miniDSE Scheme in Functional QCD

• miniDSE Scheme is a minimal Dyson–Schwinger truncation in QCD that retains key propagator and vertex dynamics for efficient and quantitative phase diagram computations.
• It constructs the quark–gluon vertex using only the dominant transverse tensor structures enforced by Slavnov–Taylor identities and a fitted vacuum gluon propagator.
• By incorporating a difference DSE for the charm-loop, miniDSE quantitatively assesses the shift of the critical endpoint in the chiral phase diagram when heavy flavors are included.

Searching arXiv for the cited paper and related miniDSE work by Lu et al. miniDSE Scheme denotes a recently proposed Dyson–Schwinger truncation for functional QCD that is designed to be computationally minimal yet quantitatively controlled. In the formulation used to study the effect of the charm quark on the QCD chiral phase diagram, the scheme solves the quark gap equations self-consistently, uses a simple but quantitatively tuned model for the gluon propagator, constructs the quark–gluon vertex from only the dominant tensor structures constrained by Slavnov–Taylor identities, and minimizes the number of free parameters essentially to $\alpha_s$ and the current quark masses. In this setting, miniDSE is applied to $2+1$ and $2+1+1$ flavor QCD at finite $T$ and $\mu_B$ to map the chiral phase diagram and to quantify the shift of the critical endpoint induced by dynamical charm (Gao et al., 5 Mar 2026).

1. Definition and position within functional QCD

In functional QCD, the full Dyson–Schwinger system is an infinite tower of coupled integral equations for all $n$-point functions, including propagators and vertices. Any practical calculation therefore requires a truncation. The miniDSE scheme is defined as a minimal but systematically improvable DSE truncation built around a small core set of equations: the quark gap equations are solved explicitly, whereas the gluon and ghost sectors are supplied through fitted vacuum input, simple thermal corrections, and a difference DSE for the heavy-flavor effect.

Relative to common truncations, miniDSE differs from rainbow–ladder by retaining two transverse quark–gluon vertex tensors rather than using only $\Gamma_\mu \sim \gamma_\mu$, by enforcing the STI for the Dirac part through ghost and quark dressings, and by using a gluon propagator fitted to functional FRG/DSE results together with an HTL thermal mass. It also differs from beyond-RL DSE systems with explicit gluon DSEs because it does not solve the gluon and ghost DSEs fully; instead it uses a fitted vacuum gluon dressing $Z_{\rm vac}(k)$, a simple HTL mass term for quark-induced thermal modifications, an explicit charm-loop correction via a difference DSE, and a ghost dressing taken from a fit to another functional study. It remains a propagator/vertex-level DSE approach rather than an effective model such as FRG flow equations for effective potentials or PNJL/PQM quark–meson models plus Polyakov loops (Gao et al., 5 Mar 2026).

The physical motivation is to maintain dynamical chiral symmetry breaking, correct UV running, and reasonable agreement with lattice thermodynamics and CEP estimates while including heavy flavors in a controlled way. The computational motivation is to reduce the coupled functional system to a tractable subset without giving up quantitative tuning.

2. Core equations and truncation structure

At finite $T$ and $\mu$, the Euclidean quark propagator for flavor $2+1$0 is parameterized in Landau gauge and Matsubara formalism as

$2+1$1

with $2+1$2, $2+1$3, and $2+1$4. For light flavors, $2+1$5. Chiral symmetry breaking is encoded in the scalar dressing $2+1$6; in the chiral limit, $2+1$7 signals dynamical chiral symmetry breaking. At $2+1$8, the propagator reduces to the standard $2+1$9-symmetric form

$2+1+1$0

with mass function $2+1+1$1.

The flavor-dependent quark DSE is

$2+1+1$2

with bare inverse propagator

$2+1+1$3

and self-energy

$2+1+1$4

where $2+1+1$5 and $2+1+1$6.

The gluon propagator is written as

$2+1+1$7

and an $2+1+1$8-symmetric approximation is employed, with no split between chromoelectric and chromomagnetic dressings. The vacuum $2+1+1$9 flavor gluon dressing is modeled by

$T$0

with dynamically generated gluon mass

$T$1

and anomalous dimension

$T$2

At finite $T$3 and $T$4, the scheme adds only the quark-induced thermal mass through an HTL-motivated term with $T$5; the strange quark is neglected in that HTL part because of its heavier mass.

The quark–gluon vertex retains only two transverse tensor structures,

$T$6

with

$T$7

These are identified as the dominant nonperturbative transverse structures in Landau gauge. The Dirac part is fixed by the STI:

$T$8

with

$T$9

and $\mu_B$0 the ghost dressing. The Pauli part is modeled as

$\mu_B$1

where

$\mu_B$2

A defining miniDSE feature is that the vertex is not solved from an independent vertex DSE; instead it is computed on the fly from the quark, gluon, and ghost dressing functions. The ghost dressing itself is taken from a fit to another functional calculation, and any $\mu_B$3 dependence of $\mu_B$4 is neglected as vacuum input (Gao et al., 5 Mar 2026).

3. Flavor content, renormalization, and parameter tuning

In the $\mu_B$5 implementation, the quark multiplet is

$\mu_B$6

with mass matrix

$\mu_B$7

The chemical-potential assignment is

$\mu_B$8

so only the light flavors carry finite baryon chemical potential, while strange and charm are taken at $\mu_B$9. Each flavor has its own propagator $n$0 with dressing functions $n$1, $n$2, and $n$3, solved from flavor-dependent quark DSEs. The gluon interaction is flavor-blind except through its dependence on $n$4 and heavy-quark loops.

Renormalization is multiplicative:

$n$5

with renormalized parameters

$n$6

and vertex renormalization

$n$7

The scheme uses a modified momentum-subtraction scheme $n$8 at finite $n$9, imposing renormalization conditions at an $\Gamma_\mu \sim \gamma_\mu$0-symmetric point with $\Gamma_\mu \sim \gamma_\mu$1, and setting the subtraction scale equal to the UV cutoff, $\Gamma_\mu \sim \gamma_\mu$2. In this setup the running coupling at the renormalization point equals the bare coupling, and effectively $\Gamma_\mu \sim \gamma_\mu$3. The renormalization conditions are

$\Gamma_\mu \sim \gamma_\mu$4

For $\Gamma_\mu \sim \gamma_\mu$5 flavors, the tuned parameters are

$\Gamma_\mu \sim \gamma_\mu$6

at $\Gamma_\mu \sim \gamma_\mu$7 GeV, reproducing a light-quark condensate $\Gamma_\mu \sim \gamma_\mu$8 MeV, pion mass $\Gamma_\mu \sim \gamma_\mu$9 MeV, and pion decay constant $Z_{\rm vac}(k)$0 MeV via the Pagels–Stokar formula. The condensate is computed as

$Z_{\rm vac}(k)$1

with

$Z_{\rm vac}(k)$2

For $Z_{\rm vac}(k)$3 flavors, the parameters are

$Z_{\rm vac}(k)$4

A central methodological point is that these parameters are re-tuned so that light-sector vacuum observables remain consistent between $Z_{\rm vac}(k)$5 and $Z_{\rm vac}(k)$6, enabling a controlled comparison of phase diagrams (Gao et al., 5 Mar 2026).

4. Phase-diagram construction and CEP identification

The principal order parameter for chiral symmetry is the light-quark condensate $Z_{\rm vac}(k)$7. At finite $Z_{\rm vac}(k)$8 and $Z_{\rm vac}(k)$9, the condensate is scanned across the $T$0 plane. The crossover region is characterized by a smooth but rapid drop of $T$1 with $T$2 at fixed $T$3; the pseudo-critical line $T$4 is defined by the maximum of the chiral susceptibility $T$5 or by the maximal slope in $T$6. The first-order region is identified by multiple coexisting solutions of the quark DSE, described as Nambu versus Wigner-like solutions, together with a discontinuous jump in the condensate. The critical endpoint is the endpoint of the first-order line, where the chiral susceptibility diverges and in practice becomes maximal and strongly peaked.

Numerically, the phase diagram is mapped by solving the quark DSEs self-consistently for each flavor on a grid in $T$7, extracting the light-quark condensate, and identifying the crossover line, first-order line, and CEP location $T$8. The truncation enters through three main approximations: the modeled gluon propagator comprising vacuum input, HTL thermal correction, and charm-loop contribution; the two-tensor vertex ansatz; and the neglect of medium dependence in the ghost sector together with omission of more complicated vertex structures.

Within this setup, the $T$9 flavor miniDSE result for the CEP is reported to be consistent with other state-of-the-art functional approaches, including FRG and more elaborate DSE truncations, at the level of $\mu$0. This establishes the intended role of miniDSE: not a full coupled functional calculation, but a compact framework that preserves quantitative contact with broader continuum-QCD benchmarks while remaining efficient enough for extensive scans in temperature, baryon chemical potential, and flavor content (Gao et al., 5 Mar 2026).

5. Charm-quark contribution and the difference DSE

Charm is treated as a dynamical quark with its own gap equation, but the paper states that because of its large mass it has essentially no direct impact on the light-quark condensate at physically relevant $\mu$1 and $\mu$2; the effect arises mainly through the gluon sector. This is a point on which misconceptions often arise: the miniDSE analysis does not attribute the phase-diagram shift primarily to direct light–charm mixing in the condensate, but to the charm contribution to gluon polarization.

To incorporate this contribution, the scheme introduces a difference DSE for the gluon propagator:

$\mu$3

where the baseline system is

$\mu$4

and the extended system is

$\mu$5

The resulting self-energy difference is the charm vacuum polarization,

$\mu$6

The full charm propagator and vertex are used, although the Pauli tensor is found negligible in the gluon self-energy and is dropped there for simplicity.

The quadratic UV divergences of this integral are treated through Brown–Pennington projection,

$\mu$7

with scalar self-energy extracted as

$\mu$8

This removes the quadratic divergence, while the remaining logarithmic divergences are absorbed into renormalization constants. For validation, the resulting $\mu$9 is compared with the analytic one-loop charm polarization in dimensional regularization using a tree-level propagator and bare vertex; good agreement is found for $2+1$00 GeV, whereas in the infrared the full nonperturbative charm propagator and vertex enhance $2+1$01.

In vacuum, the gluon dressing function peak height changes from $2+1$02 in $2+1$03 flavor QCD to $2+1$04 in $2+1$05 flavor QCD, while the position of the maximum is nearly unchanged, and the two dressings converge around $2+1$06 GeV in the ultraviolet. After retuning $2+1$07 and the quark masses, the light-quark mass functions in $2+1$08 and $2+1$09 remain nearly identical, which the paper uses as evidence that vacuum light-hadron physics is preserved. At finite $2+1$10 and $2+1$11, the slight suppression of the gluon dressing at intermediate momenta weakens the effective interaction in the quark DSE; within the miniDSE interpretation, this shifts the balance between dynamical chiral symmetry breaking and restoration, moving the CEP to smaller $2+1$12 (Gao et al., 5 Mar 2026).

6. Quantitative results, limitations, and prospective developments

The computed phase diagrams for $2+1$13 and $2+1$14 flavors are close in the crossover region, but the critical endpoint shifts measurably when charm is included through the difference DSE. The reported values are as follows.

Flavor content	$2+1$15	CEP $2+1$16
$2+1$17	$2+1$18 MeV	$2+1$19 MeV
$2+1$20	$2+1$21 MeV	$2+1$22 MeV

From these numbers, $2+1$23 increases by about $2+1$24, whereas $2+1$25 decreases by about $2+1$26. The shape and curvature of the crossover line are described as essentially unchanged, with the two phase boundaries lying almost on top of each other in the crossover region. The paper’s interpretation is that the charm loop slightly suppresses the mid-momentum gluon dressing, weakens quark binding, and therefore yields an earlier onset of the first-order chiral transition in baryon chemical potential. A plausible implication is that heavy-flavor effects are small in bulk terms but not negligible when CEP searches aim at few-percent precision.

The strengths of miniDSE are stated explicitly. It is numerically efficient because only the quark gap equations are solved while the gluon and ghost sectors are represented by simple analytic models. It makes controlled use of functional input by fitting vacuum gluon and ghost dressings to FRG/DSE results, implementing thermal quark-loop effects through HTL mass terms, and adding the charm loop via a difference DSE with a full nonperturbative charm propagator. It also preserves gauge and STI consistency through Landau gauge and an STI-constrained Dirac vertex supplemented by an explicit Pauli term. For $2+1$27 flavors, its CEP predictions and pseudo-critical temperatures are reported to agree well with more elaborate functional approaches and lattice-inferred thermodynamics.

Its limitations are equally central to its definition. The gluon sector is not solved self-consistently, so thermal and density dependence beyond HTL is modeled rather than derived, and light-quark feedback on the gluon beyond HTL is neglected. The ghost dressing is frozen to a vacuum fit with no $2+1$28 dependence. The vertex truncation retains only two transverse tensors, omitting other structures, including longitudinal pieces and structures relevant for confinement or meson channels. Meson and baryon back-coupling in the quark–gluon vertex is absent, although such effects are known in other works to shift the CEP. No Polyakov loop is included, so there is no explicit gauge-invariant order parameter for deconfinement. The use of the $2+1$29 scheme with $2+1$30 is convenient but may hide scheme dependence in $2+1$31. The authors emphasize that the charm-induced shift of order $2+1$32–$2+1$33 in $2+1$34 is comparable to anticipated truncation uncertainties (Gao et al., 5 Mar 2026).

The scheme is already being extended to the QCD equation of state, thermodynamic observables at finite density and temperature, conserved-charge fluctuations, signatures of confinement, and astrophysical applications requiring a realistic QCD equation of state. The developments mentioned or implied include a back-coupled gluon DSE, more sophisticated vertex constructions with additional tensor structures and hadronic back-coupling, Polyakov-loop or background-field extensions, and a medium-dependent ghost sector. These directions suggest an effort to move miniDSE closer to fully coupled functional QCD while preserving its minimal-parameter ethos.