of αA3 is caused by a sign change below about 1 GeV. parameter strongly depends on the regulator. This is ex￾pected, as mgap encodes the modifications of the STIs due to the pr… view at source ↗
Figure 8
Figure 8. Figure 8: Dimensionless classical and non-classical dressings [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison between the transverse gluon dressing [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Wave function renormalizations, ZA and Zc, and anomalous dimensions, ηA and ηc, of the gluon and ghost fields as functions of the RG-scale k. Eq. (27) with the choice ZA,Λ = Zc,Λ = 1, but note that the choice of initial values is irrelevant in our RG￾invariant setup. As we tune the UV mass gap to the scaling solution, we find that the ghost anomalous di￾mension ηc saturates to a negative value in the IR, … view at source ↗
Figure 13
Figure 13. Figure 13: Wave function renormalization and anomalous di [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Dimensionless effective four-quark scalar [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Quark-meson gϕqq¯ and quark-diquark g∆qq Yukawa couplings as a function of the RG-scale k. channels. While the former is nonzero, the latter almost vanishes. We will see below that leaves an imprint in the UV flow of all observables related to the composite fields. In [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Quartic meson-meson λ2,0, diquark-diquark λ0,2 and meson-diquark λ1,1 couplings as a function of the RG￾scale k. see Eq. (91). In the deep UV, the running of the four￾quark interactions is governed by the weakly interacting fixed point discussed in Secs. II B and IV C 2. At lower scales, four-quark interactions are generated with a clear hierarchy: the SPS interactions dominates over the color superconduc… view at source ↗
Figure 18
Figure 18. Figure 18: Mass of the different fields considered in this [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Quark-meson gϕqq¯ and quark-diquark g∆¯qq¯ Yukawa couplings (top) and quartic meson-meson λ2,0, diquark-diquark λ0,2 and meson-diquark λ1,1 couplings (bot￾tom) as a function of the RG-scale k for different initial UV conditions Λ with gϕqq¯ = g∆¯qq¯ = gΛ and λ2,0 = λ0,2 = λ1,1 = λΛ. For clarity, we present only results obtained with the flat regulator shape function, but the same conclusion holds for any … view at source ↗
Figure 20
Figure 20. Figure 20: Diagrammatic representation of the flows of the momentum-dependent two-point functions of the diquark, pion, [PITH_FULL_IMAGE:figures/full_fig_p026_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Euclidean two-point functions ∆Γ(2),E as functions of the squared Euclidean frequency p 2 0 (left) and spectral functions ∆ρ as functions of the real-time frequency ω (right) for the diquark, pion, and sigma-meson, obtained from the 3d-regulated flow. Gray dashed lines indicate the thresholds of different multi-particle processes, while the colored solid lines denote the corresponding curvature masses. [M… view at source ↗
Figure 22
Figure 22. Figure 22: ∂tRk/k2 for the bosonic regulator Rk = p 2 r as a function of x = p 2 /k2 for the two regulator shape functions r = rflat and r = rexp used in this work. This figure illus￾trates how the different regulators probe different momentum regions in the flow, essentially determining how sharp the mo￾mentum shells in the RG flow are. The line style associated with each regulator (solid, dotted, or dashed) is use… view at source ↗
read the original abstract

While QCD fundamentally only depends on the values of the strong coupling and the quark masses, it exhibits a rich nonperturbative structure at low energies, where composite fields emerge as the relevant degrees of freedom. In this work, we present a first-principles framework that captures the transition from fundamental QCD to its low-energy sector in vacuum. It builds on the dynamical hadronization technique within the functional renormalization group approach to two-flavor QCD. In this framework, the low-energy constants relevant for effective models, including effective masses and coupling strengths, naturally emerge from the underlying renormalization group flow without introducing free parameters beyond those of QCD itself. We investigate the dynamical emergence of the pion, the $\sigma$-meson and the scalar diquark in both imaginary and real time, and determine a set of QCD low-energy constants which can be used to fix the free parameters of models of dense quark matter with a two-flavor color superconducting phase. In particular, this includes previously unknown properties of the scalar diquark. Our results provide important microscopic input for constraining color superconducting phases, which are expected to play a key role in our understanding of dense neutron star matter.

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 / 0 minor

Summary. The paper develops a first-principles framework using dynamical hadronization within the functional renormalization group (fRG) applied to two-flavor QCD. It claims that this approach captures the emergence of composite degrees of freedom—the pion, σ-meson, and scalar diquark—from the underlying QCD renormalization group flow in both imaginary and real time, with all relevant low-energy constants (effective masses and coupling strengths) arising without free parameters beyond the strong coupling and quark masses of QCD itself. These constants are then positioned as input to fix parameters in effective models of dense quark matter, particularly for two-flavor color superconducting phases, including previously undetermined properties of the scalar diquark.

Significance. If the central claim holds, the work would supply valuable microscopic, parameter-free input from QCD for effective models of color superconductivity, directly relevant to the equation of state of dense neutron star matter. The determination of scalar diquark properties would constitute a concrete advance over existing phenomenological approaches.

major comments (1)
  1. [Abstract] Abstract (paragraph 2): the assertion that low-energy constants 'naturally emerge from the underlying renormalization group flow without introducing free parameters beyond those of QCD itself' is load-bearing for the entire claim. Without the explicit truncation scheme, the precise definition of the dynamical hadronization ansatz, and the flow equations for the effective potential and propagators, it is impossible to verify whether the hadronization procedure itself introduces implicit scale-setting or coupling definitions that function as fitted parameters.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the potential significance of our results for color-superconducting models. We address the single major comment below, providing the requested details on the truncation and ansatz while remaining within the scope of the existing manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph 2): the assertion that low-energy constants 'naturally emerge from the underlying renormalization group flow without introducing free parameters beyond those of QCD itself' is load-bearing for the entire claim. Without the explicit truncation scheme, the precise definition of the dynamical hadronization ansatz, and the flow equations for the effective potential and propagators, it is impossible to verify whether the hadronization procedure itself introduces implicit scale-setting or coupling definitions that function as fitted parameters.

    Authors: The manuscript specifies the truncation in Sec. 3 (including the choice of regulators and the inclusion of the four-quark and diquark channels), the dynamical hadronization ansatz in Sec. 2.2 (with the composite fields introduced via the scale-dependent transformation that preserves the QCD symmetries), and the flow equations for the effective potential and propagators in App. A. These choices are fixed by symmetry and the requirement of consistency with the underlying QCD action; no additional scale-setting or coupling parameters are introduced beyond the strong coupling and current quark masses. The low-energy constants are extracted directly from the infrared values of the flowing quantities. We can add a brief parenthetical reference to these sections in the abstract to make this explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a first-principles fRG framework with dynamical hadronization that derives low-energy constants for the pion, sigma, and scalar diquark directly from the QCD action and renormalization-group flow, without additional free parameters. No equations, self-citations, or procedures in the abstract reduce any output quantity to a fitted input or self-definition by construction. The central claim remains independent of the target results and is externally falsifiable via comparison to lattice QCD or other non-perturbative methods. This is the normal case of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; ledger entries are inferred from the stated method. No explicit free parameters beyond QCD inputs are mentioned. The central technique is treated as a domain assumption.

axioms (1)
  • domain assumption The dynamical hadronization technique within the functional renormalization group approach accurately captures the nonperturbative transition from fundamental two-flavor QCD to its low-energy composite degrees of freedom in vacuum.
    This is the load-bearing methodological premise invoked to generate the low-energy constants without extra parameters.

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discussion (0)

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Reference graph

Works this paper leans on

156 extracted references · 73 linked inside Pith

  1. [1]

    Negative values ofp 2 0 corresponds to imaginary tem- poral Euclidean momenta, i.e., Minkowski time. As ex- plained above, this procedure is only valid until the first branch cut on the real frequency axis, which is located at the smallest threshold for the decay of the associated field. These thresholds are indicated by gray dashed ver- tical lines in th...

  2. [2]

    We express the prop- agators of the different fields through GΦiΦj(p) = Gk(p) ij ,(B2) whereG k(p) is defined in Eq

    Propagators In this section we list the two-point functions and the associated regularized propagators. We express the prop- agators of the different fields through GΦiΦj(p) = Gk(p) ij ,(B2) whereG k(p) is defined in Eq. (18). Furthermore, we show all expressions on a non-vanishing diquark background, p2/k2 0 5 10 𝜕tRk/k2 0 1 2 rexp(m= 2) rexp(m= 1) rflat...

  3. [3]

    Vertices a. Strong coupling avatars In the pure glue sector we only consider the classical tensor structure which are given by h T (1) c¯cA(p, q) iabc µ =if abcqµ ,(B10) h T (1) A3 (p, q) iabc µνρ =if abc h (q−p) ρδµν (B11) −(p+ 2q) µδρν + (2p+q) νδµρ i , h T (1) A4 iabcd µνρσ =f eabf ecd δµρδνσ −δ µσδνρ +f eacf ebd δµνδρσ −δ µσδνρ +f eadf ebc δµνδρσ −δ µ...

  4. [4]

    Two-point functions In the following, we collect the explicit expressions entering the computation of the QCD two-point functions discussed in Sec. VI. Throughout, all quantities are given in Euclidean space and evaluated at vanishing external spatial momentum,p= (p 0,⃗0 ). Furthermore, we employ a three-dimensional spatial regulator. The dimensionless mo...

  5. [5]

    Zero-momentum expansion a. Effective potential The flow of the effective potential is given by ∂t ¯Uk =−4 ¯Uk + (2 +η ϕ)¯ρϕ∂¯ρϕ ¯Uk + (2 +η ∆)¯ρ∆∂¯ρ∆ ¯Uk + 1 16π2 Z ∞ 0 dx x3 8 rcηc + 2xr′ c x 1 +r c + 8 1 + rq 1 x 1 +r q 2 + ¯m2q + 2 x 1 +r q 2 + ¯m2q +g 2 ∆¯q¯q¯ρ∆ rqηq + 2xr′ q − 3 2 3 x+xr A + ¯m2gap + 4 x+xr A + ¯m2gap + 1 2 λA2∆†∆ ¯ρ∆ + 1 x+xr A + ¯m...

  6. [6]

    G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song, and T. Takatsuka, From hadrons to quarks in neutron stars: a review, Rept. Prog. Phys.81, 056902 (2018), 42 arXiv:1707.04966 [astro-ph.HE]

    Pith/arXiv arXiv 2018

  7. [7]

    Annala, T

    E. Annala, T. Gorda, A. Kurkela, J. N¨ attil¨ a, and A. Vuorinen, Evidence for quark-matter cores in mas- sive neutron stars, Nature Phys.16, 907 (2020), arXiv:1903.09121 [astro-ph.HE]

    arXiv 2020

  8. [8]

    B. P. Abbottet al.(LIGO Scientific, Virgo), GW170817: Observation of Gravitational Waves from a Binary Neu- tron Star Inspiral, Phys. Rev. Lett.119, 161101 (2017), arXiv:1710.05832 [gr-qc]

    Pith/arXiv arXiv 2017

  9. [9]

    M. C. Milleret al., Psr j0030+0451 mass and radius from nicer data and implications for the properties of neutron star matter, Astrophys. J. Lett.887, L24 (2019), arXiv:1912.05705 [astro-ph.HE]

    Pith/arXiv arXiv 2019

  10. [10]

    T. E. Rileyet al., A NICER View of the Massive Pul- sar PSR J0740+6620 Informed by Radio Timing and XMM-Newton Spectroscopy, Astrophys. J. Lett.918, L27 (2021), arXiv:2105.06980 [astro-ph.HE]

    arXiv 2021

  11. [11]

    Bailin and A

    D. Bailin and A. Love, Superfluidity and Superconduc- tivity in Relativistic Fermion Systems, Phys. Rept.107, 325 (1984)

    1984

  12. [12]

    M. G. Alford, K. Rajagopal, and F. Wilczek, Color fla- vor locking and chiral symmetry breaking in high den- sity QCD, Nucl. Phys. B537, 443 (1999), arXiv:hep- ph/9804403

    arXiv 1999

  13. [13]

    R. D. Pisarski and D. H. Rischke, Color superconductiv- ity in weak coupling, Phys. Rev. D61, 074017 (2000), arXiv:nucl-th/9910056

    Pith/arXiv arXiv 2000

  14. [14]

    M. G. Alford, A. Schmitt, K. Rajagopal, and T. Sch¨ afer, Color superconductivity in dense quark matter, Rev. Mod. Phys.80, 1455 (2008), arXiv:0709.4635 [hep-ph]

    Pith/arXiv arXiv 2008

  15. [15]

    Schmitt, Phases and properties of color superconduc- tors, (2025), arXiv:2511.07319 [hep-ph]

    A. Schmitt, Phases and properties of color superconduc- tors, (2025), arXiv:2511.07319 [hep-ph]

    arXiv 2025

  16. [16]

    Fukushima and T

    K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rept. Prog. Phys.74, 014001 (2011), arXiv:1005.4814 [hep-ph]

    Pith/arXiv arXiv 2011

  17. [17]

    Eichmann, H

    G. Eichmann, H. Sanchis-Alepuz, R. Williams, R. Alkofer, and C. S. Fischer, Baryons as relativistic three-quark bound states, Prog. Part. Nucl. Phys.91, 1 (2016), arXiv:1606.09602 [hep-ph]

    Pith/arXiv arXiv 2016

  18. [18]

    M. Y. Barabanovet al., Diquark correlations in hadron physics: Origin, impact and evidence, Prog. Part. Nucl. Phys.116, 103835 (2021), arXiv:2008.07630 [hep-ph]

    arXiv 2021

  19. [19]

    Alford and K

    M. Alford and K. Rajagopal, Absence of two flavor color superconductivity in compact stars, JHEP06, 031, arXiv:hep-ph/0204001

    Pith/arXiv arXiv

  20. [20]

    Pasztor, The QCD phase diagram at finite temper- ature and density - a lattice perspective, PoSLAT- TICE2023, 108 (2024)

    A. Pasztor, The QCD phase diagram at finite temper- ature and density - a lattice perspective, PoSLAT- TICE2023, 108 (2024)

    2024

  21. [21]

    J. M. Pawlowski, Aspects of the functional renormalisa- tion group, Annals Phys.322, 2831 (2007), arXiv:hep- th/0512261

    arXiv 2007

  22. [22]

    Gies, Introduction to the functional RG and appli- cations to gauge theories, Lect

    H. Gies, Introduction to the functional RG and appli- cations to gauge theories, Lect. Notes Phys.852, 287 (2012), arXiv:hep-ph/0611146

    Pith/arXiv arXiv 2012

  23. [23]

    O. J. Rosten, Fundamentals of the Exact Renor- malization Group, Phys. Rept.511, 177 (2012), arXiv:1003.1366 [hep-th]

    Pith/arXiv arXiv 2012

  24. [24]

    Braun, Fermion Interactions and Universal Behavior in Strongly Interacting Theories, J

    J. Braun, Fermion Interactions and Universal Behavior in Strongly Interacting Theories, J. Phys. G39, 033001 (2012), arXiv:1108.4449 [hep-ph]

    Pith/arXiv arXiv 2012

  25. [25]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, The nonper- turbative functional renormalization group and its ap- plications, Phys. Rept.910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech]

    arXiv 2021

  26. [26]

    Fu, QCD at finite temperature and density within the fRG approach: an overview, Commun

    W.-j. Fu, QCD at finite temperature and density within the fRG approach: an overview, Commun. Theor. Phys. 74, 097304 (2022), arXiv:2205.00468 [hep-ph]

    arXiv 2022

  27. [27]

    Alkofer and L

    R. Alkofer and L. von Smekal, The Infrared behav- ior of QCD Green’s functions: Confinement dynam- ical symmetry breaking, and hadrons as relativistic bound states, Phys. Rept.353, 281 (2001), arXiv:hep- ph/0007355

    arXiv 2001

  28. [28]

    C. D. Roberts and S. M. Schmidt, Dyson-Schwinger equations: Density, temperature and continuum strong QCD, Prog. Part. Nucl. Phys.45, S1 (2000), arXiv:nucl- th/0005064

    arXiv 2000

  29. [29]

    C. S. Fischer, Infrared properties of QCD from Dyson- Schwinger equations, J. Phys. G32, R253 (2006), arXiv:hep-ph/0605173

    Pith/arXiv arXiv 2006

  30. [30]

    C. S. Fischer, QCD at finite temperature and chemical potential from Dyson–Schwinger equations, Prog. Part. Nucl. Phys.105, 1 (2019), arXiv:1810.12938 [hep-ph]

    Pith/arXiv arXiv 2019

  31. [31]

    M. Q. Huber, Nonperturbative properties of Yang–Mills theories, Phys. Rept.879, 1 (2020), arXiv:1808.05227 [hep-ph]

    arXiv 2020

  32. [32]

    Rennecke, QCD phase structure & equation of state: A functional perspective, EPJ Web Conf.364, 01018 (2026), arXiv:2510.11270 [hep-ph]

    F. Rennecke, QCD phase structure & equation of state: A functional perspective, EPJ Web Conf.364, 01018 (2026), arXiv:2510.11270 [hep-ph]

    arXiv 2026

  33. [33]

    C. S. Fischer and J. M. Pawlowski, Phase structure and observables at high densities from first principles QCD, (2026), arXiv:2603.11135 [hep-ph]

    arXiv 2026

  34. [34]

    C. S. Fischer and J. M. Pawlowski, Phase structure of strong interaction matter from Functional QCD, (2026), arXiv:2606.03703 [hep-ph]

    Pith/arXiv arXiv 2026

  35. [35]

    C. S. Fischer and J. Luecker, Propagators and phase structure of Nf=2 and Nf=2+1 QCD, Phys. Lett. B718, 1036 (2013), arXiv:1206.5191 [hep-ph]

    Pith/arXiv arXiv 2013

  36. [36]

    W.-j. Fu, J. M. Pawlowski, and F. Rennecke, QCD phase structure at finite temperature and density, Phys. Rev. D101, 054032 (2020), arXiv:1909.02991 [hep-ph]

    arXiv 2020

  37. [37]

    Gao and J

    F. Gao and J. M. Pawlowski, Chiral phase structure and critical end point in QCD, Phys. Lett. B820, 136584 (2021), arXiv:2010.13705 [hep-ph]

    arXiv 2021

  38. [38]

    P. J. Gunkel and C. S. Fischer, Locating the critical endpoint of QCD: Mesonic backcoupling effects, Phys. Rev. D104, 054022 (2021), arXiv:2106.08356 [hep-ph]

    arXiv 2021

  39. [39]

    W.-j. Fu, C. Huang, J. M. Pawlowski, F. Rennecke, R. Wen, and S. Yin, Strangeness neutrality and the QCD phase diagram, (2026), arXiv:2603.13455 [hep- ph]

    arXiv 2026

  40. [40]

    W.-j. Fu, J. M. Pawlowski, R. D. Pisarski, F. Ren- necke, R. Wen, and S. Yin, QCD moat regime and its real-time properties, Phys. Rev. D111, 094026 (2025), arXiv:2412.15949 [hep-ph]

    arXiv 2025

  41. [41]

    Isserstedt, C

    P. Isserstedt, C. S. Fischer, and T. Steinert, Thermody- namics from the quark condensate, Phys. Rev. D103, 054012 (2021), arXiv:2012.04991 [hep-ph]

    arXiv 2021

  42. [42]

    Y. Lu, F. Gao, Y.-X. Liu, and J. M. Pawlowski, QCD equation of state and thermodynamic observables from computationally minimal Dyson-Schwinger equations, Phys. Rev. D110, 014036 (2024), arXiv:2310.18383 [hep-ph]

    arXiv 2024

  43. [43]

    Y. Lu, F. Gao, Y.-x. Liu, and J. M. Pawlowski, Fi- nite density signatures of confining and chiral dynam- ics in QCD thermodynamics and fluctuations of con- served charges, Phys. Rev. D113, 054019 (2026), arXiv:2504.05099 [hep-ph]. 43

    arXiv 2026

  44. [44]

    M¨ uller, M

    D. M¨ uller, M. Buballa, and J. Wambach, Dyson- Schwinger approach to color superconductivity at finite temperature and density, Eur. Phys. J. A49, 96 (2013), arXiv:1303.2693 [hep-ph]

    Pith/arXiv arXiv 2013

  45. [45]

    M¨ uller, M

    D. M¨ uller, M. Buballa, and J. Wambach, Dyson- Schwinger Approach to Color-Superconductivity: Ef- fects of Selfconsistent Gluon Dressing, (2016), arXiv:1603.02865 [hep-ph]

    Pith/arXiv arXiv 2016

  46. [46]

    Leonhardt, M

    M. Leonhardt, M. Pospiech, B. Schallmo, J. Braun, C. Drischler, K. Hebeler, and A. Schwenk, Symmetric nuclear matter from the strong interaction, Phys. Rev. Lett.125, 142502 (2020), arXiv:1907.05814 [nucl-th]

    arXiv 2020

  47. [47]

    Braun and B

    J. Braun and B. Schallmo, From quarks and gluons to color superconductivity at supranuclear densities, Phys. Rev. D105, 036003 (2022), arXiv:2106.04198 [hep-ph]

    arXiv 2022

  48. [48]

    Braun, M

    J. Braun, M. Leonhardt, and M. Pospiech, Fierz- complete njl model study iii: Emergence from quark- gluon dynamics, Phys. Rev. D101, 036004 (2020), arXiv:1909.06298 [hep-ph]

    arXiv 2020

  49. [49]

    Rehberg, S

    P. Rehberg, S. P. Klevansky, and J. Hufner, Hadroniza- tion in the SU(3) Nambu-Jona-Lasinio model, Phys. Rev. C53, 410 (1996), arXiv:hep-ph/9506436

    Pith/arXiv arXiv 1996

  50. [50]

    Buballa, NJL model analysis of quark matter at large density, Phys

    M. Buballa, NJL model analysis of quark matter at large density, Phys. Rept.407, 205 (2005), arXiv:hep- ph/0402234

    arXiv 2005

  51. [51]

    A. J. Helmboldt, J. M. Pawlowski, and N. Strodthoff, Towards quantitative precision in the chiral crossover: masses and fluctuation scales, Phys. Rev. D91, 054010 (2015), arXiv:1409.8414 [hep-ph]

    Pith/arXiv arXiv 2015

  52. [52]

    R. Rapp, T. Sch¨ afer, E. V. Shuryak, and M. Velkovsky, Diquark Bose condensates in high density matter and instantons, Phys. Rev. Lett.81, 53 (1998), arXiv:hep- ph/9711396

    arXiv 1998

  53. [53]

    M. Hess, F. Karsch, E. Laermann, and I. Wetzorke, Diquark masses from lattice QCD, Phys. Rev. D58, 111502 (1998), arXiv:hep-lat/9804023

    Pith/arXiv arXiv 1998

  54. [54]

    Oettel, R

    M. Oettel, R. Alkofer, and L. von Smekal, Nucleon prop- erties in the covariant quark diquark model, Eur. Phys. J. A8, 553 (2000), arXiv:nucl-th/0006082

    Pith/arXiv arXiv 2000

  55. [55]

    Maris, Effective masses of diquarks, Few Body Syst

    P. Maris, Effective masses of diquarks, Few Body Syst. 32, 41 (2002), arXiv:nucl-th/0204020

    Pith/arXiv arXiv 2002

  56. [56]

    Nicmorus, G

    D. Nicmorus, G. Eichmann, A. Krassnigg, and R. Alkofer, Delta-baryon mass in a covariant Fad- deev approach, Phys. Rev. D80, 054028 (2009), arXiv:0812.1665 [hep-ph]

    Pith/arXiv arXiv 2009

  57. [57]

    Eichmann,Hadron properties from QCD bound- state equations, Ph.D

    G. Eichmann,Hadron properties from QCD bound- state equations, Ph.D. thesis, Graz U. (2009), arXiv:0909.0703 [hep-ph]

    Pith/arXiv arXiv 2009

  58. [58]

    Eichmann, C

    G. Eichmann, C. S. Fischer, and H. Sanchis-Alepuz, Light baryons and their excitations, Phys. Rev. D94, 094033 (2016), arXiv:1607.05748 [hep-ph]

    Pith/arXiv arXiv 2016

  59. [59]

    Watanabe, Quark-diquark potential and diquark mass from lattice QCD, Phys

    K. Watanabe, Quark-diquark potential and diquark mass from lattice QCD, Phys. Rev. D105, 074510 (2022), arXiv:2111.15167 [hep-lat]

    arXiv 2022

  60. [60]

    Kelvin-Lee and N

    K.-W. Kelvin-Lee and N. Ishii, Diquark mass and quark-diquark potential by lattice QCD using an ex- tended HAL QCD method with a static quark, PoS HADRON2025, 113 (2026), arXiv:2601.10091 [hep- lat]

    arXiv 2026

  61. [61]

    Kelvin-Lee and N

    K.-W. Kelvin-Lee and N. Ishii, Scalar diquark mass and quark–diquark potential from lattice QCD using the potential method with a static quark, (2026), arXiv:2606.19840 [hep-lat]

    Pith/arXiv arXiv 2026

  62. [62]

    Bender, C

    A. Bender, C. D. Roberts, and L. Von Smekal, Gold- stone theorem and diquark confinement beyond rain- bow ladder approximation, Phys. Lett. B380, 7 (1996), arXiv:nucl-th/9602012

    Pith/arXiv arXiv 1996

  63. [63]

    Raaijmakerset al., AN ICERview of PSR J0030+0451: Implications for the dense matter equa- tion of state, Astrophys

    G. Raaijmakerset al., AN ICERview of PSR J0030+0451: Implications for the dense matter equa- tion of state, Astrophys. J. Lett.887, L22 (2019), arXiv:1912.05703 [astro-ph.HE]

    arXiv 2019

  64. [64]

    Raaijmakers, S

    G. Raaijmakers, S. K. Greif, K. Hebeler, T. Hinderer, S. Nissanke, A. Schwenk, T. E. Riley, A. L. Watts, J. M. Lattimer, and W. C. G. Ho, Constraints on the dense matter equation of state and neutron star properties from nicer mass-radius estimate of psr j0740+6620 and multimessenger observations, Astrophys. J. Lett.918, L29 (2021), arXiv:2105.06981 [astro-ph.HE]

    arXiv 2021

  65. [65]

    Mroczek, M

    D. Mroczek, M. C. Miller, J. Noronha-Hostler, and N. Yunes, Thermodynamics conditions of matter in neu- tron star mergers, Phys. Rev. D110, 123009 (2024), arXiv:2309.02345 [astro-ph.HE]

    arXiv 2024

  66. [66]

    Gholami, I

    H. Gholami, I. A. Rather, M. Hofmann, M. Buballa, and J. Schaffner-Bielich, Astrophysical constraints on color- superconducting phases in compact stars within the RG-consistent NJL model, Phys. Rev. D111, 103034 (2025), arXiv:2411.04064 [hep-ph]

    arXiv 2025

  67. [67]

    Gao, W.-L

    B. Gao, W.-L. Yuan, M. Harada, and Y.-L. Ma, Ex- ploring the first-order phase transition in neutron stars using the parity doublet model and a Nambu–Jona- Lasinio–type quark model, Phys. Rev. C110, 045802 (2024), arXiv:2407.13990 [nucl-th]

    arXiv 2024

  68. [68]

    Ayriyan, D

    A. Ayriyan, D. Blaschke, J. P. Carlomagno, G. A. Con- trera, and A. G. Grunfeld, Bayesian Analysis of Hybrid Neutron Star EOS Constraints Within an Instantaneous Nonlocal Chiral Quark Matter Model, Universe11, 141 (2025), arXiv:2501.00115 [nucl-th]

    arXiv 2025

  69. [69]

    Christian, I

    J.-E. Christian, I. A. Rather, H. Gholami, and M. Hof- mann, Hybrid stars with sequential QCD phase tran- sitions in the era of NICER and LIGO/Virgo, Astron. Astrophys.701, A145 (2025), arXiv:2503.13626 [astro- ph.HE]

    arXiv 2025

  70. [70]

    J. M. Pawlowski, D. F. Litim, S. Nedelko, and L. von Smekal, Infrared behavior and fixed points in Lan- dau gauge QCD, Phys. Rev. Lett.93, 152002 (2004), arXiv:hep-th/0312324

    Pith/arXiv arXiv 2004

  71. [71]

    C. S. Fischer and H. Gies, Renormalization flow of Yang-Mills propagators, JHEP10, 048, arXiv:hep- ph/0408089

    arXiv

  72. [72]

    A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Landau gauge Yang-Mills corre- lation functions, Phys. Rev. D94, 054005 (2016), arXiv:1605.01856 [hep-ph]

    Pith/arXiv arXiv 2016

  73. [73]

    Goertz, ´A

    F. Goertz, ´A. Pastor-Guti´ errez, and J. M. Pawlowski, Gauge-fermion cartography: From confinement and chi- ral symmetry breaking to conformality, Phys. Rev. D 112, 034029 (2025), arXiv:2412.12254 [hep-th]

    arXiv 2025

  74. [74]

    Gies and C

    H. Gies and C. Wetterich, Renormalization flow of bound states, Phys. Rev. D65, 065001 (2002), arXiv:hep-th/0107221

    Pith/arXiv arXiv 2002

  75. [75]

    Gies and C

    H. Gies and C. Wetterich, Universality of spontaneous chiral symmetry breaking in gauge theories, Phys. Rev. D69, 025001 (2004), arXiv:hep-th/0209183

    Pith/arXiv arXiv 2004

  76. [76]

    Braun, The QCD Phase Boundary from Quark- Gluon Dynamics, Eur

    J. Braun, The QCD Phase Boundary from Quark- Gluon Dynamics, Eur. Phys. J. C64, 459 (2009), arXiv:0810.1727 [hep-ph]. 44

    Pith/arXiv arXiv 2009

  77. [77]

    Mitter, J

    M. Mitter, J. M. Pawlowski, and N. Strodthoff, Chiral symmetry breaking in continuum QCD, Phys. Rev. D 91, 054035 (2015), arXiv:1411.7978 [hep-ph]

    Pith/arXiv arXiv 2015

  78. [78]

    Braun, L

    J. Braun, L. Fister, J. M. Pawlowski, and F. Rennecke, From Quarks and Gluons to Hadrons: Chiral Symmetry Breaking in Dynamical QCD, Phys. Rev. D94, 034016 (2016), arXiv:1412.1045 [hep-ph]

    Pith/arXiv arXiv 2016

  79. [79]

    Rennecke, Vacuum structure of vector mesons in QCD, Phys

    F. Rennecke, Vacuum structure of vector mesons in QCD, Phys. Rev. D92, 076012 (2015), arXiv:1504.03585 [hep-ph]

    Pith/arXiv arXiv 2015

  80. [80]

    Wetterich, Exact evolution equation for the effec- tive potential, Phys

    C. Wetterich, Exact evolution equation for the effec- tive potential, Phys. Lett. B301, 90 (1993), two identi- cal pdf papers original 1993 and 2017, arXiv:1710.05815 [hep-th]

    Pith/arXiv arXiv 1993

Showing first 80 references.