Screened Massive Expansion in Yang–Mills Theory

• Screened massive expansion is a reorganization of perturbation theory that shifts from a massless to a massive transverse gluon propagator, capturing dynamical mass generation in the infrared regime.
• The method improves the behavior of loop calculations by systematically introducing crossed graphs that cancel spurious mass divergences and reproduce lattice-like propagators.
• Extensions include finite-temperature formulations and quark sector applications, providing insights into gauge independence, complex pole structures, and renormalization-group improvements.

Screened massive expansion is a reorganization of perturbation theory for infrared Yang–Mills theory, and by extension selected sectors of QCD, in which the expansion point is shifted from a massless gluonic vacuum to a vacuum built on a massive transverse gluon propagator, while the full gauge-fixed action is kept unchanged by adding and subtracting the same transverse mass term. In this framework the added mass is not a physical parameter of the Lagrangian but a bookkeeping device that encodes infrared screening already at zeroth order, improves the behavior of loop calculations at low momentum, reproduces lattice-like gluon propagators, and generically yields complex-conjugate gluon poles (Comitini, 2019, Siringo et al., 2022).

1. Definition and motivation

The screened massive expansion was introduced to address the infrared failure of ordinary massless perturbation theory in pure Yang–Mills theory. Lattice calculations indicate that the transverse gluon propagator saturates at low momentum rather than diverging, which is naturally interpreted as dynamical mass generation at the level of correlation functions. Standard perturbation theory around the massless vacuum cannot reproduce such behavior at any finite order, because the tree-level theory contains no mass scale and the resulting self-energy remains proportional to $p^2$ in the infrared. The central premise of the screened expansion is therefore that the ordinary perturbative vacuum is a poor infrared starting point, and that a massive transverse vacuum is a better zeroth-order approximation (Comitini, 2019, Siringo, 2022).

In the formulation used for covariant gauges, the standard massless quadratic kernel gives

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$

with

$$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$

The screened expansion shifts the quadratic kernel by a transverse mass term

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$

so that the new free propagator becomes

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$

Because the same term is subtracted from the interaction, the exact Yang–Mills action is unchanged; only the perturbative organization is modified (Siringo et al., 2022, Comitini, 2019).

This is the sense in which the expansion is “screened.” The transverse gluon propagates as if it were massive already at zeroth order, mimicking the infrared screening associated with dynamical mass generation, while longitudinal gluons and ghosts remain massless at tree level. The thesis formulation also motivates this choice variationally through the Gaussian Effective Potential, which indicates that the massless perturbative vacuum of the transverse sector is unstable toward a massive one (Comitini, 2019).

2. Reorganized perturbation theory and crossed graphs

The formal reorganization is implemented by replacing the usual split $S=S_0+S_{\text{int}}$ with

$$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$

where $\delta S$ contains only the transverse gluon mass shift. Diagrammatically, the subtraction term appears as a new two-point mass counterterm vertex, often represented by a cross on a gluon line. The exact dressed transverse propagator can be written as

$\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$

where the tree-level contribution to $\Pi$ is precisely the counterterm $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$0. Defining $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$1, one obtains

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$2

In this form the explicit mass shift cancels, so any surviving gluonic mass scale is generated dynamically by loops rather than inserted as a bare physical mass (Siringo et al., 2022).

A characteristic feature of the method is the appearance of “crossed” graphs, i.e. ordinary loop topologies with one or more mass insertions on internal gluon lines. Their combinatorics is controlled by derivative identities. In the perturbative diagrammatics of the original formulation, a gluon line with $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$3 mass counterterms satisfies

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$4

while in the later Schwinger–Dyson reformulation the basic identity is

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$5

These relations generate the crossed contributions systematically and explain why the same massive reference propagator can be used without introducing a genuine mass counterterm into the renormalized Yang–Mills action (Comitini, 2019, Siringo, 2022).

The cancellation of spurious mass divergences is central. Crossed graphs cancel the unwanted divergences proportional to $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$6, so no independent mass renormalization is required inside the screened expansion. After these cancellations, the remaining ultraviolet divergences are the usual wave-function and coupling divergences of the original gauge theory (Siringo et al., 2022, Siringo, 2022).

At one loop, the Landau-gauge transverse propagator can be written in the compact form

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$7

with a free additive renormalization constant $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$8. The analytic continuation of this propagator displays several possible pole patterns depending on $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$9, including two negative real poles, one pair of complex-conjugate poles, two pairs of complex-conjugate poles, or no poles. The physically relevant region was argued to be the one with a single complex-conjugate pair, and in the optimized Landau-gauge setting the preferred value is

$$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$0

very close to the best-fit lattice value $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$1 (Comitini, 2019).

3. Nielsen identities, gauge dependence, and complex poles

A major issue for the screened massive expansion is gauge dependence. In exact Yang–Mills theory, BRST symmetry implies Nielsen identities that constrain how Green functions vary with the covariant-gauge parameter $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$2. For the exact gluon propagator, the operator form derived with the Nakanishi–Lautrup field $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$3 is

$$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$4

with $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$5, and for the transverse scalar part

$$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$6

At a pole $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$7, where $$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$8, this implies

$$t_{\mu\nu}(p)=g_{\mu\nu}-\frac{p_\mu p_\nu}{p^2},\qquad \ell_{\mu\nu}(p)=\frac{p_\mu p_\nu}{p^2}.$$9

Hence the pole position is gauge independent in the exact theory (Siringo et al., 2022).

Near a complex pole,

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$0

and the phase of the residue obeys

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$1

If $\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$2 is finite at the pole, the right-hand side vanishes and the residue phase is gauge independent. If $\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$3 is singular at the same point, the conclusion is more delicate. The framework therefore distinguishes the phase of the residue from its modulus: the modulus can be altered by a real renormalization factor, whereas the phase is the nontrivial quantity (Siringo et al., 2022).

This distinction becomes important because the screened expansion at fixed order is not strict perturbation theory. Although the full action is unchanged, a finite-order truncation around $\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$4 induces a soft breaking of BRST symmetry, so the exact Nielsen identities are not expected to hold identically at each truncated order. The one-loop analysis therefore asks how much of the exact gauge-independence structure survives approximately. The explicit one-loop gluon Nielsen function contains a transverse term proportional to $\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$5 and reduces smoothly to the standard massless perturbative result when $\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$6, while its ultraviolet divergence is absorbed by the usual gluon wave-function renormalization (Siringo et al., 2022).

The analytic structure of the propagator is one of the most distinctive outcomes of the expansion. The optimized one-loop propagator has complex-conjugate poles

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$7

and the corresponding pole structure is associated with the absence of a standard Källén–Lehmann representation and with confinement-like positivity violation. Previous numerical studies, and the explicit one-loop Nielsen analysis, found that both the pole position and the phase of the residue are nearly gauge-parameter independent. The paper on Nielsen identities does not claim a formal proof inside the approximate resummation schemes, but it interprets the explicit calculations as evidence in favor of exact pole invariance and probable phase invariance in a more complete treatment (Comitini, 2019, Siringo et al., 2022).

4. Schwinger–Dyson formulation and renormalization-group improvement

A later reformulation derives the screened massive expansion directly from Schwinger–Dyson equations. The exact gluon equation

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$8

is rewritten using the identity

$\delta\Gamma^{\mu\nu}(p)=m^2\,t^{\mu\nu}(p),$9

which gives

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$0

Introducing a fictitious expansion parameter $$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$1 and expanding around $$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$2 makes the screened expansion a controlled $$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$3-expansion around the massive reference propagator. At first order, crossed graphs are generated by the operator

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$4

so the one-loop screened Schwinger–Dyson system takes the form

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$5

with analogous expressions for the ghost self-energy and vertices (Siringo, 2022).

This derivation clarifies several points that were ambiguous in earlier presentations. It distinguishes a minimal truncation, in which the number of mass-insertion orders matches the loop order, from a vertex-counting or “democratic” scheme, in which mass insertions are counted on the same footing as ordinary vertices. The difference matters when finite crossed graphs affect infrared observables. In the pinch-technique/background-field case, for example, the doubly crossed tadpole is not mandatory in the minimal $$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$6 scheme, but it becomes natural in the vertex-counting scheme and yields a finite infrared value for the gauge-parameter-independent effective gluon propagator (Siringo, 2022).

Renormalization-group improvement was developed at one loop in the Landau gauge in both MOM and SMOM renormalization schemes. In the MOM scheme the subtraction conditions are

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$7

whereas in SMOM they are

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$8

The corresponding Taylor-scheme running couplings do not develop a Landau pole, provided the initial coupling is sufficiently small. In MOM the running coupling vanishes in the deep infrared,

$$\Delta_m^{\mu\nu}(p)=\Delta_m(p)\,t^{\mu\nu}(p)+\frac{\xi}{-p^2}\,\ell^{\mu\nu}(p), \qquad \Delta_m(p)=\frac{1}{-p^2+m^2}.$$9

whereas in SMOM it saturates to the finite constant

$S=S_0+S_{\text{int}}$0

In both schemes the ultraviolet limit matches ordinary perturbation theory, and the improved propagators retain a finite infrared gluon propagator and a massless ghost (Comitini et al., 2020).

The MOM implementation can be optimized by matching to the earlier fixed-coupling screened framework. The optimized values reported are

$S=S_0+S_{\text{int}}$1

with $S=S_0+S_{\text{int}}$2 GeV if $S=S_0+S_{\text{int}}$3 GeV. After optimization, the gluon mass parameter $S=S_0+S_{\text{int}}$4 is the only free dimensionful parameter left, and the paper argues that it then plays the same role as the usual perturbative scale $S=S_0+S_{\text{int}}$5: once $S=S_0+S_{\text{int}}$6 is fixed, the running coupling and propagators are determined at all scales (Comitini et al., 2020).

5. Finite-temperature extension and quasi-gluon dispersion

The screened massive expansion has also been extended to pure $S=S_0+S_{\text{int}}$7 Yang–Mills theory at finite temperature in the Landau gauge. In the Euclidean Matsubara formalism,

$S=S_0+S_{\text{int}}$8

the thermal bath breaks Lorentz symmetry and the gluon propagator splits into longitudinal and transverse components with projectors $S=S_0+S_{\text{int}}$9 and $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$0. The projected inverse propagators are

$$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$1

$$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$2

The finite-temperature extension keeps the same screened massive expansion and adds the thermal parts of the one-loop self-energies (Siringo et al., 2021).

A technical result of this extension is that all thermal one-loop integrals can be reduced to analytic expressions with only one remaining numerical integral over a real variable. The resulting thermal functions are analytic in the complex external energy, which makes it possible to continue the propagators directly into the complex plane and search for poles at finite temperature. This analytic control is essential for extracting quasi-gluon dispersion relations rather than restricting the analysis to Euclidean screening masses (Siringo et al., 2021).

At $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$3, the optimized zero-temperature parameters recalled in the thermal paper are

$$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$4

If these values are kept fixed for $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$5, the formalism reproduces the correct qualitative trends but not the quantitative lattice behavior. In particular, the longitudinal propagator shows a crossover around

$$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$6

which is far below the lattice deconfinement scale, while the transverse propagator decreases monotonically with temperature in qualitative agreement with lattice trends (Siringo et al., 2021).

A better description is obtained by fitting $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$7 and $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$8 separately at each temperature and for each polarization. The transverse sector is then described well down to $$S=\big(S_0+\delta S\big)+\big(S_{\text{int}}-\delta S\big),$$9 GeV, whereas the longitudinal sector remains problematic in the deep infrared and develops a turnover absent from the lattice data. The paper interprets this asymmetry as evidence that a one-mass ansatz is suboptimal at finite temperature and motivates a future extension with separate screening masses,

$\delta S$0

leading to distinct zeroth-order propagators $\delta S$1 and $\delta S$2 (Siringo et al., 2021).

Because the thermal propagators are analytic in the complex plane, one can define quasi-gluon poles through

$\delta S$3

At zero temperature the complex poles correspond to

$\delta S$4

and at zero momentum they give

$\delta S$5

Using fitted thermal parameters, the transverse branch shows a crossover behavior: below $\delta S$6 MeV both $\delta S$7 and $\delta S$8 are suppressed relative to their vacuum values, while above $\delta S$9 the energy rises again and the damping increases. At $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$0, the transverse quasi-gluon mass decreases from $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$1 MeV to about $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$2 MeV near $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$3, while the width decreases from $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$4 MeV to about $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$5 MeV near $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$6, and both grow again above the crossover (Siringo et al., 2021).

6. Extension to quarks, scope, and unresolved issues

The framework has been extended from pure Yang–Mills theory to the quark propagator in Landau-gauge QCD. The quark-sector reorganization follows the same logic: add and subtract a mass term $\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$7 so that the zeroth-order propagator is

$\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$8

while the compensating two-point vertices are

$\Delta^{-1}(p)=-p^2+m^2-\Pi(p),$9

The dressed quark propagator is written as

$\Pi$0

Because the scalar self-energy remains nonzero even when the bare quark mass $\Pi$1, the mass function satisfies

$\Pi$2

so the expansion captures dynamical infrared quark mass generation already at one loop (Comitini et al., 2021).

Three resummation prescriptions were studied. The minimalistic scheme retains the ordinary one-loop quark–gluon graph and the crossed quark-mass insertion; the vertex-wise scheme also includes the crossed gluon-mass insertion; and the complex-conjugate (CC) scheme replaces the internal gluon line by the principal part of the fully dressed gluon propagator, approximated by its complex pole structure. In Landau gauge, the vector divergence of the ordinary one-loop quark self-energy vanishes, while the scalar divergence of the ordinary graph is canceled by the crossed quark-mass insertion, so the basic one-loop combination is finite (Comitini et al., 2021).

Phenomenologically, all three schemes reproduce quenched lattice data for the quark mass function $\Pi$3 very well. The fitted infrared quark screening scale $\Pi$4 lies in the few-hundred-MeV range: about $\Pi$5–$\Pi$6 MeV in the minimalistic scheme, $\Pi$7–$\Pi$8 MeV in the vertex-wise scheme, and $\Pi$9–$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$00 MeV in the CC scheme. The gluon-sector mass parameter used in these fits is fixed from earlier pure-Yang–Mills analysis,

$$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$01

By contrast, the quark wave-function factor $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$02 remains problematic: the minimalistic and vertex-wise schemes display the wrong qualitative momentum dependence, while the CC scheme gives the correct qualitative trend only for sufficiently large momenta, roughly $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$03 GeV (Comitini et al., 2021).

These results delimit the current status of the screened massive expansion. Its strongest successes are in pure Yang–Mills theory, especially for the Landau-gauge gluon and ghost propagators, the complex pole structure of the gluon propagator, and the transverse finite-temperature sector. At the same time, several limitations are explicit in the literature. Fixed-order truncations softly break BRST symmetry, so exact Nielsen identities are not guaranteed at each step. The finite-temperature longitudinal sector is not described satisfactorily by a one-mass ansatz. In the quark sector, the mass function is robust but the $$\Delta_0^{\mu\nu}(p)=\Delta_0(p)\big[t^{\mu\nu}(p)+\xi\,\ell^{\mu\nu}(p)\big], \qquad \Delta_0(p)=\frac{1}{-p^2},$$04-function is not. These points do not negate the method’s utility; rather, they define it as an optimized infrared reorganization of perturbation theory whose consistency is judged by lattice agreement, analytic control, and approximate preservation of gauge-invariant pole data, rather than by exact order-by-order BRST identities (Siringo et al., 2022, Siringo et al., 2021, Comitini et al., 2021).